External open problems

1719 of 1739 solved in this repository.

What “solved” means here: a frozen Lean theorem in this repository is recorded against the problem. Nobody has machine-checked that the theorem says the same thing as the source, and a problem with no record here may still have been solved by someone else.

Solved (1719)

Equal mixed and ordinary palindromic costs on Thue–Morse prefixes

Proved. Lean theorem unbounded_equal_cost, frozen in this repository 2026-11-05.

Petr Ambrož, Ondřej Kadlec, Zuzana Masáková, Edita Pelantová (2019). Palindromic length of words and morphisms in class P. DOI: 10.1016/j.tcs.2019.02.024. URL: https://arxiv.org/abs/1812.00711v2.

Claim. Section 4 recalls Frid's Fibonacci prefix conjecture; Section 6 defines mixed pal-antipal length and conjectures unbounded equal-cost Thue-Morse prefixes.

Problem details · Reading note

Barker's coordination formulas for the tea slab graph

Proved. Lean theorem result, frozen in this repository 2026-11-05.

N. J. A. Sloane, Colin Barker, Stefano Spezia (2018). OEIS A299285, coordination sequence for the tea 3D uniform tiling. URL: https://oeis.org/A299285.

Claim. Barker's generating function and recurrence for the tea coordination sequence, which Sloane marks as probably only conjectures, and Spezia's closed form a(n) = (31 - 3*(-1)^n + 126*n^2 + 4*A056594(n))/16 for n > 0, stated conditionally on them.

Problem details · Reading note

Bernáth's OEIS A400645 consecutive-ratio conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-05.

Bence Bernáth (2026). OEIS A400645, signed four-lag rounded norm ratio. URL: https://oeis.org/A400645.

Claim. The consecutive ratio of the signed four-lag rounded norm sequence tends to sqrt(A289917), the largest real root of x^8-x^6-x^4-x^2+1.

Problem details · Reading note

Refuting the unrestricted well-hued graph characterization

Refuted. Lean theorem result, frozen in this repository 2026-11-05.

Geoffrey Boyer, Kirsti Kuenzel, Jeremy Lyle, Ryan Pellico (2025). Well-hued graphs with first difference two. DOI: 10.48550/arXiv.2506.04993. URL: https://arxiv.org/abs/2506.04993v1.

Claim. Section 3, Conjecture 2 proposes the unrestricted balanced multipartite characterization with independence number alpha(G).

Problem details · Reading note

A counterexample to Hausmann Question 5.4

Refuted. Lean theorem result, frozen in this repository 2026-11-05.

Will Hausmann (2026). Weighted Max Cut in Subcubic Graphs. URL: https://arxiv.org/html/2611.05178v1.

Claim. Question 5.4 asks for an independent side and residual component order at most three in every subcubic graph of odd girth exactly five.

Problem details · Reading note

Hanna's square-exponent EGF congruence modulo six

Proved. Lean theorem result, frozen in this repository 2026-11-05.

Paul D. Hanna (2013). OEIS A226838 — E.g.f.: exp( Sum_{n>=1} x^(n^2) / n^n ).. URL: https://oeis.org/A226838.

Claim. a(n) == 1 (mod 6) (conjecture - valid up to n=1024; if true for n>=0, why?).

Problem details · Reading note

Density-independent Alhasan–Fawwaz boundedness

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Sami Abdullah Abed; Irina Nikolaeva; Andrej Novikov (2026). How characterisations of traces and central elements fail: sharp violation constants via Michelson contrast and noncentrality. DOI: 10.48550/arXiv.2610.34388. URL: https://arxiv.org/abs/2610.34388v1.

Claim. Section 11.2(5) asks whether the Alhasan–Fawwaz constant is bounded on M_n(C) independently of the positive definite density.

Problem details · Reading note

Halving partitions admit no finite integral decimation family

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Max A. Alekseyev (2026). Halving Partitions and Hyperbinary Expansions. URL: https://arxiv.org/abs/2611.01243v1.

Claim. Section 6 asks whether a(n) belongs to a finite family of integer sequences closed under both binary decimations.

Problem details · Reading note

Aliabadi's finite nonabelian quotient-multiset problem

Proved. Lean theorem exists_obstructed_multiset, frozen in this repository 2026-11-04.

Mohsen Aliabadi (2026). A nonabelian twist on differences of bijections. DOI: 10.61091/um128-18. URL: https://arxiv.org/html/2605.16478v2.

Claim. Problem 7.6 asks whether every finite nonabelian group admits a multiset of cardinality its order satisfying the abelianization obstruction but not quotient-realizable.

Problem details · Reading note

Barker’s coordination formulas for the tsi alternating prism graph

Proved. Lean theorem result, frozen in this repository 2026-11-04.

N. J. A. Sloane, Colin Barker (2018). OEIS A299289, coordination sequence for the tsi 3D uniform tiling. URL: https://oeis.org/A299289.

Claim. Conjectures from Colin Barker: a(n) = (13*n^2 + 4)/2 for n>0 and even, a(n) = (13*n^2 + 3)/2 for n odd, with the corresponding generating function and recurrence.

Problem details · Reading note

The alternating vortex polygon has exactly $N/2-2$ distinct real eigenvalue pairs

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Anna M. Barry, Panayotis G. Kevrekidis (2013). Few-particle vortex cluster equilibria in Bose–Einstein condensates: existence and stability. DOI: 10.1088/1751-8113/46/44/445001. URL: https://arxiv.org/abs/1306.1545v2.

Claim. The paper studies equilibria and linear stability of few-vortex clusters in the precessing point-vortex model of a Bose–Einstein condensate and conjectures (Conjecture 4.4) that for N >= 6 the regular N-gon of alternating charges has N/2 - 2 distinct pairs of real eigenvalues.

Problem details · Reading note

The classical value of the odd polygon game is its Fibonacci bound

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Vir B. Bulchandani, Fiona J. Burnell, S. L. Sondhi (2023). Playing nonlocal games with phases of quantum matter. DOI: 10.1103/PhysRevB.107.045412. URL: https://arxiv.org/abs/2206.11252v2.

Claim. The paper introduces the P-player polygon game, gives a perfect quantum strategy from the 2P-qubit cyclic cluster state, shows that odd P admits no perfect classical strategy, exhibits a classical strategy losing on F_{P-1} of the 2^P inputs, and conjectures that the resulting lower bound p_cl*(P) >= 1 - F_{P-1}/2^P on the classical winning probability is tight for odd P.

Problem details · Reading note

Strong subadditivity of bosonic Gaussian Rényi entropy for $0<\alpha\le2$, $\alpha\ne1$

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Giancarlo Camilo, Gabriel T. Landi, Sebas Eliëns (2019). Strong subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states. DOI: 10.1103/PhysRevB.99.045155. URL: https://arxiv.org/abs/1810.07070v2.

Claim. The paper expresses the Rényi entropies of bosonic and fermionic Gaussian states through the symplectic or covariance eigenvalues, proves subadditivity for all orders (bosons) and for orders up to two (fermions), proves strong subadditivity at order two for bosons, finds numerical violations of strong subadditivity above order two for bosons and above about 1.3 for fermions, and conjectures that strong subadditivity holds for bosonic Gaussian states for orders up to two and for fermionic Gaussian states for orders up to one.

Problem details · Reading note

Four-qubit pair-correlation monogamy

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Shuming Cheng, Antony Milne, Michael J. W. Hall, Howard M. Wiseman (2016). Volume monogamy of quantum steering ellipsoids for multi-qubit systems. DOI: 10.1103/PhysRevA.94.042105. URL: https://arxiv.org/abs/1606.02390v2.

Claim. The paper proves volume monogamy relations for quantum steering ellipsoids of pure three- and four-qubit states and conjectures (Eq. (36)) that for all pure four-qubit states the squared Frobenius norms of the two-qubit correlation matrices of qubit A with B, C and D sum to at most 3.

Problem details · Reading note

The $\mathrm F_3$ steering inequality has the sharp ellipsoid-centre threshold two thirds

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Andrés F. Ducuara, Cristian E. Susa, Paul Skrzypczyk (2024). Locally inaccessible hidden quantum correlations. DOI: 10.1103/PhysRevA.110.022435. URL: https://arxiv.org/abs/2406.03945v1.

Claim. The paper proves, modulo a conjecture on quantum steering ellipsoids, the existence of two-qubit states whose hidden CHSH and F3 violations cannot be revealed by local filters on one side only; the conjecture states that the maximal CHSH and F3 violations are bounded by functions of the steering-ellipsoid centre length alone, with numerically estimated impossibility thresholds 0.5 and 0.66.

Problem details · Reading note

GNP arXiv v4 Question 8.20: Dicke extension cones

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Aabhas Gulati, Ion Nechita and Sang-Jun Park (2025). Positive maps and extendibility hierarchies from copositive matrices. DOI: 10.48550/arXiv.2509.15201.

Claim. arXiv:2509.15201v4 Question 8.20 asks whether the fully PPT Bose symmetric complete graph and star extension cones agree for mixtures of Dicke states.

Problem details · Reading note

Sharp return-probability bounds for conference graphs

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Chris Godsil (2021). Sedentary quantum walks. DOI: 10.1016/j.laa.2020.08.027. URL: https://arxiv.org/abs/1710.11192v1.

Claim. The paper studies continuous quantum walks U(t) = exp(itA) and their mixing matrices M(t) = U(t) ∘ conj(U(t)), shows the stay-at-home property for complete graphs, cones and several families of strongly regular graphs, and asks whether, for conference graphs, the diagonal entries of M(t) can be bounded away from zero.

Problem details · Reading note

Han–Pedon Conjecture C: all higher-shift metallic Hankel residue polynomials

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Guo-Niu Han, Emmanuel Pedon (2026). Hankel determinants for the (n+2)-shifted q-metallic numbers. DOI: 10.48550/arXiv.2610.36132. URL: https://arxiv.org/abs/2610.36132v1.

Claim. Conjecture C: integer-valued rational residue polynomials for all higher shifted q-metallic Hankel determinants.

Problem details · Reading note

Hardin's six-column trace-avoidance recurrence

Proved. Lean theorem result, frozen in this repository 2026-11-04.

R. H. Hardin (2011). OEIS A185765: six-column binary arrays with unequal neighboring subblock traces. URL: https://oeis.org/A185765.

Claim. Empirical: a(n)=a(n-1)+30*a(n-2)+24*a(n-3)-95*a(n-4)-15*a(n-5)+64*a(n-6)-116*a(n-7)+16*a(n-8)+16*a(n-9) for n>12

Problem details · Reading note

The covariance-corrected entropy-power uncertainty relation fails for mixed states

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Anaelle Hertz, Michael G. Jabbour, Nicolas J. Cerf (2017). Entropy-power uncertainty relations: towards a tight inequality for all Gaussian pure states. DOI: 10.1088/1751-8121/aa852f. URL: https://arxiv.org/abs/1702.07286v2.

Claim. The paper proposes the entropic uncertainty relation h(x)+h(p)-(1/2)ln(sigma_x^2 sigma_p^2/|gamma|) >= ln(pi e hbar), saturated by all pure Gaussian states, proves by variational calculus that pure Gaussian states are stationary points of the uncertainty functional, and states that the relation holds for all states under assumptions that include the concavity of the functional for mixed states.

Problem details · Reading note

Hughes iteration-depth no-gap conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Charles E. Hughes (2026). Undecidability of Adjacent Equality for Insertion, Shuffle, and Crossover Language Operations. URL: https://arxiv.org/abs/2608.27755v1.

Claim. Version 1, Section 10 conjectures that the minimum iteration-depth spectrum of literal fixed-degree insertion for arbitrary languages is an initial interval of the non-negative integers. Version 2, Section 11, Theorem 9 identifies the inserted-unary singleton insertion-degree spectrum by occupied gaps.

Problem details · Reading note

A projective toric two-design satisfying the MUB overlap condition need not be a subgroup

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Joseph T. Iosue, T. C. Mooney, Adam Ehrenberg, Alexey V. Gorshkov (2024). Projective toric designs, quantum state designs, and mutually unbiased bases. DOI: 10.22331/q-2024-12-03-1546. URL: https://arxiv.org/abs/2311.13479v3.

Claim. The paper defines projective toric t-designs on P(T^n), constructs them from difference and Sidon sets, disproves Zhu's conjecture on uniformly weighted quantum state 2-designs, shows that complete sets of mutually unbiased bases in dimension d correspond to P(T^d) 2-designs of size d^2 satisfying an overlap condition, finds by exhaustive search that no group design of size 36 in P(T^6) satisfies it, and conjectures that in any dimension a P(T^d) 2-design of size d^2 that is not a subgroup does not satisfy the overlap condition.

Problem details · Reading note

Kagey's A343881 eventual equality conjecture

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

P. Kagey; OEIS Foundation Inc. (2021). OEIS A343881, power-product minima and A064549. URL: https://oeis.org/A343881.

Claim. Conjecture: T(p,k) = A064549(k) for fixed k > 1 and sufficiently large p.

Problem details · Reading note

Bipartite encoder forms in an operation-generated class differ only by permutations and row operations

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Andrey Boris Khesin, Alexander M. Li (2024). Equivalence Classes of Quantum Error-Correcting Codes. DOI: 10.48550/arXiv.2406.12083. URL: https://arxiv.org/abs/2406.12083v1.

Claim. Conjecture 7.1 states that connected equivalence classes of normalized [n,k] encoders have sizes divisible by 3^k for positive integers n > k; Conjecture 8.3 states that bipartite forms in one equivalence class are related by output permutations, input permutations and row operations alone.

Problem details · Reading note

Weak mixing and neighborhood shadowing without small periodic sets

Proved. Lean theorem quadratic_progression_example, frozen in this repository 2026-11-04.

Jonathan Meddaugh and Elyssa Stephens (2025). Shadowing with small diameter sets of bounded cardinality. URL: https://arxiv.org/abs/2509.04312v1.

Claim. Question 5.3 asks for neighborhood two-shadowing without exact-length quasi-finite type; Question 5.6 asks for weak mixing and neighborhood N-shadowing without dense small periodic sets.

Problem details · Reading note

A closed formula for the bubble-eigenstate configuration count

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Bhaskar Mukherjee, Debasish Banerjee, K. Sengupta, Arnab Sen (2021). Minimal model for Hilbert space fragmentation with local constraints. DOI: 10.1103/PhysRevB.104.155117. URL: https://arxiv.org/abs/2106.14897v3.

Claim. The paper studies a one-dimensional spin-1/2 chain with no two adjacent up spins whose Hamiltonian H_3 flips three-site patterns, exhibits Hilbert space fragmentation with bubble eigenstates built from three-site X and Y units separated by at least two down spins, tabulates their degeneracies for L = 16, 18, 20, and states that it has not found a closed form for these degeneracies for arbitrary L.

Problem details · Reading note

A cooling-map transition has no finite ground-space thermal realization

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Claim. In the low-temperature limit, thermal operations act with an ancilla in its ground space; the paper introduces cooling maps, channels whose Kraus operators are diagonal or lower a single energy level, characterizes their state transitions, and conjectures (Conjecture 1) that cooling maps are equivalent to low-temperature thermal operations with regard to the feasibility of state transitions.

Problem details · Reading note

The complete first growth assertion of OEIS A222759

Proved. Lean theorem result, frozen in this repository 2026-11-04.

T. D. Noe (2013). OEIS A222759: first growth assertion for literal binomial remainders. URL: https://oeis.org/A222759.

Claim. It appears that, for k > 2 and n >= prime(prime(k)^3), then a(n) >= k.

Problem details · Reading note

Oudra's A248186 cubic floor formula

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Clark Kimberling; Ridouane Oudra (2023). OEIS A248186, least reciprocal-product threshold index. URL: https://oeis.org/A248186/internal.

Claim. Conjecture: a(n) = floor(r*n + r^2/n - 1), where r = 3^(-1/3), for n > 2. - _Ridouane Oudra_, Oct 06 2023

Problem details · Reading note

Ta Problem 11.6: three invariants do not determine the count

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Lực Ta (2025). Good involutions of conjugation subquandles. DOI: 10.48550/arXiv.2505.08090. URL: https://arxiv.org/abs/2505.08090v5.

Claim. Problems 11.6 and 11.7 ask respectively whether three group invariants determine the whole-group good-involution count, and whether the Corollary 6.12 bound is always strict under its two lower bounds.

Problem details · Reading note

Ta Problem 11.7: the good-involution bound need not be strict

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Lực Ta (2025). Good involutions of conjugation subquandles. DOI: 10.48550/arXiv.2505.08090. URL: https://arxiv.org/abs/2505.08090v5.

Problem details · Reading note

Exact W-state Bell-state-measurement optimum

Proved. Lean theorem result, frozen in this repository 2026-11-04.

Sudip Chakrabarty and Amit Kundu (2026). Network nonlocality with tripartite entangled sources. DOI: 10.48550/arXiv.2610.35838. URL: https://arxiv.org/abs/2610.35838v1.

Claim. Proposition 1 expresses the squared Bell-state-measurement network optimum through orthogonal-direction quadratic optimization over two complete orthonormal frames and all four central-observable indices; the W-state paragraph gives the lower bound sqrt(13)/3 and reports no larger numerical value.

Problem details · Reading note

Destruction covariance need not be a superchannel fixed set

Refuted. Lean theorem result, frozen in this repository 2026-11-04.

Goni Yoeli and Gilad Gour (2026). Instability as a Quantum Resource. DOI: 10.48550/arXiv.2602.18323. URL: https://arxiv.org/abs/2602.18323v1.

Claim. The Open directions question after Eq. (29) asks whether destruction-covariant channels themselves arise as fixed points of an idempotent superchannel.

Problem details · Reading note

A320043: an additional square value

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Stefano Spezia; OEIS Foundation Inc. (2018). OEIS A320043: Row sums of the triangle A322550. URL: https://oeis.org/A320043/internal.

Claim. Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96.

Problem details · Reading note

An infinite twin-free theta family separating the Laplacian equalities

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Monther Rashed Alfuraidan, Suliman Khan, Lorenzo Sauras-Altuzarra (2026). Metric dimension and maximum eigenvalue multiplicity of graph Laplacians. DOI: 10.48550/arXiv.2610.26875. URL: https://arxiv.org/abs/2610.26875v1.

Claim. Problem 30 asks for infinite connected twin-free graph families satisfying exactly one of mmL(G)=dim(G) and mmQ(G)=dim(G).

Problem details · Reading note

Bala's A258899 and A258903 totient-period conjectures

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Paul D. Hanna; Peter Bala (2025). OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures. URL: https://oeis.org/A258899.

Claim. For each of A258899 and A258903: Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k).

Problem details · Reading note

The exact halting time of Wolfram machine 3626

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Peter Bala; Sean A. Irvine; Stephen Wolfram (2026). OEIS A393210: halting time for Wolfram machine 3626. URL: https://oeis.org/A393210.

Claim. Conjecture: a(n) = Sum_{d | n} m(d)^2 * phi(d), where m(n) = Sum_{d | n} (-1)^(d+1) * moebius(d) = A069517(n).

Problem details · Reading note

Skewness at most two does not force randomness at most one in multicyclic enzyme networks

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Andre C. Barato, Udo Seifert (2015). Skewness and Kurtosis in Statistical Kinetics. DOI: 10.1103/PhysRevLett.115.188103. URL: https://arxiv.org/abs/1510.03279v1.

Claim. For unicyclic enzymatic schemes with an irreversible product release, the randomness parameter, skewness and kurtosis of the cycle completion time are bounded above by 1, 2 and 9; for a four-state two-cycle scheme the paper finds numerically that a constraint S <= 2 keeps the randomness parameter at most 1, and leaves open whether this ordering holds for arbitrary networks that do not consist of a single cycle.

Problem details · Reading note

Three-central edge extremality against all metrics

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Oleksiy Dovgoshey and Olga Rovenska (2026). On the Graph Corresponding to Center of Distances of Metric Space. DOI: 10.48550/arXiv.2610.11596. URL: https://arxiv.org/abs/2610.11596v1.

Claim. Conjecture 4.2 characterizes three-central ultrametric edge minima against all metric spaces of the same order.

Problem details · Reading note

Every three-site improper marginalization is birational for open-boundary binary bond-two MPS

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Critch, Andrew; Morton, Jason (2014). Algebraic geometry of matrix product states. DOI: 10.3842/SIGMA.2014.095. URL: https://arxiv.org/abs/1210.2812v4.

Claim. The paper studies the varieties of translation-invariant matrix product states with periodic and open boundary conditions, gives defining equations in small cases (among them the quartic hypersurface of open-boundary binary bond-dimension-two MPS on three sites), and states identifiability conjectures, including that each improper marginalization of open-boundary 2x2 binary MPS to three consecutive sites is a birational equivalence (Conjecture 8).

Problem details · Reading note

The NHCRB/HCRB ratio for full orthonormal qudit tomography is at most $d+1$

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Das, Aritra; Conlon, Lorcán O.; Suzuki, Jun; Yung, Simon K.; Lam, Ping K.; Assad, Syed M. (2025). Holevo Cramér-Rao bound: How close can we get without entangling measurements?. DOI: 10.22331/q-2025-09-30-1867. URL: https://arxiv.org/abs/2405.09622v2.

Claim. The paper compares the Holevo Cramér-Rao bound (collective measurements) with the Nagaoka-Hayashi Cramér-Rao bound (separable measurements) in multiparameter estimation, proves that their ratio is at most n for n parameters and at most d+2 for full orthonormal-basis tomography of a d-dimensional state, shows the value d+1 at the maximally mixed state, and proposes that the ratio for the linear generalized Gell-Mann model is at most d+1.

Problem details · Reading note

Entropy saturation of a Lüders MIC channel forces SIC existence

Proved. Lean theorem result, frozen in this repository 2026-11-03.

DeBrota, John B.; Stacey, Blake C. (2019). Lüders channels and the existence of symmetric-informationally-complete measurements. DOI: 10.1103/PhysRevA.100.062327. URL: https://arxiv.org/abs/1907.10999v2.

Claim. The paper studies the Lüders channel of a minimal informationally complete measurement (MIC), shows that it acts as the depolarizing channel (I + rho)/(d+1) exactly when the MIC is a SIC, bounds the Haar-averaged output entropy of every Lüders MIC channel on pure inputs by log(d+1) - 2 log 2/(d+1) with equality if a SIC exists, and conjectures that equality holds only if a SIC exists (Conjecture 1).

Problem details · Reading note

A non-exhaustive periodic orbit in Fernandez's three-cell degrade-and-fire chain

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Bastien Fernandez (2018). Revisiting asymptotic periodicity in networks of degrade-and-fire oscillators. DOI: 10.1140/epjst/e2018-800068-8. URL: https://arxiv.org/abs/1809.03228v1.

Claim. For networks of degrade-and-fire oscillators the paper improves the conditions for asymptotic periodicity under exhaustive firing, analyses the unidirectional 3-cell chain, shows that the simplest non-exhaustive periodic orbit of the epsilon > 1/2 transition graph cannot exist, and states the belief that the same impossibility holds for every other periodic trajectory with non-exhaustive firing in that graph.

Problem details · Reading note

Three two-strand boxes refute Jones–Wenzl generation at a root of unity

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Steven M. Flores and Eveliina Peltola (2018). Generators, projectors, and the Jones-Wenzl algebra. DOI: 10.48550/arXiv.1811.12364. URL: https://arxiv.org/abs/1811.12364v2.

Claim. Theorem 1.1 proves that the unit P_ς and the elements U_i = P_ς U_{s_1+...+s_i} P_ς form a minimal generating set of the Jones–Wenzl algebra JW_ς(ν) = P_ς TL_{n_ς}(ν) P_ς when n_ς < p̄(q); Conjecture 1.2 asks whether item 1 holds whenever max ς < p̄(q).

Problem details · Reading note

Three product observables with useless pairs and a useful triple

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Gachechiladze, Mariami; Wyderka, Nikolai; Gühne, Otfried (2018). The structure of ultrafine entanglement witnesses. DOI: 10.1088/1751-8121/aad3dc. URL: https://arxiv.org/abs/1805.06404v2.

Claim. The paper characterizes when two product observables are useful for entanglement detection (necessary and sufficient for two qubits, with counterexamples for two qutrits), proves that a pair whose local factors commute on one side is useless, and asks for three product observables each pair of which is useless while the triple is useful.

Problem details · Reading note

The generic quadratic Nambu KHK map has a Cremona decomposition

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Michele Graffeo, Giorgio Gubbiotti (2024). Growth and Integrability of Some Birational Maps in Dimension Three. DOI: 10.1007/s00023-023-01339-5. URL: https://arxiv.org/abs/2301.09129v2.

Claim. The KHK discretisation of the Euler top decomposes as l2 o crem3 o l1 with two projectivities and the standard cubic Cremona map of P^3 (Lemma 2.6); Remark 7.7 conjectures that the KHK map of a quadratic three-dimensional Nambu system with positive definite Nambu-Hamiltonians is, up to conjugation by a projectivity, of the same type (A)-(i).

Problem details · Reading note

Granvik's A292273 real-part identity

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Mats Granvik (2024). OEIS A292273, real-part formula for the even-supported signed Mobius function. URL: https://oeis.org/A292273.

Claim. Conjecture: a(n) = Re(A008683(n)*(i^n)). - Mats Granvik, Jul 06 2024

Problem details · Reading note

A null direct summand in a tensor power of a singular semigroup module

Proved. Lean theorem result, frozen in this repository 2026-11-03.

David He and Daniel Tubbenhauer (2025). Tensor powers of representations of (diagram) monoids. DOI: 10.48550/arXiv.2508.04054. URL: https://arxiv.org/abs/2508.04054v1.

Claim. Conjecture 4.2 asserts that every singular finite-semigroup module has a one-dimensional null direct summand in some positive tensor power, and its indecomposable-summand count is asymptotic to the tensor dimension.

Problem details · Reading note

Hwang–Kim: log-Euclidean to Wasserstein near order

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Jinmi Hwang; Sejong Kim (2024). Near-order relation of power means. DOI: 10.1007/s11117-024-01099-4. URL: https://arxiv.org/abs/2407.07438v1.

Claim. The final unnumbered question in Section 6 asks whether LE(ω; 𝔸) ≼ Ω(ω; 𝔸) for positive definite matrices and strictly positive probability weights.

Problem details · Reading note

Jenzer's permuted rotor-loop formula without the star condition

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Nik Jenzer (2026). Star-Ordered Rotor Loops in Clifford Algebras. DOI: 10.48550/arXiv.2610.21593. URL: https://arxiv.org/abs/2610.21593v1.

Claim. The source's scope remark asks whether the permuted loop formula holds for anticommuting unit bivectors without the star condition.

Problem details · Reading note

Refutation of the complete original Jaco addition law

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa (2014). Characteristics of Finite Jaco Graphs, J_n(1), n in N. DOI: 10.48550/arXiv.1404.0484.

Claim. Theorem 2.10 (Conjectured) gives a two-branch maximum-total-degree addition law for all distinct orders at least three.

Problem details · Reading note

The original Jaco directed path-count equivalence

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian Mukungunugwa (2014). Characteristics of Jaco Graphs, J_infinity(a), a in N. URL: https://arxiv.org/abs/1404.1714v1.

Claim. Proposition 4.4 conjectures the equivalence of both adjacent infinite-outdegree inequalities and both adjacent shortest-directed-path-count inequalities for every k at least seven in every Fibonacci bracket.

Problem details · Reading note

Kok's unique maximum-degree values and infinite realizing family

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Johan Kok (2025). Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs. DOI: 10.48550/arXiv.2507.16500.

Claim. Conjecture 2.2 identifies maximum degrees with the golden floor and literal A319433 deletion; Conjecture 2.9 bounds a primary minimal dom-path by the graph diameter plus one, while Conjecture 2.12 says that A000149 indexes a gamma-set of J_infinity(x) and is p-graphical for the finite J_n(x).

Problem details · Reading note

The three-distinguishability region in dimension four is not of product form

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Korzekwa, Kamil; Czachórski, Stanisław; Puchała, Zbigniew; Życzkowski, Karol (2019). Distinguishing classically indistinguishable states and channels. DOI: 10.1088/1751-8121/ab30f7. URL: https://arxiv.org/abs/1812.09083v3.

Claim. The paper defines M-distinguishability regions of the probability simplex (classical versions admitting M perfectly distinguishable quantum states), bounds them by permutohedra, analyses the case d=4, M=3 (edges of the permutohedron belong to the region, faces do not), and conjectures from numerical evidence that the points of the region have a product structure over the four face centres (Eq. (30)); it also studies classically indistinguishable channels.

Problem details · Reading note

Kurkov's three-parameter Stirling-minor Vandermonde sum

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Mikhail Kurkov (2026). OEIS A396563: three-parameter Stirling-minor Vandermonde sum. URL: https://oeis.org/A396563/internal.

Claim. The first-kind Stirling minor B(n,k,m) equals the normalized nested Vandermonde-square sum for n,k >= 0 and m >= 1.

Problem details · Reading note

Li–Guo Conjecture 2.22

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Wen-Jun Li; Ji-Ming Guo (2025). A new conjecture on Laplacian eigenvalues and degree sequence of a graph. DOI: 10.2298/FIL2513551L. URL: https://www.pmf.ni.ac.rs/filomat-content/2025/39-13/39-13-22-25952.pdf.

Claim. Conjecture 2.22 bounds the truncated signless Laplacian square sum by the degree sequence.

Problem details · Reading note

Mourad–Abbas Conjecture 5.1 is false

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Bassam Mourad; Hassan Abbas (2013). On the symmetric doubly stochastic matrices that are determined by their spectra. DOI: 10.48550/arXiv.1310.1273. URL: https://arxiv.org/abs/1310.1273v1.

Claim. Conjecture 5.1 places every spectrally determined symmetric doubly stochastic matrix of positive trace on specified closed segments of the full symmetric polytope.

Problem details · Reading note

Morken's first A112399 divisor-sum conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Jamie Morken (2026). First divisor-sum conjecture in OEIS A112399. URL: https://oeis.org/A112399/internal.

Claim. For every n>=1, sum_{1<=k<=n,gcd(k,n)=1} mu(k) = sum_{d|n,mu(d)!=0} sum_{1<=m<=n/d,gcd(m,d)=1} mu(m).

Problem details · Reading note

Werner's all-column formula for the affine polynomial pair

Proved. Lean theorem werner_a208509, frozen in this repository 2026-11-03.

Knud Werner (2022). Conjectured coefficient formula in OEIS A208509. URL: https://oeis.org/A208509.

Claim. Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - _Knud Werner_, Jan 11 2022

Problem details · Reading note

Absolute Hermite coefficient modes: the complete A381524 difference assertion

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Mike Sheppard (2025). OEIS A381524, smallest exponent of a maximal absolute Hermite coefficient. URL: https://oeis.org/A381524/internal?rev=18.

Claim. Conjecture: Differences are either +1 or -1.

Problem details · Reading note

The complete A381737 Conjecture 1 enumeration

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Mike Sheppard (2025). Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice. URL: https://oeis.org/A381737/internal?rev=28.

Claim. Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).

Problem details · Reading note

The complete A381737 Conjecture 3 generating function

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Mike Sheppard (2025). Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice. URL: https://oeis.org/A381737/internal?rev=28.

Claim. Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).

Problem details · Reading note

The complete A381737 Conjecture 2 recurrence

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Mike Sheppard (2025). Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice. URL: https://oeis.org/A381737/internal?rev=28.

Claim. Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).

Problem details · Reading note

The sharp upper bound on skewness minus kurtosis of a unidirectional unicyclic current

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Claim. The paper derives bounds on the skewness and kurtosis of steady-state currents for several classes of systems; for unicyclic Markovian networks it infers numerically that the extremal values are attained when all clockwise rates are equal but one, and on that topology finds the minimum skewness -(8-8N+N^2)/(16(N-1)^2) of a unidirectional N-state cycle at a = 1/(N-1); in the limit N -> infinity it gives the minimum kurtosis -(1+sqrt(5))/10 for a = A/N, A = (3-sqrt(5))/2, and the minimum of skewness plus kurtosis -8/27 for A = 1/2; it lists the product of skewness and kurtosis in [-0.054, 1] (eq. produni), the lower value approximate.

Problem details · Reading note

The sharp lower bound on skewness times kurtosis for unidirectional unicyclic currents

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Problem details · Reading note

Ptaszyński's exact lower endpoint of skewness minus kurtosis

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Problem details · Reading note

The expected two-bath Bose-selection bound is refuted for cubic spectra

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Schnell, Alexander; Ketzmerick, Roland; Eckardt, André (2018). On the number of Bose-selected modes in driven-dissipative ideal Bose gases. DOI: 10.1103/PhysRevE.97.032136. URL: https://arxiv.org/abs/1802.04088v2.

Claim. The paper characterizes the Bose-selected states of driven-dissipative ideal Bose gases by the condition mu = A eta on the rate-asymmetry matrix, shows that two ohmic baths with product coupling select at most three states (M_S <= 2 M_B - 1 for M_B baths), and expects the same bound for non-ohmic spectral densities |epsilon|^d sgn(epsilon).

Problem details · Reading note

A connected K₄-free graph with nonreal clique roots

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

Hossein Teimoori Faal (2021). Sums of Polynomials and Clique Roots. DOI: 10.48550/arXiv.2112.09744. URL: https://arxiv.org/abs/2112.09744v1.

Claim. Conjecture 4.1 (p. 5): Let G be a connected K₄-free graphs, then G has only clique roots.

Problem details · Reading note

wiseman-a344611

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Gus Wiseman (2021). OEIS A344611 — reverse-alternating sums of integer partitions. URL: https://oeis.org/A344611.

Claim. The number of partitions of 2n with nonnegative reverse-alternating sum is at most the number of partitions of 2n+1 into an odd number of parts.

Problem details · Reading note

Wolford's generalized binomial biroot convergence conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-03.

Isaac Wolford (2025). Combinatorial and Gaussian Foundations of Rational Nth Root Approximations: Theorems and Conjectures. DOI: 10.48550/arXiv.2508.14095. URL: https://arxiv.org/abs/2508.14095v3.

Claim. Conjecture 3.4: the generalized binomial biroot converges for all positive real x,c and every positive integer n.

Problem details · Reading note

Xu-Zhao Conjecture 1.3

Proved. Lean theorem full_result, frozen in this repository 2026-11-03.

Ce Xu and Jianqiang Zhao (2026). Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers. DOI: 10.48550/arXiv.2609.11072. URL: https://arxiv.org/html/2609.11072v1.

Claim. Equations (7)-(8) define the strict multiple polylogarithm and its depth-normalized reciprocal coefficients; Conjecture 1.3 asks for eventual coefficient signs and an admissible strict binomial lower bound.

Problem details · Reading note

Strict averaging need not improve detector-sensor probes

Refuted. Lean theorem result, frozen in this repository 2026-11-03.

C. Zhan, H. Gupta, M. Hillery (2024). Optimizing Initial State of Detector Sensors in Quantum Sensor Networks. DOI: 10.1145/3655028. URL: https://arxiv.org/abs/2306.17401v6.

Claim. Section 5, Conjecture 1 proposes a uniform coefficient-square state on any weight partition minimizing the common final-state overlap as an optimal initial detector-sensor state; Conjecture 2 claims that the coefficient-square average of two probes with equal error and unequal coefficient-squares has strictly smaller error.

Problem details · Reading note

Multiset rook-Eulerian interlacing on Ferrers boards

Refuted. Lean theorem result, frozen in this repository 2026-10-04. Lean theorem result, frozen in this repository 2026-11-02.

Per Alexandersson, Aryaman Jal, and Maena Quemener (2025). Real-rootedness of rook-Eulerian polynomials. DOI: 10.48550/arXiv.2502.05939. URL: https://arxiv.org/abs/2502.05939v1.

Claim. Conjecture 28: Ferrers-board multiset rook-Eulerian real-rootedness and pairwise interlacing of the decreasing first-letter refinements.

Problem details · Reading note

Refutation of order-33 minimality for domination root minus four

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Saeid Alikhani; Max Griswold (2026). On the Integer Domination Root Conjecture. DOI: 10.48550/arXiv.2608.00109. URL: https://arxiv.org/abs/2608.00109v1.

Claim. Section 4 item 2 asks whether order 33 is minimal for a connected ordinary domination root minus four; item 3 asks whether infinitely many connected finite simple graphs have that root.

Problem details · Reading note

The original Bajnok–Beatty Conjecture 4.1

Proved. Lean theorem result, frozen in this repository 2026-11-02.

B. Bajnok and W. K. Beatty (2024). On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups. URL: https://arxiv.org/abs/2406.04045v1.

Claim. Original Conjecture 1.6 states the sharp maximum order for finite noncyclic abelian groups with a signed two-element spanning set, and Theorem 1.7 supplies its lower constructions; Conjecture 4.1 gives the four-clause classification for Z2 times Z2k, with the order bound and low regimes in Theorems 4.2–4.4.

Problem details · Reading note

Bala's A295462 binary-spacing parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Peter Bala (2026). OEIS A295462: binary-spacing parity conjecture. URL: https://oeis.org/A295462.

Claim. Conjecture: a(n) is odd iff the 1 bits in the binary expansion of n are separated by at least three 0's, that is, iff n belongs to A048718.

Problem details · Reading note

The double-root row of Ballot–Kimberling–Moses Table 5

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Christian Ballot, Clark Kimberling and Peter J. C. Moses (2017). Linear Recurrences Originating From Polynomial Trees. DOI: 10.1080/00150517.2017.12427731. URL: https://www.fq.math.ca/Papers1/55-5/BallotKimberlingMoses.pdf.

Claim. Table 5 conjectures an eventual generation recurrence with coefficients 3, -3, 1 for the polynomial tree modulo x^2 - 2x + 1.

Problem details · Reading note

Chen and Chen's step-m Eulerian parity positivity

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Xin Chen and Yudian Chen (2026). OEIS A392664, A392665 and A392666: negative-step Eulerian triangles. URL: https://oeis.org/A392664.

Claim. Conjecture: T(n,k) >= 0 for all even n and all 1 <= k <= n, for m=-3,-4,-5.

Problem details · Reading note

The complete singular squared-distance Laplacian conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Joyentanuj Das; Sumit Mohanty (2024). Inverse of the Squared Distance Matrix of a Complete Multipartite Graph. DOI: 10.13001/ela.2024.8283. URL: https://arxiv.org/abs/2311.01069v2.

Claim. The concluding conjecture asks for both the exact positive eigenvalue multiplicities and the inertia of the singular squared-distance Laplacian-like matrix defined in equation (3.6).

Problem details · Reading note

A regular subset of a four-dimensional hyperdisk has no finite hyperdisk cover

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Ding, Feng; Hu, Xueyuan (2020). Masking quantum information on hyperdisks. DOI: 10.1103/PhysRevA.102.042404. URL: https://arxiv.org/abs/1909.11256v3.

Claim. The paper defines hyperdisks of pure states and their regular subsets, shows that the target states of a nondegenerate masking machine form a regular subset of a Schmidt hyperdisk, characterizes maskable sets in small dimensions, and asks whether a regular subset of a hyperdisk can be covered by finitely many hyperdisks.

Problem details · Reading note

Hanna's prime-power correction formula

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Paul D. Hanna (2025). OEIS A384819: bounded integral square-weight corrections. URL: https://oeis.org/A384819.

Claim. Conjecture: a(p^n) = p - 1 when p is prime for n >= 1.

Problem details · Reading note

The flag partition maximizes the squared sizes of affine parts outside the origin

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Arne Heimendahl; Markus Heinrich; David Gross (2022). The axiomatic and the operational approaches to resource theories of magic do not coincide. DOI: 10.1063/5.0085774. URL: https://arxiv.org/abs/2011.11651v3.

Claim. The paper shows that completely stabiliser-preserving channels strictly contain the stabiliser operations, describes stabiliser operations in a diagonal subpolytope through partitions of the nonzero vectors of F_d^n into affine subspaces, and conjectures which partition maximizes the resulting overlap bound.

Problem details · Reading note

The Huang–Stoica–Xia–Zhong conjecture on $T_2$ is refuted

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

An Huang; Bogdan Stoica; Xuyang Xia; Xiao Zhong (2020). Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs. URL: https://arxiv.org/abs/2006.06716v1.

Claim. The paper defines Ricci curvature and an action for weighted graphs, derives the equations of motion on regular trees, proves uniqueness of the constant solution under constant boundary conditions, constructs solutions on the line and on trees of even degree, and conjectures that the tree equations of motion on the 3-regular tree have no positive non-constant solution.

Problem details · Reading note

Binomial perfect-power counts over GF(2)

Proved. Lean theorem result, frozen in this repository 2026-11-02.

P. Kagey; OEIS Foundation Inc. (2022). OEIS A350532, sparse perfect powers over Z/2Z. URL: https://oeis.org/A350532.

Claim. Conjecture: T(2*n,2) = n. Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1.

Problem details · Reading note

Trinomial perfect-power counts over GF(2)

Proved. Lean theorem result, frozen in this repository 2026-11-02.

P. Kagey; OEIS Foundation Inc. (2022). OEIS A350532, sparse perfect powers over Z/2Z. URL: https://oeis.org/A350532.

Claim. Conjecture: T(2*n,2) = n. Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1.

Problem details · Reading note

The even Kimoto–Wakayama normalized Apéry congruence is refuted

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Kimoto, Kazufumi; Wakayama, Masato (2022). Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals. DOI: 10.4171/aihpd/129. URL: https://arxiv.org/abs/1905.01775v2.

Claim. The paper defines Apéry-like numbers J_k(n) from the first anomaly of the spectral zeta function of the non-commutative harmonic oscillator, normalizes them to rational numbers satisfying a common three-term inhomogeneous recurrence, relates their generating functions to automorphic integrals, and conjectures congruences modulo prime powers among the normalized numbers (Conjecture 3.6), proving weaker versions.

Problem details · Reading note

Kaygorodov–Martín González–Viruel Question 31

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Ivan Kaygorodov; Cándido Martín González; Antonio Viruel (2026). Evolution algebras: a survey of results and a list of open problems. DOI: 10.48550/arXiv.2610.22248. URL: https://arxiv.org/abs/2610.22248v1.

Claim. Conjecture 4 asks for the simple complex evolution-algebra automorphism maximum max{n!, n(2^n-1)}; Theorem 4 and Conjecture 3 concern perfect algebra automorphisms; Question 46 asks whether dense square forces uniqueness of a Hilbert natural orthonormal basis; Question 14 asks whether an infinite natural-basis quotient can coexist with finite full automorphisms; Question 31 asks about realizing arbitrary linear algebraic groups as full evolution-algebra automorphism groups or group schemes.

Problem details · Reading note

Kok's maximum degree and literal A319433 deletion

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Johan Kok (2025). Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs. DOI: 10.48550/arXiv.2507.16500.

Problem details · Reading note

Even double paths with a nonempty loop set allow nSSP

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Sara Koljančić and Polona Oblak (2026). Combinatorial aspects of the non-symmetric Strong Spectral Property for graphs. DOI: 10.48550/arXiv.2604.21552. URL: https://arxiv.org/abs/2604.21552v1.

Claim. Question 5.11 asks whether every even double path with a nonempty loop set allows the nSSP.

Problem details · Reading note

Three continuous invariant parity-odd variables are necessary for two interchangeable pairs

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Christopher G. Lester; Ward Haddadin; Ben Gripaios (2022). Lorentz and permutation invariants of particles III: constraining non-standard sources of parity violation. DOI: 10.1142/S0217751X22500932. URL: https://arxiv.org/abs/2008.05206v6.

Claim. The paper defines chirality of collider events under Lorentz and permutation symmetries, constructs sets of continuous Lorentz- and permutation-invariant parity-odd event variables that are nonzero on every chiral event of several event classes, and conjectures that its three-variable sets for events with two pairs of interchangeable momenta are minimal.

Problem details · Reading note

The Liu–DeBrota fewer-effects construction does not minimize orthogonality

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Yizhou Liu; John B. DeBrota (2021). Relating measurement disturbance, information, and orthogonality. DOI: 10.1103/PhysRevA.104.052216. URL: https://arxiv.org/abs/2105.02074v2.

Claim. The paper defines the disturbance, measurement strength and orthogonality of a POVM with Lüders updating, proves an information-disturbance trade-off relation in which orthogonality plays a role, identifies SICs as the measurements closest to a nondisturbing ideal, and conjectures that a specific block-diagonal projective POVM minimizes the orthogonality when there are fewer effects than the dimension.

Problem details · Reading note

Mathar's OEIS A174193 order-four recurrence

Proved. Lean theorem result, frozen in this repository 2026-11-02.

R. J. Mathar; OEIS Foundation Inc. (2016). OEIS A174193 — an order-four recurrence for an algebraic generating series. URL: https://oeis.org/A174193.

Claim. Mathar conjectures the displayed order-four polynomial-coefficient recurrence for the coefficients of the specified algebraic generating function.

Problem details · Reading note

The full weighted orthogonal-minor inequality

Proved. Lean theorem weighted_minors, frozen in this repository 2026-11-02.

meler (MathOverflow user 526848; real name unverified) (2024). Inequality involving minors of an orthogonal matrix. URL: https://mathoverflow.net/questions/469781.

Claim. The source asks the full weighted squared two-minor versus max-weighted squared three-minor inequality for every real orthonormal triple and every symmetric entrywise positive real matrix.

Problem details · Reading note

Affine support of the odd cyclic sign-matrix trace

Proved. Lean theorem affine_trace_support_dimension, frozen in this repository 2026-11-02.

ricardorr (MathOverflow user 49534; real-name attribution unverified) (2026). Does the support of a cyclic trace of 2x2 sign matrices always span a hyperplane in Z_2^n?. URL: https://mathoverflow.net/questions/512655.

Claim. For every positive odd n and every nonzero b in F_2^n, the affine hull over F_2 of the nonzero support of the specified cyclic integer sign-matrix trace has dimension at least n-1.

Problem details · Reading note

Inertia of a singular diagonal matrix after a structured rank-one downdate

Proved. Lean theorem mo513954_revision2, frozen in this repository 2026-11-02.

Lucid (MathOverflow user 596445) (2026). Inertia of a singular diagonal matrix after a structured rank-one downdate, revision 2. URL: https://mathoverflow.net/revisions/513954/2.

Claim. The source poses the complete structured-downdate inertia triple and continuous-family corollary and proposes the hyperplane decomposition and block congruences.

Problem details · Reading note

Curtz's six-residue denominator conjecture

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Paul Curtz (2011). OEIS A189733, denominators of the first row of a rational difference array. URL: https://oeis.org/A189733.

Claim. a(n) = denominator(B(0,n)). Conjecture: a(6*n)=1, a(1+6*n)=1+3*n, a(2+6*n)=1+2*n, a(3+6*n)=2+3*n, a(4+6*n)=1, a(5+6*n)=6+6*n. a(n) = 2*a(n-6) - a(n-12).

Problem details · Reading note

oeis-a378258-hanna-mod-six

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Paul D. Hanna (2024). OEIS A378258, a compositional square root and dyadic parity. URL: https://oeis.org/A378258.

Claim. Conjecture: a(n) is odd iff n is a power of 2.

Problem details · Reading note

oeis-a381423-hermite-differences

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Mike Sheppard (2025). OEIS A381423, exponent of the maximal signed Hermite coefficient. URL: https://oeis.org/A381423/internal.

Claim. Differences are either 1 or -3

Problem details · Reading note

Okada's pre-merged Gaussian determinants of width at least three are not almost palindromic

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Okada, So (2020). Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases. DOI: 10.48550/arXiv.2003.02112. URL: https://arxiv.org/abs/2003.02112v2.

Claim. The paper introduces merged-log-concavity of rational functions, constructs parcels and pre-merged determinants from Gaussian binomial coefficients, relates them to phase transitions of ideal boson-fermion gases, and conjectures that width-three pre-merged determinants with unit weights and base shifts are not almost palindromic.

Problem details · Reading note

Second-family singlet-like data exist in every dimension

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Pál, Károly F.; Tóth, Géza; Bene, Erika; Vértesi, Tamás (2021). Bound entangled singlet-like states for quantum metrology. DOI: 10.1103/PhysRevResearch.3.023101. URL: https://arxiv.org/abs/2002.12409v4.

Claim. The paper constructs two families of PPT entangled states on (2d x 2d)-dimensional systems whose quantum Fisher information 16 sqrt(d)/(1+sqrt(d)) exceeds every separable state; the second family is defined through orthogonal matrices with prescribed properties, constructed for d = 3 and d = 2^n, and the authors believe it exists for all d.

Problem details · Reading note

The correlation-minor bound fails without filter normal form

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Bar Y. Peled; Amit Te'eni; Avishy Carmi; Eliahu Cohen (2021). Correlation minor norms, entanglement detection and discord. DOI: 10.1038/s41598-021-82303-3. URL: https://arxiv.org/abs/2005.12079v2.

Claim. The paper defines the correlation minor norms of a bipartite state as Schatten norms of compound matrices of its correlation matrix, bounds them for separable states in filter normal form, constructs saturating states from quantum designs, and conjectures that the bounds hold without the filter-normal-form assumption.

Problem details · Reading note

The sharp lower bound on the kurtosis of a unidirectional unicyclic current

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Problem details · Reading note

The sharp lower bound on skewness plus kurtosis of a unidirectional unicyclic current

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Problem details · Reading note

A faithful mixed-state counterexample to KD-real SLD exclusion

Refuted. Lean theorem result, frozen in this repository 2026-11-02.

Matéo Spriet, Christopher Langrenez, Raymond Brummelhuis, Stephan De Bièvre (2026). What Is Special About the Kirkwood–Dirac Distributions? Only They Produce Natural Conditional Expectations. DOI: 10.48550/arXiv.2511.01996. URL: https://arxiv.org/abs/2511.01996v2.

Claim. In view of what precedes, it is tempting to conjecture that, if $\hat \rho$ and $\hat X$ are both KD real, the optimal observable $\hat L$ must not be KD-real itself, provided $\rm I_{QF}$ does not vanish.

Problem details · Reading note

Tournament solid-metric lower bounds

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Anni Hakanen and Pavan P D (2026). Solid-Resolving Sets on Directed Graphs. URL: https://arxiv.org/abs/2610.35041v1.

Claim. Conjecture 3.4 asserts the six tournament solid-metric dimension lower bounds 4, 5, 5, 5, 5, 6 at orders 6, 7, 8, 9, 10, 11.

Problem details · Reading note

The remaining correlation domain admits the weight gap exactly when $c<\sigma$

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Yuriy Stepanov; Hendrik Herrmann; Thomas Guhr (2021). Generic features in the spectral decomposition of correlation matrices. DOI: 10.1063/5.0054438. URL: https://arxiv.org/abs/2104.08966v1.

Claim. The paper bounds the largest eigenvalue of a correlation matrix and the weights of its eigenvectors on the normalized diagonal vector in terms of the mean correlation and the standard deviation of the correlation coefficients, shows that the top eigenvector carries the largest weight for large dimension on one subdomain of characteristics, constructs counterexamples on another, and asks what happens on the remaining domain.

Problem details · Reading note

Pure preparations cannot violate the HTMB inequality in finite dimension

Proved. Lean theorem result, frozen in this repository 2026-11-02.

Armin Tavakoli; Emmanuel Zambrini Cruzeiro; Roope Uola; Alastair A. Abbott (2021). Bounding and simulating contextual correlations in quantum theory. DOI: 10.1103/PRXQuantum.2.020334. URL: https://arxiv.org/abs/2010.04751v3.

Claim. The paper introduces a hierarchy of semidefinite relaxations that bounds quantum correlations in prepare-and-measure contextuality scenarios, applies it to several noncontextuality inequalities, and conjectures from a numerical bound that quantum models with pure preparations cannot violate the Hameedi-Tavakoli-Marques-Bourennane inequality.

Problem details · Reading note

Connected graphs with ordinary domination root minus four

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Saeid Alikhani; Max Griswold (2026). On the Integer Domination Root Conjecture. DOI: 10.48550/arXiv.2608.00109. URL: https://arxiv.org/abs/2608.00109v1.

Claim. Section 4 item 2 asks whether order 33 is minimal for a connected ordinary domination root minus four; item 3 asks whether infinitely many connected finite simple graphs have that root.

Problem details · Reading note

Browne's Chebyshev numerator and floor product

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Anthony Browne; Wolfdieter Lang (2016). OEIS A164660: Chebyshev row numerators and Browne's floor product. URL: https://oeis.org/search?q=id:A164660&fmt=text.

Claim. a(n+1) = Product_{k=1..n} ( 1-2*(floor(k^n/n)-floor((k^n -1)/n)) ) = (-1)^(A003557(n)) for n>0 (conjecture).

Problem details · Reading note

Blecher's reversible four-by-four algebra is not unique up to unitary equivalence

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

David P. Blecher (2025). Commutativity of operator algebras. DOI: 10.1016/j.exmath.2025.125742. URL: https://arxiv.org/abs/2509.17673v2.

Claim. The paper studies reversible operator algebras, those whose reversed multiplication with the same matrix norms is again an abstract operator algebra, shows that they are 3-commutative, constructs an anticommuting noncommutative reversible subalgebra of M_4 from the canonical anticommutation relations, and conjectures that up to unitary equivalence it is the only noncommutative reversible 4×4 matrix algebra.

Problem details · Reading note

The Bu–Gu–Jaffe mixed-state quantum inverse sumset bound is false

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Kaifeng Bu; Weichen Gu; Arthur Jaffe (2025). Quantum Ruzsa Divergence to Quantify Magic. DOI: 10.1109/TIT.2025.3543276. URL: https://arxiv.org/abs/2401.14385v3.

Claim. The paper defines a quantum Ruzsa divergence and a quantum-doubling constant through the qudit quantum convolution, proves a quantum inverse sumset theorem for pure states bounding the relative entropy to the mean state by the doubling constant, the rank of the mean state and the magic gap, and expects the analogous bound with a purity factor to hold for mixed states.

Problem details · Reading note

A closest fully separable state of an X-MEMS need not have the conjectured shape

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Mirko Consiglio; Tony J. G. Apollaro; Marcin Wieśniak (2022). Variational approach to the quantum separability problem. DOI: 10.1103/PhysRevA.106.062413. URL: https://arxiv.org/abs/2209.01430v2.

Claim. The paper introduces a variational algorithm that finds the closest fully separable state of a multiqubit state in the Hilbert–Schmidt distance, applies it to GHZ states and to maximally entangled mixed X-states (X-MEMS), and conjectures an explicit form (A11) for the closest separable state of the n-qubit X-MEMS.

Problem details · Reading note

Irwin's hook-sum conjecture for OEIS A138099

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Paul Curtz and Benedict W. J. Irwin (2016). OEIS A138099: irregular triangle and Irwin's hook-sum conjecture. URL: https://oeis.org/A138099.

Claim. Irwin conjectures that each term is the hook-sum increment of the displayed spiral tableaux.

Problem details · Reading note

Swapping an earliest adjacent pair minimizes dephasing for every number of segments

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Lars Kamin; Evgeny Shchukin; Frank Schmidt; Peter van Loock (2023). Exact rate analysis for quantum repeaters with imperfect memories and entanglement swapping as soon as possible. DOI: 10.1103/PhysRevResearch.5.023086. URL: https://arxiv.org/abs/2203.10318v1.

Claim. The paper computes exact secret-key rates of linear quantum repeaters with up to eight segments, memory dephasing and parallel entanglement distribution, defines the dephasing of the scheme that always swaps an earliest pair of adjacent segments and of the globally optimal swapping order, verifies that they coincide for up to eight segments, and conjectures that they coincide for every number of segments.

Problem details · Reading note

Question 1.10: regular Toeplitz and strictly ergodic strengthening

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Z. Kang; T. Jäger (2026). On lifts of strictly ergodic subshifts by permutative sliding block codes. DOI: 10.48550/arXiv.2609.01140. URL: https://arxiv.org/abs/2609.01140v1.

Claim. Question 1.10 asks whether a minimal or strictly ergodic subshift admits a permutative code of length at least two with a minimal or strictly ergodic complete inverse image, optionally under a Toeplitz hypothesis.

Problem details · Reading note

The minimum number of arcs requiring nSSP

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Sara Koljančić; Polona Oblak (2026). Combinatorial aspects of the nonsymmetric strong spectral property for graphs. DOI: 10.13001/ela.2026.10523. URL: https://arxiv.org/abs/2604.21552v1.

Claim. Problem 1.8 asks whether the minimum number of arcs in an n-vertex digraph requiring nSSP is 2n-1.

Problem details · Reading note

Malkoun's two-matrix elementary symmetric inequality

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Malkoun (2026). Promoting polynomial equalities for nonnegative variables to elementary symmetric polynomial inequalities. URL: https://mathoverflow.net/questions/512579.

Claim. For all complex Hermitian positive semidefinite X,Y and every 1 ≤ k ≤ n, e_k(lambda(XY+YX)) ≤ e_k(lambda(X²+Y²)).

Problem details · Reading note

Identity and transposition classes have the two largest tensor high-temperature degrees

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Denjoe O'Connor and Sanjaye Ramgoolam (2025). Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle. DOI: 10.48550/arXiv.2506.18813. URL: https://arxiv.org/abs/2506.18813v2.

Claim. The paper expresses the canonical partition functions of gauged permutation invariant tensor quantum mechanics as sums over partitions of N with factors built from least common multiples of parts, conjectures that the identity and transposition classes give the two leading high-temperature contributions for every tensor rank s, and asks for a proof.

Problem details · Reading note

The normalized nonlocality–contextuality trade-off holds for every binary n-cycle

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Lucas E. A. Porto, Gabriel Ruffolo, Rafael Rabelo, Marcelo Terra Cunha and Paweł Kurzyński (2024). Trade-off relations between Bell nonlocality and local Kochen–Specker contextuality in generalized Bell scenarios. DOI: 10.1088/1367-2630/ad7167. URL: https://arxiv.org/abs/2405.02964v2.

Claim. In generalized Bell scenarios where Alice has two dichotomic measurements and Bob an n-cycle contextuality setup, normalized generalized-Bell and noncontextuality inequalities satisfy S_GBell + S_NC <= 1 on every nonsignaling and nondisturbing behavior for n = 3, 4, 5; the paper conjectures the same for every n.

Problem details · Reading note

Every classical six-leg perfect tensor factorization has perfect two-wire factors

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Balázs Pozsgay; Ian M. Wanless (2024). Tensor network decompositions for absolutely maximally entangled states. DOI: 10.22331/q-2024-05-08-1339. URL: https://arxiv.org/abs/2308.07042v3.

Claim. The paper studies decompositions of six-leg perfect tensors (absolutely maximally entangled states) into networks of four-leg tensors, treats states from classical maps through orthogonal arrays, solves the linear case over finite fields, where the factors of a decomposition of a perfect tensor are themselves perfect, and leaves the nonlinear case of this statement open.

Problem details · Reading note

Second adjacency eigenvalue of the inner-face Apollonian stages

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Rad Sarar Anando (2026). Second adjacency eigenvalue of Apollonian networks: is theta_2(G_m)/2^(m/2) -> sqrt(2)?. URL: https://mathoverflow.net/questions/516405.

Claim. The second largest ordinary adjacency eigenvalue of the deterministic inner-face Apollonian stages, divided by 2^(m/2), converges to sqrt(2).

Problem details · Reading note

The perp hyperplane refutes the three-qubit hexagon contextuality generalization

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Metod Saniga; Frédéric Holweck; Colm Kelleher; Axel Muller; Alain Giorgetti; Henri de Boutray (2025). Hexagons govern three-qubit contextuality. DOI: 10.22331/q-2025-01-20-1601. URL: https://arxiv.org/abs/2312.07738v3.

Claim. The paper shows that classically-embedded split Cayley hexagons of order two in the three-qubit symplectic polar space W(5,2) share with the whole space, with both types of quadrics and with doilies exactly as many lines as the degree of contextuality of these line configurations, and conjectures that this holds for any contextual sub-geometry of W(5,2) whose contexts are lines.

Problem details · Reading note

Shapiro Conjecture 1(iii): identity words on finite trees

Proved. Lean theorem result, frozen in this repository 2026-11-01.

Boris Shapiro (2026). Transposition Cayley Spectra and Subgraph Counts. URL: https://arxiv.org/abs/2610.14373v1.

Claim. Conjecture 1(iii) characterizes identity edge words on every finite tree by multiplicity two and vertexwise noncrossing.

Problem details · Reading note

A three-qubit counterexample to the Shannon branch-and-bound lower bound

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Poetri Sonya Tarabunga; Martina Frau; Tobias Haug; Emanuele Tirrito; Lorenzo Piroli (2025). A nonstabilizerness monotone from stabilizerness asymmetry. DOI: 10.1088/2058-9565/adfd0d. URL: https://arxiv.org/abs/2411.05766v1.

Claim. The paper defines the basis-minimized stabilizerness asymmetry, a Rényi entropy of stabilizer-basis measurement outcomes minimized over stabilizer bases, proves its monotonicity properties, computes it by a branch-and-bound method whose lower bound it proves for integer Rényi index above one, and conjectures that the lower bound remains valid for the Shannon entropy.

Problem details · Reading note

The orthogonal absolute-product minimum is attained at one

Proved. Lean theorem result, frozen in this repository 2026-11-01.

User3 (MathOverflow user 600125) (2026). A nontrivial lower bound for the orthogonally transformed absolute product. URL: https://mathoverflow.net/questions/515549.

Claim. The question asks for a strict uniform lower bound above one, or an intrinsic characterization of the infimum of rho(abs(PQ) abs(transpose(Q) inverse(P))) over real orthogonal Q for nonsingular real square P.

Problem details · Reading note

The optimized GHZ witness bound need not equal the geometric GME measure

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Xin-Yu Xu; Qing Zhou; Shuai Zhao; Shu-Ming Hu; Li Li; Nai-Le Liu; Kai Chen (2023). General framework for genuine multipartite entanglement detection. DOI: 10.1103/PhysRevA.107.052405. URL: https://arxiv.org/abs/2211.05563v1.

Claim. The paper constructs genuine multipartite entanglement witnesses from bipartite witnesses, gives for noisy n-qudit GHZ states a witness finer than the fidelity witness, derives from it a lower bound on the geometric measure of genuine multipartite entanglement by the Legendre-transform method, and leaves open whether this lower bound equals the geometric measure.

Problem details · Reading note

Yoshida's rapid-decrease closure question

Refuted. Lean theorem result, frozen in this repository 2026-11-01.

Rintaro Yoshida (2010). On some questions of Fisk and Brändén. DOI: 10.48550/arXiv.1005.4218. URL: https://arxiv.org/abs/1005.4218v1.

Claim. Open Problem 5 asks whether the coefficient transform a_k^2-a_{k-r}a_{k+r} preserves positive rapidly decreasing sequences.

Problem details · Reading note

GHZ phase tangents refute a rank-only lower bound for normalized Pauli Fisher information

Refuted. Lean theorem result, frozen in this repository 2026-10-31.

Anirudh Acharya, Theodore Kypraios and Mădălin Guţă (2016). Statistically efficient tomography of low rank states with incomplete measurements. DOI: 10.1088/1367-2630/18/4/043018. URL: https://arxiv.org/abs/1510.03229v2.

Claim. For multiple-ion tomography with Pauli settings, the paper studies the smallest eigenvalue of G^{-1/2} Ī G^{-1/2}, where Ī is the Fisher information averaged over all 3^n settings and G the Frobenius metric of a local parametrisation of rank-r states, and conjectures that for any fixed rank it is bounded below by a constant for all states of rank r and arbitrarily many ions.

Problem details · Reading note

The Fuchs–Caves fidelity measurement need not give optimal ROC points

Refuted. Lean theorem result, frozen in this repository 2026-10-31.

András Bodor and Mátyás Koniorczyk (2017). Receiver Operation Characteristics of Quantum State Discrimination. DOI: 10.1007/s10946-017-9629-5. URL: https://arxiv.org/abs/1603.06801v1.

Claim. For two quantum states, binary discriminators give points (false-positive rate, true-positive rate) of a convex ROC domain whose upper boundary is the optimal ROC curve; the projective measurement of the Fuchs–Caves fidelity observable defines d+1 cumulative ROC points, and the paper leaves open whether these points lie on the optimal ROC curve.

Problem details · Reading note

The plus-count statistic is insufficient for exact-weight discrimination

Refuted. Lean theorem result, frozen in this repository 2026-10-31.

Leonardo Bohac (2025). Bias-Class Discrimination of Universal QRAM Boolean Memories. DOI: 10.48550/arXiv.2512.17503. URL: https://arxiv.org/abs/2512.17503v1.

Claim. The paper derives the single-query ensemble state and the single-copy Helstrom test for exact-weight bias classes of Boolean memories queried through a universal QRAM, gives a separable multi-query strategy based on the count of plus outcomes, and leaves open whether that count statistic is sufficient for optimal discrimination.

Problem details · Reading note

Factorized ladders for generalized quantum Zernike Hamiltonians

Proved. Lean theorem result, frozen in this repository 2026-10-31.

R. Campoamor-Stursberg, F. J. Herranz, D. Latini, I. Marquette, A. Blasco (2026). Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum. DOI: 10.1088/1361-6544/ae64a2. URL: https://arxiv.org/abs/2502.02491v2.

Claim. Conjecture 1(iii) asserts the existence of number and ladder operators with the factorized structure function (5.1) for every order N.

Problem details · Reading note

Trace-distance anticoherence can decrease under covariant spin noise

Refuted. Lean theorem result, frozen in this repository 2026-10-31.

Jérôme Denis, Tara Lacaille, John Martin and Eduardo Serrano-Ensástiga (2026). Total, quantum, and classical measures of anticoherence for mixed spin states. DOI: 10.48550/arXiv.2605.29436. URL: https://arxiv.org/abs/2605.29436v1.

Claim. The paper axiomatizes total t-anticoherence measures for mixed spin-j states, including monotonicity (T4) under SU(2)-covariant channels, proves (T4) for the Hilbert–Schmidt distance-based measure, and leaves open whether (T4) holds for the trace-distance-based measure.

Problem details · Reading note

Hanna's A369673 square classification and five-adic orders

Proved. Lean theorem result, frozen in this repository 2026-10-31.

Paul D. Hanna; N. J. A. Sloane; Reinhard Zumkeller (2024). OEIS A369673, square classification and five-adic orders, with A127428 and A112765. URL: https://oeis.org/A369673/internal.

Claim. A369673 conjectures C.1-C.4 classify square values and give exact five-adic orders using the factorial valuation difference A127428.

Problem details · Reading note

Linear spin measurements never violate CHSH for any spin $s\ge1$

Proved. Lean theorem result, frozen in this repository 2026-10-31.

Louis Hanotel and Elena R. Loubenets (2025). Nonviolation of the CHSH inequality under local spin-1 measurements on two spin qutrits. DOI: 10.1088/1402-4896/adcc66. URL: https://arxiv.org/abs/2503.05172v2.

Claim. For two spin-s qudits measured by the spin projections S_r=r·S with unit vectors r, the CHSH inequality has local bound 2s^2; the paper finds analytically and numerically that spin-1 measurements violate it for no tested two-qutrit state, conjectures nonviolation for every two-qutrit state, and states that whether the CHSH inequality is violated under high-spin s≥1 measurements is still open.

Problem details · Reading note

Hardin–Barker five-column clockwise-edge-difference formulas

Proved. Lean theorem result, frozen in this repository 2026-10-31.

R. H. Hardin; Colin Barker (2012). OEIS A209723: five-column arrays with distinct clockwise edge differences. URL: https://oeis.org/A209723/internal.

Claim. The five-column labelled-array count satisfies Barker's two parity formulas, the empirical third-order recurrence and the conjectured rational generating function.

Problem details · Reading note

The worst-case CNOT count for reflection-symmetric diagonal unitaries

Proved. Lean theorem result, frozen in this repository 2026-10-31.

Xinchi Huang, Taichi Kosugi, Hirofumi Nishi and Yu-ichiro Matsushita (2024). Optimized Synthesis of Circuits for Diagonal Unitary Matrices with Reflection Symmetry. DOI: 10.7566/JPSJ.93.054002. URL: https://arxiv.org/abs/2310.06676v2.

Claim. Every 2^{2n} x 2^{2n} diagonal unitary whose phases satisfy the reflection symmetry theta(j,j') = theta(2^n-1-j, 2^n-1-j') is realized up to a global phase by an ancilla-free {CNOT, R_z} circuit with 2^{2n-1}+2n-2 CNOT gates; the paper conjectures that this count is optimal in the worst case.

Problem details · Reading note

The CZX parent Lindbladian reaches its conjectured even-ring steady state

Proved. Lean theorem result, frozen in this repository 2026-10-31.

Yuhan Liu; Alberto Ruiz-de-Alarcón; Georgios Styliaris; Xiao-Qi Sun; David Pérez-García; J. Ignacio Cirac (2026). Parent Lindbladians for matrix product density operators. DOI: 10.1103/1qyd-59z7. URL: https://arxiv.org/abs/2501.10552v2.

Claim. Parent Lindbladians are built from renormalization channels of matrix product density operators; for the CZX example the four-site channel E = T∘S and the Lindbladian L = Σ_i (E_{2i-1} − 1) on a ring of N qubits are given, and the paper conjectures that the steady state reached from (1+σ_x^{⊗N})/2^N is (1 + 2^{1−N/2} U_CZX)/2^N for every even N, verified numerically for N = 4, 6, 8.

Problem details · Reading note

Mathar and Kimberling's self-convolution recurrence

Proved. Lean theorem result, frozen in this repository 2026-10-31.

R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Claim. The self-convolution sequence of A027170 satisfies the six-term integer recurrence in the A027182 FORMULA field for n at least 6.

Problem details · Reading note

The full alternating Laurent-row formula in A391810

Proved. Lean theorem full_formula_five, frozen in this repository 2026-10-31.

Paul D. Hanna (2025). OEIS A391810, nonpositive Laurent coefficients and zero row sums. URL: https://oeis.org/A391810/internal.

Claim. Formula (2): zero full row sums. Formula (4): Sum_{k=0..floor((n+1)/2)^2} k^2*T(n,k)=-(2*n-5) for n >= 2. Formula (5): the negative alternating full row sum equals 2*round(2^n/3)=2*A001045(n), for n >= 1 (all conjectures).

Problem details · Reading note

The first weighted moment of A391810

Proved. Lean theorem result, frozen in this repository 2026-10-31.

Paul D. Hanna (2025). OEIS A391810, nonpositive Laurent coefficients and zero row sums. URL: https://oeis.org/A391810/internal.

Problem details · Reading note

The full signed Laurent second moment in A391810

Proved. Lean theorem result, frozen in this repository 2026-10-31.

Paul D. Hanna (2025). OEIS A391810, nonpositive Laurent coefficients and zero row sums. URL: https://oeis.org/A391810/internal.

Problem details · Reading note

The zero higher-order EPR bound is tight at every positive order

Proved. Lean theorem result, frozen in this repository 2026-10-31.

E. Shchukin and P. van Loock (2016). Higher-order Einstein-Podolsky-Rosen correlations and inseparability conditions for continuous variables. DOI: 10.1103/PhysRevA.93.032114. URL: https://arxiv.org/abs/1511.08444v1.

Claim. For two bosonic modes and every positive integer n, ⟨(a†ⁿ ± bⁿ)(aⁿ ± b†ⁿ)⟩ ≥ n! for separable states and ≥ 0 for all states; for n = 2 the bound 0 is proved tight, and the paper conjectures that the bound 0 is tight for every n and that the separable bound cannot be violated by Gaussian states.

Problem details · Reading note

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Alastair A. Abbott, Julian Wechs, Fabio Costa and Cyril Branciard (2017). Genuinely multipartite noncausality. DOI: 10.22331/q-2017-12-14-39. URL: https://arxiv.org/abs/1708.07663v2.

Claim. In the lazy scenario, the inequality J₁(N) ≥ 0 is valid for every 2-causal N-partite correlation; the paper reports that it is a facet of the 2-causal polytope for N = 2, 3, 4 and conjectures that it is a facet for all N.

Problem details · Reading note

Andrade–Severini star-graph evolution maps are unitarily similar

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Fabiano M. Andrade; Simone Severini (2018). Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs. DOI: 10.1103/PhysRevA.98.062107. URL: https://arxiv.org/abs/1808.01154v2.

Claim. For a quantum star graph the Schrödinger approach gives the quantum evolution map U^S = S D and the Green's function approach gives U^G = D S, with S the vertex scattering matrix and D the diagonal edge-propagation matrix; the paper shows they have the same eigenvalues and states, as Claim 1, that they are unitarily similar, checked for n = 2, 3, 4, 5 with Specht's theorem, a proof for general n being not known.

Problem details · Reading note

The A126156 eventual totient-period conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Peter Bala (2026). OEIS A126156: eventual periods dividing the Euler totient. URL: https://oeis.org/A126156.

Claim. Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic with the period dividing phi(k), where phi(n) = A000010(n). An example is given below. - _Peter Bala_, Mar 01 2026

Problem details · Reading note

The exhaustive-pairs question for caterpillar principal specializations

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Aram Bingham, Kyle Celano, Patricia Commins, Rosa Orellana, Jianping Pan, and Mary Claire Simone (2026). Principal specializations of the t-chromatic symmetric function. DOI: 10.48550/arXiv.2610.31785. URL: https://arxiv.org/abs/2610.31785v1.

Claim. Remark 3.11 asks whether every positive-j listed pair has unequal second and equal third ordinary principal specializations, and whether the base and these pairs exhaust the phenomenon.

Problem details · Reading note

The unified-entropy correlation triangle inequality is not universally valid

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

G. M. Bosyk, G. Bellomo, S. Zozor, M. Portesi and P. W. Lamberti (2016). Unified entropic measures of quantum correlations induced by local measurements. DOI: 10.1016/j.physa.2016.06.131. URL: https://arxiv.org/abs/1604.00329v1.

Claim. Quantum correlation measures D^K_(q,s), K∈{A,B,AB}, are defined as the minimal change of the unified (q,s)-entropy under local rank-one projective measurements, divided by (Tr ρ^q)^s; the triangle-like inequality D^A+D^B≥D^AB is proved for von Neumann and Tsallis-2 entropies and left open for other entropic measures.

Problem details · Reading note

The relabelled chained inequality holds for every odd-input binary COD model

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Jonatan Bohr Brask and Rafael Chaves (2017). Bell scenarios with communication. DOI: 10.1088/1751-8121/aa5840. URL: https://arxiv.org/abs/1607.08182v1.

Claim. In a bipartite Bell scenario where Alice's output may influence Bob's output (causal outcome-dependent models), the chained Bell inequality with Alice's outputs relabelled for every second input is valid for 3, 5, 7 and 9 inputs by vertex enumeration, is violated for even numbers of inputs up to 8, and is conjectured to hold for every odd number of inputs.

Problem details · Reading note

Effective resistance is the optimal finite-graph moving particle constant

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Joe P. Chen (2017). The moving particle lemma for the exclusion process on a weighted graph. DOI: 10.1214/17-ECP82. URL: https://arxiv.org/abs/1606.01577v4.

Claim. For the symmetric exclusion process on a finite weighted graph and every density α∈[0,1], the cost ½ν_α[(∇_{xy}f)²] of exchanging the occupations at x and y is at most the effective resistance R_eff(x,y) times the exclusion Dirichlet energy; the paper conjectures that this moving particle lemma is optimal on any finite weighted graph.

Problem details · Reading note

Completely symmetric three-qutrit states need not be fully separable

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Lin Chen; Delin Chu; Lilong Qian; Yi Shen (2019). Separability of completely symmetric states in a multipartite system. DOI: 10.1103/PhysRevA.99.032312. URL: https://arxiv.org/abs/1812.04182v4.

Claim. A state on (C^N)^{⊗d} is completely symmetric when its entries are invariant under every permutation of the 2d index positions; the paper shows that multi-qubit and two-qutrit completely symmetric states are separable and asks whether multi-qutrit completely symmetric states are separable.

Problem details · Reading note

Cuntz's integral Fourier zero-free column question

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Michael Cuntz (2009). Integral modular data and congruences. DOI: 10.1007/s10801-008-0139-y. URL: https://arxiv.org/abs/math/0611233v2.

Claim. Section 7(5) asks whether an integral Fourier matrix has a zero-free column containing an entry other than 1 or −1.

Problem details · Reading note

The exact linear-span dimension of binary bond-two uniform matrix product states

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Claudia De Lazzari; Harshit J. Motwani; Tim Seynnaeve (2022). The Linear Span of Uniform Matrix Product States. DOI: 10.3842/SIGMA.2022.099. URL: https://arxiv.org/abs/2204.10363v3.

Claim. uMPS(m,n,d) is the closure of the image of the trace parametrization (A_0,…,A_{n−1}) ↦ Σ Tr(A_{i_1}···A_{i_d}) e_{i_1}⊗···⊗e_{i_d}; the paper computes the linear span of uMPS(2,2,d) for d ≤ 20 and conjectures its dimension: (d⁴ − 4d² + 192d + 192)/192 for even d and (d⁴ − 10d² + 192d + 201)/192 for odd d.

Problem details · Reading note

A mixed density tensor attains the nuclear-norm maximum

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Harm Derksen, Shmuel Friedland, Lek-Heng Lim and Li Wang (2017). Theoretical and computational aspects of entanglement. URL: https://arxiv.org/abs/1705.07160v1.

Claim. For a d-partite density tensor A, log₂‖A‖₁ ≤ 2 log₂ α(n,ℂ), with equality for the pure density tensor of a maximally entangled state; the paper conjectures that equality implies that A is such a pure state.

Problem details · Reading note

Easlick's complete minimal-advantage dice cycle conjecture

Proved. Lean theorem conjecture_one, frozen in this repository 2026-10-30.

Terry Easlick (2026). Computer-assisted results for n-sided non-transitive dice taking face values in [n]. URL: https://arxiv.org/abs/2610.40321v1.

Claim. Conjecture 1 asks whether six is the largest n at least three for which the global maximum length of a simple cycle at the least attainable unconditional strict winning probability above one half equals n.

Problem details · Reading note

Three-state overlap interval propagation is not always optimal

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Ernesto F. Galvão; Daniel J. Brod (2020). Quantum and classical bounds for two-state overlaps. DOI: 10.1103/PhysRevA.101.062110. URL: https://arxiv.org/abs/1902.11039v3.

Claim. For three pure states with known overlaps r_AB and r_AC, the unknown overlap r_BC lies between explicit tight bounds r_- (or 0) and r_+; for a connected graph of known overlaps the paper propagates these three-state intervals by intersecting over common neighbours, and leaves open whether this procedure is optimal.

Problem details · Reading note

The reverse Mittag-Leffler modulus characterization is false

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Roberto Garrappa, Stefan Gerhold, Marina Popolizio, Thomas Simon (2025). On some inequalities for the two-parameter Mittag-Leffler function in the complex plane. URL: https://arxiv.org/abs/2410.11852v4.

Claim. Conjecture 25 characterizes the global reverse modulus inequality by the parameter interval and complete monotonicity of the reciprocal.

Problem details · Reading note

The inverse of a critical Carleman Jordan chain is an increasing-path sum

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Nicolas Grosjean; Thierry Huillet (2016). Some combinatorial aspects of discrete non-linear population dynamics. DOI: 10.1016/j.chaos.2016.10.004. URL: https://arxiv.org/abs/1610.05919v1.

Claim. For an iterated power series map with the Carleman matrix P(k,k')=[x^{k'}]φ(x)^k, the critical case α₁=1 is treated by a normalized Jordan-chain matrix W with PW=W(I+S), and the paper conjectures that W⁻¹(k,k')=σ(k−1,k'), a sum over strictly increasing index paths of products of entries of P.

Problem details · Reading note

The rank vector $(3_2,2_6)_7$ is realized by an extreme POVM

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Erkka Haapasalo; Juha-Pekka Pellonpää (2020). Effective methods for constructing extreme quantum observables. DOI: 10.1088/1751-8121/ab8d52. URL: https://arxiv.org/abs/1809.09935v1.

Claim. An N-outcome POVM is extreme when it is not a nontrivial midpoint of two POVMs; the paper characterizes the possible rank vectors of extreme POVMs in dimensions 1 to 4 and names the rank vector (3_2, 2_6)_7 as the problematic case in dimension 7.

Problem details · Reading note

hanna-a174470

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Paul D. Hanna (2010). OEIS A174470: Thue/Roth weighted exponential integrality. URL: https://oeis.org/A174470.

Claim. Conjectured to consist entirely of integers.

Problem details · Reading note

No parity Majorana code on at most fourteen modes has distance four and a logical qubit

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Mathew B. Hastings (2017). Small Majorana fermion codes. DOI: 10.26421/QIC17.13-14-7. URL: https://arxiv.org/abs/1703.00612v1.

Claim. Majorana fermion codes are self-orthogonal binary codes; with the fermion parity in the stabilizer group, no code on at most 12 modes has distance 4 and a logical qubit, codes on 16 modes do, and the paper believes that 16 is the minimum.

Problem details · Reading note

The Jacq–Lardizabal mean-hitting-time form is false

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Thomas S. Jacq; Carlos F. Lardizabal (2017). Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster's Theorem. DOI: 10.1007/s10955-017-1876-x. URL: https://arxiv.org/abs/1701.08400v3.

Claim. For the gambler's-ruin open quantum random walk on {0,...,M} induced by matrices L, R with L*L+R*R=I, the paper computes absorption probabilities and mean hitting times through first-visit generating functions, and conjectures that for every such pair the mean hitting time has the form 1+h(M,M-k)ρ11+h(M,k)ρ22+j(M,k)Re(ρ12) with j(M,k)=-j(M,M-k).

Problem details · Reading note

Cartesian-control angle matrices are invertible for mixed qudit dimensions

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Jacky Jiang; Natalie Klco; Olivia Di Matteo (2025). Non-Abelian dynamics on a cube: Improving quantum compilation through qudit-based simulations. DOI: 10.1103/rrmd-fqx6. URL: https://arxiv.org/abs/2506.10945v1.

Claim. For multi-controlled qudit rotations the rotation angles are transformed by θ = Mβ with M_ij = Π_l (−1)^{f(b_l^(i), g_l^(j))}, f(b,g) = 1 exactly when 0 < b ≤ g; the paper conjectures that its implementation yields an invertible M when the control sequence is a Cartesian product Z_{d1} × ··· × Z_{dk}.

Problem details · Reading note

Rectangular-torus rotation descent

Proved. Lean theorem rectangular_torus_gcd_descent, frozen in this repository 2026-10-30.

Peter Kagey; William Keehn (2023). Counting tilings of the n x m grid, cylinder, and torus. URL: https://arxiv.org/abs/2311.13072v1.

Claim. Conjecture 5.1.1: a rectangular-torus plane tiling fixed by a translated transposing right action is equivalent to a gcd-square torus tiling.

Problem details · Reading note

Eighteen-mode bosonic occupation assignments exist exactly for even particle numbers

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Paweł Kurzyński (2017). Contextuality of identical particles. DOI: 10.1103/PhysRevA.95.012133. URL: https://arxiv.org/abs/1609.01250v1.

Claim. For N identical particles occupying the modes of the 18-projector Kochen–Specker set, a non-contextual model assigns occupation numbers to modes with each context summing to N; the paper shows state-dependent contextuality for every N>0, shows that no state-independent contextuality arises for N=2, and leaves open whether for some N>2 the bosonic occupation numbers cannot be assigned non-contextually.

Problem details · Reading note

Narayan's expanded-generator delta-set conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Ashwin Narayan (2026). The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence. DOI: 10.48550/arXiv.2610.02495. URL: https://arxiv.org/abs/2610.02495v1.

Claim. Conjecture 18.3 asserts that the expanded generating tuple has delta set E(h−1,d).

Problem details · Reading note

Hanna's triangular-exponent EGF congruence

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Paul D. Hanna (2013). OEIS A226839: triangular-exponent exponential generating function. URL: https://oeis.org/A226839.

Claim. E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ). a(n) == 1 (mod 3) (conjecture - valid up to n=1024; if true for n>=0, why?).

Problem details · Reading note

A301420: positive monomial coefficients for five data

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Gregory Gerard Wojnar (2018). OEIS A301420: positive coefficients of five-data generalized Chebyshev polynomials. URL: https://oeis.org/A301420.

Claim. The entry defines the sum of positive monomial coefficients for five data, and conjectures its rational generating function and order-fourteen recurrence.

Problem details · Reading note

Hanna's OEIS A352700 Catalan power-series and modular-coefficient conjectures

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Paul D. Hanna (2022). OEIS A352700: Catalan inverse-power binomial series and modulo 2 and 3 congruences. URL: https://oeis.org/A352700.

Claim. Defines A(x) = sum_{n>=0} (binomial((n+1)(2n+1), n)/(2n+1)) x^n / C(x)^(n(2n+1)+1) over the Catalan series C(x) = 1 + x C(x)^2, and conjectures six modulo 2 and modulo 3 power-series and coefficient congruences (1)-(6).

Problem details · Reading note

Kurkov's periodic boundary recurrences and the large Schroder diagonal

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Mikhail Kurkov (2024). OEIS A376317, periodic shifts of a two-index recurrence. URL: https://oeis.org/A376317.

Claim. The shifts d mod 2 and the indicator of d mod 3 being positive both give A006318.

Problem details · Reading note

The literal adjacent difference of OEIS A378495

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Peter Kagey (2024). OEIS A378495: derangements with no k-cycles. URL: https://oeis.org/A378495.

Claim. On source-agreeing columns 2<=k<=n, T(n,k) counts derangements in S_n with no k-cycles; the source sets T(n,1)=0, and two FORMULA conjectures concern T(n,n-2)-T(n,n) and T(n,n-1)-T(n,n-2).

Problem details · Reading note

The terminal difference of OEIS A378495

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Peter Kagey (2024). OEIS A378495: derangements with no k-cycles. URL: https://oeis.org/A378495.

Claim. On source-agreeing columns 2<=k<=n, T(n,k) counts derangements in S_n with no k-cycles; the source sets T(n,1)=0, and two FORMULA conjectures concern T(n,n-2)-T(n,n) and T(n,n-1)-T(n,n-2).

Problem details · Reading note

The A396659 universal edge-collision ratio

Proved. Lean theorem collision_ratio, frozen in this repository 2026-10-30. Lean theorem long_count_positive, frozen in this repository 2026-10-30.

Arkadeep Dutta (2026). OEIS A396659: edge-collision walks visiting every vertex. URL: https://oeis.org/A396659/internal.

Claim. The FORMULA conjecture asks whether T(n,n^2)/T(n,n^2+1)=(2n-1)/n for every n at least two, with a fixed starting vertex and ordered directed edges including loops.

Problem details · Reading note

The maximum number of pairwise strongly complementary permutation matrices

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Grzegorz Rajchel; Adam Gąsiorowski; Karol Życzkowski (2018). Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces. DOI: 10.1007/s11786-018-0384-y. URL: https://arxiv.org/abs/1804.10715v2.

Claim. Two n×n permutation matrices are complementary when P_ij = P_hk = Q_ik = 1 implies Q_hj = 1, and strongly complementary when moreover their nonzero entries lie in different places; the paper asks how the maximal number of pairwise strongly complementary permutation matrices depends on n.

Problem details · Reading note

The instability count of a clamped Lennard–Jones chain

Proved. Lean theorem result, frozen in this repository 2026-10-30.

A. S. Rodrigues; P. G. Kevrekidis; M. Dobson (2019). N-break states in a chain of nonlinear oscillators. DOI: 10.1103/PhysRevE.99.022201. URL: https://arxiv.org/abs/1802.03477v2.

Claim. For a clamped, pre-stretched chain of oscillators with the Lennard-Jones interaction, solutions with k broken bonds are computed for k up to 5, and the abstract conjectures that k-break solutions possess at least k-1 and at most k pairs of unstable eigenvalues.

Problem details · Reading note

Four physical levels admit a $U(1)$-symmetric injective matrix-product family

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

M. Sanz; M. M. Wolf; D. Pérez-García; J. I. Cirac (2009). Matrix product states: Symmetries and two-body Hamiltonians. DOI: 10.1103/PhysRevA.79.042308. URL: https://arxiv.org/abs/0901.2223v1.

Claim. A translationally invariant MPS with SU(2) symmetry (irreducible representation) and even physical dimension cannot be injective; for U(1) symmetries the obstruction holds when the generator has eigenvalues −m/2,…,m/2 but fails for general generators (an injective example with physical dimension D²−D), physical dimension 2 with U(1) symmetry excludes injectivity, and the case of spin 3/2 is stated as open.

Problem details · Reading note

Ancilla-free testers with a common normalization state have canonical POVM robustness

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Claim. Testers (process POVMs) are compatible when they are marginals of a joint tester; the robustness of incompatibility R_t is the least noise weight, with separate noise testers, that makes two testers compatible. For testers with a common normalization state R_t is at most the robustness R_m of their canonical POVMs, with equality for a pure normalization state; whether equality holds for all ancilla-free testers is stated as open.

Problem details · Reading note

The second exterior trace of Spezia’s serpentine matrix

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Stefano Spezia (2018). OEIS A322844 and the serpentine matrix A317614. URL: https://oeis.org/A322844.

Claim. The two COMMENTS conjectures assert the prescribed absolute second exterior trace and vanishing of every exterior trace above order two.

Problem details · Reading note

The higher exterior traces of Spezia’s serpentine matrix

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Stefano Spezia (2018). OEIS A322844 and the serpentine matrix A317614. URL: https://oeis.org/A322844.

Claim. The two COMMENTS conjectures assert the prescribed absolute second exterior trace and vanishing of every exterior trace above order two.

Problem details · Reading note

Composite prime divisors of the gcd determinant

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Zhi-Wei Sun (2013). Determinant of the n X n matrix with (i,j)-entry equal to the greatest common divisor of i-j and n. URL: https://oeis.org/A228884/internal.

Claim. OEIS A228884 COMMENTS clause (ii) conjectures that for any composite number n, all prime divisors of a(n) are smaller than n.

Problem details · Reading note

Multipartite partial tightness of the entangled-measurement bound

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Armin Tavakoli, Alastair A. Abbott, Marc-Olivier Renou, Nicolas Gisin and Nicolas Brunner (2018). Semi-device-independent characterization of multipartite entanglement of states and measurements. DOI: 10.1103/PhysRevA.98.052333. URL: https://arxiv.org/abs/1805.00377v1.

Claim. In the semi-device-independent prepare-and-measure game with n parties of local dimension d, a measurement with at least k fully separable operators has success probability at most d^{-n}(d^n − k + k/d); for n = 2 the bound is attained with k = md separable operators for every m, and the paper leaves open whether a similar partial tightness holds for general n.

Problem details · Reading note

Wang Conjecture 3.26: finite algebraic constant does not bound the relative constants

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Guoqing Wang (2025). Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups. URL: https://arxiv.org/abs/2509.18724v1.

Claim. Conjecture 3.26 asserts that finite algebraic Davenport constant forces a uniform finite bound on relative constants in the Green quotient over nongroup cyclic elements; Conjecture 5.3 asserts a separate finite-semigroup formula using relative constants and Schutzenberger translation groups.

Problem details · Reading note

Wang's Conjecture 5.3

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Guoqing Wang (2025). Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups. URL: https://arxiv.org/abs/2509.18724v1.

Problem details · Reading note

Yu's mixed-state skew-coherence polygamy conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-30.

Chang-shui Yu (2017). Quantum coherence via skew information and its polygamy. DOI: 10.1103/PhysRevA.95.042337. URL: https://arxiv.org/abs/1704.04871v1.

Claim. The coherence C(ρ) = 1 − Σ_k ⟨k|√ρ|k⟩² defined from skew information satisfies the polygamy inequality 1 − C(Ψ_AB) ≤ [1 − C(ρ_A)][1 − C(ρ_B)] for bipartite pure states; the paper notes that the inequality fails for some two-qubit mixed states and conjectures that it holds for bipartite mixed states of dimension N ≥ 6.

Problem details · Reading note

Eventual stability of Zhang's l-complexity

Proved. Lean theorem result, frozen in this repository 2026-10-30.

Pengcheng Zhang (2024). The 2-complexity of even positive integers. DOI: 10.48550/arXiv.2411.19364. URL: https://arxiv.org/abs/2411.19364v1.

Claim. Question 3 asks whether every positive multiple n of every l > 1 has some l^{m₀} n in A_l.

Problem details · Reading note

Hanna's diagonal ghost-power integrality conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2009). OEIS A156216: diagonal ghost-power integrality conjecture. URL: https://oeis.org/A156216/internal.

Claim. More generally, if exp( Sum_{n>=1} C(n) * x^n/n ) equals a power series in x with integer coefficients, then exp( Sum_{n>=1} C(n)^n * x^n/n ) also equals a power series in x with integer coefficients (conjecture).

Problem details · Reading note

Acosta Babb's weighted Bessel-kernel positivity conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Ryan L. Acosta Babb (2024). A partial converse to the Riemann–Lebesgue lemma for Bessel–Fourier series of order zero. DOI: 10.48550/arXiv.2410.17681. URL: https://arxiv.org/abs/2410.17681v1.

Claim. Conjecture 1 asserts nonnegativity of the explicit weighted order-zero Bessel kernel for 0 <= alpha <= 1/2, every natural mode count and every x in [0,1].

Problem details · Reading note

Aliabadi's uncapped subspace-product conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Mohsen Aliabadi (2025). Conditions for matchability in groups and field extensions II. DOI: 10.7151/dmgaa.1461. URL: https://www.dmgaa.uz.zgora.pl/publish/pdf.php?doi=1461.

Claim. Conjecture 38 proposes an uncapped lower bound for all subspace products under a premise concerning proper finite intermediate fields.

Problem details · Reading note

The coherence vector and spectrum satisfy the product bound in every positive dimension

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Yanet Alvarez; Marcelo Losada; Mariela Portesi; Gustavo M Bosyk (2023). Complementarity between quantum coherence and mixedness: a majorization approach. DOI: 10.1088/1572-9494/acc92f. URL: https://ctp.itp.ac.cn/EN/PDF/10.1088/1572-9494/acc92f.

Claim. The coherence vector nu(rho) of a d-level state is the majorization supremum of the averages sum_k q_k mu(psi_k) of decreasingly ordered squared-amplitude vectors over pure-state decompositions of rho, and lambda(rho) is its decreasingly ordered spectrum. The paper studies majorization bounds for nu(rho) tensor lambda(rho), finds the optimal qubit bound and numerical qutrit bounds, and conjectures nu_1(rho) lambda_1(rho) >= 1/d for every state.

Problem details · Reading note

Bala's A361034 parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Peter Bala (2023). OEIS A361034 and A361035: parity of quartic factorial quotients. URL: https://oeis.org/A361034.

Claim. Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1.

Problem details · Reading note

Bala's A361035 parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Peter Bala (2023). OEIS A361034 and A361035: parity of quartic factorial quotients. URL: https://oeis.org/A361034.

Claim. Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1.

Problem details · Reading note

Ferrers Boards with Prescribed Ordered Level Numbers

Proved. Lean theorem source2023_count, frozen in this repository 2026-10-29. Lean theorem lOperator_count, frozen in this repository 2026-10-29.

Kenneth Barrese (2023). Weighted File Placements on Singleton Boards. DOI: 10.48550/arXiv.2312.10855. URL: https://arxiv.org/abs/2312.10855v1.

Claim. Sections 2.2 and 5 ask for the number of all Ferrers boards with prescribed ordered level numbers in the fixed n-column, n-level ambient rectangle.

Problem details · Reading note

Equal third and unequal second principal specializations for a caterpillar family

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Claim. Remark 3.11 asks whether every positive-j listed pair has unequal second and equal third ordinary principal specializations, and whether the base and these pairs exhaust the phenomenon.

Problem details · Reading note

Bonetto–Jardón Kojakhmetov Conjecture 1

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Riccardo Bonetto; Hildeberto Jardón Kojakhmetov (2024). On the Eigenvalues of Graphs with Mixed Algebraic Structure. URL: https://arxiv.org/html/2408.00487v1.

Claim. Definition 3 assigns the matrix M(ε)=L_G+ε A_H to a disjoint spanning edge partition of a finite simple graph, for ε>0. Definitions 7 and 8 count real eigenvalues with multiplicity and define signature as Nplus minus Nminus. Section 3, Conjecture 1 asks whether this signature is nonincreasing in ε.

Problem details · Reading note

Prime multiplicities in Bouras's continued fractions

Proved. Lean theorem result, frozen in this repository 2026-10-29.

M. Bouras (2023). OEIS A363347 and A363102, prime multiplicities of continued-fraction denominators. URL: https://oeis.org/A363347.

Claim. Conjecture 3: Except for 5, the primes all appear exactly twice.

Problem details · Reading note

Two or three prime occurrences in Bouras A363102

Proved. Lean theorem result, frozen in this repository 2026-10-29.

M. Bouras (2023). OEIS A363347 and A363102, prime multiplicities of continued-fraction denominators. URL: https://oeis.org/A363347.

Claim. Conjecture 3: Except for 5, the primes all appear exactly twice.

Problem details · Reading note

The original ari coproduct preserves every right-factor level filtration

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Annika Burmester; Ulf Kühn (2025). On post-Lie structures for free Lie algebras. DOI: 10.48550/arXiv.2504.19661. URL: https://arxiv.org/html/2504.19661v1.

Claim. Conjecture 4.29: for every k >= 0 the ari coproduct preserves the increasing k-level filtration in its right tensor factor on all rational word polynomials.

Problem details · Reading note

The literal numerator degree in Cigler's Conjecture 16

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

Claim. Experimentally conjectured closed forms for shifted Hankel determinants of Catalan powers; Conjecture 11 evaluates a size residue class, while v2 Section 3 Conjecture 16 specifies full generating-function numerators, exact degrees and reciprocity.

Problem details · Reading note

Two-site uniform matrix product states need not have expected dimension

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Adam Czapliński; Mateusz Michałek; Tim Seynnaeve (2023). Uniform matrix product states from an algebraic geometer's point of view. DOI: 10.1016/j.aam.2022.102417. URL: https://arxiv.org/abs/1904.07563v1.

Claim. The set uMPS(D,d,N) of uniform (translation-invariant) matrix product states with bond dimension D, physical dimension d and N sites is the image of the trace-word map; its closure has dimension at most min{(d−1)D²+1, dim Cyc^N(C^d)}, and Conjecture 2.14 states that equality holds for every (D,d,N).

Problem details · Reading note

Removing intermediate memory resets can increase permutation approximation error

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Jakub Czartowski; A. de Oliveira Junior; Kamil Korzekwa (2023). Thermal Recall: Memory-Assisted Markovian Thermal Processes. DOI: 10.1103/PRXQuantum.4.040304. URL: https://arxiv.org/abs/2303.12840v2.

Claim. Markovian thermal processes on a system extended by an N-dimensional memory in its thermal state approximate thermal operations, through protocols built from two-level thermalisations with a memory reset after each transposition. At infinite temperature the paper conjectures, from numerics, that omitting the intermediate memory resets never worsens the total variation distance to the target permutation of the state.

Problem details · Reading note

Divisibility does not force entanglement breaking for qutrits

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

David Davalos; Mario Ziman; Carlos Pineda (2019). Divisibility of qubit channels and dynamical maps. DOI: 10.22331/q-2019-05-20-144. URL: https://arxiv.org/abs/1812.11437v4.

Claim. A channel is divisible when it is a composition of two non-unitary channels; the P-divisible channels are those reached, up to closure, by continuous dynamical maps from the identity whose propagators are positive and trace preserving. For qubits every channel with negative determinant is entanglement breaking, so every divisible qubit channel outside the P-divisible set is entanglement breaking; the paper leaves open whether something similar holds in higher dimensions.

Problem details · Reading note

Prefix closure for squares with three digits

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Michael Geißer, Theresa Körner, Sascha Kurz and Anne Zahn (2021). Squares with three digits. URL: https://arxiv.org/abs/2112.00444v2.

Claim. If ⟨a₁′,...,aₗ′⟩ is a solution with l ≥ 2, then ⟨a₁′,...,aₗ₋₁′⟩ is also a solution.

Problem details · Reading note

Informationally complete pairs are exactly complementary pairs with forest fiber graphs

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Stan Gudder (2019). Finite-Dimensional Convex Effect Algebras. DOI: 10.48550/arXiv.1912.05110. URL: https://arxiv.org/abs/1912.05110v1.

Claim. For random variables f, g on a finite set X and probability measures on X, the pair {f, g} is informationally complete (IC) if equal distributions of f and of g force equal measures. Strongly complementary pairs are IC and IC pairs are complementary; a complementary pair need not be IC, and finding a simple characterization of IC pairs is stated as an open problem.

Problem details · Reading note

Maximum single-party entropy need not decrease on average under LOCC

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Y. Guo; Z.-X. Jin (2025). Measure of entanglement and the monogamy relation: a topical review. DOI: 10.48550/arXiv.2512.21992. URL: https://arxiv.org/abs/2512.21992v1.

Claim. Section 9.4 conjectures that the maximal single-party reduced function does not increase on average under LOCC; Table 16 includes von Neumann entropy.

Problem details · Reading note

gurvich-krnc-vyalyi-conjecture-1-total-amoeba-confinement

Proved. Lean theorem total_confinement, frozen in this repository 2026-10-29.

Vladimir Gurvich; Matjaž Krnc; Mikhail Vyalyi (2025). Growing Trees and Amoebas’ Replications. DOI: 10.1007/s00025-025-02421-6. URL: https://arxiv.org/abs/2401.07484.

Claim. Section 2 and Conjecture 1 state that an amoeba A=(H,m) with extension parameter l is totally confined whenever the maximum degree of the finite tree H exceeds the number of roots k by more than two.

Problem details · Reading note

Hanna's mixed trigonometric and hyperbolic congruence

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2013). OEIS A227051: mixed trigonometric and hyperbolic reciprocal. URL: https://oeis.org/A227051.

Claim. NAME: E.g.f.: 1/(cos(x) - sin(x)*sinh(x)), omitting the zero-valued coefficients of odd powers of x. FORMULA: a(n) == 3 (mod 10) for n>0 (conjecture).

Problem details · Reading note

h277403-div10

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2016). OEIS A277403 — E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx.. URL: https://oeis.org/A277403.

Claim. a(n) is divisible by 10 for n>2 (conjecture).

Problem details · Reading note

Polygon and three-time Leggett–Garg inequalities are insufficient at six times

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

J. J. Halliwell; C. Mawby (2019). Fine's theorem for Leggett-Garg tests with an arbitrary number of measurement times. DOI: 10.1103/PhysRevA.100.042103. URL: https://arxiv.org/abs/1906.04865v2.

Claim. For a dichotomic variable measured at n times with compatible nonnegative pairwise probabilities, the n-gon inequalities and all n-time Leggett-Garg inequalities are necessary for a joint probability matching the pairwise data; Section 4.B conjectures that the n-gon inequalities together with all three-time Leggett-Garg inequalities are sufficient, and proves this for n = 4 and n = 5 in the symmetric case.

Problem details · Reading note

Integer existence and dyadic parity for OEIS A397594

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2026). OEIS A397594: quadrupled linear rows and dyadic parity. URL: https://oeis.org/A397594.

Claim. Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).

Problem details · Reading note

Hanna's derivative-substitution congruences for A232686

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2013). Derivative-substitution exponential generating functions A232694, A232695, A232696 and A232686. URL: https://oeis.org/A232694.

Claim. The nominated COMMENTS conjectures concern congruences of the factorial-normalized coefficients of four derivative-substitution exponential generating functions.

Problem details · Reading note

Hanna's derivative-substitution congruences for A232694

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2013). Derivative-substitution exponential generating functions A232694, A232695, A232696 and A232686. URL: https://oeis.org/A232694.

Claim. The nominated COMMENTS conjectures concern congruences of the factorial-normalized coefficients of four derivative-substitution exponential generating functions.

Problem details · Reading note

Hanna's derivative-substitution congruences for A232695

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2013). Derivative-substitution exponential generating functions A232694, A232695, A232696 and A232686. URL: https://oeis.org/A232694.

Claim. The nominated COMMENTS conjectures concern congruences of the factorial-normalized coefficients of four derivative-substitution exponential generating functions.

Problem details · Reading note

Hanna's derivative-substitution congruences for A232696

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna (2013). Derivative-substitution exponential generating functions A232694, A232695, A232696 and A232686. URL: https://oeis.org/A232694.

Claim. The nominated COMMENTS conjectures concern congruences of the factorial-normalized coefficients of four derivative-substitution exponential generating functions.

Problem details · Reading note

Refutation of Javaheri's cyclic arrangement dimension conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Mohammad Javaheri (2026). A new arrangement inequality (revised and extended version). DOI: 10.48550/arXiv.2610.33653. URL: https://arxiv.org/abs/2610.33653v1.

Claim. Conjecture 1.3: for all nonnegative real a,b with a+b != 1, the largest cyclic dimension satisfying (sum x_i)(sum x_i^a x_(i+1)^b) <= (sum x_i^((a+b+1)/2))^2 equals the largest n with n=1 or (a+b-1)^2 <= 8ab sin^2(pi/n).

Problem details · Reading note

Zero belongs to the spectrum of a finite-range translation-invariant lattice Lindbladian

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Frederik Ravn Klausen (2025). Spectra of generators of Markovian evolution in the thermodynamic limit: From non-Hermitian to full evolution via tridiagonal Laurent matrices. DOI: 10.48550/arXiv.2206.09879. URL: https://arxiv.org/abs/2206.09879v3.

Claim. A single-particle Lindbladian on the Hilbert-Schmidt operators over ℓ²(ℤ), with finite-range translation-invariant Hamiltonian and Lindblad operators, is a bounded operator and decomposes as a direct integral of finite-range Laurent operators with finite-range perturbations. Under the additional rank-one hypothesis the paper proves that if 0 lies in the spectrum then the Lindbladian is gapless or has an infinite-dimensional kernel, and lists as a further question whether 0 always lies in the spectrum.

Problem details · Reading note

Ku's maximum-density conjecture for ordinary hypercubes

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Kuo-Han Ku (2026). Monochromatic k in a row. URL: https://arxiv.org/html/2606.12880v1.

Claim. Definitions 2.1, 2.2, 2.4, 2.6 give ordinary boards, near-line avoidance, saturation and maximum density; Conjecture 5.6 asserts D(k,[k]^d)=1-2/k for every k>=2 and natural d.

Problem details · Reading note

Kurkov's weighted-ballot formula for OEIS A107667

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Paul D. Hanna; Mikhail Kurkov (2026). OEIS A107667: Kurkov's weighted-ballot formula for Hanna's triangular matrix. URL: https://oeis.org/history/view?seq=A107667&v=25.

Claim. Conjecture: T(n+m, m) = (n+m+1) * P(n+1, n+1, m) where P(n, k, m) = P(n-1, k, m) + (n+m) * P(n, k-1, m) for 1 < k < n with P(n, 1, m) = 1, P(n, n, m) = (n+m) * P(n, n-1, m). - _Mikhail Kurkov_, Jun 02 2026

Problem details · Reading note

Rational collision density of bracket expressions

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Urban Larsson and Hikaru Manabe (2026). Additive Subtraction Games. DOI: 10.48550/arXiv.2603.10414. URL: https://arxiv.org/html/2603.10414v2#Thmthm20.

Claim. Question 20(iii) separately asks whether the collision density of an arbitrary bracket expression is always rational.

Problem details · Reading note

Minimum Seidel energy of chain graphs

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Santanu Mandal, Ranjit Mehatari, and Kinkar Chandra Das (2022). On the spectrum and energy of Seidel matrix for chain graphs. DOI: 10.48550/arXiv.2205.00310. URL: https://arxiv.org/abs/2205.00310v1.

Claim. Conjecture 4.14 asserts the minimum Seidel energy of connected chain graphs and characterizes equality by Gamma(n,k).

Problem details · Reading note

Seven and five versus binary carryless multiplication

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Antti Karttunen (2025). OEIS A115770: seven and five versus binary carryless multiplication. URL: https://oeis.org/A115770.

Claim. A115770 names the natural numbers k satisfying 7*k = A048720(11,k) and conjectures that they are exactly those satisfying 5*k = A048720(13,k).

Problem details · Reading note

Equivalent ordinary and binary carryless multiplier conditions

Proved. Lean theorem result, frozen in this repository 2026-10-29.

OEIS Foundation (2026). OEIS A115801: ordinary and binary carryless multiplier conditions. URL: https://oeis.org/A115801.

Claim. A115801 defines the nonnegative integers k satisfying 9*k = A048720(25,k) and conjectures that the same numbers satisfy 15*k = A048720(19,k). A048720 is binary polynomial multiplication over GF(2).

Problem details · Reading note

The A115805 to A115803 carryless containment question

Proved. Lean theorem result, frozen in this repository 2026-10-29.

OEIS Foundation (2026). OEIS A115803 and A115805: carryless multiplier containment. URL: https://oeis.org/A115803.

Claim. A115805 records the condition 13*k = A048720(29,k), while A115803 records 11*k = A048720(31,k); the qualified target is containment of the former solution set in the latter. A048720 is binary polynomial multiplication over GF(2).

Problem details · Reading note

Containment of the A115809 carryless multiplier condition

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Antti Karttunen (2006). OEIS A115809: seventeen-times binary carryless multiplier condition. URL: https://oeis.org/A115809.

Claim. A115809 defines integers k satisfying 17*k = A048720(49,k) and asks whether it is a subset of A115801, whose defining condition is 9*k = A048720(25,k).

Problem details · Reading note

A397436: the unrestricted product supercongruence

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Peter Bala; OEIS Foundation Inc. (2026). OEIS A397436 revision 44, Conjecture 3 on integer-parameter binomial products. URL: https://oeis.org/A397436/internal.

Claim. The unrestricted same-r supercongruence for inclusive truncated products of signed generalized binomial coefficients.

Problem details · Reading note

Pandey–Patra subgraph core conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Dinesh Pandey; Kamal Lochan Patra (2021). Some new central parts of connected graphs. URL: https://arxiv.org/abs/2110.00738v1.

Claim. Conjecture 3.25 states that the subgraph core of every finite simple connected graph is contained in one block.

Problem details · Reading note

Natural valuations cannot preserve every abelian complexity function

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Pierre Popoli and Jeffrey Shallit and Manon Stipulanti (2024). Additive word complexity and Walnut. DOI: 10.4230/LIPIcs.FSTTCS.2024.32. URL: https://arxiv.org/abs/2410.02409v1.

Claim. Question 31 asks whether a natural valuation can make the additive complexity of a given infinite word equal its original-symbol abelian complexity at every length.

Problem details · Reading note

A strict Bayesian feedback advantage for the qutrit Werner–Holevo channel

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Milajiguli Rexiti; Samad Khabbazi Oskouei; Stefano Mancini (2024). On the effectiveness of Bayesian discrete feedback for quantum information reclaiming. DOI: 10.1007/s11128-024-04467-x. URL: https://arxiv.org/abs/2402.18941v1.

Claim. A channel is measured on its environment through a Kraus decomposition and each outcome is corrected by the inverse of the unitary part of its Kraus operator, giving entanglement fidelity (1/d^2) sum_x (tr|T_x|)^2. For two uses of the channel, Markovian feedback corrects each step independently and Bayesian feedback corrects the accumulated branch; their fidelities involve tr(|T2||T1|) and tr|T2|T1||. The paper finds no Bayesian advantage for qubit channels and for the qutrit dephasing channel, an advantage for the qutrit amplitude damping channel, and asks for other examples of channels where Bayesian feedback is more effective.

Problem details · Reading note

Successive new Laplacian layer maxima of token graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

M. A. Reyes and C. Dalfó and M. À. Fiol (2024). On the spectra and spectral radii of token graphs. DOI: 10.1007/s40590-023-00583-3. URL: https://arxiv.org/html/2310.16929v1.

Claim. Definition 3.1 takes extrema of consecutive Laplacian spectral multiset differences; Conjecture 3.3 proposes monotonicity of their maxima.

Problem details · Reading note

Rouleux's affine-determinant conjecture for alternating polynomials

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Michel Rouleux (2019). Multimode entanglement for fermions. DOI: 10.48550/arXiv.1910.04790. URL: https://arxiv.org/abs/1910.04790v1.

Claim. The affine determinant det(x1 − x0, …, xd − x0) of d+1 points of C^d is an antisymmetric, non-multilinear form of degree d in its d+1 variables. The paper conjectures that it is the only non-degenerate antisymmetric tensor of degree d on C^d depending on d+1 variables, and that no nontrivial antisymmetric tensor of degree d depends on more than d+1 variables.

Problem details · Reading note

Measurement can extract more than half the ergotropy of a non-stationary state

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

Andrea Solfanelli; Lorenzo Buffoni; Alessandro Cuccoli; Michele Campisi (2019). Maximal energy extraction via quantum measurement. DOI: 10.1088/1742-5468/ab3721. URL: https://arxiv.org/abs/1905.10262v2.

Claim. The metrotropy, the maximal energy extractable from a finite quantum system by a nonselective rank-one projective measurement, satisfies M ≤ W/2 for stationary states, where W is the ergotropy, and M ≤ W in general. Whether M ≤ W/2 holds for non-stationary states is stated as open.

Problem details · Reading note

Sun Conjecture 4.2(i) absolute-difference permanent congruences

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Zhi-Wei Sun (2022). Arithmetic properties of some permanents. DOI: 10.48550/arXiv.2108.07723. URL: https://arxiv.org/abs/2108.07723v7.

Claim. Conjecture 4.2(i), equations (4.3) and (4.4): the two absolute-difference permanent congruences for every odd prime.

Problem details · Reading note

Tebni's OEIS A350759 prime congruences

Proved. Lean theorem result, frozen in this repository 2026-10-29.

Mélika Tebni (2022). OEIS A350759, COMMENTS prime congruences. URL: https://oeis.org/A350759.

Claim. Both prime congruences for the integer sequence with exponential generating function exp(-1+(1+x)*(1-log(1+x))).

Problem details · Reading note

Ulgenes's truncated-binomial Hadamard limit

Proved. Lean theorem result, frozen in this repository 2026-10-29.

David Ulgenes (2026). OEIS A398064, truncated-binomial limit for Hadamard's Gamma function. URL: https://oeis.org/A398064.

Claim. Conjecture: H(x+1) = lim_{n->oo} n^{x+1} * Sum_{k=0..n} (-1)^k/(n+k) * binomial(x, k) such that H(1/4) = 0.79564036... . Changing the upper bound of the sum to oo yields Euler's product representation of the Gamma function.

Problem details · Reading note

The Zhou–Wu–Chau endpoint lower bound on overall error is false

Refuted. Lean theorem result, frozen in this repository 2026-10-29.

S.S. Zhou; Shengjun Wu; H.F. Chau (2016). State-independent error-disturbance trade-off for measurement operators. DOI: 10.1016/j.physleta.2016.03.046. URL: https://arxiv.org/abs/1509.03812v1.

Claim. For rank-one projective measurements A, A' and B on C^d, the state-independent error of replacing A by A', the disturbance that A' causes to B, and their overall error Delta(A,A',B) are maxima over states of infinity-norm distances between outcome distributions. The paper proves trade-offs for qubits and for mutually unbiased A and B, and conjectures, with the error between A and B relaxed to ignore outcome labels, that eps(A,A')+eta(A',B) and Delta(A,A',B) are both at least min{eps(A,B), eta(A,B)} in every dimension.

Problem details · Reading note

The Shapley allocation of a graph energy game can leave its core

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Gerardo Arizmendi and Octavio Arizmendi (2020). The graph energy game. DOI: 10.1016/j.dam.2023.01.030. URL: https://arxiv.org/abs/2012.14114v2.

Claim. Conjecture 3.10: For the energy game, the Shapley value is in the core.

Problem details · Reading note

Connectivity of prefix normal generators

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Duncan Adamson; Moritz Dudey; Pamela Fleischmann; Annika Huch (2025). Word Chain Generators for Prefix Normal Words. DOI: 10.1007/978-3-032-09524-4_5. URL: https://arxiv.org/abs/2508.19619v1.

Claim. The v1 Conclusion asks whether every prefix normal generator is reachable from every other by swapping neighbouring entries; Lemma 17 supplies preservation when an adjacent ascent is exchanged.

Problem details · Reading note

Andoyo Question 5.2: infinitude of exact prime fibers

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Jason D. Andoyo (2026). (a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes. DOI: 10.62072/acm.2026.09011. URL: https://arxiv.org/html/2601.10561v1.

Claim. Section 5, Question 5.3 asks whether distinct nonzero integer seed pairs can have equal complete Pisano sets.

Problem details · Reading note

Andoyo's equal Pisano sets question

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Jason D. Andoyo (2026). (a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes. DOI: 10.62072/acm.2026.09011. URL: https://arxiv.org/html/2601.10561v1.

Claim. Section 5, Question 5.3 asks whether distinct nonzero integer seed pairs can have equal complete Pisano sets.

Problem details · Reading note

Global-coalition graph universality

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Marcin Anholcer; Nazli Besharati; Doost Ali Mojdeh; Mohammad Reza Samadzadeh (2025). Global coalitions in graphs. URL: https://arxiv.org/abs/2509.15386v4.

Claim. Section 7 Problem 58 asks whether every graph is realizable as the global coalition graph of an actual global coalition partition.

Problem details · Reading note

Three-adic nondivisibility

Proved. Lean theorem result, frozen in this repository 2026-10-28.

P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Claim. The entries conjecture the five-adic real-coordinate formula, the initial type-one prime list, and two three-adic imaginary-coordinate properties.

Problem details · Reading note

Paired three-adic valuations

Proved. Lean theorem result, frozen in this repository 2026-10-28.

P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Claim. The entries conjecture the five-adic real-coordinate formula, the initial type-one prime list, and two three-adic imaginary-coordinate properties.

Problem details · Reading note

Type-one primes through 101

Proved. Lean theorem result, frozen in this repository 2026-10-28.

P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Claim. The entries conjecture the five-adic real-coordinate formula, the initial type-one prime list, and two three-adic imaginary-coordinate properties.

Problem details · Reading note

Bala's OEIS A363411 and A363412 valuation conjectures

Proved. Lean theorem result, frozen in this repository 2026-10-28.

P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Claim. The entries conjecture the five-adic real-coordinate formula, the initial type-one prime list, and two three-adic imaginary-coordinate properties.

Problem details · Reading note

The unrestricted rank-only Jensen converse is false

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Mátyás Barczy; Zsolt Páles (2025). Jensen convex functions and doubly stochastic matrices. URL: https://arxiv.org/abs/2510.03715v1.

Claim. Remark 2.2(i) asks whether the all-input comparison in equation (2.1) forces midpoint convexity for every real doubly stochastic matrix of rank at least two.

Problem details · Reading note

Quantum twins in three finite classical graphs

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Arkadiusz Bochniak; Igor Chełstowski; Paweł Kasprzak; Piotr M. Sołtan (2025). Quantum Mycielskians: symmetries, twin vertices and distinguishing labelings. DOI: 10.48550/arXiv.2502.10521. URL: https://arxiv.org/abs/2502.10521v2.

Claim. Definition 3.8 defines quantum twin vertices through two distinct actions agreeing after adjacency; Question 3.17 asks whether K₂ □ K₅, C₄ □ C₃ and the tesseract possess them.

Problem details · Reading note

Carnero Bravo's hereditary lexicographic product question

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Andrés Carnero Bravo (2025). On independence complexes of graph products. URL: https://arxiv.org/abs/2505.06457v2.

Claim. Theorem 21 gives a suspended independence-complex conclusion for induced subgraphs of lexicographic products; Question 22 asks for unsuspended membership in the hereditary class W; Question 32 asks whether the hereditary sphere-wedge statement of Theorem 31 extends to every king-grid width.

Problem details · Reading note

The Caro–Graswald maximal index is $2d-5$ for $d\ge5$, and $2$ for $d=3$

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Matthias C. Caro; Benedikt R. Graswald (2021). Necessary criteria for Markovian divisibility of linear maps. DOI: 10.1063/5.0031760. URL: https://arxiv.org/abs/2009.06666v3.

Claim. The paper derives necessary criteria for a linear map to be infinitesimal Markovian divisible, including a bound of the determinant of a quantum channel by a product of its smallest singular values. Optimizing the number of singular values in that bound reduces to a matrix inequality for a probability vector p: with a_ij = (p_i + p_j)/2 and g_ij = sqrt(p_i p_j), the paper conjectures that the largest h(d) with A + G <= d, where A and G sum the h(d) largest entries of a and g, is h(d) = 2d - 5.

Problem details · Reading note

Costa–Catarino bidimensional Fibonacci matrices

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Eudes Antonio Costa; Paula M.M.C. Catarino (2026). On Matrices with Bidimensional Fibonacci Numbers. DOI: 10.2478/amsil-2025-0014. URL: https://journals.us.edu.pl/index.php/AMSIL/article/download/22306/18885.

Claim. The unnumbered conjecture in section 4, page 73, asserts that every bidimensional Fibonacci matrix of order 2 has nonzero determinant at all nonnegative indices.

Problem details · Reading note

Coşkun–Eden Remark 5.9: the octonion clause

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

O. Coşkun and A. Eden (2025). Structure of Vidinli Algebras: Simple Nonassociative Jordan–Lie Algebras with Azumaya Properties. DOI: 10.48550/arXiv.2511.09395. URL: https://arxiv.org/abs/2511.09395v1.

Claim. Definition 5.1 gives the exceptional Vidinli product; Remark 5.9 asks whether it is a linear push-forward of octonion multiplication.

Problem details · Reading note

Edge irregularity strength of cycle-star graphs

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Umme Salma; H. M. Nagesh; Narahari N. (2024). On edge irregularity strength of cycle-star graphs. DOI: 10.22199/issn.0717-6279-5801. URL: https://arxiv.org/abs/2405.12263v1.

Claim. Conjecture 3.1 asserts es(CS_{k,n-k}) = ceil((n+1)/2) for k+1 <= n <= 2k-4 and es(CS_{k,n-k}) = n-k+2 for n >= 2k-3, whenever k >= 8 and n-k >= 1.

Problem details · Reading note

Refutation of the unrestricted strong distance-two period bound

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Gregory Demo, David Lund, Oleksiy Al-saadi (2026). On the Periodicity of k-distance Graphs. URL: https://arxiv.org/abs/2606.25493v1.

Claim. Conjecture 1: Any connected graph G=(V,E) has rho^2(G) <= |V|-3. Preliminaries and Definition 1 specify exact-distance iteration and the least positive return up to graph isomorphism, with zero when no return exists.

Problem details · Reading note

At most binomially many inputs certify failure of d-message simulation

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Brian Doolittle; Eric Chitambar (2021). Certifying the classical simulation cost of a quantum channel. DOI: 10.1103/PhysRevResearch.3.043073. URL: https://arxiv.org/abs/2102.12543v1.

Claim. A classical channel from n inputs to n' outputs is d-simulable when it is a shared-randomness mixture of encoders into d messages followed by decoders; these channels form the signaling polytope, and the signaling dimension is the least such d. The paper characterizes the signaling polytope for two-message (one-bit) simulation of four-output channels, shows that six inputs suffice there to certify that one bit is not enough, and conjectures that binom(n', d) inputs always suffice to certify signaling dimension larger than d.

Problem details · Reading note

The uniform single-left-multiplier Toeplitz factorization assertion

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Ignacio García-Marco, Irene Márquez-Corbella, Daniel Seco (2025). On the minimum number of Toeplitz factors of a matrix. DOI: 10.48550/arXiv.2506.16432. URL: https://arxiv.org/abs/2506.16432v1.

Claim. Section 5.2, Conjecture 5.1 proposes a positive dimension-independent number C such that every complex square matrix is a product of an invertible left matrix and C ordinary Toeplitz matrices; its subsequent numerical consequence is a separate assertion.

Problem details · Reading note

The original unnormalized achievement-set residue converse

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Szymon Głąb and Franciszek Prus-Wiśniowski (2025). Achievement sets — current results and open problems. URL: https://arxiv.org/abs/2512.17285v1.

Claim. Section 2 Problem 9 asks the unnormalized positive-measure residue converse for indexed multigeometric achievement sets; it is the same question as Section 5.1 Problem 1 of arXiv 2508.19454v1.

Problem details · Reading note

Refutation of the unconditional entropic Glibichuk–Konyagin conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

M. K. Goh (2026). On an entropic analogue of additive energy. DOI: 10.2140/ent.2026.5.243. URL: https://arxiv.org/abs/2406.18798v4.

Claim. Conjecture 15 (89) asks for an unconditional entropic Glibichuk–Konyagin lemma.

Problem details · Reading note

Every equal-modulus HMR qutrit channel has a unitary and doubly extreme decomposition

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Uffe Haagerup; Magdalena Musat; Mary Beth Ruskai (2021). Extreme Points and Factorizability for New Classes of Unital Quantum Channels. DOI: 10.1007/s00023-021-01071-y. URL: https://arxiv.org/abs/2006.03414v4.

Claim. The paper introduces a two-parameter family of unital, trace-preserving completely positive maps on 3 by 3 matrices with four Kraus operators. Almost every member is factorizable and extreme among the unital trace-preserving completely positive maps, but extreme neither among the unital nor among the trace-preserving ones. At alpha = beta = 1/sqrt(2) the channel is a convex combination of a unitary conjugation and a Choi-rank-3 map extreme in both larger sets; the paper conjectures the same decomposition whenever |alpha| = |beta| = 1/sqrt(2).

Problem details · Reading note

Hedemann's proposed diagonal-correlance normalization is exceeded by seven qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Samuel R. Hedemann (2020). Correlance and discordance: computable measures of nonlocal correlation. DOI: 10.1007/s11128-020-02676-8. URL: https://arxiv.org/abs/2001.03453v2.

Claim. The paper defines the correlance of an N-mode state as the normalized squared Hilbert-Schmidt distance tr[(rho - varsigma)^2] between the state and the tensor product of its single-mode reductions, and related measures of nonlocal correlation. For diagonal input it normalizes by the maximum over all diagonal states and asserts that the equal mixture of the first and last basis projectors attains it in all systems, giving 1/2 - 1/2^N; the proof in the appendix is a sketch, and the paper says a rigorous proof is still lacking.

Problem details · Reading note

The minimum output overlap of orthogonal C–Q inputs

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Kiera Hutton; Arthur Mehta; Andrej Vukovic (2025). Deciding whether a C–Q channel preserves a bit is QCMA-complete. DOI: 10.48550/arXiv.2508.10664. URL: https://arxiv.org/abs/2508.10664v1.

Claim. Conjecture 1.5 asks whether the average overlap of outputs from k orthogonal pure inputs is at least the minimum average overlap over k distinct computational basis inputs.

Problem details · Reading note

Strictly positive coefficients are not necessary for generalized action graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

S. Klanderman, K. McDicken, A. Tebbe (2026). Conditions for Building Generalized Action Graphs From Sequences. DOI: 10.46787/pump.v9i.6304. URL: https://journals.calstate.edu/pump/article/view/6304.

Claim. Another open question is whether the hypotheses of Theorem 1.1 are in fact necessary conditions for generalized action graphs to exist.

Problem details · Reading note

Kotesovec's weighted distinct-partition logarithmic limits

Proved. Lean theorem result, frozen in this repository 2026-10-28. Lean theorem result, frozen in this repository 2026-10-28. Lean theorem result, frozen in this repository 2026-10-28. Lean theorem result, frozen in this repository 2026-10-28. Lean theorem result, frozen in this repository 2026-10-28.

OEIS Foundation Inc.; Vaclav Kotesovec (2018). OEIS A022629 and the weighted distinct-partition logarithmic conjectures. URL: https://oeis.org/A022629.

Claim. The logarithms of the weighted distinct-partition coefficients have the stated ratio asymptotics for natural exponents one through five.

Problem details · Reading note

Every finite-dimensional signalling channel has a pure product input with positive PDM negativity

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Xiangjing Liu; Yixian Qiu; Oscar Dahlsten; Vlatko Vedral (2025). Quantum causal inference with extremely light touch. DOI: 10.1038/s41534-024-00956-0. URL: https://arxiv.org/abs/2303.10544v4.

Claim. For coarse-grained measurements at two times, the pseudo-density matrix of a channel with Choi-Jamiolkowski matrix M is the Jordan product R = (M rho + rho M)/2 with rho = rho_1 tensor identity, and the PDM negativity f(R) = ||R||_tr - 1 vanishes for every input when the bipartite channel does not signal from B to A. The paper conjectures that every channel signalling from B to A admits a pure product input state with f(R_{B1A2}) > 0, and proves this for two-qubit unitary evolutions.

Problem details · Reading note

Non-isotropic mixed-unitary qubit channels do not commute with marginal pinching

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Zi-Wen Liu; Ryuji Takagi; Seth Lloyd (2019). Diagonal quantum discord. DOI: 10.1088/1751-8121/ab0774. URL: https://arxiv.org/abs/1708.09076v2.

Claim. Diagonal discord is the relative-entropy distance of a bipartite state from its pinching pi_A in the eigenbasis of the A marginal. The paper classifies local discord-nongenerating operations into mixed-unitary, isotropic and semiclassical channels, studies which local operations commute with pi_A, derives a qubit criterion for mixed-unitary channels, and conjectures that for qubits exactly the isotropic channels commute with pi_A, so that non-isotropic qubit mixed-unitary channels all fail the commutation condition.

Problem details · Reading note

Matveeva's literal nonzero-parameter global-boundedness belief

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Anastasia Matveeva (2025). On the integrality of some P-recursive sequences. URL: https://arxiv.org/abs/2511.02121v1.

Claim. The literal nonzero-parameter belief in Sections 4.2 and 7 is refuted by the rational sequence s_n = n + 1.

Problem details · Reading note

Gibonacci power-window gcd normalization

Proved. Lean theorem result, frozen in this repository 2026-10-28.

aBa Mbirika; Jürgen Spilker (2022). GCD of sums of k consecutive squares of generalized Fibonacci numbers. DOI: 10.1080/00150517.2022.12427444. URL: https://arxiv.org/abs/2204.12626v2.

Claim. Question 6.2 asks whether dividing integer Gibonacci seeds by their gcd divides the gcd of all shifted e-th-power windows by the e-th power of that seed gcd, for every e >= 3.

Problem details · Reading note

Forberg's exponential formula for the quarter-squares

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Richard R. Forberg; OEIS Foundation Inc. (2013). Forberg's empirical exponential formula for OEIS A002620. URL: https://oeis.org/A002620/internal.

Claim. For positive integers n, the floor of n/(exp(4/n)-1) equals the shifted quarter-square a(n-1).

Problem details · Reading note

Mathar's first recurrence for OEIS A152661

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Emeric Deutsch; R. J. Mathar (2008). OEIS A152661: permutations whose first two elements have the same parity. URL: https://oeis.org/A152661.

Claim. The source defines the permutation count for n >= 2 and records Mathar's conjectural first recurrence and prior even/odd formulas.

Problem details · Reading note

The A193020 minus-four eventual identity

Proved. Lean theorem result, frozen in this repository 2026-10-28.

John W. Layman (2011). OEIS A193020, distinct binary self-shuffles. URL: https://oeis.org/A193020.

Claim. For positive n, A193020 defines a(n) as the number of distinct shuffles of the binary expansion of n with itself. Revision 13 states separate eventual assertions including a(2^k+1)=k^2 and a(2^k-4)=k*(k-1)/2, each for sufficiently large k, with the literal lower-bound caveat.

Problem details · Reading note

The A193020 plus-one eventual identity

Proved. Lean theorem result, frozen in this repository 2026-10-28.

John W. Layman (2011). OEIS A193020, distinct binary self-shuffles. URL: https://oeis.org/A193020.

Problem details · Reading note

Common positive threshold minima for reciprocal-sigma sums

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Jaroslav Krizek; Robert Israel (2015). OEIS A265714, minima of reciprocal-sigma divisor sums. URL: https://oeis.org/history/view?seq=A265714&v=23.

Claim. For each positive integer threshold, the least positive argument whose reciprocal-sigma divisor sum reaches the threshold also has floor equal to the threshold.

Problem details · Reading note

Kurkov's full lower-bit recurrence for A284005

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Mikhail Kurkov; Antti Karttunen; OEIS contributors (2019). OEIS A284005: Kurkov's full lower-bit recurrence. URL: https://oeis.org/A284005.

Claim. Kurkov's November 10, 2019 conjecture: for n>1, with L=floor(log_2(n)), f(n)=n-2^L and T(n,k)=floor(n/2^k) mod 2, a(n)=2*a(f(n))+Sum_{k=0..L-1} a(f(n)+2^k*(1-T(n,k))), for A284005 with a(0)=1, a(1)=2 and a(n)=(1+popcount(n))*a(floor(n/2)) for n>1.

Problem details · Reading note

The one-common-Zeckendorf-term greedy sequence is a permutation

Proved. Lean theorem result, frozen in this repository 2026-10-28.

URL: https://oeis.org/A370630/internal.

Problem details

Rather's book domination-root Conjecture 5.4

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Bilal Ahmad Rather (2026). On roots of domination polynomials for friendship and book graphs. URL: https://arxiv.org/abs/2604.08998v2.

Claim. The literal arXiv v1/v2 Conjecture 5.4 classifies all real roots with multiplicities for every shared-edge four-cycle book of order n >= 1; the order-one book contradicts its odd clause.

Problem details · Reading note

Every q-transversal matroid is representable over a finite extension

Proved. Lean theorem result, frozen in this repository 2026-10-28.

M. Saaltink (2025). A theory of q-transversals. DOI: 10.48550/arXiv.2503.12201. URL: https://arxiv.org/abs/2503.12201v1.

Claim. Conjecture 3. All q-transversal matroids are representable.

Problem details · Reading note

Three volume ratios for generalized Pauli eigenvalue regions

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Katarzyna Siudzińska (2020). Geometry of generalized Pauli channels. DOI: 10.1103/PhysRevA.101.062323. URL: https://arxiv.org/abs/2002.04657v2.

Claim. The paper studies the geometry of generalized Pauli channels built from mutually unbiased bases, with the Hilbert-Schmidt line and volume elements of their Choi states. For N = d + 1 bases it computes, for d = 2, 3, 4, 5, the volumes of completely positive maps relative to a box of positivity estimates, of channels with nonnegative eigenvalues relative to all channels, and of entanglement-breaking channels relative to those, and conjectures closed formulas for all d.

Problem details · Reading note

Unrestricted two-state dequantumness power can exceed its maximal-quantumness restriction

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Hongting Song; Nan Li (2022). Quantumness and Dequantumness Power of Quantum Channels. DOI: 10.3390/e24081146. URL: https://europepmc.org/article/PMC/PMC9407519.

Claim. The quantumness of an ensemble {(p_i, rho_i)} is minus the weighted sum of tr[rho_i, rho_j]^2. The dequantumness power D of a channel is the largest decrease of quantumness over input ensembles, and D' restricts the inputs to ensembles of maximal quantumness; D_2 and D_2' are the two-ingredient versions. For amplitude damping the paper finds D_2 = D_2' and conjectures that this equality may fail for other channels.

Problem details · Reading note

Spezia's Pell circulant permanent-to-determinant ratio

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Stefano Spezia (2026). OEIS A392131 and A392132: Pell circulant determinant and permanent. URL: https://oeis.org/A392131/internal.

Claim. Conjecture: lim_{n->oo} A392132(n)/a(n) = 1.

Problem details · Reading note

Sra's reciprocal-permanent Lorentzian question

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Suvrit Sra (2026). A Koteljanskii inequality for permanents. DOI: 10.48550/arXiv.2609.39979. URL: https://arxiv.org/abs/2609.39979v1.

Claim. Section 4(b) asks about alpha-permanent log-supermodularity for symmetrizable inverse M matrices; Section 4(d) asks whether the reciprocal-permanent polynomial is Lorentzian.

Problem details · Reading note

Sra's symmetric alpha-permanent question

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Suvrit Sra (2026). A Koteljanskii inequality for permanents. DOI: 10.48550/arXiv.2609.39979. URL: https://arxiv.org/abs/2609.39979v1.

Claim. Section 4(b) asks about alpha-permanent log-supermodularity for symmetrizable inverse M matrices; Section 4(d) asks whether the reciprocal-permanent polynomial is Lorentzian.

Problem details · Reading note

Level-one QAOA reaches a genuine four-state superposition on three qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Michael Streif; Martin Leib (2020). Forbidden subspaces for level-1 quantum approximate optimization algorithm and instantaneous quantum polynomial circuits. DOI: 10.1103/PhysRevA.102.042416. URL: https://arxiv.org/abs/2007.13563v1.

Claim. The paper characterizes diagonal problem Hamiltonians for which the level-1 QAOA state has unit overlap with a target subspace spanned by computational basis states. It treats one and two target states, and, assuming that the only mixing angles solving its necessary condition are odd multiples of pi/4, states that level-1 QAOA cannot reach a genuine superposition of more than 2 and fewer than 2^n target states; it conjectures that those angles are the only solutions.

Problem details · Reading note

Tebni's prime and dyadic row-divisibility conjectures for A341724

Proved. Lean theorem result, frozen in this repository 2026-10-28.

M. Tebni (2023). OEIS A341724, Column EGF and prime and dyadic row-divisibility conjectures. URL: https://oeis.org/A341724.

Claim. E.g.f. of column k: x^k / ((1-2*sinh(-x))*k!). For 0 < k < p and p prime, T(p,k) == 0 (mod p). For 0 <= k < n and n = 2^m (m natural number), T(n,k) == 0 (mod n).

Problem details · Reading note

Infinitely many extreme classes survive the commuting-unitary quotient in matrix D-majorization

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Frederik vom Ende (2022). Strict positivity and D-majorization. DOI: 10.1080/03081087.2020.1860887. URL: https://arxiv.org/abs/2004.05613v4.

Claim. For a positive definite D, a matrix A is D-majorized by B when some quantum channel maps B to A and fixes D. The paper relates strict positivity of channels to this order. It shows that D is the unique minimal element of its trace hyperplane, and that the pure basis state for a minimal entry of d is maximal among positive semidefinite matrices of that trace. In the vector case the set M_D(A) of matrices D-majorized by A is a polytope; in the matrix case it has infinitely many extreme points, and the paper asks whether finitely many remain after identifying extreme points related by unitaries commuting with D.

Problem details · Reading note

The unrestricted exact Wang–Tian saturation formula is false

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Qinglin Wang; Yingzhi Tian (2026). Extremal Problems for the Family of k-Strongly Connected Digraphs. DOI: 10.48550/arXiv.2605.01269. URL: https://arxiv.org/abs/2605.01269v1.

Claim. Section 4, Conjecture 1 states the unrestricted exact eventual extremal arc formula for the family of k-strongly connected strict digraphs.

Problem details · Reading note

Complete monotonicity of the Xu–Yang Mills remainders

Proved. Lean theorem result, frozen in this repository 2026-10-28.

Hui-Zuo Xu and Zhen-Hang Yang (2026). Higher order monotonicity involving the Mills ratio with applications. DOI: 10.1186/s13660-026-03505-9. URL: https://doi.org/10.1186/s13660-026-03505-9.

Claim. Conjecture 7.1 proposes complete monotonicity of the signed asymptotic remainders of Y at a = -1.

Problem details · Reading note

A mixed four-dimensional state has equal coherence roofs

Refuted. Lean theorem result, frozen in this repository 2026-10-28.

Ming-Jing Zhao; Rajesh Pereira; Teng Ma; Shao-Ming Fei (2021). Coherence of assistance and assisted maximally coherent states. DOI: 10.1038/s41598-021-85273-8. URL: https://arxiv.org/abs/2103.08818v1.

Claim. For a real symmetric concave function f on the probability simplex vanishing at the vertices, the pure-state coherence C_f is f of the coherence vector. Its convex roof C_c and its coherence of assistance C_a take the minimum and maximum of the decomposition averages over pure-state decompositions of a mixed state. The paper proves that equal roofs force the pure-state function to be a quadratic form on the range of the state, deduces C_c < C_a for full-rank states, and asks whether C_c < C_a holds for all mixed states.

Problem details · Reading note

Majorization need not survive normalization in a common charge sector

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Raúl Arias; Giuseppe Di Giulio; Esko Keski-Vakkuri; Erik Tonni (2023). Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement. DOI: 10.1007/JHEP03(2023)175. URL: https://arxiv.org/abs/2301.02117v1.

Claim. The paper studies the capacity of entanglement and the Schur-concave entanglement monotones built from it along renormalization-group flows, in charge sectors of theories with a U(1) symmetry and after quantum quenches; it decomposes a reduced density matrix that commutes with the subsystem charge into normalized charge blocks and asks whether a majorization order between two density matrices survives this decomposition.

Problem details · Reading note

Bala's A193236 congruences modulo five and seven

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Peter Bala; Paul D. Hanna (2021). OEIS A193237 and A193236, congruences for triangular reciprocal cube roots. URL: https://oeis.org/A193237/internal.

Claim. Conjectures: a(5*n+4) == 0 (mod 5) and a(7*n+5) == 0 (mod 7) (both checked up to n = 200). - _Peter Bala_, Feb 26 2021

Problem details · Reading note

Bala's A193237 congruences modulo five and seven

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Peter Bala; Paul D. Hanna (2021). OEIS A193237 and A193236, congruences for triangular reciprocal cube roots. URL: https://oeis.org/A193237/internal.

Claim. Conjectures: a(5*n+4) == 0 (mod 5) and a(7*n+5) == 0 (mod 7) (both checked up to n = 200). - _Peter Bala_, Feb 26 2021

Problem details · Reading note

A max multiflow need not reproduce the Ingleton entropy combination

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Ning Bao; Jonathan Harper (2020). Bit threads on hypergraphs. DOI: 10.48550/arXiv.2012.07872. URL: https://arxiv.org/abs/2012.07872v1.

Claim. The paper extends bit threads and multiflows from graphs to hypergraph geometries. It defines the subflow of a union of boundary regions from the thread numbers of a multiflow, shows that a max multiflow saturates every single-region entropy, and conjectures that a linear combination of entropies that is nonnegative and somewhere nonzero on subflows equals its subflow value on every max multiflow. It names the Ingleton combination as satisfying these hypotheses.

Problem details · Reading note

Tight purification circuits need not have monotone output error

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Ben Barber; Neil I. Gillespie; J. M. Taylor (2023). Post-selection-free preparation of high-quality physical qubits. DOI: 10.22331/q-2023-05-04-994. URL: https://arxiv.org/abs/2209.05391v3.

Claim. The paper defines (n,k,e) purification circuits, CNOT and Toffoli circuits that map every n-bit string of weight at most e to a string that is zero in k nominated positions, proves that they exist exactly when the sum of the binomial coefficients up to e is at most 2^(n-k), analyses their output error under independent preparation, idle, CNOT and Toffoli errors, and asks whether, for circuits meeting this bound with equality and preparation error below 1/2, the output error is an increasing function of each of the four error rates.

Problem details · Reading note

Unequal minima for sumsets with at most two distinct terms

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Jagannath Bhanja (2025). Sumsets with a minimum number of distinct terms. URL: https://arxiv.org/abs/2307.03977v2.

Claim. Question 1 asks whether ordinary and at-most-r-distinct h-fold sumsets have equal minimum cardinalities over k-element subsets of every abelian group.

Problem details · Reading note

The overpartition threshold identity

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Subhash Chand Bhoria, Pramod Eyyunni, Subhrangsu Santra (2026). On the number of missing integers in partitions. DOI: 10.48550/arXiv.2604.12557. URL: https://arxiv.org/abs/2604.12557v1.

Claim. Question 3 asks for a concise overpartition analogue of the all-threshold missing-size statistic in Theorem 2.9.

Problem details · Reading note

Edge marginals at least one half do not imply GHS for general exponential random graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Alessandra Bianchi; Francesca Collet; Elena Magnanini (2022). The GHS and other correlation inequalities for the two-star model. DOI: 10.30757/ALEA.v19-64. URL: https://arxiv.org/abs/2107.08889v2.

Claim. The paper proves the FKG, GKS and GHS inequalities for the two-star exponential random graph model with nonnegative parameters, extends FKG and GKS to general exponential random graphs with nonnegative non-edge coefficients, and conjectures that in this general setting the GHS inequality holds exactly when every edge marginal is at least one half.

Problem details · Reading note

Brown–George's argument-dependent degree-root bound

Proved. Lean theorem result, frozen in this repository 2026-10-27.

J. I. Brown and I. C. George (2025). On the roots of degree polynomials. DOI: 10.48550/arXiv.2505.04882. URL: https://arxiv.org/abs/2505.04882v1.

Claim. Conjecture 4.1 asserts that every nonzero degree root z of a graph of order n satisfies |z| ≤ (n − 1)^{|arg(z)/π|}.

Problem details · Reading note

Burri's SW–Petz decoder comparison

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Laura Burri (2025). Entanglement fidelity of Petz decoder for one-shot entanglement transmission. URL: https://arxiv.org/abs/2502.17411v1.

Claim. Question 1 asks which of the actual Schumacher-Westmoreland and ordinary Petz decoders performs better; two source-faithful instances have opposite strict entanglement-fidelity comparisons.

Problem details · Reading note

Classical Markov monogamy for every finite alphabet and every permutation

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Matheus Capela; Lucas C. Céleri; Rafael Chaves; Kavan Modi (2022). Quantum Markov monogamy inequalities. DOI: 10.1103/PhysRevA.106.022218. URL: https://arxiv.org/abs/2108.11533v4.

Claim. The paper proves a monogamy inequality for coherent information in four-time quantum Markov processes, gives six- and eight-time inequalities, and conjectures that for every Markov chain X_n to X_1 to Y_1 to Y_n and every permutation f the diagonal sum of the mutual informations I(X_i:Y_i) dominates the permuted sum, classically (Conjecture 1, checked for n up to 4) and with coherent information quantumly (Conjecture 2).

Problem details · Reading note

The fixed CDK image of improper partition matrices

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Shane Chern, Shishuo Fu (2026). Signed counting of partition matrices. DOI: 10.48550/arXiv.2508.21318. URL: https://arxiv.org/abs/2508.21318v2.

Claim. Question 5.5 asks for a characterization of the fixed CDK image of improper partition matrices.

Problem details · Reading note

The symmetric coin maximizes full-mode entanglement of the Hadamard walk on every finite circle

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Emil K. F. Donkersloot; René Sondenheimer; Jan Sperling (2026). Multipartite entanglement dynamics in quantum walks. DOI: 10.48550/arXiv.2603.24679. URL: https://arxiv.org/abs/2603.24679v1.

Claim. The paper computes the geometric measure of multipartite entanglement of single-walker quantum walks in linear optical networks, where one photon in M modes is a generalized W state, studies the Hadamard walk on a circle of P positions with localized initial coins, and, from numerics for all tested graph sizes and times, expects the symmetric coin (|-> + i|+>)/sqrt 2 to always yield the maximum entanglement for the partition into individual modes.

Problem details · Reading note

The largest root of Dukes's permaspin cubic dominates the two explicit eigenvalues

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Mark Dukes (2023). An Ising model having permutation spin motivated by a permutation complexity measure. DOI: 10.1016/j.physa.2023.129090. URL: https://arxiv.org/abs/2211.08908v3.

Claim. The paper introduces the permaspin model, an Ising-type model whose spins are permutations with an interaction energy given by a permutation statistic, solves the one-dimensional model without external field through the statistic's generating function, writes the partition function of the one-dimensional 3-permaspin model in an external field through a 6x6 symmetric transfer matrix whose eigenvalues are c(1-d)^2, c(1-d^2) and the three roots of an explicit cubic, observes numerically that the largest root of the cubic is the largest eigenvalue, and states the free energy conditionally on that observation.

Problem details · Reading note

The Dzhenzher–Sakbaev random semigroup SLLN holds on every Banach space

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Sviatoslav V. Dzhenzher; Vsevolod Zh. Sakbaev (2026). The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces. DOI: 10.1007/s43036-025-00481-7. URL: https://arxiv.org/abs/2507.07658v3.

Claim. The paper proves a strong law of large numbers for products of exponentials of i.i.d. bounded random operators: on a uniformly smooth Banach space the products exp(A_1 t/n)...exp(A_n t/n) converge almost surely in the strong operator topology to exp(t E A), uniformly for t in any segment; it proves the weak-operator version on every Banach space, deduces the strong version on l_1 by Schur's theorem, and conjectures the strong version on every Banach space.

Problem details · Reading note

Positive joint weights and all-vertex physical weights in the tree Lee–Yang conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-12. Lean theorem result, frozen in this repository 2026-10-27.

B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Claim. Section 2.2, Conjecture 1 asserts that tree Ising polynomials with arbitrary positive configuration weights have all their x roots on the unit circle.

Problem details · Reading note

Fisk Question 10: coefficient-Toeplitz minors

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Steve Fisk (2008). Questions about determinants and polynomials. URL: https://arxiv.org/abs/0808.1850v1.

Claim. Question 10 asserts that every minor of the coefficient-Toeplitz matrix of a positive upper bivariate polynomial is Hurwitz stable.

Problem details · Reading note

The sharp weighted Bessel K tail bound

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Robert E. Gaunt (2019). Inequalities for some integrals involving modified Bessel functions. DOI: 10.1090/proc/14433. URL: https://arxiv.org/abs/1806.00524.

Claim. Conjecture 2.4 asks for a sharp lower bound on a weighted modified-Bessel K tail.

Problem details · Reading note

Gupta's uniform residual degree bound

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Chakshu Gupta (2026). One construction for the Miura-ori flip-graph degree sequence. DOI: 10.48550/arXiv.2607.05567. URL: https://arxiv.org/abs/2607.05567v3.

Claim. Conjecture 8.3 bounds the residual split-count polynomial by total degree d-3; the finite Envelope Lemma and normalized-height finiteness supply the height substrate.

Problem details · Reading note

Hanna's A323689 Gaussian coefficient parity

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Paul D. Hanna (2019). OEIS A323689: Gaussian generating function and square parity. URL: https://oeis.org/A323689.

Claim. Odd terms occur only at n = k^2 for k >= 0 (conjecture); a(n^2) = 1 (mod 2) for n >= 0.

Problem details · Reading note

Hanna's A324300 Gaussian coefficient parity

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Paul D. Hanna (2019). OEIS A324300: Gaussian generating function and square parity. URL: https://oeis.org/A324300.

Claim. Odd terms occur only at n = k*(k+2) for k >= 0 (conjecture); a(n*(n+2)) = 1 (mod 2) for n >= 0.

Problem details · Reading note

Hanna's second congruence for A365771

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Paul D. Hanna (2023). OEIS A365771: Catalan-binomial residues modulo five. URL: https://oeis.org/A365771.

Claim. The three conjectures of A365771 give residues 4 at 5m, 2 at 5m+1 for positive sums m of distinct powers of five, and 0 outside the stated exceptional set.

Problem details · Reading note

Hanna's third congruence for A365771

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Paul D. Hanna (2023). OEIS A365771: Catalan-binomial residues modulo five. URL: https://oeis.org/A365771.

Claim. The three conjectures of A365771 give residues 4 at 5m, 2 at 5m+1 for positive sums m of distinct powers of five, and 0 outside the stated exceptional set.

Problem details · Reading note

Mixed imitation and best-response dynamics need not equilibrate on general graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Nazanin Hasheminejad; Pouria Ramazi (2023). Equilibration of Coordinating Imitation and Best-Response Dynamics. DOI: 10.1109/LCSYS.2023.3292049. URL: https://arxiv.org/abs/2306.03243v3.

Claim. The paper studies networks of agents who play two-strategy coordination games with their neighbours and update asynchronously, each either imitating its highest-earning neighbour or best-responding; it proves that linear graphs, rings, starlike networks under a condition on the branching agent, and sparse trees equilibrate under every persistent activation sequence, and leaves open whether dense trees or general graphs equilibrate under every activation sequence.

Problem details · Reading note

Kagey's order-eight recurrence for binary null polynomials modulo three

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Peter Kagey (2019). OEIS A329479: Binary null polynomials modulo three. URL: https://oeis.org/A329479/internal.

Claim. Conjectured recurrence: a(n) = 2a(n-1) + 2a(n-2) - 5a(n-3) - 2a(n-4) + 10a(n-5) - 4a(n-6) - 4a(n-7) + 8a(n-8).

Problem details · Reading note

The simple evolution-algebra automorphism maximum

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Problem details · Reading note

Kim–Stanton Conjecture 8.3: integer polynomial denominators

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Jang Soo Kim and Dennis Stanton (2021). Three families of q-Lommel polynomials. DOI: 10.48550/arXiv.2105.10096. URL: https://arxiv.org/abs/2105.10096v3.

Claim. Conjectures 8.3 and 8.4 prescribe integer polynomial numerators for contiguous basic hypergeometric ratios with the stated floor exponent denominator.

Problem details · Reading note

Kim–Stanton Conjecture 8.4: integer polynomial denominators

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Jang Soo Kim and Dennis Stanton (2021). Three families of q-Lommel polynomials. DOI: 10.48550/arXiv.2105.10096. URL: https://arxiv.org/abs/2105.10096v3.

Claim. Conjectures 8.3 and 8.4 prescribe integer polynomial numerators for contiguous basic hypergeometric ratios with the stated floor exponent denominator.

Problem details · Reading note

manabe-2024-restricted-amalgamation-nim-conjecture-1

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Hikaru Manabe (2024). An Amalgamation Nim with Restriction. DOI: 10.48550/arXiv.2411.14488. URL: https://arxiv.org/abs/2411.14488v1.

Claim. Conjecture 1 relates both parities of the Grundy value to the same adjacent pair of XOR values in restricted three-pile amalgamation Nim.

Problem details · Reading note

Biased post-processed interferometric improvements are not faithful

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Michele Masini; Thomas Theurer; Martin B. Plenio (2021). Coherence of operations and interferometry. DOI: 10.1103/PhysRevA.103.042426. URL: https://arxiv.org/abs/2102.04863v2.

Claim. The paper defines dynamical coherence measures from discrimination games in multi-path interferometers: pre-processed improvements for the ability of an operation to detect coherence and post-processed improvements N for its ability to create coherence, optimized over maximally incoherent operations; for phase vectors with at least two different components it proves that the pre-processed improvements are faithful exactly for unbiased priors and that the post-processed improvements are faithful for unbiased priors, and leaves open whether analogous results hold for the post-processed improvements.

Problem details · Reading note

Refutation of both Toda-prime denominator assertions

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Stephen McKean (2025). Toda primes. URL: https://arxiv.org/abs/2511.19744v1.

Claim. Definition 1.1 defines Toda primes; Notation 5.5 defines Bernoulli-denominator prime sets and first indices; Conjecture 5.6 asserts denominator-based invariance and monotonicity.

Problem details · Reading note

Neighborhood two-shadowing without quasi-finite type

Proved. Lean theorem powers_two_return_distance_example, frozen in this repository 2026-10-27.

Jonathan Meddaugh and Elyssa Stephens (2025). Shadowing with small diameter sets of bounded cardinality. URL: https://arxiv.org/abs/2509.04312v1.

Claim. Question 5.3 asks for neighborhood two-shadowing without exact-length quasi-finite type; Question 5.6 asks for weak mixing and neighborhood N-shadowing without dense small periodic sets.

Problem details · Reading note

Gaussian quotient monotonicity does not force two exceptional signs

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Madjid Mirzavaziri, Daniel Yaqubi (2026). Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler. DOI: 10.48550/arXiv.2610.19249. URL: https://arxiv.org/abs/2610.19249v1.

Claim. Problem 7.2 asks whether weak increase of the first Gaussian quotient through its middle forces the two-exception sign pattern after a second q-integer division.

Problem details · Reading note

The attained coherence optimum for arbitrary finite qubit Lüders measurements

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Sanuja D. Mohanty; Gautam Sharma; Sk Sazim; Biswajit Pradhan; Arun K. Pati (2022). Creation of quantum coherence with general measurement processes. DOI: 10.1007/s11128-021-03394-5. URL: https://arxiv.org/abs/2004.10875v2.

Claim. The paper studies how nonselective measurements, acting by the Lüders rule with the square roots of POVM effects, create l1-norm coherence. For one-parameter qubit POVMs it computes the maximum output coherence (C+|r|)/2 and characterizes the states without coherence gain and those with maximal gain. For general two-outcome POVMs it obtains no closed form, leaves open which states admit no gain and which attain the maximal gain, and lists analytical results for two-outcome and n-outcome POVMs as future work.

Problem details · Reading note

Integral adjacency support gives transfer through uniform clone matchings

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Hermie Monterde; Hiranmoy Pal; Steve Kirkland (2025). Laplacian quantum walks on blow-up graphs. DOI: 10.1016/j.laa.2025.10.001. URL: https://arxiv.org/abs/2504.11585v1.

Claim. The paper studies Laplacian quantum walks on the n-fold blow-up of a graph, whose clones of a vertex are pairwise false twins. It characterizes periodicity, strong cospectrality and Laplacian perfect and pretty good state transfer in blow-ups, and constructs blow-ups with Laplacian perfect state transfer. For n divisible by four and a vertex with integral Laplacian eigenvalue support, adding a matching among the clones of that vertex gives Laplacian perfect state transfer at pi/2 between the ends of every inserted edge. It asks whether this holds for the adjacency matrix.

Problem details · Reading note

The separable CHSH optimum under imprecise qubit measurements is attained

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Simon Morelli; Hayata Yamasaki; Marcus Huber; Armin Tavakoli (2022). Entanglement detection with imprecise measurements. DOI: 10.1103/PhysRevLett.128.250501. URL: https://arxiv.org/abs/2202.13131v2.

Claim. The paper bounds entanglement witnesses when lab measurements deviate from the targets by a fidelity-based inaccuracy epsilon; for the CHSH witness with targets sigma_X and sigma_Z on two qubits it gives an explicit separable model attaining 4(1-2eps)sqrt(eps(1-eps)) + sqrt(2-16eps(1-eps)(1-2eps)^2) for eps up to 1/2-1/(2 sqrt 2) and 2 beyond, and conjectures that this model is optimal.

Problem details · Reading note

OEIS A381190 trapezohedral domination series

Proved. Lean theorem actual_series, frozen in this repository 2026-10-27.

Eric W. Weisstein; Joerg Arndt; Peter Luschny; Emily Egolf; Veronika Furst (2026). OEIS A381190, connected minimal dominating sets in the n-trapezohedral graph. URL: https://oeis.org/A381190.

Claim. Conjectured g.f.: -2*x^3*(4*x^5 + 8*x^4 + 4*x^3 - 9*x^2 - 8*x - 3)/(x^3 + x^2 - 1)^2.

Problem details · Reading note

Howroyd's full Andrasfai mutual-visibility generating function

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Andrew Howroyd; Eric W. Weisstein (2026). OEIS A391632: mutual-visibility subsets of Andrasfai graphs. URL: https://oeis.org/A391632/internal.

Claim. Conjectured g.f.: x*(4 - 19*x + 33*x^2 - 40*x^3 + 8*x^4)/(1 - 10*x + 29*x^2 - 32*x^3 + 8*x^4).

Problem details · Reading note

The minimum skewness of a unidirectional unicyclic current

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Krzysztof Ptaszyński (2022). Bounds on skewness and kurtosis of steady-state currents. DOI: 10.1103/PhysRevE.106.024119. URL: https://arxiv.org/abs/2204.08421v4.

Problem details · Reading note

The Harrow channel pair's parallel Chernoff exponent is not one

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Farzin Salek; Masahito Hayashi; Andreas Winter (2022). Usefulness of adaptive strategies in asymptotic quantum channel discrimination. DOI: 10.1103/PhysRevA.105.022419. URL: https://arxiv.org/abs/2011.06569v3.

Claim. The paper compares adaptive and non-adaptive strategies for discriminating two channels asymptotically. For classical-quantum channels it shows that adaptive strategies give no advantage in the Chernoff, Hoeffding and Stein settings. For quantum channels it bounds the parallel Chernoff exponent through a positive-definiteness condition. For the Harrow et al. channel pair, which adaptive strategies discriminate perfectly with two uses, it proves the parallel exponent finite, at most 13.83, and conjectures that it equals 1.

Problem details · Reading note

Local-channel monotonicity of Rényi conditional quantum mutual information

Refuted. Lean theorem result, frozen in this repository 2026-10-13. Lean theorem result, frozen in this repository 2026-10-27.

K. P. Seshadreesan, M. Berta, M. M. Wilde (2015). Rényi squashed entanglement, discord, and relative entropy differences. DOI: 10.1088/1751-8113/48/39/395303. URL: https://arxiv.org/abs/1410.1443v3.

Claim. Conjecture 1 The Rényi conditional quantum mutual information obeys monotonicity with respect to CPTP maps on system A, i.e., for ρABE ∈ S(HABE) and α ∈ (0, 1) ∪ (1, 2], Iα(A; B|E)ρ ≥ Iα(A′; B|E)τ, where τA′BE ≡ MA→A′(ρABE) and MA→A′ is a CPTP map acting on system A alone.

Problem details · Reading note

The signalling game has equal classical, quantum and no-signalling values

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Akshay Seshadri; Felix Leditzky; Vikesh Siddhu; Graeme Smith (2023). On the separation of correlation-assisted sum capacities of multiple access channels. DOI: 10.1109/TIT.2023.3274434. URL: https://arxiv.org/abs/2205.13538v3.

Claim. The paper bounds the classical, quantum-assisted and no-signalling-assisted sum capacities of multiple access channels built from nonlocal games by the games' winning probabilities, introduces the signalling game in which each of two players must guess the other's question, proves that its classical value with uniformly drawn questions is one over the larger question-set size, and, from linear programs for question-set sizes 2 to 10, expects the classical, quantum and no-signalling values of this game to coincide.

Problem details · Reading note

Odd order does not ensure the conjectured minimum of Hadamardness

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Satvik Singh; Ion Nechita (2022). Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators. DOI: 10.1088/1751-8121/ac7017. URL: https://arxiv.org/abs/2112.11123v1.

Claim. The paper studies bipartite unitary and orthogonal evolution operators with diagonal symmetries and their entangling power; for the real case it introduces the Hadamardness measure Tr[(CC*)^2] of a sign matrix C, shows it is at least d^3 with equality exactly for Hadamard matrices, tabulates its minimum for d = 3, 5, 6, and conjectures that for every odd d the minimum is d^3 + d(d-1).

Problem details · Reading note

Parity-position sequences in the Sobolewski–Ulas recurrence

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Bartosz Sobolewski and Maciej Ulas (2022). Solutions of certain meta-Fibonacci recurrences. DOI: 10.4064/aa220613-19-1. URL: https://arxiv.org/abs/2204.04011v1.

Claim. Question 7.5 asks whether the increasing occurrence enumerations of each parity of h_{1,1} are k-regular for some k.

Problem details · Reading note

A steady prior does not imply tabletop time reversibility

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

M. Song, H. Kwon and V. Scarani (2025). Exact and approximate conditions of tabletop reversibility: when is Petz recovery cost-free?. DOI: 10.1088/2058-9565/aea92d. URL: https://arxiv.org/abs/2510.26895v1.

Claim. Section V, open question 2 asks whether a steady prior always makes the fixed dilation tabletop time-reversible; an explicit qutrit instance refutes this assertion even for a positive definite prior.

Problem details · Reading note

Higher exterior traces of Spezia's perimeter matrix

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Stefano Spezia (2019). OEIS A330700, second exterior-trace conjecture. URL: https://oeis.org/A330700.

Claim. For k > 2, the trace of the k-th exterior power of M(n) is equal to zero.

Problem details · Reading note

Bound-coherent states exist in every dimension at least three

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Benjamin Stratton; Chung-Yun Hsieh; Paul Skrzypczyk (2026). Unspeakable coherence concentration. DOI: 10.1103/fj8w-v1bb. URL: https://arxiv.org/abs/2512.04255v1.

Claim. The paper studies the concentration of unspeakable coherence, measured mode by mode with respect to a truncated number operator, from two identical copies of a state into one by allowed unitaries that commute with the total number operator; it characterises the qubit case completely, gives upper bounds beyond qubits, constructs a qubit concatenation protocol, exhibits a family of qubit states with bound coherence, and leaves open whether bound-coherent states exist in higher dimensions.

Problem details · Reading note

Tebni's odd-prime congruence for OEIS A364822

Proved. Lean theorem result, frozen in this repository 2026-10-27.

M. Tebni (2023). OEIS A364822, Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)). URL: https://oeis.org/A364822.

Claim. Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)). Conjectures: For p prime (p > 2), a(p) == 2 (mod p). For n = 2^m (m natural number), a(n) == 1 (mod n).

Problem details · Reading note

Tebni's power-of-two congruence for OEIS A364822

Proved. Lean theorem result, frozen in this repository 2026-10-27.

M. Tebni (2023). OEIS A364822, Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)). URL: https://oeis.org/A364822.

Claim. Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)). Conjectures: For p prime (p > 2), a(p) == 2 (mod p). For n = 2^m (m natural number), a(n) == 1 (mod n).

Problem details · Reading note

Testart Question 1: the native tree side and a type-C permutation model

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Benjamin Testart (2026). On minimal pattern-containing inversion sequences. URL: https://arxiv.org/abs/2602.12130v2.

Claim. Section 7 Question 1 asks for bijections from the Section 6 families IP or T to known type-C poly-Bernoulli models.

Problem details · Reading note

Every localized quantum Bernoulli noise walk has the standard Gaussian limit

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Caishi Wang; Ce Wang; Yuling Tang; Suling Ren (2018). Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise. DOI: 10.1155/2018/2507265. URL: https://onlinelibrary.wiley.com/doi/10.1155/2018/2507265.

Claim. The paper studies the quantum walk on the integers whose coin space is the space of square-integrable Bernoulli functionals, driven by the annihilation and creation operators of quantum Bernoulli noise. It links the moments of the walk's distribution to these operators, proves the Gaussian limit for the localized initial states built from the vacuum and one excited coordinate, derives a quantum central limit theorem, and conjectures that every localized initial state has the classical random walk's limit distribution.

Problem details · Reading note

Wiseman A387118 total partition identity

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Gus Wiseman (2025). OEIS A388712, A387118 and A388713. URL: https://oeis.org/A388712.

Claim. The three conjectures equate choosable initial-interval partition counts and superdiagonal reversed-partition counts, with complementation or a common statistic.

Problem details · Reading note

Wiseman A388713 maximum-refined partition identity

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Gus Wiseman (2025). OEIS A388712, A387118 and A388713. URL: https://oeis.org/A388712.

Claim. The three conjectures equate choosable initial-interval partition counts and superdiagonal reversed-partition counts, with complementation or a common statistic.

Problem details · Reading note

The exact triplewise joint-measurability threshold of biased trine qubit measurements

Proved. Lean theorem result, frozen in this repository 2026-10-27.

Shiv Akshar Yadavalli; Nikola Andrejic; Ravi Kunjwal (2024). Qualitative equivalence between incompatibility and Bell nonlocality. DOI: 10.1103/PhysRevA.110.L060201. URL: https://arxiv.org/abs/2008.10100v3.

Claim. The paper shows that every non-trivial joint measurability structure with finitely many vertices has a quantum realization by measurements that enable a Bell violation, so a non-trivial structure is necessary and sufficient for a Bell violation. Its Supplemental Material studies trine qubit measurements with a purity parameter and a common bias. For the first family it states the pairwise joint-measurability condition from earlier work, computes the triplewise threshold numerically by a semidefinite program, and conjectures that threshold to be the piecewise function 1+b, 2/3, 1-b.

Problem details · Reading note

The quantum coherence fraction can increase under incoherent operations

Refuted. Lean theorem result, frozen in this repository 2026-10-27.

Yao Yao; Dong Li; C. P. Sun (2019). Quantum coherence fraction. DOI: 10.1103/PhysRevA.100.032324. URL: https://arxiv.org/abs/1906.09789v1.

Claim. The paper introduces the quantum coherence fraction, the largest overlap of a density matrix with a maximally coherent state, studies its properties and its relation to the robustness of coherence and to genuinely incoherent operations, proves that it equals a modified robustness for qubit and qutrit states, and conjectures, by analogy with the fully entangled fraction, that it may not be monotone under general incoherent operations.

Problem details · Reading note

The spectral degree exponent is not submodular

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Massimo A. Achterberg, Piet Van Mieghem (2025). The spectral degree exponent of a graph. DOI: 10.48550/arXiv.2502.01815. URL: https://arxiv.org/abs/2502.01815v1.

Claim. Definition 2.1 defines the spectral degree exponent by equality of the largest adjacency eigenvalue and the degree power mean; Section 7 asks whether it is submodular.

Problem details · Reading note

Word-trace domination does not control fractional Schatten norms

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Mohammad A. Alhejji; Emanuel Knill (2024). Towards a Resolution of the Spin Alignment Problem. DOI: 10.1007/s00220-024-04980-1. URL: https://arxiv.org/abs/2307.06894v3.

Claim. The paper generalizes spin alignment problems, proves alignment results for Schatten norms of integer order through an overlap lemma on traces of ordered products, and asks whether domination of all ordered-product traces implies the Schatten-norm inequality for every real order p in [1, ∞).

Problem details · Reading note

Altizio's geometric-sum disk classification

Proved. Lean theorem result, frozen in this repository 2026-10-26.

David Altizio (2025). Zeros of Stern polynomials in the complex plane. DOI: 10.48550/arXiv.2511.03847. URL: https://arxiv.org/abs/2511.03847v1.

Claim. Remark 4 asks whether the bound |(z)_n| ≥ n|z| on |z − 2| ≤ 1 holds exactly for n = 5, 6, 7.

Problem details · Reading note

Bădică–Bădică: the Jaccard triangle inequality on arbitrary lattices

Proved. Lean theorem result, frozen in this repository 2026-10-26.

C. Bădică and A. Bădică (2026). On the Triangle Inequality for the Jaccard Distance in Arbitrary Lattices. DOI: 10.48550/arXiv.2608.18194. URL: https://arxiv.org/abs/2608.18194v1.

Claim. Section 8 asks whether supermodularity and log-submodularity, or supermodularity of f and 1/f, suffice for the generalized Jaccard triangle inequality on arbitrary lattices.

Problem details · Reading note

A closest product vector to a q-Dicke state need not be q-symmetric

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Angel Ballesteros; Ivan Gutierrez-Sagredo; Jose de Ramón; J. Javier Relancio (2025). Deformations of the symmetric subspace of qubit chains. DOI: 10.1088/1751-8121/ae05da. URL: https://arxiv.org/abs/2503.23554v3.

Claim. The paper defines the q-symmetric subspace of an N-qubit chain through the coproduct of the quantum group U_q(su(2)), constructs q-Dicke states, and in its applications section conjectures that, as it asserts for two qubits, the closest product state to an arbitrary q-Dicke state in the geometric measure of entanglement is q-symmetric.

Problem details · Reading note

A counterexample to the gamma-gamma NBU assertion

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Á. Baricz, S. Ponnusamy and M. Vuorinen (2011). Functional inequalities for modified Bessel functions. DOI: 10.1016/j.exmath.2011.07.001. URL: https://arxiv.org/abs/1009.4814v2.

Claim. Question 8 asks whether the gamma-gamma distribution has the new-is-better-than-used property.

Problem details · Reading note

Classical permutations admit a strict two-query sequential advantage

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Jessica Bavaresco; Mio Murao; Marco Túlio Quintino (2022). Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies. DOI: 10.1063/5.0075919. URL: https://arxiv.org/abs/2105.13369v3.

Claim. The paper studies minimum-error discrimination of ensembles of unitary channels with several uses, proves that parallel strategies are optimal for uniformly distributed group ensembles, exhibits sequential and indefinite-causal-order advantages for other ensembles, and asks whether discrimination of classical permutation channels, the classical analogue of unitary channels, can also be enhanced by sequential strategies.

Problem details · Reading note

bouras372763-once

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

M. Bouras (2024). OEIS A372763: denominators of a continued fraction. URL: https://oeis.org/A372763.

Claim. Conjecture 2: Except for 2 and 3, all primes appear in the sequence once.

Problem details · Reading note

Linear growth for every Gamma-label split

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Hùng Việt Chu, Steven J. Miller, and Garrett Tresch (2025). Problems Regarding a Pair of Diophantine Equations. DOI: 10.48550/arXiv.2512.12681. URL: https://arxiv.org/abs/2512.12681v1.

Claim. Section 3.4 asks whether every prescribed split of Gamma labels can be realized by a strictly increasing sequence with polynomial growth.

Problem details · Reading note

Deterministic-input violations exist at every positive energy budget below one half

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Raffaele D'Avino; Gabriel Senno; Mir Alimuddin; Antonio Acín (2025). Entanglement in the energy-constrained prepare-and-measure scenario: applications to randomness certification and channel discrimination. URL: https://arxiv.org/abs/2510.27559v3.

Claim. The paper studies energy-constrained prepare-and-measure behaviors with a shared state between the devices, shows that entanglement enlarges the set of behaviors compared with separable shared states, derives that separable devices deterministic on input 0 satisfy E_1 >= 2(1-2 omega)^2 - 1, and conjectures that for every positive energy bound some entangled behavior deterministic on input 0 violates this inequality.

Problem details · Reading note

A uniform mass gap for the π/2-flux Hofstadter model

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Simone Fabbri; Giovanna Marcelli; Alessandro Giuliani (2026). The conductivity matrix at the topological phase transition. URL: https://arxiv.org/abs/2609.03832v1.

Claim. The paper studies the conductivity matrix of two-dimensional lattice fermion models at a topological phase transition, under a gap assumption on the Bloch bands for nonzero mass; for its pi/2-flux Hofstadter example it supports the gap assumption only numerically, stating that for every m in [-2,2] minus {0} the fourth band stays above and the third band below the Fermi energy over the Brillouin zone, and expects an analytic proof.

Problem details · Reading note

Maximal connectivity of full mutual-visibility reconfiguration graphs

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Julián A. Fresán-Figueroa, Ilán A. Goldfeder, Aldo Lozano-Piña (2026). Reconfiguration graphs of mutual-visibility sets. DOI: 10.48550/arXiv.2610.24324. URL: https://arxiv.org/html/2610.24324v1.

Claim. The question after Theorem 3.6 asks whether the full mixed reconfiguration graph of mutual-visibility sets is maximally connected for every connected host.

Problem details · Reading note

The spatiotemporal Born rule for the LvN distribution has exactly two cases

Proved. Lean theorem result, frozen in this repository 2026-10-26.

James Fullwood; Zhihao Ma; Zhen Wu (2026). Quasiprobabilistic nature of the spatiotemporal Born rule. DOI: 10.1103/cw25-pqx1. URL: https://arxiv.org/abs/2507.16919v2.

Claim. The paper compares the Lüders-von Neumann distribution of two-point sequential measurements with the Terletsky-Margenau-Hill quasiprobability distribution, shows that a spatiotemporal Born rule exists for the Lüders-von Neumann distribution when the initial state is maximally mixed or the channel is discard-and-prepare, and asks whether one of these two conditions is also necessary in arbitrary dimension.

Problem details · Reading note

A counterexample to the universal Kummer modulus lower bound

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

R. Garrappa, S. Gerhold, M. Popolizio and T. Simon (2025). On some inequalities for the two-parameter Mittag-Leffler function in the complex plane. DOI: 10.1016/j.jmaa.2025.129588. URL: https://arxiv.org/abs/2410.11852.

Claim. Remark 6 asks whether the real-axis Kummer value universally bounds the complex modulus from below.

Problem details · Reading note

Gu's pseudo-coarse-grained dissipation-time trade-off is false

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Jie Gu (2023). Speed limit, dissipation bound, and dissipation-time trade-off in thermal relaxation processes. DOI: 10.1103/PhysRevE.108.L052103. URL: https://arxiv.org/abs/2304.08752v2.

Claim. The letter proves a speed limit for Markov jump processes with time-independent rates in terms of the infinity-Rényi divergence and the total rate, and conjectures, with numerical evidence, a divergence bound and a dissipation-time trade-off bound on the non-adiabatic entropy production obtained from two-state pseudo-coarse-grainings.

Problem details · Reading note

Hanna's three congruences for A365771

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Paul D. Hanna (2023). OEIS A365771: Catalan-binomial residues modulo five. URL: https://oeis.org/A365771.

Claim. The three conjectures of A365771 give residues 4 at 5m, 2 at 5m+1 for positive sums m of distinct powers of five, and 0 outside the stated exceptional set.

Problem details · Reading note

Identical TW measurements attain $2(d-1)$ for every $d\ge2$

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Hsin-Yu Hsu; Gelo Noel M. Tabia; Kai-Siang Chen; Mu-En Liu; Tamás Vértesi; Nicolas Brunner; Yeong-Cherng Liang (2026). Trading symmetry for Hilbert-space dimension in Bell-inequality violation. DOI: 10.1088/2058-9565/ae6bb1. URL: https://arxiv.org/abs/2601.02893v4.

Claim. The paper asks whether symmetric quantum strategies, with identical measurements on a swap-invariant state, reach the maximal quantum violation of party-permutation-invariant Bell inequalities in the minimal Hilbert-space dimension; it gives a family of identical Fourier-type measurements for the CGLMP inequality, finds Bell inequalities where symmetric strategies in minimal dimension are suboptimal, and conjectures that its symmetric form of the two-setting SATWAP inequalities reaches the maximal quantum value 2(d-1) with these measurements and a symmetric maximally entangled state for every d.

Problem details · Reading note

The always-failing purification policy attains the sharp availability lower bound

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Álvaro G. Iñesta; Bethany Davies; Sounak Kar; Stephanie Wehner (2026). Entanglement buffering with multiple quantum memories. DOI: 10.1038/s41534-025-01161-3. URL: https://arxiv.org/abs/2502.20240v1.

Claim. The paper models an entanglement buffer with one long-lived memory and n short-lived generation memories, with probabilistic generation, consumption, depolarising decay and probabilistic purification governed by coefficients a_k, b_k, c_k, d_k; it derives closed-form formulas for the availability and the average consumed fidelity, proves that the availability is non-increasing in the purification probability q, gives policy-independent bounds, and conjectures that the availability at q = 1 of the always-failing policy (c_k = d_k = 0) is a tight lower bound for every purification policy.

Problem details · Reading note

Complex testers do not detect every entangled Werner state

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Maria Anastasia Jivulescu; Cécilia Lancien; Ion Nechita (2022). Multipartite Entanglement Detection Via Projective Tensor Norms. DOI: 10.1007/s00023-022-01187-9. URL: https://arxiv.org/abs/2010.06365v1.

Claim. The paper introduces entanglement testers, trace-norm-to-Euclidean contractions of operator norm 1 whose tensor products detect entanglement when the projective tensor norm of the image exceeds 1; it shows that the realignment and SIC POVM criteria are testers, that both detect the Werner states exactly for mu < 1/d, and asks whether the testers are complete without a permutation of indices: does every entangled state have testers whose image has projective norm greater than 1?

Problem details · Reading note

Johnson–Wakefield Question 6.19: nonvanishing in rank three

Proved. Lean theorem result, frozen in this repository 2026-10-26.

John Johnson; Max Wakefield (2024). A non-associative incidence near-ring with a generalized Möbius function. DOI: 10.26493/1855-3974.2894.b07. URL: https://arxiv.org/abs/2204.07158v1.

Claim. Definition 5.1 defines J by rectangle sums, Definition 6.2 defines its Möbius polynomial, and Question 6.19 asks for a connected nonmodular rank-three matroid with vanishing evaluation at minus one.

Problem details · Reading note

One generating measurement does not make pseudo-broadcasting equivalent to non-contextuality

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Pauli Jokinen; Mirjam Weilenmann; Martin Plávala; Juha-Pekka Pellonpää; Jukka Kiukas; Roope Uola (2024). No-Broadcasting Characterizes Operational Contextuality. DOI: 10.1103/PhysRevLett.133.240201. URL: https://arxiv.org/abs/2406.07305v1.

Claim. The letter shows that measurement and preparation non-contextuality of a subtheory of quantum theory is equivalent to 1-to-n pseudo-broadcastability for all n, and conjectures that for a subtheory whose effects are generated by k measurements, 1-to-k pseudo-broadcasting already coincides with non-contextuality.

Problem details · Reading note

Kania–Kąkol Problem 8.2: the full Talagrand construction under CH

Proved. Lean theorem actual_CH_Talagrand_scheme_and_endpoints, frozen in this repository 2026-10-26.

Tomasz Kania and Jerzy Kakol (2026). Talagrand compacta, 2DCP, and pointwise quotients. URL: https://arxiv.org/abs/2607.06808v2.

Claim. Problem 8.2 asks whether Theorem A can be obtained from CH in place of diamond(S); actual_CH_Talagrand_scheme_and_endpoints now has a successful targeted Lean 4.35 build proving the full literal scheme, A-G, finite disjoint images and closed copies, no 2DCP, full signed dual-ball exclusion, Grothendieck convergence and point-space consequences on exactly one actual finalPrefix; scoped axiom inspection and three independent reviews have passed; PR 22319 tracks delivery CI, and no official acceptance is claimed.

Problem details · Reading note

Infinite natural-basis quotient with finite full automorphisms

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Problem details · Reading note

The ordinary nilpotency spectrum of evolution algebras

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Problem details · Reading note

Perfect homogeneous Wronskian extensions exist exactly at order one

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Markuss G. Ķēniņš; Arthemy V. Kiselev (2026). Explicit class of finite-dimensional polynomial algebras with Wronskians over R^d as N-ary Lie brackets: beyond sl(2). URL: https://arxiv.org/abs/2605.27305v2.

Claim. The paper studies finite-dimensional polynomial N-ary SH-Lie algebras in d variables with the complete generalised Wronskian of order k as bracket, shows that with a monomial of degree k+1 on top of the polynomials of degree at most k the algebra is perfect only for d=1 or d=2, k=1, gives perfect examples with homogeneous polynomials for d=2,3 and k=1, and asks whether for every d>=2 and k>=1 some homogeneous polynomial of degree k+1 makes the algebra perfect.

Problem details · Reading note

Kimberling's A341249 strict bound is false

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Clark Kimberling (2021). OEIS A341249: a(n) = floor(r*floor(s*n)), where r = 2 + sqrt(2) and s = sqrt(2).. URL: https://oeis.org/A341249.

Claim. Conjecture: 1 < r*s*n - a(n) < 2 for n >= 1.

Problem details · Reading note

Natural-basis dependence of balance in degenerate evolution algebras

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Problem details · Reading note

Finite generation of one-parameter polynomial Vassiliev algebras

Proved. Lean theorem result, frozen in this repository 2026-10-26.

E. Lanina; A. Sleptsov (2026). Algebraic structures of Vassiliev invariants for knot families. DOI: 10.1016/j.nuclphysb.2025.117268. URL: https://arxiv.org/abs/2508.02385v2.

Claim. The paper studies rational Vassiliev invariants restricted to knot families on which they are polynomials in the family parameters, shows that a k-parameter family has at most k algebraically independent such invariants, gives a two-parameter family whose algebra of invariants is infinitely generated, and conjectures that the algebra of polynomial Vassiliev invariants of every one-parameter family is finitely generated.

Problem details · Reading note

Reverse level order labeling gives one nonzero entry in every nonfinal column tail

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Olga Leskovjanová; Ladislav Mišta Jr. (2025). Minimal criteria for continuous-variable genuine multipartite entanglement. DOI: 10.22331/q-2025-08-27-1837. URL: https://arxiv.org/abs/2401.04376v3.

Claim. The paper constructs genuine multipartite entanglement criteria for N-mode Gaussian states from tree graphs, labels each tree by a reverse level order labeling rooted at a center vertex or an endpoint of the central edge, and conjectures, after checking all trees up to N = 10, that this labeling always gives an adjacency matrix whose column tails A_{i+1:N,i} each have exactly one nonzero entry, so that the corresponding quadrature combinations have at most two nonzero terms.

Problem details · Reading note

A general obstruction to the pinnacle middle-order lattice

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

J. Liang and B. E. Sagan (2026). Log-concavity and log-convexity via distributive lattices. DOI: 10.1007/s11083-026-09735-2. URL: https://arxiv.org/abs/2408.02782v2.

Claim. Question 8.6(a) asks whether the middle order restricted to each ordered pinnacle class is a distributive lattice.

Problem details · Reading note

Lin–Miao–Yu–Sheng Conjecture 5.11: second-norm uniqueness

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Zhen Lin, Lianying Miao, Guanglong Yu and Han Sheng (2022). On the Ky Fan k-norm of the LI-matrix of graphs. DOI: 10.13001/ela.2022.5943. URL: https://arxiv.org/abs/2012.11162v2.

Claim. Conjecture 5.11 asserts uniqueness of the connected graph G_{m,n} maximizing the second and third Ky Fan norms of the centered Laplacian.

Problem details · Reading note

The Ma–Weigert derivative-sign regions form a descending chain

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Rourou Ma; Julian Weigert (2025). Complete monotonicity of log-functions. DOI: 10.1007/s13324-025-01136-9. URL: https://arxiv.org/abs/2505.04225v2.

Claim. The paper studies the functions p(log x_1, ..., log x_s)/(x_1 ... x_s) with p a polynomial of total degree at most n, shows that their completely monotone members form a semialgebraic cone, proves that in one variable and degree two the regions cut out by the sign conditions on single derivatives form a descending chain, and conjectures that for every degree these regions form a descending chain converging to the completely monotone cone.

Problem details · Reading note

Malkoun's Conjecture D fails for Young subgroups

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Joseph Malkoun (2025). Finite graphs and configurations of points. DOI: 10.1016/j.geomphys.2026.105993. URL: https://arxiv.org/abs/2508.13472v3.

Claim. The paper generalizes the Atiyah problem on configurations of points by graph amplitudes and states Conjecture D: for two equivalence relations on [n] with class-preserving permutation groups P_1, P_2 and every Hermitian positive semidefinite H, Re f(H) >= |P_1 ∩ P_2| prod_i H_ii, where f(H) sums prod_i H_{sigma_1(i), sigma_2(i)} over P_1 × P_2; it proves the case of nested relations and that Conjecture D implies its tree Conjecture C.

Problem details · Reading note

Non-local partial entanglement annihilation is not closed under aligned tensor products

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Bivas Mallick; Nirman Ganguly; A. S. Majumdar (2026). On the characterization of partially entanglement breaking and annihilating channels. DOI: 10.1063/5.0274921. URL: https://arxiv.org/abs/2411.19315v2.

Claim. The paper characterizes r-partially entanglement breaking channels, shows that they are in general not closed under tensor products for r>1, introduces local and non-local r-partially entanglement annihilating channels acting on a bipartite system C^d ⊗ C^d through the Schmidt number of their outputs, and leaves open whether the non-local r-partially entanglement annihilating channels are closed under tensor products.

Problem details · Reading note

Mazorchuk–Xu: every first-column deficit has a counting polynomial

Proved. Lean theorem mazorchukXu_conjecture, frozen in this repository 2026-10-26.

Volodymyr Mazorchuk, Shuoyang Xu (2026). On the number of cokernel-closed additive subcategories for uniformly oriented A_n quivers. DOI: 10.48550/arXiv.2610.07999. URL: https://arxiv.org/html/2610.07999v2.

Claim. Section 5.7 conjectures that every fixed first-column deficit i gives a polynomial of exact degree 2i and leading coefficient 1/i!.

Problem details · Reading note

Connected Hamming-weight gate patterns have an inverse-polynomial spectral gap

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Léo Monbroussou; Eliott Z. Mamon; Jonas Landman; Alex B. Grilo; Romain Kukla; Elham Kashefi (2025). Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning. DOI: 10.22331/q-2025-05-15-1745. URL: https://arxiv.org/abs/2309.15547v4.

Claim. The paper studies Hamming-weight preserving variational circuits of RBS/FBS gates, reduces their gradient variance to a product T = T_J ... T_1 of column-stochastic matrices obtained by squaring the gate entries at angle pi/4 in the weight-k subspace, and conjectures that for connected gate patterns with polynomially many gates and fixed weight k the spectral gap 1 - |lambda_2(T_n)| is at least inverse-polynomial in n.

Problem details · Reading note

Noninteger parameters do not ensure injectivity of the Szalay–Tóth partition generators

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Szilárd Szalay; Géza Tóth (2025). Alternatives of entanglement depth and metrological entanglement criteria. DOI: 10.22331/q-2025-04-18-1718. URL: https://arxiv.org/abs/2408.15350v3.

Claim. The paper labels multipartite entanglement properties by integer partitions, introduces generator functions on partitions such as the power sum s_q and the entanglement dimension Dim_d, notes integer collisions of both, and expects s_q to be injective for noninteger q and Dim_d to be injective for noninteger d.

Problem details · Reading note

Wiseman's nonchoosable partition identity

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Gus Wiseman (2025). OEIS A388712, A387118 and A388713. URL: https://oeis.org/A388712.

Claim. The three conjectures equate choosable initial-interval partition counts and superdiagonal reversed-partition counts, with complementation or a common statistic.

Problem details · Reading note

Wu’s adjacent-index Fibonacci divisibility conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Chai Wah Wu; Alex Ratushnyak (2026). OEIS A221018, first FORMULA conjecture on adjacent-index Fibonacci divisibility. URL: https://oeis.org/A221018/internal.

Claim. a(n) % 10 == 0, 2, or 8 (conjectured). - _Chai Wah Wu_, Aug 06 2026

Problem details · Reading note

A finite set supplies one state entangled for all fixed tensor partitions in every frame

Proved. Lean theorem result, frozen in this repository 2026-10-26.

Baichu Yu; Pooja Jayachandran; Adam Burchardt; Yu Cai; Nicolas Brunner; Valerio Scarani (2021). Absolutely entangled sets of pure states for bipartitions and multipartitions. DOI: 10.1103/PhysRevA.104.032414. URL: https://arxiv.org/abs/2106.16216v3.

Claim. The paper studies absolutely entangled sets of pure states, sets for which every global unitary leaves at least one state entangled; it defines them for multipartitions of a non-prime dimension, constructs a set that is absolutely entangled with respect to every partition, and asks whether some absolutely entangled set has, for every unitary, one state entangled according to all partitions.

Problem details · Reading note

Zhang's three-operator Clarkson–McCarthy conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-26.

Teng Zhang (2024). Clarkson–McCarthy type inequalities, part I: ℓ_p–ℓ_p and ℓ_q–ℓ_p Schatten p-estimates. URL: https://arxiv.org/abs/2410.21961v4.

Claim. The paper proves Clarkson–McCarthy type inequalities for Schatten p-norms, including the sharp Euler-type estimate with constant 2^(p-2) for three operators, and conjectures, from numerical evidence, the sharp three-operator inequality ||A+B+C||_p^p+||A||_p^p+||B||_p^p+||C||_p^p <= (3^(p-1)+1)/2^p (||A+B||_p^p+||B+C||_p^p+||C+A||_p^p) for p >= 2, reversed for 0 < p <= 2.

Problem details · Reading note

Both proposed spectral-radius bounds for asynchronous iteration are false

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Sinan G. Aksoy; Ryan Bennink; Yuzhou Chen; José Frías; Yulia R. Gel; Bill Kay; Uwe Naumann; Carlos Ortiz Marrero; Anthony V. Petyuk; Sandip Roy; Ignacio Segovia-Dominguez; Nate Veldt; Stephen J. Young (2023). Seven open problems in applied combinatorics. DOI: 10.4310/JOC.2023.v14.n4.a8. URL: https://arxiv.org/abs/2303.11464v1.

Claim. The section on the price of asynchrony models delayed block updates of x ↦ Mx by a linear operator M^(δ) on the stacked history, shows that its spectral radius does not depend on the history length and equals ρ(M)^(1/(k+1)) for uniform delay k, and asks whether ρ(M^(δ)) ≤ ρ(M)^(1/(k+1)) for every delay pattern with entries in [k], or at least whether ρ(M^(δ)) ≤ f(ρ(M), k) for some f with values in (0,1).

Problem details · Reading note

Refutation of order log-convexity for the modified Bessel product

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Á. Baricz and S. Ponnusamy (2013). On Turán type inequalities for modified Bessel functions. URL: https://arxiv.org/abs/1010.3346.

Claim. The final conjecture asserts log-convexity of the modified Bessel product in its order on (-1,infinity) for every positive argument.

Problem details · Reading note

Global ZZ gates admit schedules within the parity bound

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Pascal Baßler; Markus Heinrich; Martin Kliesch (2024). Time-optimal multi-qubit gates: Complexity, efficient heuristic and gate-time bounds. DOI: 10.22331/q-2024-03-13-1279. URL: https://arxiv.org/abs/2307.11160v2.

Claim. The paper casts the time-optimal synthesis of global ZZ gates as a linear program over the cut-polytope vertices m m^T, proves gate-time bounds and a tightness theorem for M = -C m m^T, and conjectures that the optimal gate time is at most the largest coupling times n (odd n) or n-1 (even n).

Problem details · Reading note

The source-digraph polytope

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Ämin Baumeler; Eleftherios-Ermis Tselentis; Stefan Wolf (2025). Graphical Tests of Causality. DOI: 10.1016/j.jlamp.2026.101133. URL: https://arxiv.org/abs/2511.22552v1.

Claim. After Definition 13, page 24: We know that Q̃_n^src = Q_n^src for n ≤ 4. In general, this identity is unsolved.

Problem details · Reading note

Bergman's Question 19: the field condition in Question 16

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

George M. Bergman (2014). Thoughts on Eggert's Conjecture. DOI: 10.1090/conm/609/12101. URL: https://arxiv.org/abs/1206.0326v2.

Claim. Questions 15 and 19 ask about powers in contracted semigroup spans and the field-condition extension of the nilpotent-algebra dimension bound from Question 16.

Problem details · Reading note

The classical network coherence maximum is $1/4$ for every $n\ge2$

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Fatemeh Bibak; Flavio Del Santo; Borivoje Dakić (2024). Quantum coherence in networks. DOI: 10.1103/PhysRevLett.133.230201. URL: https://arxiv.org/abs/2407.11122v1.

Claim. The paper defines a network figure of merit C_n distinguishing coherent n-party distributions from classical mixtures of two product distributions, computes the quantum value 1/2 - 2^{-n}, and from numerical optimization for n = 2, …, 10 conjectures that the classical maximum of C_n is 1/4 for all n.

Problem details · Reading note

Strongly polynomial graph families with constant depth

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Tristram Bogart and Santiago Jiménez-Salazar (2026). Strongly regular graphs, strongly polynomial sequences, and two-level polynomials. DOI: 10.48550/arXiv.2609.32077. URL: https://arxiv.org/abs/2609.32077v1.

Claim. Question 3.19 asks for strongly polynomial graph families with constant depth and a degree function of positive degree.

Problem details · Reading note

Six broadcast-dominating packing classes on the two-way infinite path

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Boštjan Brešar, Jasmina Ferme, Wenjie Hu (2026). Partitioning an S-packing coloring into broadcast dominating sets. DOI: 10.48550/arXiv.2610.03477. URL: https://arxiv.org/abs/2610.03477v1.

Claim. Introduces packing k-domatic colourings and the invariant χ_{ρ,k}; proves χ_{ρ,k}(P_n) > k for k ≥ 12 and n ≥ k (Theorem 3.5) and asks in Problem 2 whether χ_{ρ,k}(P_n) ≤ k + 1 for k ≥ 3 and n ≥ 2k.

Problem details · Reading note

No reverse perfect state transfer in the seven-state bipartite walk

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Qiuting Chen; Chris Godsil; Mariia Sobchuk; Hanmeng Zhan (2024). Hamiltonians of bipartite walks. DOI: 10.37236/12040. URL: https://arxiv.org/abs/2207.01673v1.

Claim. The paper studies bipartite discrete quantum walks with transition matrix U=(2P-I)(2Q-I) built from the two vertex-class partitions of the edges of a bipartite graph, and in its open questions displays the transition matrix of a seven-edge example with perfect state transfer from e1 to e6 at step 1 and states the suspicion that there is no perfect state transfer from e6 to e1.

Problem details · Reading note

Locality tensor products need not be locality vector spaces

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Pierre J. Clavier; Loic Foissy; Diego A. López; Sylvie Paycha (2022). Tensor products and the Milnor-Moore theorem in the locality setup. DOI: 10.1007/s40879-025-00822-z. URL: https://arxiv.org/abs/2205.14616v1.

Claim. The paper develops locality vector spaces and locality tensor products V⊗_⊤W = K(V×_⊤W)/(I_bil ∩ K(V×_⊤W)), and states as Conjectural statement 6.16 that I_bil(V1,V2) ∩ K(V1×_⊤V2) is locality compatible with the relation ⊤_{V1×_⊤V2}, a condition under which the locality tensor product is a locality vector space.

Problem details · Reading note

A circulant counterexample to Davies–Marigliano Conjecture 4.4

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Isobel Davies and Orlando Marigliano (2021). Coloured Graphical Models and their Symmetries. URL: https://arxiv.org/abs/2012.01905v5.

Claim. Conjecture 4.4 equates full-automorphism unordered-pair orbit counts and distinct adjacency eigenvalue counts for every uncoloured circulant graph.

Problem details · Reading note

Vacuum and one-photon Wigner-majorization chains

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Jan de Boer, Giuseppe Di Giulio, Esko Keski-Vakkuri, Erik Tonni (2025). Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states. DOI: 10.1103/w561-h3z5. URL: https://arxiv.org/abs/2412.19698v3.

Claim. The vacuum/one-photon mixtures form a Wigner-majorization chain increasing on [1/2,1] and decreasing on [0,1/2] under Proposal 1.

Problem details · Reading note

A nonfactorizable doubled triangle

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Gemma De las Cuevas; Andreas Klingler; Tim Netzer (2024). Polynomial decompositions with invariance and positivity inspired by tensors. DOI: 10.1016/j.laa.2024.05.025. URL: https://arxiv.org/abs/2109.06680v2.

Claim. The paper defines invariant decompositions of polynomials over weighted simplicial complexes with group actions, introduces factorizable (Omega,G) structures through a positive equivariant solution of a product system with right-hand side K_alpha^{-1}, uses factorizability as a hypothesis of its rank inequalities, shows that a free vertex action and the double edge with copy-swapping action are factorizable, and asks whether non-factorizable (Omega,G) structures exist.

Problem details · Reading note

The Katz cut-off limit for path graphs

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Emily J. Evans, Amanda E. Francis and Rebecca D. Jones (2025). Katz similarity index comparisons. DOI: 10.48550/arXiv.2509.02874. URL: https://arxiv.org/abs/2509.02874v1.

Claim. The cut-off values α_n converge to 1/√5 from above.

Problem details · Reading note

Fan–Wu–Fei channel coherence domination is false

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Jiaorui Fan; Zhaoqi Wu; Shao-Ming Fei (2024). Quantifying coherence of quantum channels based on the generalized α-z-relative Rényi entropy. DOI: 10.1007/s11128-024-04309-w. URL: https://arxiv.org/abs/2506.10796v1.

Claim. Section 3 conjectures C_{α,1}(φ) ≥ C̃_{α,1}(φ) for all quantum channels φ, with α ∈ (0,1) ∪ (1,2].

Problem details · Reading note

Jacobi and reducedness obstruct the second E6 and truck Poisson bracket conjectures

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Kirsty Gledhill; Amihay Hanany (2023). Poisson brackets for some Coulomb branches. DOI: 10.1007/JHEP03(2023)154. URL: https://arxiv.org/abs/2210.02966v2.

Claim. The paper computes Poisson brackets between generators of Coulomb branches of 3d N=4 unitary quivers, and for several families of magnetic quivers of 5d theories at infinite coupling conjectures the brackets between generators viewed as representations of the global symmetry, including the E6,n^II and truck families with spinor generators s^α, s_α and the U(1) invariant C_1.

Problem details · Reading note

Conditional randomness implies uniform attenuation without sharpness

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Stan Gudder (2021). Mutually Unbiased Quantum Observables. DOI: 10.48550/arXiv.2110.03099. URL: https://arxiv.org/abs/2110.03099v1.

Claim. The paper defines sequential products A∘B=A^{1/2}BA^{1/2} of effects on a finite-dimensional Hilbert space, finite observables, sequential products of observables and the conditioned observable (B|A)_y=Σ_x A_x∘B_y, introduces uniform attenuation (Condition (1)) and conditional randomness (Condition (2)) of two observables, proves that Condition (1) implies Condition (2) and that the two are equivalent for sharp observables, and states that it does not know whether the converse of Theorem 2.1(i) or (ii) holds.

Problem details · Reading note

Gupta's rectangular-grid dispersion inequality

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Chakshu Gupta (2026). Height functions on the m × n Miura-ori flip graph: degree sequence and diameter. DOI: 10.48550/arXiv.2606.22614. URL: https://arxiv.org/abs/2606.22614v2.

Claim. Inequality (4), p. 14: max_ϕ disp(ϕ) ≤ D(m, n) for integer 1-Lipschitz functions on the rectangular grid.

Problem details · Reading note

Refutation of reciprocal q-gamma coefficient positivity

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

S. I. Kalmykov and D. B. Karp (2018). Inequalities for series in q-shifted factorials and q-gamma functions. DOI: 10.1016/j.jmaa.2017.11.041. URL: https://arxiv.org/abs/1702.03788.

Claim. Conjecture 1 extends Theorem 3 from positive integral alpha to all positive real alpha, including the coefficient positivity assertion under its remaining parameter restrictions.

Problem details · Reading note

Refutation of reciprocal q-gamma function positivity

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

S. I. Kalmykov and D. B. Karp (2018). Inequalities for series in q-shifted factorials and q-gamma functions. DOI: 10.1016/j.jmaa.2017.11.041. URL: https://arxiv.org/abs/1702.03788.

Claim. Conjecture 1 extends Theorem 3 from positive integral alpha to all positive real alpha, including the coefficient positivity assertion under its remaining parameter restrictions.

Problem details · Reading note

The function clause of the shifted Gamma Turanian conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

S. I. Kalmykov and D. B. Karp (2017). Log-concavity and Turán-type inequalities for the generalized hypergeometric function. DOI: 10.1007/s10476-017-0503-z. URL: https://arxiv.org/abs/1611.06724.

Claim. Conjectures 1 and 2 assert coefficient nonnegativity for shifted hypergeometric Turanians when mu is at least one.

Problem details · Reading note

Refutation of the sum-condition Luke lower bound

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

D. Karp and S. M. Sitnik (2009). Inequalities and monotonicity of ratios for generalized hypergeometric function. DOI: 10.1016/j.jat.2008.10.002. URL: https://arxiv.org/abs/math/0703084.

Claim. Conjectures 1 and 2 propose Luke lower and upper bounds under a positive sum of denominator-minus-numerator parameters.

Problem details · Reading note

Refutation of the sum-condition Luke upper bound

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

D. Karp and S. M. Sitnik (2009). Inequalities and monotonicity of ratios for generalized hypergeometric function. DOI: 10.1016/j.jat.2008.10.002. URL: https://arxiv.org/abs/math/0703084.

Claim. Conjectures 1 and 2 propose Luke lower and upper bounds under a positive sum of denominator-minus-numerator parameters.

Problem details · Reading note

Log-concavity of modified Bessel ratios

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

D. Karp, A. Vishnyakova, and Y. Zhang (2025). Unimodality preservation by ratios of functional series and integral transforms. DOI: 10.1007/s00025-025-02431-4. URL: https://arxiv.org/abs/2408.01755.

Claim. Conjecture 1 asserts unimodality of modified Bessel ratios and log-concavity for positive ordered orders.

Problem details · Reading note

Coefficientwise log-concavity of the triple rising-factorial series

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Dmitrii Karp and Yi Zhang (2023). Log-concavity and log-convexity of series containing multiple Pochhammer symbols. DOI: 10.1007/s13540-023-00238-0. URL: https://arxiv.org/abs/2305.09029.

Claim. Conjecture 5.2 asserts coefficient-wise log-concavity of both the reciprocal-Pochhammer series and the related gamma series for a log-concave coefficient sequence.

Problem details · Reading note

Coefficientwise log-concavity and log-convexity of Catalan-power series

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Dmitrii Karp and Yi Zhang (2023). Log-concavity and log-convexity of series containing multiple Pochhammer symbols. DOI: 10.1007/s13540-023-00238-0. URL: https://arxiv.org/abs/2305.09029.

Claim. Conjecture 5.2 asserts coefficient-wise log-concavity of both the reciprocal-Pochhammer series and the related gamma series for a log-concave coefficient sequence.

Problem details · Reading note

Coefficientwise log-concavity of the two Karp--Zhang series

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Dmitrii Karp and Yi Zhang (2023). Log-concavity and log-convexity of series containing multiple Pochhammer symbols. DOI: 10.1007/s13540-023-00238-0. URL: https://arxiv.org/abs/2305.09029.

Claim. Conjecture 5.2 asserts coefficient-wise log-concavity of both the reciprocal-Pochhammer series and the related gamma series for a log-concave coefficient sequence.

Problem details · Reading note

Natural orthonormal bases of a dense-square Hilbert evolution algebra

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Problem details · Reading note

Interior perfect state transfer without endpoint transfer

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang (2019). Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials. DOI: 10.1080/03081087.2018.1442810. URL: https://arxiv.org/abs/1708.03283v2.

Claim. Rational weights conjecture: a weighted path on at least four vertices, with or without potentials, whose edge weights are all rational has no adjacency matrix perfect state transfer between its end vertices at readout time π. Proposition prop:rat proves it for n = 4 and for n ≥ 5 with n ≡ 3 or 5 (mod 8). Proposition prop:internal proves that endpoint PST implies PST between each pair of mirror interior vertices, with a converse when no eigenvector vanishes at the interior vertex, and leaves the general converse open.

Problem details · Reading note

Self-conjugate even polynomial Conjecture 2

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

William F. Klostermeyer (2026). Prime Nullspace Matrices and Even Dominating Sets. URL: https://arxiv.org/html/2610.31475v1#Thmconj2.

Claim. Conjecture 2 asserts a sparse powers-of-two characterization of self-conjugate even polynomials over GF(2), with parity determined by the constant term.

Problem details · Reading note

A block measurement refutes mutual-unbiasedness necessity for two-classical dephasing

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Davide Lonigro; Dariusz Chruściński (2022). On the classicality of quantum dephasing processes. DOI: 10.3389/frqst.2022.1090022. URL: https://arxiv.org/abs/2211.02014v2.

Claim. The paper studies the multitime statistics of pure dephasing systems probed by repeated sharp measurements, proves that a diagonal initial state measured in a phase-modified Fourier basis of Eq. eq:mubs, which is unbiased with respect to the dephasing basis, gives a 2-classical process for every environment, and conjectures that, apart from a maximally mixed initial state, this is the only measurement choice yielding a genuinely time-dependent 2-classical process.

Problem details · Reading note

The Meer–Weeks fifth-moment bounds for distributions bounded on one side

Proved. Lean theorem result, frozen in this repository 2026-10-25.

David J. Meer; Eric R. Weeks (2024). Bounded distributions place limits on skewness and larger moments. DOI: 10.1371/journal.pone.0297862. URL: https://arxiv.org/abs/2308.05006v2.

Claim. The paper proves lower bounds on the skewness of distributions bounded on one side in terms of the coefficient of variation, derives the corresponding bidisperse limits for all standardized moments D_n, conjectures that the bidisperse limit D_5 ≥ z^3 - z + z^(-1) - z^(-3) holds for every distribution bounded from below (and its analogue for distributions bounded from above), and suggests that the two-sided bidisperse kurtosis limits hold for every distribution bounded on both sides.

Problem details · Reading note

A three-point law exceeds the Meer–Weeks two-sided kurtosis upper limit

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

David J. Meer; Eric R. Weeks (2024). Bounded distributions place limits on skewness and larger moments. DOI: 10.1371/journal.pone.0297862. URL: https://arxiv.org/abs/2308.05006v2.

Problem details · Reading note

Two strongly cospectral vertices of $K_{2,3}$ exceed the multiplicity bound

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Hermie Monterde (2022). Strong cospectrality and twin vertices in weighted graphs. DOI: 10.13001/ela.2022.6721. URL: https://arxiv.org/abs/2111.01265v2.

Claim. The paper studies strong cospectrality of vertices with respect to generalized adjacency and normalized adjacency matrices of weighted graphs, relates it to twin vertices, recalls the Arnadottir-Godsil bound |V(X)|/n on pairwise adjacency strongly cospectral vertices of normal Cayley graphs, where n is the largest eigenvalue multiplicity of A, and asks whether this bound holds for other classes of graphs and in general for generalized adjacency matrices.

Problem details · Reading note

Optimal unambiguous discrimination without repetition for equal-overlap states

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Shayeef Murshid; Tathagata Gupta; Vincent Russo; Somshubhro Bandyopadhyay (2026). Quantum nonlocality without entanglement and state discrimination measures. DOI: 10.22331/q-2026-07-23-2174. URL: https://arxiv.org/abs/2506.20560v3.

Claim. The paper constructs product ensembles that LOCC discriminates optimally for unambiguous but not for minimum-error discrimination, and conjectures for the no-repetition ensembles E_{N,k} that the local and global unambiguous optima coincide with value (1-s)^k for k < N and (1-s)^(N-1)(1+(N-1)s) for k = N.

Problem details · Reading note

Nathanson Problem 9: existence of a nontrivial proper spectrum

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Melvyn B. Nathanson (2026). Problems and results on intersections of product sets and sumsets in semigroups. DOI: 10.48550/arXiv.2604.04781. URL: https://arxiv.org/abs/2604.04781v2.

Claim. Problem 9 asks for a nontrivial proper product intersection spectrum in every infinite torsion group; Problem 12 asks for affine invariance of integer sum intersection spectra.

Problem details · Reading note

The proposed pruning ranges do not prepare every qutrit Dicke state

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Rafael I. Nepomechie; David Raveh (2024). Qudit Dicke state preparation. DOI: 10.26421/QIC24.1-2-2. URL: https://arxiv.org/abs/2301.04989v4.

Claim. The paper constructs a deterministic circuit U_n = W_2 ⋯ W_n that prepares every qudit Dicke state from the identity-permutation input, decomposes the qutrit W operators into controlled X and R^y gates I_{m,l} and II_{m,l1,l2}, and conjectures, from experiments, that for fixed occupations the qutrit circuit can be pruned to the gates in the index ranges of its equations W3genericsimple, W3specialsimple and kkdefs while still preparing the Dicke state.

Problem details · Reading note

Kagey's eventual equality of power-sequence minima

Proved. Lean theorem result, frozen in this repository 2026-10-25.

P. Kagey; OEIS Foundation Inc. (2021). OEIS A343825, prime-power sequence minima and A071537. URL: https://oeis.org/A343825.

Claim. Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p.

Problem details · Reading note

The steering substitution need not preserve powers

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Alexandre C. Orthey; Remigiusz Augusiak (2025). Certifying classes of d-outcome measurements with quantum steering. DOI: 10.1088/1367-2630/adda73. URL: https://arxiv.org/abs/2410.20477v2.

Claim. The paper constructs steering inequalities that self-test classes of d-outcome measurements built from Heisenberg-Weyl operators with admissible phases, proves the self-testing statement, and for the robustness analysis with d > 3 assumes Conjecture 1: the transformed operators satisfy B̃_k^(n) = B̃_k^n for every prime d and every n in {1, …, d-1}, noted as true for d = 3.

Problem details · Reading note

The noisy qutrit Werner–Holevo channel is a GKSL exponential only at zero noise on $[0,2/3]$

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Shayan Roofeh; Vahid Karimipour (2024). Noisy Werner-Holevo channel and its properties. DOI: 10.1103/PhysRevA.109.052620. URL: https://arxiv.org/abs/2310.15353v7.

Claim. The paper studies the qutrit channel Λ_x = (1-x) id + x Λ_WH, computes its spectrum and determinant (negative for 2/3 < x ≤ 1, so not infinitely divisible there), and leaves open whether Λ_x is the exponential of Lindbladian dynamics for 0 ≤ x ≤ 2/3.

Problem details · Reading note

Subsystem size does not order coherent information in noisy graph states

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Aqil Sajjad; Eneet Kaur; Kenneth Goodenough; Don Towsley; Saikat Guha (2026). Lower bounds on bipartite entanglement in noisy graph states. DOI: 10.22331/q-2026-05-21-2111. URL: https://arxiv.org/abs/2404.09014v2.

Claim. The paper computes the coherent information of graph states whose qubits are dephased before the CZ gates, in terms of the biadjacency matrix of a bipartition, and conjectures from its examples that the subsystem with more qubits always has the larger coherent information.

Problem details · Reading note

The temporal Bell endpoint contraction identity for every open XX chain

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Andrea Tononi; Maciej Lewenstein (2025). Temporal Bell inequalities in non-relativistic many-body physics. DOI: 10.1088/2058-9565/adcbce. URL: https://arxiv.org/abs/2409.17290v4.

Claim. The paper evaluates temporal Clauser-Horne correlators for the XX spin chain in a transverse field through the Jordan-Wigner fermions and their propagator, and conjectures a contraction identity Re[<0|f_N f_1 σ^x_N(t) f_1†|0> + <0|f_N f_1 σ^x_N(t) f_N†|0>] = Re[G_NN(t) - G_1N(t)], verified for 2 ≤ N ≤ 8 and assumed for every N.

Problem details · Reading note

Simple-spectrum Gibbs-stochastic matrices for every positive Gibbs vector

Proved. Lean theorem result, frozen in this repository 2026-10-25.

Frederik vom Ende (2024). Almost All Quantum Channels Are Diagonalizable. DOI: 10.1142/S1230161224500124. URL: https://arxiv.org/abs/2403.19643v3.

Claim. The paper shows that diagonalizable maps are dense in several analytically path-connected classes of channels once one element with only simple eigenvalues is found (Lemma lemma_hartfiel), and leaves as an open problem a construction, valid for all Gibbs states, of Gibbs-stochastic matrices with distinct eigenvalues.

Problem details · Reading note

Compatible projective tests do not imply compatible quantum channels

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Qing-Hua Zhang; Ion Nechita (2022). A Fisher information-based incompatibility criterion for quantum channels. DOI: 10.3390/e24060805. URL: https://arxiv.org/abs/2204.09963v1.

Claim. The paper defines compatibility of quantum channels through a joint channel with the given marginals, proves that compatible channels send every choice of orthonormal bases to compatible projective-test POVMs, conjectures the converse for two channels on the same space, and derives a Fisher information-based incompatibility criterion applied to partially depolarizing channels.

Problem details · Reading note

The controlled-rotation channel assemblages do not obey the proposed LOSR angle ordering

Refuted. Lean theorem result, frozen in this repository 2026-10-25.

Beata Zjawin; David Schmid; Matty J. Hoban; Ana Belén Sainz (2023). The resource theory of nonclassicality of channel assemblages. DOI: 10.22331/q-2023-10-10-1134. URL: https://arxiv.org/abs/2209.10177v3.

Claim. The paper develops an LOSR resource theory for channel assemblages, gives semidefinite programs that test freeness and LOSR convertibility, and, from SDP observations on the family R of assemblages produced by a Bell-state-controlled y rotation, conjectures that R^θ1 converts to R^θ2 under LOSR if and only if θ1 ≥ θ2.

Problem details · Reading note

The extremal matching number of almost regular triangle-free graphs

Proved. Lean theorem result, frozen in this repository 2026-10-24.

M. Ahanjideh; T. Ekim; M. A. Yıldız (2022). Maximum size of a triangle-free graph with bounded maximum degree and matching number. DOI: 10.1007/s10878-024-01123-z. URL: https://arxiv.org/abs/2207.02271v1.

Claim. Conjecture 6.5 states that Z(d) = floor(5(d+1)/4) for every odd d at least 21.

Problem details · Reading note

arxiv-2506-09912-shkolnikov-sandpile-limit-divisible

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Mikhail Shkolnikov (2025). Algebraic Limits of Sandpiles. DOI: 10.48550/arXiv.2506.09912. URL: https://arxiv.org/abs/2506.09912v3.

Claim. Section 3 states that the direct limit of rectangular sandpile groups is expected to be a rational subtorus or a divisible group.

Problem details · Reading note

A polynomial bound for binary-polynomial Collatz length

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Luis H. Gallardo and Olivier Rahavandrainy (2025). A variant of Collatz's Conjecture over Binary Polynomials. DOI: 10.48550/arXiv.2510.07530. URL: https://arxiv.org/abs/2510.07530v1.

Claim. However, several computations suggest that here also r_A is bounded by a polynomial in deg(A) (see Section 3).

Problem details · Reading note

bahturin-centrone-pereira-2026-peak-group-refutation

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

Bahturin, Yuri; Centrone, Lucio; Pereira, Kauê (2026). Quantum regularity of finite dimensional semisimple algebras. DOI: 10.48550/arXiv.2608.21936. URL: https://arxiv.org/abs/2608.21936v1.

Claim. Conjecture 31 asserts that every finite peak group is either solvable or a Frobenius group with the specified complement structure, exclusively.

Problem details · Reading note

The complex matrix exponential does not determine the halved commutator

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

Michal Bathory (2025). Commutator calculus and symbolic differentiation of matrix functions. DOI: 10.1137/25M1760209. URL: https://arxiv.org/abs/2505.07987v3.

Claim. The paper develops a calculus of functions of the left and right multiplication operators of a real symmetric matrix, expresses derivatives of matrix functions through the anticommutator and commutator operators, proves identities such as ad_G = (log L_A − log R_A)/2 for A = e^G, and conjectures that, with a suitable generalization of the defining functional calculus, these identities hold for every complex square matrix G.

Problem details · Reading note

Braun–Bruegge Conjecture 5.1

Proved. Lean theorem result, frozen in this repository 2026-10-24.

B. Braun, K. Bruegge (2023). Facets of Symmetric Edge Polytopes for Graphs with Few Edges. URL: https://arxiv.org/abs/2201.13303v4.

Claim. Conjecture 5.1: for the stated even and odd sums, F is maximized at the two endpoint triples and both coordinate transfers weakly increase F.

Problem details · Reading note

The distinct upper degrees of deterministic simplicial complexes shift by dimension

Proved. Lean theorem result, frozen in this repository 2026-10-24.

S. N. Dorogovtsev; P. L. Krapivsky (2025). Deterministic simplicial complexes. DOI: 10.1088/1742-5468/ae2092. URL: https://arxiv.org/abs/2507.07402v1.

Claim. The paper studies simplicial complexes that grow deterministically from one vertex, every existing simplex joining a new vertex at each step; it computes the numbers of simplices, the upper-degree distributions and Hodge Laplacian spectra, studies constrained versions with bounded dimension, and states several relations observed for early generations as conjectures, among them the relation K^(d)(n) = 0 ∪ K^(0)(n−d) between the sets of distinct upper degrees.

Problem details · Reading note

The binary kite construction works exactly when $n$ is not one modulo three

Proved. Lean theorem result, frozen in this repository 2026-10-24.

David Gross; Paulina Goedicke (2025). Thirty-six officers, artisanally entangled. DOI: 10.48550/arXiv.2504.15401. URL: https://arxiv.org/abs/2504.15401v2.

Claim. The paper constructs doubly perfect functions and the associated two-unitary complex Hadamard matrices, characterizes quadratic exponentials as doubly perfect exactly when N and N+J have trivial kernel, proves existence on the binary phase space Z_2^{2n} for every n>1 via trace-orthonormal bases, and reports from computer experiments that kite-shaped binary matrices A=B with A_ij=1 iff i+j<=n+1 give a solution for all n not congruent to 1 modulo 3.

Problem details · Reading note

Stationary characterization of Gini mean comparisons

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

R. Grünwald; Zs. Páles (2024). Comparison of Gini means with fixed number of variables. URL: https://arxiv.org/abs/2408.07658v1.

Claim. Section 4, Conjecture 1 proposes that the parameter-sum inequality and two extreme endpoint tests characterize comparison of Gini means on a bounded positive interval for n ≥ 3.

Problem details · Reading note

Coefficients of the shifted Gamma product Turanian

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

S. I. Kalmykov and D. B. Karp (2017). Log-concavity and Turán-type inequalities for the generalized hypergeometric function. DOI: 10.1007/s10476-017-0503-z. URL: https://arxiv.org/abs/1611.06724.

Claim. Conjectures 1 and 2 assert coefficient nonnegativity for shifted hypergeometric Turanians when mu is at least one.

Problem details · Reading note

Coefficients of the two-block hypergeometric Turanian

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

S. I. Kalmykov and D. B. Karp (2017). Log-concavity and Turán-type inequalities for the generalized hypergeometric function. DOI: 10.1007/s10476-017-0503-z. URL: https://arxiv.org/abs/1611.06724.

Claim. Conjectures 1 and 2 assert coefficient nonnegativity for shifted hypergeometric Turanians when mu is at least one.

Problem details · Reading note

First-parameter coefficient signs of the normalized incomplete beta function

Proved. Lean theorem result, frozen in this repository 2026-10-24.

D. B. Karp (2016). Normalized incomplete beta function: log-concavity in parameters and other properties. DOI: 10.1007/s10958-016-2958-z. URL: https://arxiv.org/abs/1509.05120v1.

Claim. Conjecture 1 asserts strict positivity of every coefficient in the normalized incomplete beta Turanian for positive real parameters and shifts.

Problem details · Reading note

The right-special rotor word is the reversed prefix after the initial zero

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Govind S. Krishnaswami; Anirudh Rameshan (2026). Symbol sequences from three-rotor coincidences and their word-complexity. DOI: 10.48550/arXiv.2606.25766. URL: https://arxiv.org/abs/2606.25766v1.

Claim. The paper encodes classical three-rotor trajectories by rotor-coincidence symbol sequences over a four-letter alphabet, proves word-complexity n+3 for ultra-high-energy quasiperiodic orbits through a correspondence with Sturmian cutting sequences, and conjectures that the unique right-special (n-1)-word is the unique n-word starting with 0 read backwards without the 0.

Problem details · Reading note

A qubit refutation of the maximal reverse Pinsker bound

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

D. Lanier; J. Béguinot; O. Rioul (2025). From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum f-Divergence. DOI: 10.48550/arXiv.2501.14340. URL: https://arxiv.org/abs/2501.14340v1.

Claim. We conjecture that (49) is in fact not necessary for (50) to hold.

Problem details · Reading note

Louvet symmetric absolutely-PPT spectrum conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-24.

J. Louvet, E. Serrano-Ensástiga, T. Bastin, J. Martin (2025). Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace. DOI: 10.1103/PhysRevA.111.042418. URL: https://arxiv.org/abs/2411.16461v2.

Claim. Conjecture 1 states that symmetric N-qudit states with the specified p-spectrum are symmetric absolutely PPT for p between the conjectured threshold and one.

Problem details · Reading note

Five is not the minimal quaternary extension-counterexample length

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

Ali Assem Mahmoud (2026). Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes. DOI: 10.48550/arXiv.2607.26214. URL: https://arxiv.org/abs/2607.26214v1.

Claim. Theorem 5.3 gives the weight-isometric, monomially inequivalent pairs of [[5,2]] codes S_A = <ZZZZI, XXIIX, IIXXX>, S_B = <ZZIZX, XIZXI, IXXYI> and of [[6,3]] codes S+ = <XXIIXX, IIXXXX, ZZZZIX>, S- = <XXZXII, ZZXYII, IZZZXX>; Question 5.4 asks whether the codespaces of S_A and S_B, or those of S+ and S-, are mapped to one another by some product unitary U1 x ... x Un composed with a qubit permutation. Open problem 2 of §6.2 expects q+1 to be the minimal length at which the quantum extension property QEP(q,n) fails, for every prime power q.

Problem details · Reading note

Refutation of Miska–Ulas Conjecture 4.8

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

Piotr Miska; Maciej Ulas (2022). On the Diophantine equation σ₂(X̄ₙ) = σₙ(X̄ₙ). DOI: 10.1142/S1793042124500635. URL: https://arxiv.org/abs/2203.03942v1.

Claim. Conjecture 4.8 proposes simultaneous bounds on the common value of σ₃ and σₙ and on the last coordinate of a nondecreasing positive integer solution.

Problem details · Reading note

Bala's A347854 fractional-factorial integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Bala's A347855 auxiliary b integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Bala's A347855 auxiliary c integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Bala's A347855 fractional-factorial integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Bala's A347856 fractional-factorial integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Bala's A347857 fractional-factorial integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Bala's A347858 fractional-factorial integrality

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Peter Bala (2021). OEIS A347854–A347858: fractional factorial ratios. URL: https://oeis.org/A347854.

Claim. The sequence is conjectured to be integral.

Problem details · Reading note

Hanna's A382124 congruence

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Paul D. Hanna (2025). OEIS A382124 and A382125, sigma-product exponential series. URL: https://oeis.org/A382124.

Claim. Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0. Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.

Problem details · Reading note

Hanna's A382125 residue pattern

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Paul D. Hanna (2025). OEIS A382124 and A382125, sigma-product exponential series. URL: https://oeis.org/A382124.

Claim. Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0. Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.

Problem details · Reading note

Commuting product encoders define exactly the commuting-reflection qubit state class

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Jef Pauwels; Otfried Gühne (2026). The Entanglement Content of Quantum Measurement Bases. URL: https://arxiv.org/abs/2608.21185v1.

Claim. Four-qubit marginal entropy: the Higuchi–Sudbery candidate, its known restrictions, and the unrestricted maximum question. Section VI asks for characterizations of further encoding classes, among them commuting families whose local factors are not Hermitian.

Problem details · Reading note

No finite measure-and-prepare process perfectly clones the amplitude-damping family

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Pavel Sekatski; Yelena Guryanova; Naga Bhavya Teja Kothakonda; Michalis Skotiniotis (2025). Cloning Quantum Channels. DOI: 10.48550/arXiv.2509.08059. URL: https://arxiv.org/abs/2509.08059v1.

Claim. The paper treats deterministic cloning of quantum channels with quantum processes, gives necessary conditions for super-replication, shows that amplitude-damping channels have at most linear cloning rate, compares coherent and measure-and-prepare processes, observes that a coherent process attains Choi fidelity one for amplitude-damping channels at N=M, and conjectures that no measure-and-prepare process does.

Problem details · Reading note

Distinct optimal entanglement witnesses can have an optimal sum

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Jiheon Seong; Anindita Bera; Beatrix C. Hiesmayr; Dariusz Chruściński; Joonwoo Bae (2026). Mirrored entanglement witnesses for multipartite and high-dimensional quantum systems. DOI: 10.1088/2058-9565/ae30a3. URL: https://arxiv.org/abs/2510.06863v2.

Claim. The paper develops mirrored entanglement witnesses for multipartite and high-dimensional systems, generalizes mirroring to pairs W, M with W+M=K block-positive, and leaves open whether a pair of optimal witnesses can be mirrored through another optimal witness.

Problem details · Reading note

Iterated moment minor inequalities

Proved. Lean theorem result, frozen in this repository 2026-10-24.

S. Simić (2018). Second and third order moment inequalities for probability distributions. DOI: 10.1007/s10474-018-0848-1. URL: https://arxiv.org/abs/1509.08519v2.

Claim. Conjectures 1 and 2 are the eight-parameter and displaced-parameter iterated moment minor inequalities.

Problem details · Reading note

A negative third difference of exponential remainder ratios

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

S. M. Sitnik (1993). Inequalities for the exponential remainder of the Taylor series. URL: https://arxiv.org/abs/1608.08784.

Claim. Problem 11 conjectures positivity of every positive-order forward difference of the consecutive exponential Taylor remainder ratio at each positive argument.

Problem details · Reading note

The component count is the graph count plus loop-free strange components

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Gian Luca Spitzer; Ion Nechita (2026). Quantum Graph Theory by Example. DOI: 10.22331/q-2026-10-06-2228. URL: https://arxiv.org/abs/2603.23651v3.

Claim. The paper studies quantum graphs on M_n invariant under diagonal unitary and orthogonal groups, describes the graphs X_{A,B,C} through strange graphs whose edges are classical or carry a phase, proves that X_{A,B,C} has at least as many connected components as its strange graph and that connectedness agrees for n at least 3, and asks for the exact number of connected components of X_{A,·,C} in terms of the strange graph, in particular the role of isolated strange edges with phase π.

Problem details · Reading note

The sharp worst-case MATEN process fidelity is $4^{-N}$ for $N\ge2$

Proved. Lean theorem result, frozen in this repository 2026-10-24.

James Sud; Jeffrey Marshall; Zhihui Wang; Eleanor Rieffel; Filip A. Wudarski (2022). Dual-map framework for noise characterization of quantum computers. DOI: 10.1103/PhysRevA.106.012606. URL: https://arxiv.org/abs/2112.04414v1.

Claim. The paper introduces a dual-map framework for noisy expectation values, defines the marginal approximation to the effective noise (MATEN) as the tensor product of the single-qubit marginals of the noise channel, evaluates it on QAOA circuits and on Rigetti hardware, computes that the maximally entangled process matrix gives fidelity 4^-N with its marginal approximation, and conjectures that this process matrix is the worst case for the MATEN protocol.

Problem details · Reading note

Exact classification of elementary channel divisibility

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Frederik vom Ende (2024). A Sufficient Criterion for Divisibility of Quantum Channels. DOI: 10.1063/5.0231812. URL: https://arxiv.org/html/2407.17103v3.

Claim. Definition 1 specifies elementary channels; Proposition 1 and Theorem 1 give necessary and sufficient Kraus-space criteria. Section V asks whether their converses hold.

Problem details · Reading note

Comparing the Wu–Wu unified imaginarity monotones in every finite dimension

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Chuanfa Wu; Zhaoqi Wu (2025). Two imaginarity monotones induced by unified (α,β)-relative entropy. DOI: 10.1088/1572-9494/adbdbd. URL: https://arxiv.org/abs/2506.09799v1.

Claim. The paper defines the imaginarity quantifiers M^H, the unified (α,β)-relative entropy between a state and its complex conjugate, and M^E, its minimum over real states, proves that both are imaginarity monotones for α in (0,1) and β in (0,1], verifies M^H >= M^E on a qubit family, and conjectures M^H >= M^E for every quantum state.

Problem details · Reading note

The entropic tetrahedron is not a genuine four-qudit entanglement measure

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

Songbo Xie; Daniel Younis; Yuhan Mei; Joseph H. Eberly (2024). Multipartite Entanglement: A Journey Through Geometry. DOI: 10.3390/e26030217. URL: https://arxiv.org/abs/2304.03281v2.

Claim. The paper defines the four-qubit entropic fill from the volume of an entropic tetrahedron whose face areas are the one-to-other entanglement entropies, proves that it vanishes exactly on biseparable four-qubit pure states, states that the tetrahedron measure is a genuine four-qudit entanglement measure, and conjectures that it applies to qudit systems.

Problem details · Reading note

Zhang–Jing Theorem 2.3: the upper power interval

Refuted. Lean theorem result, frozen in this repository 2026-10-24.

Feng Zhang; Hefang Jing (2025). More inequalities for positive semidefinite matrices. DOI: 10.13001/ela.2025.9201. URL: https://doi.org/10.13001/ela.2025.9201.

Claim. Question 2.14 asks whether positivity of the functional-calculus block is equivalent to normality under three scalar conditions on f and g.

Problem details · Reading note

Zhang–Tian Conjecture 5.1: Steiner diameter and anchored radius of trees

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Qingnan Zhang; Yingzhi Tian (2025). On the Steiner k-diameter and Steiner (k, k′)-radius of trees. DOI: 10.48550/arXiv.2511.22492. URL: https://arxiv.org/abs/2511.22492v1.

Claim. Conjecture 5.1 bounds every finite tree's Steiner k-diameter by k/(k−k′) times its Steiner (k,k′)-radius minus k′(k′−1)/(k−k′).

Problem details · Reading note

Exact Hilbert–Schmidt distances for noisy four-qubit Smolin states

Proved. Lean theorem result, frozen in this repository 2026-10-24.

Xuanran Zhu; Keyi Huang; Chao Zhang; Xinfang Nie; Qi Zhao; Dawei Lu (2026). Universal entanglement detection with geometrically optimal witnesses. DOI: 10.1103/82gn-78lv. URL: https://arxiv.org/abs/2610.17592v1.

Claim. The paper constructs entanglement witnesses from the Hilbert-Schmidt distance to the separable states, evaluates this distance numerically for several families, and conjectures for the noisy four-qubit Smolin state that the distance to the separable states of the partitions where it is entangled is 0 for p at most 1/3 and the distance to the noisy state at p = 1/3, equal to (√3/4)(p − 1/3), above it.

Problem details · Reading note

The supremum of the elliptic remainder parameter exponent

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Q. Bao, M.-K. Wang and S.-L. Qiu (2022). Monotonicity properties of the generalized elliptic integrals with respect to the parameter. DOI: 10.7153/mia-2022-25-64. URL: https://arxiv.org/abs/2112.13993.

Claim. Open problem 3.5(2) asks for the supremum of the remainder parameter exponent lambda10 for every positive truncation order.

Problem details · Reading note

The averaged-kernel characterization fails

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

D. Bilyk, D. Ferizović, A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk (2022). Potential theory with multivariate kernels. DOI: 10.1007/s00209-022-03000-z. URL: https://arxiv.org/abs/2104.03410.

Claim. Normalized surface measure minimizes a continuous symmetric rotationally invariant multivariate energy if and only if its averaged pair kernel is conditionally positive definite.

Problem details · Reading note

Bourin's matrix subadditivity Question 1

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Jean-Christophe Bourin (2008). Matrix subadditivity inequalities and block-matrices. DOI: 10.1142/S0129167X09005509. URL: https://arxiv.org/abs/0805.1954v1.

Claim. Question 1 asks whether a positive linear map reflecting positive contractions satisfies the concave functional-calculus inequality for every symmetric norm.

Problem details · Reading note

The pure-state scale does not confine purely dissipative spectra to the unit disk

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Dariusz Chruściński; Sergey Denisov; Wojciech Tarnowski; Karol Życzkowski (2025). Spectral delineation of Markov generators: classical vs quantum. DOI: 10.1088/1751-8121/ae58d2. URL: https://arxiv.org/abs/2504.07903v1.

Claim. The paper rescales generators of classical and quantum Markov semigroups so that their spectra lie in the disk of radius one centred at -1, proves such confinement for purely dissipative GKLS generators under a quantum rescaling parameter, and conjectures (Conjecture 3) that for generators of positive trace-preserving dynamics the scale given by the maximal absolute diagonal value on pure states already confines the spectrum.

Problem details · Reading note

A seventeen-vertex tree admits normalized-Laplacian perfect state transfer

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Gabriel Coutinho; Pedro Ferreira Baptista (2023). Quantum walks in the normalized Laplacian. DOI: 10.1080/03081087.2023.2236282. URL: https://arxiv.org/abs/2209.04668v1.

Claim. The paper characterizes perfect state transfer in the continuous quantum walk generated by the normalized Laplacian, restricts the eigenvalue supports of transferring vertices in trees, and conjectures that no tree other than the paths on two and three vertices admits perfect state transfer in this model.

Problem details · Reading note

Conditional subentropy and guessing min-entropy

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Nilanjana Datta, Tony Dorlas, Richard Jozsa, Fabio Benatti (2014). Properties of subentropy. DOI: 10.48550/arXiv.1310.1312. URL: https://arxiv.org/abs/1310.1312v1.

Claim. Conjecture 2 bounds conditional subentropy of every finite classical-quantum state by minus the natural logarithm of its optimal guessing probability.

Problem details · Reading note

Model-level ensemble equivalence does not imply measure-level equivalence

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

F. Giuffrida, T. Squartini, P. Grünwald and D. Garlaschelli (2025). Description length of canonical and microcanonical models. DOI: 10.1103/1sd2-rxmy. URL: https://arxiv.org/abs/2307.05645v4.

Claim. Section IV.C asks whether model-level equivalence implies measure-level ensemble equivalence, the converse of implication (71).

Problem details · Reading note

A corner in the group-invariant asymptotic overlap

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Fumio Hiai and Milan Mosonyi and Masahito Hayashi (2009). Quantum hypothesis testing with group symmetry. DOI: 10.1063/1.3234186. URL: https://arxiv.org/abs/0904.0704v2.

Claim. Section 7 asks whether the asymptotic logarithmic overlap is differentiable throughout (0,1) for group-invariant state discrimination.

Problem details · Reading note

Product formulas for periodic Temperley–Lieb seed recurrences

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Yacine Ikhlef and Alexi Morin-Duchesne (2024). Fusion of irreducible modules in the periodic Temperley–Lieb algebra. DOI: 10.21468/SciPostPhys.17.5.132. URL: https://arxiv.org/abs/2312.14837v2.

Claim. Appendix C, Conjectures C.1 and C.2, gives product evaluations of the seed recurrences at y = εq^m.

Problem details · Reading note

Coefficient-wise Wright log-concavity of normalized incomplete beta functions

Proved. Lean theorem result, frozen in this repository 2026-10-23.

D. B. Karp (2016). Normalized incomplete beta function: log-concavity in parameters and other properties. DOI: 10.1007/s10958-016-2958-z. URL: https://arxiv.org/abs/1509.05120v1.

Claim. Conjecture 1 asserts strict positivity of every coefficient in the normalized incomplete beta Turanian for positive real parameters and shifts.

Problem details · Reading note

The Karp–Prilepkina Meijer zero-count conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

D. B. Karp and E. G. Prilepkina (2016). Completely monotonic gamma ratio and infinitely divisible H-function of Fox. DOI: 10.1007/s40315-015-0128-9. URL: https://arxiv.org/abs/1501.05388.

Claim. Conjecture 1 bounds the number of interior zeros of the Meijer G-function by that of the power-sum control function when all a_i are nonnegative and the parameter excess is positive.

Problem details · Reading note

Partial trace can increase Fisher discord

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

H. Li, S. Luo, Y. Zhang (2025). Fisher discord as a quantifier of quantum complexity. DOI: 10.1103/6kv3-hr92. URL: https://arxiv.org/abs/2510.15375v1.

Claim. Section VI asks whether Fisher discord, the difference between symmetric-logarithmic-derivative Fisher information and Wigner–Yanase skew information, does not increase under partial trace.

Problem details · Reading note

Lin's matrix majorization conjecture fails at exponent four

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Minghua Lin (2017). On a determinantal inequality arising from diffusion tensor imaging. DOI: 10.1142/S0219199716500449. URL: https://arxiv.org/abs/1604.04141v1.

Claim. Conjecture 4.3 asserts that lambda(A² + |BA|^p) majorizes lambda(A² + |AB|^p) for positive semidefinite A and B and every real p > 0.

Problem details · Reading note

The Luiz–Vieira generalized Blanuša closing conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

A. G. Luiz and F. A. S. Vieira (2025). Independent [k]-Roman Domination on Graphs. DOI: 10.7151/dmgt.2580. URL: https://arxiv.org/abs/2406.11688v1.

Claim. The authors conjecture that gamma_[kR](G) = i_[kR](G) for k >= 2 on all generalized Blanusa snarks.

Problem details · Reading note

Strict monotonicity of Wallis gamma remainders

Proved. Lean theorem result, frozen in this repository 2026-10-23.

M. Mahmoud and C. Mortici (2026). Structural Properties, Complete Monotonicity, Bounds, and Analytic Approximations with Bounded Remainders for a Wallis-Type Gamma Ratio. DOI: 10.1007/s00025-026-02697-2. URL: https://link.springer.com/article/10.1007/s00025-026-02697-2.

Claim. Section 4 asks whether mu and chi are strictly increasing and theta is strictly decreasing for every positive real argument.

Problem details · Reading note

The prime-dimensional stabilizer commutation-rank bound

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Owidiusz Makuta; Błażej Kuzaka; Remigiusz Augusiak (2023). Fully non-positive-partial-transpose genuinely entangled subspaces. DOI: 10.22331/q-2023-02-09-915. URL: https://arxiv.org/abs/2203.16902v2.

Claim. The paper constructs genuinely entangled subspaces from qudit stabilizers, relates genuine entanglement of a stabilizer subspace to the span of the commutation vectors of its generators, proves the case of two generators, and conjectures (Conjecture 1) that for prime local dimension d a genuinely entangled stabilizer subspace has commutation-space dimension at least the ceiling of (N-1)/(d-1).

Problem details · Reading note

A self-dual irreducible extreme thermal process can factor on proper subspaces

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Paweł Mazurek; Michał Horodecki (2018). Decomposability and convex structure of thermal processes. DOI: 10.1088/1367-2630/aac057. URL: https://arxiv.org/abs/1707.06869v2.

Claim. Thermal processes on diagonal states are the column-stochastic matrices preserving the Gibbs vector; the paper describes their extreme points for three levels and conjectures that an extremal process that factors into extremal processes on proper subspaces and is not a direct sum cannot satisfy detailed balance.

Problem details · Reading note

The full zero-sum prefix-count characterization

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

John C (2026). When does the zero-sum sign count satisfy a linear recurrence?, MathOverflow 516232 revision 2. URL: https://mathoverflow.net/questions/516232/when-does-the-zero-sum-sign-count-s-nd-varepsilon-in-dn-sum-vare.

Claim. For every positive integer sequence with a constant integer recurrence, all finite integer digit sets have C-finite zero-sum prefix counts iff the minimal characteristic polynomial has no unit-circle roots, simple expanding roots, and root-of-unity ratios between every pair of expanding roots.

Problem details · Reading note

The non-unital channel conditions imply both network bounds

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Kaushiki Mukherjee; Nirman Ganguly (2025). Network nonlocality breaking channels. URL: https://arxiv.org/abs/2510.26417v1.

Claim. The paper studies channels that destroy the violation of n-local inequalities in linear quantum networks, gives sufficient conditions on the parameters of a class of non-unital qubit channels under which k uses of the channel destroy the non n-local resource of linear n-local networks (Theorem 3), and conjectures from numerical comparison with the channel conditions that every non-unital channel of the class is non n-locality breaking (Conjecture 1).

Problem details · Reading note

Every linear N-qubit doily has an odd number of negative lines

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Axel Muller; Metod Saniga; Alain Giorgetti; Henri de Boutray; Frédéric Holweck (2022). Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five. DOI: 10.1016/j.jocs.2022.101853. URL: https://arxiv.org/abs/2206.03599v2.

Claim. N-qubit doilies are copies of the generalized quadrangle of order two whose points are N-qubit observables and whose lines are commuting triples; a line is negative if its three observables multiply to minus the identity. The paper counts linear and quadratic doilies for all N, classifies them for N = 4 and 5, and conjectures that every linear doily has an odd number of negative lines for all N ≥ 2.

Problem details · Reading note

The A145110 conjectured Redheffer support counts

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Mats Granvik (2008). OEIS A145110, Number of elements in the Redheffer matrix that contribute to the Moebius function. URL: https://oeis.org/A145110.

Claim. a(11) to a(30) are conjectured to be 11, 23, 13, 18, 19, 25, 17, 29, 19, 30, 26, 26, 23, 45, 27, 30, 32, 39, 29, 48. [From _Mats Granvik_, Dec 21 2008]

Problem details · Reading note

Granvik's Möbius diagonal and Mertens row

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Mats Granvik; Olivier Gérard; Michael Somos (2011). OEIS A192763 and the Granvik Mertens array. URL: https://oeis.org/A192763.

Claim. The main diagonal is the Mobius function times the natural numbers A055615 (conjecture). For k>1 the first row is the Mertens function + 2 = A002321 + 2 (conjecture).

Problem details · Reading note

Hanna's OEIS A356508 product-partition congruence

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Paul D. Hanna (2022). OEIS A356508, product-partition generating-function congruence. URL: https://oeis.org/A356508.

Claim. Conjecture: a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002), otherwise a(n) == 0 (mod 4) when n > 0.

Problem details · Reading note

Hanna's prime bilateral integrality conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Paul D. Hanna (2025). OEIS A380681–A380686, prime parameters and integral bilateral series. URL: https://oeis.org/A380681.

Claim. Conjecture: if F(x) satisfies p*x = Sum_{n=-oo..+oo} (-1)^n * x^(p*n) * (F(x) + x^n)^(p*n) with F(0) = 1 for some fixed integer p, then F(x) is an integer series in x iff p is prime. The g.f. A(x) of this sequence is the case when p = 2; other cases include A380682 (p=3), A380683 (p=5), A380684 (p=7), A380685 (p=11), and A380686 (p=13).

Problem details · Reading note

The Poncini–Rasmussen central factorial transfer expansion

Proved. Lean theorem result, frozen in this repository 2026-10-23.

Xavier Poncini and Jørgen Rasmussen (2023). Integrability of planar-algebraic models. DOI: 10.1088/1742-5468/acdce7. URL: https://arxiv.org/abs/2206.14462v2.

Claim. Conjecture C.12 gives the eight-vertex transfer matrix as a central factorial polynomial expansion of its principal Hamiltonian.

Problem details · Reading note

Bell-diagonal entanglement distance is not twice the squared concurrence

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

Arthur Vesperini; Ghofrane Bel-Hadj-Aissa; Roberto Franzosi (2023). Entanglement and Quantum Correlation Measures from a Minimum Distance Principle. DOI: 10.1038/s41598-023-29438-7. URL: https://arxiv.org/abs/2205.07143v2.

Claim. The paper defines a quantum correlation distance (QCD) and, by minimizing it over decompositions of a mixed state and local rotations of each component, an entanglement distance (ED); it reports numerical evidence that on Bell-diagonal states the ED equals twice the squared concurrence.

Problem details · Reading note

Taylor-coefficient weak majorization for Hermitian matrices

Refuted. Lean theorem result, frozen in this repository 2026-10-23.

T. Zhang (2026). Weak majorization inequalities for the cubic and quartic coefficients of e^{(A+B)t} versus e^{At}e^{Bt}. DOI: 10.48550/arXiv.2601.07286. URL: https://arxiv.org/abs/2601.07286.

Claim. Conjecture 5.5 compares the nonincreasing eigenvalue lists of (A+B)^k and the Hermitian part of the ordered binomial coefficient for all Hermitian A and B and every k at least five, without a positivity assumption.

Problem details · Reading note

Aliabadi–Zerbib Question 2.7 has a negative answer

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Mohsen Aliabadi; Shira Zerbib (2024). Matchings in matroids over abelian groups. DOI: 10.1007/s10801-024-01308-z. URL: https://arxiv.org/abs/2202.07719v3.

Claim. Question 2.7 asks whether a matroid is matched to a sparse paving matroid of the same rank over a torsion-free abelian group when the ground sizes are equal or differ by one.

Problem details · Reading note

Floors of reciprocal generalized Fibonacci tails

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Tewodros Amdeberhan, Xi Chen, Victor H. Moll, Bruce E. Sagan (2014). Generalized Fibonacci polynomials and Fibonomial coefficients. DOI: 10.1007/s00026-014-0242-9. URL: https://arxiv.org/abs/1306.6511v2.

Claim. Conjecture 4.6, first clause: If s > t ≥ 1 with (s, −t) ≠ (2, −1) and n, r ∈ P then ⌊(∑_{k=n}^∞ 1/{rk}_{s,−t})^{−1}⌋ = {rn}_{s,−t} − {r(n − 1)}_{s,−t} − 1.

Problem details · Reading note

Advantage-quantum measurements are not quantum n-simulable

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Hayato Arai (2023). The Diversity of Entanglement Structures with Self-Duality and Non-Orthogonal State Discrimination in General Probabilistic Theories. URL: https://arxiv.org/abs/2302.12361v1.

Claim. The thesis classifies the two-outcome dual-operator-valued measures (DOVMs) of bipartite entanglement structures by the eigenvalues of their effects, defines quantum n-simulability of a DOVM on its domain of non-negative probabilities, proves that beyond-quantum (BQ) DOVMs are not simulable, and leaves open whether advantage-quantum (AQ) DOVMs are.

Problem details · Reading note

Half-density modes of the asymmetric K-exclusion process

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Arvind Ayyer and Samarth Misra (2024). An exactly solvable asymmetric K-exclusion process. DOI: 10.1088/1751-8121/ad5edd. URL: https://arxiv.org/abs/2310.03343v2.

Claim. Section 4.5 asks which adjacent occupation is most likely at half-integer density for small and large positive t.

Problem details · Reading note

Refutation of the normalized Bessel Redheffer lower bound

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Á. Baricz and K. Mehrez (2018). Redheffer type bounds for Bessel and modified Bessel functions of the first kind. DOI: 10.1007/s00010-018-0545-8. URL: https://arxiv.org/abs/1701.08446.

Claim. Conjecture 1 in section 3 asserts the lower bound in inequality (3.2) for the normalized Bessel ratio before its first positive zero.

Problem details · Reading note

The whole prescribed packing-maximum conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Natasha Blitvić, Mohammed Slim Kammoun, and Einar Steingrímsson (2026). Permutations, Pattern Covers, Posets and Positivity: Part I: Patterns and Posets. DOI: 10.48550/arXiv.2610.29846. URL: https://arxiv.org/abs/2610.29846v1.

Claim. Conjecture 7.1: prescribed interleaved minimal prefix and suffix attain the maximum packing count for m≤K/2.

Problem details · Reading note

Hyperbolic bounds for the Kilbas–Saigo function

Proved. Lean theorem result, frozen in this repository 2026-10-22.

L. Boudabsa and T. Simon (2021). Some properties of the Kilbas–Saigo function. DOI: 10.3390/math9030217. URL: https://arxiv.org/abs/2012.05666.

Claim. Conjecture 4.5 gives two-sided hyperbolic bounds on the negative half-line for 0 < alpha <= 1, m > 0 and l > m - 1/alpha.

Problem details · Reading note

Positive couplings do not force a real spectrum for U_q(sl_2)-invariant spin chains

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

A. G. Bytsko (2011). Non-Hermitian spin chains with inhomogeneous coupling. DOI: 10.1090/S1061-0022-2011-01148-3. URL: https://arxiv.org/abs/0911.4476v3.

Claim. The paper studies open U_q(sl_2)-invariant spin chains H = sum a_n P^{S,s}_{n,n+1} with |q| = 1, builds universal metric operators for them, determines the ranges of gamma with real spectra for short chains, and conjectures (Conjecture 3) that for |gamma| < pi/(2S) the spectrum is real whenever all couplings a_n are positive.

Problem details · Reading note

The upper-boundary joint instrument is unique

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Claudio Carmeli; Teiko Heinosaari; Alessandro Toigo (2025). Postponing the choice: advantage of deferred measurements in quantum information processing. DOI: 10.1088/1367-2630/adf87c. URL: https://arxiv.org/abs/2502.19871v1.

Claim. The paper determines the compatibility region of a noisy sharp meter Q_s and a depolarizing channel I_t, describes the joint instruments attaining its boundary, shows that on the upper boundary the instrument tilde-J^(t) is the unique W-covariant joint instrument, and asks whether other joint instruments exist there.

Problem details · Reading note

Complete homogeneous symmetric norms of regular graphs

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Ángel Chávez, Sarah Fullerton, Matilda LaFortune, Keyron Linarez, Nethmin Liyanage, Justin Son and Tyler Ting (2023). Extremal polynomial norms of graphs. DOI: 10.48550/arXiv.2311.04689. URL: https://arxiv.org/abs/2311.04689v3.

Claim. Problem 1 asks for the complete homogeneous symmetric norms of finite path adjacency matrices in even degrees at least four.

Problem details · Reading note

Entanglement assistance raises the communication value by at most the output dimension

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Eric Chitambar; Ian George; Brian Doolittle; Marius Junge (2023). The Communication Value of a Quantum Channel. DOI: 10.1109/TIT.2022.3218540. URL: https://arxiv.org/abs/2109.11144v1.

Claim. The paper defines the communication value cv of a quantum channel as the largest unnormalized guessing score of the classical channels it induces, its entanglement-assisted version cv*, characterizes both by conic programs over the separable and positive cones, and conjectures cv*(N) <= d_B cv(N) for every channel.

Problem details · Reading note

The uncertainty condition suffices for the mixed discrete Gaussian

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Nicolae Cotfas (2026). Frame-dependent coherence of Gaussian quantum states. DOI: 10.48550/arXiv.2610.05623. URL: https://arxiv.org/abs/2610.05623v1.

Claim. The uncertainty condition (11) suffices for positivity of the mixed discrete-variable Gaussian matrix in Eqs. (21)–(22).

Problem details · Reading note

Cullinan–Young Remark 8 leading coefficient

Proved. Lean theorem result, frozen in this repository 2026-10-22.

J. Cullinan and E. Young (2026). The Gibbs phenomenon for the Krawtchouk polynomials. DOI: 10.1007/s10958-026-08499-9. URL: https://arxiv.org/abs/2603.05408v1.

Claim. Remark 8 states that numerical experimentation suggests the leading coefficient of the even Krawtchouk sign interpolant I_N is (-1)^((N-2)/2) C((N-2)/2)/(N-1)!.

Problem details · Reading note

A non-extreme minimum-rank superchannel extension

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

P. C. Daly (2022). Axiomatic Approach to Quantum Superchannels. DOI: 10.48550/arXiv.2210.00370. URL: https://arxiv.org/abs/2210.00370v1.

Claim. Remark 4.1 asks whether superchannel extensions with minimum reduced Choi rank are extreme points of the extension set.

Problem details · Reading note

Daugherty et al., Latin tableaux Conjecture 6.3

Proved. Lean theorem conjecture63_integer, frozen in this repository 2026-10-22.

Spencer Daugherty, Michaela Fahnestock, Shawn Huxtable, Michael Larson, Nourah Al Munayes and Storey Peacock (2026). Counting and Constructing Latin tableaux. URL: https://arxiv.org/abs/2610.29291v1.

Claim. Conjecture 6.3 states that for every integer k >= 5, the number LT(lambda) of Latin tableaux of shape lambda = (k,k-1,...,5,4,3,3) and conjugate type lambda-prime is 9*4^(k-3).

Problem details · Reading note

The geometric measure of the flip-conjugate symmetric subspace

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Sophia Denker; Satoya Imai; Otfried Gühne (2025). Chiral Symmetries and Multiparticle Entanglement. URL: https://arxiv.org/abs/2506.15609v1.

Claim. The paper constructs highly entangled three-qudit subspaces from chiral and flip-conjugate symmetries, shows that all states of the flip-conjugate symmetric subspace are local-unitary equivalent, proves the bound G >= 1 - d/(d^2-1) for their geometric measure, and conjectures from numerics for d <= 12 that G = 1 - d^2/((d^2-1)(d+1)) for every state of the subspace.

Problem details · Reading note

The bound L_k(M) ≤ k L(M) is not tight for k ≥ 3

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Péter Diviánszky; István Márton; Erika Bene; Tamás Vértesi (2023). Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant. DOI: 10.1038/s41598-023-39529-0. URL: https://arxiv.org/abs/2211.17185v2.

Claim. The paper certifies qubits in prepare-and-measure scenarios through the matrix functions L_k(M), proves L_k(M) <= k L(M), notes that the bound is tight for k = 2 by the CHSH expression, and conjectures that it is not tight for k >= 3.

Problem details · Reading note

Continued-fraction tails with u = 2 and v = 1 miss the Beatty sequence

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Robbert Fokkink, Gandhar Joshi (2025). On Cloitre's hiccup sequences. DOI: 10.1007/s11139-025-01305-1. URL: https://arxiv.org/abs/2507.16956v4.

Claim. Section 4 generalizes Conjecture 20 to two-parameter continued-fraction tails and their asymptotic Beatty sequence; Theorem 24 proves one parameter case.

Problem details · Reading note

Fuchs's physical-basis logarithm question

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Christopher A. Fuchs (1996). Distinguishability and Accessible Information in Quantum Theory. URL: https://arxiv.org/abs/quant-ph/9601020v1.

Claim. After equation (3.244), Fuchs asks whether a basis can make the matrix appearing in the logarithm diagonal.

Problem details · Reading note

The work stored in a stabilizer battery is not a function of the degree difference

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Debkanta Ghosh; Tanoy Kanti Konar; Gianluca Francica; Amit Kumar Pal; Aditi Sen De (2025). Quantum advantage unlocked: Charging quantum batteries with K-regular graph stabilizers. URL: https://arxiv.org/abs/2512.22908v2.

Claim. The paper studies quantum batteries whose battery and charging Hamiltonians are sums of the stabilizer generators of K-regular circulant graphs, with the battery prepared in its ground state and charged by unitary evolution; a remark reports that for N ≤ 10 the stored work appears to depend only on the degree difference |K − K′| when N is not a multiple of four.

Problem details · Reading note

Giacomelli's cone of infinitely log-concave initial data

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

P. Giacomelli (2026). Log-Concavity and Infinite Log-Concavity of Linear Recurrent Sequences with Linear Coefficients via Companion Matrix Methods. DOI: 10.48550/arXiv.2604.14391. URL: https://arxiv.org/abs/2604.14391v1.

Claim. Section 7 (1) asks whether the cone of infinitely log-concave initial data is convex for d ≥ 3.

Problem details · Reading note

Christoffel words need not attain the Lyndon-factor minimum

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Amy Glen, Jamie Simpson, and W. F. Smyth (2017). Counting Lyndon factors. DOI: 10.37236/6915. URL: https://arxiv.org/abs/1701.00928v1.

Claim. Open Problem 1 asks whether at every length the least number of distinct Lyndon factors among Lyndon words is attained by a Christoffel word.

Problem details · Reading note

Jaffe's unrestricted primitive Frobenius successor-ratio conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Claim. Conjecture 7.3 bounds g(a,c(a,b))/g(a,b) by 5 for every coprime pair of integers a,b at least 3, where c(a,b) is the least integer greater than b and coprime to a, and g uses primitive nonnegative coefficients.

Problem details · Reading note

First letters of permutations with a fixed inversion number

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Peter Kagey (2021). Expected value of letters of permutations with a given number of k-cycles. DOI: 10.48550/arXiv.2112.05281. URL: https://arxiv.org/abs/2112.05281v1.

Claim. Conjecture 5.1: for fixed k and n > k, the expected first letter conditional on k inversions equals M(n + 1, k)/M(n, k).

Problem details · Reading note

Refutation of reciprocal Gamma coefficient positivity

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

S. I. Kalmykov and D. B. Karp (2013). Log-concavity for series in reciprocal gamma functions and applications. DOI: 10.1080/10652469.2013.764874. URL: https://arxiv.org/abs/1206.4814.

Claim. The Moreover clause of Conjecture 1 asserts positive coefficients for the shifted reciprocal Gamma Turanian under Theorem 2's sequence hypotheses.

Problem details · Reading note

Monotonicity and exact ranges of the shifted Kummer ratio

Proved. Lean theorem result, frozen in this repository 2026-10-22.

D. Karp and S. M. Sitnik (2010). Log-convexity and log-concavity of hypergeometric-like functions. DOI: 10.1016/j.jmaa.2009.10.057. URL: https://arxiv.org/abs/0902.3073.

Claim. The unnumbered Conjecture after Theorem 4 asks for monotonicity and exact ranges of the shifted upper-parameter Kummer ratio in inequalities (10) and (11).

Problem details · Reading note

Kaygorodov–Martín González–Viruel Conjecture 5

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Problem details · Reading note

The largest automorphism groups of perfect evolution algebras

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Problem details · Reading note

No single polynomial inequality characterizes separability in every dimension

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

O. Krüger; R. F. Werner (2005). Some Open Problems in Quantum Information Theory. URL: https://arxiv.org/abs/quant-ph/0504166v1.

Claim. Problem 3 of the open-problem list asks whether polynomial entanglement invariants are complete and, as a closely related problem, whether the set of separable states can be described by a single polynomial invariant f with f(ρ) ≥ 0 equivalent to separability, possibly depending on the dimensions.

Problem details · Reading note

Lemos–Moura–Ribas–Silva Conjecture 3.4

Proved. Lean theorem result, frozen in this repository 2026-10-22.

A. Lemos, A. O. Moura, S. Ribas and A. T. Silva (2024). A note on weighted consecutive Davenport constant. DOI: 10.48550/arXiv.2404.11312. URL: https://arxiv.org/abs/2404.11312v1.

Claim. Conjecture 3.4: For every finite group G, it holds C(G) = |G|.

Problem details · Reading note

li-morgan-siegrist-2026-inflection-points-conjecture-6-1

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Yulong Li; Rachel Morgan; Michael Siegrist (2026). Concavity structure of fractional differential equations and zeros of Mittag–Leffler functions. DOI: 10.1007/s41478-026-01192-9. URL: https://link.springer.com/article/10.1007/s41478-026-01192-9.

Claim. Conjecture 6.1 asserts a unique sign-changing zero of the second derivative under the hypotheses of Theorem 5.2.

Problem details · Reading note

Purify-and-clone fails one-site worst-case optimality

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Z. Li; E. Theil; A. W. Harrow; I. Chuang (2026). An Exponential Sample-Complexity Advantage for Coherent Quantum Inference. DOI: 10.48550/arXiv.2605.21457. URL: https://arxiv.org/abs/2605.21457v2.

Claim. Section S3 A conjectures optimality of the purify-and-clone channel for m > n; the one-site loss is equation (S13), with one common target state chosen before the maximum over output sites.

Problem details · Reading note

Generalized doubly nonnegative cone inclusion

Proved. Lean theorem result, frozen in this repository 2026-10-22.

M. Nishijima and K. Nakata (2024). Generalizations of doubly nonnegative cones and their comparison. DOI: 10.15807/jorsj.67.84. URL: https://arxiv.org/abs/2204.12119.

Claim. Conjecture 5.1 states that the NN-type generalized doubly nonnegative cone is contained in the BD-type cone for a direct product of a nonnegative orthant and second-order cones.

Problem details · Reading note

The Somos prime integrality family in OEIS A091190/A091200

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2004). OEIS A091190 and A091200: the Somos prime integrality family. URL: https://oeis.org/A091190.

Claim. More generally, given A(x) satisfies x*A(x)^p = B(x*A(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime.

Problem details · Reading note

Hanna's overpartition logarithmic derivative congruences

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2012). OEIS A219431: logarithmic derivative of square-indexed overpartitions. URL: https://oeis.org/A219431.

Claim. Conjecture: a(2*n+1) == 2 (mod 4), a(2*n) == 0 (mod 4).

Problem details · Reading note

Hanna's prime substitution integrality conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2016). OEIS A271931, A271932 and A271933: prime substitution integrality. URL: https://oeis.org/A271931.

Claim. More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).

Problem details · Reading note

Dequatre's OEIS A343502 divisor-count prime conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-22.

C. H. R. Dequatre (2021). OEIS A343502, Numbers k such that tau(tau(k)) and tau(k+1) are both prime. URL: https://oeis.org/A343502.

Claim. %C A343502 Conjecture: all members except 2 and 3 are composite.

Problem details · Reading note

The Karttunen–Munn prime-values conjecture for OEIS A344695

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Antti Karttunen; Peter Munn (2021). OEIS A344695: gcd of the divisor sum and Dedekind psi. URL: https://oeis.org/A344695.

Claim. Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes).

Problem details · Reading note

Hanna's A363558 power-of-two coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363558 prime-index coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363559 power-of-two coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363559 prime-index coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363561 power-of-two coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363561 prime-index coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363569 power-of-two coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A363569 prime-index coefficient formula

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Claim. Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i.

Problem details · Reading note

Hanna's A365574 Lagrange parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2023). OEIS A365574: generating function and parity conjecture. URL: https://oeis.org/A365574.

Claim. Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1.

Problem details · Reading note

Hanna's OEIS A375449 diagonal derivative parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2024). OEIS A375449 and A375452, diagonal derivative parity conjectures. URL: https://oeis.org/A375449.

Claim. For n > 1, the coefficients in each of the two diagonal derivative generating functions are odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.

Problem details · Reading note

Hanna's OEIS A375452 diagonal derivative parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2024). OEIS A375449 and A375452, diagonal derivative parity conjectures. URL: https://oeis.org/A375449.

Claim. For n > 1, the coefficients in each of the two diagonal derivative generating functions are odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.

Problem details · Reading note

Integrality and dyadic parity of OEIS A377106

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2024). OEIS A377106: cubic substitution and power-of-two parity. URL: https://oeis.org/A377106.

Claim. Conjecture: a(n) is odd iff n = 2^k for some k >= 0.

Problem details · Reading note

oeis-a378258-hanna-square-root-parity

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2024). OEIS A378258, a compositional square root and dyadic parity. URL: https://oeis.org/A378258.

Claim. Conjecture: a(n) is odd iff n is a power of 2.

Problem details · Reading note

Hanna's A379764 theta-four coefficient conjectures

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2025). OEIS A379764, theta-four bilateral generating function and coefficient congruences. URL: https://oeis.org/A379764.

Claim. Conjecture: for n > 0, a(n) is odd iff n is a square; and a(n) == (-1)^(n-1) (mod 4) iff n is a square, otherwise a(n) is divisible by 4 if n is nonsquare.

Problem details · Reading note

Hanna's A380066 bilateral product congruence

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2025). OEIS A380066, bilateral product generating function. URL: https://oeis.org/A380066.

Claim. G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(n-1) * (A(x) + x^n)^(n-1). Conjecture: for n > 0, a(n) == 2 (mod 4).

Problem details · Reading note

Hanna's square-residue conjecture for OEIS A380680

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2025). OEIS A380680, bilateral rational-quotient square series. URL: https://oeis.org/A380680.

Claim. Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.

Problem details · Reading note

Hanna's A380711 absolute cubic congruence

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2025). OEIS A380711: generating function and parity conjecture. URL: https://oeis.org/A380711.

Claim. Conjecture: a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) for n >= 0.

Problem details · Reading note

Hanna's Lambert W coefficient congruences

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2025). OEIS A386655–A386658, Lambert W exponential generating functions. URL: https://oeis.org/A386655.

Claim. Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating.

Problem details · Reading note

Hanna's power-of-two parity conjecture for OEIS A393173

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Paul D. Hanna (2026). OEIS A393173, squared-index inverse-diagonal parity. URL: https://oeis.org/A393173.

Claim. G.f. A(x) satisfies [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1. Conjecture: for n > 0, a(n) is odd iff n is a power of 2.

Problem details · Reading note

Alternative q-Charlier Fibonacci positivity: Petrov–Scott Conjecture 6.3

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Leonid Petrov and Jeanne Scott (2026). Random Fibonacci words via clone Schur functions. DOI: 10.1017/fms.2025.10165. URL: https://arxiv.org/abs/2412.21126v2.

Claim. Conjecture 6.3: There exist values of ρ and q for which the following specialization (which we call the alternative q-Charlier) is Fibonacci positive.

Problem details · Reading note

Sharp parameter thresholds for the zero-balanced Gauss quotient

Proved. Lean theorem result, frozen in this repository 2026-10-22.

S.-L. Qiu, X.-Y. Ma and X.-Y. Xiang (2024). Turán-type inequalities for Gaussian hypergeometric functions, and Baricz's conjecture. DOI: 10.48550/arXiv.2408.15723. URL: https://arxiv.org/abs/2408.15723.

Claim. Conjecture 6.2 classifies strict parameter decrease of the zero-balanced Gauss quotient in (1.6) and the shifted-parameter comparison in (1.7).

Problem details · Reading note

Ramane–Kitturmath's regular graph energy conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-22.

H. S. Ramane; D. V. Kitturmath (2023). On the conjecture of Sombor energy of a graph. DOI: 10.1016/j.exco.2023.100115. URL: https://doi.org/10.1016/j.exco.2023.100115.

Claim. There is no regular graph with adjacency energy 2k√2, where k∈Z⁺.

Problem details · Reading note

Squares do not maximize the mean exit time of lattice clusters

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Silvia Rognone; Vincenzo Nicosia (2023). Clusters in randomly-coloured spatial networks. DOI: 10.1088/2632-072X/aeb36a. URL: https://arxiv.org/abs/2308.07880v1.

Claim. The paper studies monochromatic clusters of randomly coloured square lattices, defines the mean exit time of a simple random walk from a cluster, computes it for the tetrominoes, rectangles and sampled clusters, and conjectures that at each fixed size N the square maximizes the mean exit time.

Problem details · Reading note

Steering does not guarantee atemporality

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Minjeong Song; Varun Narasimhachar; Bartosz Regula; Thomas J. Elliott; Mile Gu (2024). Causal Classification of Spatiotemporal Quantum Correlations. DOI: 10.1103/PhysRevLett.133.110202. URL: https://arxiv.org/abs/2306.09336v2.

Claim. The paper classifies two-qubit pseudo-density operators by their compatibility with spatial and temporal distribution mechanisms, characterizes temporal compatibility by R_AB = (id_A ⊗ Λ)K_AB for a quantum channel Λ, introduces an atemporality measure f, and asks whether atemporality is guaranteed by steering or Bell nonlocality.

Problem details · Reading note

Uhlmann's unrestricted diagonal-majorization question

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Jeffrey Uhlmann (2023). Partial Proof of a Conjecture with Implications for Spectral Majorization. DOI: 10.46571/JCI.2023.2.4. URL: https://arxiv.org/abs/2309.01302v1.

Claim. Section 4 considers whether the diagonal-spectrum majorization property extends to unrestricted positive definite similarities; a real symmetric positive definite matrix in dimension seven refutes that extension.

Problem details · Reading note

Compatibility under every Heisenberg channel equals compatibility in every subspace

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Roope Uola; Tristan Kraft; Sébastien Designolle; Nikolai Miklin; Armin Tavakoli; Juha-Pekka Pellonpää; Otfried Gühne; Nicolas Brunner (2021). Quantum measurement incompatibility in subspaces. DOI: 10.1103/PhysRevA.103.022203. URL: https://arxiv.org/abs/2010.04048v2.

Claim. The paper studies the joint measurability of POVM assemblages after compression to lower-dimensional subspaces, constructs incompatible qutrit assemblages compatible in every qubit subspace, and leaves open whether demanding compatibility under every Heisenberg channel to a smaller dimension gives a strictly smaller class than compatibility in every subspace.

Problem details · Reading note

Endpoint homomorphisms of everywhere overlapping flat pairs

Proved. Lean theorem result, frozen in this repository 2026-10-22.

Frits Verhagen; Marco Tomamichel; Erkka Haapasalo (2026). Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of α-z Relative Entropies. DOI: 10.1007/s00220-026-05675-5. URL: https://arxiv.org/abs/2507.07520v3.

Claim. Remark 15 suspects that the additional endpoint and negative tropical homomorphisms on everywhere overlapping flat pairs are not monotone.

Problem details · Reading note

Refutation and exact classification of exponential derivative ratios

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

C.-F. Wei and B.-N. Guo (2014). Complete monotonicity of functions connected with the exponential function and derivatives. DOI: 10.1155/2014/851213. URL: https://doi.org/10.1155/2014/851213.

Claim. Section 5, Conjecture 12 asserts complete monotonicity of both consecutive signed derivative ratios for every nonnegative integer index.

Problem details · Reading note

The phase-shifted extension of Conjecture 2

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Benjamin Wong; Sergey Bravyi; David Gosset; Yinchen Liu (2026). Lee-Yang tensors and Hamiltonian complexity. DOI: 10.48550/arXiv.2602.03605. URL: https://arxiv.org/abs/2602.03605v1.

Claim. Section 5.6 asks whether Conjectures 1,2,3 extend to phase-shifted EPR-like Hamiltonians; the four-cycle with pi flux refutes the extension of Conjecture 2.

Problem details · Reading note

Zhang–Wu–Fei: relative-entropy superadditivity after unitary conjugation

Refuted. Lean theorem result, frozen in this repository 2026-10-22.

Lin Zhang, Junde Wu, and Shao-Ming Fei (2013). Comment on 'Convergence of macrostates under reproducible processes' [Phys. Lett. A 374: 3715–3717 (2010)]. DOI: 10.1016/j.physleta.2013.05.047. URL: https://arxiv.org/abs/1210.4720v4.

Claim. Section 4, conjecture (4.2), asserts relative-entropy superadditivity after independent joint and local unitary conjugations for every full-ranked reference state.

Problem details · Reading note

Derivative signs of the power-weighted hyperbolic cotangent remainder

Proved. Lean theorem result, frozen in this repository 2026-10-22.

L. Zhu (2020). Completely monotonic integer degrees for a class of special functions. DOI: 10.3934/math.2020224. URL: https://www.aimspress.com/article/doi/10.3934/math.2020224.

Claim. Conjecture 6.4 asserts positivity at derivative order m+2n-2 and failure of positivity everywhere at order m+2n-1 for the signed Bernoulli remainder, for integers m >= 1 and n >= 2.

Problem details · Reading note

The path does not minimize Kemeny's edge derivative

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

C. Albright; K. P. Hadaway; A. H. Pomerance; J. Jeffries; K. J. Lorenzen; A. K. Nix (2023). On the Edge Derivative of the Normalized Laplacian with Applications to Kemeny's Constant. DOI: 10.48550/arXiv.2211.01495. URL: https://arxiv.org/abs/2211.01495v2.

Claim. Conjecture 3.5 asserts that the path-ends non-edge minimizes the edge derivative of Kemeny's constant among simple graphs of a fixed order.

Problem details · Reading note

Bob cannot perfectly hedge more than n/2 of n parallel games

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Srinivasan Arunachalam; Abel Molina; Vincent Russo (2018). Quantum hedging in two-round prover-verifier interactions. DOI: 10.4230/LIPIcs.TQC.2017.5. URL: https://arxiv.org/abs/1310.7954v5.

Claim. For a two-round game in which Bob returns one half of an entangled two-qubit state to Alice for a projective measurement, the paper characterizes when Bob can win at least one of n parallel repetitions with certainty, and suggests from numerical evidence that Bob cannot perfectly hedge more than n/2 of the games.

Problem details · Reading note

Perfect restricted bases of size s + 1 force cyclicity

Proved. Lean theorem result, frozen in this repository 2026-10-21.

B. Bajnok, C. Berson and H. A. Just (2022). On Perfect Bases in Finite Abelian Groups. DOI: 10.2140/involve.2022.15.525. URL: https://arxiv.org/abs/2211.13675v1.

Claim. Question 6 asks whether any noncyclic group has a perfect restricted s-basis of size s + 1; the settlement proves that every such finite abelian group is cyclic.

Problem details · Reading note

A CSS-T doubling map with better distance than the identity

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Elena Berardini; Reza Dastbasteh; Josu Etxezarreta Martinez; Shreyas Jain; Olatz Sanz Larrarte (2025). Asymptotically Good CSS-T Codes and a New Construction of Triorthogonal Codes. DOI: 10.1109/JSAIT.2025.3582156. URL: https://arxiv.org/abs/2412.08586v2.

Claim. From a binary CSS code (C1, C2) and a linear map φ on C1 satisfying a triple-product parity condition, the paper builds the CSS-T code (C1^{Nφ}, C2^{Nφ}) with C^{Nφ} = {(x, φ(x))}, proves that φ = I gives a [[2n, k, ≥ d]] CSS-T code, and asks for maps φ giving better parameters than the identity or a permutation.

Problem details · Reading note

All-ring coordinates for unrestricted face-magic hypercubes

Proved. Lean theorem face_magic_module_structure, frozen in this repository 2026-10-21.

Manjul Bhargava (2026). The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes. URL: https://arxiv.org/html/2609.26762v1.

Claim. Section 6.2 asks for the unrestricted face-magic hypercube module in even positive characteristic and for real even-subset-sum collisions.

Problem details · Reading note

Complete homogeneous symmetric norms of complete multipartite graphs

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Claim. Problem 1 asks for the complete homogeneous symmetric norms of finite path adjacency matrices in even degrees at least four.

Problem details · Reading note

A sign obstruction to Chirvasitu–Cusick Conjecture 6.1

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

A. Chirvasitu; T. W. Cusick (2024). Quadratic rotation symmetric Boolean functions. DOI: 10.1016/j.dam.2023.10.010. URL: https://arxiv.org/abs/2304.12734v1.

Claim. Section 6 defines the binomial rules matrices R(i,j) and Conjecture 6.1 proposes their scaled-root-of-unity spectrum and order 2 lcm(4, i-j, i+j).

Problem details · Reading note

The temporal Hardy maximum 1/4 is attained with spin measurements for every spin

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Sujit K. Choudhary; Sandeep K. Goyal; Thomas Konrad; Sibasish Ghosh (2013). Persistence of Hardy's nonlocality in time. URL: https://arxiv.org/abs/1302.5296v1.

Claim. For the temporal version of Hardy's argument on a single d-level system with projective measurements, the paper proves that the success probability is at most 1/4 in every dimension, shows that spin-1 and spin-3/2 spin measurements attain it, and conjectures that the maximum 1/4 can be observed for any spin system.

Problem details · Reading note

The tribonacci and four-Bonacci local Dold observations

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Sompong Chuysurichay; Sawian Jaidee; Chatchawan Panraksa; Thomas Ward (2026). The local Dold congruence for Fibonacci and Lucas sequences. URL: https://arxiv.org/abs/2609.24544v1.

Claim. For a prime p other than 2 and 5, the local Fibonacci repair factor is z(p)/gcd(z(p),p-1), where z(p) is the Fibonacci rank of apparition; it equals 1 exactly for p congruent to plus or minus 1 modulo 5.

Problem details · Reading note

Cowtan–Burton Conjecture D.1: LDPC monic CSS pushouts

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Alexander Cowtan; Simon Burton (2024). CSS code surgery as a universal construction. DOI: 10.22331/q-2024-05-14-1344. URL: https://arxiv.org/abs/2301.13738v6.

Claim. Appendix D, Conjecture D.1 asks whether pushouts of basis-preserving monic spans of LDPC CSS complexes remain LDPC.

Problem details · Reading note

Dmitrović: constant-one mixed-norm point evaluation

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

D. Dmitrović (2024). Norm of composition operator on mixed norm spaces. DOI: 10.7153/oam-2024-18-42. URL: https://doi.org/10.7153/oam-2024-18-42.

Claim. The conjecture on page 705 asserts the constant-one point evaluation estimate for all positive mixed-norm exponents.

Problem details · Reading note

ellis-thanatipanonda-2022-cut-family-a-conjecture-1

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Paul Ellis; Thotsaporn Aek Thanatipanonda (2022). The Arithmetic-Periodicity of cut for C = {1, 2c}. DOI: 10.1016/j.dam.2022.08.027. URL: https://arxiv.org/abs/2203.02457v1.

Claim. Conjecture 1 states that every Family A cut-set has nim-sequence (0, 1)^c (+2).

Problem details · Reading note

A singular nonzero point of the odd spectral-invariant variety

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

M. Faust, L. Friedman, G. O'Malley, R. Ramos and A. Sharma (2026). Algebraic Properties of the Ideal of Spectral Invariants for the Discrete Laplacian. DOI: 10.48550/arXiv.2602.08304. URL: https://arxiv.org/abs/2602.08304v1.

Claim. Conjecture 3 asserts that V(I') has only regular nonzero points for odd periods.

Problem details · Reading note

Feldman–Wilce Question 1: finite, unbounded cliques

Proved. Lean theorem exists_countablyCompact_closed_fibreGraph, frozen in this repository 2026-10-21.

David V. Feldman and Alexander Wilce (2026). A Note on Compactness and Clique Size. DOI: 10.48550/arXiv.2608.13320. URL: https://arxiv.org/abs/2608.13320.

Claim. Question 1 asks for a countably compact T1 or Hausdorff space with an all-product closed orthogonality relation having finite but unbounded cliques; Section 4 gives a conditional finite-block construction.

Problem details · Reading note

gawron-ulas-2022-factorial-sign-conjecture-4-4

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Filip Gawron and Maciej Ulas (2022). Signs behaviour of sums of weighted numbers of partitions. DOI: 10.1007/s11139-022-00602-3. URL: https://arxiv.org/abs/2202.05221v1.

Claim. Conjecture 4.4 asserts eventual period-four signs for k ≤ m − 2 and eventual alternating signs for k ≥ m − 1 for the factorial part sets U_m.

Problem details · Reading note

The upper local degree comparison fails on P6

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

J. Guerrero (2022). Laplacian Energies of Vertices. DOI: 10.1016/j.laa.2024.11.016. URL: https://arxiv.org/abs/2201.01252v1.

Claim. Remark 6 conjectures d_min LE_G(v) ≤ E_G(v) ≤ d_max LE_G(v) for every vertex of a simple connected graph.

Problem details · Reading note

The multilayer percolation determinant can have three zeros in [0, 1]

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Saikat Guha; Donald Towsley; Philippe Nain; Çağatay Çapar; Ananthram Swami; Prithwish Basu (2016). Spanning connectivity in a multilayer network and its relationship to site-bond percolation. DOI: 10.1103/PhysRevE.93.062310. URL: https://arxiv.org/abs/1402.7057v4.

Claim. For M independently site-percolated layers of a configuration-model graph with degree distribution p_k, the paper characterizes the multilayer site-percolation threshold q_c(M) as the smallest zero in [0, 1] of det A(q) for an explicit M × M matrix A(q), and conjectures that det A(q) has a unique zero in [0, 1] for all M ≥ 1 whenever C = Σ_k (k − 1) k p_k / z > 1.

Problem details · Reading note

Guo's partial-negativity subadditivity conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Y. Guo (2024). Complete Genuine Multipartite Entanglement Monotone. DOI: 10.1016/j.rinp.2024.107430. URL: https://arxiv.org/abs/2301.00334v3.

Claim. Equation (24) conjectures subadditivity of the reduced function ĥ(ρ) = √(δ1 δ2), where δ1 and δ2 are the two largest eigenvalues of ρ.

Problem details · Reading note

The d-orthostochastic dimension is not eventually bounded by 3

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Eugene Gutkin (2013). On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices. DOI: 10.1016/j.geomphys.2013.06.009. URL: https://arxiv.org/abs/1301.2537v1.

Claim. The paper defines F^d-bistochastic matrices as the bistochastic matrices P with p_i^j = ||v_i^j||^2 for a matrix V of vectors in F^d satisfying its equations (1)–(2), introduces d_min(F, n) as the least d for which every bistochastic n × n matrix is F^d-bistochastic, and conjectures that 2 ≤ d_min(n) ≤ 3 for all large n when F = R.

Problem details · Reading note

Maximal mutual correlation dominates the square root of geometric discord for all two-qubit states

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Marek Gwóźdź; Lech Jakóbczyk (2014). Spontaneous emission can locally create quantum discord out of classical correlations. DOI: 10.1007/s11128-013-0639-0. URL: https://arxiv.org/abs/1306.0130v2.

Claim. For two qubits, the paper relates the normalized geometric discord to the maximal mutual correlation, proves C_M(ρ) ≥ √D_G^A(ρ) for Bell-diagonal states, and conjectures from numerical simulations that it holds for all mixed two-qubit states.

Problem details · Reading note

Distance Seidel energy after deleting an edge

Proved. Lean theorem result, frozen in this repository 2026-10-21.

T. Haritha and A. V. Chithra (2022). Distance Seidel matrix of a connected graph. DOI: 10.1080/09728600.2025.2479147. URL: https://arxiv.org/abs/2210.05940v2.

Claim. Problem 5.1 asks whether deleting any edge of K_{a,b}, with a,b > 1, strictly increases the distance Seidel energy.

Problem details · Reading note

Jain: multiplicity two for absolute-cosine Hadamard powers

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Tanvi Jain (2017). Hadamard powers of some positive matrices. DOI: 10.1016/j.laa.2016.06.030. URL: https://doi.org/10.1016/j.laa.2016.06.030.

Claim. The final paragraph of section 3 conjectures multiplicity two for every nonzero non-top eigenvalue of odd absolute-cosine Hadamard powers.

Problem details · Reading note

Maximal mutual correlation dominates the trace-norm geometric discord of two qubits

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Lech Jakóbczyk (2016). Maximal mutual correlation: an algebraic measure of correlations in bipartite quantum systems. URL: https://arxiv.org/abs/1607.06585v1.

Claim. The paper studies the maximal mutual correlation C_M of bipartite quantum states, proves that C_M dominates the trace-norm geometric discord D_1 for Bell-diagonal two-qubit states, and conjectures that the inequality holds for all two-qubit states.

Problem details · Reading note

A connected cubic counterexample to Jean–Seo Conjecture 1

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Devin C. Jean; Suk J. Seo (2022). Optimal Error-detection system for Identifying Codes. DOI: 10.48550/arXiv.2208.06052. URL: https://arxiv.org/abs/2208.06052v1.

Claim. Conjecture 1 asserts that every cubic graph has minimum error-detecting identifying-code density at least 3/4.

Problem details · Reading note

Johnston–Martín Duro–Rumynin Question 2.5

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Dylan Johnston; Diego Martín Duro; Dmitriy Rumynin (2025). Global Representation Ring and Knutson Index. DOI: 10.1080/00927872.2025.2498055. URL: https://arxiv.org/abs/2403.18498v6.

Claim. Section 2 defines regular elements, the Knutson index and S-subindex; Question 2.5 asks whether one finite subset computes every index.

Problem details · Reading note

Koutschan–Krattenthaler–Schlosser Conjecture 23, equation (10.12)

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Christoph Koutschan; Christian Krattenthaler; Michael J. Schlosser (2024). Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations. DOI: 10.1016/j.jsc.2024.102352. URL: https://arxiv.org/abs/2401.08481v1.

Claim. Conjecture 23, equations (10.10)–(10.12), asserts the displayed identities between the four-parameter determinants for all n ≥ 3.

Problem details · Reading note

Koutschan–Krattenthaler–Schlosser Conjecture 23, equation (10.11)

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Claim. Conjecture 23, equations (10.10)–(10.12), asserts the displayed identities between the four-parameter determinants for all n ≥ 3.

Problem details · Reading note

Koutschan–Krattenthaler–Schlosser Conjecture 23, equation (10.10)

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Claim. Conjecture 23, equations (10.10)–(10.12), asserts the displayed identities between the four-parameter determinants for all n ≥ 3.

Problem details · Reading note

Kumar–Peng–Tran Conjecture 4.4 fails at modulus 17

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Aarya Kumar, Siyu Peng, and Vincent Tran (2022). Integer Complexity Generalizations in Various Rings. DOI: 10.48550/arXiv.2211.04379. URL: https://arxiv.org/abs/2211.04379v1.

Claim. Conjecture 4.4 states that E(m, 2) is the third most inefficient for odd m.

Problem details · Reading note

An order-19 counterexample to Conjecture 5.4

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Zhen Lin; Jiajia Wang; Min Cai (2023). The Laplacian spectral ratio of connected graphs. DOI: 10.48550/arXiv.2302.10491. URL: https://arxiv.org/abs/2302.10491v1.

Claim. Conjecture 5.4 proposes star and parity-dependent broom bounds for the Laplacian spectral ratio of trees with at least eight vertices.

Problem details · Reading note

The Maity–Furuichi–Dutta noncommuting Tsallis bound ordering is false

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta (2026). r-deformed α-z-Rényi relative entropy. URL: https://arxiv.org/abs/2607.01805v1.

Claim. Definitions 1–2 and equations (6), (8)–(9), (47) define the deformed logarithm and relative entropies; Section 4 Case 5 asserts the ordering B₁ ≤ B₂ for noncommuting density operators.

Problem details · Reading note

Malkoun: the unrestricted subgroup permanent lower bound

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Malkoun (2023). Do these cousins of permanents satisfy the following inequality?. URL: https://mathoverflow.net/questions/456248/do-these-cousins-of-permanents-satisfy-the-following-inequality.

Claim. The main question asks for a positive constant for every pair of permutation subgroups, uniformly over all complex Hermitian positive semidefinite matrices of the fixed size.

Problem details · Reading note

Minimal rational faces of the fan of interlace certs

Proved. Lean theorem result, frozen in this repository 2026-10-21.

A. Mandel; S. Robins (2019). Dragging the roots of a polynomial to the unit circle. URL: https://arxiv.org/abs/1908.03208.

Claim. Section 10, Problem 3 asks for the minimal faces of the fan of interlace certs containing rational points and for interlace rational polynomials in those faces.

Problem details · Reading note

Strength three of every Mariot–Manzoni MOCA array

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Luca Mariot and Luca Manzoni (2022). Building Correlation Immune Functions from Sets of Mutually Orthogonal Cellular Automata. DOI: 10.48550/arXiv.2207.08280. URL: https://arxiv.org/abs/2207.08280v1.

Claim. Section 5 asks whether the binary orthogonal array obtained from every set of k-MOCA always has strength at least three.

Problem details · Reading note

Strict decrease in Mendonça's min-plus configuration triangle

Proved. Lean theorem result, frozen in this repository 2026-10-21.

J. R. G. Mendonça (2026). OEIS A400584 and The min-plus configuration triangle: Technical note supporting OEIS A400584. URL: https://oeis.org/A400584.

Claim. Conjecture: T(n,k) > T(n,k+1) for n >= 3 and 1 <= k < 2^n.

Problem details · Reading note

Novaes's strong-absorption spectral and Wigner moment conjectures

Proved. Lean theorem result, frozen in this repository 2026-10-21.

M. Novaes (2023). Time delay statistics for chaotic cavities with absorption. DOI: 10.1007/s10955-023-03183-7. URL: https://arxiv.org/abs/2305.18235v1.

Claim. Section 5: Eqs. (54) and (55), strong-absorption moment expansions for every positive channel count and order; Eq. (58), the first two cumulant coefficients for every M ≥ 1 and n > 1.

Problem details · Reading note

oeis-a347285-de-vlieger-prime-power-descent

Proved. Lean theorem result, frozen in this repository 2026-10-21.

M. De Vlieger; OEIS Foundation Inc. (2021). OEIS A347285 descending prime-power exponents. URL: https://oeis.org/A347285.

Claim. For every positive row index n, the sum S of absolute first differences satisfies n - S = 1.

Problem details · Reading note

Huber's minimum-volume cuboid conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-21.

F. Huber (2024). OEIS A375786: minimum volume of a cuboid with the surface of an n-cube. URL: https://oeis.org/A375786.

Claim. Conjecture: From the integer-sided cuboids with same surface 6*n^2 always the one with the smallest edge length has the minimum volume. If there are several integer-sided cuboids having the smallest edge length, then the one with the smallest second smallest edge length has the minimum volume (checked up to a(1000)).

Problem details · Reading note

Huber's prime conjecture for OEIS A394407

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Felix Huber (2026). OEIS A394407: first occurrences of the totient-to-divisor-count ratio. URL: https://oeis.org/A394407.

Claim. Every prime other than 3 occurs in A394407, the positive integers whose totient-to-divisor-count ratio has no smaller positive occurrence.

Problem details · Reading note

Question 3 on outgoing digit-sum arcs

Proved. Lean theorem result, frozen in this repository 2026-10-21.

P. N. Phunphayap; P. Noppakaew; P. Pongsriiam (2023). Directed graphs of integers with arcs determined by an arithmetic function. DOI: 10.61091/um124-06. URL: https://arxiv.org/abs/2303.13017v1.

Claim. Question 3 asks for a necessary and sufficient condition for infinitely many outgoing digit-sum arcs not divisible by the digit sum of n, and for their natural density.

Problem details · Reading note

Equal multipartite powers of diagonal phase gates

Proved. Lean theorem result, frozen in this repository 2026-10-21.

M. Samanta; S. Mondal; A. Sen (De) (2025). Disparity between multipartite entangling and disentangling powers of unitaries: Even vs Odd. DOI: 10.48550/arXiv.2505.18539. URL: https://arxiv.org/abs/2505.18539v1.

Claim. Appendix B.3 expects equality of multipartite entangling and disentangling powers for arbitrary diagonal unitaries on any number of qubits.

Problem details · Reading note

Shala Question 7.4: weakly but not strongly coset-like atomistic lattices

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Besfort Shala (2024). The Probabilistic Zeta Function of a Finite Lattice. URL: https://arxiv.org/abs/2112.13766v6.

Claim. Question 7.4 asks whether there exists a weakly coset-like atomistic lattice that is not strongly coset-like.

Problem details · Reading note

Maximal input entropy for binary two-deletion and two-insertion channels

Proved. Lean theorem result, frozen in this repository 2026-10-21.

S. Singhvi; O. Sabary; D. Bar-Lev; E. Yaakobi (2025). Conditional Entropies of k-Deletion/Insertion Channels. DOI: 10.1109/TIT.2025.3581849. URL: https://arxiv.org/abs/2407.10026v2.

Claim. Section VII-A asks for the maximizing words and maximal input entropies of the binary two-deletion and two-insertion channels.

Problem details · Reading note

Boundedness of the Damascus violation sets

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Chanatip Sujsuntinukul and Christophe Chesneau (2026). Variants of the Damascus inequality. DOI: 10.1007/s11139-026-01345-1. URL: https://arxiv.org/abs/2601.00916v4.

Claim. Section 5 asks whether the Damascus violation sets are bounded for arbitrary dimension; the paper proves boundedness in dimension three.

Problem details · Reading note

A ring of four complete blocks refutes the flat-graph clique-root conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

H. Teimoori Faal (2021). On Clique Roots of Flat Graphs. DOI: 10.48550/arXiv.2112.09721. URL: https://arxiv.org/abs/2112.09721v1.

Claim. Conjecture 1 (p. 6): The class of 2-connected K₅-free flat graphs without isolated triangles has only clique roots. In particular, this class has always the clique root r = −1 of multiplicity 2.

Problem details · Reading note

Two unbounded classicalization gaps with a qubit

Proved. Lean theorem result, frozen in this repository 2026-10-21.

Zhen-Peng Xu, Satoya Imai, and Otfried Gühne (2023). Fate of multiparticle entanglement when one particle becomes classical. DOI: 10.1103/PhysRevA.107.L040401. URL: https://arxiv.org/abs/2206.12834v2.

Claim. Section V.B asks whether both classicalization entanglement-change gaps can be arbitrarily large while the measured party is a qubit.

Problem details · Reading note

λ_f = min{1, 1/[4f(0)]} fails for a regular symmetric operator monotone function

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Sixia Yu; C. H. Oh (2013). Robertson-Schrödinger Uncertainty Relation Refined by Skew Information. URL: https://arxiv.org/abs/1303.6404v1.

Claim. For regular symmetric operator monotone functions f, the paper refines the Robertson-Schrödinger uncertainty relation by f-skew information with a constant λ_f, proves 1 − f(0) ≤ λ_f ≤ min{1, 1/[4f(0)]}, and conjectures that λ_f = min{1, 1/[4f(0)]} always holds.

Problem details · Reading note

Zhang–Jing Question 2.14: block positivity and normality

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Feng Zhang; Hefang Jing (2025). More inequalities for positive semidefinite matrices. DOI: 10.13001/ela.2025.9201. URL: https://doi.org/10.13001/ela.2025.9201.

Claim. Question 2.14 asks whether positivity of the functional-calculus block is equivalent to normality under three scalar conditions on f and g.

Problem details · Reading note

The overlap entropy bound is not c-optimal for qutrits

Refuted. Lean theorem result, frozen in this repository 2026-10-21.

Steeve Zozor; Gustavo Martín Bosyk; Mariela Portesi (2014). General entropy-like uncertainty relations in finite dimensions. DOI: 10.1088/1751-8113/47/49/495302. URL: https://arxiv.org/abs/1311.5602v3.

Claim. For two nondegenerate observables with eigenbasis overlap c, the paper proves a lower bound on the sum of two generalized (h,φ)-entropies, shows it is c-optimal for c > 1/√2 when N = 2 or N ≥ 4, and suspects that it is also c-optimal when N = 3.

Problem details · Reading note

Ruleset domination does not imply nimber domination

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Tomoaki Abuku; Shun-ichi Kimura; Hironori Kiya; Urban Larsson; Indrajit Saha; Koki Suetsugu; Takahiro Yamashita (2024). Enforce and selective operators of combinatorial games. DOI: 10.1007/s00182-024-00910-6. URL: https://arxiv.org/abs/2311.01006v2.

Claim. Problem 42 asks whether ruleset domination implies equality of all enforce nimbers and the dominating ruleset's nimbers.

Problem details · Reading note

adamson-flaherty-potapov-spirakis-2023-two-swap-greedy

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Duncan Adamson; Nathan Flaherty; Igor Potapov; Paul G. Spirakis (2024). Structural and Combinatorial Properties of 2-swap Word Permutation Graphs. DOI: 10.1007/978-3-031-55601-2_5. URL: https://arxiv.org/abs/2307.01648v3.

Claim. The closing conjecture asserts that the greedy algorithm of Lemma 4 can find a shortest path between every pair of vertices of a 2-swap word graph.

Problem details · Reading note

The Alcaraz–Ramos lowest-mode conformal prediction fails

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Francisco C. Alcaraz, Lucas M. Ramos (2024). Conformally invariant free parafermionic quantum chains with multispin interactions. DOI: 10.1103/PhysRevE.109.044138. URL: https://arxiv.org/abs/2402.09936v3.

Claim. The homogeneous N=3 chain is predicted to satisfy M ε_i → π(2√3)(7/6+i−1) along M=3m for each fixed i≥1.

Problem details · Reading note

Collective dephasing of two qutrits admits nonstationary states with constant positive negativity

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Mazhar Ali (2019). Freezing dynamics of entanglement and nonlocality for qutrit-qutrit (3 ⊗ 3) quantum systems. DOI: 10.1142/s0217732319501025. URL: https://arxiv.org/abs/1810.01123v1.

Claim. For two qutrits under Markovian collective dephasing, the paper studies the negativity of a mixed family and of random pure states, observes freezing dynamics, and conjectures that time-invariant entanglement, in which the state changes while its entanglement stays constant, may not exist under this dynamics.

Problem details · Reading note

Strong domination under clique gluing

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Saeid Alikhani, Nima Ghanbari, Michael A. Henning (2026). Strong domination number of graphs from primary subgraphs. DOI: 10.2989/16073606.2025.2574046. URL: https://www.tandfonline.com/doi/full/10.2989/16073606.2025.2574046.

Claim. Problem 1, Inequality (4): γst(G) ≥ γst(G1) + γst(G2) − 1 − ΣX,Y for every r ≥ 2.

Problem details · Reading note

The Alsing–Birrittella logarithmic-negativity sandwich

Proved. Lean theorem result, frozen in this repository 2026-10-20.

P. M. Alsing; R. J. Birrittella (2024). Numerical Evidence for a Bipartite Pure State Entanglement Witness from Approximate Analytical Diagonalization. DOI: 10.3390/foundations5020019. URL: https://arxiv.org/abs/2404.13725v1.

Claim. Equation (22) numerically supports the logarithmic-negativity sandwich for positive Hermitian amplitude ensembles and upper equality for real amplitudes.

Problem details · Reading note

Every (n,d)↦1 random access code has a quantum advantage with two measurement settings

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Andris Ambainis; Dmitry Kravchenko; Sk Sazim; Joonwoo Bae; Ashutosh Rai (2024). Quantum advantages in (n,d)↦1 random access codes. DOI: 10.1088/1367-2630/ad9bdf. URL: https://arxiv.org/abs/1510.03045v4.

Claim. For (n,d)↦1 random access codes with shared randomness, the paper proves that majority-encoding-identity-decoding is an optimal classical strategy, constructs quantum strategies with two measurement settings that beat it in the cases computed, and conjectures that quantum advantages appear in (n,d)↦1 random access codes in general.

Problem details · Reading note

Eventual positivity of the complete D-prime family

Proved. Lean theorem result, frozen in this repository 2026-10-20.

George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

Claim. Positivity questions for two families of q-series C'(k,m,n) and D'(k,m,n) generating weighted two-color partitions; positivity proofs for small parameters and Conjectures 1–5 on positivity, sign patterns and eventual positivity.

Problem details · Reading note

Alternating binary words need not minimize positional autocorrelation

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Arash Atashpendar, David Mestel, A. W. Roscoe, and Peter Y. A. Ryan (2018). A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics. URL: https://arxiv.org/abs/1807.11609v1.

Claim. Conjecture IV.4 proposes alternating words as autocorrelation minimizers; Section V asks whether positional autocorrelation determines finite deletion-channel entropy ordering.

Problem details · Reading note

Badhani–Das noisy-unitary conditional min-entropy

Proved. Lean theorem result, frozen in this repository 2026-10-20.

H. Badhani and S. Das (2026). Conditional channel entropy sets fundamental limits on thermodynamic quantum information processing. DOI: 10.48550/arXiv.2604.01217. URL: https://arxiv.org/abs/2604.01217v2.

Claim. Section VI.1 conjectures that Proposition 10 extends to all unitaries with added white noise, with qubit conditioning.

Problem details · Reading note

The gcd-three equator formula for optimized positive maps is false

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

A. Bera; G. Sarbicki; D. Chruściński (2023). Optimizing positive maps in the matrix algebra M_n. DOI: 10.48550/arXiv.2309.09621. URL: https://arxiv.org/abs/2309.09621v1.

Claim. Conjecture 2, Eq. (13), gives the greatest positivity parameter n − k + (n−3)/(n − (2/3)k) at equal coefficient norms when GCD(n,k) = 3.

Problem details · Reading note

Bhatia–Jain: Stolarsky mean matrices in class A

Proved. Lean theorem result, frozen in this repository 2026-10-20.

R. Bhatia and T. Jain (2016). Mean matrices and conditional negativity. DOI: 10.13001/1081-3810.3082. URL: https://journals.uwyo.edu/index.php/ela/article/view/1495.

Claim. Section 2.9, Conjecture 1 asks for the class A range of Stolarsky mean matrices.

Problem details · Reading note

Arithmetic complexity of singleton partitions

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Alexander Bors, Daniel Panario, and Qiang Wang (2023). Functional graphs of generalized cyclotomic mappings of finite fields. DOI: 10.4171/MEMS/23. URL: https://arxiv.org/abs/2304.00181v1.

Claim. Question 6.3.1 asks whether the lower limit of AC(T_m)/(log² m/log log m) is positive.

Problem details · Reading note

All-odd randomized subtraction games do not converge

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Nicolás Capitelli, Francisco Somma (2024). On sequences arising from randomizing subtraction games. DOI: 10.48550/arXiv.2405.19593. URL: https://arxiv.org/abs/2405.19593v1.

Claim. Section 3 conjectures nonconvergence for all-odd finite subtraction sets with gcd one; Section 1 defines the uniformly randomized legal-move winning probability.

Problem details · Reading note

Coulter–Do Conjecture 5.8

Proved. Lean theorem conjecture58, frozen in this repository 2026-10-20.

Coulter, Xavier; Do, Norman (2025). From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys–Murphy elements. DOI: 10.48550/arXiv.2506.04002. URL: https://arxiv.org/abs/2506.04002v1.

Claim. Conjecture 5.4(a) asserts commutation of the b-deformed Jucys–Murphy operators on the cyclic orbit of the identity pair partition.

Problem details · Reading note

The unitary scaling landscape has a nonglobal local minimum

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Alexis De Vos; Stijn De Baerdemacker (2014). Scaling a unitary matrix. DOI: 10.1142/S1230161214500139. URL: https://arxiv.org/abs/1401.7883v2.

Claim. The paper studies the scaling of a unitary matrix by diagonal unitaries to a matrix with all line sums equal to one, introduces the landscape Ψ(λ, ρ) = n² − |sum(B)|² with B_jk = e^(i(λ_j+ρ_k)) A_jk, and conjectures that all its stationary points are maxima, saddle points or global minima, i.e. that no local minima exist, and that the global minima satisfy Ψ = 0.

Problem details · Reading note

Dement's odd-index square conjecture for OEIS A113249

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Creighton Dement (2005). OEIS A113249: odd-index squares in a fourth-order recurrence family. URL: https://oeis.org/A113249.

Claim. Conjecture: a(m, 2*n+1) is a perfect square for all m, n.

Problem details · Reading note

The three-state root-fidelity Holevo bound

Proved. Lean theorem result, frozen in this repository 2026-10-20.

M. Fannes, F. de Melo, W. Roga, K. Życzkowski (2011). Matrices of fidelities for ensembles of quantum states and the Holevo quantity. URL: https://arxiv.org/abs/1104.2271v1.

Claim. Conjecture 1 bounds the Holevo quantity of every three-state ensemble by the entropy of its root-fidelity matrix.

Problem details · Reading note

Fu–Luo decoherence convexity is false

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

S. Fu and S. Luo (2021). Quantifying Decoherence via Increases in Classicality. DOI: 10.3390/e23121594. URL: https://www.mdpi.com/1099-4300/23/12/1594.

Claim. The source conjectures that D(rho_S|Pi,U) is convex in rho.

Problem details · Reading note

Furuichi–Eghbali Amlashi: the hyperbolic ratio below one

Proved. Lean theorem result, frozen in this repository 2026-10-20.

S. Furuichi and M. Eghbali Amlashi (2024). On bounds of logarithmic mean and mean inequality chain. DOI: 10.7153/mia-2024-27-27. URL: https://arxiv.org/abs/2203.01134.

Claim. Remark 3.2(i) of arXiv v2 conjectures that t (cosh t)^(1-r) / sinh t is not positive definite for r < 1.

Problem details · Reading note

Gallardo–Rahavandrainy Conjecture 3.11

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Luis H. Gallardo and Olivier Rahavandrainy (2023). On Collatz Conjecture for binary polynomials. DOI: 10.48550/arXiv.2308.01181. URL: https://arxiv.org/abs/2308.01181v2.

Claim. If A = (M² + M + 1)ⁿ with n ≥ 2, then ℓ_A equals n + 1.

Problem details · Reading note

The Han–Wang–Yu gluing spectral hull conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

X. B. Han; S. Wang; C. Yu (2025). PageRank of Gluing Networks and Corresponding Markov Chains. DOI: 10.3390/math13132080. URL: https://www.mdpi.com/2227-7390/13/13/2080.

Claim. Glue Markov chains M_i = (S_i, A_i) to a Markov chain M = (S, A), then all the eigenvalues of A lie in the convex hull of the eigenvalues of all A_i on the complex plane.

Problem details · Reading note

hanna-a107099-half-iterate-mod-three

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Paul D. Hanna (2005). OEIS A107099, compositional half-iterate of x + 4*x^3. URL: https://oeis.org/A107099.

Claim. Coefficients [x^n] A(x) = 0 (mod 3) except at n = 3^k (conjecture).

Problem details · Reading note

Hanna’s ternary-digit residue conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Paul D. Hanna (2014). OEIS A246056 and A246423, two binomial-square generating functions. URL: https://oeis.org/A246056.

Claim. A246056 conjectures the ternary-digit residue rule modulo three; A246423 conjectures residue two exactly at powers of two of exponent at least two modulo four.

Problem details · Reading note

Hanna’s power-of-two residue conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Paul D. Hanna (2014). OEIS A246056 and A246423, two binomial-square generating functions. URL: https://oeis.org/A246056.

Claim. A246056 conjectures the ternary-digit residue rule modulo three; A246423 conjectures residue two exactly at powers of two of exponent at least two modulo four.

Problem details · Reading note

Hanna’s OEIS A271960 dyadic vanishing conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Paul D. Hanna (2016). OEIS A271960: a squared compositional generating function and dyadic vanishing. URL: https://oeis.org/A271960.

Claim. Conjecture: [x^(2^n-2^k)] (x/A(x))^(2^n) = 0 for k = 0..n-2 and n >= 2.

Problem details · Reading note

Hanna's square-composition congruence for OEIS A372527

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Paul D. Hanna (2024). OEIS A372527, square-composition generating function. URL: https://oeis.org/A372527.

Claim. Conjecture: a(2^k + 1) == 2 (mod 4) for k >= 0, and a(n) == 0 (mod 4) for n > 0 when n is not of the form 2^k + 1.

Problem details · Reading note

The entanglement of formation has a supporting affine functional at every pure state

Proved. Lean theorem result, frozen in this repository 2026-10-20.

A. S. Holevo; M. E. Shirokov (2026). Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation. URL: https://arxiv.org/abs/2609.12667v3.

Claim. With E_F the convex roof of the marginal entropy over finite pure ensembles, the paper states that neither a proof nor a counterexample is known for the existence of a supporting affine functional for the entanglement of formation at every pure state of a finite-dimensional bipartite system.

Problem details · Reading note

The qudit CHSH orthogonality graph is (2d − 1)(d − 1)-regular for every prime d

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Mark Howard; Eoin Brennan; Jiri Vala (2013). Quantum Contextuality with Stabilizer States. DOI: 10.3390/e15062340. URL: https://arxiv.org/abs/1501.04342v1.

Claim. The paper decomposes qudit Bell and noncontextuality inequalities into rank-one stabilizer projectors, builds the orthogonality graph of these projectors, and conjectures from the cases d = 2, 3, 5, 7 that the graph Γ_CHSH obtained from the qudit CHSH inequality is (2d − 1)(d − 1)-regular.

Problem details · Reading note

Conjecture 29 on the path Steiner coefficient peak is false

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

S. Jana; I. Mahato; S. Sivasubramanian (2025). Unimodality and peak location of the characteristic polynomials of two distance matrices of trees. DOI: 10.1007/s40314-025-03163-5. URL: https://arxiv.org/abs/2407.03309v1.

Claim. Conjecture 29 places the largest nonleading coefficient magnitude of the path's principal 2-Steiner distance characteristic polynomial at n−1 for every n>5.

Problem details · Reading note

A stronger cumulative Sturmian progression bound is false

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Gandhar Joshi (2025). Automatic sequences from algebraic, dynamical, and computational perspectives. DOI: 10.21954/ou.ro.00106631. URL: https://oro.open.ac.uk/106631/.

Claim. Section 6.1 asks whether d ≤ qₙ implies A(d) ≤ A(qₙ) for every positive n and every Sturmian slope, without the hypothesis g(d) ≤ α of Proposition 4.4.4.

Problem details · Reading note

The two Karlsson families are the only affine suborbits of BC_9^(2)

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Bengt R. Karlsson (2016). BCCB complex Hadamard matrices of order 9, and MUBs. DOI: 10.1016/j.laa.2016.04.012. URL: https://arxiv.org/abs/1604.03770v1.

Claim. The paper constructs the two-parameter non-affine orbit BC_9^(2) of symmetric block-circulant complex Hadamard matrices of order 9 with circulant blocks, the locus x+y+u+w=1 of four unimodular parameters, identifies the affine one-parameter suborbits BC_9A^(1) and BC_9B^(1), and conjectures (Conjecture 3) that these are the only affine suborbits.

Problem details · Reading note

Divisibility of connected encoder equivalence classes

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Andrey Boris Khesin, Alexander M. Li (2024). Equivalence Classes of Quantum Error-Correcting Codes. DOI: 10.48550/arXiv.2406.12083. URL: https://arxiv.org/abs/2406.12083v1.

Problem details · Reading note

Kotěšovec's second A368959 conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Václav Kotěšovec (2024). OEIS A368959: nonzero complete monomial exponential sums. URL: https://oeis.org/A368959.

Claim. Conjecture 1: This is a subset of A001694. Conjecture 2: This sequence contains all the terms from A338325, plus 243, 972, 1944, 2401, ...

Problem details · Reading note

Kotěšovec's first A368959 conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Václav Kotěšovec (2024). OEIS A368959: nonzero complete monomial exponential sums. URL: https://oeis.org/A368959.

Claim. Conjecture 1: This is a subset of A001694. Conjecture 2: This sequence contains all the terms from A338325, plus 243, 972, 1944, 2401, ...

Problem details · Reading note

One ancilla qubit and bounded rounds cannot simulate every channel with constant success

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Marcin Kotowski; Michał Kotowski (2026). Randomized simulation of quantum channels using small ancilla. URL: https://arxiv.org/abs/2606.08784v1.

Claim. The paper defines simulation of a channel with classical randomization and postselection, and asks (Question 6.1) whether every channel can be simulated with success probability Omega(1) using one additional qubit and O(1) rounds of projective measurements and adaptive unitaries.

Problem details · Reading note

Strict containment for graph quantum magic squares on the five-cycle

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Francesca La Piana (2025). Graph Quantum Magic Squares and Free Spectrahedra. DOI: 10.48550/arXiv.2512.08797. URL: https://arxiv.org/abs/2512.08797v3.

Claim. Question 1.3 asks whether graph quantum permutation matrices generate all graph quantum magic squares; Section 4.1 names C5 as a candidate for strict containment.

Problem details · Reading note

Li–Tan circulant positivity is false at order four

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Mao-Sheng Li; Yi-Xi Tan (2024). On the Bargmann invariants for quantum imaginarity. DOI: 10.48550/arXiv.2412.08022. URL: https://arxiv.org/abs/2412.08022v1.

Claim. Section III conjectures that the circulant formed from geometric means of moduli and arithmetic means of principal phases preserves positive semidefiniteness of correlation matrices in general dimension.

Problem details · Reading note

Independent extreme-point tests in polyhedral Banach spaces

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

J. Manna; K. Mandal; K. Paul; D. Sain (2025). An improvement of the Blanco-Koldobsky-Turnšek characterization of isometries. URL: https://arxiv.org/abs/2511.20345v1.

Claim. The final Open Question asks whether preservation at any chosen n independent extreme points forces a bounded endomorphism of every n-dimensional polyhedral Banach space to be a scalar multiple of an isometry.

Problem details · Reading note

Nikiforov: asymptotic graph Ky Fan gaps

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Vladimir Nikiforov (2015). Extrema of graph eigenvalues. URL: https://arxiv.org/abs/1502.00359v1.

Claim. Conjecture 4.7 asks for infinitely many strict asymptotic gaps in graph Ky Fan singular-value sums.

Problem details · Reading note

A strict gap between measured and Uhlmann average fidelities

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Theshani Nuradha; Hemant K. Mishra; Felix Leditzky; Mark M. Wilde (2025). Multivariate Fidelities. DOI: 10.1088/1751-8121/adc645. URL: https://arxiv.org/abs/2404.16101v3.

Claim. Multivariate SDP fidelity maximizes jointly positive block coherences and is bounded by average pairwise root fidelity; Remark 5.45 asks for a strict measured-versus-Uhlmann gap, and Section 6 item 2 asks for operational interpretations.

Problem details · Reading note

Two indexed wreath eigenvalues coincide

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Jan Petr; Pavel Turek (2025). The wreath matrix. DOI: 10.48550/arXiv.2501.07269. URL: https://arxiv.org/abs/2501.07269v2.

Claim. Question 5.5 asks whether the eigenvalues λ₁, λ₂, …, λₖ from Theorem 1.5 are all distinct.

Problem details · Reading note

A pinned triangle refutes the spanning-tree dimension conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Matteo Polettini; Pedro E. Harunari; Sara Dal Cengio; Vivien Lecomte (2024). Coplanarity of rooted spanning-tree vectors. DOI: 10.1007/s11005-025-02026-8. URL: https://arxiv.org/abs/2407.16093v1.

Claim. The authors conjecture that the 3ⁿ deletion-constriction vectors through n reversible edges span a space of dimension n+1.

Problem details · Reading note

Unique integral circulant energy maximum at order $p^2q^3$

Proved. Lean theorem result, frozen in this repository 2026-10-20.

D. G. Roldán (2026). Graph Energy Maximisation for Integral Circulant Graphs of Order n = p^2q^3. DOI: 10.48550/arXiv.2604.09491. URL: https://arxiv.org/abs/2604.09491v1.

Claim. Conjecture 1.3 states that D* = {1, p^2, pq, q^2, p^2q^2, pq^3} uniquely maximises the integral circulant energy at n = p^2q^3 for distinct odd primes p and q.

Problem details · Reading note

Conjecture 1 on paired coalition numbers of perfect binary trees is false

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Mohammad Reza Samadzadeh; Doost Ali Mojdeh; Reza Nadimi (2024). Paired coalition in graphs. DOI: 10.1080/09728600.2024.2389070. URL: https://arxiv.org/abs/2402.10842v4.

Claim. Conjecture 1 assigns paired coalition number 5 at height 2, 3 at odd heights, and 0 otherwise for perfect binary trees.

Problem details · Reading note

svinin-2015-discrete-polynomial-identity

Proved. Lean theorem result, frozen in this repository 2026-10-20.

A. K. Svinin (2015). On integrals for some class of ordinary difference equations admitting a Lax pair representation. DOI: 10.1088/1751-8113/49/9/095201. URL: https://arxiv.org/abs/1505.06394v2.

Claim. Conjecture 1 asserts the discrete-polynomial convolution identity (48) in every degree for the case (h,n)=(1,1).

Problem details · Reading note

The MUB and SIC correlation trace norms agree up to the factor K

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Armin Tavakoli; Simon Morelli (2024). Enhanced Schmidt number criteria based on correlation trace norms. DOI: 10.1103/PhysRevA.110.062417. URL: https://arxiv.org/abs/2402.09972v2.

Claim. For a bipartite state of Schmidt number at most r, the trace norm of the table of joint probabilities of local SIC measurements or of complete sets of MUBs is bounded (Results 1 and 2), and for incomplete measurements (n > d equiangular effects, or m < d + 1 MUBs) the bounds of Eqs. (51) and (53) hold. Conjecture 2 asserts ‖P_n‖_tr ≤ [d(d − 1) + r(n − d)]/[n(n − 1)] for equiangular measurements (Eq. (52)), and Conjecture 3 asserts ‖Q_m‖_tr ≤ 1 + (m − 1)r/d for m MUBs (Eq. (54)); the introduction leaves deciding these conjectures as an open problem. Conjecture 1 asserts that, when both a SIC and a complete set of MUBs exist in the local dimensions, the two correlation trace norms satisfy ‖Q‖_tr = K‖P‖_tr with K = √(d_A(d_A+1)) √(d_B(d_B+1)).

Problem details · Reading note

The separable modified Wasserstein distance can exceed the De Palma–Trevisan one

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Géza Tóth; József Pitrik (2026). Quantum Wasserstein distance and its relation to several types of fidelities. URL: https://arxiv.org/abs/2506.14523v6.

Claim. For Hermitian observables H_1,...,H_N, the paper compares the De Palma-Trevisan quantum Wasserstein distance with its variant optimized over separable couplings, and conjectures (Conjecture 1) that the modified separable distance never exceeds the modified De Palma-Trevisan distance for arbitrary mixed states.

Problem details · Reading note

Ulfarsson's alternating mesh-pattern basis conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Henning Ulfarsson (2024). BiSC: An algorithm for discovering generalized permutation patterns. DOI: 10.48550/arXiv.2411.17778. URL: https://arxiv.org/abs/2411.17778v1.

Claim. Conjecture 2.4.2: every mesh-pattern basis of the even permutations consists exactly of fully shaded odd permutations.

Problem details · Reading note

The balanced-cut gradient bound of Wölk and Gühne fails for eight qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Sabine Wölk; Otfried Gühne (2016). Characterizing the width of entanglement. DOI: 10.1088/1367-2630/aa5015. URL: https://arxiv.org/abs/1507.07226v2.

Claim. The paper introduces the width of entanglement in spin chains, derives criteria from the collective spin and from a magnetic-gradient operator, and conjectures (Appendix D) that every state with entanglement only within the two halves of the chain satisfies a linear upper bound on the gradient second moment in terms of the squared total spin, attained by mixtures of adjacent singlets and the classical state.

Problem details · Reading note

Independence polynomials do not determine T2-threshold hypergraphs

Refuted. Lean theorem result, frozen in this repository 2026-10-20.

Erchuan Zhang (2023). On the Independence Polynomial and Threshold of an Antiregular k-Hypergraph. DOI: 10.1016/j.dam.2025.07.017. URL: https://arxiv.org/abs/2303.01024v1.

Claim. Section 5 asks whether two T2-threshold k-hypergraphs with equal independence polynomials must be isomorphic.

Problem details · Reading note

All qubit tuples maximizing the imaginary Bargmann invariant

Proved. Lean theorem result, frozen in this repository 2026-10-20.

Lin Zhang; Bing Xie (2026). A Survey of Bargmann Invariants: Geometric Foundations and Applications. DOI: 10.48550/arXiv.2601.01858. URL: https://arxiv.org/abs/2601.01858v2.

Claim. The survey asks for the complete set of unit qubit tuples attaining the maximal imaginary Bargmann invariant tau_n for every n at least three.

Problem details · Reading note

Two centered abelian period lengths

Proved. Lean theorem result, frozen in this repository 2026-10-19.

S. Avgustinovich, J. Karhumäki and S. Puzynina (2012). On abelian versions of critical factorization theorem. DOI: 10.1051/ita/2011121. URL: https://www.numdam.org/item/ITA_2012__46_1_3_0/.

Claim. Open problem 6.2 asks whether two centered abelian period lengths can occur at every position of a non-periodic biinfinite word.

Problem details · Reading note

A uniform family of reduced-coordinate cube recursions

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Mateo Matijasevick and Santiago Rodriguez and Gregorio Salazar (2026). Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields. DOI: 10.48550/arXiv.2610.00765. URL: https://arxiv.org/abs/2610.00765v1.

Claim. Section 7 question 2 asks for a natural rational-recursion family with reduced numerator plus denominator a cube, outside the conjugacy class of the cubic power map.

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The two identities of Remark 4.3

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Ludovick Bouthat, Ángel Chávez and Sam Sheng (2026). A general framework for inequalities on simple graphs. DOI: 10.48550/arXiv.2606.17356. URL: https://arxiv.org/abs/2606.17356v1.

Claim. Remark 4.3 proposes trigonometric and bilateral binomial expressions for the path CHS values, followed by a one-term simplification.

Problem details · Reading note

The one-term formula at the boundary

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Ludovick Bouthat, Ángel Chávez and Sam Sheng (2026). A general framework for inequalities on simple graphs. DOI: 10.48550/arXiv.2606.17356. URL: https://arxiv.org/abs/2606.17356v1.

Claim. Remark 4.3 proposes trigonometric and bilateral binomial expressions for the path CHS values, followed by a one-term simplification.

Problem details · Reading note

Complete homogeneous symmetric norms of paths

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Claim. Problem 1 asks for the complete homogeneous symmetric norms of finite path adjacency matrices in even degrees at least four.

Problem details · Reading note

The Cobucci–Tavakoli ten-setting GHZ witness in odd dimensions

Proved. Lean theorem result, frozen in this repository 2026-10-19.

G. Cobucci, A. Tavakoli (2024). Detecting the dimensionality of genuine multi-particle entanglement. DOI: 10.1126/sciadv.adq4467. URL: https://arxiv.org/abs/2402.06234.

Claim. Supplementary Material V conjectures the ten-setting GHZ witness bounds (V3)–(V5) for every odd local dimension.

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Positivity of the JNR triple sum

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Josh Cork and Linden Disney-Hogg (2025). Locations of JNR skyrmions. DOI: 10.1007/s11040-025-09513-3. URL: https://arxiv.org/abs/2505.00075v2.

Claim. Remark 13 conjectures strict positivity of the JNR triple sum (18) without the balance condition (15).

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Czerwińska–Franco–Merino: radial convexity and matrix monotonicity

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

M. M. Czerwińska; J. A. Franco; A. Merino (2026). Towards a Characterization of Matrix Monotonicity via Radial Convexity. DOI: 10.48550/arXiv.2610.17393. URL: https://arxiv.org/abs/2610.17393.

Claim. Let f : (0, ∞) → R. Then f is radially convex of order n (with respect to the Frobenius norm) if and only if f is matrix monotone of degree n.

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Datta–Dorlas–Jozsa–Benatti: conditional subentropy and guessing

Proved. Lean theorem cq_conditional_subentropy_guessing, frozen in this repository 2026-10-19.

Nilanjana Datta, Tony Dorlas, Richard Jozsa, Fabio Benatti (2014). Properties of subentropy. DOI: 10.1063/1.4882935. URL: https://arxiv.org/abs/1310.1312.

Claim. Subentropy has a finite simple-spectrum formula with continuous extension to repeated and zero eigenvalues; Conjecture 1 asks for bipartite subadditivity.

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Datta–Dorlas–Jozsa–Benatti: subadditivity of subentropy

Proved. Lean theorem subentropy_subadditive, frozen in this repository 2026-10-19.

Nilanjana Datta, Tony Dorlas, Richard Jozsa, Fabio Benatti (2014). Properties of subentropy. DOI: 10.1063/1.4882935. URL: https://arxiv.org/abs/1310.1312.

Claim. Subentropy has a finite simple-spectrum formula with continuous extension to repeated and zero eigenvalues; Conjecture 1 asks for bipartite subadditivity.

Problem details · Reading note

Gan–Kim: weak log-majorization monotonicity of Wasserstein power means

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Luyining Gan; Sejong Kim (2024). Weak log-majorization between the geometric and Wasserstein means. DOI: 10.48550/arXiv.2301.07934. URL: https://arxiv.org/abs/2301.07934.

Claim. Conjecture 5.7 proposes weak log-majorization monotonicity and convergence of the power-transformed Wasserstein mean as its exponent decreases to zero.

Problem details · Reading note

Gaussian atemporality implies entanglement

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Minjeong Song; Jayne Thompson; Matthew S. Winnel; Biveen Shajilal; Timothy C. Ralph; Syed M. Assad; Mile Gu (2025). Gaussian Atemporality: When Gaussian Quantum Correlations Imply Common Cause. DOI: 10.48550/arXiv.2508.11804. URL: https://arxiv.org/abs/2508.11804v2.

Claim. Section III.D states: From Fig. 3, Gaussian-atemporality appears to imply entanglement.

Problem details · Reading note

Hanna's A274479 coefficient congruence

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Paul D. Hanna (2016). OEIS A274479, a generating-function coefficient conjecture. URL: https://oeis.org/A274479.

Claim. G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 4*x^2) ). a(n) = 1 (mod 3) for n>=1 (conjecture).

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Hanna's OEIS A338632 congruence

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Paul D. Hanna (2026). OEIS A338632, continued-fraction generating function with odd numerators. URL: https://oeis.org/A338632.

Claim. G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - ...))))), a continued fraction relation. a(n) = 2 (mod 4) for n > 1 (conjecture).

Problem details · Reading note

Hanna's A377100 coefficient congruence

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Paul D. Hanna (2024). OEIS A377100, a generating-function coefficient conjecture. URL: https://oeis.org/A377100.

Claim. G.f. A(x) satisfies A(x) = A(x^3)/A(x^2) + A(x)^2. Conjecture: a(n) == 1 (mod 3) for n >= 1, which has been verified for the initial 2305 terms.

Problem details · Reading note

Hanna's bilateral theta congruence, OEIS A380060

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Paul D. Hanna (2025). OEIS A380060, bilateral unit-quotient square series. URL: https://oeis.org/A380060.

Claim. Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.

Problem details · Reading note

Hanna's A382318 congruence modulo three

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Paul D. Hanna (2025). OEIS A382318, g.f. A(x) satisfies A(x) = x + ( Sum_{n>=1} A(x^n) )^3. URL: https://oeis.org/A382318.

Claim. Conjectures: a(3*n) == A382319(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 1.

Problem details · Reading note

Hanna's A382319 congruence modulo three

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Paul D. Hanna (2025). OEIS A382319, g.f. A(x) satisfies A(x) = x/(1-x) + Sum_{n>=1} A(x^n)^3. URL: https://oeis.org/A382319.

Claim. Conjecture: a(n) == A382318(3*n) (mod 3) for n >= 0.

Problem details · Reading note

Hanna's bilateral reversion parity conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-19.

P. D. Hanna (2024). OEIS A379202, a bilateral Laurent generating equation. URL: https://oeis.org/A379202.

Claim. Conjecture: a(n) is even for n > 1.

Problem details · Reading note

Hayajneh–Hayajneh–Kittaneh: mixed-power monotonicity for unitarily invariant norms

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Mostafa Hayajneh; Saja Hayajneh; Fuad Kittaneh (2018). On some classical trace inequalities and a new Hilbert–Schmidt norm inequality. DOI: 10.7153/mia-2018-21-81. URL: https://doi.org/10.7153/mia-2018-21-81.

Claim. Conjecture 1 proposes monotonicity of the mixed-power expression in every unitarily invariant norm for positive semidefinite A and Hermitian B.

Problem details · Reading note

The Hemmer–Straub–Westrem Ev character identity

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Hemmer, David J.; Straub, Armin; Westrem, Karlee J. (2025). New identities in the character table of symmetric groups involving Riordan numbers. DOI: 10.37236/14401. URL: https://arxiv.org/abs/2509.02796v1.

Claim. Conjecture 2.3 equates the alternating Ev sum over even three-row characters with the Ev sum of the two-row character; Lemma 3.1 supplies the Frobenius character interpretation.

Problem details · Reading note

Huang–Ma Remark 2.7: the binary delta invariant

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Min Huang and Qiling Ma (2026). A mutation invariant for skew-symmetrizable matrices. DOI: 10.1016/j.jalgebra.2026.07.040. URL: https://arxiv.org/abs/2602.03194v1.

Claim. Remark 2.7 conjectures that for fixed n and a fixed positive skew-symmetrizer, the delta invariant is binary.

Problem details · Reading note

Finite Otto congruence quotients are cancellative

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Trevor Jack (2026). Answering Five Open Problems Involving Semigroup Conjugacy. DOI: 10.1007/s00233-026-10614-w. URL: https://arxiv.org/abs/2411.16284.

Claim. Section 2 defines Otto conjugacy and Section 9, Problem 9.4 asks whether a finite semigroup can have an Otto congruence with a noncancellative quotient.

Problem details · Reading note

The Jeudy–Rouleux global-permutation Lie-algebra expectation

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

C. Jeudy; M. Rouleux (2024). Generalized exchange operators for a system of spin-1 particles. DOI: 10.1088/1742-6596/2912/1/012043. URL: https://arxiv.org/abs/2411.03952v1.

Claim. Section 3.1 expects the spin-1 three-particle Schrödinger Hamiltonian to belong to the Lie algebra of the global-label Q-representation.

Problem details · Reading note

Absolute two-incoherence beyond the purity bound

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Nathaniel Johnston; Shirin Moein; Rajesh Pereira; Sarah Plosker (2022). Absolutely k-incoherent quantum states and spectral inequalities for factor width of a matrix. DOI: 10.1103/PhysRevA.106.052417. URL: https://arxiv.org/abs/2205.05110v2.

Claim. Section 5 asks whether the sufficient purity condition for absolute 2-incoherence is necessary in dimension four or higher.

Problem details · Reading note

Khavinson–Pereira–Putinar–Saff–Shimorin: nonunivalence radius

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

D. Khavinson, R. Pereira, M. Putinar, E. B. Saff, S. Shimorin (2011). Borcea's variance conjectures on the critical points of polynomials. DOI: 10.1007/978-3-0348-0142-3_16. URL: https://arxiv.org/abs/1010.5167.

Claim. Section 5, Conjecture 4 proposes a nonunivalence radius in terms of the minimum root moment; this assertion is refuted by the polynomial z minus z to the fourth power at order one.

Problem details · Reading note

Local distinguishability at every boundary projection state

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Young-Hoon Kiem; Seung-Hyeok Kye (2026). Local geometry for Schmidt number witnesses. DOI: 10.48550/arXiv.2608.14199. URL: https://arxiv.org/abs/2608.14199v1.

Claim. Sections 2, 3 and 5 define local distinguishability and the trace-one blockpositive sets, and conjecture local distinguishability of adjacent blockpositive sets around every boundary projection state.

Problem details · Reading note

A square discriminant refutes the odd clause of Levens Conjecture 6.4

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Owen Levens (2025). Polynomials Arising from Sorted Binomial Coefficients. DOI: 10.48550/arXiv.2511.03082. URL: https://arxiv.org/abs/2511.03082v1.

Claim. Conjecture 6.4 asserts the full symmetric group for even Pascalian indices and the mth hyperoctahedral group at index 2m+1.

Problem details · Reading note

Lin: both spectral properties of the map Ψ

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Minghua Lin (2016). New properties for certain positive semidefinite matrices. DOI: 10.48550/arXiv.1609.08231. URL: https://arxiv.org/abs/1609.08231v1.

Claim. Example 3.6 asks whether the map Ψ(T)=2(tr T)I−T gives the a-property and g-property for every positive semidefinite two-by-two block matrix.

Problem details · Reading note

Lin: the g-property of trace augmentation

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Minghua Lin (2016). A singular value inequality related to a linear map. DOI: 10.13001/1081-3810.3108. URL: https://arxiv.org/abs/1609.08231.

Claim. Conjecture 3.1 asks whether trace augmentation gives the g-property for every positive semidefinite two-by-two block matrix.

Problem details · Reading note

The powered binary Rényi entropy is not monotone in the order

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Yupan Liu (2026). Computational hardness of estimating quantum entropies via binary entropy bounds. DOI: 10.4230/LIPIcs.STACS.2026.66. URL: https://arxiv.org/abs/2601.03734v2.

Claim. With the binary Renyi entropy H^R_alpha(x) = ln(x^alpha + (1-x)^alpha)/(1-alpha), the paper asks whether (H^R_alpha(x)/ln 2)^{2/alpha} is monotonically non-decreasing in alpha for every fixed x in [0,1], as suggested by numerical evidence and generalizing its comparison with order 2.

Problem details · Reading note

The Lotka–Volterra integrals are birational coordinates in every dimension

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Ian Marquette; Peter H. van der Kamp; G. R. W. Quispel (2026). Subalgebras of integrals, commutants, and superintegrable deformations of Lotka-Volterra systems. URL: https://arxiv.org/abs/2602.17945v1.

Claim. For the n-dimensional Lotka-Volterra system with integrals F_k and their coordinate reversals G_k, the change of variables from x_1,...,x_n to the n-1 distinct integrals and the distinguished variable x_n is conjectured to be birational in every dimension.

Problem details · Reading note

Martinez's mirror-step gambler's ruin limit

Proved. Lean theorem result, frozen in this repository 2026-10-19.

L. Martinez (2024). A symbolic computational approach to the generalized gambler's ruin problem in one and two dimensions. DOI: 10.48550/arXiv.2412.07667. URL: https://arxiv.org/abs/2412.07667v1.

Claim. Conjecture 5.1 gives a radical expression for the infinite-interval limit of the probability of reaching N from 1 when the left and mirror step probabilities are (1-p)/2 and the right step probability is p.

Problem details · Reading note

mathar-oeis-a073759-least-prime-factor

Proved. Lean theorem result, frozen in this repository 2026-10-19.

R. J. Mathar (2026). OEIS A073759: largest number neither a divisor of nor relatively prime to n. URL: https://oeis.org/A073759.

Claim. Conjecture: a(n) = 2*A046667(n) if n>=5. - _R. J. Mathar_, May 15 2026

Problem details · Reading note

Uniform prime-square approximation for the full local moment

Proved. Lean theorem result, frozen in this repository 2026-10-19.

unkown (2026). Uniform dominance of a prime-square term in an Euler-product moment. URL: https://mathoverflow.net/questions/516091/uniform-dominance-of-a-prime-square-term-in-an-euler-product-moment.

Claim. The question asks whether the complete infinite valuation-factor moment admits a uniform logarithmic prime-square approximation for large real s and primes in a moving exponential window.

Problem details · Reading note

Nathanson's Problem 12: affine invariance of sum intersection sets

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Melvyn B. Nathanson (2026). Problems and results on intersections of product sets and sumsets in semigroups. DOI: 10.48550/arXiv.2604.04781. URL: https://arxiv.org/abs/2604.04781v2.

Claim. Problem 9 asks for a nontrivial proper product intersection spectrum in every infinite torsion group; Problem 12 asks for affine invariance of integer sum intersection spectra.

Problem details · Reading note

Rather's real-rootedness question for general position polynomials

Proved. Lean theorem result, frozen in this repository 2026-10-19.

B. A. Rather (2026). Explicit Formulas and Unimodality Phenomena for General Position Polynomials. DOI: 10.48550/arXiv.2603.06930. URL: https://arxiv.org/abs/2603.06930v3.

Claim. Section 8 item 1 asks for real-rootedness results for general position polynomials; the connected classification is real-rooted if and only if complete.

Problem details · Reading note

The optimal W-state PPT measurement is outside SEP

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

L. Rossetti; S. Mancini; A. Winter; J. Schindler (2025). Observational entropy of quantum correlations and entanglement. DOI: 10.1109/TIT.2026.3700031. URL: https://arxiv.org/abs/2510.10058v1.

Claim. The membership of the optimal three-qubit W-state PPT POVM in SEP is posed as an open question.

Problem details · Reading note

Complex hopping can increase parity-twisted ground-state coherences

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Aleksandrs Sokolovs (2026). Parity superselection obstructs monogamy of mutual information in free fermions. URL: https://arxiv.org/abs/2603.14154v2.

Claim. For ground states of fermionic Hamiltonians with real non-positive hopping, the parity-twisted partial trace satisfies |rho^tw_ij| <= rho^spin_ij = |rho^spin_ij| on D-parity-changing elements (Theorem 1); whether this coherence damping extends to complex hopping or frustrated lattices is stated as open.

Problem details · Reading note

A Clifford unitary that changes the operator norm of the MUB autocorrelation matrix

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Blake C. Stacey (2025). SICs, the Stabilizer Polytope and Majorization. URL: https://arxiv.org/abs/2509.16819v1.

Claim. For a pure state in prime dimension d, the matrix A(psi)_{mk} = sum_j p_{m,j} p_{m,j+k} built from the Weyl-Heisenberg MUB probabilities is asserted to be changed by Clifford unitaries only by permutations of its rows and columns, so that every unitarily invariant norm of A is Clifford-invariant and gives a candidate magic monotone.

Problem details · Reading note

Tărnăuceanu's reciprocal subgroup-order conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-19.

M. Tărnăuceanu (2024). On the sum of inverses of subgroup orders in finite groups. URL: https://www.math.uaic.ro/~martar/pdf/articole/articol%20141.pdf.

Claim. Conjecture 3.1 asks whether no constant c > 1 is a lower bound for the subgroup-order reciprocal sum on all finite non-cyclic groups.

Problem details · Reading note

Resolved-count coherent-state weak values

Proved. Lean theorem result, frozen in this repository 2026-10-19.

H. M. Wiseman, A. M. Steinberg, M. Hallaji (2023). Obtaining a single-photon weak value from experiments using a strong (many-photon) coherent state. DOI: 10.1116/5.0137579. URL: https://arxiv.org/abs/2211.15761v2.

Claim. Section V conjectures that the weak value post-selected on detecting m photons exceeds the no-click weak value by precisely m times the single-photon weak value.

Problem details · Reading note

Zhang–Li rank-one Hadamard product variational problem

Proved. Lean theorem result, frozen in this repository 2026-10-19.

Lei-Hong Zhang and Ren-Cang Li (2026). On a Conjecture Related to Eigenvalue Perturbations. DOI: 10.48550/arXiv.2609.25552. URL: https://arxiv.org/abs/2609.25552.

Claim. Problem 3 asks for a relation between a nonsingular Hadamard norm product minimum and the squared permutation minimum; §5 identifies the rank-one nonsingular case as unresolved.

Problem details · Reading note

Zhang–Singh–Pati: strong subadditivity of subentropy

Refuted. Lean theorem result, frozen in this repository 2026-10-19.

Lin Zhang, Uttam Singh, Arun Kumar Pati (2017). Average subentropy, coherence and entanglement of random mixed quantum states. DOI: 10.1016/j.aop.2016.12.024. URL: https://arxiv.org/abs/1510.08859.

Claim. Section 7.1 asks whether subentropy satisfies strong subadditivity, Q(ρ_ABC) + Q(ρ_B) ≤ Q(ρ_AB) + Q(ρ_BC).

Problem details · Reading note

The exact five-dimensional bad science optimum

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Alex Albors, Hisham Bhatti, Lukshya Ganjoo, Raymond Guo, Dmitriy Kunisky, Rohan Mukherjee, Alicia Stepin, Tony Zeng (2026). On the Structure of Bad Science Matrices. DOI: 10.2140/involve.2026.19.443. URL: https://arxiv.org/abs/2408.00933.

Claim. Section 1.4 asks whether the displayed five-by-five unit-row matrix with beta equal to (2 + 3 sqrt(3))/4 is optimal.

Problem details · Reading note

The Allen–Meyer family does not give the entire three-qutrit negativity region

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Grant W. Allen; David A. Meyer (2017). Polynomial Monogamy Relations for Entanglement Negativity. DOI: 10.1103/PhysRevLett.118.080402. URL: https://arxiv.org/abs/1502.04807v2.

Claim. For three qubits the paper determines the region of achievable triples of squared negativities (N²_A|C, N²_A|B, N²_A|BC), with N(ρ) = ‖ρ^{T_A}‖₁ − 1, and describes it by the states d|000⟩ + a|101⟩ + b|110⟩ + c|111⟩; it generalizes them to the family (13) for D-dimensional qudits and conjectures that this family gives the entire achievable set for D > 2.

Problem details · Reading note

The quasidistillable candidates are not two-qutrit negativity maximizers for fixed marginals

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Giuseppe Baio; Dariusz Chruściński; Paweł Horodecki; Antonino Messina; Gniewomir Sarbicki (2019). Bounds on the entanglement of two-qutrit systems from fixed marginals. DOI: 10.1103/PhysRevA.99.062312. URL: https://arxiv.org/abs/1904.07650v2.

Claim. For two qutrits with fixed ordered marginals, the authors maximize the negativity (Eq. (4)) within the circulant family (22), obtain the quasidistillable candidates (27)–(29), and conjecture from numerical optimization that these candidates are the maximally entangled states with respect to the fixed marginals.

Problem details · Reading note

The l1-norm correlated coherence is not monogamous for tripartite pure states

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Marcos L. W. Basso; Jonas Maziero (2020). Monogamy and trade-off relations for correlated quantum coherence. DOI: 10.1088/1402-4896/abafcb. URL: https://arxiv.org/abs/2006.14037v2.

Claim. For the l1-norm coherence C in a fixed product basis, the correlated coherence of a bipartite marginal is C(ρ_XY) − C(ρ_X) − C(ρ_Y), and C^c(ρ_A|BC) − C^c(ρ_AB) − C^c(ρ_AC) = C(ρ_ABC) + Σ_α C(ρ_α) − Σ_{α<β} C(ρ_αβ) (Eq. (33)); the paper proves the monogamy relation under conditions (22)–(24) and conjectures that it holds for every tripartite pure state.

Problem details · Reading note

The cross-integral bound for continuous-time walks on two cycles

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Shyam Dhamapurkar; Xiu-Hao Deng (2023). Quantum walk mixing is faster than classical on periodic lattices. DOI: 10.1016/j.physa.2023.129252. URL: https://arxiv.org/abs/2309.16352v1.

Claim. For the continuous-time quantum walk on Z_n, the time-dependent part of the transition probability is n(t) = sum over 0 <= j, k <= n - 1 with j != k and j + k != n of exp(-i t sin(pi (j+k)/n) sin(pi (j-k)/n)) w^((q-p)(j-k)), w = exp(2 pi i/n) (Eq. (B8)). The paper conjectures (Conjecture 1, Eq. (14)) a bound on the time integral of products of such terms for odd, pairwise coprime n_1 > ... > n_d, and uses the case d = 2, |int_0^T n_1(t) n_2(t) dt| <= 32 n_1 (n_2 log n_2)^2 + 32 n_2 (n_1 log n_1)^2 (Eq. (15)), in its mixing-time theorem.

Problem details · Reading note

The exact three-coherence threshold of the Ramsey moment ratio R3

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Benjamin Dive; Nikolaos Koukoulekidis; Stefanos Mousafeiris; Florian Mintert (2020). Characterisation of multi-level quantum coherence without ideal measurements. DOI: 10.1103/PhysRevResearch.2.013220. URL: https://arxiv.org/abs/1901.08599v3.

Claim. For the interference pattern p(t) = ⟨χ|e^{−iHt} ρ e^{iHt}|χ⟩ of a commensurate Hamiltonian, the moment ratio R3 = M3/M1² is convex in ρ and its maximum over states that are not (k + 1)-coherent is a threshold for k-coherence; Table I gives the values 1 (k = 1), 5/4 (k = 2) and, for k = 3, only the analytic upper bound 179/96 and the numerical value 1.77.

Problem details · Reading note

Positivity of the multi-entropy replica invariant

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

A. Gadde, S. Jain, V. Krishna, H. Kulkarni, and T. Sharma (2024). Monotonicity conjecture for multi-party entanglement. Part I. DOI: 10.1007/JHEP02(2024)025. URL: https://arxiv.org/abs/2308.16247v1.

Claim. Section 7 asks whether the replica invariant E_n^(q) is positive on all normalized pure states.

Problem details · Reading note

The two-qubit binegativity lower bound at fixed negativity

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Mark W. Girard; Gilad Gour (2017). The binegativity of two qubits. DOI: 10.48550/arXiv.1701.02724. URL: https://arxiv.org/abs/1701.02724v3.

Claim. For a two-qubit state sigma with partial transpose sigma^Gamma, the negativity is N(sigma) = 2 Tr[(sigma^Gamma)_-] and the binegativity is N_2(sigma) = Tr[(sigma^Gamma)_-] + 2 Tr[(((sigma^Gamma)_-)^Gamma)_-], where X_+ and X_- are the positive and negative components of a self-adjoint X. Based on randomly generated states, the paper conjectures that nu (c + nu)(nu + 1)/((c + nu)^2 + 2c(1 - c)) <= N_2(sigma) <= (nu/2)(c + nu)^2/(c^2 + nu^2) for all states with negativity nu and concurrence c (Eq. (9)), and, with nu alone fixed and c_nu = sqrt(2 nu (nu + 1)) - nu, that (nu/2)(1 + c_nu/sqrt((1 - c_nu)^2 + c_nu^2)) <= N_2(sigma) (Eq. (8)).

Problem details · Reading note

Optimal hyperbolic cotangent constant

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Hiroshi Hirai, Harold Nieuwboer, and Michael Walter (2023). Interior-point methods on manifolds: theory and applications. DOI: 10.48550/arXiv.2303.04771. URL: https://arxiv.org/abs/2303.04771.

Claim. Lemma 5.13 and Remark 5.14 concern the hyperbolic cotangent inequality and its optimal coefficient; Theorem 5.15 relates this estimate to self-concordance of squared distance on positive definite matrices.

Problem details · Reading note

Arimoto information combining convexity and concavity

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Christoph Hirche (2020). Rényi Bounds on Information Combining. DOI: 10.48550/arXiv.2004.14408. URL: https://arxiv.org/abs/2004.14408v1.

Claim. Conjectures IV.6 and IV.18 ask for separate convexity and concavity thresholds for binary Arimoto and Cachin Rényi information combining.

Problem details · Reading note

Cachin Rényi information combining convexity and concavity

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Christoph Hirche (2020). Rényi Bounds on Information Combining. DOI: 10.48550/arXiv.2004.14408. URL: https://arxiv.org/abs/2004.14408v1.

Claim. Conjectures IV.6 and IV.18 ask for separate convexity and concavity thresholds for binary Arimoto and Cachin Rényi information combining.

Problem details · Reading note

A weight density below Janssen's moment lower bound for the annealed Potts critical parameter

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

A. J. E. M. Janssen (2025). Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights. URL: https://arxiv.org/abs/2508.21409v3.

Claim. With F0(t) = E[W/E[W] (e^{tW}-1)/(e^{tW}+q-1)] first positive and then negative in its second derivative, t_c is the unique positive zero of K(t) = E[ln((e^{tW}+q-1)/q)]/E[W] - ((q-1)/(2q)) t F0(t) - t/q; the paper suggests from limited numerics that 2 (mu_3/mu_4) ln(q-1) < t_c < 2 (mu_0/mu_1) ln(q-1) for q > 2 and general weight densities.

Problem details · Reading note

The Koch-Narayan bipartite edge bound is false

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Garrison Koch; Darren Narayan (2025). Maximal bipartite graphs with a unique minimum dominating set. DOI: 10.48550/arXiv.2511.01719. URL: https://arxiv.org/abs/2511.01719v1.

Claim. Conjecture 1 gives a universal upper bound on the number of edges of a finite bipartite graph without isolated vertices and with a unique minimum dominating set, gamma>=2 and n>=3gamma.

Problem details · Reading note

The strengthened Marcus–de Oliveira conjecture has a three-dimensional counterexample

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

A. Kovačec (2026). A conjecture more precise and stronger than the one by Marcus and de Oliveira. URL: https://www.mat.uc.pt/preprints/ps/p2611.pdf.

Claim. Conjecture 7 asserts ∆(A, B) ⊆ closure(J(A, B)) whenever the exterior boundary of the σ-point graph is a Jordan curve.

Problem details · Reading note

Polynomial content has no maximizer on suitable unit-area polygons

Proved. Lean theorem result, frozen in this repository 2026-10-18.

A. Kraus and B. Simanek (2024). New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar. DOI: 10.1007/s10659-024-10087-4. URL: https://arxiv.org/abs/2309.16450.

Claim. Conjecture 2.1 asserts that for every polynomial degree there is a polygon side count for which the unit-area polynomial content has no maximizer.

Problem details · Reading note

Lee–Liu integer {2}-domination of three-row grids

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Jia-Ying Lee; Chia-An Liu (2025). The integer {2}-domination number of grids. DOI: 10.48550/arXiv.2502.00134. URL: https://arxiv.org/abs/2502.00134.

Claim. Defines integer {2}-domination on Cartesian grids; Conjecture 2.5 asserts γ{2}(G3,n) = ⌊3n/2⌋ + 1 for every positive n, Remark 2.6 reports algorithmic evidence, Theorem 3.6 gives four-row values through n = 100, and Conjecture 3.7 asserts γ{2}(G4,n) = 2n − ⌊(n − 9)/10⌋ for n ≥ 49.

Problem details · Reading note

The Mišta–Fiurášek qudit partial-estimation trade-off

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Ladislav Mišta Jr., Jaromír Fiurášek (2006). Optimal partial estimation of quantum states from several copies. DOI: 10.1103/PhysRevA.74.022316. URL: https://arxiv.org/abs/quant-ph/0606112.

Claim. Introduction, Eq. (2): the conjectured optimal output-fidelity versus estimation-fidelity trade-off for N identical pure qudits, with one output qudit.

Problem details · Reading note

Monkman–Sirker configurational entropy is Schur concave

Proved. Lean theorem result, frozen in this repository 2026-10-18.

D. Monkman; J. Sirker (2023). Symmetry-resolved entanglement: general considerations, calculation from correlation functions, and bounds for symmetry-protected topological phases. DOI: 10.48550/arXiv.2307.05820. URL: https://arxiv.org/abs/2307.05820.

Claim. Section 3.3 conjectures that the configurational entropy Sc = S − SN is Schur concave in the Bernoulli parameters Ck. Here S is the sum of binary entropies and SN is the Shannon entropy of their independent Bernoulli sum. The authors investigate M = 2 and M = 3.

Problem details · Reading note

A coherence lower bound on the magic of pure qutrits

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Chiranjib Mukhopadhyay; Sk Sazim; Arun Kumar Pati (2018). Coherence makes quantum systems 'magical'. DOI: 10.1088/1751-8121/aac8e8. URL: https://arxiv.org/abs/1801.04807v3.

Claim. For a qutrit density matrix rho with rho_12 = lambda_1 + i mu_1, rho_13 = lambda_2 + i mu_2, rho_23 = lambda_3 + i mu_3, the paper gives the nine entries of the discrete Wigner function and the sum negativity M_SN = sum |W_u| - 1 (Eq. (13)). From numerical simulation it states, for pure qutrit states, M_SN >= (C/2) sqrt(1 - C/2) with C the l1-norm coherence (Eq. (18)), and for qutrit-qubit states the trade-off 16 E_AB^2 + 9 M_SN_A^2 < 4 (Eq. (19)).

Problem details · Reading note

OEIS A400361: the exact second-order cumulative formula

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Ctibor O. Zizka; OEIS Foundation Inc. (2026). OEIS A400361 revision 17, the second-order Pillai minus floor-gcd conjecture. URL: https://oeis.org/A400361.

Claim. The unrestricted second-order cumulative formula with the literal digamma, totient, and zeta-derivative constant.

Problem details · Reading note

Refutation of Onnis's square convergence prohibition

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Luca Onnis (2022). On a variant of the happy numbers and their generalizations, arXiv:2203.03381v1. URL: https://arxiv.org/abs/2203.03381v1.

Claim. There are no perfect square that are term of the sequence with length of convergence equal to 3.

Problem details · Reading note

The most classical orthonormal basis of spin 3/2

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Marcin Rudziński; Adam Burchardt; Karol Życzkowski (2024). Orthonormal bases of extreme quantumness. DOI: 10.22331/q-2024-01-25-1234. URL: https://arxiv.org/abs/2306.00532v3.

Claim. For a spin-j state psi identified with a symmetric state F(psi) of 2j qubits by |j,m> -> |D_{2j,j-m}> (Eq. (8)), the t-anticoherence is A_t(psi) = ((t+1)/t)[1 - tr(rho_t(F(psi))^2)] with rho_t the t-qubit reduced state (Eqs. (12)-(14)), and the quantumness of an orthonormal basis given by the columns of a unitary U is B_t(U) = (1/N) sum_i A_t(psi_i) (Eq. (15)). The paper conjectures (Conjecture 1) that for N = 4 (j = 3/2) the matrix U_4^c of Eq. (21), with B_1 = 1/9, gives the most spin-coherent (classical) basis.

Problem details · Reading note

Explicit flat-spectrum probe probabilities

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Eddie Schoute, Dmitry Grinko, Yiğit Subaşı, Tyler Volkoff (2024). Quantum Programmable Reflections. DOI: 10.48550/arXiv.2411.03648. URL: https://arxiv.org/abs/2411.03648v2.

Claim. Section 4.2.1, Conjecture 4.6 asks for probe probabilities giving a flat spectrum for every copy number; equation (183) states the equivalent Clebsch–Gordan system.

Problem details · Reading note

Tang–Zhang: the sharp order −2 Schoenberg inequality

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Q. Tang, T. Zhang (2025). Sharp Schoenberg type inequalities and the de Bruin–Sharma problem. DOI: 10.48550/arXiv.2508.10341. URL: https://arxiv.org/abs/2508.10341v3.

Claim. Section 1.3, Conjecture 1.13 states the real-order reciprocal critical-point moment inequality and its equality criterion; Section 6, Conjecture 6.3 states the sharp order −2 critical-point inequality (6.9) and its collinear sufficient equality condition; Proposition 6.4 proves the n=2 case.

Problem details · Reading note

Tang–Zhang: real-order reciprocal critical-point moments

Refuted. Lean theorem result, frozen in this repository 2026-10-18.

Q. Tang, T. Zhang (2025). Sharp Schoenberg type inequalities and the de Bruin–Sharma problem. DOI: 10.48550/arXiv.2508.10341. URL: https://arxiv.org/abs/2508.10341v3.

Problem details · Reading note

Gurland's original logarithmic remainder

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Halina Wiśniewska (2025). Some inequalities for Gurland's ratio of the gamma functions. URL: https://arxiv.org/html/2512.07028v1.

Claim. Open Problem 1 asks whether the original modified-Gamma-ratio logarithmic remainder tends to zero for every positive real pair.

Problem details · Reading note

Every incoherent channel with coherence-breaking index at least two is amended by one unitary

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Long-Mei Yang; Bin Chen; Tao Li; Shao-Ming Fei; Zhi-Xi Wang (2018). Amending Coherence-Breaking Channels via Unitary Operations. DOI: 10.1007/s10773-018-3890-4. URL: https://arxiv.org/abs/1810.09642v1.

Claim. A coherence-breaking channel maps every state to an incoherent state, and the coherence-breaking index n(Φ) is the least k with Φ^k coherence-breaking; the paper amends qubit channels with n(Φ) = 1, 2 by Λα ◦ Φ and Φ ◦ Λα ◦ Φ for unitary Λα, and conjectures that every d-dimensional channel with n(Φ) ≥ 2 is amended by Φ ◦ Λα ◦ ⋯ ◦ Λα ◦ Φ with n copies of Φ.

Problem details · Reading note

The two-qubit non-selective measurement optimum N/2 + 1/4 is global

Proved. Lean theorem result, frozen in this repository 2026-10-18.

Margherita Zuppardo; Ray Ganardi; Marek Miller; Somshubhro Bandyopadhyay; Tomasz Paterek (2019). Entanglement gain in measurements with unknown results. DOI: 10.1103/PhysRevA.99.042319. URL: https://arxiv.org/abs/1806.05465v1.

Claim. A non-selective measurement in an orthonormal basis {|ψ_j⟩} maps a pure state |φ⟩ to ρ′ = Σ_j |⟨ψ_j|φ⟩|² |ψ_j⟩⟨ψ_j|; for two qubits and the measurement bases (28), the largest final negativity satisfies N^max_f − N_i = 1/4 − N_i/2 (Eq. (33)), and the authors leave open whether this is optimal among all measurement bases.

Problem details · Reading note

The fixed-polynomial hypergraph Alon-Tarsi bound

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

M. Anholcer, B. Bosek, G. Gutowski, M. Lasoń, J. Przybyło, O. Serra, M. Tuczyński, L. Vena and M. Zając (2026). Alon-Tarsi for Hypergraphs. DOI: 10.37236/15173. URL: https://arxiv.org/html/2501.00157v2.

Claim. Conjectures 4 and 28 give a fixed-polynomial Alon-Tarsi bound over arbitrary fields; one fully unbalanced characteristic-three multihypergraph refutes both.

Problem details · Reading note

The quantum crystal potential is not a Yangian combination on the bosonic states

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Thiago Araujo; Domenico Orlando; Susanne Reffert (2021). Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality. DOI: 10.1103/PhysRevD.103.026020. URL: https://arxiv.org/abs/2005.09103v2.

Claim. The quantum crystal melting Hamiltonian H = −J(Γ + Π) + V(√q O₊ + O₋/√q) acts on plane-partition states; a fermion-boson correspondence (46) defines bosonic states through symmetric-group characters, and Section VI conjectures that Γ, Π and O± are linear combinations of the affine Yangian Y[gl̂(1)] generators, with the bosonic states forming its MacMahon representation, so that O± lie in the Cartan part B.

Problem details · Reading note

A four-level system whose triplets all violate the three-level conditions, yet shows the Markovian Mpemba effect

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Ido Avitan; Roee Factor; David Gelbwaser-Klimovsky (2026). Necessary conditions for the Markovian Mpemba effect. URL: https://arxiv.org/abs/2603.04567v1.

Claim. For a three-level system with detailed-balance rates, the Markovian Mpemba effect requires condition (3) on the rates, one inequality for each value of k. For an N-level system, applying condition (3) to every triplet of levels with the original rates, the authors conjecture that if all triplets violate (3) then the system does not experience the Mpemba effect.

Problem details · Reading note

A non-semi-positive semi-nonnegative qutrit map can signal information backflow

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Ningping Cao; Maxwell Fitzsimmons; Zachary Mann; Rajesh Pereira; Raymond Laflamme (2023). Quantum Maps Between CPTP and HPTP. URL: https://arxiv.org/abs/2308.01894v1.

Claim. A Hermiticity-preserving trace-preserving map is semi-positive (SP) if some invertible density has an invertible density image, and semi-nonnegative (SN) if some density has a density image (Definitions 1–2); SN is equivalent to Ψ ∘ Φ = Ξ for CPTP Φ, Ξ (Theorem 2). The authors conjecture that non-SP SNTP maps still signal no information backflow in higher dimensions (Outlook).

Problem details · Reading note

Once-subdivided complete graphs have infinite edge multiset dimension

Proved. Lean theorem result, frozen in this repository 2026-10-17.

Mohammad Farhan, Sandi Klavžar, Dorota Kuziak, Ismael G. Yero (2026). Multiset resolvability parameters in graphs: A survey with new results and open problems. URL: https://arxiv.org/pdf/2607.10311v1.

Claim. Section 6.1 defines edge multiset resolution using minimum shortest endpoint distances; Problem 24 asks whether every once-subdivided complete graph of order at least three has infinite edge multiset dimension.

Problem details · Reading note

Weak-four metric dimension of complete graph direct squares

Proved. Lean theorem result, frozen in this repository 2026-10-17.

Mohammad Farhan, Dorota Kuziak, Ismael G. Yero (2026). The weak k-metric dimension of the direct product of complete graphs. DOI: 10.48550/arXiv.2605.22307. URL: https://arxiv.org/html/2605.22307v1.

Claim. Section 3 paragraph S3.p17.1 asks for the exact weak-four metric dimension of K_n times K_n for every n at least four; Section 6 first bullet asks whether Theorem 3.3 is sharp.

Problem details · Reading note

A partition strictly between group isomorphism and simultaneous power equivalence

Proved. Lean theorem finite_source_endpoint, frozen in this repository 2026-10-17.

Arthur Fernandes, Claudio Qureshi, Lucas Reis and Sávio Ribas (2026). On a class of combinatorial group invariants. DOI: 10.48550/arXiv.2609.20516. URL: https://arxiv.org/html/2609.20516v1.

Claim. Problem 6.2 asks for a partition strictly in one of three displayed intervals; the first is between group isomorphism and simultaneous integer-power equivalence.

Problem details · Reading note

The activity bound on cross-correlation asymmetry fails for general networks

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Jie Gu (2024). Thermodynamic bounds on the asymmetry of cross-correlations with dynamical activity and entropy production. DOI: 10.1103/PhysRevE.109.L042101. URL: https://arxiv.org/abs/2312.00354v2.

Claim. For a Markov jump process on N states, the normalized asymmetry of cross-correlation obeys |χ_ab| ≤ γ/tan(π/N) · √(σ/(2A)) (Eq. (8)), proved for unicyclic networks; the author conjectures Eq. (8) for arbitrary networks and proves the weaker Eq. (9) in general.

Problem details · Reading note

Square-root Jensen–Shannon balls need not admit majorization-minimizers

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Eric P. Hanson (2020). Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels. URL: https://arxiv.org/abs/2010.02408v1.

Claim. Total-variation balls in the probability simplex admit majorization-minimizers and -maximizers (Chapter 3, Eq. (3.1)), p-norm balls with 1 < p < ∞ do not (Section 3.7), and Chapter 8, Question 1 asks whether balls in the Hellinger distance, the Bures distance, the square-root Jensen–Shannon divergence, Wasserstein distances or other metrics admit majorization-minimizers.

Problem details · Reading note

The output maximal coherence of channels is not tensorization-stable

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Jaehak Lee; Kyunghyun Baek; Jiyong Park; Jaewan Kim; Hyunchul Nha (2022). Fundamental limits on concentrating and preserving tensorized quantum resources. DOI: 10.1103/PhysRevResearch.4.043070. URL: https://arxiv.org/abs/2104.12307v2.

Claim. With the maximal coherence η(ρ) = max_{i≠j} |ρ_ij|/√(ρ_ii ρ_jj) of Eq. (15) and the output maximal coherence η̃(Φ) = max_ρ η(Φ(ρ)) of Eq. (28), the authors conjecture the tensorization property η̃(Φ₁ ⊗ Φ₂) = max{η̃(Φ₁), η̃(Φ₂)} (Conjecture 1, Eq. (29)).

Problem details · Reading note

Sublattice dismantling of the priority lattice

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Adrián Lillo and Mercedes Rosas (2026). The Priority Lattice. DOI: 10.48550/arXiv.2603.28905. URL: https://arxiv.org/abs/2603.28905v1.

Claim. Section 6, item 3 asks for isomorphic principal intervals and conjectures left-factorial and quadratic counts; item 4 asks whether every priority lattice is sublattice dismantlable.

Problem details · Reading note

Glauber annealing from a thermal state need not preserve neighbourhood passivity

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Yutong Luo; Yi-Zheng Zhen; Xiangjing Liu; Daniel Ebler; Oscar Dahlsten (2023). Bound on annealing performance from stochastic thermodynamics, with application to simulated annealing. DOI: 10.1103/PhysRevE.108.054119. URL: https://arxiv.org/abs/2311.10424v1.

Claim. For simulated annealing with Glauber acceptance, P_ij = γ_i/(|N(j)|(γ_i + γ_j)) on neighbours (Eqs. (30)–(32)), the relaxation rate is µ = (E′_p − E_p)/(E′_γ − E_p) (Eq. (46)), and a partial swap model p′ = p + µ^PS(γ′ − p) has λ_2(P′^PS) = 1 − µ^PS (Eq. (43), Proposition 2). With τ_rel(P) = 1/(1 − λ_2(P)), the authors conjecture that τ_rel ≤ τ^PS_rel implies µ ≥ µ^PS (Eq. (50)). They also define the neighbourhood passivity P_i of Eq. (41), with Gibbs weights at the next temperature, and expect P_i ≥ 0 to be preserved along the annealing from a thermal state.

Problem details · Reading note

Two-qubit conditional domains need not be C*-subalgebras

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Arthur J. Parzygnat (2021). Conditional Distributions for Quantum Systems. DOI: 10.4204/EPTCS.343.1. URL: https://arxiv.org/abs/2102.01529v3.

Claim. For a density matrix ν on M_m ⊗ M_n with invertible marginals ρ and σ, the conditional domains B_ν = {B : [ρ, tr_B(ν(1 ⊗ B))] = 0} and A_ν = {A : [σ, tr_A((A ⊗ 1)ν)] = 0} are operator systems (Definition 5.5, Lemma 5.8); they need not be C*-subalgebras for m = n = 3, and the author suspects that they always are for 4 × 4 density matrices (footnote 13).

Problem details · Reading note

Unitary entanglement-assisted encoding beats every unassisted strategy in an energy-restricted four-message task

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Carles Roch i Carceller; Armin Tavakoli (2025). The role of entanglement in energy-restricted communication and randomness generation. DOI: 10.1088/2058-9565/ae4d80. URL: https://arxiv.org/abs/2510.27473v1.

Claim. In entanglement-assisted prepare-and-measure scenarios with an energy restriction ⟨0|τ_x^C|0⟩ ≥ 1 − ω on the message, unitary encodings of a shared pure state give no advantage over unassisted quantum communication for the probabilistic transmission of a bit (Result 4); whether unitary encoding remains resourceless for general tasks is left open.

Problem details · Reading note

The steady-state block of a Markov generator with a commuting symmetry need not admit a Markov form

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Lucas Sá; Pedro Ribeiro; Tomaž Prosen; Denis Bernard (2025). Symmetry Classes of Classical Stochastic Processes. DOI: 10.1007/s10955-025-03423-y. URL: https://arxiv.org/abs/2406.17955v2.

Claim. A Markov generator is a real matrix with nonnegative off-diagonal entries and zero column sums (Eqs. (2)–(4)). For a generator with a commuting symmetry, the authors conjecture that the irreducible block containing the invariant measure can be recast into Markov form (§2, after Eq. (5)), and prove this under assumption (A3) (Appendix A).

Problem details · Reading note

A disordered exclusion ring whose ε² current response vanishes although neither symmetry condition holds

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Issei Sakai; Takuma Akimoto (2026). Symmetry breaking of current response in disordered exclusion processes. DOI: 10.1103/21vm-n8gp. URL: https://arxiv.org/abs/2512.03316v2.

Claim. For the asymmetric simple exclusion process on a ring with quenched site- or bond-dependent rates p_i(ε) = (1 + ε) w_{i,i+1}/2 and q_i(ε) = (1 − ε) w_{i,i−1}/2, a uniform bond-bias ratio p_i(ε)/q_{i+1}(ε) makes the bias-reversal and particle-hole symmetries equivalent, and either a uniform ratio (condition (i)) or the inversion-correspondence condition (ii) makes the stationary current odd in ε. The authors conjecture that an ε² contribution to the current appears whenever neither condition holds.

Problem details · Reading note

A SIC-compound in dimension 8 from Hoggar SICs

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Armin Tavakoli; Ingemar Bengtsson; Nicolas Gisin; Joseph M. Renes (2020). Compounds of symmetric informationally complete measurements and their application in quantum key distribution. DOI: 10.1103/PhysRevResearch.2.043122. URL: https://arxiv.org/abs/2007.01007v1.

Claim. A SIC-compound is a set of d³ unit vectors in ℂ^d that splits into d SICs and into d² orthonormal bases; one exists for d = 2 and d = 4 and none for d = 3 with the Weyl–Heisenberg group, the introduction reports evidence against existence for d = 5, 6, 7, 8, and whether one can be formed from the exceptional (Hoggar) SICs of dimension 8 is left open.

Problem details · Reading note

Conjecture 2 of Tavakoli–Morelli: the Schmidt-number bound for equiangular measurements

Proved. Lean theorem result, frozen in this repository 2026-10-17.

Armin Tavakoli; Simon Morelli (2024). Enhanced Schmidt number criteria based on correlation trace norms. DOI: 10.1103/PhysRevA.110.062417. URL: https://arxiv.org/abs/2402.09972v2.

Problem details · Reading note

Conjecture 3 of Tavakoli–Morelli: the Schmidt-number bound for incomplete sets of MUBs

Proved. Lean theorem result, frozen in this repository 2026-10-17.

Armin Tavakoli; Simon Morelli (2024). Enhanced Schmidt number criteria based on correlation trace norms. DOI: 10.1103/PhysRevA.110.062417. URL: https://arxiv.org/abs/2402.09972v2.

Problem details · Reading note

The Eq. (10) set does not generate the 3 × 3 stochastic matrices

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Frederik vom Ende; Fereshte Shahbeigi (2024). Generating Sets of Stochastic Matrices. URL: https://arxiv.org/abs/2411.18946v2.

Claim. Theorem 3 asserts that the set G of Eq. (10), the swap of the first two levels together with the convex hull of the identity and three explicit matrices, generates the 3 × 3 column-stochastic matrices with G^20 = s(3); the authors conjecture that the bound N_G(s(3)) ≤ 20 is optimal.

Problem details · Reading note

The coherent discrimination bias is not a metric on quantum channels

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Mark M. Wilde (2020). Coherent Quantum Channel Discrimination. DOI: 10.1109/ISIT44484.2020.9174425. URL: https://arxiv.org/abs/2001.02668v1.

Claim. In coherent channel discrimination the prover's success probability p^coh_s(N⁰, N¹) is the supremum over pure inputs and prover unitaries of Eq. (17), independent of the isometric extensions; it satisfies 1/2 ≤ p^coh_s ≤ 1 (Proposition 2), and Section VI asks whether p^coh_s − 1/2 is a metric on quantum channels.

Problem details · Reading note

Logical Z operators with Z support on every stabilizer coset need not exist

Refuted. Lean theorem result, frozen in this repository 2026-10-17.

Yunzhe Zheng; Dong E. Liu (2025). From Magic State Distillation to Dynamical Systems. DOI: 10.22331/q-2025-09-15-1858. URL: https://arxiv.org/abs/2412.04402v2.

Claim. For a stabilizer code with stabilizer elements s_j and logical operators Z̄_i, if every s_j Z̄_i contains a Pauli Z tensor factor then z = 0 is a fixed plane of the distillation map (Proposition 3, Eq. (23)); the authors conjecture that this condition can be met for any stabilizer code by choosing the logical Z operators.

Problem details · Reading note

Negativity at most (χ − 1)/2 does not imply positivity under the χ-reduction map

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Jofre Abellanet-Vidal; Guillem Müller-Rigat; Albert Rico; Anna Sanpera (2026). Bounding the entanglement of a state from its spectrum. URL: https://arxiv.org/abs/2604.02420v2.

Claim. The negativity is N(ρ) = (||ρ^Γ||_1 − 1)/2 (Definition 2, Eq. (7)), and the reduction map is RED_κ(ρ) = Tr(ρ)1 − ρ/κ (Eq. (35)). Schmidt number at most χ implies N(ρ) ≤ (χ − 1)/2 (Lemma 3), and [1 ⊗ RED_κ](ρ) ≱ 0 implies Schmidt number greater than κ (Theorem 7). §5.4 leaves open whether NEGFS_{(χ−1)/2} ⊂ PREDFS_χ, and states that the inclusion is also unknown for the non-FS case.

Problem details · Reading note

Maximum purity in the 4-by-4 absolutely PPT sector

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Jennifer Ahiable; Naga Bhavya Teja Kothakonda; Andreas Winter (2026). The geometry of absolute separability and other convex matrix properties from spectrum. DOI: 10.48550/arXiv.2608.03390. URL: https://arxiv.org/abs/2608.03390v2.

Claim. Conjecture 6.7 states equality of APPT maximum purity and the inscribed polytope maximum, with the spectra of equation (44). Theorem 6.2 proves that the linear condition (31) implies absolute PPT; Section 7 asks to show that (31) is sufficient for absolute separability.

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Bonin's shadow Markoff positivity region

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Nathan Bonin (2024). Conditions of positivity on a shadow Markoff tree. DOI: 10.48550/arXiv.2403.07668. URL: https://arxiv.org/abs/2403.07668.

Claim. Conjecture 1 identifies the quadrilateral of initial shadows yielding nonnegative integer shadow Markoff trees.

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Log-negativity entangling capacity depends on the input resource for CPTP maps

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Earl T. Campbell (2010). Optimal entangling capacity of dynamical processes. DOI: 10.1103/PhysRevA.82.042314. URL: https://arxiv.org/abs/1007.1445.

Claim. Section VII asks whether log-negativity entangling capacity is resource independent for all unitaries or all CPTP maps.

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The eigenvalues of symmetric two-qubit entanglement witnesses satisfy µ3 ≥ −√(µ1µ2)

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Gabriel Champagne; Nathaniel Johnston; Mitchell MacDonald; Logan Pipes (2022). Spectral properties of symmetric quantum states and symmetric entanglement witnesses. DOI: 10.1016/j.laa.2022.05.004. URL: https://arxiv.org/abs/2108.10405v1.

Claim. A symmetric entanglement witness is a Hermitian W on Cd ∨ Cd with (v ⊗ v)*W(v ⊗ v) ≥ 0 for all v ∈ Cd (§2.3); positive semidefinite W are not excluded (footnote 1). For d = 2, Theorem 3 shows that real µ1 ≥ µ2 ≥ µ3 with µ2 ≥ 0 and µ3 ≥ −√(µ1µ2) are the eigenvalues of a (decomposable) symmetric witness, that every decomposable symmetric witness satisfies these two inequalities, and that every symmetric witness satisfies µ2 ≥ 0 and a weaker bound (c). The authors conjecture that µ2 ≥ 0 and µ3 ≥ −√(µ1µ2) characterize the eigenvalues of every symmetric witness in L(C2 ∨ C2) (after Theorem 3 and §6(c)).

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The naive orthostochastic equations are insufficient at order five

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Justin Chen; Papri Dey (2023). The 4 × 4 orthostochastic variety. DOI: 10.1080/10586458.2021.1982427. URL: https://arxiv.org/abs/2001.10691.

Claim. Question 5 asks whether the naive equations define the orthostochastic variety inside the Birkhoff polytope; Proposition 6 answers negatively for n ≥ 6, and Section 5 leaves n = 4, 5 open.

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The one-negative-eigenvalue bound for Hildebrand's APPT matrices fails in dimension four

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Matthew Chen; Natalie Klco (2026). Susceptibility of qudit-qudit entanglement to quantum noise: insights from the negativity rank. URL: https://arxiv.org/abs/2610.10820v1.

Claim. For the ordered eigenvalues λ1 ≥ … ≥ λ9 ≥ 0 of a qutrit-qutrit density matrix, ρ is absolutely PPT iff the two 3 × 3 Hildebrand matrices L1, L2 of Eq. (5) are positive semidefinite (Theorem II.1, citing Hildebrand), and L1, L2 have at most one negative eigenvalue (Theorem II.2). §V asks whether this limited rank available for inequality violation persists for the corresponding inequalities in qudit-qudit systems with d > 3.

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The whole positive coefficient anti-recurrence conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Robbert Fokkink and Gandhar Joshi (2025). Anti-Recurrence Sequences. DOI: 10.5281/zenodo.18305140. URL: https://arxiv.org/abs/2506.13337v1.

Claim. Conjecture 4 in arXiv v1 and Conjecture 2 in INTEGERS 26, A24: for every positive integral vector of dimension k > 1, the exact discrepancy A_n-(k*tau+1)n is tau-automatic, where tau is the coefficient sum.

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The fixed-support Davenport limit of integer boxes

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Benjamin Girard and Alain Plagne (2026). The Davenport constant of low-dimensional boxes. DOI: 10.48550/arXiv.2610.16562. URL: https://arxiv.org/abs/2610.16562v1.

Claim. Section 2 asks whether the fixed-support box Davenport ratio has a limit in every positive dimension; Theorem 5 gives the three-dimensional asymptotic constant 16.

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The tentative Hart–Rowley prefix/suffix equality

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Sarah B. Hart and Peter J. Rowley (2025). A note on involution prefixes in Coxeter groups. DOI: 10.48550/arXiv.2502.00777. URL: https://arxiv.org/abs/2502.00777v1.

Claim. Section 3, page 5 tentatively suggests equality of the greedy prefix and suffix involution lengths of every element.

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Conditional hockey-stick contraction coefficients reduce to orthogonal states and do not depend on γ

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Christoph Hirche; Ian George; Theshani Nuradha; Mark M. Wilde (2026). Conditional contraction coefficients and their applications to quantum networks. URL: https://arxiv.org/abs/2608.27171v1.

Claim. For a quantum channel N and a generalized divergence D, the conditional contraction coefficient is η^Δ_D(N) := sup over |R| and ρ_RA ≠ σ_RA of [D((id ⊗ N)(ρ_RA)‖(id ⊗ N)(σ_RA)) − D(ρ_R‖σ_R)] / [D(ρ_RA‖σ_RA) − D(ρ_R‖σ_R)] (Eq. (2.14)), optimized over pairs with a strictly positive finite denominator (Remark 2). The hockey-stick divergence is E_γ(ρ‖σ) := Tr(ρ − γσ)_+ (Eq. (5.1)). For the trace distance (γ = 1), η^c_Tr(N) and η^Δ_Tr(N) are optimized by orthogonal states (Proposition 8). §9 asks whether, for the hockey-stick divergence, the optimization can be restricted to orthogonal states, which would imply monotonicity of the coefficient in γ.

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Same-phase periodic SSH ground states can have positive averaged excess pseudo entropy

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Pramod Kamal Kharel; Manghang Limbu; Nabaraj Khatri; Ashish Khanal; Kiran Adhikari (2026). Pseudo entropy and topological phases of matter. URL: https://arxiv.org/abs/2606.29311v1.

Claim. For pure states |ψ⟩, |φ⟩ with ⟨φ|ψ⟩ ≠ 0, the transition matrix is T = |ψ⟩⟨φ|/⟨φ|ψ⟩, the reduced transition matrix is T_A = Tr_B T, and the pseudo entropy is S(T_A) = −Tr(T_A log T_A) (Eqs. (1)–(3)); the averaged excess entropy is ∆S = Re S(T_A) − (S(ρ_A^ψ) + S(ρ_A^φ))/2 (Eq. (21)). For half-filled ground states of the periodic SSH Hamiltonian (Eq. (15)) at w = 1 in the same phase, the authors report that the conjectured non-positivity ∆S ≤ 0 holds and state in §VI that under periodic boundary conditions it is fully satisfied with no exceptions at all system sizes.

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Mobile general position on the seven-cycle and ten-cycle cylinders

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Sandi Klavžar; Aditi Krishnakumar; Dorota Kuziak; Ethan Shallcross; James Tuite; Ismael G. Yero (2026). Moving through Cartesian products, coronas and joins in general position. DOI: 10.1016/j.dam.2025.10.041. URL: https://arxiv.org/abs/2505.00535.

Claim. Section 3, after Proposition 3.7, conjectures that the mobile general position numbers of C₇ □ Pₛ and C₁₀ □ Pₛ are four for every s ≥ 5.

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A four-state qubit ensemble where an adaptive post-measurement query beats a pre-measurement query

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Hanwool Lee; Teiko Heinosaari (2026). Perfect Discrimination of Non-Orthogonal Quantum States via Adaptive Post-Measurement Queries. URL: https://arxiv.org/abs/2608.01167v1.

Claim. In state discrimination with a one-bit query, the pre-measurement success probability is P_pre = max over binary partitions and subensemble measurements (Eq. (3)), and the adaptive post-measurement success probability is P^adap_post = max over a measurement and outcome-dependent partitions (Eq. (5)); P^adap_post = (n − 1) P_aux for the auxiliary ensemble of pairwise mixtures (Theorem 1). §VI asks whether an ensemble of four qubit states with P^adap_post > P_pre exists, and suggests that this may never occur for qubit ensembles with n ≥ 4, leaving a proof open.

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The spacetime density matrix violates the Audenaert-type Schatten subadditivity

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Alexey Milekhin; Zofia Adamska; John Preskill (2025). Observable and computable entanglement in time. URL: https://arxiv.org/abs/2502.12240v2.

Claim. The spacetime density matrix T_AB of subsystems A at time 0 and B at time t is defined by Tr(T_AB (O_A ⊗ 1_B)(1_A ⊗ O_B)) = ⟨O_A(0) O_B(t)⟩ = Tr(ρ O_A U† O_B U) (Eqs. (4), (9)); its partial traces are the reduced density matrices (Eq. (15)). Conjecture 1 asserts the analogue of Audenaert's inequality, 1 + ||T_AB||_p ≥ ||Tr_A T_AB||_p + ||Tr_B T_AB||_p for every finite-dimensional T_AB and p > 1 (Eq. (25)), with the Schatten norm ||M||_p = (Tr (M†M)^{p/2})^{1/p} (footnote a).

Problem details · Reading note

Novaes's strong-absorption cumulant conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-16.

M. Novaes (2023). Time delay statistics for chaotic cavities with absorption. DOI: 10.1007/s10955-023-03183-7. URL: https://arxiv.org/abs/2305.18235v1.

Claim. Section 5: Eqs. (54) and (55), strong-absorption moment expansions for every positive channel count and order; Eq. (58), the first two cumulant coefficients for every M ≥ 1 and n > 1.

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Direct four-copy universal entanglement concentration beats the bound of Eq. (8)

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Alexandre C. Orthey; Aby Philip; Tulja Varun Kondra; Alexander Streltsov (2026). Optimality of universal conclusive entanglement concentration protocols. DOI: 10.1103/bz3j-9njg. URL: https://arxiv.org/abs/2509.09423v2.

Claim. Definition 1: an n-to-1 LOCC map E1 is a universal conclusive entanglement concentration protocol if ⟨ϕ+|E1(ρ⊗n)|ϕ+⟩ = Tr E1(ρ⊗n) for every ρ in the input set. For four copies of an arbitrary two-qubit pure state c1|00⟩ + c2|01⟩ + c3|10⟩ + c4|11⟩ (Eq. (1)), Theorem 3 bounds the success probability of concatenated protocols based on two-qubit operations by 2|c2⁴c3⁴| + 2|c1⁴c4⁴| − 4Re[c1²c4²(c2*)²(c3*)²] (Eq. (8)), and §VIII conjectures that protocols processing four copies directly cannot surpass this bound.

Problem details · Reading note

Relative entropy coherence is bounded by l1 coherence

Proved. Lean theorem result, frozen in this repository 2026-10-16.

Swapan Rana, Preeti Parashar, Maciej Lewenstein (2016). Trace-distance measure of coherence. DOI: 10.1103/PhysRevA.93.012110. URL: https://arxiv.org/abs/1511.01854v2.

Claim. Conjecture 7. For all states ρ, C_{l1}(ρ) ≥ C_r(ρ).

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Noisy entanglement curvature can exceed quantum Fisher information

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Zain H. Saleem; Anil Shaji; Anjala M. Babu; Da-Wei Luo; Quinn Langfitt; Ting Yu; Stephen K. Gray (2025). Quantum Fisher Information and the Curvature of Entanglement. DOI: 10.48550/arXiv.2504.13729. URL: https://arxiv.org/abs/2504.13729.

Claim. Section IV.B conjectures CoE ≤ QFI for the amplitude-damped two-qubit evolution (41), with equality at specific points.

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Formation is not constant on the Song–Chen–Đoković subspace

Refuted. Lean theorem result, frozen in this repository 2026-10-16.

Zhiwei Song, Lin Chen and Dragomir Z. Djokovic (2024). The existence of distinguishable bases in three-dimensional subspaces of qutrit-qudit systems under one-way local operations and classical communication. DOI: 10.1109/TIT.2024.3435412. URL: https://arxiv.org/abs/2307.01980v3.

Claim. Eqs. (2)–(3) define a qutrit-qudit subspace; p. 3 asks whether every mixed state supported there has the EOF of the pure state |Ψ₁⟩.

Problem details · Reading note

Antipin's four-qubit subspaces are not genuinely entangled for every parameter

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

K. V. Antipin (2021). Construction of genuinely multipartite entangled subspaces and the associated bounds on entanglement measures for mixed states. DOI: 10.1088/1751-8121/ac37e5. URL: https://arxiv.org/abs/2104.09664v1.

Claim. §III.B.2 constructs a four-qubit subspace from the seven vectors (35a)–(35g), √λ_i|p_i⟩⊗|q_i⟩ + √(1 − λ_i)|p′_i⟩⊗|q′_i⟩ in ℂ⁴ ⊗ ℂ⁴ with 0 < λ_i < 1, by the substitutions (39) for the first and (40) for the second ququart into two qubits each; with every λ_i = 1/2 the span is stated to be a genuinely entangled subspace, and the paper conjectures that eqs. (35) together with eqs. (39) and (40) determine a GES for any set of values of λ_i.

Problem details · Reading note

Ayyer–Mallick maximal degeneracy of the annihilation-process generator

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

A. Ayyer and K. Mallick (2010). Exact results for an asymmetric annihilation process with open boundaries. DOI: 10.1088/1751-8113/43/4/045003. URL: https://arxiv.org/abs/0910.0693.

Claim. After Eq. (79): each eigenvalue seems to have exactly one eigenvector independent of the number of times the eigenvalue appears.

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Without shared entanglement the odd-prime communication task succeeds with probability at most 2/(d + 1)

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Pharnam Bakhshinezhad, Mohammad Mehboudi, Carles Roch i Carceller, Armin Tavakoli (2024). Scalable entanglement certification via quantum communication. DOI: 10.1103/PRXQuantum.5.020319. URL: https://arxiv.org/abs/2401.00796v2.

Claim. In a symmetric prepare-and-measure scenario where Alice and Bob each encode x, y ∈ {0, …, d − 1}² into d-dimensional messages and Charlie, with input z ∈ {0, …, d} and output c, may measure jointly, the average success probability R_d of computing w_z = x₁ + y₁ − 2z(x₀ − y₀) (z ≠ d) and w_d = x₀ − y₀ mod d (Eqs. (27)–(28)) is 1 with shared entanglement; the authors conjecture that without shared entanglement the largest value is R_d = 2/(d + 1) for every odd prime d (Conjecture 1, Eq. (29)).

Problem details · Reading note

Exact quantum bound of the Bernards–Gühne nonlocality-depth family

Proved. Lean theorem result, frozen in this repository 2026-10-15.

F. Bernards; O. Gühne (2023). Bell inequalities for nonlocality depth. DOI: 10.1103/PhysRevA.107.022412. URL: https://arxiv.org/abs/2205.04250v2.

Claim. For (k,m) models with m > 2, the classical bound F_n <= n 2^(n-2) is conjectured and proving it or finding a counterexample remains open.

Problem details · Reading note

Optimal calibrated visibility for joint quantum work measurements

Proved. Lean theorem result, frozen in this repository 2026-10-15.

K. Beyer; R. Uola; K. Luoma; W. T. Strunz (2022). Joint measurability in nonequilibrium quantum thermodynamics. DOI: 10.1103/PhysRevE.106.L022101. URL: https://arxiv.org/abs/2111.02854.

Claim. The ansatz for the effects W^U_ab in Eq. (6) is a priori not the most general one [54]. However, numerical simulations suggest that our visibility bounds are tight and we conjecture that our construction is optimal to satisfy condition (ii) for the given model Eq. (5)

Problem details · Reading note

The threshold prior of the G-test is O/(M + O)

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Harry Buhrman; Dmitry Grinko; Philip Verduyn Lunel; Jordi Weggemans (2026). Permutation tests for quantum state identity. URL: https://arxiv.org/abs/2405.09626v2.

Claim. For a subgroup G of S_n the G-test measures Π_G = (1/|G|) Σ_{π∈G} ψ(π) (Eqs. (61)–(62)); it has completeness 1 (Eq. (63)) and soundness P^G_s(µ) = 1 − (1/(n choose µ)) Σ_λ K_{λ,µ} r^G_λ (Eq. (65)). For G = S_n, Theorem 5 gives the threshold prior p∗(µ) = 1/(1 + (n choose µ)) below which the trivial test is optimal. §6 leaves open the threshold prior p∗(G, µ) below which the trivial test outperforms the G-test.

Problem details · Reading note

A maximally entangled (3, 3, 7) state of length 9 refutes the length formula Σ d_i − 2

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Roman V. Buniy, Robert P. Feger, Thomas W. Kephart (2024). Decoherence and the classes of maximally entangled states. DOI: 10.1142/S0219749924500357. URL: https://arxiv.org/abs/2210.07618v1.

Claim. A maximally entangled state of an n-partite system is a state at which every algebraic entanglement invariant (kernel dimensions of the flattenings and contraction maps built from its coordinates; for n = 3 the four invariants of Appendix A) takes its least value, and these states form the maximally entangled class C_D. The length of a state is its number of nonzero coordinates in a product basis, and L_D is the least length in C_D. From exact values for 3-partite dimensions up to 5 (Table XII) the authors conjecture L_(d₁,d₂,d₃) = d₁ + d₂ + d₃ − 2 when min(d₁, d₂, d₃) ≥ 3 (Conjecture III.4).

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No randomisation-consistent time reversal on the full channel set

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Giulio Chiribella and Erik Aurell and Karol Życzkowski (2021). Symmetries of quantum evolutions. DOI: 10.1103/PhysRevResearch.3.033028. URL: https://arxiv.org/abs/2101.04962.

Claim. Section V.B conjectures that no symmetry of the full set of quantum channels reverses every unitary channel, without assuming reversal of composition order.

Problem details · Reading note

The lower boundary of the entropy region of qutrit channels and their complements is the conjectured curve

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Jakub Czartowski; Daniel Braun; Karol Życzkowski (2019). Trade-off relations for operation entropy of complementary quantum channels. DOI: 10.1142/S0219749919500461. URL: https://arxiv.org/abs/1908.03492v1.

Claim. The map entropy S_map(Φ) of a channel is the von Neumann entropy of its normalized Choi–Jamiołkowski state, and S_map(Φ) = S(Φ̃(ρ*)), S_map(Φ̃) = S(Φ(ρ*)) for the complementary channel Φ̃ and ρ* = I/N (Proposition 1). For qutrit channels with a qutrit environment the authors conjecture that the lower boundary of the allowed set A_3 of pairs (S_map(Φ), S_map(Φ̃)) is the curve p(a) of Eq. (20), a ∈ (0, 1/3), together with its reflection through S = S̃.

Problem details · Reading note

The whole finite-cyclic SGB extension

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Shrabani Das; Ahmad Erfanian; Rajat Kanti Nath (2026). Certain topological indices and spectral properties of SGB-graphs of finite cyclic groups. DOI: 10.48550/arXiv.2602.07587. URL: https://arxiv.org/abs/2602.07587v1.

Claim. Section 5 closing extension of the complete (a)-(c) list to any cyclic group.

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Symmetric GHZ resources for complete symmetric Boolean functions

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Michael de Oliveira, Luís Soares Barbosa, Ernesto F. Galvão (2024). Quantum advantage in temporally flat measurement-based quantum computation. DOI: 10.22331/q-2024-04-09-1312. URL: https://arxiv.org/abs/2212.03668.

Claim. Section 4.2, Conjecture 2 asks for an asymptotic lower bound Ω(n^{⌊k/2⌋−1}) on the GHZ resource for deterministic evaluation of the complete symmetric function of constant degree k with symmetric measurement assignments.

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The Geiger–Kedem spin-1 spin-entropy depends on the relative phases

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Davi Geiger; Zvi M. Kedem (2022). Spin Entropy. DOI: 10.3390/e24091292. URL: https://arxiv.org/abs/2111.11605v3.

Claim. The spin-entropy of a pure spin state is the Shannon entropy of the S_z probabilities plus the differential entropy of the polarization-angle density ρ_s(φ) = |λ_s(φ)|² (Eq. (4)). For spin 1, with the state (15) and the density ρ_1 of Proposition 2, the authors conjecture (Conjecture 1) that the spin-entropy depends only on the variables θ_α, θ_β of the z-axis component, so that Eq. (17) simplifies to its zero-phase form Eq. (18).

Problem details · Reading note

The exact critical concurrence for universal teleportation through qubit channels

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Arkaprabha Ghosal, Debarshi Das, Subhashish Banerjee (2021). Characterizing qubit channels in the context of quantum teleportation. DOI: 10.1103/PhysRevA.103.052422. URL: https://arxiv.org/abs/2102.02054v2.

Claim. When Alice sends one half of |Ψa⟩ = √a|00⟩ + √(1−a)|11⟩ (1/2 < a < 1) to Bob through a qubit channel, the final state is useful for universal quantum teleportation (|t11| = |t22| = |t33| > 1/3 in the canonical form) for unital channels only if the concurrence exceeds 1/2; for non-unital channels the authors conjecture a critical concurrence 0 < C_Critical ≤ 0.41 (Conjecture 1) and state that they could not determine its precise value.

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Compatible quantum channels need not be Jordan compatible

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Mark Girard; Martin Plávala; Jamie Sikora (2021). Jordan products of quantum channels and their compatibility. DOI: 10.1038/s41467-021-22275-0. URL: https://arxiv.org/abs/2009.03279v1.

Claim. Two channels Φ_1, Φ_2 are Jordan compatible if some Hermitian A on X ⊗ X_1 ⊗ X_2 with Tr_{X_1}(A) = Tr_{X_2}(A) = J(1_{L(X)}) makes the generalized Jordan product Φ_1 ⊙_A Φ_2, whose Choi matrix is (1 ⊗ Φ_1 ⊗ Φ_2)(A), completely positive (Definitions 25 and 27). Jordan compatible channels are compatible (Proposition 26), and compatible invertible channels are Jordan compatible (Theorem 30). The authors conjecture that no pair of compatible channels fails to be Jordan compatible.

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ColorSym optimality is false

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Jean-Christophe Godin; Rémi Grisot; Olivier Togni (2025). On List Coloring with Separation of the Complete Graph and Set System Intersections. DOI: 10.5614/ejgta.2025.13.1.5. URL: https://arxiv.org/abs/2209.03436.

Claim. Section 3.1 conjectures that Algorithm ColorSym produces an (L, b)-coloring whenever K_n is (L, b)-colorable.

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An unsharp qubit Lüders image violates ambient Sob sum closure

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Stanley Gudder (2023). Dual Quantum Instruments and Sub-observables. URL: https://arxiv.org/abs/2302.12243v1.

Claim. Section 3 conjectures that an instrument's image of effects need not be a Sob effect algebra and asks whether an arbitrary Luders instrument has this property.

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The polar rank-one encoder of Li–Jiang is not exactly optimal at fixed noise

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Bikun Li; Liang Jiang (2026). High-Rank Encoding Can Improve Approximate Quantum Error Correction. URL: https://arxiv.org/abs/2609.00778v1.

Claim. Exact optimality of this pair at fixed p > 0 remains open.

Problem details · Reading note

A shorter relaxation time does not imply a faster energy relaxation in Glauber annealing

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Problem details · Reading note

Mitra's instrument-independent unsharpness measure is not monotone under white noise for qubit observables

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Arindam Mitra (2022). Quantifying unsharpness of observables in an outcome-independent way. DOI: 10.1007/s10773-022-05219-2. URL: https://arxiv.org/abs/2201.02578v3.

Claim. For an observable A = {A_i}, the X-matrix is X^A = Σ_i ‖A_i‖A_i and the instrument-independent unsharpness measures are E(A) = ‖I − X^A‖ and E′(A) = Tr[(I/d)(I − X^A)]. Theorems 3–4 show that E and E′ are non-decreasing under the white noise A^λ_i = λA_i + (1 − λ)I/n exactly when Σ^A_1 ≥ Σ^A_2 and Σ′^A_1 ≥ Σ′^A_2, respectively; Propositions 1–2 prove both for dichotomic qubit observables, and Conjecture 1 asserts both for every qubit observable.

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Finite Gilbreath extension component maximum

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Leila Muney (2026). Holes in Valid-Extension Sets of Finite Gilbreath Sequences. URL: https://arxiv.org/html/2606.23721v2.

Claim. Section 16(4) asks whether the maximum number of maximal step-two runs in K_S is 2^(n-4) for every n at least five.

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The cell-order dynamics of two-dimensional granular systems can have two physical steady states

Refuted. Lean theorem result, frozen in this repository 2026-10-15.

Alex D. C. Myhill, Raphael Blumenfeld (2023). Steady states of two-dimensional granular systems are unique, stable, and sometimes satisfy detailed balance. DOI: 10.1088/1751-8121/ace56d. URL: https://arxiv.org/abs/2306.10526v2.

Claim. For the cell-order dynamics of two-dimensional granular systems (Eq. (1)), with fluxes η_{i,j} = p_{i,j}Q_iQ_j − q_{i,j}Q_{i+j−2} and steady-state equations Eq. (2), the authors prove that a detailed-balance steady state is unique when it exists, and that for C = 6, 7 no other steady state exists when one does. From numerical solution of 614400 systems up to C = 8 they conjecture that for any choice of constant rates there is only one physical (real and positive) steady state regardless of the highest cell order C.

Problem details · Reading note

For two qubits the SVD entanglement entropy is at least the smaller of the two entanglement entropies

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Arthur J. Parzygnat, Tadashi Takayanagi, Yusuke Taki, Zixia Wei (2023). SVD entanglement entropy. DOI: 10.1007/JHEP12(2023)123. URL: https://arxiv.org/abs/2307.06531v2.

Claim. For two states the SVD entanglement entropy is the von Neumann entropy of the normalized absolute value √(τ†τ)/Tr √(τ†τ) of the reduced transition matrix τ = Tr_B|ψ¹⟩⟨ψ²|/⟨ψ²|ψ¹⟩; it lies between 0 and log d_A. The authors give numerical evidence that the lower bound min{S(ρ¹_A), S(ρ²_A)} might hold for qubits (failing for qudits), state that rigorous proofs for qubits are lacking, and prove it only when the second state is a local unitary image of the first (Eq. (3.37)), a restricted version of Eq. (3.38).

Problem details · Reading note

The coefficient system of Quesne's Kepler–Coulomb extensions on the sphere has the conjectured closed forms for every m

Proved. Lean theorem result, frozen in this repository 2026-10-15.

C. Quesne (2023). Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere. DOI: 10.1016/j.aop.2023.169265. URL: https://arxiv.org/abs/2206.01033v3.

Claim. For the generating-function construction of quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere, the m + 1 coefficients a_0, …, a_m solve the linear system (4.13)–(4.17). The author conjectures that for every m > 1 the solution has the closed forms (4.19)–(4.24), with a common denominator Q^{2µ} + b_1κQ^{2µ−2} + ⋯ + b_µκ^µ and positive constants b_j and c_j^{(k)}. The conjecture is confirmed for m ≤ 7, and proving it by finding the constants is stated as open.

Problem details · Reading note

Schur-channel diamond-distance equality

Proved. Lean theorem result, frozen in this repository 2026-10-15.

Nengkun Yu, Runyao Duan and Quanhua Xu (2017). Bounds on the distance between a unital quantum channel and the convex hull of unitary channels. DOI: 10.1109/TIT.2016.2641447. URL: https://arxiv.org/abs/1201.1172.

Claim. Section V, Eq. (8), asks whether every Schur channel has equal diamond distances to all mixed-unitary channels and to Schur mixed-unitary channels.

Problem details · Reading note

Zero forcing the ordinary powers of equal-legged spiders

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Aida Abiad; Mary Flagg; Sina Ghasemi Nezhad; Sjanne Zeijlemaker (2026). The Zero Forcing Number of Graph Powers. DOI: 10.48550/arXiv.2610.03892. URL: https://arxiv.org/abs/2610.03892v1.

Claim. Conjecture 12 asserts Z((P_b Cartesian P_ell)^k)=b*k for every b>=1, k>=1 and ell>=k+1, without a b<=ell condition. Section 3, Conjecture 10 asks whether Z(S_m(ell)^k)=(m-1)k for m>=2, k>=1, ell>=k+1.

Problem details · Reading note

The Grassmann entwining Yang–Baxter map S over Γ(3) does not preserve the measure y₁₁⁻³

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

P. Adamopoulou, G. Papamikos (2025). Entwining Yang–Baxter maps over Grassmann algebras. DOI: 10.1016/j.physd.2024.134469. URL: https://arxiv.org/abs/2311.18673v1.

Claim. Over the Grassmann algebra Γ(n) the authors construct birational parametric entwining Yang–Baxter maps R_{a,b}, S_{a,b}, T_{a,b} from refactorisations of Lax supermatrices of a Grassmann NLS equation (Eqs. (17)–(19)), and expand them into 2^{n+2}-dimensional maps with commutative variables. They show for n = 1 and n = 2 that S_{a,b} preserves the measure m = y₁^{−1} and m = y₁₁^{−2} (the Jacobian determinant equals m(z)/m(S(z))), and conjecture that in Γ(n) R_{a,b} is volume preserving while S_{a,b} and T_{a,b} preserve measures of the form y₁₁^{−n} and x₁₁^{−n}.

Problem details · Reading note

Positional autocorrelation does not determine finite deletion-channel entropy ordering

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Claim. Conjecture IV.4 proposes alternating words as autocorrelation minimizers; Section V asks whether positional autocorrelation determines finite deletion-channel entropy ordering.

Problem details · Reading note

The peakon transfer matrices expand in the Hamiltonians h^(k) with explicit universal coefficients

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Jean Avan, Luc Frappat, Eric Ragoucy (2026). Quadratic Poisson brackets for the Camassa–Holm peakons. DOI: 10.1088/1751-8121/ae7683. URL: https://arxiv.org/abs/2512.11066v3.

Claim. For the deformed Camassa–Holm peakon Lax matrix L(z) = S + Λ(z)T + ξ(z)A, the transfer matrices τ_m(z) = tr L(z)^m are polynomials in Λ(z) of degree m without constant term, with degree-one term h^(m) = m tr((T + γA/2)(S + A)^(m−1)) (Proposition 5.1). Conjecture 5.2 asserts that τ_m(z) expands in products h^(k_1)⋯h^(k_j) over partitions of m with coefficients a_j(k), a_1(m) = 1, and the values (5.21); the other coefficients are left undetermined.

Problem details · Reading note

Four MUBs detect a PPT state in dimension eight

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

J. Bae; A. Bera; D. Chruściński; B. C. Hiesmayr; D. McNulty (2022). How many mutually unbiased bases are needed to detect bound entangled states?. DOI: 10.1088/1751-8121/acaa16. URL: https://arxiv.org/abs/2108.01109.

Claim. Equation (4) defines the MUB entanglement witness; Conjecture 1 predicts that the minimal number of MUBs for bound entanglement detection in dimension d = 2^r, r > 1, is d/2 + 1.

Problem details · Reading note

The repetition-count variance of the noisy voter model

Proved. Lean theorem result, frozen in this repository 2026-10-14.

I. Benjamini, H. Helman Tov and M. Zhukovskii (2023). Global information from local observations of the noisy voter model on a graph. DOI: 10.1214/23-AOP1637. URL: https://arxiv.org/abs/2207.01224.

Claim. May Var(ζ₁ + … + ζ_t) be sublinear? We do not have an answer. However, we prove that it is either linear in t or O(ln t).

Problem details · Reading note

Campbell’s classification of irreducible reciprocal pairs is false

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Kirsty Campbell (2020). Group-Graph Reciprocal Pairs. DOI: 10.48550/arXiv.2005.13566. URL: https://arxiv.org/abs/2005.13566.

Claim. The Conjecture on page 3 classifies irreducible group-graph reciprocal pairs as the two trivial cases, the four-cycle pair, or the k-star pairs of Theorem 1.

Problem details · Reading note

A product decomposition of the generalized CHSH groups

Proved. Lean theorem result, frozen in this repository 2026-10-14.

D. Cui; A. Mehta; H. Mousavi; S. S. Nezhadi (2020). A generalization of CHSH and the algebraic structure of optimal strategies. DOI: 10.22331/q-2020-10-21-346. URL: https://arxiv.org/abs/1911.01593.

Claim. Characterize the groups G_n in terms of direct and semidirect products of small well-known groups (Section 1.4, item 2).

Problem details · Reading note

A full-rank qutrit refutes the purification lower bound on Krylov operator complexity

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Rathindra Nath Das, Takato Mori (2026). Krylov complexity of purification. DOI: 10.1103/qgcx-wxpd. URL: https://arxiv.org/abs/2408.00826v4.

Claim. For a density matrix ρ evolving as ρ(t) = U_t ρ U_t† with U_t = e^{−iHt}, the authors conjecture (Eq. (4)) that the Krylov operator complexity of the mixed state is bounded below by the spread complexity of the time-dependent purification |Ψ^{U*}_ρ(t)⟩ = (U_t ⊗ U_t^*)|Ψ_ρ⟩ and above by the operator complexity of the time-independent purification: C_S(|Ψ^{U*}_ρ(t)⟩) ≤ C_K(ρ(t)) ≤ C_K(|Ψ^I_ρ(t)⟩), with equality in the pure-state limit; they confirm it analytically for one-qubit states.

Problem details · Reading note

The minimum average trace distance of a rebit strategy for Elliott's exclusion task is exactly 1/12

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Thomas J. Elliott (2025). Strict advantage of complex quantum theory in a communication task. DOI: 10.1103/PhysRevA.111.062401. URL: https://arxiv.org/abs/2505.16523v2.

Claim. A four-input communication task admits a complex-qubit strategy but no real-qubit strategy; non-real Bargmann invariants witness the ensemble's basis-independent imaginarity. Appendix D shows that a rebit strategy reaches average trace distance d = 1/12 from the target distribution and weakly conjectures that 1/12 is the minimum.

Problem details · Reading note

Periodicity of generalized pseudostandard words

Proved. Lean theorem result, frozen in this repository 2026-10-14.

J. Florian, Ľ. Dvořáková (2016). On periodicity of generalized pseudostandard words. DOI: 10.37236/4771. URL: https://arxiv.org/abs/1508.02020.

Claim. Conjecture 29 characterizes periodic d-ary generalized pseudostandard words by two conditions on their normalized directive tails.

Problem details · Reading note

fokkink-rust-2024-splythoff-grundy-rows

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Robbert Fokkink; Dan Rust (2024). Queen reflections: a modification of Wythoff Nim. DOI: 10.1007/s00182-022-00824-1. URL: https://arxiv.org/abs/1904.08339.

Claim. Section 9 asks whether every Splythoff Nim row and column contains each Sprague-Grundy value exactly once.

Problem details · Reading note

The unital-block overlap lower bound fails for qutrit channels

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Matthew Girling, Cristina Cîrstoiu, and David Jennings (2022). Estimation of correlations and non-separability in quantum channels via unitarity benchmarking. DOI: 10.1103/PhysRevResearch.4.023041. URL: https://arxiv.org/abs/2104.04352.

Claim. Appendix C.5 conjectures that the unital-block overlap of any two quantum channels is at least minus one; two qutrit coarse-graining channels have overlap minus four thirds.

Problem details · Reading note

A stochastic qutrit channel that is not unital

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Matthew A. Graydon and Joshua Skanes-Norman and Joel J. Wallman (2022). Designing Stochastic Channels. URL: https://arxiv.org/abs/2201.07156v1.

Claim. Section II asks whether all stochastic channels are unital for d = 3; Eq. (11) defines stochasticity by a positive identity-vector eigenvalue of the normalized Choi state, and Appendix A proves qubit unitality.

Problem details · Reading note

Gerzon equality fixes the equiangular parameters in every dimension at least two

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Gary R. W. Greaves, Joseph W. Iverson, John Jasper, Dustin G. Mixon (2021). Frames over finite fields: Equiangular lines in orthogonal geometry. DOI: 10.48550/arXiv.2012.13642. URL: https://arxiv.org/abs/2012.13642v2.

Claim. Definition 2.8 defines equiangular systems; Propositions 2.9 and 2.10 give Gerzon equality and the ETF parameter relation; Example 5.8 asks whether the dimension congruence restrictions on the parameter identity can be removed.

Problem details · Reading note

The purity of a state under loss is not always log-convex in the transmission

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Anaelle Hertz, Noah Lupu-Gladstein, Khabat Heshami, Aaron Z. Goldberg (2026). Equalities and inequalities from entanglement, loss, and beam splitters. DOI: 10.1007/s40509-026-00402-0. URL: https://arxiv.org/abs/2501.02047v1.

Claim. For the vacuum-environment loss channel ρ_T = Tr₂[B(T)(ρ₁ ⊗ |0⟩⟨0|)B(T)†], the paper proves convexity of the purity in the transmission T and conjectures that it is log-convex on [0, 1] for every state (Conjecture 7, equivalently inequality (26)); Appendix B extends the conjecture to mixtures of identical two-copy states (Conjecture 19, (B4)).

Problem details · Reading note

The aligned general-dyne measurement attains the global steady-state optimum

Proved. Lean theorem result, frozen in this repository 2026-10-14.

K. H. Kua; A. Serafini; M. G. Genoni (2025). Daemonic ergotropy of Gaussian quantum states and the role of measurement-induced purification via general-dyne detection. DOI: 10.1088/2058-9565/ae2200. URL: https://arxiv.org/abs/2506.22288.

Claim. Section VI, Eq. (66), gives the aligned pure optimum and conjectures its optimality over all steady-state general-dyne monitoring.

Problem details · Reading note

The Fibonacci numbers are not 2-super-accessible

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Bruce M. Landman, Aaron Robertson, Quinn Robertson (2023). Ramsey properties for integer sequences with restricted gaps. DOI: 10.2140/moscow.2023.12.181. URL: https://math.colgate.edu/~aaron/pap/moreDoAFinal.pdf.

Claim. Section 5 question 3 asks: Is the set of Fibonacci numbers 2-super-accessible? We know it is 2-accessible but that it is not 2-large.

Problem details · Reading note

Every data qubit of the delayed remote-controlled circuit violates the dimension witness S ≤ 4

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Peicun Lin, Dongyang Wang, Yong Liu, Anqi Huang (2026). Nonclassicality of a delayed remote-controlled quantum computing model. DOI: 10.48550/arXiv.2606.31103. URL: https://arxiv.org/abs/2606.31103v1.

Claim. A delayed 1-U-M remote-controlled quantum computing circuit (a Bell pair on program and remote-control qubits, controlled gates A = diag(1, e^{i(β+δ)}) and B = [[0, −1], [e^{i(β−δ)}, 0]] on the data qubit, a Hadamard and computational measurement on the program qubit) is mapped to a prepare-and-measure scenario. Simulated annealing on a 30 × 30 grid of initial data qubits finds violations of the dimension witness S ≤ 4 (Eq. (2)) at all 900 points, and the authors conjecture that every initial data qubit admits parameters and a data measurement violating S ≤ 4.

Problem details · Reading note

CHSH nonlocality is tightly bounded by the steering-ellipsoid centre

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Antony Milne; David Jennings; Sania Jevtic; Terry Rudolph (2014). Quantum correlations of two-qubit states with one maximally mixed marginal. DOI: 10.1103/PhysRevA.90.024302. URL: https://arxiv.org/abs/1404.3951v2.

Claim. Let rho be a general two-qubit state with steering ellipsoid centred at c. The CHSH nonlocality is tightly bounded as beta(rho) <= max[2 sqrt(2(1 - c)), 2] (Conjecture 1, Section V), and the fully entangled fraction is tightly bounded as f(rho) <= 1 - c/2 (Conjecture 2, Section V); c is the length of the centre vector.

Problem details · Reading note

Two completely useless channels with a useful concatenation switch branch

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

A. Mitra; H. Badhani; S. Ghosh (2023). Improvement in quantum communication using quantum switch. DOI: 10.1088/1402-4896/acbcf7. URL: https://arxiv.org/abs/2108.14001.

Claim. Section III.B, Conjecture 1 asserts that concatenation of two completely useless channels is always completely useless.

Problem details · Reading note

Hanna's universal logarithmic integer-series closure

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Paul D. Hanna (2009). Hanna's logarithmic integer-series power closure conjecture. URL: https://oeis.org/A167138.

Claim. Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

Problem details · Reading note

Hanna's full logarithmic arithmetic-subsequence closure

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Paul D. Hanna (2009). Hanna's logarithmic integer-series arithmetic-subsequence conjecture. URL: https://oeis.org/A167540.

Claim. For integers m>0, b>=0, Sum_{n>=0} L(m*n+b) * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

Problem details · Reading note

OEIS A167857: three distinct prime divisors

Proved. Lean theorem result, frozen in this repository 2026-10-14.

T. D. Noe (2009). OEIS A167857: Numbers whose divisors are represented by an integer polynomial. URL: https://oeis.org/A167857.

Claim. Are there terms with three distinct prime divisors?

Problem details · Reading note

The literal A361552 equation and ordinary partition parity

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Paul D. Hanna (2023). OEIS A361552: implicit pentagonal unit series and partition parity. URL: https://oeis.org/A361552.

Claim. Conjecture: a(n)/2 == A000041(n-1) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n-1, for n >= 1.

Problem details · Reading note

The full Laurent rows in A391810 sum to zero

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Paul D. Hanna (2025). OEIS A391810, nonpositive Laurent coefficients and zero row sums. URL: https://oeis.org/A391810/internal.

Problem details · Reading note

The Haar mean of two-qubit non-local nonstabilizerness is 10/3 − π

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Dongheng Qian; Jing Wang (2025). Quantum non-local nonstabilizerness. DOI: 10.1103/PhysRevA.111.052443. URL: https://arxiv.org/abs/2502.06393v4.

Claim. Appendix F conjectures that some separable multi-qubit states cannot be brought into STAB by local unitaries. Appendix C derives the density (C5) of the non-local nonstabilizerness of Haar-random two-qubit pure states and leaves the closed form of its mean ȳ = ∫ yP(y)dy open, reporting ȳ ≈ 0.1917.

Problem details · Reading note

Circulant bracelet sufficiency fails in dimension twelve

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Grzegorz Rajchel-Mieldzioć, Kamil Korzekwa, Zbigniew Puchała, Karol Życzkowski (2021). Algebraic and geometric structures inside the Birkhoff polytope. DOI: 10.1063/5.0046581. URL: https://arxiv.org/abs/2101.11288.

Claim. Section V asks whether the bracelet condition suffices for unistochasticity of circulant matrices for d > 4; Definition 7 gives the bracelet inequalities and Theorem 17 settles d = 4. Definition 6 defines doubly circulant unistochastic matrices; Conjecture 19 asserts circulant unitary lifts for every d.

Problem details · Reading note

A circulant unistochastic matrix without a circulant unitary lift

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Problem details · Reading note

The termwise local-perception CHSH witness is at most 2 for every measurement setting

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Rohit Kishan Ray (2025). Local perception operators and classicality: new tools for old tests. DOI: 10.48550/arXiv.2511.00314. URL: https://arxiv.org/abs/2511.00314v1.

Claim. For dichotomic observables A_x, B_y with local means a_x = Tr(ρ_A A_x), b_y = Tr(ρ_B B_y) and correlators C_xy = Tr(ρ A_x ⊗ B_y), the symmetric local-perception witness W^s_LPO(ρ) = sup over settings of Σ α_xy a_x b_y C_xy (Eq. (21); for CHSH α = [[1, 1], [1, −1]], Eq. (19)) satisfies W^s_LPO,⊥(ρ) ≤ 2‖r_A‖‖r_B‖ ≤ 2 under mutually orthogonal settings (Theorem 4, Eq. (31)); the author conjectures that for all other measurement settings the upper bound saturates at 2.

Problem details · Reading note

Global dynamic cooling strictly beats equal parallel groups except at (N, M) = (6, 2)

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Mattia Reda, Massimiliano Sacchi, Chiara Macchiavello, Giacomo Guarnieri (2026). Optimal Dynamic Cooling of Multiple Qubits. DOI: 10.48550/arXiv.2607.23641. URL: https://arxiv.org/abs/2607.23641v1.

Claim. Cooling M of N identical thermal qubits by one global unitary to the lowest common local temperature is solved: the optimum with identical diagonal one-qubit target marginals equals the unconstrained passive minimum for every N > M (Theorem 1). §IX compares it with a parallel benchmark of M groups of ⌊N/M⌋ qubits, each running the optimal single-target protocol: the global optimum is never worse, strictly better for M + 2 ≤ N ≤ 3M − 1, ties at (N, M) = (6, 2), and the uniqueness of that tie is left as a conjecture.

Problem details · Reading note

Positive Kraus operators do not exclude a side-entanglement advantage

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

M. Rexiti; L. Memarzadeh; S. Mancini (2022). Discrimination of dephasing channels. DOI: 10.1088/1751-8121/ac6d2c. URL: https://arxiv.org/abs/2201.00388.

Claim. Section 7 conjectures that Kraus operators with identity polar unitaries give no benefit from side entanglement.

Problem details · Reading note

The ideal heralding rate of Fourier photon distillation decreases strictly to 1/4 from n = 5

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Jason Saied, Jeffrey Marshall, Namit Anand, Eleanor G. Rieffel (2025). General protocols for the efficient distillation of indistinguishable photons. DOI: 10.1103/PhysRevApplied.23.034079. URL: https://arxiv.org/abs/2404.14217v4.

Claim. For n-photon distillation protocols built from an n × n unitary whose first row is identically 1/√n (for example the discrete Fourier transform), the ideal heralding rate is h_n(0) = (−1/n)^{n−1}(n − 1)! Σ_{t=0}^{n−1} (n − t)(−n)^t/t! (Theorem III.5, Eq. (III.9)), with lim h_n(0) = 1/4; the authors conjecture that for n ≥ 5 the sequence h_n(0) decreases monotonically to 1/4 (Conjecture III.7).

Problem details · Reading note

Even-legged hyperlinks reconstruct the entanglement entropies with N-independent leg-factors

Proved. Lean theorem result, frozen in this repository 2026-10-14.

Silvia N. Santalla, Sudipto Singha Roy, Germán Sierra, Javier Rodríguez-Laguna (2026). The hyperlink representation of entanglement and the inclusion-exclusion principle. DOI: 10.1103/gmjw-rb13. URL: https://arxiv.org/abs/2601.17926v2.

Claim. The entanglement hyperlinks J_I = Σ_{A⊆I} (−1)^{|I|−|A|} S_A of a pure state reconstruct its entanglement entropies through all ranks (proved). Appendix B conjectures the even-legged reconstruction S_A = Σ_{|I| even} Λ_{|I|,|I∩A|} J_I with leg-factors independent of N, lists Λ_{2l,p} for 2l ≤ 8 in (B5), and reports checks up to N = 9.

Problem details · Reading note

A 1225-vertex graph refutes the star bound on the minimal-energy uniform overlap of quantum batteries

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Matthieu Sarkis, Oskar A. Prośniak, Samuel Nigro, Alexandre Tkatchenko (2026). Star Topology Optimizes the Charging Power of Quantum Batteries. DOI: 10.48550/arXiv.2603.10853. URL: https://arxiv.org/abs/2603.10853v1.

Claim. For a quantum battery of N fermionic cells coupled by a graph G, the early-time charging power at zero local field is maximized by the star graph (Theorem 1). With a local field, the power is a weighted mean of the uniform overlaps w_k = (1ᵀu_k)² of the adjacency eigenvectors; Conjecture 1 asserts that the overlap of the lowest adjacency eigenspace satisfies w_min(G) ≤ N/2 − √(N − 1), the star's value, for every non-empty graph on N vertices.

Problem details · Reading note

Sun's weighted distinct-sum bound

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Zhi-Wei Sun (2021). On permutations of {1,...,n} and related topics. DOI: 10.1007/s10801-021-01028-8. URL: https://arxiv.org/abs/1811.10503v6.

Claim. Conjecture 4.9 gives a lower bound for weighted sums of distinct elements of a finite subset of a field.

Problem details · Reading note

The traceless BPS cohomology at (p, N) = (5, 4) is not free over the diagonal creators

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Miguel Tierz (2026). BPS spectra of Tr[Ψ^p] matrix models for odd p. URL: https://arxiv.org/abs/2604.27164v2.

Claim. The BPS cohomology of the fermionic matrix model with supercharge Q_p = Tr(Ψ^p) factors as Λ(χ) ⊗ B_0 with B_0 the traceless cohomology, a graded module over the exterior algebra Λ(C^{N−1}) of the traceless diagonal creators. v1 conjectures that this module is free; v2 states the freeness B_0 ≅ Λ(C^{N−1}) ⊗ M^{(p,N)} as a sufficient explanation, not proved, of the (1 + x) factors of the computed BPS polynomials.

Problem details · Reading note

A rotating replacement state refutes purely non-unital eternal non-Markovianity impossibility

Refuted. Lean theorem result, frozen in this repository 2026-10-14.

Shrikant Utagi and Subhashish Banerjee and R. Srikanth (2024). On the eternal non-Markovianity of non-unital quantum channels. DOI: 10.1142/S0219749923500399. URL: https://arxiv.org/abs/2203.10625.

Claim. Section IV conjectures the impossibility of purely non-unital eternal non-Markovianity in arbitrary finite dimensions.

Problem details · Reading note

Dicke optimality for detector-sensor identification

Proved. Lean theorem result, frozen in this repository 2026-10-14.

C. Zhan, H. Gupta, M. Hillery (2024). Optimizing Initial State of Detector Sensors in Quantum Sensor Networks. DOI: 10.1145/3655028. URL: https://arxiv.org/abs/2306.17401v6.

Problem details · Reading note

abiad-flagg-ghasemi-nezhad-zeijlemaker-2026-grid-power-zero-forcing

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Aida Abiad; Mary Flagg; Sina Ghasemi Nezhad; Sjanne Zeijlemaker (2026). The Zero Forcing Number of Graph Powers. DOI: 10.48550/arXiv.2610.03892. URL: https://arxiv.org/abs/2610.03892v1.

Problem details · Reading note

The trace exponential is not exponentially convex on density matrices

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Jop Briët, Peter Harremoës (2009). Properties of classical and quantum Jensen-Shannon divergence. DOI: 10.1103/PhysRevA.79.052311. URL: https://arxiv.org/abs/0806.4472v4.

Claim. However, based on a large amount of numerical evidence, we conjecture the function A → Tr(e^A) to be exponentially convex for density matrices A.

Problem details · Reading note

Refutation of the symmetric binary-outcome detection-threshold conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

T. Cope (2021). The Binary-Outcome Detection Loophole. DOI: 10.1088/1367-2630/ac10a5. URL: https://arxiv.org/abs/2005.03344v3.

Claim. The binary-outcome detection threshold has the bound 2(m_A+m_B-8)/(m_A m_B-16), and Section 4.3 conjectures eta* = 4/(m+4) for the symmetric m-input case.

Problem details · Reading note

The optimal T-parameter is not monotone in edge density

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

A. N. Day; A. Sarkar (2021). On a Conjecture of Nagy on Extremal Densities. DOI: 10.1137/19M1296525. URL: https://arxiv.org/abs/1910.13465.

Claim. Conjecture 2. For each graph G_l, we have that q(G_l, β) is an increasing function of β.

Problem details · Reading note

Characterizing rigid alternative means

Proved. Lean theorem result, frozen in this repository 2026-10-13.

R. Dumitru; J. Franco; A. Merino (2026). On The Linearization of Alternative Means. DOI: 10.48550/arXiv.2606.25150. URL: https://arxiv.org/abs/2606.25150v1.

Claim. Can we characterize all functions f : (0, ∞) → (0, ∞) such that Aσ̂_f B is linearizable if and only if AB = BA and |Spec(A⁻¹♯B)| ≤ 2 ?

Problem details · Reading note

Centred convex bodies and Gaussian dilation

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

D. Faifman; I. Polterovich (2026). The Faber–Krahn position of convex bodies and Gaussian measure inequalities. DOI: 10.48550/arXiv.2607.21539. URL: https://arxiv.org/abs/2607.21539v2.

Claim. Question 5.3. Let K ⊂ R^n be a convex body whose Lebesgue barycenter is at the origin. Is t ↦ γ_n(e^t K) log-concave? More generally, is t ↦ γ_n(e^{tS} K) log-concave for every symmetric matrix S?

Problem details · Reading note

The diploid cellular automaton 10-245 has exact density 1/4 + λ/2

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Henryk Fukś, Nazim Fatès (2026). Analysis of the asymptotic density of endogamous diploid cellular automata. DOI: 10.1140/epjs/s11734-026-02447-z. URL: https://arxiv.org/abs/2607.01917v1.

Claim. Diploid elementary cellular automata mix two rules cell by cell with probability λ. For rule 10-245, third-order local structure approximation gives the asymptotic density 1/4 + λ/2 from the symmetric Bernoulli start, and §5 asks whether this expression can be formally demonstrated to be exact.

Problem details · Reading note

Expected largest block-palindrome width

Proved. Lean theorem result, frozen in this repository 2026-10-13.

D. Gabrić and J. Shallit (2023). Smallest and Largest Block Palindrome Factorizations. DOI: 10.1007/978-3-031-33180-0_14. URL: https://arxiv.org/abs/2302.13147.

Claim. Conjecture 5 states that the expected width of the largest block-palindrome factorization tends to 1 plus twice the A274199 generating series. Lemma 3 gives the source's auxiliary tail bound.

Problem details · Reading note

Godrèche–Luck moving-average record bound

Proved. Lean theorem result, frozen in this repository 2026-10-13.

C. Godrèche and J.-M. Luck (2020). Records for the moving average of a time series. DOI: 10.1088/1742-5468/ab5d08. URL: https://arxiv.org/abs/1907.07598.

Claim. Finally, the data shown in figure 1 suggest that (6.33) provides an absolute upper bound for $Q_n$.

Problem details · Reading note

The reduced function of the partial negativity is not concave

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Y. Guo (2023). Partial-Norm of Entanglement: Entanglement Monotones That are not Monogamous. DOI: 10.1088/1367-2630/acf152. URL: https://arxiv.org/abs/2212.06521v6.

Claim. The partial negativity of a pure state with decreasing Schmidt coefficients λ₁ ≥ λ₂ ≥ ⋯ is λ₁λ₂, with reduced function ĥ(ρ) = √(δ₁δ₂) for the two largest eigenvalues of the reduced state; the paper conjectures that ĥ is concave.

Problem details · Reading note

Hanaoka–Konno Conjecture 4.2: self-similarity of the Riesz walk

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Ryota Hanaoka, Norio Konno (2021). Return probability and self-similarity of the Riesz walk. DOI: 10.26421/QIC21.5-6-5. URL: https://arxiv.org/abs/2010.04518v3.

Claim. Section 2 defines the Riesz walk; Conjecture 4.2 states equality of its probabilities in the same rescaled intervals at times 2t and 8t.

Problem details · Reading note

Hiai's spectral geometric Log-Euclidean comparison fails

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Fumio Hiai (2026). Log-majorizations between quasi-geometric type means for matrices. URL: https://arxiv.org/abs/2510.04691v3.

Claim. Problem 3.20. The case remaining open is whether SGα,p ≺log LEα for α > 1. Note that this is affirmative if SGα,p ≺log SGα,q if α > 1 and p ≥ q (see Problem 3.12).

Problem details · Reading note

Generic logarithmic Lie-length of su(n)

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Yifan Jia; Angela Capel (2024). A generic quantum Wielandt's inequality. DOI: 10.22331/q-2024-05-02-1331. URL: https://arxiv.org/abs/2301.08241.

Claim. Conjecture 6 asks whether a random Lie-generating set of su(n) has Lie-length Theta(log n) almost surely.

Problem details · Reading note

Pick functions do not preserve embeddable matrices

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Claim. The entries marked Open indicate cases for which preservation under Pick functions remains an open question.

Problem details · Reading note

Pick functions do not preserve strong infinite divisibility

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

C. R. Johnson; S. Mondal; M. J. Tsatsomeros (2026). Powers, Roots, and Positivity. DOI: 10.48550/arXiv.2609.06298. URL: https://arxiv.org/abs/2609.06298v1.

Claim. The entries marked Open indicate cases for which preservation under Pick functions remains an open question.

Problem details · Reading note

Non-tightness of the antidistinguishability overlap bound

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Nathaniel Johnston, Vincent Russo, Jamie Sikora (2025). Tight bounds for antidistinguishability and circulant sets of pure quantum states. DOI: 10.22331/q-2025-02-04-1622. URL: https://arxiv.org/abs/2311.17047v2.

Claim. Theorem 5.1 gives a spectral sufficient condition for antidistinguishability; Corollary 5.4 gives an overlap bound whose tightness for n ≥ 5 is questioned in Section 6.

Problem details · Reading note

kechrimparis-bae-2020-isolated-qubit-ensembles

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Spiros Kechrimparis; Joonwoo Bae (2020). Optimal measurement preserving qubit channels. DOI: 10.1088/1367-2630/abcc9b. URL: https://arxiv.org/abs/2006.16366.

Claim. The authors ask whether isolated qubit ensembles exist and expect that none exist; OMP means that a POVM is optimal both before and after applying the channel.

Problem details · Reading note

A design with the maximally mixed state is not the most discriminable qubit ensemble at N = N′(2,3) + 1

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

M. Kvashchuk, P. Chernyshova, L. E. A. Porto, T.-A. Ohst, L. B. Vieira, M. T. Quintino (2026). The most discriminable quantum states in the multicopy regime. URL: https://arxiv.org/abs/2604.26927v3.

Claim. Weighted state k-designs on N′(d,k) states, the smallest size admitting one, are used to build k-copy discriminable ensembles; with one extra state the paper adjoins the maximally mixed state. It asks (§12, question 4) whether this construction is optimal for N = N′(d,k) + 1 in every dimension, citing numerical evidence for qubits.

Problem details · Reading note

Exact rational attainment of finite-group subgroup conjugacy ratios

Proved. Lean theorem rational_attainment, frozen in this repository 2026-10-13.

Mihai-Silviu Lazorec and Marius Tărnăuceanu (2025). On the number of conjugacy classes of subgroups of a finite group. DOI: 10.48550/arXiv.2508.14267. URL: https://arxiv.org/html/2508.14267v2.

Claim. Proposition 2.2(ii) and its proof give the whole subgroup-lattice count 2n+p−1 and inner-conjugacy class count 2n for the cyclic split extensions G_{p,q,n}; Section 4 Problem 4.1 asks whether every rational a/b with integers 1≤a<b is attained by an actual finite group.

Problem details · Reading note

Lin's operator Wielandt norm conjecture fails

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Minghua Lin (2013). On an operator Kantorovich inequality for positive linear maps. DOI: 10.1016/j.jmaa.2013.01.015. URL: https://arxiv.org/abs/1212.5690v1.

Claim. Conjecture 3.4. Under the same condition as in Theorem 3.1. Then for every 2-positive linear map Φ, ‖Φ(X*AY)Φ(Y*AY)⁻¹Φ(Y*AX)Φ(X*AX)⁻¹‖ ≤ ((M−m)/(M+m))².

Problem details · Reading note

QTD squared can exceed base-2 quantum Jensen–Shannon divergence

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Yupan Liu (2025). Quantum state testing beyond the polarizing regime and quantum triangular discrimination. DOI: 10.48550/arXiv.2303.01952. URL: https://arxiv.org/abs/2303.01952v5.

Claim. We observe that Theorem 1.4(ii) is not tight. Numerical simulations indicate that the tight bound is QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁) for any states ρ₀ and ρ₁.

Problem details · Reading note

Normalization of weak shuffle products

Proved. Lean theorem result, frozen in this repository 2026-10-13.

C. Mammez (2022). Weak stuffle algebras. DOI: 10.24330/ieja.1060709. URL: https://arxiv.org/abs/2003.09142.

Claim. Conjecture 3.10 asks whether Proposition 3.9 holds for every countable alphabet: over a field of characteristic 0, every weak shuffle product on the noncommutative polynomials is isomorphic, as an algebra, to the weak shuffle product with its exceptional off-diagonal coefficients normalized.

Problem details · Reading note

n + 1 displaced on-off measurements stay incompatible under pure loss of transmissivity 1/n

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Mohammad Mehboudi, Fatemeh Rezaeinia, Saleh Rahimi-Keshari (2025). Measurement incompatibility under loss. DOI: 10.1103/s9sp-2fd7. URL: https://arxiv.org/abs/2411.05920v3.

Claim. A pure-loss channel of transmissivity τ ≤ 1/n makes every set of n measurements jointly measurable (Result 2). Result 3 asserts that some n + 1 measurements, displaced on-off photodetections, stay incompatible at τ = 1/n; the paper proves this numerically for 2 ≤ n ≤ 13 and conjectures it for arbitrary n.

Problem details · Reading note

Mathar's D-finite recurrence for OEIS A308815

Proved. Lean theorem result, frozen in this repository 2026-10-13.

R. J. Mathar; OEIS Foundation Inc. (2020). OEIS A308815 — D-finite recurrence for alternating Catalan numbers. URL: https://oeis.org/A308815.

Claim. Mathar conjectures the order-three polynomial-coefficient recurrence for the alternating Catalan numbers defined by the entry's PARI formula.

Problem details · Reading note

OEIS A400644 rounded three-term norm ratio

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Bence Bernáth (2026). OEIS A400644, rounded three-term norm ratio. URL: https://oeis.org/A400644.

Claim. The consecutive ratio of the rounded three-term Euclidean-norm recurrence tends to the largest positive root of x^6-x^4-x^2-1 (conjecture).

Problem details · Reading note

Circular hypergraphs do not maximise orientation-count entropy

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Christos Pelekis (2019). On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs. DOI: 10.4418/2019.74.1.8. URL: https://arxiv.org/abs/1808.04166.

Claim. Conjecture 1.3 asserts degree balance for every orientation-count entropy maximiser; Section 4 asserts circular-hypergraph maximality.

Problem details · Reading note

Entropy maximisers need not have balanced degrees

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Christos Pelekis (2019). On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs. DOI: 10.4418/2019.74.1.8. URL: https://arxiv.org/abs/1808.04166.

Claim. Conjecture 1.3 asserts degree balance for every orientation-count entropy maximiser; Section 4 asserts circular-hypergraph maximality.

Problem details · Reading note

Diophantine exponents of aperiodic episturmian words

Proved. Lean theorem result, frozen in this repository 2026-10-13.

J. Peltomäki (2024). Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents. DOI: 10.1016/j.ejc.2024.103942. URL: https://arxiv.org/abs/2103.08351v3.

Claim. Section 8, Conjecture 8.4 asks whether an aperiodic episturmian word has finite Diophantine exponent exactly when its partial quotients are bounded.

Problem details · Reading note

The boundary α-z Rényi relative entropies of magic are not additive on three qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Roberto Rubboli, Ryuji Takagi, Marco Tomamichel (2024). Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond. DOI: 10.22331/q-2024-10-04-1492. URL: https://arxiv.org/abs/2307.08258v4.

Claim. The α-z Rényi relative entropies of magic D_{α,z}(ρ) = inf over stabilizer states of D_{α,z}(ρ‖σ) are additive for all single-qubit states when |(1−α)/z| = 1. §5 states as open whether they are additive for all two- and three-qubit states in that range, and Table 1 and §10 conjecture that they are.

Problem details · Reading note

A passive incoherent memory realizes a non-classical dephasing superchannel in dimension 3

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

R. Salazar; F. Shahbeigi (2025). Dephasing noise simulation for coherence-generating devices. DOI: 10.1088/1402-4896/adda94. URL: https://arxiv.org/abs/2403.19247v1.

Claim. A dephasing superchannel multiplies Choi matrices entrywise by a Gram matrix C. The paper calls an implementation's memory passive when its marginal after encoding does not depend on the input. It proves that for qubit channels a passive memory realizes exactly the classical Gram matrices Σ q_i C₁⁽ⁱ⁾ ⊗ C₂⁽ⁱ⁾, and leaves open whether a passive memory is equivalent to a classical memory in every dimension.

Problem details · Reading note

First-level SDP inexactness on the weighted-crown interval

Proved. Lean theorem result, frozen in this repository 2026-10-13.

J. Takahashi; C. Rayudu; C. Zhou; R. King; K. Thompson; O. Parekh (2023). An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut. DOI: 10.48550/arXiv.2307.15688. URL: https://arxiv.org/abs/2307.15688v3.

Claim. Appendix C.1 leaves first-level SDP exactness on the weighted-crown interval (n+2)^2/(4(n+1)) < x < (n+2)/3 open.

Problem details · Reading note

The Wagner graph and the rank-5 eight-vertex graph have no qutrit violation, so d* = 4

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Uğur Tamer, Berke Ünal, Alper Dizdar, Zafer Gedik, Özgür E. Müstecaplıoğlu (2026). The Quad-C5 Graph: Maximum Contextuality Gap on Eight Vertices. URL: https://arxiv.org/abs/2605.12828v3.

Claim. An exhaustive search over the 11,117 connected eight-vertex graphs ranks exclusivity graphs by the gap between the Lovász number and the independence number. For the dimension-restricted value η_d (Eq. (13)) and the minimum dimension d* (Eq. (25)), the Wagner graph and the rank-5 graph show no three-dimensional violation in the numerical search; the paper states that d* = 4 for them is numerically indicated and that a rigorous three-dimensional upper bound is not established.

Problem details · Reading note

Three-weight Pauli channels have Bayesian inverses only at the maximally mixed state

Proved. Lean theorem result, frozen in this repository 2026-10-13.

O. Ting; J. Fullwood; Z. Wu (2026). Operational time-reversal symmetry for unital qubit channels. DOI: 10.48550/arXiv.2605.10375. URL: https://arxiv.org/abs/2605.10375v1.

Claim. For a Pauli channel with precisely three non-zero weights, either the maximally mixed state is the only state with respect to which a Bayesian inverse exists, or such states form a measure-zero subset of the Bloch ball; the authors found no example away from the maximally mixed state.

Problem details · Reading note

The sharp six-level bound for extreme two-qutrit absolutely PPT states

Proved. Lean theorem result, frozen in this repository 2026-10-13.

Nalan Wang; Lin Chen; Zhiwei Song (2026). Extreme points of absolutely PPT states with exactly three distinct eigenvalues. DOI: 10.1088/1402-4896/ae9148. URL: https://arxiv.org/abs/2603.20717v1.

Claim. Fourth, what is the achievable upper bound of distinct eigenvalues of extreme two-qutrit AP states?

Problem details · Reading note

The incompatibility probability of random qubit measurements

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Lin Zhang; Hua Xiang; Xiao Li-Jost; Shao-Ming Fei (2019). Incompatibility probability of random quantum measurements. DOI: 10.1103/PhysRevE.100.062139. URL: https://arxiv.org/abs/1912.12321.

Claim. The incompatibility probability of a pair of random qubit measurements is conjectured to equal one quarter.

Problem details · Reading note

Ziętak's strict spectral approximant convergence conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-13.

Krystyna Ziętak (2017). From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix. DOI: 10.4064/bc112-0-16. URL: https://doi.org/10.4064/bc112-0-16.

Claim. Conjecture 1.1 asserts convergence of Schatten-p residuals to the lexicographically minimal singular-value residual.

Problem details · Reading note

apte-parekh-sud-2025-token-graph-cut-bound

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

Anuj Apte; Ojas Parekh; James Sud (2025). Conjectured Bounds for 2-Local Hamiltonians via Token Graphs. URL: https://arxiv.org/abs/2506.03441v2.

Claim. Appendix Conjecture 7 bounds the least token-graph adjacency eigenvalue by minus half the sum of maximum cut and size-restricted matching weights.

Problem details · Reading note

The normal-modulus triangle constant in dimension two

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

J.-C. Bourin; E.-Y. Lee (2026). Triangle inequalities for the operator symmetric modulus. DOI: 10.1090/proc/17861. URL: https://arxiv.org/abs/2602.19607v1.

Claim. Corollary 4.2 gives the constant 1/4 for normal matrices in every dimension, proves sharpness for n ≥ 3, and the following paragraph asks for the best constant when n = 2.

Problem details · Reading note

A SUNTF_R(2, 5, 3) exists

Proved. Lean theorem result, frozen in this repository 2026-10-12.

W. J. Brinkley, M. Fickus, J. Jasper and D. G. Mixon (2026). Incoherent filter banks. DOI: 10.48550/arXiv.2610.05302. URL: https://arxiv.org/abs/2610.05302v1.

Claim. We do not yet know whether a SUNTF_R(2, 5, 3) exists, and we leave this for future research.

Problem details · Reading note

The theta representation set: Question (iv), first convexity clause

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

Marcel K Carli Silva; Gabriel Coutinho; Thiago Oliveira; Levent Tunçel (2026). On some k-fold generalizations of Lovász theta and their sandwich theorems. URL: https://arxiv.org/html/2609.06312v1.

Claim. Theorem 16 defines the trace-k orthonormal representation set; Section 7 item (iv) asks whether it is convex and separately asks for its closed convex lower hull.

Problem details · Reading note

PPT-squared and IPT-squared equivalence

Proved. Lean theorem result, frozen in this repository 2026-10-12.

Giulio Chiribella; Kenneth R. Davidson; Vern I. Paulsen; Mizanur Rahaman (2023). Positive maps and entanglement in real Hilbert spaces. DOI: 10.1007/s00023-023-01325-x. URL: https://arxiv.org/abs/2207.02510v2.

Claim. Section 9.2 asks whether the complex PPT-squared and real and complex IPT-squared conjectures are equivalent.

Problem details · Reading note

Convexity of the spatial numerical range in non-unital normed algebras

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

H. V. Dedania and A. B. Patel (2024). Spatial numerical range in non-unital, normed algebras and their unitizations. URL: https://arxiv.org/abs/2306.16172v1.

Claim. Section 1, page 2, asks whether the spatial numerical range of an element of a non-unital normed algebra is always convex.

Problem details · Reading note

Concatenation can increase the cHaar moment norm

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

Matthew Duschenes, Diego García-Martín, Zoë Holmes, and M. Cerezo (2026). Moments of Quantum Channel Ensembles. DOI: 10.1103/grdp-jph9. URL: https://arxiv.org/abs/2511.12700v2.

Claim. Appendix B, Conjecture 1 asserts that the cHaar moment-operator Hilbert-Schmidt norm is non-increasing with concatenations; the fourth and fifth powers at d = dE = t = 2 contradict this clause.

Problem details · Reading note

Logarithmic scalar curvature is strictly Schur-increasing

Proved. Lean theorem result, frozen in this repository 2026-10-12.

Paolo Gibilisco; Tommaso Isola (2005). On the monotonicity of scalar curvature in classical and quantum information geometry. DOI: 10.1063/1.1834693. URL: https://arxiv.org/abs/math-ph/0407007v2.

Claim. Conjecture 4.1 asks whether the scalar curvature of the logarithmic alpha-geometry is Schur-increasing on the positive simplex; this note records the classical p = infinity endpoint and quantum Conjecture 4.2 at p = 3.

Problem details · Reading note

Quantum cube-root scalar curvature is strictly Schur-increasing

Proved. Lean theorem result, frozen in this repository 2026-10-12.

Paolo Gibilisco; Tommaso Isola (2005). On the monotonicity of scalar curvature in classical and quantum information geometry. DOI: 10.1063/1.1834693. URL: https://arxiv.org/abs/math-ph/0407007v2.

Problem details · Reading note

Endpoint comparison of Gini means on a bounded interval

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

R. Grünwald; Zs. Páles (2024). Comparison of Gini means with fixed number of variables. URL: https://arxiv.org/abs/2408.07658v1.

Claim. Section 4, Conjecture 1 proposes that the parameter-sum inequality and two extreme endpoint tests characterize comparison of Gini means on a bounded positive interval for n ≥ 3.

Problem details · Reading note

Pure programs need more dimension than mixed covariant programs

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

Martina Gschwendtner, Andreas Bluhm, Andreas Winter (2021). Programmability of covariant quantum channels. DOI: 10.22331/q-2021-06-29-488. URL: https://arxiv.org/abs/2012.00717v2.

Claim. Exact covariant programming has mixed optimum d_c; the Section 5 question asks whether pure programs attain the same dimension.

Problem details · Reading note

A hidden central direction refutes Heib–Bruschi Conjecture 80

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

T. Heib; D. E. Bruschi (2026). On the structural properties of Lie algebras via associated labeled directed graphs. DOI: 10.48550/arXiv.2601.16161. URL: https://arxiv.org/abs/2601.16161v1.

Claim. Conjecture 80 states that a minimal-graph-admissible Lie algebra with nonzero center has a proper non-empty vertex subset of its minimal graph with the ideal-graph-property that spans a solvable ideal. Conjecture 83 states that a minimal-graph-admissible Lie algebra with trivial center either has no proper non-empty vertex subset with the ideal-graph-property in its minimal graph, or is a direct sum of centerless components each of whose minimal graphs has no such subset.

Problem details · Reading note

Herrera's function-approximation conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

D. Herrera (2023). On the commutator modulus of continuity for operator monotone functions. URL: https://arxiv.org/abs/2311.17448v1.

Claim. Conjecture 2 asserts that E_f(c) ≤ C f(c) for every non-negative operator monotone f on [0,∞) and every c > 0 when C = 1.

Problem details · Reading note

A quantum behavior with rank of contextuality three

Proved. Lean theorem result, frozen in this repository 2026-10-12.

K. Horodecki; J. Zhou; M. Stankiewicz; R. Salazar; P. Horodecki; R. Raussendorf; R. Horodecki; R. Ramanathan; E. Tyhurst (2022). The rank of contextuality. DOI: 10.1088/1367-2630/acdf78. URL: https://arxiv.org/abs/2205.10307.

Claim. Definition III.1 defines the rank of contextuality; Section IX asks for a quantum behavior whose rank is not a power of two.

Problem details · Reading note

A resonant memory parameter refutes Krawec's recycled-coin uniformity conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

W. O. Krawec (2015). History dependent quantum walk on the cycle with an unbalanced coin. DOI: 10.1016/j.physa.2015.02.061. URL: https://arxiv.org/abs/1411.6298v1.

Claim. The recycled-coin quantum walk on the d-cycle uses a memory parameter φ ∈ [0, 8). The paper conjectures that its time-averaged distribution is uniform for every non-integer φ: if 12 ∤ d, only φ = 0, 2, 4, 6 give a non-uniform distribution; if 12 ∣ d, only φ = 0, 1, 2, 4, 5, 6 do. It reports numerical verification for d ≤ 300 on the grid φ = m/10.

Problem details · Reading note

Tightness of the lossy-qubit partial joint-measurability bound η ≤ 2/(2 + sin θ)

Proved. Lean theorem result, frozen in this repository 2026-10-12.

E. P. Lobo; M. Balanzó-Juandó; S. Pironio (2026). Generalized measurement incompatibility. DOI: 10.48550/arXiv.2605.16151. URL: https://arxiv.org/abs/2605.16151v1.

Claim. Measurements are G-jointly measurable when a common parent reproduces them while determining only the outcomes in the subsets G_y; for two lossy projective qubit measurements whose axes are separated by an angle θ ∈ [0, π/2], with G = ({+,−}, ∅), the paper proves G-joint measurability for η ≤ 2/(2 + sin θ) and reports that this bound is tight up to machine precision.

Problem details · Reading note

L∞-contraction for a weight

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Claim. Section 6, Problem 22 asks whether every L∞-bounded analytic semigroup in the sectorial-form setting admits a bounded strictly positive weight with bounded inverse making it L∞-contractive.

Problem details · Reading note

Mahalingam–Pandoh maximum number of scattered palindromes

Proved. Lean theorem result, frozen in this repository 2026-10-12.

K. Mahalingam, P. Pandoh (2021). Counting scattered palindromes in a finite word. URL: https://arxiv.org/abs/2108.02729v1.

Claim. Conjecture 5.6 proposes the sharp maximum of distinct nonempty scattered palindromic subwords and the alphabet sizes at which it is attained.

Problem details · Reading note

The multiplicative Hornich–Hlawka inequality in inner product spaces

Proved. Lean theorem result, frozen in this repository 2026-10-12.

D.-Ş. Marinescu; C. P. Niculescu (2024). A survey of the Hornich-Hlawka inequality. URL: https://arxiv.org/abs/2407.03278v1.

Claim. Section 6 derives the scalar multiplicative Hornich–Hlawka inequality from Theorem 8 and states that no extension to inner product spaces is known.

Problem details · Reading note

Variable-exponent Lebesgue spaces and the Hornich–Hlawka inequality

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

D.-Ş. Marinescu; C. P. Niculescu (2024). A survey of the Hornich-Hlawka inequality. URL: https://arxiv.org/abs/2407.03278v1.

Claim. Section 6 derives the scalar multiplicative Hornich–Hlawka inequality from Theorem 8 and states that no extension to inner product spaces is known.

Problem details · Reading note

Mináč–Nguyen–Tân Question 3.5

Proved. Lean theorem result, frozen in this repository 2026-10-12.

Ján Mináč, Tung T. Nguyen, Nguyen Duy Tân (2026). Sums of units in finite rings and applications to Cayley graphs. URL: https://arxiv.org/abs/2607.00404v1.

Claim. Question 3.5 asks whether the exact n-sum property of a finite field transfers to every finite group algebra with units restricted by standard augmentation.

Problem details · Reading note

Mitra broadcasting-channel extrema

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

A. Mitra (2021). Layers of classicality in the compatibility of measurements. DOI: 10.1103/PhysRevA.104.022206. URL: https://arxiv.org/abs/2101.05752v4.

Claim. Section III.A asks whether the channels broadcasting an observable have a highest element or a lowest element under precomposition by nondisturbing channels.

Problem details · Reading note

Nonnegative Chebyshev expansions of snake polynomials

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

L. Kryvonos (ed.) (2023). Open problems on polynomials, their zero distribution and related questions: 2023. URL: https://arxiv.org/abs/2312.07754v1.

Claim. Geno Nikolov conjectures that snake polynomials for continuous even convex nonnegative majorants have nonnegative expansions in Chebyshev polynomials of the first kind.

Problem details · Reading note

The ratio-limit assertion for A394578

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

Clark Kimberling (2026). A394578: a(n) = 2^(2*n-1) * p(n, n+1/2), where p(n,x) is the Lagrange interpolating polynomial through (1,0) and (k,1) for k=2..n+1. URL: https://oeis.org/A394578/internal.

Claim. The entry asserts that the consecutive ratio tends to one quarter for its moving-point Lagrange interpolation sequence.

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OEIS A400585: the odd-cofactor summatory asymptotic

Proved. Lean theorem result, frozen in this repository 2026-10-12.

Jishnu Babu Ranitha; Vaclav Kotesovec (2026). A400585 — odd-cofactor binomial divisor sums and a cubic summatory conjecture. URL: https://oeis.org/A400585.

Claim. Revision 22 defines the odd-cofactor divisor sum and conjectures its summatory asymptotic 7*zeta(3)*n^3/48.

Problem details · Reading note

Four clues are necessary for unique four by four SudoQ solvability

Proved. Lean theorem result, frozen in this repository 2026-10-12.

J. Paczos, M. Wierzbiński, G. Rajchel-Mieldzioć, A. Burchardt, K. Życzkowski (2021). Genuinely quantum SudoQ and its cardinality. DOI: 10.1103/PhysRevA.104.042423. URL: https://arxiv.org/abs/2106.02967v2.

Claim. Section III.A conjectures that no 4×4 SudoQ with three clues is uniquely solvable; Theorem 13 gives uniquely solvable grids with four clues.

Problem details · Reading note

M-resistant qudit states exist for every admissible loss parameter

Proved. Lean theorem result, frozen in this repository 2026-10-12.

G. M. Quinta; R. André; A. Burchardt; K. Życzkowski (2019). Cut-resistant links and multipartite entanglement resistant to particle loss. DOI: 10.1103/PhysRevA.100.062329. URL: https://arxiv.org/abs/1908.10390v1.

Claim. For any N and m, there exists an m-resistant N-qudit state in some local dimension d.

Problem details · Reading note

The K-number and extreme-point K-sets of real finite-dimensional sup-norm spaces

Proved. Lean theorem result, frozen in this repository 2026-10-12.

D. Sain; J. Manna; K. Paul (2024). On local preservation of orthogonality and its application to isometries. URL: https://arxiv.org/abs/2407.08900v1.

Claim. The open question at the end of Section 2 asks for the minimal K-sets in real finite-dimensional sup-norm spaces and their K-number; Proposition 2.1 and Remark 2.20 give the bounds n ≤ κ(ℓ∞ⁿ) ≤ 2^(n−1).

Problem details · Reading note

The Sun–Luo–Ye projection-matrix AM-GM conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

R. Sun; Z.-Q. Luo; Y. Ye (2020). On the efficiency of random permutation for ADMM and coordinate descent. DOI: 10.1287/moor.2019.0990. URL: https://arxiv.org/abs/1503.06387v3.

Claim. Section 3.3, Conjecture 1 proposes a matrix AM-GM inequality in the positive-semidefinite order for finite families of real projection matrices.

Problem details · Reading note

Interaction-picture asymptotics

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Problem details · Reading note

Operational comparison of qubit incompatibility without equal lengths

Proved. Lean theorem result, frozen in this repository 2026-10-12.

K. Torii; R. Takakura; R. Imamura (2025). Comparing quantum incompatibility of device sets from an operational perspective. DOI: 10.1103/s6mc-lnvl. URL: https://arxiv.org/abs/2509.08284v2.

Claim. Device sets are compared by the state sets on which their incompatibility can be detected; for pairs of unbiased qubit observables with |a1|=|a2| and |b1|=|b2| in (1/√2,1], operational equivalence forces equality up to signs and exchange (Proposition 7), and whether this holds without the equal-length constraints is left open.

Problem details · Reading note

The quantum value of balanced hypercube rendezvous is strictly below its no-signalling value

Proved. Lean theorem result, frozen in this repository 2026-10-12.

C. Wang (2026). Quantum-assisted rendezvous on hypercubes: a quantum Bernoulli noise approach. URL: https://arxiv.org/abs/2610.12949v1.

Claim. In balanced one-step rendezvous on the d-dimensional hypercube with Hamming step t, the no-signalling value is C(2t,t)/C(d,t) (Lemma 5.1), and no finite-dimensional quantum strategy attains it when t ≥ 2 and d > 2t (Theorems 5.5 and 6.4). The paper leaves open whether the quantum value, the supremum over finite-dimensional strategies, is then strictly below the no-signalling value.

Problem details · Reading note

Critical values of parametrized quantum circuits

Refuted. Lean theorem result, frozen in this repository 2026-10-12.

D. Wierichs; K. Kottmann; N. Killoran (2025). Unitary synthesis with optimal brick wall circuits. DOI: 10.48550/arXiv.2511.16736. URL: https://arxiv.org/abs/2511.16736v1.

Claim. Brick wall circuits with the optimal number of parameters parametrize SU(2^n); their Jacobian has full rank almost everywhere for n ≤ 5, and universality would follow from Conjecture 3: the critical values of a parametrized quantum circuit with full rank almost everywhere form an at most (d−2)-dimensional subset of the target group.

Problem details · Reading note

Normal limits for Speyer g-polynomials of growing uniform-matroid rank

Proved. Lean theorem result, frozen in this repository 2026-10-12.

Rong Zhang, James Jing Yu Zhao (2024). Analytic properties of Speyer’s g-polynomial of uniform matroids. DOI: 10.1007/s10114-026-4502-5. URL: https://arxiv.org/abs/2408.15506v1.

Claim. Proposition 1.1 gives the uniform-matroid polynomial; Theorem 4.2 gives its normality criterion; Conjecture 4.8 asks for central and local normal limits when rank diverges.

Problem details · Reading note

The linear spectral condition (31) and absolute separability

Proved. Lean theorem result, frozen in this repository 2026-10-11.

Problem details · Reading note

Sixfold evil and odious representation counts

Proved. Lean theorem result, frozen in this repository 2026-10-11.

Jean-Paul Allouche and Jeffrey Shallit (2021). Additive properties of the evil and odious numbers and similar sequences. DOI: 10.7169/facm/2108. URL: https://arxiv.org/abs/2112.13627v3.

Claim. Conjecture 12 asserts eventual strict increase of sixfold evil and odious representation counts, with thresholds 37 and 5.

Problem details · Reading note

Validity of the five three-outcome PI Bell inequalities

Proved. Lean theorem result, frozen in this repository 2026-10-11.

A. Aloy; G. Müller-Rigat; J. Tura; M. Fadel (2024). Deriving three-outcome permutationally invariant Bell inequalities. DOI: 10.3390/e26100816. URL: https://arxiv.org/abs/2406.11792v1.

Claim. Section III.A proposes five Table III Bell inequalities for every N > 3 and asks for proofs of their validity for arbitrary party number.

Problem details · Reading note

baek-jo-nha-2026-maximal-coherence-monotonicity

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

K. Baek; Y. Jo; H. Nha (2026). Maximal coherence of quantum measurement and the resource theory of sharpness. URL: https://arxiv.org/abs/2607.05847v1.

Claim. Section III.C asks whether maximal distance-based measurement coherence is monotone under general CP unital maps for non-dichotomic POVMs.

Problem details · Reading note

Bergman's Question 15

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

George M. Bergman (2014). Thoughts on Eggert's Conjecture. DOI: 10.1090/conm/609/12101. URL: https://arxiv.org/abs/1206.0326v2.

Claim. Questions 15 and 19 ask about powers in contracted semigroup spans and the field-condition extension of the nilpotent-algebra dimension bound from Question 16.

Problem details · Reading note

Right multiplication maximizes CHSH over the nonlocal-box orbit

Proved. Lean theorem result, frozen in this repository 2026-10-11.

P. Botteron; A. Broadbent; R. Chhaibi; I. Nechita; C. Pellegrini (2024). Algebra of Nonlocal Boxes and the Collapse of Communication Complexity. URL: https://arxiv.org/abs/2312.00725v3.

Claim. Conjecture 15 asks whether right multiplication maximizes CHSH over every parenthesization of copies of a box in the PR-SR-I triangle.

Problem details · Reading note

A qubit dephasing midpoint refutes convexity of Wasserstein complexity

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

G. Bunth; J. Pitrik; T. Titkos; D. Virosztek (2024). On the metric property of quantum Wasserstein divergences. DOI: 10.1103/PhysRevA.110.022211. URL: https://arxiv.org/abs/2402.13150v4.

Claim. Section 4.1 Proposition 3, equation (35), states D²(ρ,ρ)=4(1−√(1−|bρ|²)), corresponding for eigenvalues (1±r)/2 to 8−4(tr√ρ)²; Section 5.1 defines the Wasserstein complexity and states that its convexity is open.

Problem details · Reading note

Refutation of the fixed-resistance Fiedler invariance conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

A. Cruz Ortega; F. Menéndez-Conde (2023). Conectividad algebraica máxima para caminos con resistencia total fija. DOI: 10.15381/pesquimat.v26i2.24218. URL: https://arxiv.org/abs/2306.08187v1.

Claim. Section 5 conjectures that the maximum algebraic connectivity at fixed total effective resistance is attained at automorphism-invariant edge weights.

Problem details · Reading note

The Thue–Morse two-block fixed point has frequency one half

Proved. Lean theorem result, frozen in this repository 2026-10-11.

F. M. Dekking and M. Keane (2022). Two-block substitutions and morphic words. DOI: 10.48550/arXiv.2202.13548. URL: https://arxiv.org/abs/2202.13548v1.

Claim. Conjecture 4 of v1 asserts that the frequency of 1 in the Thue-Morse two-block fixed point with prefix 00 exists and equals one half.

Problem details · Reading note

Refutation of the symmetric Bell cross-polytope conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Sébastien Designolle, Tamás Vértesi, Sebastian Pokutta (2023). Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms. DOI: 10.1103/PhysRevA.109.022205. URL: https://arxiv.org/abs/2310.20677v3.

Claim. Section V conjectures that the symmetrised local polytope is affinely equivalent to the cross-polytope in dimension ceil(m/2).

Problem details · Reading note

Ferenczi–Zamboni Conjecture 1 on longest clustering words

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Sebastien Ferenczi; Luca Q. Zamboni (2024). Clustering and Arnoux-Rauzy words. DOI: 10.1016/j.aam.2023.102621. URL: https://arxiv.org/abs/2305.18986v1.

Claim. Section 2 defines the Burrows–Wheeler transform and perfect clustering; Section 4.1 defines Arnoux–Rauzy languages and standard words; Proposition 14 constructs a primitive perfectly clustering word conjugate to a standard Arnoux–Rauzy word; Conjecture 1 asserts that it is the longest clustering word or the longest such conjugate.

Problem details · Reading note

Symmetric Order and No Connections Do Not Force Full Palindromic Complexity

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Sébastien Ferenczi, Luca Q. Zamboni (2025). Clustering, order conditions, and languages of interval exchanges. URL: https://arxiv.org/abs/2507.17370v2.

Claim. Conjecture 1: A language on k letters satisfying a symmetric order condition and with no connection has palindromic complexity 1 for n even, k for n odd.

Problem details · Reading note

An odd midpoint refutes Fiore-Nasr-Stone Conjecture 6.8

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Evelyn Fiore, George D. Nasr, and Cooper Stone (2026). Dice Relabeling using Square-Sided Dice. URL: https://arxiv.org/html/2606.20311v1.

Claim. Definition 6.1 defines the full cyclotomic pairs; Conjecture 6.8 explicitly includes negativity at the midpoint when p is not two.

Problem details · Reading note

Foucart–Liao's Chebyshev-center regularization question

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Simon Foucart; Chunyang Liao (2024). S-Procedure Relaxation: a Case of Exactness Involving Chebyshev Centers. DOI: 10.1007/978-3-031-66497-7_1. URL: https://arxiv.org/abs/2310.09677v1.

Claim. Section 5, Question 2 asks whether the Chebyshev center is always obtained via quadratic regularization in the real optimal-recovery setting.

Problem details · Reading note

Refutation of Fucci–Stanfill coefficient nonvanishing

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Guglielmo Fucci; Jonathan Stanfill (2024). The exotic structure of the spectral ζ-function for the Schrödinger operator with Pöschl–Teller potential. DOI: 10.1007/s00023-025-01587-7. URL: https://arxiv.org/abs/2411.17860v1.

Claim. Remark B.3 conjectures that all g_{m,j}(p/q,β) are nonzero for the considered coprime 0 < p < q and nondegenerate β; the coefficient g_{3,1}(2/3,π/4) vanishes.

Problem details · Reading note

Subadditivity of the entropy of quantum states over time

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

James Fullwood and Arthur J. Parzygnat (2025). On Dynamical Measures of Quantum Information. DOI: 10.3390/e27040331. URL: https://doi.org/10.3390/e27040331.

Claim. Remark 1 conjectures that the entropy functional S(X) = -tr(X log|X|), restricted to quantum states over time (half the anticommutator of the input state tensor the identity with the Jamiołkowski matrix of a channel), is subadditive; Section 5 states the conjecture as non-negativity of the dynamical mutual information. Fullwood and Yang (arXiv:2608.28946) prove it for single-qubit inputs and state it open in general.

Problem details · Reading note

Ghosh–Ramachandran's capacity-one area decay question

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

S. Ghosh; K. Ramachandran (2026). A note on the Erdős minimal area problem. DOI: 10.1090/proc/17897. URL: https://arxiv.org/abs/2604.03036v3.

Claim. The question after Proposition 3.6 asks whether Fekete polynomials minimize unit-lemniscate area when the compact set containing their zeros is connected.

Problem details · Reading note

Ghosh–Ramachandran's connected-set Fekete area question

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

S. Ghosh; K. Ramachandran (2026). A note on the Erdős minimal area problem. DOI: 10.1090/proc/17897. URL: https://arxiv.org/abs/2604.03036v3.

Claim. The question after Proposition 3.6 asks whether Fekete polynomials minimize unit-lemniscate area when the compact set containing their zeros is connected.

Problem details · Reading note

Han–Liu's twirling phase-retrievability conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Deguang Han and Kai Liu (2023). Zero-Error Correctibility and Phase Retrievability for Twirling Channels. DOI: 10.1016/S0034-4877(24)00012-0. URL: https://arxiv.org/abs/2311.16349v1.

Claim. Section 5 conjectures pr(Phi_pi) = max{beta(Phi_pi), k} when I_k <= d < I_(k+1), for d inequivalent irreducible summands with positive multiplicities; Theorem 3.4 proves beta(Phi_pi) = max_i m_i.

Problem details · Reading note

Connectedness of weak Pareto solutions for continuous convex data

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Vu Trung Hieu (2026). A connectedness result for convex polynomial multi-objective optimization without boundedness. URL: https://arxiv.org/abs/2610.01029v1.

Claim. Section 4.3 asks whether the weak Pareto solution set remains connected for general continuous convex data on a closed convex feasible set; Theorem 1 proves connectedness for convex polynomial data without boundedness or constraint qualification.

Problem details · Reading note

Order-three Rényi information-combining equality

Proved. Lean theorem result, frozen in this repository 2026-10-11.

Christoph Hirche; Xinyue Guan; Marco Tomamichel (2023). Chain Rules for Rényi Information Combining. DOI: 10.1109/ISIT54713.2023.10206941. URL: https://arxiv.org/abs/2305.02589v1.

Claim. Conjecture V.5 asserts equality in both BSC-PSC branches at orders two and three for independent binary classical-quantum inputs; Proposition V.3, equation V.13, proves order two, and the remark after Proposition V.4 leaves the explicit order-three expression to be calculated.

Problem details · Reading note

The tree–path conjecture for the game cordiality number

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Elliot Krop; Aryan Mittal; Michael C. Wigal (2024). The Cordiality Game and the Game Cordiality Number. DOI: 10.1007/s00373-024-02798-1. URL: https://arxiv.org/abs/2403.18060v1.

Claim. Section 1 defines the cordiality game and game cordiality number; Section 3 conjectures that every tree of order n has game cordiality number at most that of the path on n vertices.

Problem details · Reading note

Principal filters of the priority lattice

Proved. Lean theorem result, frozen in this repository 2026-10-11.

Adrián Lillo and Mercedes Rosas (2026). The Priority Lattice. DOI: 10.48550/arXiv.2603.28905. URL: https://arxiv.org/abs/2603.28905v1.

Claim. Section 6, item 3 asks for isomorphic principal intervals and conjectures left-factorial and quadratic counts; item 4 asks whether every priority lattice is sublattice dismantlable.

Problem details · Reading note

Principal ideals of the priority lattice

Proved. Lean theorem result, frozen in this repository 2026-10-11.

Adrián Lillo and Mercedes Rosas (2026). The Priority Lattice. DOI: 10.48550/arXiv.2603.28905. URL: https://arxiv.org/abs/2603.28905v1.

Claim. Section 6, item 3 asks for isomorphic principal intervals and conjectures left-factorial and quadratic counts; item 4 asks whether every priority lattice is sublattice dismantlable.

Problem details · Reading note

Miña-Díaz–Rubin's Chebyshev–Faber equipotential question

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Erwin Miña-Díaz; Olof Rubin (2025). Chebyshev polynomials on equipotential curves. URL: https://arxiv.org/abs/2505.03967v1.

Claim. Problem 2.2 asks whether equality of a monic Chebyshev polynomial and a monic Faber polynomial on K persists on every exterior equipotential curve.

Problem details · Reading note

Non-backtracking Laplacian gap and minimum degree

Proved. Lean theorem result, frozen in this repository 2026-10-11.

R. Mulas; D. Zhang; G. Zucal (2024). There is no going back: Properties of the non-backtracking Laplacian. DOI: 10.1016/j.laa.2023.10.014. URL: https://arxiv.org/abs/2303.00373v2.

Claim. Conjecture 5.3: If G is not the cycle graph, then ε ≤ 1/(δ−1).

Problem details · Reading note

Sharpness of the high-degree fermionic Gaussian constants

Proved. Lean theorem result, frozen in this repository 2026-10-11.

Amir-Reza Negari; Farzin Salek; Zoltán Zimborás; Aram Harrow; Patrick Hayden; Jens Eisert (2026). Approximation theorems for fermionic Gaussian states. DOI: 10.48550/arXiv.2610.01860. URL: https://arxiv.org/abs/2610.01860v1.

Claim. Section VI asks whether the constants in the high-degree energy and free-energy estimates are optimal.

Problem details · Reading note

The vertex position ratio is unbounded on connected graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

M. Thankachy, E. J. Thomas, U. Chandran, J. Tuite, G. Di Stefano, G. Erskine (2024). On the Vertex Position Number of Graphs. DOI: 10.7151/dmgt.2491. URL: https://arxiv.org/abs/2209.00359v2.

Claim. Problem 2.4: Is the ratio vp(G)/vp⁻(G) bounded for connected graphs?

Problem details · Reading note

Wei–Liu CCZ magic-capacity threshold

Refuted. Lean theorem result, frozen in this repository 2026-10-11.

Fuchuan Wei and Zi-Wen Liu (2024). Noise robustness and threshold of many-body quantum magic. URL: https://arxiv.org/abs/2410.21215v1.

Claim. The local-depolarizing CCZ magic-capacity threshold is conjectured to be 1/3; stabilizer normal forms and robustness faithfulness specify its meaning.

Problem details · Reading note

Cube polynomial of the associated Mersenne graphs

Proved. Lean theorem result, frozen in this repository 2026-10-11.

J. Wei and Y. Yang (2024). Associated Mersenne graphs. DOI: 10.48550/arXiv.2407.08237. URL: https://arxiv.org/abs/2407.08237v1.

Claim. Questions 6.2 and 6.3 concern degree counts and cube polynomials of the Associated Mersenne graphs.

Problem details · Reading note

Degree generating function of Associated Mersenne graphs

Proved. Lean theorem result, frozen in this repository 2026-10-11.

J. Wei and Y. Yang (2024). Associated Mersenne graphs. DOI: 10.48550/arXiv.2407.08237. URL: https://arxiv.org/abs/2407.08237v1.

Claim. Questions 6.2 and 6.3 concern degree counts and cube polynomials of the Associated Mersenne graphs.

Problem details · Reading note

Asgarli–Yip two-dimensional pseudo-Paley cliques

Proved. Lean theorem result, frozen in this repository 2026-10-10.

Shamil Asgarli; Chi Hoi Yip (2024). The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes. DOI: 10.48550/arXiv.2110.07176. URL: https://arxiv.org/abs/2110.07176v5.

Claim. Definition 1.1 defines pseudo-Paley graphs; Conjecture 5.9 classifies two-dimensional cliques containing one in PP(p^4,p+1,I); Proposition 5.10 gives a conditional implication to Conjecture 5.8.

Problem details · Reading note

A completely contractive data-processing inequality for the trace-norm CMI

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão (2020). Matrix Product Density Operators: when do they have a local parent Hamiltonian?. URL: https://arxiv.org/abs/2010.14682v3.

Claim. Conjecture III.1 asserts a uniform completely contractive data-processing bound for conditional mutual information under every channel with scalar whole-space correctable algebra. Conjecture III.2 asserts the analogous uniform bound for the trace-norm CMI of Definition 2 under every channel with local trace-norm contraction ratio below one.

Problem details · Reading note

Infinitely many non-star trees admit Laplacian peak state transfer

Proved. Lean theorem result, frozen in this repository 2026-10-10.

Gabriel Coutinho, Krystal Guo, Vincent Schmeits (2025). Peak state transfer in continuous quantum walks. DOI: 10.48550/arXiv.2505.11986. URL: https://arxiv.org/abs/2505.11986v4.

Claim. Section 3 defines the bounding matrix and peak state transfer; Lemma 5.2 characterizes phase alignment; Open Problem 7.1 asks for infinitely many non-star trees admitting Laplacian peak state transfer.

Problem details · Reading note

garcia-2026-erdos64-agl29-orientation

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2609.04686. URL: https://arxiv.org/abs/2609.04686v1.

Problem details

garcia-2026-erdos64-agl31-orientation

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2609.04686. URL: https://arxiv.org/abs/2609.04686v1.

Problem details

Quantum max-flow equals min-cut in the symmetric bridge regions

Proved. Lean theorem result, frozen in this repository 2026-10-10.

Fulvio Gesmundo; Vladimir Lysikov; Vincent Steffan (2025). Quantum max-flow in the bridge graph. DOI: 10.1007/s00031-024-09863-2. URL: https://arxiv.org/abs/2212.09794v2.

Claim. Conjectures 1.4 and 3.14 predict quantum max-flow equals quantum min-cut in the symmetric bridge regions; Proposition 3.18 reduces them to Conjecture 3.15 at width three.

Problem details · Reading note

Weak-product closure of the edge-complexity equality class

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

V. Gupta, A. Iosevich, J. Iosevich, B. Song, H. Tian (2026). Edge complexity of graphs. DOI: 10.48550/arXiv.2607.15598. URL: https://arxiv.org/abs/2607.15598v1.

Claim. The Remark after Corollary 3.10 asks whether the energy-equality class is closed under unrestricted weak products.

Problem details · Reading note

Exact preparation-noncontextuality threshold of the twenty-four-cell

Proved. Lean theorem result, frozen in this repository 2026-10-10.

Jonte R. Hance (2026). Why three? A two-level system with four mutually unbiased questions. URL: https://arxiv.org/abs/2609.10078v1.

Claim. Whether the further preparation equivalences lower the twenty-four-cell noise threshold is open.

Problem details · Reading note

Hou–Zhao: the dimension of a sum-free function's code

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2609.31489. URL: https://arxiv.org/abs/2609.31489v1.

Problem details

Hou–Zhao: minimum-weight independence of sum-free codes

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2609.31489. URL: https://arxiv.org/abs/2609.31489v1.

Problem details

kasel-2026-displacement-ladder

Proved. Lean theorem result, frozen in this repository 2026-10-10.

William Kasel (2026). Structural rigidity in the Erdős–Graham two-set permutation problem. URL: https://arxiv.org/abs/2609.02939v1.

Claim. The full order gadget is nevertheless conjectured infeasible at every M ≥ 16.

Problem details · Reading note

A matrix refutation of Krishna's C*-algebraic Novak conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

K. Mahesh Krishna (2021). C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture. DOI: 10.48550/arXiv.2108.06662. URL: https://arxiv.org/abs/2108.06662v1.

Claim. Conjecture 4.3 asserts positivity of the ordered cosine-product matrix over every unital C*-algebra; Theorem 4.4 proves the commutative case.

Problem details · Reading note

The C*-algebraic de Bruin–Sharma Conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

K. Mahesh Krishna (2022). C*-algebraic Schoenberg Conjecture. DOI: 10.48550/arXiv.2206.06653. URL: https://arxiv.org/abs/2206.06653v1.

Claim. Section 2 defines the ordered derivative P′ and states Conjectures 2.1, 2.3 and 2.4; Theorem 2.2 proves the degree-two case of Conjecture 2.1.

Problem details · Reading note

The C*-algebraic Dubinin–Sugawa norm mean value conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

K. Mahesh Krishna (2022). C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture. DOI: 10.48550/arXiv.2206.08154. URL: https://arxiv.org/abs/2206.08154v1.

Claim. Conjecture HIGHERMEAN (journal Conjecture 2.3) and Conjecture DUALSMALE (Conjecture 3.1) ask for a critical point controlling the Smale ratios, respectively the Dubinin-Sugawa dual ratio, in a commutative C*-algebra; the paper proves both degree-2 cases.

Problem details · Reading note

The Higher Order C*-algebraic Smale Mean Value Conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

K. Mahesh Krishna (2022). C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture. DOI: 10.48550/arXiv.2206.08154. URL: https://arxiv.org/abs/2206.08154v1.

Problem details · Reading note

The C*-algebraic Kushel–Tyaglov Conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

K. Mahesh Krishna (2022). C*-algebraic Schoenberg Conjecture. DOI: 10.48550/arXiv.2206.06653. URL: https://arxiv.org/abs/2206.06653v1.

Claim. Section 2 defines the ordered derivative P′ and states Conjectures 2.1, 2.3 and 2.4; Theorem 2.2 proves the degree-two case of Conjecture 2.1.

Problem details · Reading note

The C*-algebraic Schoenberg Conjecture is false

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

K. Mahesh Krishna (2022). C*-algebraic Schoenberg Conjecture. DOI: 10.48550/arXiv.2206.06653. URL: https://arxiv.org/abs/2206.06653v1.

Claim. Section 2 defines the ordered derivative P′ and states Conjectures 2.1, 2.3 and 2.4; Theorem 2.2 proves the degree-two case of Conjecture 2.1.

Problem details · Reading note

lesaulnier-vijay-2011-beta3-lower-density

Refuted. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.1004.1740. URL: https://arxiv.org/abs/1004.1740.

Problem details

manattu-lakshmanan-2025-cubic-ar-graphs

Proved. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2502.11760. URL: https://arxiv.org/abs/2502.11760v1.

Problem details

Sublinear edge completion for consecutive cycle sums

Proved. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2610.08889. URL: https://arxiv.org/abs/2610.08889.

Problem details

The valuation-step greedy sequence is a permutation

Proved. Lean theorem result, frozen in this repository 2026-10-10.

URL: https://oeis.org/A382357.

Problem details

Every integer occurs in the factorial Heraclitus transform

Proved. Lean theorem result, frozen in this repository 2026-10-10.

URL: https://oeis.org/A393434.

Problem details

Twin-prime sigma-gcd three forces twice a square

Proved. Lean theorem result, frozen in this repository 2026-10-10.

URL: https://oeis.org/A394399.

Problem details

Prime twin-center sigma-gcd values are two or three

Proved. Lean theorem result, frozen in this repository 2026-10-10.

OEIS Foundation Inc. (2026). OEIS A394757. URL: https://oeis.org/A394757.

Claim. OEIS A394757 records the twin-prime center sequence whose terms satisfy the prime sigma-gcd condition and states that all terms are divisible by 18.

Problem details · Reading note

oeis-a397578-sigma-neighbour-peaks

Proved. Lean theorem result, frozen in this repository 2026-10-10.

URL: https://oeis.org/A397578.

Problem details

oeis-a399155-value-finiteness

Proved. Lean theorem result, frozen in this repository 2026-10-10.

URL: https://oeis.org/A399155.

Problem details

A sharp Weyl–Bloch ℓ1 entanglement criterion for two qutrits

Proved. Lean theorem result, frozen in this repository 2026-10-10.

Tobias C. Sutter, Christopher Popp and Beatrix C. Hiesmayr (2026). Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits. DOI: 10.1103/z7jn-4try. URL: https://arxiv.org/abs/2508.18393v2.

Claim. Section VI states as open whether an entanglement criterion based on the entry-wise 1-norm of the Weyl–Bloch vector can be derived for general bipartite qudits, and reports numerically that separable pure two-qutrit states seem to satisfy ‖β‖₁ ≤ 25 while pure states reach ≈ 25.9735.

Problem details · Reading note

Tyagi's sharper Sprague–Grundy bound for Divisor Nim

Proved. Lean theorem result, frozen in this repository 2026-10-10.

DOI: 10.48550/arXiv.2610.06925. URL: https://arxiv.org/abs/2610.06925v1.

Problem details

Token-1 spectral contraction for every finite positive-velocity chain

Proved. Lean theorem result, frozen in this repository 2026-10-10.

T. Wallner, D. Krupke, A. Schmidt, and S. P. Fekete (2026). Pass the Bucket: Efficient, Robust, Local Load Balancing for Teams of Heterogeneous Robots. DOI: 10.48550/arXiv.2608.27085. URL: https://arxiv.org/abs/2608.27085v1.

Claim. Conjecture 1 asserts strict spectral contraction and decay of centered trajectories under repeated Token-1 damped transfers.

Problem details · Reading note

No generalized Choi map detects absolutely PPT entanglement

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Srinivasan Arunachalam, Nathaniel Johnston and Vincent Russo (2015). Is absolute separability determined by the partial transpose?. URL: https://arxiv.org/abs/1405.5853v3.

Claim. Section 7 states as open that none of the positive generalized Choi maps Φ_{b,c} of Section 5.3 detects entanglement in absolutely PPT states of M_3 ⊗ M_3.

Problem details · Reading note

A two-element T1-groupoid whose reduct is not a reduced T1-groupoid

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

V. G. Bardakov; B. B. Chuzhinov; I. A. Emelyanenkov; M. E. Ivanov; T. A. Kozlovskaya; V. E. Leshkov (2024). Set-Theoretical Solutions of the n-Simplex Equation. DOI: 10.1134/S1055134424010012. URL: https://arxiv.org/abs/2206.08906v1.

Claim. The n-simplex equation R_1 R_2 ... R_(n+1) = R_(n+1) ... R_2 R_1 for a map R of X^n acts on X^N, N = n(n+1)/2, with the multi-indices given by the rows of the matrix MI_n; a solution is simple when it has the form (x_1, ..., x_n) -> (x_s(1), ..., x_s(n)). Question 2 (Question 4.22 of the journal version) asks for which n > 2 there are non-identity permutations without fixed points that give solutions of the n-simplex equation. Section 9.2 defines the first tetrahedral 4-groupoid (T1-groupoid) and the reduced T1-groupoid and states that it is not clear whether some T1-groupoid yields, after forgetting the operations ◁ and ▷, a system that is not a reduced T1-groupoid.

Problem details · Reading note

The ceiling formula for monochromatic sublattices is false

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.02833. URL: https://arxiv.org/abs/2610.02833v1.

Problem details

The proposed union-closed counting threshold is false

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.02833. URL: https://arxiv.org/abs/2610.02833v1.

Problem details

Pretty good state transfer in double blow-ups of paths

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Bikash Bhattacharjya, Hermie Monterde, Hiranmoy Pal (2024). Quantum walks on blow-up graphs. DOI: 10.1088/1751-8121/ad6653. URL: https://arxiv.org/abs/2308.13887v2.

Claim. Conjecture 1 excludes PGST between twin copies of every vertex divisible by 2^(t-1) in the double blow-up of P_(2^t*r-1), for t >= 2 and an odd prime r; P_11 at source vertex 4 refutes it.

Problem details · Reading note

The exact compatibility degree of four dichotomic qubit measurements

Proved. Lean theorem result, frozen in this repository 2026-10-09.

A. Bluhm, E. Evert, I. Klep, V. Magron, I. Nechita (2025). Inclusion constants for free spectrahedra with applications to quantum incompatibility. DOI: 10.48550/arXiv.2512.17706. URL: https://arxiv.org/abs/2512.17706v1.

Claim. For γ = 4/(1+√3) and every θ ∈ [0, π/2] the feasibility SDP (37) at X(θ) has a solution (Theorem 5.6, by a two-piece construction split at π/8); Appendix A.1 conjectures that one explicit single-formula tuple (C₁, C₂ given, C₃–C₆ by Eq. (47)) is feasible on the whole interval. Definitions 2.13–2.16 define joint-measurement compatibility and white-noise degrees; Conjecture 6.10 states sℂ(2,4) = 2/√13; Proposition 6.12 gives the qubit vector program.

Problem details · Reading note

Caldwell and coauthors' super Catalan action-graph identity

Proved. Lean theorem result, frozen in this repository 2026-10-09.

D. Caldwell, A. Cochran, N. Glisson, B. Jennings, K. McDicken, L. Proctor, S. Klanderman, A. Tebbe (2024). Catalan Number Sequences and Generalized Action Graphs. DOI: 10.48550/arXiv.2507.22719. URL: https://arxiv.org/abs/2507.22719v1.

Claim. Conjecture 5.6. The subsequent super Catalan number can be computed from the n-table of its previous action graph via S(0, n + 1) = Σ_{ℓ=0}^{n} (2/2^ℓ) Σ_{v=0}^{n} Kℓ,v,n.

Problem details · Reading note

Chen–He–Lai–Li Conjecture 6.1: the local average at a stem

Proved. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.11446. URL: https://arxiv.org/abs/2610.11446v1.

Problem details

Christiansen's fermionic correlational bound

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Martin Ravn Christiansen (2025). A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators. URL: https://arxiv.org/abs/2505.21167v1.

Claim. Conjecture 3 asks for a universal positive C bounding the two-body Rayleigh expectation by N(1-(N-2)/2 sum lambda_k^4+C(N lambda_max^2)^2) for even N with N lambda_max^2 <= 1.

Problem details · Reading note

A rank-one partner breaks the negative-q bound for a traceless matrix

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

D. Chruściński, G. Kimura, H. Ohno, T. Singal (2022). Bounding the Frobenius norm of a q-deformed commutator. DOI: 10.48550/arXiv.2202.11520. URL: https://arxiv.org/abs/2202.11520v2.

Claim. For q > 0 the bound ‖AB − qBA‖²_F ≤ (1 + q²)‖A‖²_F‖B‖²_F holds when A or B is normal and can fail in general; Conjecture 1 asserts that it holds and is sharp when A or B is traceless, proved there for n = 2; Conjecture 2 asserts for q ≤ 0, A or B traceless, the sharp bound ‖[A,B]_q‖²_F ≤ max[g(n)(1−q)², 1+q²]‖A‖²_F‖B‖²_F with g(n) = (n²−3n+3)/(n(n−1)).

Problem details · Reading note

Extended local permutations do not suffice for block-shift form

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Gunther Dirr; Uwe Helmke; Martin Kleinsteuber; Thomas Schulte-Herbrüggen (2008). Relative C-numerical ranges for applications in quantum control and quantum information. DOI: 10.1080/03081080701535898. URL: https://arxiv.org/abs/math-ph/0702005v1.

Claim. Conjecture 3.6: Every element of E(tloc) is similar to a block-shift matrix via an element of Πex_loc := Πloc·Πout.

Problem details · Reading note

Minimal bipartite Erdős–Gyárfás counterexamples are 2-edge-connected

Proved. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2609.28594. URL: https://arxiv.org/abs/2609.28594v1.

Problem details

The positive-branch Wigner-distance deficit does not factor through the equatorial input

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Soumyojyoti Dutta; Tushar (2026). A Phase-Space Geometric Measure of Magic in Qubit Systems. DOI: 10.48550/arXiv.2603.20792. URL: https://arxiv.org/abs/2603.20792v3.

Claim. Conjecture 5.5 asks that for equatorial magic ρ and qubit σ with s(σ) > 0 the tensor deficit factor as C(ρ)·f(σ) for a universal f; Conjecture 5.6 asks for equatorial multiplicativity of the qubit Wigner distance; Conjecture 5.7 asks for self-tensor superadditivity on the nonpositive Bloch-product branch.

Problem details · Reading note

Early state exclusion in seven-site chains with symmetric spectrum

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Mia Gabriella Escobar; Valentin Garcia; Anastasiia Minenkova (2025). Early State Exclusion in 7-Qubit Spin Chains. DOI: 10.48550/arXiv.2507.18767. URL: https://arxiv.org/abs/2507.18767v1.

Claim. Section 4 conjectures that a 7 x 7 Jacobi matrix with symmetric spectrum realizing perfect state transfer does not have early state exclusion if and only if its positive eigenvalues are integer multiples of the smallest positive eigenvalue; Theorems 3.2 and 3.4 prove the spectra {0, +-1, +-2m, +-(2m+1)} (no early state exclusion) and {0, +-(2m+1), +-(2m+2), +-(2m+3)} (early state exclusion exactly 2m times).

Problem details · Reading note

Garg's Fibonacci prefix antipower conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Swapnil Garg (2021). Antipowers in Uniform Morphic Words and the Fibonacci Word. DOI: 10.46298/dmtcs.7134. URL: https://arxiv.org/abs/1907.10816v4.

Claim. Let $F_n$ be an even Fibonacci number. Then, there is an $(F_n-1)$-antipower with block length $\frac{F_n}{2}+F_{n-1}$ that is a prefix of $\mathbf f$.

Problem details · Reading note

Every vertex an identity Seidel switch

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Severino V. Gervacio (2026). On identity Seidel switches. URL: https://arxiv.org/abs/2601.04530v1.

Claim. Problem 6.1 asks for a complete characterization of finite nonempty simple graphs in which every vertex is a vertex-ISS.

Problem details · Reading note

Hoster–Stump Conjecture 1.5: simplicial-poset Chow polynomials

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Elena Hoster and Christian Stump (2025). Chow polynomials of simplicial posets with positive h-vector are real-rooted. URL: https://arxiv.org/abs/2508.15538v1.

Claim. Conjecture 1.5 asserts real-rootedness of the Chow, dual Chow and augmented Chow polynomials of every simplicial poset, together with their interlacing.

Problem details · Reading note

The Gorenstein pairs for alternating adjacent-sum polytopes

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Xinru Jiang, Suzhen Wen, Yueming Zhong (2026). Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series. URL: https://arxiv.org/abs/2607.14887v1.

Claim. Question Q1 asks for all Gorenstein pairs (s,r) with s ≥ 2 for the alternating adjacent-sum polytope in dimension 2r.

Problem details · Reading note

The Fourier LDOI LOCC lower bound is not tight

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Nathaniel Johnston, Vincent Russo (2026). Distinguishability of locally diagonal orthogonally invariant quantum states. URL: https://arxiv.org/abs/2604.12808v1.

Claim. Example 17 asks whether opt_LOCC(E) = 1/2 − (n − 2)/(2n²) for the uniform Fourier LDOI ensemble with n ≥ 3; it proves opt_PPT(E) = opt_SEP(E) = 1/2.

Problem details · Reading note

First occurrences in Kimberling's array

Proved. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.05776. URL: https://arxiv.org/abs/2610.05776v1.

Problem details

Optimality of the coefficient in the entanglement-breaking norm inequality

Proved. Lean theorem result, frozen in this repository 2026-10-09.

E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Claim. Theorem 1: the Bloch data (A, c) of an entanglement-breaking channel on M_d satisfy ‖A‖_*² + (d(d−1)/2)|c|² ≤ (d−1)²; open question 3 asks whether the coefficient d(d−1)/2 is optimal.

Problem details · Reading note

Kumar's remaining rectangular weak two-metric dimensions

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Aryan Kumar (2026). Weak k-metric dimension of Hamming graphs: rectangular products and near-maximum parameters. DOI: 10.48550/arXiv.2609.32161. URL: https://arxiv.org/abs/2609.32161.

Claim. Section 6 proves an eventual near-maximum formula with an explicit sufficient cutoff; Section 7 asks for the sharp nonbinary threshold.

Problem details · Reading note

Pure-target quantum Černý complexity

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Pui Hang Lee, Pui Yee Lee and Bjørn Kjos-Hanssen (2026). Quantum Černý complexity of binary words. URL: https://arxiv.org/abs/2609.40154.

Claim. The quantum Černý complexity qc(w) is the least d for which two quantum channels on d x d density matrices and a start state make w the unique shortest synchronizing word; 2 <= qc(w) <= ceil(sqrt(|w|+1)), qc(0^m) = ceil(sqrt(m+1)), qc(01^n0) = 2; open problem 1 conjectures qc(01101001) = 2.

Problem details · Reading note

The coefficient matrix of a 2p-term separable qubit state has 2^(p-1) forms

Proved. Lean theorem result, frozen in this repository 2026-10-09.

D. Li (2025). A necessary and sufficient condition for genuinely entangled n-qubit states with six non-zero coefficients. DOI: 10.48550/arXiv.2510.16561. URL: https://arxiv.org/abs/2510.16561v1.

Claim. Section 6 asks how many forms of the corresponding 2 by p coefficient matrix exist for a separable, not trivially separable n-qubit state with m = 2p non-zero coefficients (p prime), after recording two forms for m = 4 and four forms for m = 6.

Problem details · Reading note

The averaged Zeitlin Ricci curvature tends to minus half of the harmonic number minus one

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Leandro Lichtenfelz, Klas Modin, Stephen C. Preston (2026). Ricci curvature for hydrodynamics on the sphere. DOI: 10.1007/s00220-025-05533-w. URL: https://arxiv.org/abs/2508.09833v1.

Claim. Conjecture 1 states the four spin-weighted Wigner six-j identities (2.10)-(2.13); their validity supplies the hypothesis of Theorem 3. Conjecture 2 states that for each fixed l > 1 the averaged Ricci curvature of the Zeitlin metric on SU(N) becomes negative for large N and tends to -(H_l - 1)/2.

Problem details · Reading note

Mane's negative-index factorization

Proved. Lean theorem result, frozen in this repository 2026-10-09.

S. R. Mane (2026). Identically vanishing k-generalized Fibonacci polynomials. URL: https://arxiv.org/abs/2507.11596v4.

Claim. Conjecture 6.6 asserts a negative-index root-amplitude upper bound and attainment at every negative multiple of k; the attainment clause fails at k = 3 and n = -15.

Problem details · Reading note

The negative-index root-amplitude attainment clause is refuted

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

S. R. Mane (2026). Identically vanishing k-generalized Fibonacci polynomials. URL: https://arxiv.org/abs/2507.11596v4.

Claim. Conjecture 6.6 asserts a negative-index root-amplitude upper bound and attainment at every negative multiple of k; the attainment clause fails at k = 3 and n = -15.

Problem details · Reading note

O'Bryant §5.7: the golden minimizer

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2411.08139. URL: https://arxiv.org/abs/2411.08139v2.

Problem details

O'Bryant's seven-point positive-real sum-product cell

Proved. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2411.08139. URL: https://arxiv.org/abs/2411.08139v2.

Problem details

Minimal matrices of fixed RSK shape need not be Hankel

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Nimisha Pahuja (2026). Minimal Inversions in Integer Matrices of Fixed RSK Shape. URL: https://arxiv.org/abs/2602.14931v1.

Claim. Conjecture 1.1: every minimal nonnegative integer matrix of a partition shape with n positive parts is symmetric and Hankel.

Problem details · Reading note

Pham–Vu Question 3.9: diameter of graphs with well-covered powers

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.09300. URL: https://arxiv.org/abs/2610.09300v1.

Problem details

A commuting obstruction to the nonlinear measurement monotone

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Albert Rico; Karol Życzkowski (2024). Discrete dynamics in the set of quantum measurements. DOI: 10.1088/1751-8121/ad7dc2. URL: https://arxiv.org/abs/2308.05835v2.

Claim. Conjecture 1 asserts that blockwise bistochastic dynamics allow an input ordering whose every centered prefix has 2-norm at least that of any prescribed output ordering.

Problem details · Reading note

Conjecture 3.1: APN functions and large binary flats

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2609.22394. URL: https://arxiv.org/abs/2609.22394v2.

Problem details

Sahbi's Conjecture 4.1: bipartite equality for all positive k

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.03785. URL: https://arxiv.org/abs/2610.03785v2.

Problem details

Sahbi's Conjecture 4.2: bipartite two-independence and upper star formation

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.03785. URL: https://arxiv.org/abs/2610.03785v2.

Problem details

Sahbi's Conjecture 7.2: triangle-free equality at two

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.03785. URL: https://arxiv.org/abs/2610.03785v2.

Problem details

Shah–Kiselev Conjecture 3.1: the late growing ratio

Proved. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.2610.08636. URL: https://arxiv.org/abs/2610.08636v1.

Problem details

shapiro-2026-unweighted-fourth-moment-inversion

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Boris Shapiro (2026). The (n−2,2)-Spectrum of a Graph. URL: https://arxiv.org/abs/2605.17501v2.

Claim. The unweighted fourth-moment inversion question asks whether Laplacian and cubic data recover the four-edge support-forest counts.

Problem details · Reading note

Štampach–Waclawek's alternative Birman weight expansion

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

František Štampach; Jakub Waclawek (2026). Optimal discrete p-Hardy–Rellich–Birman inequalities. URL: https://arxiv.org/abs/2605.25238v1.

Claim. Conjecture 5.6(iii) asserts a convergent expansion in negative powers of n with entirely non-negative coefficients for the alternative Birman weight at every positive order and p > 1.

Problem details · Reading note

Sun's Conjecture 3.8: circular permutations with primitive adjacent products

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

DOI: 10.48550/arXiv.1309.1679. URL: https://arxiv.org/abs/1309.1679.

Problem details

Finite-Fock beam-splitter states need not be extreme Wigner-positive states

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Zacharie Van Herstraeten; Jack Davis; Nuno C. Dias; João N. Prata; Nicolas J. Cerf; Ulysse Chabaud (2025). Extreme non-negative Wigner functions. URL: https://arxiv.org/abs/2512.14831v3.

Claim. Every beam-splitter state formed from two Fock-bounded pure inputs is an extreme Wigner-positive state.

Problem details · Reading note

The kernel criterion for witness optimality does not imply the trace criterion

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

Frederik vom Ende and Simon Cichy (2025). Simple Sufficient Criteria for Optimality of Entanglement Witnesses. URL: https://arxiv.org/abs/2505.15615v2.

Claim. Section IV asks whether the trace criterion of Corollary 2 (saturation of ⟨Ω|W|Ω⟩ ≥ −tr(W)/min{m,n} by a maximally entangled Ω) is equivalent to the kernel criterion of Theorem 2 (a vector of full Schmidt rank in the kernel of W + tr₂(W) ⊗ 1, or of W + 1 ⊗ tr₁(W)).

Problem details · Reading note

Disorder-free localization on even hypercubes

Proved. Lean theorem result, frozen in this repository 2026-10-09.

Ce Wang (2026). Spectral Criterion for Disorder-Free Localization of Quantum Walks on Hypercube: QBN Approach. URL: https://arxiv.org/abs/2609.07267v1.

Claim. Conjecture 4.1 asserts a nonuniform fixed point for the all-plus spin-coupled Grover walk on every even hypercube of dimension at least four.

Problem details · Reading note

In-between-state optimality for transmissivity sensing with a beta prior

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

B. Zhou; B. A. Bash; S. Guha; C. N. Gagatsos (2023). Bayesian minimum mean square error for transmissivity sensing. DOI: 10.1103/PhysRevResearch.5.043033. URL: https://arxiv.org/abs/2304.05539v1.

Claim. Section V conjecture: in-between states, up to phase, minimize the two-point-prior MMSE at real mean photon number. Sections VI-VII conjecture: for the beta prior PDF the in-between states (Fock states at integer mean photon number) are optimal.

Problem details · Reading note

The isotropic aggregate of a two-qudit state in dimension five is below six

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Claim. Conjecture 2: for an odd prime d and |Ψ⟩ ∈ H_d^{⊗n}, 0 ≤ κ(Ψ,𝒯) ≤ 1 for every stochastic Lagrangian subspace 𝒯 ∈ Σ(d), together with three aggregate inequalities that follow from it; κ(Ψ,𝒯) = tr[R(𝒯)(|Ψ⟩⟨Ψ|)^{⊗3}], R(𝒯) = r(𝒯)^{⊗n}, r(𝒯) = Σ_{(x;y)∈𝒯} |x⟩⟨y|.

Problem details · Reading note

At dimension five the full and non-symmetric aggregates of a two-qudit state fall below their conjectured lower bounds

Refuted. Lean theorem result, frozen in this repository 2026-10-09.

H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Problem details · Reading note

A Unique Indecomposable Basis Without Intersection Closure

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

Pranab Basu, Navin Kashyap (2019). The Lattice Structure of Linear Subspace Codes. URL: https://arxiv.org/abs/1911.00721v1.

Claim. A linear code 𝒰 in ℙ_q(n) has a unique indecomposable basis if and only if 𝒰 is closed under intersection.

Problem details · Reading note

A single-formula feasibility witness for the qubit simplex-and-interval inclusion

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Problem details · Reading note

Half letter frequency characterizes balanced borders

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Miklós Bóna, Balázs Maga, Jacob Richey (2026). Letter frequency in shifts of finite type with one forbidden word. URL: https://arxiv.org/abs/2606.06655v2.

Claim. A word $w$ has $\rho^w = 1/2$ if and only if $w$ has balanced borders.

Problem details · Reading note

Refutation of Braunstein–Ghosh–Severini Conjecture 6.7

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

Samuel L. Braunstein; Sibasish Ghosh; Simone Severini (2006). The Laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states. DOI: 10.1007/s00026-006-0289-3. URL: https://arxiv.org/abs/quant-ph/0406165v2.

Claim. Conjecture 6.7 asserts that the star maximizes entanglement of formation among connected graph density matrices.

Problem details · Reading note

Dense Independent Sets of the Square Grid Contain a Full Cross

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Yair Caro; Mirko Petruševski; Riste Škrekovski; Zsolt Tuza (2025). The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs. URL: https://arxiv.org/abs/2510.01897v1.

Claim. Studies odd independent sets (independent sets S such that every vertex outside S has no neighbour or an odd number of neighbours in S) in grids and chessboard graphs, proves the density bounds 3/8 ≤ ρ_od ≤ 5/13 for the infinite square grid, and asks in Problem 29 whether every independent set of size (3/8 + ε_n)n² in P_n □ P_n, ε_n → 0, contains all four neighbours of some vertex.

Problem details · Reading note

Choi–Kiem–Kye maximal bi-rank edge states

Proved. Lean theorem result, frozen in this repository 2026-10-08.

J. Choi, Y.-H. Kiem, and S.-H. Kye (2020). Entangled edge states of corank one with positive partial transposes. DOI: 10.1063/1.5122836. URL: https://arxiv.org/abs/1903.10745v2.

Claim. Conjecture 6.4 states that for every n ≥ 3 the open set of parameters yielding PPT entangled edge states of bi-rank (n²−1,n²−2n+3) is nonempty.

Problem details · Reading note

Orthocross MIC elements are never orthogonal, and their Gram entries tend to zero

Proved. Lean theorem result, frozen in this repository 2026-10-08.

John B. DeBrota; Christopher A. Fuchs; Blake C. Stacey (2020). The Varieties of Minimal Tomographically Complete Measurements. DOI: 10.1142/S0219749920400055. URL: https://arxiv.org/abs/1812.08762v5.

Claim. For an orthonormal basis of C^d, the orthocross MIC has elements E_alpha = Omega^(-1/2) Pi_alpha Omega^(-1/2), where Pi_alpha are the d^2 projectors onto |j>, (|j> + |k>)/sqrt(2) and (|j> + i|k>)/sqrt(2) (j < k) and Omega is their sum; its Gram matrix is G_ij = tr(E_i E_j); the authors conjecture that for any orthocross MIC the entries of G^(-1) are integers or half-integers; Conjecture 1: A rank-1 MIC in dimension 3 can have no more than 7 pairs of orthogonal elements; Conjecture 2: the entries in G for orthocross MICs can become arbitrarily small with increasing d, but no two elements of an orthocross MIC can be exactly orthogonal.

Problem details · Reading note

Filimonova–Puzynina's rank-one iteration-bound question

Proved. Lean theorem effective_iteration_bound, frozen in this repository 2026-10-08.

Arina Filimonova; Svetlana Puzynina (2026). On abelian periodicity of purely morphic words. URL: https://arxiv.org/abs/2605.30306v1.

Claim. Section 4 asks for an upper bound on the iteration K in the primitive binary rank-one criterion for abelian periodicity.

Problem details · Reading note

The Choi spectrum of unital product-unitary-equivariant maps

Proved. Lean theorem result, frozen in this repository 2026-10-08.

A. García-Velo, A. Ibort (2026). Schwarz maps with symmetry. DOI: 10.48550/arXiv.2601.02282. URL: https://arxiv.org/abs/2601.02282v1.

Claim. Remark V.2: the eigenvalues of the Choi matrix of a unital, Hermiticity-preserving U(n₁)⊗U(n₂)-equivariant map are the four values of Lemma V.8 with multiplicities 1, n₁²−1, n₂²−1 and (n₁²−1)(n₂²−1); Lemma V.8 states them only for n₁, n₂ ∈ {2, 3}, and the general case is stated as a conjecture.

Problem details · Reading note

Gazeau Problem 4.3: no normal Hilbert realization

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

Jean-Pierre Gazeau (2026). Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator. URL: https://arxiv.org/abs/2609.28511v1.

Claim. Problem 4.3 asks for essential self-adjointness, skew-adjointness, or normality of the ordinary Schwartz block action under a natural Hilbert structure.

Problem details · Reading note

Ghosh, Birbonshi and Ojha: the unique polynomial root

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Arobinda Ghosh, Riddhick Birbonshi, Sarita Ojha (2026). On the numerical radius of a class of weighted shift operators. URL: https://arxiv.org/abs/2608.17486v1.

Claim. Section 2 defines the degree-eight polynomial g and asks whether it has a unique root in (1/3,1) for all real s,t>0 and 0<q<1.

Problem details · Reading note

Kay's transfer-rate conjecture for perfect state transfer chains

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

Alastair Kay (2010). A Review of Perfect State Transfer and its Application as a Constructive Tool. DOI: 10.1142/S0219749910006514. URL: https://arxiv.org/abs/0903.4274v3.

Claim. Lemma 6 (subsection Transfer Rate): for eigenvalues fulfilling the perfect state transfer condition, the rate M/2t_0 is perfectly achieved for an integer M if and only if the sums R_k of (-1)^n/B'(lambda_n) over the residue classes of (t_0/pi)(lambda_n - lambda_1) modulo M, k = 0, ..., M - 1, are all equal. The review conjectures that this condition cannot be fulfilled for any M > 2 and states that it has a proof for M > N/2.

Problem details · Reading note

A three-level QSP segment that collapses without a common monomial factor

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

L. Laneve, S. Wolf (2024). On multivariate polynomials achievable with quantum signal processing. DOI: 10.22331/q-2025-02-20-1641. URL: https://arxiv.org/abs/2407.20823v2.

Claim. Conjecture 8: if a three-level Protocol C segment A_m W̃ ⋯ A_1 W̃ maps a polynomial state of effective dimension ≤ 2 to one of effective dimension ≤ 2 while every intermediate state has effective dimension > 2, then the segment acts as a^k b^h · U on some pair of subspaces of dimension d ≥ 2; stated as the missing step towards Conjecture 7.

Problem details · Reading note

lee-2026-reduction-operator-invertibility

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Eunghyun Lee (2026). Integrability of multispecies long-range swap models with species-dependent interpolation. DOI: 10.1088/1742-5468/ae80b5. URL: https://arxiv.org/abs/2604.12136v1.

Claim. Remark 3.1 conjectures that every reduction operator 𝔄_k is invertible for every species-dependent parameter μ_i in [0,1].

Problem details · Reading note

Tightness of the 0-fidelity lower bound and the worst-case processes

Proved. Lean theorem result, frozen in this repository 2026-10-08.

K. Mayer (2021). A short note on the 0-fidelity. DOI: 10.48550/arXiv.2109.09629. URL: https://arxiv.org/abs/2109.09629v1.

Claim. For every n-qubit process, the process fidelity and the 0-fidelity defined by product single-qubit SIC states satisfy 1 − (3/2)(1 − F_0) ≤ F ≤ F_0; an SDP for n ≤ 4 suggests that the lower bound is tight for all n, which is conjectured, and a description of the worst-case process is stated as an open problem.

Problem details · Reading note

Weighted Böttcher–Wenzel inequality, case (ii)

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Aina Mayumi; Gen Kimura; Hiromichi Ohno; Dariusz Chruściński (2024). Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics. DOI: 10.1016/j.laa.2024.07.013. URL: https://arxiv.org/abs/2403.04199v2.

Claim. Conjecture 1, case (ii), equation (15), bounds the weighted commutator norm by sqrt((lambda_min + lambda_max)/lambda_min) times the weighted norm of A and the Frobenius norm of B.

Problem details · Reading note

McNulty's path-robustness conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Daniel McNulty (2025). A Graph-Theoretic Approach to Quantum Measurement Incompatibility. URL: https://arxiv.org/abs/2511.15954.

Claim. Conjecture 1 asserts that every path P_{2n}, path P_{2n+1} and even cycle C_{2n+2} has incompatibility robustness (2/(2n+2)) csc(pi/(2n+2)), for n >= 1.

Problem details · Reading note

The fully entangled fraction is at most one minus half the steering-ellipsoid centre

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Problem details · Reading note

OEIS A399232 radical identity

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Rui Ferreira (2026). OEIS A399232, odd-power residue sum and radical identity. URL: https://raw.githubusercontent.com/oeis/oeisdata/2ce625d68f85e35e26e15c0d9681f69280c095e7/seq/A399/A399232.seq.

Claim. a(n) appears to satisfy rad(2*n+1) = (2*n+1)^2 / ((2*n+1)^2 - a(n)).

Problem details · Reading note

Maximum length of a two-dense divisor block

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Omar E. Pol (2026). OEIS A400194, maximum length of a two-dense divisor block. URL: https://oeis.org/A400194.

Claim. Conjecture: a(n) = A001511(n) * A400195(n).

Problem details · Reading note

qian-wang-2025-separable-states-stab-local-unitaries

Proved. Lean theorem result, frozen in this repository 2026-10-08.

Dongheng Qian; Jing Wang (2025). Quantum non-local nonstabilizerness. DOI: 10.1103/PhysRevA.111.052443. URL: https://arxiv.org/abs/2502.06393v4.

Problem details · Reading note

Concealment-equivalent channels with different adjoint kernels

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

Mohd Asad Siddiqui; Zizhu Wang (2026). Operational Concealment of Measurement Incompatibility by Quantum Channels: Rank Loss versus Contraction. DOI: 10.48550/arXiv.2607.11762. URL: https://arxiv.org/abs/2607.11762v2.

Claim. For channels with a common output space, equality of adjoint kernels (restricted to Hermitian operators) implies equality of the sets of concealed POVM pairs; whether equality of adjoint kernels is also necessary for concealment-equivalence is stated as open.

Problem details · Reading note

Faal's Open Problem 1: non-chordal graphs with unstable restricted clique polynomials

Proved. Lean theorem result, frozen in this repository 2026-10-08.

H. Teimoori Faal (2026). A Bivariate B-Restricted Clique Polynomial: From Local Neighborhoods to Global Expansion. URL: https://arxiv.org/abs/2602.24151v1.

Claim. Open Problem 1 asks whether an r-connected K_{r+3}-free non-chordal graph can have a B-restricted clique polynomial that fails to be real-stable.

Problem details · Reading note

Visomirski–Griffin frozen-wave conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

Matthew Visomirski; Christopher Griffin (2026). Coherent Structures and Travelling Waves in Spatial Replicators from a Biased Volterra Lattice. DOI: 10.1016/j.chaos.2026.118260. URL: https://arxiv.org/abs/2601.06314v1.

Claim. Conjecture 2 asserts that the alternating sine initial data on every biased even cycle evolve to a frozen wave maintaining ecological niches.

Problem details · Reading note

Spectral bound for conditionally 2-positive maps

Proved. Lean theorem result, frozen in this repository 2026-10-08.

F. vom Ende, D. Chruściński, G. Kimura, P. Muratore-Ginanneschi (2025). Universal Bound on the Eigenvalues of 2-Positive Trace-Preserving Maps. DOI: 10.1016/j.laa.2025.10.022. URL: https://arxiv.org/abs/2506.02145v1.

Claim. Theorem 1 proves the spectral trace bound for 2-positive trace-preserving maps; question (10) asks for the bound for arbitrary conditionally 2-positive maps.

Problem details · Reading note

Comparability of conical-design concurrence bounds

Proved. Lean theorem result, frozen in this repository 2026-10-08.

H.-F. Wang; W. Zhou; L. Chen; S.-M. Fei (2026). Estimating the concurrence for quantum states via symmetric measurements. URL: https://arxiv.org/abs/2606.31010v2.

Claim. The lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable.

Problem details · Reading note

A negative Clifford third-moment expectation at dimension eleven

Refuted. Lean theorem result, frozen in this repository 2026-10-08.

H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Problem details · Reading note

The exact maximum purity of absolutely PPT qutrit–qudit states

Proved. Lean theorem result, frozen in this repository 2026-10-07.

Problem details · Reading note

The n = 3 collapse of the genuine four-party multi-entropy for stabilizer states

Proved. Lean theorem result, frozen in this repository 2026-10-07.

S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v2.

Claim. The paper conjectures that for stabilizer states and q ≥ 4 the q-partite genuine multi-entropy at replica index n < q collapses to multi-entropies of at most q − 2 parties; at q = 4 the n = 3 case reads GM^{(4)}_{n=3} = −(a − 1/9) I_3, and whether a counting argument exists for n = 3 is stated as open.

Problem details · Reading note

Positive polygonal couplings realize every discrete Fourier transform

Proved. Lean theorem result, frozen in this repository 2026-10-07.

Edgar Barriga et al. (2026). N-dimensional discrete Fourier transform via bosonic Hamiltonian. URL: https://arxiv.org/abs/2609.05644.

Claim. A waveguide array with propagation constants beta_l and couplings C_jk evolves by U = exp(-i C z); it realizes the N-dimensional discrete Fourier transform F_N (entries omega^(xy)/sqrt N, omega = exp(-2 pi i/N)) when input and output phase shifters give Phi_out U Phi_in = F_N. The graph has the waveguides as vertices and the nonzero couplings as edges. Section III.D conjectures edge ranges for the solutions; for N = 10 the third conjecture, with l = 5, gives at least N l / 2 = 25 edges. Section II.B asks whether the polygonal model (zero-diagonal real circulant couplings with C_l = C_{N-l}) can always yield F_N.

Problem details · Reading note

Even-depth permuted brickwork has a negative moment direction

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Claim. Like the brickwork, this architecture can be shown to have a PSD vectorization if the depth is odd. It is unclear if the vectorization is PSD at even depths.

Problem details · Reading note

The k-positive region of the diagonal-perturbed Tomiyama maps

Proved. Lean theorem result, frozen in this repository 2026-10-07.

A. Bera, B. Bhattacharya, D. Chruściński (2026). Tomiyama-type maps with a diagonal perturbation. DOI: 10.48550/arXiv.2604.18600. URL: https://arxiv.org/abs/2604.18600v1.

Claim. Conjecture 2.4: the k-positive maps of the family Φ_{α,β} = (1−α−β) id + α τ₀ + β Δ on M_d form the quadrilateral conv{Ψ₀, Ψ₁, Ψ₂, 𝒯_k}; proved by the authors for β ≤ 0, for k dividing d and for k = d − 1, and stated as open in general.

Problem details · Reading note

Essentially Hermitian off-diagonal blocks

Proved. Lean theorem result, frozen in this repository 2026-10-07.

J.-C. Bourin and E.-Y. Lee (2021). Eigenvalue inequalities for positive block matrices with the inradius of the numerical range. DOI: 10.1142/S0129167X22500094. URL: https://arxiv.org/abs/2111.15180v1.

Claim. Let $X\in\mathbb{M}_n$. If the inequality $\left\| \begin{bmatrix} A &X \\ X^* &B\end{bmatrix}\right\|_\infty \le \|A+B\|_\infty$ holds for all positive block-matrix with $X$ as off-diagonal block, then $X$ is essentially Hermitian.

Problem details · Reading note

Packing k-Domatic Colourings of Paths Need More Than k + 1 Colours

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

Boštjan Brešar, Jasmina Ferme, Wenjie Hu (2026). Partitioning an S-packing coloring into broadcast dominating sets. DOI: 10.48550/arXiv.2610.03477. URL: https://arxiv.org/abs/2610.03477v1.

Problem details · Reading note

The large-squeezing law of the genuine multi-entropy of fully symmetric Gaussian states

Proved. Lean theorem result, frozen in this repository 2026-10-07.

H. A. Camargo, M. Nishida (2026). Genuine Multi-Entropy of Fully Symmetric Gaussian States. DOI: 10.48550/arXiv.2609.30754. URL: https://arxiv.org/abs/2609.30754v1.

Claim. For fully symmetric pure Gaussian states with diagonal a ≥ 1, the tripartite genuine Rényi multi-entropy is conjectured to satisfy GM^{(3)}_n ∼ ((2−n)/(2n)) log a as a → ∞ for every replica index n and every number of modes N; GM^{(3)}_2 = 0 is proved for every pure bosonic Gaussian state.

Problem details · Reading note

Zero-magic-gap three-dimensional subspaces in every power-of-two dimension

Proved. Lean theorem result, frozen in this repository 2026-10-07.

S. Cepollaro, G. Cuffaro, M. B. Weiss, S. Cusumano, A. Hamma, S. Lloyd (2025). Stabilizer Entropy of Subspaces. DOI: 10.48550/arXiv.2512.23013. URL: https://arxiv.org/abs/2512.23013v1.

Claim. Numerically, when the single-qudit dimension d_B is a power of 2, a three-dimensional subspace with zero average stabilizer entropy gap is always achievable; the authors state that it is unclear why.

Problem details · Reading note

Chen–Kato–Brandão completely contractive CMI data processing

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão (2020). Matrix Product Density Operators: when do they have a local parent Hamiltonian?. URL: https://arxiv.org/abs/2010.14682v3.

Problem details · Reading note

A traceless counterexample to the q-deformed Böttcher–Wenzel bound

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

D. Chruściński, G. Kimura, H. Ohno, T. Singal (2022). Bounding the Frobenius norm of a q-deformed commutator. DOI: 10.48550/arXiv.2202.11520. URL: https://arxiv.org/abs/2202.11520v2.

Problem details · Reading note

A qutrit SIC fiducial that is not a Clifford-stabilizer state

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

Muhammad Erew and Moshe Goldstein (2025). Extremizing Measures of Magic on Pure States by Clifford-stabilizer States. DOI: 10.48550/arXiv.2512.19657. URL: https://arxiv.org/abs/2512.19657v2.

Claim. The ququint state and Wigner convention are taken from the paper; the constrained direction analysis is left open there. Conjecture 1 asserts that every SIC-POVM fiducial state is a Clifford-stabilizer state (uniquely stabilized by a subgroup of the finite eigenphase-extended Clifford group).

Problem details · Reading note

Garcia–Volčič Conjecture 4.5

Proved. Lean theorem result, frozen in this repository 2026-10-07.

S. R. Garcia and J. Volčič (2025). A noncommutative generalization of Hunter's positivity theorem. DOI: 10.1090/proc/17480. URL: https://arxiv.org/abs/2503.12376v2.

Claim. Let $n,d\ge 2$. For all tuples of hermitian operators $X_1,\dots,X_n$ on a Hilbert space, $\ker \left(H_{2d}(X_1,\dots,X_n)-\mu_{n,d}(X_1^{2d}+\cdots+X_n^{2d})\right) = \ker X_1\cap\cdots\cap \ker X_n.$

Problem details · Reading note

Goel's OQ4: Fibonacci period-to-rank ratios omit five

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

Aradhya Goel (2026). Sophie Germain Primes and the Totient of Fibonacci Numbers. URL: https://arxiv.org/abs/2604.17847v3.

Claim. OQ4 asks whether the prime-domain values of π(q)/z(2q+1) are exactly the odd integers.

Problem details · Reading note

Hiai–Ruskai Conjecture 6.3: exact qubit CQ contraction coefficients

Proved. Lean theorem result, frozen in this repository 2026-10-07.

Fumio Hiai; Mary Beth Ruskai (2016). Contraction coefficients for noisy quantum channels. DOI: 10.1063/1.4936215. URL: https://arxiv.org/abs/1508.03551v1.

Claim. Conjecture 6.3. Equality holds in (45c) through (45e) above.

Problem details · Reading note

higuchi-sudbery-2000-four-qubit-average-entropy-maximum

Proved. Lean theorem result, frozen in this repository 2026-10-07.

A. Higuchi; A. Sudbery (2000). How entangled can two couples get?. DOI: 10.1016/S0375-9601(00)00480-1. URL: https://arxiv.org/abs/quant-ph/0005013.

Claim. Four-qubit marginal entropy: the Higuchi–Sudbery candidate, its known restrictions, and the unrestricted maximum question.

Problem details · Reading note

Infinite odd fibers of Thue-Morse global maxima

Proved. Lean theorem result, frozen in this repository 2026-10-07.

Gandhar Joshi and Dan Rust (2025). Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words. DOI: 10.1016/j.tcs.2025.115391. URL: https://arxiv.org/html/2501.05830v2.

Claim. Conjecture 4.24 gives (A(d)-1)/d < sqrt(5)/tau for every positive d; Conjecture 3.19 gives i(F_(2n+1))=F_(2n+3)-2 and i(F_(2n))=F_(4n)-1 for every n>=1; Conjecture 3.8 gives the first longest Thue-Morse progression starts at differences 2^e+1 and 2^e-1, contextually for every e>=2; Question 3.7 second clause asks whether infinitely many lengths have infinitely many positive odd differences with that exact global maximum.

Problem details · Reading note

Kulhandjian–Hanzo Conjecture 25: the odd-prime distance bound

Proved. Lean theorem result, frozen in this repository 2026-10-07.

M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Claim. Conjecture 24 asserts that Proposition 23 holds for every d >= 2: for q = 2 the cross-correlation u = M_Q tau_H of the code Q(2,d) is the all-ones vector, every Z_i Z_j is a non-stabilizer element of the centralizer, and the minimum distance is 2.

Problem details · Reading note

Lidar's QPU-restricted IQP anticoncentration conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

D. A. Lidar (2025). Digital-Analog-Digital Quantum Supremacy. DOI: 10.48550/arXiv.2512.07127. URL: https://arxiv.org/abs/2512.07127v1.

Claim. Let $\{\mH_n\}$ be a family of hardware graphs as in \cref{def:QPU-R-E}. For each $n$, draw $\mG\sim \mathrm{Unif}[\mF_d(\mH_n)]$ and $\theta\sim \mathrm{Unif}[0,2\pi)^n$. For any fixed choice of single-qubit angles $\{v_i\}$ in \cref{eq:HZ'}, there exist constants $a,b>0$ (depending only on $d$) such that for every $s\in\{0,1\}^n$, $\Pr_{\mG,\theta} \bigl[ P_{\UIQP^{(\theta)}}(s)\ge a 2^{-n} \bigr] \ge b.$

Problem details · Reading note

A bad Anderson potential without nontrivial shared symmetry

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

O. Lindblad and E. Guerrero (2025). Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model. URL: https://arxiv.org/abs/2512.00278v1.

Claim. Conversely, we also ask whether every bad potential shares a nontrivial symmetry with the laplacian.

Problem details · Reading note

A qutrit obstruction to positive/CP factorization of the channel transpose

Refuted. Lean theorem result, frozen in this repository 2026-10-07.

Samuel A. Márquez González (2026). Feasibility Ordering of Entanglement-Source Placement for Qubit Channels. DOI: 10.48550/arXiv.2609.18803. URL: https://arxiv.org/abs/2609.18803v1.

Claim. Section VI, Eq. (47), asks whether every unital qudit channel is CP-filter equivalent to its channel transpose with CP inverses; Eq. (48) asks the weaker question whether the transpose factors as a positive map after the channel after a completely positive map.

Problem details · Reading note

The Tsirelson bound of the d-outcome Bell functional of Perito et al.

Proved. Lean theorem result, frozen in this repository 2026-10-07.

I. Perito, R. D'Avino, M. Jung, P. Mironowicz, A. Acín, R. Augusiak (2026). Bell inequalities tailored to optimal global randomness certification. DOI: 10.48550/arXiv.2606.21362. URL: https://arxiv.org/abs/2606.21362v3.

Claim. Conjecture 1: the Tsirelson bound of the d-outcome Bell functional I_d = (1/2)Σ_{x<2}Σ_{y<d} ω^{xy} A_x⊗B_y + h.c. is 2/sin(π/(2d)) and is attained by A_0 = Z, A_1 = X, the maximally entangled state and B_y = Σ_k λ_{y,k} X^{k+1}Z^k; the appendix proves that these B_y are unitary with d-th-root-of-unity eigenvalues, and the bound is checked by NPA for d ≤ 6.

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All odd kinetic frequency moments

Proved. Lean theorem result, frozen in this repository 2026-10-07.

P. Tolias; T. Dornheim; J. Vorberger (2025). Kinetic contribution to the arbitrary order odd frequency moments of the dynamic structure factor. DOI: 10.1002/ctpp.70090. URL: https://arxiv.org/abs/2508.17810v1.

Claim. Therefore, our conjecture states that the following result holds for the interacting uniform electron gas M_{\mathrm{S,KIN}}^{(2k+1)}(q)=\left(\frac{\hbar{q}^2}{2m}\right)^{2k+1}\frac{1}{2k+2}\sum_{i=0}^{k}\binom{2k+2}{2i+1}\left(\frac{2}{\hbar{q}}\right)^{2i}\langle{p}^{2i}\rangle_{0}\,, where $k$ is an arbitrary non-negative integer.

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Two singlets give more than twice the statistical strength of CHSH

Proved. Lean theorem result, frozen in this repository 2026-10-07.

W. van Dam; R. D. Gill; P. D. Grünwald (2005). The Statistical Strength of Nonlocality Proofs. DOI: 10.1109/TIT.2005.851738. URL: https://arxiv.org/abs/quant-ph/0307125v2.

Claim. There is an experiment on pairs of Bell singlets, of the 2×4×4 type, more than twice as strong as CHSH, and involving joint measurements on the pairs.

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The order-seven unicyclic CDSO minimizer has no universal vertex

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

A. M. Albalahi, S. Das, A. Ali, J. Barman, A. E. Hamza (2025). On the hyperbolic Sombor index and its counterpart. DOI: 10.47443/dml.2025.176. URL: https://www.dmlett.com/archive/v16/DML25_v16_pp108-115.pdf.

Claim. A graph minimizing (maximizing, respectively) the CDSO index (HSO index, respectively) among fixed-order connected graphs with cyclomatic number ℓ(≥ 1) has a vertex adjacent to all other vertices.

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Refuted. Lean theorem result, frozen in this repository 2026-10-06.

Saeid Alikhani, Nima Ghanbari, Mohammad Ali Dehghanizadeh (2024). Elliptic Sombor energy of a graph. DOI: 10.48550/arXiv.2404.18622. URL: https://arxiv.org/abs/2404.18622v1.

Claim. There is no graph with integer-valued elliptic Sombor energy.

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Period-doubling PPL-difference: Conjecture 17

Proved. Lean theorem result, frozen in this repository 2026-10-06.

Anna E. Frid, Enzo Laborde, and Jarkko Peltomäki (2021). On prefix palindromic length of automatic words. DOI: 10.1016/j.tcs.2021.08.016. URL: https://arxiv.org/abs/2009.02934v2.

Claim. Section 5.1, Conjecture 17 asks whether the period-doubling PPL-difference is not 2-automatic and its prefix palindromic length is not 2-regular.

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An asymmetric mixed memory outside Gayrard's hierarchy

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

Véronique Gayrard (2025). Mixed memories in Hopfield networks. URL: https://arxiv.org/abs/2504.04879v2.

Claim. Given any $n\in\mathbb N$ odd, $\xi^{(N)}(m)$ is an $n$-mixed memory of type $F$ if and only if $m\in\mathcal M_{n,F}$.

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Ghanbari's integer Sombor energy conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

Nima Ghanbari (2022). On the Sombor characteristic polynomial and Sombor energy of a graph. DOI: 10.1007/s40314-022-01957-5. URL: https://arxiv.org/abs/2108.08552v1.

Claim. There is no graph with integer-valued Sombor energy.

Problem details · Reading note

Proved. Lean theorem result, frozen in this repository 2026-10-06.

B. Grandjean; Y.-C. Liang; J.-D. Bancal; N. Brunner; N. Gisin (2012). Bell inequalities for three systems and arbitrarily many measurement outcomes. DOI: 10.48550/arXiv.1204.3829. URL: https://arxiv.org/abs/1204.3829v2.

Claim. The inequality (1), Mermin-CGLMP, defines a facet of the full local polytope for every K >= 2.

Problem details · Reading note

The affine Lie algebra refutes Heib–Bruschi Conjecture 83

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

T. Heib; D. E. Bruschi (2026). On the structural properties of Lie algebras via associated labeled directed graphs. DOI: 10.48550/arXiv.2601.16161. URL: https://arxiv.org/abs/2601.16161v1.

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Singlet padding refutes the universal precession GME threshold

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

Khoi-Nguyen Huynh-Vu; Lin Htoo Zaw; Valerio Scarani (2024). Certification of genuine multipartite entanglement in spin ensembles with measurements of total angular momentum. DOI: 10.1103/PhysRevA.109.042402. URL: https://arxiv.org/abs/2311.00806v2.

Claim. Conjecture 3: Consider a spin ensemble. Perform the precession protocol with odd K ≥ 3 on the total angular momentum of the system. If the score P_K > 𝐏_K^conj is obtained, then the spin ensemble is GME.

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First longest Thue-Morse progressions at power-of-two differences

Proved. Lean theorem result, frozen in this repository 2026-10-06.

Problem details · Reading note

The q = 2 Singer-quadric qubit codes have minimum distance 2 for every d ≥ 2

Proved. Lean theorem result, frozen in this repository 2026-10-06.

Problem details · Reading note

Data processing fails for the cumulant-based Rényi functional at every order above one

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

Claim. Conjecture 15 asserts that Q = -S_0^Q decreases under commutativity-preserving channels for noncommuting regularized states with output support inclusion; the abstract and conclusion state that the quantum data-processing inequality of the cumulant-based functional for alpha > 1 under arbitrary CPTP maps remains open.

Problem details · Reading note

A local measure-and-reset refutes the LOCC monotonicity of the MUB correlation $I_2$

Refuted. Lean theorem result, frozen in this repository 2026-10-06.

S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Claim. The correlation I2 = C(a,b) + C(a',b'), summed over equal outcomes of local measurements in two mutually unbiased bases, is conjectured to be monotonically non-increasing under LOCC operations.

Problem details · Reading note

The Jordan obstruction for the Jones–Temperley–Lieb endpoint-sector seeds

Proved. Lean theorem result, frozen in this repository 2026-10-06.

H. Negami (2026). Quantum gates from the middle convolution of twisted Burau representations. DOI: 10.48550/arXiv.2610.00293. URL: https://arxiv.org/abs/2610.00293v1.

Claim. Problem 6.8, last sentence: prove the Jordan obstruction for the Jones-Temperley-Lieb endpoint-sector seeds of Remark 6.7(b), whose ambient Long-Moody braid matrices were found numerically to have non-trivial Jordan blocks.

Problem details · Reading note

Three-column permutation arrays with compatible row orders

Proved. Lean theorem count_arrays, frozen in this repository 2026-10-06.

R. H. Hardin; Colin Barker (2013). OEIS A222001, Number of n X 3 arrays with permutation rows and compatible descent and lexicographic orders. URL: https://oeis.org/A222001.

Claim. For every n >= 1, the number of n X 3 permutation-row arrays with nondecreasing downstep counts and lexicographically nonincreasing rows is 2+(n+1)*(n+2)*(n+3)/6.

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Rivin's cycle-geodesic midpoint formula

Proved. Lean theorem result2, frozen in this repository 2026-10-06.

Igor Rivin (2026). Permanents of matrix ensembles: computation, distribution, and geometry. URL: https://arxiv.org/abs/2602.10141v3.

Claim. Open Problems 2 and 3: the midpoint formula and a closed form for the universal function.

Problem details · Reading note

Rivin's cycle-geodesic universal function

Proved. Lean theorem result3, frozen in this repository 2026-10-06.

Igor Rivin (2026). Permanents of matrix ensembles: computation, distribution, and geometry. URL: https://arxiv.org/abs/2602.10141v3.

Claim. Open Problems 2 and 3: the midpoint formula and a closed form for the universal function.

Problem details · Reading note

Abdesselam's Problem 2: padded general Ginibre inequalities for the XY model

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Abdelmalek Abdesselam (2022). Non-Abelian correlation inequalities and stable determinantal polynomials. DOI: 10.48550/arXiv.2207.07603. URL: https://arxiv.org/abs/2207.07603v2.

Claim. Problem 2 asks whether the XY model satisfies the PGG inequalities; Problem 3 asks whether Theorem 2.3, the PGG inequalities for stable determinantal polynomials, holds for every positive real exponent eta in place of positive half integers.

Problem details · Reading note

Every positive-uniform adjacent class inclusion is proper

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Duncan Adamson, Amanita Dietz, Pamela Fleischmann, Annika Huch, and Silas Cato Sacher (2026). 2-word-π-representable Graphs. URL: https://arxiv.org/abs/2605.27183v1.

Claim. Definitions 1 and 14 use adjacency iff equality of two-letter projections in two k-uniform words; Conjecture 31 asks for proper adjacent inclusion at every positive k.

Problem details · Reading note

All-direction maximal clique partitions of mixed-alphabet Hamming graphs

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Nasra Daher Ahmed, Ravi Kunjwal (2026). Characterizing unitaries via quasi-process functions. DOI: 10.48550/arXiv.2610.00579. URL: https://arxiv.org/html/2610.00579v1.

Claim. Conjecture V.1 asks for an all-direction partition into maximal cliques of every mixed-alphabet Hamming graph with more than three coordinates and all alphabet sizes at least two.

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Alai's Staircase for faithful GHZ–Mermin measurement dependence

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Aaron Alai (2026). Exact minimum measurement dependence for faithful local deterministic models of multipartite GHZ–Mermin correlations. DOI: 10.48550/arXiv.2608.00124. URL: https://arxiv.org/abs/2608.00124v1.

Claim. Conjecture 1 (Staircase), Section IX, PDF page 4: the faithful GHZ–Mermin minimum is R/(2(R+1)) for every n ≥ 3.

Problem details · Reading note

A 23-edge coupling realizes the ten-mode Fourier transform

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Edgar Barriga et al. (2026). N-dimensional discrete Fourier transform via bosonic Hamiltonian. URL: https://arxiv.org/abs/2609.05644.

Problem details · Reading note

Generating Function of Perfect Matchings Avoiding {123, 132, 213}

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Sucharita Biswas, Umesh Shankar, Sivaramakrishnan Sivasubramanian (2026). Matchings and shape-Wilf-Equivalence of sets of patterns of length three I: Triples. DOI: 10.48550/arXiv.2609.08562. URL: https://arxiv.org/abs/2609.08562v1.

Claim. Classifies triples of patterns of length three up to shape-Wilf-equivalence and enumerates the perfect matchings avoiding each triple class except P1 = {123, 132, 213} and P13 = {132, 213, 321}; Section 6, Question 1 asks for these two enumerations.

Problem details · Reading note

The all-order increasing-function endpoint-jet assertion

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Maxim R. Burke; Maleeha Haris; Madhavendra (2025). Repeated integrals of increasing functions. DOI: 10.48550/arXiv.2512.02151. URL: https://arxiv.org/abs/2512.02151v1.

Claim. Conjecture 1.2 asserts the original endpoint-jet property P_n for every nonnegative integer n; the source proves the cases n=0,1,2,3.

Problem details · Reading note

Calderon's reciprocal moments in imaginary quadratic orders

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Kevin Calderon (2026). Ljunggren–Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients. DOI: 10.48550/arXiv.2608.00347. URL: https://arxiv.org/abs/2608.00347v1.

Claim. Conjecture 6.3 asks for reciprocal first and second moment congruences for inert primes in imaginary quadratic orders.

Problem details · Reading note

Subsystem Lanczos positivity for general initial states

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Pawel Caputa, Giuseppe Di Giulio, Tran Quang Loc (2026). Complexity Inequalities for Quantum Subsystems. DOI: 10.48550/arXiv.2606.20790. URL: https://arxiv.org/abs/2606.20790v2.

Claim. Based on all the examples discussed in this manuscript, we conjecture that $\left(b_n^{(A)}\right)^2>0$ for every $n$, and hence that all the coefficients $b_n^{(A)}$ are real.

Problem details · Reading note

Weak Abelian Periodicity Does Not Force Finitely Many Weakly Abelian Unbordered Factors

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Émilie Charlier, Tero Harju, Svetlana Puzynina, Luca Q. Zamboni (2015). Abelian bordered factors and periodicity. DOI: 10.48550/arXiv.1501.07464. URL: https://arxiv.org/abs/1501.07464v1.

Claim. Infinite words whose long factors are abelian or weakly abelian bordered; Theorem 3 shows that finitely many weakly abelian unbordered factors force bounded weak abelian periodicity with geometric conditions on the graph, and Section 7, Question 2 asks for the converse.

Problem details · Reading note

The within-DOF minimum clause of equipartition duality

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Ivo D. Dinov (2026). Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invariant Entropy, and Directional Degrees of Freedom. DOI: 10.48550/arxiv.2607.07851. URL: https://arxiv.org/abs/2607.07851v1.

Claim. Conjecture 3.22(ii) asserts that the phase-equipartitioned product state minimizes every within-DOF uncertainty at a fixed action marginal and entropy lower bound.

Problem details · Reading note

Optimal Cycle–Clique Unions Without Regular Induced Subgraphs of Prime Order

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Paul W. Dyson, Brendan D. McKay (2026). Ramsey numbers for regular induced subgraphs. DOI: 10.48550/arXiv.2604.08215. URL: https://arxiv.org/abs/2604.08215v3.

Claim. Exact values and bounds for the least n such that every n-vertex graph has a regular induced subgraph of order k; Section 4 gives prime-order constructions from disjoint unions of products C_r[K_s] and states their optimality within that class for p ≥ 13 as a belief.

Problem details · Reading note

Wilf Classes of Rotation Avoidance for Patterns of Length Four

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Ömer Eğecioğlu, Collier Gaiser, Mei Yin (2026). Pattern avoidance in permutations and their rotations. DOI: 10.48550/arXiv.2607.20750. URL: https://arxiv.org/abs/2607.20750v1.

Claim. Exact counts of permutations whose first k rotations avoid a pattern of length three, and Open Question 6.1 on the Wilf classes of rotation avoidance for patterns of length four.

Problem details · Reading note

Signed cardinality is not 0-mesic on [3] × [12]

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, Amanda Welch (2024). Toggling, rowmotion, and homomesy on interval-closed sets. DOI: 10.48550/arXiv.2307.08520. URL: https://arxiv.org/abs/2307.08520v2.

Claim. Conjecture 4.9 states that the number of maximal elements minus the number of minimal elements is 0-mesic under rowmotion on interval-closed sets of every product of two finite chains.

Problem details · Reading note

Frid's Fibonacci-prefix palindromic-length conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Anna E. Frid (2018). Representations of palindromes in the Fibonacci word. URL: https://numeration2018.sciencesconf.org/data/pages/num18_abstracts.pdf.

Claim. Conjecture 2 asks for the minimal palindromic decomposition length of the Fibonacci prefix with Zeckendorf length (100)^(2k-1)101; the same abstract states the matching upper bound.

Problem details · Reading note

The Three-Colour Restricted Schur Number S_3(k; 2) Exceeds k^3 + 3k^2 + k − 1

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Collier Gaiser (2026). Restricted generalized Schur numbers. DOI: 10.48550/arXiv.2608.08789. URL: https://arxiv.org/abs/2608.08789v1.

Claim. Introduces the restricted generalized Schur numbers S_r(k; l), determines S_2(k; l) for large k, proves S_3(k; 2) ≥ k^3 + 3k^2 + k − 1 (Proposition 6.1) and asks in Open Question 6.2 whether equality holds for all large k.

Problem details · Reading note

Every gate ordering in an open-boundary integrable quantum circuit

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Miguel Garcia Fernandez, Chiara Paletta, Ana L. Retore (2026). Open-boundary integrable quantum circuits with different geometries. DOI: 10.48550/arXiv.2607.02093. URL: https://arxiv.org/abs/2607.02093v1.

Claim. The paper builds open-boundary Yang-Baxter integrable quantum circuits for every configuration of -kappa inhomogeneities (Theorems 1 and 2) and conjectures that the minimum circuit depth with kappa_- such sites is (N + 3)/2 - kappa_- for odd N (Conjecture 1) and (N + 4)/2 - kappa_- for even N (Conjecture 2).

Problem details · Reading note

Connected progenitor graphs give perfect adaptive fusion strategies

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Kenneth Goodenough, Andrew Landahl, Joon Lee, Antonio Russo and Kevin Thompson (2026). Optimal Fusion Strategies for Quantum Computation. URL: https://arxiv.org/abs/2609.02559.

Claim. For an [[n,1,d]] stabilizer code, fusion failures on a set W of qubits cause a logical failure iff the substrings of the failure-axis Pauli string on W contain a non-trivial logical operator (Proposition 3.1); the fusion distance equals the size of the largest minimal logical operator, and for graph codes it is the maximum LC-degree of the encoding vertex of the progenitor graph; the Discussion conjectures that every [[n,1,d]] code with a connected progenitor graph has a perfect adaptive strategy.

Problem details · Reading note

Graham–O'Bryant Conjecture 5.2: ordinary residue-set refutation

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Ron Graham; Kevin O'Bryant (2005). A discrete Fourier kernel and Fraenkel's tiling conjecture. DOI: 10.4064/aa118-3-4. URL: https://mathweb.ucsd.edu/~ronspubs/05_02_fraenkel_tiling.pdf.

Claim. Conjecture 5.2 asserts that distinct positive units modulo q>(7/4)^n, with sum at most q and every complete inverse-sine row at least 2/sin(pi/q), force q=2^n-1 and the ordinary power-of-two residue set.

Problem details · Reading note

Binary automatic apwenian sequences have finite level generators

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Ying-Jun Guo, Guo-Niu Han (2025). On a family of automatic apwenian sequences. DOI: 10.1016/j.disc.2025.114399. URL: https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf.

Claim. Families of 2-automatic apwenian sequences generated by directive sequences of morphisms, the classification of purely automatic apwenian sequences over {0,1,2} as the period-doubling-like sequences, and Conjectures 1 and 2 on general alphabets.

Problem details · Reading note

Koprowski–Lewis Conjecture 3.1: the original masked hypermatrix count

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Brandon Koprowski, Joel Brewster Lewis (2026). Enumeration of Nondegenerate 2 x (k+1) x k Hypermatrices. DOI: 10.48550/arXiv.2602.22129. URL: https://arxiv.org/abs/2602.22129v1.

Claim. Section 4 supplies the orbit, Bruhat, triangular-invariance and augmented-cell count bridges; Section 5.1 identifies the actual eligible-permutation sum weighted by inv(sigma)+inv(pi)-h and asks for its general factorization.

Problem details · Reading note

The Thue–Morse prefix 01101001 has quantum Černý complexity 3, not 2

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Pui Hang Lee, Pui Yee Lee and Bjørn Kjos-Hanssen (2026). Quantum Černý complexity of binary words. URL: https://arxiv.org/abs/2609.40154.

Problem details · Reading note

Maity–Krishna's mutual abelian-border limit question

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Anuran Maity, K. V. Krishna (2025). Mutually Abelian-Bordered Binary Words. DOI: 10.1007/978-3-032-17801-5_6. URL: https://arxiv.org/abs/2509.20773v1.

Claim. Do the limits $\lim_{n\to\infty} \mathcal{M}(n)/2^{2n}$ and $\lim_{n\to\infty} \overline{\mathcal{M}}(n)/2^{2n}$ exist?

Problem details · Reading note

A qutrit obstruction to strong CP-filter equivalence

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Samuel A. Márquez González (2026). Feasibility Ordering of Entanglement-Source Placement for Qubit Channels. DOI: 10.48550/arXiv.2609.18803. URL: https://arxiv.org/abs/2609.18803v1.

Problem details · Reading note

Mathar's recurrence for OEIS A113409

Proved. Lean theorem result, frozen in this repository 2026-10-05.

R. J. Mathar; Paul Barry; G. C. Greubel (2012). OEIS A113409, A transform of the central binomial coefficients A001405. URL: https://oeis.org/A113409.

Claim. Conjecture: (n+2)*a(n)-2*(n+1)*a(n-1) +(n-4)*a(n-2) +2*a(n-3) +4*(2-n)*a(n-4)=0. - _R. J. Mathar_, Nov 07 2012

Problem details · Reading note

Integer diminished Sombor energy of edgeless graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

F. Movahedi (2025). Diminished Sombor matrix, spectral radius, and energy of the graphs. DOI: 10.48550/arXiv.2508.06531. URL: https://arxiv.org/abs/2508.06531v1.

Claim. There does not exist a graph whose diminished Sombor energy is an integer value.

Problem details · Reading note

Barker's and Zizka's conjectures for the ON cells of Rule 13

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Robert Price (2015). OEIS A266285, ON cells of the Rule 13 elementary cellular automaton: closed-form, recurrence, generating-function and parity conjectures. URL: https://oeis.org/A266285.

Claim. A266285 is the number of ON (black) cells in the n-th iteration of the Rule 13 elementary cellular automaton started from a single ON cell; the entry records Colin Barker's conjectures a(n) = ((-1)^n*(3-2*n)+4*n+1)/4, a(n) = 2*a(n-2)-a(n-4) for n>3 and g.f. (1+x+2*x^3)/((1-x)^2*(1+x)^2), and Ctibor O. Zizka's conjectures a(2*n) = n+1 and a(2*n+1) = 3*n+1.

Problem details · Reading note

Barker's conjectures for the ON cells of Rule 41

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Robert Price (2016). OEIS A266614, ON cells of the Rule 41 elementary cellular automaton: recurrence and generating-function conjectures. URL: https://oeis.org/A266614.

Claim. A266614 is the number of ON (black) cells in the n-th iteration of the Rule 41 elementary cellular automaton started from a single ON cell; the entry records Colin Barker's conjectures a(n) = a(n-2)+a(n-4)-a(n-6) for n>5 and g.f. (1+x^2+3*x^3-2*x^4+5*x^5)/((1-x)^2*(1+x)^2*(1+x^2)).

Problem details · Reading note

The value-one terms of OEIS A334595 are exactly the powers of two

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Peter Kagey (2020). OEIS A334595: the right edge of a binary XOR triangle. URL: https://oeis.org/A334595.

Claim. Conjecture: a(n) = 1 if and only if n is a power of two.

Problem details · Reading note

Pathak's generic-state negativity conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

T. Pathak (2026). Mixed-State Entanglement in a Minimal Model of Quantum Chaos. DOI: 10.48550/arXiv.2603.14292. URL: https://arxiv.org/abs/2603.14292v1.

Claim. 2𝓔(t) = I_{A:B}^{(α)}(t), hold for generic states at all times t.

Problem details · Reading note

Peltomäki's silver-slope abelian period set conjecture

Proved. Lean theorem result, frozen in this repository 2026-10-05.

J. Peltomäki (2020). Abelian periods of factors of Sturmian words. DOI: 10.1016/j.jnt.2020.04.007. URL: https://arxiv.org/abs/1905.06138.

Claim. Let $\alpha = [0; \overline{2}]$. The abelian period set of a Sturmian word of slope $\alpha$ is $\mathcal{Q}^+_\alpha \cup \mathcal{M}_\alpha$.

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Pan–Skandera–Wang's Permanental Inequality at Every Initial Split

Proved. Lean theorem result, frozen in this repository 2026-10-05.

Sihong Pan, Mark Skandera, Jiayuan Wang (2026). Permanental Inequalities and Unit Interval Orders. DOI: 10.4204/EPTCS.445.17. URL: https://arxiv.org/abs/2606.13162v1.

Claim. Conjecture 7.8. For all n and all w ∈ A_n we have w ≤ f_n(w) in the Bruhat order.

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Unrestricted qubit generalized-fidelity data processing

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Reza Rajaei (2026). Generalized Fidelity and the Data Processing Inequality. URL: https://arxiv.org/html/2609.09753v1.

Claim. Generalized Bures--Wasserstein data processing fails in dimensions at least three; the unrestricted qubit case remains unresolved, with two sufficient conditions proved.

Problem details · Reading note

DomiRank entropy need not decrease with competition

Refuted. Lean theorem result, frozen in this repository 2026-10-05.

Yingying Zhang and Chengye Zhao (2026). Theoretical Analysis of DomiRank Centrality: Automorphism, Entropy, and Graph Transformations. DOI: 10.48550/arXiv.2610.00107. URL: https://arxiv.org/abs/2610.00107v1.

Claim. the DomiRank entropy of a connected non-regular graph decreases monotonically with σ

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Positivity of the Two-Color Partition Series C'(2,4,n)

Proved. Lean theorem result, frozen in this repository 2026-10-04.

George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

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Positivity of the Two-Color Partition Series D'(2,2,n)

Proved. Lean theorem result, frozen in this repository 2026-10-04.

George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

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The Negative Coefficients of the Two-Color Partition Series D'(2,3,n)

Proved. Lean theorem result, frozen in this repository 2026-10-04.

George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

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A three-dimensional minimal idempotent subspace

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Andres Barei (with Muse Spark via meta.ai; reviewed by Nicolás Jaramillo Torres) (2026). On solvable evolution algebras and a conjecture by García-Martínez and Pérez-Rodríguez. URL: https://ai.meta.com/research/publications/on-solvable-evolution-algebras-and-a-conjecture-by-garcia-martinez-and-perez-rodriguez/.

Claim. Remark 3.5 records an unresolved search for idempotent subspaces of dimension greater than two without a nonzero proper idempotent subspace.

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Beccaria–Hagendorf endpoint-density conjecture for the open staggered M1 chain

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Matteo Beccaria; Christian Hagendorf (2012). A staggered fermion chain with supersymmetry on open intervals. DOI: 10.1088/1751-8113/45/36/365201. URL: https://arxiv.org/abs/1206.4194v2.

Claim. Section 3.2.3 conjectures the endpoint-density identity for staggering II on every open M1 chain with N = 3n.

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The damped angular-momentum kernel positivity conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Dorje C. Brody; Eva-Maria Graefe; Rishindra Melanathuru (2026). Phase-space measurements and decoherence for angular momentum systems. DOI: 10.48550/arXiv.2605.02696. URL: https://arxiv.org/abs/2605.02696v1.

Claim. Equation (53) conjectures that coefficientwise domination by the Husimi kernel restores positivity for J > 1/2.

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Every upset in a neighbourhood convex geometry is convex

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Daniela Bubboloni; José Cáceres (2026). The neighbourhood convexity. DOI: 10.48550/arXiv.2608.25912. URL: https://arxiv.org/html/2608.25912v1.

Claim. For a finite nonempty simple undirected graph, the paper defines neighbourhood convexity and conjectures that every upset of the neighbourhood preorder is neighbourhood-convex whenever the neighbourhood convexity is a convex geometry. Lemma 29(iii) already proves this for principal upsets; the conjecture concerns all upsets.

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Local versus joint Gaussian thermometry for unequal-frequency modes

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

M. F. B. Cenni; L. Lami; A. Acín; M. Mehboudi (2022). Thermometry of Gaussian quantum systems using Gaussian measurements. URL: https://arxiv.org/abs/2110.02098v4.

Claim. Section 3.2, Eq. (47), conjectures that for product thermal states the optimal joint Gaussian measurement has the same Fisher information as the optimal local measurement.

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The equiprobable pretty-good-measurement comparison of Cha and Lee

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Hyunho Cha, Jungwoo Lee (2026). Structural perspectives from quantum states and measurements in optimal state discrimination. DOI: 10.1007/s11128-026-05335-6. URL: https://arxiv.org/abs/2507.05778v2.

Claim. For an ensemble of N states with optimal minimum-error POVM, I_+ is the set of labels whose optimal effect is nonzero, and P^PGM_+ is the pretty-good-measurement score restricted to I_+ with the original weights. For equal priors the paper conjectures (|I_+| - 1)(P^PGM_+ - 1/N) <= (N - 1)(P^PGM - 1/N), the equiprobable form of 'the restricted PGM bound is at most Renes's PGM bound'.

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A non-Mersenne exception to the Rule 30 / Rule 22 sign pattern

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

E. Chan-López and A. Martín-Ruiz (2026). Symmetric Nonlinear Cellular Automata as Algebraic References for Rule 30. DOI: 10.48550/arXiv.2604.00165. URL: https://arxiv.org/abs/2604.00165v3.

Claim. The all-row sign-pattern question of Remark 3 and section 9 for the full-row Rule 30 and Rule 22 support difference.

Problem details · Reading note

The HOMFLYPT Yang–Baxter Künneth subcomplex need not split

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

A. Christiana; B. Clingenpeel; H. Guo; J. Oh; J. H. Przytycki; X. Wang; H. Yun (2025). Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial. URL: https://arxiv.org/abs/2502.20659v1.

Claim. Conjecture 5.6 asserts that the short exact sequence of chain complexes for the ordered-block Künneth subcomplex splits.

Problem details · Reading note

Reduced Denominators for Hankel Determinants of Boundary-Weighted Motzkin Paths

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Claim. Conjectures on the generating functions of Hankel determinants of Motzkin path polynomials: reduced denominators and numerator degrees for the columns of the Motzkin triangle (Conjectures 1.2, 1.3) and for a boundary weight s at height zero (Conjecture 2.1).

Problem details · Reading note

Shifted Hankel Determinants of Odd Catalan Powers

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

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Reduced Denominators for Hankel Determinants of Motzkin-Triangle Columns

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Problem details · Reading note

Hankel Determinants of Backward Shifts of Partial Theta Coefficients

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Johann Cigler (2024). Hankel determinants of backward shifts of the coefficients of a partial theta function. DOI: 10.48550/arXiv.2407.05768. URL: https://arxiv.org/abs/2407.05768v2.

Claim. Section 1 conjectures that the quotients r_{m,n}(q) of shifted Hankel determinants of the coefficients q^{binom(n,2)} are monic integer polynomials of degree mn(n+m+2)/2 with r_{m,n}(1) = 1 and r_{m,n}(0) = (-1)^{mn}.

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Signed Dyck Paths in the Strip of Height 4k − 2 Count Walks on the 4k-Cycle

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Johann Cigler (2026). Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1. DOI: 10.48550/arXiv.2608.03363. URL: https://arxiv.org/abs/2608.03363v2.

Claim. Weighted bounded Dyck paths with Narayana weights and q = -1 weights; Conjecture 2 (a finite-height product formula) and Conjecture 3 (expansions in even strips).

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The L n-omino instance sequence

Proved. Lean theorem result, frozen in this repository 2026-10-04.

D. M. Condon, E. B. Dugan, L. M. Goldman, E. R. Williams (2026). Polyomino Density. DOI: 10.48550/arXiv.2608.29231. URL: https://arxiv.org/abs/2608.29231v1.

Claim. The instance sequence for the L n-omino is S(n, 1, 2) in general (Section 6.3).

Problem details · Reading note

Cui–Sun Conjecture 2.1: integral Legendre telescopers

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Li-Li Cui and Zhi-Hong Sun (2026). Curious identities involving Legendre polynomials and Apéry-like numbers. DOI: 10.48550/arXiv.2607.12330. URL: https://arxiv.org/abs/2607.12330v1.

Claim. Conjecture 2.1 asks for integral coefficient polynomials of every exact lower degree giving odd-power Legendre telescopers for all positive m and p.

Problem details · Reading note

Equal totals characterize margins of alternating signed square matrices

Proved. Lean theorem result, frozen in this repository 2026-10-04.

A. Ernst, S. Lia, C. O'Brien, J. Sheekey, J. Zumbrägel (2026). Generalising Latin square orthogonality and Frobenius-König with alternating sign matrices. DOI: 10.48550/arXiv.2606.25884. URL: https://arxiv.org/abs/2606.25884v1.

Claim. Section 8, Problem 8.3 asks which signed row and column margins are realizable in W_n.

Problem details · Reading note

Class 69 Mesh Patterns Are Not Equidistributed on Involutions

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Q. Fang, S. Fu, S. Kitaev, H. Li, X. Su, Z. Sun (2026). On mesh patterns of short length: Equidistribution and enumeration. DOI: 10.48550/arXiv.2606.14367. URL: https://arxiv.org/abs/2606.14367v1.

Claim. Concluding remarks, Conjecture 1: the four Class 69 length-2 mesh patterns are equidistributed on involutions.

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Fukai–Yoshida–Katsura dissipative free-fermion root expectation

Proved. Lean theorem result, frozen in this repository 2026-10-04.

K. Fukai; H. Yoshida; H. Katsura (2026). Dissipative free fermions in disguise. DOI: 10.48550/arXiv.2603.22163. URL: https://arxiv.org/abs/2603.22163v1.

Claim. The expectation after Eq. (17) places every root of the plus dissipative independence polynomial in the open upper half-plane for ECF graphs, under the standing shared-root exclusion after Eq. (26).

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Binegativity monotonicity for two qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Mark W. Girard; Gilad Gour (2017). The binegativity of two qubits. DOI: 10.48550/arXiv.1701.02724. URL: https://arxiv.org/abs/1701.02724v3.

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Purely Automatic Apwenian Sequences over an Alphabet with One Odd Letter

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Ying-Jun Guo, Guo-Niu Han (2025). On a family of automatic apwenian sequences. DOI: 10.1016/j.disc.2025.114399. URL: https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf.

Problem details · Reading note

The Number of Distinct Terms of the q-Derivatives of y₀ in the Grammar G_Sec

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Guo-Niu Han, Kathy Q. Ji, Huan Xiong (2026). q-Derivative Grammar. DOI: 10.48550/arXiv.2604.23959. URL: https://arxiv.org/abs/2604.23959v2.

Claim. A formal q-analogue of Chen's context-free grammars with rules, orders and a q-derivative operator; applications to q-tangent and q-secant numbers, and an appendix of conjectured counts of distinct terms such as Conjecture III.7 for the grammar G_Sec.

Problem details · Reading note

Periodicity of the Shift n + 2 Hankel Determinants of q-Metallic Numbers

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Guo-Niu Han, Emmanuel Pedon (2025). Hankel continued fractions and Hankel determinants for q-deformed metallic numbers. DOI: 10.48550/arXiv.2502.05993. URL: https://arxiv.org/abs/2502.05993v2.

Claim. H-fraction expansions of the q-metallic numbers and their shifts up to n+1, periodicity, values, Gale-Robinson recurrences and contiguity relations of their shifted Hankel determinants; Conjecture E on the shifts n+2 and beyond.

Problem details · Reading note

Unboundedness of the Large-Shift Hankel Determinants of q-Metallic Numbers

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Guo-Niu Han, Emmanuel Pedon (2025). Hankel continued fractions and Hankel determinants for q-deformed metallic numbers. DOI: 10.48550/arXiv.2502.05993. URL: https://arxiv.org/abs/2502.05993v2.

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Spin-1 tensor spin-K/2 precession separable bound

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Claim. Conjecture 2, Eq. (32): for the precession protocol and spins {1, K/2} with K ≥ 7, Psep_K({1, K/2}) = ½ [1 + 2^{−(K−1)} binom(K−1, (K−1)/2) (K−1)/(K+1)].

Problem details · Reading note

Pure-state ECQC fails in dimension five

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

H. Iqbal (2025). On the CQC Conjecture: A sufficient condition and an extension. DOI: 10.1007/s11128-026-05258-2. URL: https://arxiv.org/abs/2509.08286v2.

Claim. ECQC bounds the smallest sum of d classical mutual informations from d+1 bipartite MUB measurements by quantum mutual information; the closing remarks ask for an analytic pure-state proof.

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Ishida's cyclic stable-marriage arc-lemma upper bound

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Yoshiteru Ishida (2026). A Quadratic Lower Bound for the Shield Number of the Stable Marriage Problem: Rearrangement, Extremal Construction, and Biclique Realizability. DOI: 10.48550/arXiv.2609.17418. URL: https://arxiv.org/abs/2609.17418v1.

Claim. The cyclic ranks (1) define C_n; the general upper bound β(C_n) ≤ ⌊(n − 1)²/4⌋ is left open in Section 7.1 and the conclusion.

Problem details · Reading note

Monogamy of the qutrit CGLMP inequality

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Meenu Kumari, Shohini Ghose, Robert B. Mann (2017). Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities. DOI: 10.1103/PhysRevA.96.012128. URL: https://arxiv.org/abs/1704.06516v2.

Claim. For a two-qutrit state, the CGLMP expression I_3 = P(A_1=B_1) + P(B_1=A_2+1) + P(A_2=B_2) + P(B_2=A_1) - P(A_1=B_1-1) - P(B_1=A_2) - P(A_2=B_2-1) - P(B_2=A_1-1) is evaluated with A_k = U_FT U(phi_k) and B_l = U_FT^* U(phi'_l), U(phi) = diag(exp(-i phi(j))), followed by a computational-basis measurement, and B_CGLMP(rho) is its maximum over the twelve angles (Eqs. (CGLMP1), (CGLMP2)). Based on numerical studies the paper conjectures (Eq. (35)) that the CGLMP inequality is monogamous: for every three-qutrit state at most one of rho_AB, rho_BC, rho_AC has B_CGLMP > 2.

Problem details · Reading note

Labelle's Toda numerator unimodality conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Labelle, A. (2025). On a specialization of Toda eigenfunctions. DOI: 10.48550/arXiv.2502.10655. URL: https://arxiv.org/abs/2502.10655v3.

Claim. Conjecture 7.3 states that the polynomial (q)_α² J_α is unimodal.

Problem details · Reading note

The one-tenth synchronous doily value

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Tony Lau (2026). Beyond the Magic Square Game: Widening the Gap for Two Bell States. DOI: 10.48550/arXiv.2603.20748. URL: https://arxiv.org/abs/2603.20748v2.

Claim. Section 5 names the candidate classical value 22/25 for the 1/10-synchronous doily game.

Problem details · Reading note

Optimality of cat pairs for metrology with noisy measurements

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Yink Loong Len, Tuvia Gefen, Alex Retzker, Jan Kołodyński (2022). Quantum metrology with imperfect measurements. DOI: 10.1038/s41467-022-33563-8. URL: https://arxiv.org/abs/2109.01160v2.

Claim. For a pure encoded probe state and an imperfect measurement, the quantum Fisher information is multiplied by gamma = max over orthogonal pairs of sum_x Re<xi_perp|M_x|xi>^2 / <xi|M_x|xi>. The supplement conjectures that for every classical noise channel M_x = sum_i p(x|i) Pi_i applied independently to N probes, an optimal pair is a cat pair cos(theta)|j>^N + sin(theta)|k>^N, -sin(theta)|j>^N + cos(theta)|k>^N.

Problem details · Reading note

A Finite Andrews–Gordon Companion Identity

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Runqiao Li, Ali K. Uncu (2025). A MacMahon Analysis View of Cylindric Partitions. DOI: 10.48550/arXiv.2501.19272. URL: https://arxiv.org/abs/2501.19272v1.

Claim. Finite multiple-sum identities for cylindric partitions obtained by MacMahon's Partition Analysis; proofs of finite Andrews–Gordon companion identities for k = 2, 3, 4 and Conjecture 1.3 for k ≥ 5.

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The largest Bloch weight in the HKNN state

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Zimeng Li; Ning Wu (2026). Exact momentum-space analysis of small spin-1/2 J1-J2 rings. DOI: 10.48550/arXiv.2604.23149. URL: https://arxiv.org/abs/2604.23149v1.

Claim. The Bloch state whose N/2 down spins occupy successive sites has the largest weight in the HKNN ground state for arbitrary even N.

Problem details · Reading note

Subsystem monotonicity of generalized concentratable entanglement

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura (2025). Generalized Concentratable Entanglement via Parallelized Permutation Tests. DOI: 10.1103/jtlj-qs3y. URL: https://arxiv.org/abs/2406.18517v1.

Claim. For an n-qubit pure state the generalized concentratable entanglement is C^{(K)}(s) = (1 - 2^{-|s|} sum over all subsets alpha of s of Tr(rho_alpha^K)) / (K - 1), with Tr(rho_emptyset^K) = 1, for any real K > 1. The paper conjectures, from Haar-random numerics, that C^{(K)}(s') <= C^{(K)}(s) whenever s' is a subset of s and that C^{(K)} is subadditive on disjoint sets, both for every real K > 1; both statements are proved for K = 2 by Beckey et al.

Problem details · Reading note

Nonvanishing Indices of the Binary Digit Hankel Determinants at t = −2

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Bartosz Sobolewski, Maciej Ulas (2026). Hankel determinants of weighted binary sums of digits. DOI: 10.48550/arXiv.2607.09376. URL: https://arxiv.org/abs/2607.09376v1.

Claim. Recursions and closed forms for Hankel determinants of weighted binary digit sums; vanishing of the determinants at t = 2ζ on a structured index set and a conjectured exact description of the nonvanishing indices (Conjecture 5.7).

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Zeros of Binary Digit Hankel Determinants at Twice a Root of Unity

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Bartosz Sobolewski, Maciej Ulas (2026). Hankel determinants of weighted binary sums of digits. DOI: 10.48550/arXiv.2607.09376. URL: https://arxiv.org/abs/2607.09376v1.

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Van Herstraeten–Guha–Cerf Conjecture 2: coherent inputs for phase-covariant attenuators

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

Zacharie Van Herstraeten; Saikat Guha; Nicolas J. Cerf (2024). Classical capacity of quantum non-Gaussian attenuator and amplifier channels. DOI: 10.1142/S0219749924400033. URL: https://arxiv.org/abs/2312.15623v2.

Claim. The minimum output entropy of a phase-covariant attenuator channel ℳη,𝐩 is achieved by coherent states, ∀η,𝐩.

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Volkoff's exact-ground-state pair-tunnelling ansatz

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

T. J. Volkoff (2016). Optimal and near-optimal probe states for quantum metrology of number conserving two-mode bosonic Hamiltonians. URL: https://arxiv.org/abs/1610.05807v1.

Claim. We conjecture that for each N ≥ 4 there exists a value c_N for which |ω_+(c_N)⟩ or |ω_−(c_N)⟩ is the exact ground state.

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The exact A|BC constant of the GHZ measure

Proved. Lean theorem result, frozen in this repository 2026-10-04.

Shengjun Wu; Kaichen Zhong; Jeffery Wu (2026). A measure for genuine tripartite entanglement. DOI: 10.48550/arXiv.2605.02876. URL: https://arxiv.org/abs/2605.02876v3.

Claim. Equation (37) asks for an analytic proof of the sharp A|BC product-density bound E_GHZ = 1/2.

Problem details · Reading note

Unimodality of the X-state conditional entropy

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

M. A. Yurischev (2017). Extremal properties of conditional entropy and quantum discord for XXZ, symmetric quantum states. DOI: 10.1007/s11128-017-1701-0. URL: https://arxiv.org/abs/1702.03728v3.

Claim. For two-qubit X states the conditional entropy is written through f_1(x) = -h_2((1+p_2 x)/2, (1-p_2 x)/2) + h_4((1+p_2 x + sqrt(r_1))/4, (1+p_2 x - sqrt(r_1))/4, (1-p_2 x + sqrt(r_2))/4, (1-p_2 x - sqrt(r_2))/4) on x in [0,1] (Eq. (A1)), with r_1 = (p_1 + p_5 x)^2 + 4 w^2 (1 - x^2), r_2 = (p_1 - p_5 x)^2 + 4 w^2 (1 - x^2), w = (|p_3+p_4| + |p_3-p_4|)/4 and Shannon entropies in bits. The Introduction supposes the unimodal property of the conditional entropy, and the Appendix conjectures that f_1 and f_2 have at most one local extremum in (0,1) for every choice of parameters with nonnegative Shannon arguments. Weak unimodality on [a,b] means weak increase up to some x_m in [a,b] and weak decrease after it, with the analogous form for the minimum.

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In-between-state optimality for Bayesian transmissivity sensing

Refuted. Lean theorem result, frozen in this repository 2026-10-04.

B. Zhou; B. A. Bash; S. Guha; C. N. Gagatsos (2023). Bayesian minimum mean square error for transmissivity sensing. DOI: 10.1103/PhysRevResearch.5.043033. URL: https://arxiv.org/abs/2304.05539v1.

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Refutation of the dual-frame cross-Gramian Welch bound

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

R. Aceska and M. Kaczanowski (2022). Cross-Frame Potential. DOI: 10.1080/01630563.2022.2128818. URL: https://arxiv.org/abs/2205.05613v3.

Claim. Conjecture 41: Let F be a frame for Fⁿ, and let G be one of its dual frames. Then µ(Gr(F, G)) ≥ √((nk − n²)/(k²(k − 1))). (21) Conjecture 42: If a frame F for F^n forms an exclusive Grassmannian pair with one of its duals, then that dual must be the canonical dual frame of F.

Problem details · Reading note

An exclusive Grassmannian dual need not be canonical

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

R. Aceska and M. Kaczanowski (2022). Cross-Frame Potential. DOI: 10.1080/01630563.2022.2128818. URL: https://arxiv.org/abs/2205.05613v3.

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Non-shortening in the one-wall hypereclectic spin chain

Proved. Lean theorem result, frozen in this repository 2026-10-03.

C. Ahn; L. Corcoran; M. Staudacher (2022). Combinatorial Solution of the Eclectic Spin Chain. DOI: 10.1007/JHEP03(2022)028. URL: https://arxiv.org/abs/2112.04506v1.

Claim. We claim that the rank of A^(k) is always maximal: rank(A^(k)) = min(d_{S−k}, d_S).

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The Archer–Bourne Square Recurrence Fails at n = 11

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Kassie Archer, Noel Bourne (2026). Pattern avoidance in compositions and powers of permutations. DOI: 10.46298/dmtcs.17199. URL: https://arxiv.org/abs/2505.05218v3.

Claim. Section 5 conjectures a counting equality for permutations avoiding 312 and 321 whose cubes avoid 2143 and compositions with at most one part outside 1 and 3, and a tetranacci-type recurrence for permutations avoiding 312 and 54321 whose squares avoid 132.

Problem details · Reading note

Monotonicity of the residual-correlation sum of four qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Yan-Kui Bai; Dong Yang; Z. D. Wang (2007). Multipartite quantum correlation and entanglement in four-qubit pure states. DOI: 10.1103/PhysRevA.76.022336. URL: https://arxiv.org/abs/quant-ph/0703098v2.

Claim. For a four-qubit pure state, tau_k = 2(1 - tr rho_k^2) is the linear entropy of qubit k and C_ij = max(sqrt(lambda_1) - sqrt(lambda_2) - sqrt(lambda_3) - sqrt(lambda_4), 0) is the concurrence of the pair ij, where lambda_1 >= ... >= lambda_4 are the eigenvalues of rho_ij (sigma_y x sigma_y) rho_ij^* (sigma_y x sigma_y). The sum of the residual correlations is M = sum_k tau_k - 2 sum_{p>q} C_pq^2 (Eq. (7)), and the authors conjecture that M is an entanglement monotone, that is, does not increase on average under LOCC.

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Bernards–Gühne hybrid classical bound

Proved. Lean theorem result, frozen in this repository 2026-10-03.

F. Bernards; O. Gühne (2023). Bell inequalities for nonlocality depth. DOI: 10.1103/PhysRevA.107.022412. URL: https://arxiv.org/abs/2205.04250v2.

Claim. For (k,m) models with m > 2, the classical bound F_n <= n 2^(n-2) is conjectured and proving it or finding a counterexample remains open.

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Božović Conjecture 12: k-limited domination of Sierpiński graphs

Proved. Lean theorem result, frozen in this repository 2026-10-03.

D. Božović (2026). On k-limited domination: complexity and Sierpiński graphs. DOI: 10.48550/arXiv.2610.01584. URL: https://arxiv.org/abs/2610.01584v1.

Claim. Conjecture 12 (§6): Let n, m, and k be integers such that n ≥ 1 and 1 ≤ k ≤ m − 1. Then γ_k^L(S(n, m)) = (m − k)·m^(n−1).

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Greatest Pell indices dividing initial partial sums

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Nikhil Byrapuram; Adam Ge; Selena Ge; Tanya Khovanova; Sylvia Zia Lee; Rajarshi Mandal; Gordon Redwine; Soham Samanta; Daniel Wu; Danyang Xu; Ray Zhao (2024). Fibonacci Partial Sums Tricks. DOI: 10.1080/00150517.2025.2556152. URL: https://arxiv.org/html/2409.01296v1.

Claim. Conjecture 18: For the Pell sequence, we have m^P_{4k+1}=m^P_{4k+2}=1, m^P_{4k-1}=2k, and m^P_{4k}=2k+1.

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Expansions of Narayana-Weighted Dyck Paths in an Even Strip

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Claim. Weighted bounded Dyck paths with Narayana weights and q = -1 weights; Conjecture 2 (a finite-height product formula) and Conjecture 3 (expansions in even strips).

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A Product Formula for Narayana Strip Series of Heights 4m and 4m + 1

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Claim. Weighted bounded Dyck paths with Narayana weights and q = -1 weights; Conjecture 2 (a finite-height product formula) and Conjecture 3 (expansions in even strips).

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The unrestricted multivariate hard-core occupancy bound is false

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Ewan Davies; Juspreet Singh Sandhu; Jaehyeon Seo; Brian Tan (2026). Degree-sequence bounds for independent sets via multivariate local occupancy. DOI: 10.48550/arXiv.2605.05149. URL: https://arxiv.org/abs/2605.05149v1.

Claim. Section 1 proposes that the degree-sequence hard-core occupancy bound E|I| >= sum_v lambda_v/(1+(d_v+1)lambda_v) holds on the positive orthant without the small-fugacity restriction of Theorem 1.

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Equatorial multiplicativity of the qubit Wigner distance

Proved. Lean theorem resultEquatorial, frozen in this repository 2026-10-03.

Soumyojyoti Dutta; Tushar (2026). A Phase-Space Geometric Measure of Magic in Qubit Systems. DOI: 10.48550/arXiv.2603.20792. URL: https://arxiv.org/abs/2603.20792v3.

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Self-tensor superadditivity on the nonpositive Bloch-product branch

Proved. Lean theorem resultSelfTensor, frozen in this repository 2026-10-03.

Soumyojyoti Dutta; Tushar (2026). A Phase-Space Geometric Measure of Magic in Qubit Systems. DOI: 10.48550/arXiv.2603.20792. URL: https://arxiv.org/abs/2603.20792v3.

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Fishburn Permutations Avoiding 2413 and 2431, or 2431 and 3241

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

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The Number of Indecomposable 321- and 1342-Avoiders with k Inversions

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Atli Fannar Franklín (2024). Pattern avoiding permutations enumerated by inversions. DOI: 10.48550/arXiv.2410.07467. URL: https://arxiv.org/abs/2410.07467v4.

Claim. Section 1 conjectures on indecomposable pattern-avoiding permutations counted by inversions, including the count of I_k(321, 1342).

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Negativity and the partial transpose distance to the PPT states

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Ray Ganardi; Marek Miller; Tomasz Paterek; Marek Żukowski (2022). Hierarchy of correlation quantifiers comparable to negativity. DOI: 10.22331/q-2022-02-16-654. URL: https://arxiv.org/abs/2111.11887v2.

Claim. For a bipartite state rho with partial transpose rho^{T_B} on the second system, the negativity is N(rho) = (||rho^{T_B}||_1 - 1)/2 and the partial transpose distance is d_T(rho, sigma) = ||rho^{T_B} - sigma^{T_B}||_1 / 2. Conjecture 1: for every density matrix rho, the infimum of d_T(rho, sigma) over the PPT states sigma equals N(rho).

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Gingrich's proposed three-qubit monotone built from the Kempe invariant

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Robert M. Gingrich (2002). Properties of entanglement monotones for three-qubit pure states. DOI: 10.1103/PhysRevA.65.052302. URL: https://arxiv.org/abs/quant-ph/0106042v2.

Claim. For a three-qubit pure state sum t_ijk |ijk>, the polynomial invariants are P_{sigma,tau} = sum t_{i_1 j_1 k_1} ... t_{i_n j_n k_n} conj(t_{i_1 j_sigma(1) k_tau(1)}) ... conj(t_{i_n j_sigma(n) k_tau(n)}) over permutations sigma, tau of n elements (Eq. (8)), with I_1 = P_{e,(12)}, I_2 = P_{(12),e}, I_3 = P_{(12),(12)} and the Kempe invariant I_4 = P_{(123),(132)}. The paper proposes sigma_ABC = 3 - (I_1 + I_2 + I_3) I_4 as a fifth entanglement monotone, a function E with E(psi) >= sum_k p_k E(A_k psi / sqrt(p_k)) for every complete local instrument A_k on one party, states that numerical results suggest it is one, and assumes it for the rest of the paper.

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The upper binegativity bound for two qubits

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Mark W. Girard; Gilad Gour (2017). The binegativity of two qubits. DOI: 10.48550/arXiv.1701.02724. URL: https://arxiv.org/abs/1701.02724v3.

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The discriminant clears average mixing denominators

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Chris Godsil (2013). Average mixing of continuous quantum walks. DOI: 10.1016/j.jcta.2013.05.006. URL: https://arxiv.org/abs/1103.2578v3.

Claim. Section 11, Question 1 asks whether the discriminant D of the adjacency minimal polynomial always makes D times the average mixing matrix integral. Lemma 1.1 gives the spectral Schur-square formula; Lemma 3.1 proves the D squared bound.

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The local-bound clause of the Grandjean et al. conjecture on (B1)

Proved. Lean theorem result, frozen in this repository 2026-10-03.

B. Grandjean; Y.-C. Liang; J.-D. Bancal; N. Brunner; N. Gisin (2012). Bell inequalities for three systems and arbitrarily many measurement outcomes. DOI: 10.1103/PhysRevA.85.052113. URL: https://arxiv.org/abs/1204.3829v2.

Claim. The local-bound clause of the conjecture on (B1): the deterministic minimum is 6(K-1) for every K >= 2. The facet clause is separate.

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Refutation of the rational Q-system infinite-root primitive count

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

J. Hou, Y. Jiang, Y. Miao (2024). Rational Q-systems at Root of Unity I. Closed Chains. DOI: 10.21468/SciPostPhys.16.5.129. URL: https://arxiv.org/abs/2310.14966.

Claim. Appendix C, conjecture (C.2), gives the primitive-state count with infinite Bethe roots.

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The single-pass butterfly fails the relative exterior-square two-design bound

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Iordanis Kerenidis (2026). Scalable Quantum Machine Learning: Trainability, Expressivity and Efficiency. DOI: 10.48550/arXiv.2607.24014. URL: https://arxiv.org/abs/2607.24014v2.

Claim. Conjecture 20: the unitary butterfly at uniformly random parameters is a relative-error two-design in the second exterior representation, with error O(1/n) in completely positive order.

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Realizability of the correlators of boxes violating chained monogamy

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Waldemar Kłobus; Michał Oszmaniec; Remigiusz Augusiak; Andrzej Grudka (2016). Communication strength of correlations violating monogamy relations. DOI: 10.1007/s10701-015-9983-5. URL: https://arxiv.org/abs/1408.1223v2.

Claim. For boxes p(a,b,e|A_i,B_j) with M settings for A and B, one setting E for the third party, all one- and three-party expectation values zero and a common value of <B_0E>_{A_i}, the chained Bell expression I^M_AB = sum_k (<A_kB_k> + <A_(k+1)B_k>) with A_M = -A_0 obeys the monogamy relation |I^M_AB| + 2|<B_0E>| <= 2M for nonsignaling boxes. Section 5 introduces the coordinates x_A^i, y_A^i, x_B^i, y_B^i, derives the necessary inequalities (ElPrat) for boxes with R_M = I^M_AB + 2<B_0E> = 2M + Delta, Delta in [0,2], and conjectures that every correlator vector satisfying (ElPrat) is realized by some signaling box with R_M = 2M + Delta.

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Cubic Grover walks cannot have twice an odd period

Proved. Lean theorem result, frozen in this repository 2026-10-03.

S. Kubota, H. Sekido and K. Yoshino (2025). Regular graphs to induce even periodic Grover walks. DOI: 10.1016/j.disc.2024.114345. URL: https://arxiv.org/abs/2307.13227v1.

Claim. Question 4.11 asks whether 2l-periodic 3-regular graphs exist when l is an odd multiple of 3.

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Skew-Merged Permutations Are Exactly Those With One or Two Rook Placements

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Joel Brewster Lewis, Robert Won (2026). Non-attacking rook placements on crossword grids. DOI: 10.48550/arXiv.2609.03081. URL: https://arxiv.org/abs/2609.03081v1.

Claim. Question 4.2. For which positive integers r does there exist a crossword grid which admits exactly r rook placements? … Conjecture 3.10. A positive integer r occurs as the rook placement count of a permutation grid if and only if r ≠ 4, r ≠ 12, and r ≢ 3 (mod 4).

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A stabilizer measurement exceeds every maximally entangled one

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Gabriele Lo Monaco; Salvatore Lorenzo; Alessandro Ferraro; Mauro Paternostro; G. Massimo Palma; Luca Innocenti (2025). The non-stabilizerness cost of quantum state estimation. DOI: 10.48550/arXiv.2510.00157. URL: https://arxiv.org/abs/2510.00157v2.

Claim. Conjecture 1: For any pure |ψ⟩, the maximal value of s_μ can be obtained with maximally entangled S.

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Joint Symmetry of the Mesh Patterns 123 and 321 with Shading {0,1,2}² ∪ {(3,3)}

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Shuzhen Lv, Philip B. Zhang (2025). Joint Equidistributions of Mesh Patterns 123 and 321 with Symmetric and Minus-Antipodal Shadings. DOI: 10.48550/arXiv.2501.00357. URL: https://arxiv.org/abs/2501.00357v3.

Claim. Conjecture 2 (pair S21) on the joint equidistribution of the mesh patterns 123 and 321 with a common symmetric shading.

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The Rule 84 center column modulo three

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Tigran Nersissian (2026). Diagonal Bases and Diagonal Periods of Elementary Cellular Automata. DOI: 10.48550/arXiv.2609.25078. URL: https://arxiv.org/abs/2609.25078v1.

Claim. Definition 1 fixes the canonical polynomial lift and orbit; Remark 6 and Section 19 leave the Rule 84 modulo-three center-column pattern without an all-time proof.

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Refutation of Nikolov–Savov Conjecture 3.7 at k = 4

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

N. Nikolov and M. Savov (2024). Properties and conjectures regarding discrete renewal sequences. DOI: 10.53656/math2024-2-1-pro. URL: https://arxiv.org/abs/2307.00545v2.

Claim. Conjecture 3.7. For any k ≥ 3, Q_k is a maximal element in 𝒜_k and it is largest in 𝒜̂_k.

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Wythoff column residue pairs

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Clark Kimberling (2025). OEIS A035513: Wythoff array read by falling antidiagonals. URL: https://oeis.org/A035513.

Claim. Conjecture: If m >= 2, then {(T(n,1), T(n,2)) mod m} has cardinality m^2.

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Fibonachos ties and first-player majority

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Peter Kagey (2025). OEIS A382814: first-player score in the Fibonachos game. URL: https://oeis.org/A382814.

Claim. a(n) = n/2 exactly at 2,8,10,32; for n>32 the first player wins exactly in odd Fibonacci blocks.

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Rosenzweig–Stanfill Open Problem 1.3

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Bart Rosenzweig and Jonathan Stanfill (2026). On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians. DOI: 10.48550/arXiv.2606.04225. URL: https://arxiv.org/abs/2606.04225v1.

Claim. Open Problem 1.6(iv) asks whether expression (1.13) vanishes for every alpha in (0, pi) whenever m is nonnegative and even.

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Shao–Zhao cycle Italian-domination recurrence

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Pingping Shao; Chengye Zhao (2026). Counting Weight-k Italian Dominating Sets on Trees and Cycles. DOI: 10.48550/arXiv.2610.00108. URL: https://arxiv.org/abs/2610.00108v1.

Claim. Definition 2.1 defines Italian dominating functions, Definition 1.1 defines their weight-count polynomial, and Section 9 asks for a complete linear recurrence for their total count on cycles.

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The 12̇-Machine Sorts a Central Binomial Number of Permutations

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Michael Yang, Hansen Shieh, Ashley Yu (2025). Stack-Sorting with Dotted-Pattern-Avoiding Stacks. URL: https://arxiv.org/abs/2411.11914v2.

Claim. Valley runs, Proposition 3.5, and Conjecture 6.2 for the 21-dot machine; peak runs, Proposition 3.1, and Conjecture 6.1 for the 12-dot machine.

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A disconnected counterexample to the token-radius conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

X. Song and C. Dalfó and M. À. Fiol and S. Zhang (2026). The Algebraic Connectivity and Laplacian Spectral Radius of Token Graphs. DOI: 10.48550/arXiv.2610.00500. URL: https://arxiv.org/abs/2610.00500v1.

Claim. Conjecture 1.1 asserts that equality of the Laplacian spectral radius of a graph and all its token graphs in the stated range characterizes stars.

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Thanatipanonda's shifted binomial power linearization

Proved. Lean theorem result, frozen in this repository 2026-10-03.

Thotsaporn Thanatipanonda (2014). Beyond Zudilin's Conjectured q-analog of Schmidt's problem. DOI: 10.1080/10236198.2014.917635. URL: https://arxiv.org/abs/1403.4962v1.

Claim. Conjecture 4.1: integral linearization of shifted binomial powers with an integer recursion independent of the power index.

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Tian–Dokyeesun–Klavžar Conjecture 3.4 refutation

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Jing Tian; Pakanun Dokyeesun; Sandi Klavzar (2025). On the variety of general position problems under vertex and edge removal. DOI: 10.48550/arXiv.2510.01294. URL: https://arxiv.org/html/2510.01294v2.

Claim. Conjecture 3.4 asserts gp_o(G-x) <= gp_o(G) + deg_G(x) whenever x is not a cut vertex of a simple connected graph G.

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Monotonicity of the negativity of randomized graph states

Refuted. Lean theorem result, frozen in this repository 2026-10-03.

Jun-Yi Wu; Matteo Rossi; Hermann Kampermann; Simone Severini; Leong Chuan Kwek; Chiara Macchiavello; Dagmar Bruß (2014). Randomized Graph States and their Entanglement Properties. DOI: 10.1103/PhysRevA.89.052335. URL: https://arxiv.org/abs/1403.3828v3.

Claim. For a graph G and a probability p, the randomized graph state is rho_G^p = sum over spanning subgraphs F of G of p^|E_F| (1-p)^(|E_G| - |E_F|) |F><F|, where |F> = prod over the edges of F of CZ |+>^n is the graph state of F. Section V defines the negativity N(rho_AB) = (||rho_AB^Gamma_A|| - 1)/2 with the trace norm ||X|| = Tr sqrt(X^dagger X), reports numerically that it is monotone in p for the complete graphs K_n and the star graphs S_n with n <= 4, and states that it is an open question whether the monotonic behaviour of the negativity in p is a common feature to all randomized graph states.

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Optimal quantum states depend on the Rényi orders

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner (2015). Optimality of entropic uncertainty relations. DOI: 10.1142/S0219749915500458. URL: https://arxiv.org/abs/1509.00398v1.

Claim. Conjecture V.8 (Independence of the optimal states of (α,β)): If ρ is an optimal state for any unitary operator and any α,β>½ satisfying the duality relation (2), then ρ is also an optimal state for all other dual pairs.

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The minimum of the long-time algebra OTOC

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Faidon Andreadakis; Emanuel Dallas; Paolo Zanardi (2024). Long-time Quantum Scrambling and Generalized Tensor Product Structures. DOI: 10.1103/PhysRevA.109.052424. URL: https://arxiv.org/abs/2312.13386v2.

Claim. For an algebra A = direct sum over J = 1..d_Z of 1_{n_J} tensor L(C^{d_J}) on a space of dimension d = sum_J n_J d_J, Proposition 1 (ii) gives the long-time average of the A-OTOC of a Hamiltonian satisfying the non-resonance condition as 1 - (1/d) sum over X in {A, A'} of [Tr(R^(0),X R^(1),X') - (1/2) Tr(R_D^(0),X R_D^(1),X')], with R^(0),X_lk = ||P_X(|phi_k><phi_l|)||_2^2 and R^(1),X_kl = <P_X(Pi_k), P_X(Pi_l)> for the eigenbasis phi_k. Proposition 2 proves that the product eigenbasis minimizes it over Hamiltonians that are block diagonal in the sectors, with minimum 1 - (sum_J d_J + sum_J n_J - d_Z)/d, and over all such Hamiltonians when d_Z = 1. The Conjecture of Section IV states that this value is the minimum over the whole family of isomorphic algebras, attained by the algebra whose distinguished basis is the eigenbasis.

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Fixed-point-free permutation solutions of the n-simplex equation

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Problem details · Reading note

Refutation of Beluhov's invariant-kernel dimension conjecture at type (1,3,6)

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Nikolai Beluhov (2026). Diamond Determinants and Somos Sequences. URL: https://arxiv.org/abs/2602.24239v2.

Claim. Conjecture 3: For every proper type 𝐧 of order n, it holds that dim Ω⊠ = ⌊n/2⌋.

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A type-4c diagonal state refutes the geometric tangle ansatz

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Claim. The authors state the canonical five-term decomposition, the Bloch-norm coordinates of the one-qubit reduced states, the tangle tau(psi) = 4 |Hdet(t_ijk)| = 4 lambda_0^2 lambda_4^2, and conjecture a geometric ansatz for normalized GHZ states excluding type 5.

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The original simplex coverage optimizer

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Matteo Bertuzzo, Alberto Ravagnani, Eitan Yaakobi (2026). The DNA Coverage Depth Problem: Duality, Weight Distributions, and Applications. DOI: 10.48550/arXiv.2603.06489. URL: https://arxiv.org/html/2603.06489v1.

Claim. Section 3 Conjecture 3.2 asks whether the q-ary simplex code attains minimum actual iid uniform physical-column full-recovery expectation at n=(q^k-1)/(q-1).

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The no-signalling optimum of the relaxed multipartite Hardy test

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Some Sankar Bhattacharya; Arup Roy; Amit Mukherjee; Ramij Rahaman (2015). Witnessing Genuine Mutipartite Non-locality. DOI: 10.48550/arXiv.1507.07327. URL: https://arxiv.org/abs/1507.07327v1.

Claim. For N parties with inputs u, v and outcomes 1, ..., d, the relaxed Hardy conditions (relH) are P(1...1|u...u) = q > 0, P(1...1 not-d_r 1...1|v at r) = 0 for all r, and P(1...1 d_i 1...1 d_j 1...1|v at i and j) = 0 for all i != j with j fixed. Under no-signalling the paper observes the optimal q to be 1/3 for n = 3, 4 and d = 2, ..., 5, and conjectures that the optimal q is 1/3 for any dimension and any number of parties.

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The dimer bounds for k at most 4 on regular biconnected graphs

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Paolo Butera; Paul Federbush; Mario Pernici (2015). Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs. DOI: 10.1016/j.physa.2015.05.106. URL: https://arxiv.org/abs/1502.06734v2.

Claim. For a finite regular graph with N(i) the number of configurations of i dimers (i-edge matchings) and nu the matching number, the paper argues that the bounds Delta^k ln(i! N(i)) <= 0 for k = 2, ..., nu and i = 0, ..., nu - k (Eq. (1), with Delta the forward difference) correspond to the positivity of the virial coefficients, reports tests on lattice graphs and on regular biconnected graphs, and asks whether these bounds for k <= 4 always hold for regular biconnected graphs.

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Inversion Sequences Avoiding {010, 100, 102, 210} and {011, 201, 210}

Proved. Lean theorem result, frozen in this repository 2026-10-02.

David Callan, Toufik Mansour (2023). Inversion Sequences Avoiding Quadruple Length-3 Patterns. DOI: 10.5281/zenodo.8399694. URL: https://math.colgate.edu/~integers/x78/x78.pdf.

Claim. Class 152: {010, 100, 102, 210} ∼_I {011, 201, 210}; Class 207: {100, 101, 110, 201} ∼_I {101, 110, 120, 210}.

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Weak Ascent Sequences Avoiding {100, 101, 110, 201} and {021, 101, 201, 210}

Proved. Lean theorem result, frozen in this repository 2026-10-02.

David Callan, Toufik Mansour (2025). Ascent Sequences and Weak Ascent Sequences Avoiding a Quadruple of Length-3 Patterns. DOI: 10.5281/zenodo.17144266. URL: https://math.colgate.edu/~integers/z80/z80.pdf.

Claim. Class 215 of the WA-Wilf classification of quadruples of length-3 patterns is left open.

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The maximal three-qubit violation of Cabello's nonlocality argument

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

José L. Cereceda (2017). Cabello's nonlocality for generalized three-qubit GHZ states. DOI: 10.1007/s40509-016-0093-7. URL: https://arxiv.org/abs/1609.04763v2.

Claim. For three qubits with local ±1-valued projective observables U_k, D_k (eigenvectors u_k^±, d_k^±), Cabello's argument uses P = P(U_1,U_2,U_3|+++), the vanishing of P(D_1,U_2,U_3|+++), P(U_1,D_2,U_3|+++), P(U_1,U_2,D_3|+++), and Q = P(D_1,D_2,D_3|---); it contradicts local realism when Q < P. For the generalized GHZ states t|v_1v_2v_3> + |w_1w_2w_3> the maximum of C = P - Q under the three constraints is 9/64, at t = 1. The paper conjectures that for Q > 0 this maximum, 9/64, is the maximum over all possible states and choices of observables.

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Simple Permutations Sortable by Two Parallel Pop Stacks with Bypass

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Lapo Cioni, Luca Ferrari, Rebecca Smith (2025). Sorting permutations using a pop stack with a bypass. DOI: 10.1016/j.disc.2025.114964. URL: https://arxiv.org/abs/2503.08285v1.

Claim. Simple permutations sortable by two pop stacks in parallel with bypass are counted by F_{2n-5} - 1 (n odd) and F_{2n-5} (n even).

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Strong noncommutativity without complete incompatibility in every dimension at least four

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Stephan De Bièvre (2023). Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality. DOI: 10.1063/5.0110267. URL: https://arxiv.org/abs/2207.07451v1.

Claim. For every dimension d ≥ 4, noncommutativity of every pair of nontrivial coordinate projectors does not imply complete incompatibility of two orthonormal bases (§5, conjecture following Proposition 7).

Problem details · Reading note

Fishburn Permutations Avoiding 2143, 1423 and 3124

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

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Fishburn Permutations Avoiding 1243 and 2134

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

Problem details · Reading note

Fishburn Permutations Avoiding 1243 and 3124

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

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Fishburn Permutations Avoiding Pairs Counted by (n − 1)2^(n−2) + 1

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

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Fishburn Permutations Avoiding Pairs Counted by Odd-Indexed Fibonacci Numbers

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

Problem details · Reading note

Fishburn Permutations Avoiding Three Patterns from {1324, 2143, 1423, 3124}

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

Problem details · Reading note

Fishburn Permutations Avoiding 2143 and 3124

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Claim. Conjectures 10.4, 10.5, 10.7, 10.9, 10.10, 10.11, 10.12 and 10.13 on Fishburn permutations avoiding classical patterns.

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Unique minimum of the nearest-neighbor QES chain potential

Proved. Lean theorem result, frozen in this repository 2026-10-02.

A. Enciso; F. Finkel; A. González-López; M. A. Rodríguez (2007). A novel quasi-exactly solvable spin chain with nearest-neighbors interactions. DOI: 10.1016/j.nuclphysb.2007.07.001. URL: https://arxiv.org/abs/0704.3046v1.

Claim. The first one is the requirement that ξ be the unique minimum of the potential U in the domain C. Although our numerical calculations suggest that this is indeed the case, we have not been able to provide a rigorous proof of this fact.

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Divergence of the last site of the nearest-neighbor QES chain

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Problem details · Reading note

A mixed-sign Gaussian refutes Galindo–Rowell Conjecture 10.6

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

C. Galindo; E. C. Rowell (2026). Unitary Yang–Baxter Operators: Towards a Classification. URL: https://arxiv.org/abs/2608.16865.

Claim. Conjecture 10.6 states that for d≥2 with 4∤d, a primitive d-th root w, a unit α, and a coefficient vector with a 0 = 1 and all coefficient norms equal to 1, whose cyclic Pauli operator is Yang–Baxter and projectively unitary, there is one sign ε∈{+1,−1} with a_{s+t}=a_sa_tw^{εαst}.

Problem details · Reading note

The complete graph maximizes the trace of the average mixing matrix

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Chris Godsil; Krystal Guo; Mariia Sobchuk (2023). Diagonal entries of the average mixing matrix. DOI: 10.48550/arXiv.1910.02039. URL: https://arxiv.org/abs/1910.02039v1.

Claim. For a graph X with adjacency matrix A = sum_r theta_r E_r (E_r the projection onto the theta_r-eigenspace), the average mixing matrix of the continuous quantum walk is the sum of the Schur squares E_r o E_r; the trace of the average mixing matrix of K_n is (n^2 - 2n + 2)/n, regular graphs have trace at most that of K_n (Corollary 6.3), and the paper conjectures (Conjecture 9.1 of the journal version) that the complete graph on n vertices attains the maximum trace with respect to the adjacency matrix for all n.

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The five-cycle graph state is strongly 1-resistant

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Zicheng Han; Wanchen Zhang; Xiande Zhang (2026). A five-qubit 1-resistant graph state and stabilizer marginal certificates. URL: https://arxiv.org/abs/2606.08561v1.

Claim. The Discussion asks whether C5 and C6 are strongly m-resistant for m=1 and m=2 respectively.

Problem details · Reading note

Refutation of the Okada regular-trace discriminant identity

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Hivert, Florent; Scott, Jeanne (2026). Diagrammatic Okada monoid and cellularity of the Okada algebra. DOI: 10.48550/arXiv.2609.01440. URL: https://arxiv.org/abs/2609.01440v1.

Claim. Conjecture 7.9 identifies the regular-trace discriminant with a product of powers of cell-form Gram determinants.

Problem details · Reading note

The minimal translation-invariant MPS bond dimension of the W-state

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Petr Klimov; Richik Sengupta; Jacob Biamonte (2023). On Translation-Invariant Matrix Product States and advances in MPS representations of the W-state. DOI: 10.48550/arXiv.2306.16456. URL: https://arxiv.org/abs/2306.16456v2.

Claim. A TI MPS representation with PBC of bond dimension d of the W-state of order n is a pair of complex d x d matrices A_0, A_1 with Tr(A_{i_1} ... A_{i_n}) = 1/sqrt(n) when exactly one i_j is 1 and 0 otherwise. Theorem 1 gives such a representation of bond dimension floor(n/2) + 1 for every n. The paper conjectures that no TI MPS representation with PBC of the W-state of order n has bond dimension smaller than floor(n/2) + 1, and asks whether d(n) = floor(n/2) + 1.

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Arbitrary even-order convexity of split covariance intersection

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Hao Li (2026). Conjecture About Arbitrary Even-Order Convexity of w-Optimization of the Split CIF. URL: https://arxiv.org/abs/2606.27260v1.

Claim. The split covariance intersection w-optimization has nonnegative derivatives of every positive even order for both log determinant and trace.

Problem details · Reading note

Lichtenfelz–Modin–Preston Conjecture 1: four Zeitlin six-j identities

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Leandro Lichtenfelz, Klas Modin, Stephen C. Preston (2026). Ricci curvature for hydrodynamics on the sphere. DOI: 10.1007/s00220-025-05533-w. URL: https://arxiv.org/abs/2508.09833v1.

Problem details · Reading note

The closed form of the higher-antibracket coefficients

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Claim. In the algebra of linear endomorphisms of Q[x] vanishing at 1, with Phi^{n,i}(x^i) = x^{n-i}/(n-i)!, Phi^n = sum_i (-1)^{n-i} Phi^{n,i} and rho_k(Psi) = (x^k/k! - x^{k+1} d/(k+1)!) o Psi - Psi o x d^{k+1}/(k+1)!, there is for every n > 0 a unique sequence of rationals c_1^n, ..., c_n^n with Phi^{n+1} = (c_1^n rho_1^n + c_2^n rho_1^{n-2} rho_2 + ... + c_n^n rho_n) Phi^1 (Theorem 6.4); these are the coefficients of Theorem 2.3 for higher Koszul brackets. Conjecture 2.4 gives a closed formula for c_i^n, verified by the authors for n <= 12, and asserts (-1)^n c_i^n > 0.

Problem details · Reading note

Level Sequences Avoiding 101 and 102

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Toufik Mansour (2026). Wilf Classes for Level Sequences Avoiding Patterns of Length Three. DOI: 10.3390/math14111983. URL: https://www.mdpi.com/2227-7390/14/11/1983.

Claim. Problem 1: level sequences avoiding 101 and 102 are counted by the Catalan numbers.

Problem details · Reading note

Binomial and kappa states do not exhaust truncated entropy minimizers

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Laleh Memarzadeh; Stefano Mancini (2016). Minimum output entropy of a non-Gaussian quantum channel. DOI: 10.1103/PhysRevA.94.022341. URL: https://arxiv.org/abs/1605.04525v1.

Claim. Conjecture 1: truncated loss-dephasing minimum output entropy is attained by a binomial or kappa state.

Problem details · Reading note

Moradi–Rampersad–Shallit Problem 1: shifted Fibonacci–Thue–Morse state complexity

Refuted. Lean theorem problem1_refuted, frozen in this repository 2026-10-02.

Delaram Moradi; Narad Rampersad; Jeffrey Shallit (2026). Complexity of Linear Subsequences of Fibonacci-Automatic Sequences. DOI: 10.48550/arXiv.2603.21645. URL: https://arxiv.org/abs/2603.21645v1.

Claim. Problem 1 asks whether the number of states in a minimal automaton generating (t(i+c))_{i≥0} is Θ(c).

Problem details · Reading note

A400168 has a term greater than one

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Antti Karttunen (2026). OEIS A400168: arithmetic derivative primorial length minus the prime-power-divisor maximum. URL: https://oeis.org/A400168.

Claim. Question: Are there terms greater than 1?

Problem details · Reading note

Max-minus-min homomesy for rectangle interval-closed rowmotion

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Claim. Conjecture 4.9 states that the number of maximal elements minus the number of minimal elements is 0-mesic under rowmotion on interval-closed sets of every product of two finite chains.

Problem details · Reading note

Rosenzweig–Stanfill even residue vanishing

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Claim. Open Problem 1.6(iv) asks whether expression (1.13) vanishes for every alpha in (0, pi) whenever m is nonnegative and even.

Problem details · Reading note

Selective LOCC can increase Shirokov's Moreau-Yosida entanglement of formation

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

M. E. Shirokov (2026). The Moreau-Yosida approximation of the EoF: basic properties and accuracy estimates. URL: https://arxiv.org/abs/2609.30246v1.

Claim. we may conjecture, at the moment, that the function E_F^lambda does not increase under selective LOCC-operations

Problem details · Reading note

The minimal separable numerical range ratio of one two-qubit observable

Proved. Lean theorem result, frozen in this repository 2026-10-02.

Timo Simnacher; Jakub Czartowski; Konrad Szymański; Karol Życzkowski (2021). Confident entanglement detection via the separable numerical range. DOI: 10.1103/PhysRevA.104.042420. URL: https://arxiv.org/abs/2107.04365v1.

Claim. For Hermitian A_1, ..., A_k the restricted numerical range L_X(A_1, ..., A_k) is the set of vectors (Tr rho A_1, ..., Tr rho A_k) over states rho in X; L is the numerical range over all states and L_Sep the separable numerical range. The minimal volume ratio mu_{n,d,k} is the minimum over A_1, ..., A_k of vol L_Sep / vol L (Definition 2). Proposition 7 proves mu_{2,1} >= sqrt(2) - 1, and Conjecture 8 states that for a single measurement on a two-qubit system the minimal volume ratio is mu_{2,1} = 1/2, which the projector onto (|00> + |11>)/sqrt(2) attains.

Problem details · Reading note

The improved entanglement-swapping product bound fails in dimension four

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Diego S. Starke; Marcos L. W. Basso; Lucas C. Céleri; Jonas Maziero (2025). Entanglement swapping for partially entangled qudits and the role of quantum complementarity. DOI: 10.48550/arXiv.2508.00813. URL: https://arxiv.org/abs/2508.00813v2.

Claim. The improved entanglement-swapping upper-bound conjecture, Eq. (59), bounds the average post-measurement l1 entanglement by the product of the input l1 entanglements divided by d-1.

Problem details · Reading note

Xu's oblique-discord mutual-information conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-02.

Jianwei Xu (2015). Oblique discord. DOI: 10.1142/S0217979216502568. URL: https://arxiv.org/abs/1506.00404v1.

Claim. The Conjecture of Eq. (22) asserts I(rho^AB) >= I(Phi_A rho^AB) for every normalized oblique operation of Eq. (13) and every bipartite density state.

Problem details · Reading note

Schröder Enumeration of the Sorting Class of the Stack Avoiding 23-1

Proved. Lean theorem result, frozen in this repository 2026-10-02.

William Zhao (2024). Stack-sorting with Stacks Avoiding Vincular Patterns. DOI: 10.1016/j.disc.2025.114834. URL: https://arxiv.org/abs/2410.17057v1.

Claim. Conjectures 4.14 and 5.2 on preimage counts of vincular-pattern-avoiding stack-sorting maps, Conjecture 3.30 on the Schröder enumeration of a sorting class, and Conjecture 5.1 on three stacks with equal sorting classes.

Problem details · Reading note

Equal Sorting Classes and Outputs of the Stacks Avoiding 312, 31-2 and 3-12

Proved. Lean theorem result, frozen in this repository 2026-10-02.

William Zhao (2024). Stack-sorting with Stacks Avoiding Vincular Patterns. DOI: 10.1016/j.disc.2025.114834. URL: https://arxiv.org/abs/2410.17057v1.

Problem details · Reading note

Abdesselam's Problem 3: the PGG inequalities at arbitrary positive exponents

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Abdelmalek Abdesselam (2022). Non-Abelian correlation inequalities and stable determinantal polynomials. DOI: 10.48550/arXiv.2207.07603. URL: https://arxiv.org/abs/2207.07603v2.

Problem details · Reading note

Adamaszek's Conjecture 7.4 on necklace cycle lengths

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Michal Adamaszek (2012). Hard squares on cylinders revisited. URL: https://arxiv.org/abs/1202.1655v2.

Claim. Section 7 defines (k, n)-necklaces, the four pair conditions, and JUMP–TURN–FIX; Conjecture 7.4 asks whether every cycle length in Neck(k, n) divides n−3k.

Problem details · Reading note

Lazy and invertible cellular automata do not generate all cellular automata

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Edgar Alcalá-Arroyo; Alonso Castillo-Ramirez (2025). On the order of lazy cellular automata. DOI: 10.1016/j.tcs.2026.115965. URL: https://arxiv.org/abs/2510.14841v3.

Claim. A cellular automaton over A^G is lazy when a local map on a neighborhood containing the identity returns the central value on every pattern except one, the unique active transition p. The paper bounds the order of a lazy cellular automaton in terms of the fibers of p, shows the bound is attained for quasi-constant p, and asks in Problem 2 whether the invertible and the lazy cellular automata generate the monoid CA(G, A) of all cellular automata.

Problem details · Reading note

Avoiders of 231 and 312 Fixed by the Third Iterate of the Fundamental Bijection

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Kassie Archer, Robert P. Laudone (2024). Pattern avoidance and the fundamental bijection. DOI: 10.48550/arXiv.2407.06338. URL: https://arxiv.org/abs/2407.06338v1.

Claim. For σ ∈ {231, 312}, the generating functions F_σ^k(x) are rational.

Problem details · Reading note

Permutations Whose First Iterates under the Fundamental Bijection Avoid 132

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Kassie Archer, Robert P. Laudone (2024). Pattern avoidance and the fundamental bijection. DOI: 10.48550/arXiv.2407.06338. URL: https://arxiv.org/abs/2407.06338v1.

Claim. For σ ∈ {231, 312}, the generating functions F_σ^k(x) are rational.

Problem details · Reading note

The finite-length Pareto conjecture

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Daniella Bar-Lev (2026). Coded Information Retrieval for Block-Structured DNA-Based Data Storage. DOI: 10.48550/arXiv.2603.17154. URL: https://arxiv.org/html/2603.17154v2.

Claim. Conjecture 1 asserts the universal hyperbolic tradeoff s1/E1+s2/E2<=1; Conjecture 2 asserts weak Pareto improvement from length n to n+1. Both use actual minimum iid uniform whole-file retrieval and positive partitions with maximum size at least two.

Problem details · Reading note

The universal hyperbolic retrieval bound

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Daniella Bar-Lev (2026). Coded Information Retrieval for Block-Structured DNA-Based Data Storage. DOI: 10.48550/arXiv.2603.17154. URL: https://arxiv.org/html/2603.17154v2.

Problem details · Reading note

Weak Ascent Sequences Avoiding 210 and the Semi-Baxter Numbers

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Beáta Bényi, Toufik Mansour, José L. Ramírez (2024). Pattern Avoidance in Weak Ascent Sequences. DOI: 10.46298/dmtcs.12273. URL: https://arxiv.org/abs/2309.06518v4.

Claim. The sequence w_210(n) coincides with the sequence A117106.

Problem details · Reading note

Cao–Zhang–Shi–Zhao MABK positivity for the two remaining blocks

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Shen Cao; Xingjian Zhang; Fei Shi; Qi Zhao (2026). Size-Independent Robustness in Multipartite Bell Self-Testing. DOI: 10.48550/arXiv.2608.30851. URL: https://arxiv.org/abs/2608.30851v1.

Claim. Non-negativity of the explicit five-term lambda expression on [0, kappa]^n for h = 1, 2 and n >= 6.

Problem details · Reading note

CayleyPy-4 Conjecture 11: consecutive four-cycle diameter

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

A. Chervov and others (2026). CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks. DOI: 10.48550/arXiv.2603.22195. URL: https://arxiv.org/abs/2603.22195v1.

Claim. The paper reports diameter formulas, found with the CayleyPy library, for Cayley and Schreier graphs of symmetric groups; Conjecture 16 gives the diameters of the inverse-closed Schreier coset graphs S_n / (S_l x S_(n-l)) generated by consecutive k-cycles for k = 3, 4, 5, and its k = 3 clause is the ceiling of L(N - L)/2; Conjecture 15 gives the diameters without inverses as quasipolynomials in n and t = n - (l + 1), with k = 5 clause L floor((t + 1)/4) + [(t + 1) = 0 mod 4] for all L >= 4 and N >= L + 9; Conjecture 11 gives the consecutive-four-cycle inverse-closed Cayley diameter n(n-1)/6 - 1 in residues zero and one modulo three and n(n-1)/6 + 2/3 in residue two, for all n >= 6.

Problem details · Reading note

A unit-gain circulant on three vertices with universal perfect state transfer

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Claim. For the continuous-time quantum walk U(t) = exp(-i t A) of a graph with Hermitian adjacency matrix A, universal perfect state transfer means that for all vertices u, v some time gives |U(t)_{v,u}| = 1; a graph has property T if every nonzero entry of A has modulus 1. The paper notes that K_2 and Circ(0,-i,i) are the only known complex unit gain graphs with universal perfect state transfer and conjectures that Circ(0,-i,i) is the only circulant with property T which has universal perfect state transfer.

Problem details · Reading note

Coulter–Do Conjecture 5.4(a)

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Claim. Conjecture 5.4(a) asserts commutation of the b-deformed Jucys–Murphy operators on the cyclic orbit of the identity pair partition.

Problem details · Reading note

The sharp Hoggar-SIC sum negativity is seven eighths

Proved. Lean theorem result, frozen in this repository 2026-10-01.

John B. DeBrota; Christopher A. Fuchs (2017). Negativity Bounds for Weyl–Heisenberg Quasiprobability Representations. DOI: 10.1007/s10701-017-0098-z. URL: https://arxiv.org/abs/1703.08272v2.

Claim. Section 6 conjectures that the Hoggar-SIC Q-minus representation in dimension 8 has global sum negativity 7/8; equations (8)–(12) define the negative part, sum negativity, maximum over density matrices, and SIC Q-reps.

Problem details · Reading note

A binary-stabilized six-qubit state that is not locally a stabilizer state

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Éloi Descamps, Borivoje Dakić (2024). On the stabilizer formalism and its generalization. DOI: 10.1088/1751-8121/ad8607. URL: https://arxiv.org/abs/2309.09815v1.

Claim. A state of N qubits is stabilized by a stabilizing set A of single-qubit unitaries when it is, up to a complex factor, the unique common +1 eigenvector of finitely many tensor products of elements of A; standard stabilizer states are those stabilized by P = {1, -1, i, -i}·{1, X, Y, Z}. The paper conjectures that every state stabilized by the binary operators A(theta, phi) = cos(theta) Z + sin(theta)(cos(phi) X + sin(phi) Y) and the identity is locally equivalent to a standard stabilizer state.

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Nonnesting Permutations Avoiding 1132 and 2213

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Claim. In Table 4 we list some cases that seem to give interesting enumeration sequences. All the conjectures have been checked for n up to 8.

Problem details · Reading note

Nonnesting Permutations Avoiding 1233 and 1322

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Claim. In Table 4 we list some cases that seem to give interesting enumeration sequences. All the conjectures have been checked for n up to 8.

Problem details · Reading note

Nonnesting Permutations Avoiding 1322

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Claim. In Table 4 we list some cases that seem to give interesting enumeration sequences. All the conjectures have been checked for n up to 8.

Problem details · Reading note

A misclassified configuration for a ternary two-rule density classifier

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Henryk Fukś; Roman Procyk (2019). Explorations of Ternary Cellular Automata and Ternary Density Classification Problems. DOI: 10.5506/APhysPolBSupp.12.75. URL: https://arxiv.org/abs/2002.08924v1.

Claim. Ternary nearest-neighbour cellular automata act on periodic configurations by (F(x))_i = f(x_{i-1}, x_i, x_{i+1}) with indices modulo L, and a rule is named by the Wolfram number with coefficients a_{9x_0+3x_1+x_2} = f(x_0,x_1,x_2). Conjecture 1 asserts that for F = 6478767664173 and G = 7580606234490, every finite configuration of length L containing at least one zero, with density rho = (1/2L) sum x_i, satisfies G^L F^L(x) = 0^L, 1^L or 2^L according as rho lies in [0, 2/3), (2/3, 3/4) or (3/4, 1).

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Four-dimensional determinant-one representations and cyclic singlets

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Stefan Groot Nibbelink; Orestis Loukas; Andreas Mütter; Erik Parr; Patrick K. S. Vaudrevange (2020). Tension between a vanishing cosmological constant and non-supersymmetric heterotic orbifolds. DOI: 10.1002/prop.202000044. URL: https://arxiv.org/abs/1710.09237v1.

Claim. There does not exist any finite group H that has a four-dimensional representation D_4 with (i) det(D_4(h)) = 1 for all h in H, (ii) no trivial singlet of H, (iii) but whose branchings to all Z_N subgroups of H always contain the trivial Z_N-singlet (Conjecture, checked for the 1,594 CARAT groups and all groups of order at most 500).

Problem details · Reading note

The six-cycle graph state is not strongly 2-resistant

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Zicheng Han; Wanchen Zhang; Xiande Zhang (2026). A five-qubit 1-resistant graph state and stabilizer marginal certificates. URL: https://arxiv.org/abs/2606.08561v1.

Claim. The Discussion asks whether C5 and C6 are strongly m-resistant for m=1 and m=2 respectively.

Problem details · Reading note

Negativity and structured negativity differ for a pure two-qutrit state

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

A. Kumari; S. Adhikari (2022). Structured negativity: A physically realizable measure of entanglement based on structural physical approximation. DOI: 10.48550/arXiv.2209.03909. URL: https://arxiv.org/abs/2209.03909v1.

Claim. Negativity is the normalized negative partial-transpose eigenvalue sum; structured negativity is d(d^3+1) max{d/(d^3+1) - lambda_min(SPA-PT(rho)), 0}. The authors conjecture coincidence when q=d(d-1)/2.

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A signed sum of three-qubit Pauli words of norm 21

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Ola Liabøtrø (2017). Improved Classical and Quantum Random Access Codes. DOI: 10.1103/PhysRevA.95.052315. URL: https://arxiv.org/abs/1607.02667v2.

Claim. For maximal (4^m - 1, m, p) quantum random access codes with the paper's projective measurements, the worst-case success probability is governed by the most negative eigenvalue of the signed Pauli sums Sigma(beta) = sum_{k=1}^{4^m-1} (-1)^{beta(k)} sigma_{c_1(k)} x ... x sigma_{c_m(k)}; the paper checks m = 1, 2 exhaustively and conjectures ||Sigma(beta)|| <= (sqrt(3) + 1)^m - 1 for all m and all beta (Eq. (78)).

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Metya–Mukherjee's exact independent-convolution L1 minimum

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Nilava Metya and Satyaki Mukherjee (2026). Approximate Uniformity in Finite Convolution Models. DOI: 10.48550/arXiv.2609.38243. URL: https://arxiv.org/abs/2609.38243v1.

Claim. Conjecture 4.7 specifies the exact full L1 minimum 1/(2n-1) for all n>=3 over ordinary convolution of two independent real closed-simplex laws; equation (22) supplies an attaining family.

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Conjecture 15 fails for a commutativity-preserving qubit channel

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

A. Meunson; T. Deesuwan (2026). Cumulant-based quantum relative Rényi functional. DOI: 10.48550/arXiv.2606.31205. URL: https://arxiv.org/abs/2606.31205v1.

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Molinari's honeycomb trapezium determinant

Proved. Lean theorem result, frozen in this repository 2026-10-01.

L. G. Molinari (2022). Graphene nanocones and Pascal matrices. DOI: 10.48550/arXiv.2206.14428. URL: https://arxiv.org/abs/2206.14428v2.

Claim. Conjecture 2: the Hückel determinant of a honeycomb trapezium equals the displayed reduced Pascal determinant.

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The twin end vertex of P'_9 is not sharply 1/9-sedentary

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Hermie Monterde (2023). New results in vertex sedentariness. DOI: 10.1016/j.disc.2025.114959. URL: https://arxiv.org/abs/2401.00362v1.

Claim. We conjecture that u is sharply (1/n)-sedentary in P_n' for all odd n >= 5 (Example 18; P_n' is the path P_n with a vertex v = n + 1 added as a non-adjacent twin of the end vertex u = 1, and u is sharply C-sedentary when inf_{t>0} |U(t)_{u,u}| = C for U(t) = e^{itA}).

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A directed graph with democracy coefficient above one

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Giannis Moutsinas; Choudhry Shuaib; Weisi Guo; Stephen Jarvis (2021). Graph hierarchy: a novel framework to analyse hierarchical structures in complex networks. DOI: 10.1038/s41598-021-93161-4. URL: https://arxiv.org/abs/1908.04358v4.

Claim. For a weighted simple directed graph with in-degree vector d and M the transpose of the in-degree Laplacian diag(d) - A, the forward hierarchical levels g are the minimum-norm minimizer of the residual of M x = d, and the forward democracy coefficient is 1 minus the arc-weighted mean of g_j - g_i over the arcs from i to j. The paper conjectures (Conjecture 3.6 of the journal version) that the forward and backward democracy coefficients of a weakly connected directed graph are at most 1, with equality exactly for balanced graphs.

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Cardinality-sum-avoiding subset generating function

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Gus Wiseman; Chai Wah Wu (2023). OEIS A367400: cardinality-sum-avoiding subsets. URL: https://oeis.org/A367400.

Claim. G.f.: (-x^3 + x^2 + 1)/(x^4 - 2*x^3 + x^2 - 2*x + 1).

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The greedy-brick all-index identity

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Aleksei Udovenko (2026). OEIS A395531, greedy-brick row births. URL: https://oeis.org/A395531/internal.

Claim. Conjecture: when n = a(m) for some m, we have a(n) = n(n+3)/2 - m. Verified experimentally up to m=1172, n=688491.

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Eventual periodicity of every A398589 row

Proved. Lean theorem eventual_periodicity, frozen in this repository 2026-10-01.

Joshua B. Weinstein (2026). OEIS A398589: eventual periodicity of self-banning rows. URL: https://oeis.org/A398589.

Claim. Conjecture: All of the rows in this sequence are eventually periodic.

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Refutation of Ostrovskii–Shcherbakov Conjecture 6.2

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Dmitrii M. Ostrovskii and Pavel S. Shcherbakov (2025). Amplitude maximization in stable systems, Schur positivity, and some conjectures on polynomial interpolation. DOI: 10.48550/arXiv.2508.13554. URL: https://arxiv.org/abs/2508.13554v2.

Claim. Conjecture 6.2 asserts the shifted Schur-ratio bound for self-conjugate grids in the closed unit disk, together with a hook-Schur bound under nonnegative real parts.

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Original cyclic-morphism string-attractor minimum

Proved. Lean theorem result, frozen in this repository 2026-10-01.

France Gheeraert, Giuseppe Romana, Manon Stipulanti (2023). String attractors of some simple-Parry automatic sequences. URL: https://arxiv.org/abs/2302.13647v2.

Claim. Conjecture 42 gives the minimum string-attractor size of every original fixed-point prefix under weak cyclic maximality.

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The extended MUB mixed-norm equality is false

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Antonio F. Rotundo; René Schwonnek (2023). An entropic uncertainty principle for mixed states. DOI: 10.1103/PhysRevResearch.6.033043. URL: https://arxiv.org/abs/2303.11382v1.

Claim. Conjecture 1 (Extended MUB regime): a doubly stochastic matrix has mixed operator norm d^(1-lambda-mu) whenever ((1-mu)/mu)((1-lambda)/lambda) >= sigma_2^2.

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The minimum of purity plus time-reversal overlap

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Eduardo Serrano-Ensástiga; Olivier Giraud; John Martin (2026). Multiqubit monogamy relations beyond shadow inequalities. DOI: 10.1103/9fkf-hm8l. URL: https://arxiv.org/abs/2507.12680v2.

Claim. For a bipartition A|Abar of an N-qubit system into k = |A| and N - k qubits, with rho_A the reduced state of a pure state, rho~ = sigma_y^(tensor k) rho* sigma_y^(tensor k) and R_rho = Tr(rho rho~), the authors conjecture that the minimum of Tr(rho_A^2) + R_(rho_A) over pure states is 2^(k-N) if 2k > N and 2^(1-k) if 2k <= N (Conjecture 2; proved there for k = 1, numerical for 2k <= N, k >= 2).

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Suvagiya Conjecture 28 refutation

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Vaibhav Suvagiya (2026). Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models. DOI: 10.48550/arXiv.2607.17343. URL: https://arxiv.org/html/2607.17343v2.

Claim. Conjecture 28 asserts that for every integer m >= 4 the minimum spectral radius over all edge signings of C_(8m)(1,2) is the largest real root of x^4-2x^3-6x^2+12x-4.

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Non-negativity of the photon-addition coefficients

Proved. Lean theorem result, frozen in this repository 2026-10-01.

Zacharie Van Herstraeten; Nicolas J. Cerf; Saikat Guha; Christos N. Gagatsos (2024). Majorization theoretical approach to entanglement enhancement via local filtration. DOI: 10.1103/PhysRevA.110.042430. URL: https://arxiv.org/abs/2312.02066v2.

Claim. With N_kk = sum_n lambda^n C(n+k,k)^2 and coefficients c_n^(kk) defined by C(n+k+1,k)^2 = sum_{i=0}^{n+1} c_{n-i}^(kk) (i+1)^2, c_{-1}^(kk) = 1, the authors conjecture that all c_n^(kk) are non-negative for every k >= 2, i.e. that the matrix D with q^(kk) = D q^(11) is column stochastic, implying the majorization relation sigma^(k,k) < sigma^(1,1) (proved for k = 2, ..., 8, with closed forms printed for k = 2, 3, 4).

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Condensation need not preserve LC-equivalence

Refuted. Lean theorem result, frozen in this repository 2026-10-01.

Lina Vandré; Jarn de Jong; Frederik Hahn; Adam Burchardt; Otfried Gühne; Anna Pappa (2024). Distinguishing Graph States by the Properties of Their Marginals. DOI: 10.48550/arXiv.2406.09956. URL: https://arxiv.org/abs/2406.09956v2.

Claim. Conjecture 16: Given two graphs G and G′ and a condensation set C such that each node in C is connected to at most one node in the neighborhood in V \ C. If G and G′ are LC-equivalent, it follows that G_c and G′_c are LC-equivalent.

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Three weak measurements of projections below minus one eighth

Refuted. Lean theorem result, frozen in this repository 2026-09-30.

Alastair A. Abbott; Ralph Silva; Julian Wechs; Nicolas Brunner; Cyril Branciard (2019). Anomalous Weak Values Without Post-Selection. DOI: 10.22331/q-2019-10-14-194. URL: https://arxiv.org/abs/1805.09364v3.

Claim. Without post-selection the mean product of the pointer positions of n sequential weak measurements of A_1, ..., A_n on rho is 2^(-(n-1)) Tr[{A_1, {A_2, ..., {A_(n-1), A_n}...}} rho]; for two projection observables (eigenvalues 0 and 1) it is at least -1/8, and the authors conjecture the bound -1/8 for every number n of projection observables.

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The averaged enumerator of B-form codes

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Nikolaos Angelinos; Debarghya Chakraborty; Anatoly Dymarsky (2022). Optimal Narain CFTs from codes. DOI: 10.48550/arXiv.2206.14825. URL: https://arxiv.org/abs/2206.14825v1.

Claim. For prime p, the full enumerator polynomial of the code with generating matrix (I | B^T), evaluated at x_ab = t_a t_b with t_a = t_-a and averaged over the p^(c(c-1)/2) antisymmetric matrices B with zero diagonal, is conjectured to equal t_0^(2c) + (sum_k (sum_(a,b) cos(2 pi k a b / p) t_a t_b)^c - p t_0^c (sum_a t_a)^c) / p^c (eq. barP).

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The Yang–Baxter cellular automata over F_{2^n} have period dividing 2^n

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Aoi Araoka; Tetsuji Tokihiro (2026). Integrable Cellular Automata on Finite Fields of Order 2^n. DOI: 10.1007/s11040-026-09569-9. URL: https://arxiv.org/abs/2602.17148v1.

Claim. For a map f of a finite field of characteristic 2, the R-matrix R(x,y) = (y + f(x+y), x - f(x+y)) satisfies the Yang-Baxter equation exactly when f(x) + f(x + f(y)) = f(x + f(y + f(x))); the cellular automaton that runs R along N cells with the helical boundary condition b(t+1) = y_N(t) is conjectured, for bijective f, to have period dividing the order of the field (proved in the paper for orders 4 and 8).

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Dalfó-Fiol-Reyes Conjecture 2.4

Refuted. Lean theorem result, frozen in this repository 2026-09-30.

C. Dalfo and M. A. Fiol and M. A. Reyes (2026). A note on three-quarters circulant digraphs. DOI: 10.61091/um128-09. URL: https://arxiv.org/html/2609.33718v1.

Claim. Conjecture 2.4 proposes, for every k at least one, the lattice generated by (k+2,1-k) and (2,k), a three-quarters digraph of diameter k on k^2+4k-2 vertices with steps 1 and -k-4.

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The inverse Gram matrix of an orthocross MIC is half-integral

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Problem details · Reading note

A rank-one qutrit MIC with nine orthogonal pairs

Refuted. Lean theorem result, frozen in this repository 2026-09-30.

Problem details · Reading note

Nonnesting Permutations Avoiding 1231, 1312, 2231 and 3221

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Claim. In Table 4 we list some cases that seem to give interesting enumeration sequences. All the conjectures have been checked for n up to 8.

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A perfect classical strategy for the Torpedo Game

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Pierre-Emmanuel Emeriau; Mark Howard; Shane Mansfield (2022). Quantum Advantage in Information Retrieval. DOI: 10.1103/PRXQuantum.3.020307. URL: https://arxiv.org/abs/2007.15643v2.

Claim. In the dimension-d Torpedo Game, Alice receives x, z in Z_d, sends one dit, and Bob, asked q in {infinity, 0, ..., d-1}, must answer a != x (q = infinity) or a != qx - z; the classical value is theta^C_2 = 3/4, theta^C_3 = 11/12, perfect classical strategies were found for 5 <= d <= 23, and the authors conjecture theta^C_(d>=5) = 1 (eq. (11)).

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Kauffman–Mukherjee–Vojtěchovský Problem 5.24

Refuted. Lean theorem result24, frozen in this repository 2026-09-30.

Louis H. Kauffman, Sujoy Mukherjee, Petr Vojtěchovský (2026). Algebraic invariants of multi-virtual links. DOI: 10.1016/j.jalgebra.2026.03.018. URL: https://arxiv.org/abs/2504.09368v1.

Claim. Problems 5.22 and 5.24 ask about maximal R-cliques in finite connected racks.

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Kauffman–Mukherjee–Vojtěchovský Problem 5.22

Refuted. Lean theorem result22, frozen in this repository 2026-09-30.

Louis H. Kauffman, Sujoy Mukherjee, Petr Vojtěchovský (2026). Algebraic invariants of multi-virtual links. DOI: 10.1016/j.jalgebra.2026.03.018. URL: https://arxiv.org/abs/2504.09368v1.

Claim. Problems 5.22 and 5.24 ask about maximal R-cliques in finite connected racks.

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A bound on n² times a falling factorial

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Taha A. Malik; Rafael Lopez-Mobilia (2020). A new phenomenological definition of entropy and application to black holes. URL: https://arxiv.org/abs/2004.07168v1.

Claim. For every integer k' >= 1 and every integer n in [0, k'], (n^2/k') (1/k')^n k'!/(k'-n)! <= 1; stated as a conjecture after eq. (003.21), checked numerically, with the note 'We have not found a proof for this conjecture'.

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A Bernoulli–harmonic–zeta identity for the free scalar on hyperbolic space

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Tatsuma Nishioka; Yoshiki Sato (2021). Free energy and defect C-theorem in free scalar theory. DOI: 10.1007/JHEP05(2021)074. URL: https://arxiv.org/abs/2101.02399v5.

Claim. For every k >= 0, -2^(-2k-2)/(k+1) H_(2k+1) - sum_(m=1)^k 2^(-2k-2)(2^(2m)-2)/(k-m+1) B_(2m)/(2m) + sum_(j=0)^(2k+1) (-1)^j/2^(2k-j) C(2k+1,j) H_j zeta(-j) + (1-2^(-2k-1)) H_(2k+1) B_(2k+2)/(k+1) = 0 (eq. (C.12), conjectured: checked up to k = 100, no proof known to the authors).

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Least-two-element subset count

Proved. Lean theorem result, frozen in this repository 2026-09-30.

Clark Kimberling (2022). OEIS A077866: Expansion of (1-x)^(-1)/(1-x-2*x^2+2*x^3). URL: https://oeis.org/A077866.

Claim. Conjecture: let b(n) be the number of subsets S of {1,2,...,n} having more than one element such that (sum of least two elements of S) = max(S). Then b(0) = b(1) = b(2) = 0 and b(n+3) = a(n) for n >= 0.

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Four colours in small online majority edge-colouring games

Refuted. Lean theorem result, frozen in this repository 2026-09-30.

Paweł Pękała (2026). On-line majority edge-colourings of graphs. URL: https://arxiv.org/abs/2609.37973.

Claim. Problem 11 asks whether, for n in {5,6,7}, Algorithm has an online majority edge-colouring strategy using at most four colours against every Presenter strategy when the final graph has n vertices and minimum degree exactly 2.

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The four-qubit Fredkin gate generates more than two ebits across AD:BC

Refuted. Lean theorem result, frozen in this repository 2026-09-30.

Xinyu Qiu; Zhiwei Song; Lin Chen (2025). Multipartite entangling power by von Neumann entropy. DOI: 10.1103/PhysRevA.111.022407. URL: https://arxiv.org/abs/2410.15253v1.

Claim. The entanglement generation K_{L:L^c}(U) of an n-partite unitary U is the supremum, over pure inputs psi_i of each party A_i with a local auxiliary system R_i, of the von Neumann entropy in ebits of the reduced state on L of U(psi_1 ⊗ ... ⊗ psi_n); for the four-qubit Fredkin gate F_4 the paper shows K_{AD:BC}(F_4) ∈ [2, log_2 5] and conjectures K_{AD:BC}(F_4) = 2.

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Cyclic Permutations Avoiding 4123 with 1324-Avoiding Cycle Forms

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, Millie Jeske (2024). Pattern-restricted cyclic permutations with a pattern-restricted cycle form. DOI: 10.48550/arXiv.2408.15000. URL: https://arxiv.org/abs/2408.15000v1.

Claim. we conjecture that |A°_n(4123;1324)| is the (n−2)nd Tetranacci number, that |A°_n(2431;1324)| is the (n−1)st Pell number, and |A°_n(4132;1324)| is the (3n)th Padovan number.

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Cyclic Permutations Avoiding 4132 with 1324-Avoiding Cycle Forms

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Claim. we conjecture that |A°_n(4123;1324)| is the (n−2)nd Tetranacci number, that |A°_n(2431;1324)| is the (n−1)st Pell number, and |A°_n(4132;1324)| is the (3n)th Padovan number.

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A local counterexample to the triangle inequality ineq_l1

Refuted. Lean theorem result, frozen in this repository 2026-09-29.

Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner (2024). Exploring the local landscape in the triangle network. DOI: 10.48550/arXiv.2405.08939. URL: https://arxiv.org/abs/2405.08939v1.

Claim. Local models in the triangle network reach a fully symmetric four-outcome distribution with p(A=B=C) = 1/4; the paper conjectures that local fully symmetric distributions have maximal p(A=B=C) very close to 1/4, and asks as an open problem whether a local fully symmetric distribution with p(A=B=C) > 1/4 exists. From neural-network estimates it states that approximately s111(p) - 0.475 Delta_{l=1}(p) <= 0.289 (eq. ineq_l1) should hold for all local models.

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A fully symmetric triangle-local distribution with p(A = B = C) above 1/4

Refuted. Lean theorem result, frozen in this repository 2026-09-29.

Problem details · Reading note

CayleyPy Growth Conjecture 1 is false

Refuted. Lean theorem cayleyPy_conjecture1_refuted, frozen in this repository 2026-09-29.

A. Chervov and others (2025). CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs. DOI: 10.48550/arXiv.2509.19162. URL: https://arxiv.org/abs/2509.19162v2.

Claim. Conjectures 1 and 2 assert eventual quasipolynomial Cayley diameter or marked-element distance for every polynomial-time constructible generator family; both are false as stated.

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Tight penalty coefficients for the QUBO reformulations of max k-cut

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Adrian Harkness; Hamidreza Validi; Ramin Fakhimi; Illya V. Hicks; Samuel Stein; Tamás Terlaky; Luis F. Zuluaga (2025). Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing. DOI: 10.48550/arXiv.2511.01108. URL: https://arxiv.org/abs/2511.01108v3.

Claim. For max k-cut with real edge weights, every optimal solution of the one-hot QUBO reformulation is an optimal k-cut when c_v > max(d+_v/k, -(3/2) d-_v) (Theorem 1), and of the reduced R-QUBO reformulation when c_v > d+_v - 2 d-_v (Theorem 2); Conjectures 1 and 2 state the same for c_v > max(d+_v/k, -d-_v/2) and for c_v > d+_v - d-_v.

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No β-invertible S₃ symmetry in the ℤ_N × ℤ_N SymTFT when 3 ∣ N

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Da-Chuan Lu; Zhengdi Sun; Zipei Zhang (2024). Exploring G-ality defects in 2-dim QFTs. DOI: 10.1007/JHEP11(2025)081. URL: https://arxiv.org/abs/2406.12151v3.

Claim. For A = Z_N x Z_N the paper searches for beta-invertible S_3 subgroups of the anyon permutation symmetries O(A + A^) of the SymTFT, finds none for N < 20 containing a factor 2 or 3, proves this for even N, and conjectures that there is no beta-invertible S_3 symmetry when N is a multiple of 3.

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Local-unitary inequivalence of the Question 5.4 stabilizer pairs

Refuted. Lean theorem result, frozen in this repository 2026-09-29.

Ali Assem Mahmoud (2026). Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes. DOI: 10.48550/arXiv.2607.26214. URL: https://arxiv.org/abs/2607.26214v1.

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The one-bit bound of the double CGLMP game

Proved. Lean theorem result, frozen in this repository 2026-09-29.

István Márton; Erika Bene; Péter Diviánszky; Tamás Vértesi (2023). Beating one bit of communication with and without quantum pseudo-telepathy. DOI: 10.48550/arXiv.2308.10771. URL: https://arxiv.org/abs/2308.10771v1.

Claim. Two parallel copies of the CGLMP_d game have one-bit classical bound 12 for d = 2, ..., 10, and the authors conjecture this for every d >= 2; the truncations [CGLMP_d^{⊗2}]_s and [CGLMP_d^{⊗2}]_a have conjectured local bounds 7 and 4, verified up to d = 20.

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Barker's and Hasler's conjectures for the rows of Rule 201

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Robert Price (2016). OEIS A267681 and A267680, rows of the Rule 201 elementary cellular automaton: recurrence, generating-function and closed-form conjectures. URL: https://oeis.org/A267681.

Claim. A267681 is the decimal representation of the n-th iteration of the Rule 201 elementary cellular automaton started from a single ON cell, and A267680 its binary representation; the entries record Colin Barker's conjectures a(n) = 5*a(n-1)-20*a(n-3)+16*a(n-4) for n>4 with g.f. (1-5*x+21*x^2+14*x^3-40*x^4)/((1-x)*(1-2*x)*(1+2*x)*(1-4*x)) for A267681, a(n) = 101*a(n-1)-10100*a(n-3)+10000*a(n-4) for n>4 with g.f. (1-101*x+10101*x^2+89910*x^3-101000*x^4)/((1-x)*(1-10*x)*(1+10*x)*(1-100*x)) for A267680, and M. F. Hasler's conjecture a(n) = 2*4^n - (n%2*2 + [n]*5)*2^(n-1) - 1 for A267681.

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OEIS A400429 second semi-meander diagonal

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Claim. Conjecture: D_2(n) = (n^2 + 2*n + n mod 2 - 20)/2, for n >= 4.

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Enumeration of the Arrow Pattern (32; 1 → 3)

Proved. Lean theorem result, frozen in this repository 2026-09-29.

Robin D.P. Zhou, Xinyang Yu (2026). Arrow-Wilf equivalences and enumerative results for short arrow patterns. DOI: 10.48550/arXiv.2609.29392. URL: https://arxiv.org/abs/2609.29392v1.

Claim. This suggests that these two patterns may be arrow-Wilf-equivalent. … We leave this as an open problem.

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Davis's coprime width-descent difference formula

Proved. Lean theorem result, frozen in this repository 2026-09-28.

Robert Davis (2017). Width-k Generalizations of Classical Permutation Statistics. DOI: 10.48550/arXiv.1701.04788. URL: https://cs.uwaterloo.ca/journals/JIS/VOL20/Davis/davis6.pdf.

Claim. Conjecture 9 states the coprime width-descent difference formula for G_{n,k}(q).

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Mathar's Conjecture 1 on two non-bonding dominoes

Proved. Lean theorem result, frozen in this repository 2026-09-28.

Richard J. Mathar (2024). Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards. URL: https://arxiv.org/abs/2404.18806v1.

Claim. Two dominoes are non-bonding when they share at most one corner point, i.e. every square of one has L1 distance at least 2 from every square of the other; D(r,c,d) counts arrangements of d non-overlapping non-bonding dominoes on an r x c board. Conjecture 1 (equation (21)): D(r,c,2) = 2c^2r^2 - 2(cr^2 + c^2r) + (r^2 + c^2)/2 - 22cr + (59/2)(c + r) - 30 for r, c >= 3.

Problem details · Reading note

Meeussen's formula for two-layer solid partitions A381265

Proved. Lean theorem result, frozen in this repository 2026-09-28.

Wouter Meeussen (2025). OEIS A381265, solid partitions with two layers and second layer a plane partition of 3: formula conjecture. URL: https://oeis.org/A381265.

Claim. The entry counts the solid (3D) partitions with two layers whose second layer is a plane partition of 3 (data a(3) = 6, 21, 57, ...) and records the conjecture that the count equals 3*(2*A000219 - A000990 - 2*A000041 + 1), tested up to n = 20.

Problem details · Reading note

Parity of the A380558 square-reversion series

Proved. Lean theorem result, frozen in this repository 2026-09-28.

Paul D. Hanna (2025). OEIS A380558, g.f. satisfying A(x - A(x)) = x^2/(1 - x^2). URL: https://oeis.org/A380558.

Claim. Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.

Problem details · Reading note

Palindromic numerators of OEIS A398690

Proved. Lean theorem result, frozen in this repository 2026-09-28.

F. Chapoton (2026). OEIS A398690, triangle of polynomials related to Verlinde numbers. URL: https://oeis.org/A398690.

Claim. Conjecture: The polynomials are palindromic.

Problem details · Reading note

Question 6 of Alofi and Dukes: do the stochastically recurrent states of K_{m,n} dominate?

Proved. Lean theorem result, frozen in this repository 2026-09-27.

Amal Alofi, Mark Dukes (2024). A note on the lacking polynomial of the complete bipartite graph. DOI: 10.1016/j.disc.2024.114323. URL: https://arxiv.org/abs/2411.02667v1.

Claim. For the stochastic sandpile model of Chan, Marckert and Selig, the paper characterises the stochastically recurrent states of K_{2,n} and K_{m,n} with n = 2, bounds n^(m-1) m^(n-1) <= |Sto(K_{m,n})| <= n^(m-1) m^n, and asks in Question 6 whether |Sto(K_{m,n})| is larger or smaller than n^(m-1) m^n / 2.

Problem details · Reading note

Problem 5 of Brešar, Hedžet and Herrman: m(C_n × P_m, 2)

Proved. Lean theorem result, frozen in this repository 2026-09-27.

Boštjan Brešar, Jaka Hedžet, Rebekah Herrman (2024). Bootstrap percolation and P_3-hull number in direct products of graphs. DOI: 10.7151/dmgt.2603. URL: https://arxiv.org/abs/2403.10957v1.

Claim. The paper studies r-neighbour bootstrap percolation on direct products of graphs, proves m(G x H, r) <= |V(G)| when the minimum degree of G is at least r and H is connected (Proposition 3), and asks in Problem 5 to determine m(C_n x P_m, 2).

Problem details · Reading note

CayleyPy-4 Conjecture 15 fails at k = 5, L = 4, N = 13

Refuted. Lean theorem result, frozen in this repository 2026-09-27.

A. Chervov and others (2026). CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks. DOI: 10.48550/arXiv.2603.22195. URL: https://arxiv.org/abs/2603.22195v1.

Problem details · Reading note

CayleyPy-4 Conjecture 16 fails at k = 4, L = 2, N = 5

Refuted. Lean theorem result, frozen in this repository 2026-09-27.

A. Chervov and others (2026). CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks. DOI: 10.48550/arXiv.2603.22195. URL: https://arxiv.org/abs/2603.22195v1.

Problem details · Reading note

The forbidden-subgraph conjecture for MMI in graph states holds

Proved. Lean theorem result, frozen in this repository 2026-09-27.

Jesus Fuentes, Cynthia Keeler, William Munizzi, Jason Pollack (2025). Monogamy of Mutual Information in Graph States. DOI: 10.48550/arXiv.2511.19585. URL: https://arxiv.org/abs/2511.19585v1.

Claim. The paper reads MMI violation in graph states as a forbidden-subgraph phenomenon and conjectures (Conjecture 1, Forbidden-Subgraph) that any graph state that violates MMI has a graph representation that is LC-equivalent to a graph H containing a K_4 subgraph, the four-vertex star, with H a generalized star graph with respect to some partition; it verifies the conjecture exhaustively up to 8 qubits.

Problem details · Reading note

The minimum depth of open integrable circuits

Refuted. Lean theorem result, frozen in this repository 2026-09-27.

Problem details · Reading note

The box partition function of the six-vertex model

Refuted. Lean theorem result, frozen in this repository 2026-09-27.

Moritz Kade (2025). Integrable systems: From the ice rule to supersymmetric fishnet Feynman diagrams. DOI: 10.18452/33769. URL: https://arxiv.org/abs/2509.03416v1.

Claim. Section 2.3 of the thesis introduces box boundary conditions for the six-vertex model, with arrow-reflecting diagonal K-matrices on all four walls, and conjectures a determinant formula for the partition function of the square box with 2M horizontal and 2M vertical spectral lines.

Problem details · Reading note

Mathar's recurrence for the surviving walks A026023

Proved. Lean theorem result, frozen in this repository 2026-09-27.

R. J. Mathar (2012). OEIS A026023, walks on the nonnegative integers from 3: recurrence conjecture. URL: https://oeis.org/A026023.

Claim. The entry defines a(n) as the number of sequences s(0), ..., s(n) of nonnegative integers with |s(i) - s(i-1)| = 1 and s(0) = 3, records that a(n)/2^n is the probability that a random walker started at x = 4 is not adsorbed at x = 0 by time n, and records Mathar's conjecture (n+4)*(n-1)*a(n) + (n-1)*(n+1)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2) - 4*(n-1)*(n+1)*a(n-3) = 0.

Problem details · Reading note

Mathar's recurrence for the lattice-path returns A068551

Proved. Lean theorem result, frozen in this repository 2026-09-27.

R. J. Mathar (2012). OEIS A068551, a(n) = 4^n - binomial(2n, n): recurrence conjecture. URL: https://oeis.org/A068551.

Claim. The entry defines a(n) = 4^n - binomial(2n, n), records that it counts the returns to the x axis in all lattice paths with steps (1,1) and (1,-1) from the origin to (2n, 0), and records Mathar's conjecture n*a(n) + 2*(3-4*n)*a(n-1) + 8*(2*n-3)*a(n-2) = 0.

Problem details · Reading note

Mathar's recurrence for the zero-quantum transition pairs A108958

Proved. Lean theorem result, frozen in this repository 2026-09-27.

R. J. Mathar (2012). OEIS A108958, number of unordered pairs of distinct length-n binary words having the same number of 1's: recurrence conjecture. URL: https://oeis.org/A108958.

Claim. The entry defines a(n) = Sum_{k=0..n} binomial(binomial(n,k),2) and records Mathar's conjecture n(n-2)a(n) + 2(-3n^2+7n-3)a(n-1) + 4(n-1)(2n-3)a(n-2) = 0.

Problem details · Reading note

The first columns of A098052 and A098530

Proved. Lean theorem result, frozen in this repository 2026-09-27.

Wouter Meeussen (2004). OEIS A098052, T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways; with its twin A098530. URL: https://oeis.org/A098052.

Claim. The entries tabulate solid partitions of n by their number of extensions to n+1 (A098052) and solid partitions of n+1 by their number of shrinkings to n (A098530), and conjecture that the first column of each is A007426 = tau_4.

Problem details · Reading note

Pronko's raising and lowering operators of the periodic Motzkin chain

Proved. Lean theorem result, frozen in this repository 2026-09-27.

A. G. Pronko (2025). Periodic Motzkin chain: Ground states and symmetries. DOI: 10.48550/arXiv.2504.00835. URL: https://arxiv.org/abs/2504.00835v3.

Claim. The paper studies the periodic Motzkin spin-1 chain H^periodic = sum_{i=1}^{N-1} Pi_{i,i+1} + Pi_{N,1}, conjectures (Conjecture 1) that its ground states are the 2N+1 path sums v_{S^z}, and conjectures (Conjecture 2) that the operators Sigma^{+-}, sums of products s_1^{r_1} ... s_N^{r_N} over r_i in {-2,...,2} with r_1 + ... + r_N = +-1, commute with the Hamiltonian and send v_{S^z} to nonzero multiples of v_{S^z+-1}, and v_{+-N} to 0.

Problem details · Reading note

Richman's congruence for the deranged-matching numbers

Proved. Lean theorem result, frozen in this repository 2026-09-27.

Harry Richman (2023). OEIS A053871, a(n) = 2*(n-1)*(a(n-1) + a(n-2)), starting a(0) = 1; a(1) = 0; congruence conjecture of Harry Richman. URL: https://oeis.org/A053871.

Claim. The FORMULA section of A053871 conjectures that (-1)^m a(m) and (-1)^n a(n) are congruent modulo q whenever q is odd and m and n are congruent modulo q.

Problem details · Reading note

Walks of near-maximal algebraic area

Proved. Lean theorem result, frozen in this repository 2026-09-27.

Andrei Zabolotskii (2025). OEIS A385672, Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k. URL: https://oeis.org/A385672.

Claim. The entry tabulates the number of n-step square-lattice walks by their algebraic area, the integral of y dx, and conjectures T(2n, n^2 - k) = 2 A029552(k) and T(2n+1, n^2 + n - k) = 4 A098613(k) for k < n.

Problem details · Reading note

The toppling count of the n × 1 torus sandpile

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Joerg Arndt (2017). OEIS A293452, Triangle T(n,k) read by rows: T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872. URL: https://oeis.org/A293452.

Claim. The entry tabulates the number of toppling iterations of the sandpile of A249872 on the n X k torus and conjectures T(n,1) = A023855(n).

Problem details · Reading note

The diameter of the inverse-closed consecutive 3-cycle Schreier coset graph

Proved. Lean theorem result, frozen in this repository 2026-09-26.

A. Chervov and others (2026). CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks. DOI: 10.48550/arXiv.2603.22195. URL: https://arxiv.org/abs/2603.22195v1.

Problem details · Reading note

A nice-error-basis code whose stabilizer group is not normal

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Jonas Eidesen (2025). Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes. DOI: 10.48550/arXiv.2506.01843. URL: https://arxiv.org/abs/2506.01843v3.

Claim. The paper models quantum errors by projectively faithful irreducible projective representations of finite groups, compares stabilizer codes, Clifford codes and weak stabilizer codes, characterizes weak stabilizer Clifford codes for nice error bases, and asks (Question 11.3) whether a code with |G| = |L(W)| |S(W)| and a non-normal stabilizer group S(W) exists when |G| = (dim V)^2.

Problem details · Reading note

The extinction set of the wheel W_18 contains 4

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Manuel A. Espinosa-García, Ana Paulina Figueroa, Julián A. Fresán-Figueroa, Gerardo L. Maldonado, L. Ariadna Sánchez-Solís (2026). Extinction thresholds in a graph-based model of HIV infection dynamics. DOI: 10.48550/arXiv.2608.00340. URL: https://arxiv.org/abs/2608.00340v1.

Claim. The paper studies Mukwembi's cellular-automaton model of HIV infection on graphs, defines extinction sets and the thresholds hiv(G) and HIV(G), determines them for several graph families, and conjectures (Conjecture 1) from computations that the extinction set of the wheel W_n is {3} together with all R >= n - 1 for even n >= 12 and {4} together with all R >= n - 1 for odd n >= 17.

Problem details · Reading note

Feldman's Conjecture 9.3: balanced zigzag choices

Proved. Lean theorem feldman_conjecture_nine_three, frozen in this repository 2026-09-26.

David V. Feldman (2026). The Missing Zigzag: Cycles of Semitone Trichords and a Conservation Law in Equal Temperament. URL: https://arxiv.org/html/2609.26114v1.

Claim. Conjecture 9.3 gives the initial value and both parity recurrences for the balance-only count of labelled zigzag choices at n=3t, t>=2.

Problem details · Reading note

Gürses and Pekcan's four-soliton conjecture for higher order Hirota forms

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Metin Gürses, Aslı Pekcan (2025). Higher order Hirota bilinear forms. DOI: 10.48550/arXiv.2511.18466. URL: https://arxiv.org/abs/2511.18466v1.

Claim. The paper studies Hirota bilinear forms with higher powers of the D-operators, proves that D_x(D_x^3 + a1 D_t + a2 D_y)^(2k+1){f.f} = 0 has three-soliton solutions for every k, and conjectures that it has four-soliton solutions only for k = 0.

Problem details · Reading note

Ishikawa and Koutschan's Pfaffian conjecture for the Motzkin triangle, part (i)

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Masao Ishikawa, Christoph Koutschan (2012). Zeilberger's Holonomic Ansatz for Pfaffians. DOI: 10.48550/arXiv.1201.5253. URL: https://arxiv.org/abs/1201.5253v2.

Claim. The paper proves by the holonomic ansatz that Pf((j-i) M_{i+j-3})_{1<=i,j<=2n} equals the product of 4k+1 over k < n for the Motzkin numbers M_n, and conjectures product formulas for the Pfaffians Pf((j-i) M^(k)_{i+j-2}) of the columns of the Motzkin triangle and of sums of two consecutive entries.

Problem details · Reading note

Colliding blocks: the collision count leaves floor(π√n) only at records

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Peter Kagey (2020). OEIS A331859, The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall. URL: https://oeis.org/A331859.

Claim. The entry counts the elastic collisions of the colliding-blocks system with masses n and 1 and a wall, gives the formula a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1, and conjectures that the n with a(n) != floor(Pi*sqrt(n)) are positions of records of the sequence.

Problem details · Reading note

The Zero Forcing Number of P(n,3) Is Eight for Every n At Least Thirteen

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Arnav Krishnan (2026). A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3). DOI: 10.48550/arXiv.2607.19412. URL: https://arxiv.org/abs/2607.19412v1.

Claim. Conjecture 5. Z(P(n,3)) = 8 for every n ≥ 13.

Problem details · Reading note

Every Natural Number Counts the Rook Placements of a Crossword Grid

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Joel Brewster Lewis, Robert Won (2026). Non-attacking rook placements on crossword grids. DOI: 10.48550/arXiv.2609.03081. URL: https://arxiv.org/abs/2609.03081v1.

Problem details · Reading note

A Permutation Grid with 155 Rook Placements

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Joel Brewster Lewis, Robert Won (2026). Non-attacking rook placements on crossword grids. DOI: 10.48550/arXiv.2609.03081. URL: https://arxiv.org/abs/2609.03081v1.

Problem details · Reading note

Lovas and Andai's conjecture on the least trace-distance contraction over a classical channel

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Attila Lovas, Attila Andai (2016). Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient. DOI: 10.48550/arXiv.1607.01215. URL: https://arxiv.org/abs/1607.01215.

Claim. The paper computes volumes of spaces of qubit channels, shows that the trace-distance contraction coefficient takes every value strictly between |a - f| and a stated upper value over the qubit channels above a fixed classical channel, and conjectures that its infimum there is |a - f|.

Problem details · Reading note

A closed form for the generalized Stirling numbers of boson ordering

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Robert S. Maier (2024). Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers. DOI: 10.48550/arXiv.2308.10332. URL: https://arxiv.org/abs/2308.10332v4.

Claim. The paper expresses normal-ordering identities for boson creation and annihilation operators through the generalized Stirling numbers of Hsu and Shiue and associated generalized Eulerian numbers, gives closed forms for several parameter pairs, and conjectures (Conjecture 5.3) a closed form for the case alpha = -1, beta = 2 with a generalized binomial coefficient.

Problem details · Reading note

The Cyclic Latin Eulerian Numbers Are Not Fully Symmetric

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Madjid Mirzavaziri, Daniel Yaqubi (2026). Cyclic Latin Eulerian Numbers. DOI: 10.48550/arXiv.2609.28808. URL: https://arxiv.org/abs/2609.28808v1.

Claim. computations for n ≤ 5 (Table 1) reveal a surprising fact: ⟨⟨n k⟩⟩_c is actually invariant under the full symmetric group action on k … Identifying the hidden mechanism behind this full invariance remains an open problem.

Problem details · Reading note

Every Interior Multiple of n Is a Latin-Square Ascent Total

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Madjid Mirzavaziri, Daniel Yaqubi (2026). Latin Eulerian Numbers. DOI: 10.48550/arXiv.2609.25100. URL: https://arxiv.org/abs/2609.25100v1.

Claim. every multiple of n strictly between n and (n − 1)^2 − 1 is attained by some Latin square … verified for n ≤ 11 but open in general

Problem details · Reading note

Nilforoushan-Parvaresh Conjecture 8.1

Proved. Lean theorem actual_exactKDeckImage_weightedLyndon_isTheta, frozen in this repository 2026-09-26.

Arman Nilforoushan; Farzad Parvaresh (2026). Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters. DOI: 10.48550/arXiv.2609.23106. URL: https://arxiv.org/html/2609.23106v1.

Claim. Conjecture 8.1: D_{q,k}(n) = Theta(n^{E_q(k)}) for every fixed q >= 2 and k >= 1, where E_q(k) = sum_{j=1}^k j L_q(j) - 1 and L_q(j) is the number of length-j Lyndon words over a q-letter alphabet. Lemma 4.4 states classical unique nonincreasing Lyndon factorization over a finite ordered alphabet.

Problem details · Reading note

Three tetrahelix coordinate generating functions

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Peter Kagey (2026). OEIS A400216, A400217 and A400218, Boerdijk-Coxeter tetrahelix coordinates. URL: https://oeis.org/A400216.

Claim. Three coordinate generating functions of one ordered Boerdijk-Coxeter tetrahelix are marked conjectured.

Problem details · Reading note

Oh and Scrimshaw's two-shifted q-Motzkin Hankel conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Se-jin Oh, Travis Scrimshaw (2018). Identities from representation theory. DOI: 10.48550/arXiv.1805.00113. URL: https://arxiv.org/abs/1805.00113v1.

Claim. The paper derives determinant and Pfaffian identities from representation theory and, in an appendix on other q-determinants, conjectures on numerical evidence that the two-shifted Hankel determinants of Cigler's q-Motzkin numbers factor as q^(c_n) f_n(q) for an explicit f_n.

Problem details · Reading note

The nullspace of the periodic XX chain on twice a prime number of sites

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Thore Posske (2026). OEIS A392387, Nullspace dimension of the periodic XX Heisenberg chain with n > 1 sites. URL: https://oeis.org/A392387.

Claim. The entry counts the subsets K of {1,...,n} whose cosines at the angles (2j + (1 + (-1)^|K|)/2)Pi/n, j in K, sum to zero, identifies this number with the dimension of the zero-energy subspace of the periodic spin-1/2 XX Heisenberg chain on n sites, and conjectures a(2p) = 2(6^((p-1)/2)+1) for odd primes p.

Problem details · Reading note

Pronko's anti-adjoint expansion conjecture for the periodic Fredkin chain

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Andrei G. Pronko (2025). Symmetries of the periodic Fredkin chain. DOI: 10.1088/1751-8121/ae1644. URL: https://doi.org/10.1088/1751-8121/ae1644.

Claim. The paper defines the periodic Fredkin Hamiltonian and the operators Sigma plus and minus, states as Conjecture 1 that they annihilate every non-cyclic invariant eigenstate, and states as Conjecture 2 that they are finite anti-adjoint expansions of the total spin operators.

Problem details · Reading note

Pronko's commutation of the Dyck-class operator Ξ with the periodic Fredkin Hamiltonian

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Andrei G. Pronko (2025). Symmetries of the periodic Fredkin chain. DOI: 10.1088/1751-8121/ae1644. URL: https://doi.org/10.1088/1751-8121/ae1644.

Problem details · Reading note

The constraint polynomials of the two-photon Rabi model at and beyond the critical coupling

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Cid Reyes-Bustos, Masato Wakayama (2026). Two-photon quantum Rabi models – Spectral degeneracy and symmetries. DOI: 10.48550/arXiv.2609.00750. URL: https://arxiv.org/abs/2609.00750v1.

Claim. The paper studies the spectrum of the two-photon asymmetric quantum Rabi model, proves divisibility relations between constraint polynomials for shared-parity degeneracies, and conjectures that the constraint polynomial at the critical coupling x = 1 factors as a product of linear factors y + 2n(2n + 2rho - 1) and that for x > 1 it has positive coefficients.

Problem details · Reading note

Refutation of the numerator conjecture of OEIS A362534

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Mohamed Sabba (2023). OEIS A362534, Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems. URL: https://oeis.org/A362534.

Claim. The entry defines the symmetry-constrained bound f(n) and the adiabatic bound g(n) on z-magnetization transfer in AXn spin-1/2 systems, lists the numerators of f(n)/g(n), and conjectures that the numerator of g(n) is the denominator of f(n)/g(n).

Problem details · Reading note

Skalski and Stroiński's conjecture on the smallest set of uniqueness for the Ising cone

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Tomasz Skalski, Tomasz Stroiński (2025). Level sets and maximum likelihood estimation for the Ising model. DOI: 10.48550/arXiv.2511.20925. URL: https://arxiv.org/abs/2511.20925v1.

Claim. The paper bounds the smallest set of uniqueness u(k,q) for the nonnegative cone of the Walsh space B^k_q on the cube {-1,+1}^k and conjectures u(k,2) = k + 1.

Problem details · Reading note

The normalized expected range of a knight's random walk is an integer

Proved. Lean theorem result, frozen in this repository 2026-09-26.

N. J. A. Sloane (2019). OEIS A309221, Expected number of distinct squares visited by a knight's random walk on an infinite chessboard after n steps, normalized to give an integer. URL: https://oeis.org/A309221.

Claim. The entry normalizes the expected number of distinct squares visited by a knight's random walk after n steps (A326954/A326955) as a(n) = E(n)*2^(3n-3) and notes that its integrality is only a conjecture.

Problem details · Reading note

Song and Lin's parity conjecture for zero transfer on oriented circulant graphs

Refuted. Lean theorem result, frozen in this repository 2026-09-26.

Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Claim. The paper characterizes zero transfer of the continuous-time quantum walk on mixed graphs through the Hermitian adjacency matrix, proves that connected oriented circulant graphs of prime order have no zero transfer, and conjectures from computations up to order 20 that for order n ≡ 2 (mod 4) zero transfer between a vertex v and 0 forces v to be odd.

Problem details · Reading note

Sun Conjecture 5.2: both lowercase strict Turan clauses

Proved. Lean theorem g_strict_turan, frozen in this repository 2026-09-26. Lean theorem v_strict_turan, frozen in this repository 2026-09-26.

Zhi-Hong Sun (2026). Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers. DOI: 10.48550/arXiv.2608.13192. URL: https://arxiv.org/html/2608.13192v1.

Claim. Conjecture 5.2 states both strict lowercase g and v Turan inequalities for every positive index on the complete real domains x <= -1 and x <= -1/8.

Problem details · Reading note

The Arrow Patterns (12; 3 → 3) and (23; 1 → 1) Are Arrow-Wilf-Equivalent

Proved. Lean theorem result, frozen in this repository 2026-09-26.

Robin D.P. Zhou, Xinyang Yu (2026). Arrow-Wilf equivalences and enumerative results for short arrow patterns. DOI: 10.48550/arXiv.2609.29392. URL: https://arxiv.org/abs/2609.29392v1.

Claim. This suggests that these two patterns may be arrow-Wilf-equivalent. … We leave this as an open problem.

Problem details · Reading note

A canonical fixed-fiber bijection for circular two-choice parking

Proved. Lean theorem result, frozen in this repository 2026-09-25.

Asma Recioui, Hacène Belbachir, Abdelhakim Ait-Zai (2026). Circular s-choice parking functions: an exact closed formula via rotational symmetry. DOI: 10.48550/arXiv.2609.23607. URL: https://arxiv.org/html/2609.23607v1.

Claim. Open Problem 1 asks for a canonical bijection between every fixed-increment, fixed-vacancy anchor class in the circular two-choice d=1 model and the classical parking functions.

Problem details · Reading note

The Miró-Roig--Tran Strict-Sign Conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-25.

Rosa M. Miró-Roig and Quang Hoa Tran (2020). On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms. DOI: 10.1016/j.jalgebra.2019.12.029. URL: https://arxiv.org/abs/2001.06143v1.

Claim. The conjecture immediately after Proposition 3.12 states that the displayed alternating binomial coefficient is strictly negative for every integer n at least two.

Problem details · Reading note

The Pan–Skandera–Wang Map Is Bruhat-Increasing for Every n

Proved. Lean theorem result, frozen in this repository 2026-09-25.

Sihong Pan, Mark Skandera, Jiayuan Wang (2026). Permanental Inequalities and Unit Interval Orders. DOI: 10.4204/EPTCS.445.17. URL: https://arxiv.org/abs/2606.13162v1.

Claim. Conjecture 7.8. For all n and all w ∈ A_n we have w ≤ f_n(w) in the Bruhat order.

Problem details · Reading note

The Order Gadget of the Erdős–Graham Two-Set Problem Is Infeasible at Every Scale

Proved. Lean theorem result, frozen in this repository 2026-09-24.

William Kasel (2026). Structural rigidity in the Erdős–Graham two-set permutation problem. URL: https://arxiv.org/abs/2609.02939v1.

Claim. The full order gadget is nevertheless conjectured infeasible at every M ≥ 16.

Problem details · Reading note

The Nested-Square Generating Function Has Even Coefficients at Odd Indices

Proved. Lean theorem result, frozen in this repository 2026-09-24.

Paul D. Hanna (2026). OEIS A392210, G.f.: x + sq(x/(1-x) + sq(x/(1-2*x) + sq(x/(1-3*x) + ...))), an infinitely nested square. URL: https://oeis.org/A392210.

Claim. A392210 %C: It appears that a(2*n-1) is even for n > 1.

Problem details · Reading note

Hanna's Series A(x - A(x)) = x^2 + x A(x) Is Rational Modulo Eight

Proved. Lean theorem result, frozen in this repository 2026-09-24.

Paul D. Hanna (2026). OEIS A392203, G.f. A(x) satisfies A(x - A(x)) = x^2 + x*A(x). URL: https://oeis.org/A392203.

Claim. A392203 %C: It appears that a(2*n) == 1 (mod 8) and a(2*n+1) == 3 (mod 8) for n >= 1.

Problem details · Reading note

The Least Base Maximizing the Digit Sum of n Is ceiling((n+1)/2) for n > 8

Proved. Lean theorem result, frozen in this repository 2026-09-24.

Jean-Marc Rebert; Sean A. Irvine (2026). OEIS A394431, The smallest base b < n where the sum of the digits for the number n in the base b is the largest. URL: https://oeis.org/A394431.

Claim. A394431 %F: Conjecture: a(n) = ceiling((n+1)/2) for n>8. - _Sean A. Irvine_, Mar 25 2026

Problem details · Reading note

Kemp–Ramgoolam arbitrary-depth determinant and monomial-basis conjectures

Proved. Lean theorem source_monomial_basis_and_expansions, frozen in this repository 2026-09-24.

Garreth Kemp and Sanjaye Ramgoolam (2026). Gauge-string duality, monomial bases and graph determinants. URL: https://arxiv.org/html/2603.05259v2.

Claim. Sections 2, 3, 6, and 7 state the arbitrary-depth threshold, monomial-basis, and sibling-only determinant conjectures resolved by the formal owners.

Problem details · Reading note

Ratajczak's A110491 T-transform determinant conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-24.

Lechoslaw Ratajczak (2021). OEIS A110491, T-transform determinant conjecture for A093178. URL: https://oeis.org/A110491.

Claim. Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j.

Problem details · Reading note

A277829 and A278770: squared-Catalan Hankel limits

Proved. Lean theorem catalan_square_hankel_log_limits, frozen in this repository 2026-09-24.

Vaclav Kotesovec (2016). OEIS A277829 and A278770: squared-Catalan Hankel determinants. URL: https://oeis.org/A277829.

Claim. A277829 and A278770 conjecture that the logarithms of their squared-Catalan Hankel determinants, divided by n^2, tend to 2*log(2).

Problem details · Reading note

Barry's A331474 Hankel generating-function conjecture

Proved. Lean theorem a331474_hankel_generating_function, frozen in this repository 2026-09-24.

Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Claim. Hankel transform of A331473. G.f.: (1 + 3x + 16x^2 - 8x^3 + 36x^4 - 12x^5 + x^6 - x^7)/(1 + 7x^2 + x^4)^2 (conjecture).

Problem details · Reading note

Raney Conjecture 7.1: zero-run record families

Proved. Lean theorem raney_zero_run_coefficient_families, frozen in this repository 2026-09-24.

Sen-Peng Eu, Zai-Ting Huang, and Louis Kao (2026). Zero-Run Spectra of the (3,2) Raney numbers Modulo Primes. URL: https://arxiv.org/html/2609.25742v1.

Claim. Conjecture 7.1: for fixed positive k,r and a prime p not dividing kr, the tied left-to-right record values of the zero-run sequence of the Raney numbers modulo p belong to a finite union of families (a*p^m+b)/c, with a,b,c depending only on k,r,p.

Problem details · Reading note

Balanced Compositions and Nonnegative Walks from Height Two

Proved. Lean theorem result, frozen in this repository 2026-09-23.

Gus Wiseman (2018). OEIS A026010, a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. URL: https://oeis.org/A026010.

Claim. Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - _Gus Wiseman_, Mar 17 2018

Problem details · Reading note

A Multiple with Fewer than Twice as Many Divisors Divides the Square

Proved. Lean theorem result, frozen in this repository 2026-09-23.

J. Lowell (2013). OEIS A225004, a(n) is the largest multiple of n with fewer than twice as many divisors as n. URL: https://oeis.org/A225004.

Claim. Conjecture: a(n) is always a divisor of n^2.

Problem details · Reading note

Fan Conjecture 5.9: moving three-point circle capacity

Proved. Lean theorem result, frozen in this repository 2026-09-23.

Qiuling Fan (2026). Riesz capacity ratios with negative exponents. DOI: 10.48550/arXiv.2609.11186. URL: https://arxiv.org/abs/2609.11186v1.

Claim. Conjecture 5.9 asks whether the three-point negative-exponent Riesz capacity is maximized at the isosceles endpoint of the stated unit-circle motion.

Problem details · Reading note

The Knight-Dialler Recurrence

Proved. Lean theorem result, frozen in this repository 2026-09-23.

Colin Barker (2019). OEIS A327692, Number of length-n phone numbers that can be dialed by a chess knight on a 0-9 keypad that starts on any number and takes n-1 steps. URL: https://oeis.org/A327692.

Claim. Conjectures from _Colin Barker_, Oct 01 2019: (Start) G.f.: 2*x*(5 + 10*x - 7*x^2 - 8*x^3 + 2*x^4) / (1 - 6*x^2 + 4*x^4). a(n) = 6*a(n-2) - 4*a(n-4) for n>6. (End)

Problem details · Reading note

Zero Slope Fits a Sample Best Exactly When Its Weighted Index Sum Is Balanced

Proved. Lean theorem result, frozen in this repository 2026-09-23.

R. H. Hardin; Gus Wiseman (2023). OEIS A222955, Number of n X 1 0..1 arrays with every row and column least squares fitting to a zero slope straight line. URL: https://oeis.org/A222955.

Claim. Conjecture: A binary word is counted iff it has the same sum of positions of 1's as its reverse, or, equivalently, the same sum of partial sums as its reverse.

Problem details · Reading note

Nikandish Problem 4.2

Proved. Lean theorem result, frozen in this repository 2026-09-23.

R. Nikandish (2026). Annihilating-Ideal Graphs and Orthogonality Graphs over F_2. DOI: 10.48550/arXiv.2609.22769. URL: https://arxiv.org/abs/2609.22769v1.

Claim. Section 4, Problem 4.2 asks for the exact clique number of the orthogonality graph on all nonzero subspaces of F_2^n for general n.

Problem details · Reading note

Kurkov's A003319 double-array conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-23.

Mikhail Kurkov (2024). OEIS A003319 and Kurkov's A370380/A370381 double-array conjecture. URL: https://oeis.org/A003319.

Claim. Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - _Mikhail Kurkov_, Apr 26 2024

Problem details · Reading note

Reversal and the Weighted Sum of a Partition

Proved. Lean theorem result, frozen in this repository 2026-09-23.

Gus Wiseman (2023). OEIS A362559, Number of integer partitions of n whose weighted sum is divisible by n. URL: https://oeis.org/A362559.

Claim. Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Problem details · Reading note

Sahbi Conjecture 6.4

Proved. Lean theorem subQuorumChromaticNumber_hypercube, frozen in this repository 2026-09-23.

Rafik Sahbi (2026). Sub-quorum colorings of graphs. DOI: 10.48550/arXiv.2609.25128. URL: https://arxiv.org/html/2609.25128v1.

Claim. Conjecture 6.4 asks whether the sub-quorum chromatic number of Q_n is 2^(n-1) for every integer n>=2.

Problem details · Reading note

Squarefree Numbers Have No Partition Whose Parts and Multiplicities Agree in Mean

Proved. Lean theorem result, frozen in this repository 2026-09-23.

Gus Wiseman (2023). OEIS A360070, Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities. URL: https://oeis.org/A360070.

Claim. Conjecture: No term > 1 is squarefree.

Problem details · Reading note

Problem 15.5 answered negatively by a five-element Clifford semigroup

Refuted. Lean theorem result, frozen in this repository 2026-09-22.

João Araújo; Wolfram Bentz; Peter J. Cameron; Kevin Hendrey; Michael Kinyon (2026). Complete Mappings of Semigroups. DOI: 10.48550/arXiv.2608.25092. URL: https://arxiv.org/abs/2608.25092.

Claim. Problem 15.5 asks whether an orthodox semigroup with an ordering whose product is idempotent must have a complete mapping.

Problem details · Reading note

Pairwise Coprime Ordered Tuples of Divisors

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Gus Wiseman (2021). OEIS A062319, Number of divisors of n^n, or of A000312(n). URL: https://oeis.org/A062319.

Claim. Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. - _Gus Wiseman_, May 02 2021

Problem details · Reading note

Fibonacci Numbers That Are Pythagorean Perimeters

Refuted. Lean theorem result, frozen in this repository 2026-09-22.

Felix Huber; Artur Jasinski (2023). OEIS A134492, a(n) = Fibonacci(6*n). URL: https://oeis.org/A134492.

Claim. %N a(n) = Fibonacci(6*n). %C Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - _Felix Huber_, Nov 03 2023 %C All terms are divisible by 8. - _Alonso del Arte_, Jul 27 2013 %A _Artur Jasinski_, Oct 28 2007

Problem details · Reading note

Runs in the Second Binary Digit of the Fibonacci Numbers

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Andres Cicuttin (2016). OEIS A272170, Second most significant bit of Fibonacci numbers > 1 written in base 2. URL: https://oeis.org/A272170.

Claim. %N Second most significant bit of Fibonacci numbers > 1 written in base 2. %C It is conjectured that there are no more than two consecutive "0's" or "1's" (tested up to n=10^5). The sequence looks quasiperiodic and its Fourier spectrum seems to have a fractal structure. - _Andres Cicuttin_, Apr 21 2016

Problem details · Reading note

Fried's Fibonacci Shift Conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Sela Fried (2025). Proofs of Several Conjectures From the OEIS. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Fried/fried15.pdf.

Claim. The end of Section 6 conjectures that for every n in N the number F(n+2) + 2nF(n+1) is not a Fibonacci number.

Problem details · Reading note

Generalized Glasby–Paseman Conjecture 1.1(d)

Proved. Lean theorem maximum_asymptotic, frozen in this repository 2026-09-22.

Seok Hyun Byun and Svetlana Poznanović (2026). Unimodality and log-concavity of generalized Glasby-Paseman sequences. URL: https://arxiv.org/abs/2604.14639v1.

Claim. Conjecture 1.1(d) specifies the exact asymptotic maximum of the powered-sum ratio (1.4) for every fixed positive real a and positive integer l; the paper proves the case l=2 and a=1.

Problem details · Reading note

Closed Form for the Greedy Fibonacci-Average Sequence

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Sela Fried (2025). Proofs of Several Conjectures From the OEIS. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Fried/fried15.pdf.

Claim. The end of Section 6 conjectures that for every n in N the number F(n+2) + 2nF(n+1) is not a Fibonacci number.

Problem details · Reading note

Mi-Xu-Yang Conjecture 5.2 refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-22.

Jared Mi; Jeffrey Xu; Jason Yang (2025). Partially Restricted Stacks as Functions. DOI: 10.54550/ECA2025V5S3R22. URL: https://ecajournal.haifa.ac.il/Volume2025/ECA2025_S2A22.pdf.

Claim. Conjecture 5.2 states that pi takes 2n-5 sorts under s composed with t exactly when pi = 2 sigma 1 n and sigma avoids 213.

Problem details · Reading note

A132393: Kurkov's nested Stirling identity

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Mikhail Kurkov (2026). OEIS A132393: Kurkov's nested sum for unsigned Stirling numbers. URL: https://oeis.org/A132393.

Claim. Conjecture: T(n+m+1,m+1) = (n+m)! * (Product_{i=1..m} (n+i)^i) * Sum_{j_1=0..n} Sum_{j_2=0..j_1} ... Sum_{j_m=0..j_{m-1}} (-1)^(n*m + j_1 + j_2 + ... + j_m) * (Product_{1 <= q < p <= m} (j_q - j_p + p - q)^2) * Product_{t=1..m} (n+t-1)_{j_t} / ((n-j_t+t)^(m+1) * (j_t+m-t)!) where (x)_{n} is the falling factorial defined as (x)_{n} = Product_{k=0..n-1} (x-k). - _Mikhail Kurkov_, May 24 2026

Problem details · Reading note

A398542 fixed-bottom polynomial conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Charles Cornell Norton (2026). OEIS A398542: fixed-bottom polynomial conjecture. URL: https://oeis.org/A398542.

Claim. For every nonempty 132-avoiding bottom b of length m, its actual two-cell completion count is p(k)*binomial(2*k,k)+q(k)*4^k, with degree bounds m-1 and m-2.

Problem details · Reading note

The Alternating Stirling Weighted Sum of Even-Index Fibonacci and Lucas Numbers

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Daniele Parisse (2024). On Hypersequences of an Arbitrary Sequence and Their Weighted Sums. DOI: 10.5281/zenodo.13331499. URL: https://math.colgate.edu/~integers/y70/y70.pdf.

Claim. Section 4 states two conjectures: the alternating factorial-Stirling weighted sum of the even-index Fibonacci numbers equals the signed plain weighted sum of the preceding Stirling row, and the same identity with the Lucas numbers in place of the Fibonacci numbers.

Problem details · Reading note

The Order of a Pisano Period Is Not Confined to Two

Refuted. Lean theorem result, frozen in this repository 2026-09-22.

Brennan Benfield, Oliver Lippard (2025). Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers. URL: https://arxiv.org/abs/2407.20048v2.

Claim. Conjecture 5.3. For an $(a,b)$-Fibonacci sequence, the range of $\omega_{(a,b)}(m)$ for $m>1$ is given by (i) $\{1\}$ if $(a,b) \in \{(0,1),(1,0),(-1,-1),(\pm2,-1)\}$, or if $m=2$ and $(a,b) \in \{(-1,0),(1,-1),(0,-1)\}$, (ii) $\{2\}$ if $m>2$ and $(a,b) \in \{(-1,0),(1,-1),(0,-1)\}$, (iii) $\{1,2,4\}$ if $b=1$ and $a\neq0$, (iv) $\{1,2\}$ if $b=-1$ and $a\not\in\{-2,-1,0,1,2\}$, (v) $\{0,1,2\}$ if $b \neq \pm1$ and $|a|-|b| = 1$, (vi) infinite in all other cases.

Problem details · Reading note

Pronko's non-cyclic Fredkin eigenstate annihilation conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-22.

Andrei G. Pronko (2025). Symmetries of the periodic Fredkin chain. DOI: 10.1088/1751-8121/ae1644. URL: https://doi.org/10.1088/1751-8121/ae1644.

Problem details · Reading note

Barry's Riordan-Pascal kernel determinant

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Paul Barry (2013). A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays. URL: https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.pdf.

Claim. Conjecture 32 identifies the finite kernel matrix with a Riordan Gram matrix and gives determinant one.

Problem details · Reading note

Chaffin-Linderman-Sloane-Wilks Gijswijt Merge Refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-21.

Benjamin Chaffin, John P. Linderman, N. J. A. Sloane, Allan R. Wilks (2013). On curling numbers of integer sequences. DOI: 10.48550/arXiv.1212.6102. URL: https://arxiv.org/abs/1212.6102v3.

Claim. The last sentence of Section 5 asks whether Theorem 23 survives when the starting sequence may contain a 1 but does not end with 1; Section 6 lists it as open question 12.

Problem details · Reading note

The first q = 2 Schreier-multiset recurrence

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Hùng Việt Chu; Yubo Geng; Julian King; Steven J. Miller; Garrett Tresch; Zachary Louis Vasseur (2026). Linear Recurrences from Counting Schreier-Type Multisets. DOI: 10.5281/zenodo.19949535. URL: https://math.colgate.edu/~integers/aa53/aa53.pdf.

Claim. Section 5 conjectures the recurrence for the cardinalities of A^(2)_(1,2,n).

Problem details · Reading note

Ducci Modular Cycle Factorization

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Tasha Fellman, Dominic Klyve (2023). Existence of Cycles in Ducci's Four-Number Game with Modular Multiplication. DOI: 10.5281/zenodo.10160456. URL: https://math.colgate.edu/~integers/x86/x86.pdf.

Claim. Section 7 states Conjecture 1, that every cocomposite cycle is a product of a constant cycle and two generating cycles, verified for all moduli up to 161.

Problem details · Reading note

Erdos Deep Triple Classification Refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-21.

Peter J. Dukes, Tao Gaede (2023). Families of Modular Arithmetic Progressions with an Interval of Distance Multiplicities. DOI: 10.48550/arXiv.2208.05527. URL: https://math.colgate.edu/~integers/x25/x25.pdf.

Claim. Conjecture 1 proposes that the length triples of Erdos-deep families of three modular arithmetic progressions are exactly two infinite families and thirteen sporadic triples.

Problem details · Reading note

Flórez-Junes-Ramírez xz-plane path-count refutation

Refuted. Lean theorem result1, frozen in this repository 2026-09-21.

Rigoberto Flórez, Leandro Junes, José L. Ramírez (2018). Further Results on Paths in an n-Dimensional Cubic Lattice. URL: https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.pdf.

Claim. Section 6 states Conjectures 1 and 2 for xz-plane and yz-plane paths in the printed C_3^+(k) family.

Problem details · Reading note

Flórez-Junes-Ramírez yz-plane path-count refutation

Refuted. Lean theorem result2, frozen in this repository 2026-09-21.

Rigoberto Flórez, Leandro Junes, José L. Ramírez (2018). Further Results on Paths in an n-Dimensional Cubic Lattice. URL: https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.pdf.

Claim. Section 6 states Conjectures 1 and 2 for xz-plane and yz-plane paths in the printed C_3^+(k) family.

Problem details · Reading note

Mbirika-Schrader-Spilker Associated Pell GCD Entry-Point Refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-21.

aBa Mbirika; Janee Schrader; Jürgen Spilker (2023). Pell and Associated Pell Braid Sequences as GCDs of Sums of k Consecutive Pell, Balancing, and Related Numbers. DOI: 10.48550/arXiv.2301.05758. URL: https://cs.uwaterloo.ca/journals/JIS/VOL26/Mbirika/mbir5.pdf.

Claim. Conjecture 32: gcd(Q_k, k) > 1 if and only if a prime divisor p of k has entry point e_Q(p) dividing k.

Problem details · Reading note

Merca's lifted residue sum at prime order

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Mircea Merca (2011). Inequalities and Identities Involving Sums of Integer Functions. URL: https://cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.pdf.

Claim. Conjecture 1 asserts an exact residue sum under two coprimality conditions at even multiplicative order; Conjecture 2 asserts a lifted residue-sum identity at even multiplicative order for prime moduli.

Problem details · Reading note

Merca's even-order residue-sum conjecture refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-21.

Mircea Merca (2011). Inequalities and Identities Involving Sums of Integer Functions. URL: https://cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.pdf.

Problem details · Reading note

Bala's Fibonacci-power strong divisibility conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Peter Bala (2022). OEIS A000071: Bala's strong divisibility conjecture for Fibonacci minus one at odd-base powers. URL: https://oeis.org/A000071.

Claim. Conjecture: for k a positive odd integer, the sequence {a(k^n): n >= 1} is a strong divisibility sequence; that is, for n, m >= 1, gcd(a(k^n), a(k^m)) = a(k^gcd(n,m)).

Problem details · Reading note

Optimality of the Ordered Zeckendorf Long Game Strategy

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Ivan Bortnovskyi and Michael Lucas and Steven J. Miller and Iana Vranesko and Ren Watson and Cameron White (2025). The Ordered Zeckendorf Game. DOI: 10.48550/arXiv.2508.20222.

Claim. The Long Game Strategy is conjectured to achieve the greatest length among all legal terminating plays of the ordered Zeckendorf game.

Problem details · Reading note

Quasi-Injective Composition Refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-21.

Prapanpong Pongsriiam (2021). Quasi-Injectivity of Some Arithmetic Functions. URL: https://cs.uwaterloo.ca/journals/JIS/VOL24/Pongsriiam/pong23.pdf.

Claim. Question 17 asks whether, for each m at least two, some function is quasi-injective of order m-1 but not of order m.

Problem details · Reading note

Quasi-Injective Order Separation

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Prapanpong Pongsriiam (2021). Quasi-Injectivity of Some Arithmetic Functions. URL: https://cs.uwaterloo.ca/journals/JIS/VOL24/Pongsriiam/pong23.pdf.

Claim. Question 17 asks whether, for each m at least two, some function is quasi-injective of order m-1 but not of order m.

Problem details · Reading note

Stirling Lucky Displacement Spectrum

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Laura Colmenarejo, Aleyah Dawkins, Jennifer Elder, Pamela E. Harris, Kimberly J. Harry, Selvi Kara, Dorian Smith, Bridget Eileen Tenner (2024). On the lucky and displacement statistics of Stirling permutations. DOI: 10.48550/arXiv.2403.03280. URL: https://cs.uwaterloo.ca/journals/JIS/VOL27/Tenner/tenner12.html.

Claim. Corollary 41 bounds the number of nonzero entries of a displacement composition between n and 2n-1, and Problem 48 asks whether every count in that range is attained.

Problem details · Reading note

Degrees of the order-eleven quasi-Fibonacci polynomials

Proved. Lean theorem result, frozen in this repository 2026-09-21.

Roman Wituła; Damian Słota (2007). Quasi-Fibonacci Numbers of Order 11. URL: https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.pdf.

Claim. The printed page-19 Problem asks whether all five order-eleven quasi-Fibonacci polynomials have degree n for every n at least 5.

Problem details · Reading note

ayad-bouchenna-base-b-reverse-multiple-divisors

Proved. Lean theorem result, frozen in this repository 2026-09-20.

Mohamed Ayad; Rachid Bouchenna (2025). Which Integer Divides the Reverse of Any of Its Multiples?. DOI: 10.5281/zenodo.15283699. URL: https://math.colgate.edu/~integers/z37/z37.pdf.

Claim. Problem 1: Let B ≥ 2 be any number base. Are the divisors of B² − 1 the only positive integers satisfying the property P*_B ?

Problem details · Reading note

Refutation of the regular elementary symmetric image table

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

Cristina Ballantine, George Beck, Mircea Merca, Bruce E. Sagan (2024). Elementary symmetric partitions. DOI: 10.48550/arXiv.2409.11268. URL: https://arxiv.org/abs/2409.11268v3.

Claim. Conjecture 16 gives a residue-class table for r_{d,2}(n) minus r_{d,3}(n) when d is 2, 3, 4, or 5.

Problem details · Reading note

Chauve--Zhang condensed-neighborhood unary minimality

Refuted. Lean theorem resultCondensed, frozen in this repository 2026-09-20.

Cedric Chauve, Louxin Zhang (2025). On the size of the neighborhoods of a word. DOI: 10.48550/arXiv.2505.13796. URL: https://arxiv.org/abs/2505.13796v2.

Claim. The closing question asks whether unary words minimize condensed and super condensed neighborhoods.

Problem details · Reading note

Chauve--Zhang super-condensed-neighborhood unary minimality

Refuted. Lean theorem resultSuperCondensed, frozen in this repository 2026-09-20.

Cedric Chauve, Louxin Zhang (2025). On the size of the neighborhoods of a word. DOI: 10.48550/arXiv.2505.13796. URL: https://arxiv.org/abs/2505.13796v2.

Claim. The closing question asks whether unary words minimize condensed and super condensed neighborhoods.

Problem details · Reading note

Cohen Conjecture 29

Refuted. Lean theorem result29, frozen in this repository 2026-09-20.

Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.

Claim. Section 2.5 states Conjectures 28 and 29 and lists thresholds for twin-prime and consecutive cousin-prime pairs between consecutive cubes.

Problem details · Reading note

Cohen Conjecture 28

Refuted. Lean theorem result28, frozen in this repository 2026-09-20.

Joel E. Cohen (2025). Conjectures about Primes and Cyclic Numbers. DOI: 10.48550/arXiv.2508.08335. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.

Claim. Section 2.5 states Conjectures 28 and 29 and lists thresholds for twin-prime and consecutive cousin-prime pairs between consecutive cubes.

Problem details · Reading note

Refutation of the CRIM square-rectair formula at its boundary

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

Ina Bašić; Eric Gottlieb; Matjaž Krnc (2026). CRIM: A Natural Game on Integer Partitions. DOI: 10.48550/arXiv.2606.16828.

Claim. Section 3 defines row deletion and conjugate-row-conjugate column deletion; printed Conjectures 2 and 3 give Sprague–Grundy formulas for square and near-square rectairs.

Problem details · Reading note

Crown order-polytope f-vector log-concavity

Proved. Lean theorem crownGeometricFVector_log_concave, frozen in this repository 2026-09-20.

Teemu Lundström and Leonardo Saud Maia Leite (2025). Order polytopes of crown posets. DOI: 10.48550/arXiv.2504.05123. URL: https://arxiv.org/abs/2504.05123v3.

Claim. Original v3 target, Conjecture 3.7: for every positive integer n, the full geometric f-vector of the order polytope of the crown on 2n vertices is log-concave. Theorem 3.6 supplies the face-count formula, not a proof of this conjecture.

Problem details · Reading note

Refutation of the printed partition-polynomial derivative question

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

Madeline Locus Dawsey; Tyler Russell; Dannie Urban (2022). Derivatives and Integrals of Polynomials Associated with Integer Partitions. DOI: 10.48550/arXiv.2108.00943. URL: https://cs.uwaterloo.ca/journals/JIS/VOL25/Dawsey/dawsey3.pdf.

Claim. Definition (1) introduces the partition polynomial; Question 9 asks whether derivative values at 1 distinguish any two unequal partitions.

Problem details · Reading note

Refutation of the printed barrycade omitted-element relation

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

Michał Dębski; Jarosław Grytczuk; Paweł Naroski; Bartłomiej Pawlik; Jakub Przybyło; Małgorzata Śleszyńska-Nowak (2026). Finite and infinite barrycades. DOI: 10.48550/arXiv.2609.18476. URL: https://arxiv.org/abs/2609.18476v1.

Claim. The paper defines greedy quasi-barrycade rows and prints Conjecture 2, including A2(i) = A1(i) + 1 for every i >= 3.

Problem details · Reading note

El Zein-Mortada Saturated Packing Refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

Ayman El Zein, Maidoun Mortada (2026). Impact of local girth on the S-packing coloring of k-saturated subcubic graphs. DOI: 10.48550/arXiv.2603.25113. URL: https://arxiv.org/abs/2603.25113v1.

Claim. Conjecture 3 states that every 2-saturated subcubic graph is (1,1,2)-packing colorable.

Problem details · Reading note

Gil–Liang–Odetola–Weiner antidiagonal traffic

Proved. Lean theorem result, frozen in this repository 2026-09-20.

Juan Gil, Zhenni Liang, Ayodeji Odetola, Michael Weiner (2026). Points of maximal traffic on a grid with obstruction. URL: https://arxiv.org/abs/2609.01562.

Claim. Conjecture 7.4 asks whether every antidiagonal obstruction has traffic maxima at (1,1) and (n-1,n-1) for every n at least 496; Section 7 gives the binomial-ratio comparison.

Problem details · Reading note

Mohan-Neetu Conjecture 6.1

Refuted. Lean theorem result61, frozen in this repository 2026-09-20.

Mohan, Neetu (2026). On small doubling in right-ordered groups and Baumslag-Solitar groups - II. DOI: 10.48550/arXiv.2607.11194. URL: https://arxiv.org/abs/2607.11194v1.

Claim. Section 6 prints Conjectures 6.1, 6.2 and 6.3 for finite subsets of torsion-free groups.

Problem details · Reading note

Mohan-Neetu Conjecture 6.2

Refuted. Lean theorem result62, frozen in this repository 2026-09-20.

Mohan, Neetu (2026). On small doubling in right-ordered groups and Baumslag-Solitar groups - II. DOI: 10.48550/arXiv.2607.11194. URL: https://arxiv.org/abs/2607.11194v1.

Claim. Section 6 prints Conjectures 6.1, 6.2 and 6.3 for finite subsets of torsion-free groups.

Problem details · Reading note

Mohan-Neetu Conjecture 6.3

Refuted. Lean theorem result63, frozen in this repository 2026-09-20.

Mohan, Neetu (2026). On small doubling in right-ordered groups and Baumslag-Solitar groups - II. DOI: 10.48550/arXiv.2607.11194. URL: https://arxiv.org/abs/2607.11194v1.

Claim. Section 6 prints Conjectures 6.1, 6.2 and 6.3 for finite subsets of torsion-free groups.

Problem details · Reading note

Refutation of Layman's A196226 divisor-power congruences

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

John W. Layman (2011). OEIS A196226, divisor-sum residue and divisor-power congruence conjectures. URL: https://oeis.org/A196226.

Claim. If m is a sufficiently large term of A196226, then sigma(k,m) is congruent to 2^k + 1 + m/2 modulo m; the entry states the cases k = 2, 3, and 4 with thresholds 14, 22, and 38.

Problem details · Reading note

Krizek's A300657 noncomposite characterization

Proved. Lean theorem result, frozen in this repository 2026-09-20.

Jaroslav Krizek (2018). OEIS A300657, sum of divisor-sum residues. URL: https://oeis.org/A300657.

Claim. %N a(n) = Sum_{d|n} sigma(d) mod d. %C Conjecture: a(n) = A054024(n) only for the noncomposite numbers A008578.

Problem details · Reading note

Perrier's period-one generating functions

Proved. Lean theorem result, frozen in this repository 2026-09-20.

Rachel Perrier (2026). Multidimensional Continued Fractions and Riordan Arrays. DOI: 10.54550/ECA2026V6S4R33. URL: https://ecajournal.haifa.ac.il/Volume2026/ECA2026_S4R33.pdf.

Claim. Section 5, printed pp. 14-15: We conjecture that solving these recurrences yields the displayed rational functions R_1 and R_k; shifted generating functions are defined on printed pp. 8 and 12.

Problem details · Reading note

Sharma's primitive-polynomial Conjecture 4.2

Refuted. Lean theorem result, frozen in this repository 2026-09-20.

Avnish K. Sharma (2026). Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures. URL: https://arxiv.org/abs/2608.07262v2.

Claim. Conjecture 4.2 asserts that for every odd prime p, every finite field K of cardinality p^2, and every multiplicative generator λ of K, X^p+X+λ is the monic minimal polynomial of a multiplicative generator in a degree-p extension.

Problem details · Reading note

Takemura's fixed-point first appearance degree

Proved. Lean theorem resultDegree, frozen in this repository 2026-09-20.

Kenichi Takemura (2025). Alternating Power Difference and Matrix Symmetry: Closed-Form Formulas for the First Appearance Degree m_1. DOI: 10.48550/arXiv.2512.18169. URL: https://arxiv.org/abs/2512.18169v1.

Claim. Definition 1 gives APD_m(f) = sum over sigma in S_n of sgn(sigma) f(sigma)^m. Definition 2 calls the least m >= 1 with nonzero APD the first appearance degree. The fixed-point function is the number of fixed points of sigma. Conjecture 1 states m_1(I_n) = n - 1 for n >= 2, and Conjecture 2 states APD_{n-1}(I_n) = n!.

Problem details · Reading note

Takemura's fixed-point first appearance value

Proved. Lean theorem resultValue, frozen in this repository 2026-09-20.

Problem details · Reading note

Zhan–Bie consecutive cyclic-stack fibres

Proved. Lean theorem zhan_bie_conjectures_3_4, frozen in this repository 2026-09-20.

Alex Zhan and Stella Bie (2026). Cyclic-Pattern-Avoiding Stacks. DOI: 10.5281/zenodo.18154216. URL: https://math.colgate.edu/~integers/aa15/aa15.pdf.

Claim. Conjectures 3 and 4 give the exact full fibre over the layered target for the consecutive cyclic [123]-avoiding stack: cardinality one in even size and ceiling(n/2) in odd size, for every n at least four.

Problem details · Reading note

Agoh's alternating-numerator characterization

Refuted. Lean theorem result, frozen in this repository 2026-09-19.

Takashi Agoh (2026). An Intrinsic Factor of an Alternating Sum Involving Certain Rational Functions. DOI: 10.5281/zenodo.18154119. URL: https://math.colgate.edu/~integers/aa9/aa9.pdf.

Claim. The final conjecture characterizes the all-order alternating numerator factor property by monomials or simple real roots different from one.

Problem details · Reading note

Bradshaw's Conjecture 20

Refuted. Lean theorem result, frozen in this repository 2026-09-19.

Z. P. Bradshaw (2025). On a Family of Solutions to Arithmetic Differential Equations Involving the Collatz Map. URL: https://cs.uwaterloo.ca/journals/JIS/VOL28/Bradshaw/bradshaw3.pdf.

Claim. Conjecture 20: the solutions to the generalized Collatz commutation problem are squarefree for every pair of parameters of equal parity.

Problem details · Reading note

Centered reduced residue progression maximum

Proved. Lean theorem result, frozen in this repository 2026-09-19.

Chaninat Phothila; Natthakan Thoket; Narakorn Rompurk Kanasri (2026). Length of the Longest Arithmetic Progressions in a Certain Reduced Residue System. DOI: 10.5281/zenodo.18154061. URL: https://math.colgate.edu/~integers/aa6/aa6.pdf.

Claim. The unnumbered conjecture on printed page 14 gives the exact longest arithmetic-progression length for an even squarefree modulus with at least three prime factors when the complementary factor is less than twice the greatest prime factor.

Problem details · Reading note

Gobel-Misra Colored-Path Determinant Refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-19.

Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Claim. Conjecture 5.3 asserts that determinant-equal colored paths differ only by the listed symmetries and configurations.

Problem details · Reading note

Convergence of the Shieh-Yang-Yu dotted-pattern machine

Proved. Lean theorem result, frozen in this repository 2026-09-19.

Michael Yang, Hansen Shieh, Ashley Yu (2025). Stack-Sorting with Dotted-Pattern-Avoiding Stacks. URL: https://arxiv.org/abs/2411.11914v2.

Claim. Valley runs, Proposition 3.5, and Conjecture 6.2 for the 21-dot machine; peak runs, Proposition 3.1, and Conjecture 6.1 for the 12-dot machine.

Problem details · Reading note

Somer-Krizek uniform-subsequence conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-19.

Lawrence Somer; Michal Krizek (2025). Generalization of a Theorem of Velez on Uniform Distribution in Second-Order Linear Recurrences. DOI: 10.5281/zenodo.14679256. URL: https://math.colgate.edu/~integers/z1/z1.pdf.

Claim. Conjecture 4.2 proposes that every uniformly distributed k-th order integer recurrence modulo p^e, with least period p^e E and last coefficient coprime to p, remains uniformly distributed on every coprime-step subsequence, with E/gcd(g,E) copies of each residue in the stated finite range.

Problem details · Reading note

Wu's k-gonal-pyramidal complement formula

Proved. Lean theorem wu_conjecture_one, frozen in this repository 2026-09-19.

Chai Wah Wu (2025). Algorithms for Complementary Sequences. DOI: 10.5281/zenodo.17535229. URL: https://math.colgate.edu/~integers/z95/z95.pdf.

Claim. Conjecture 1. For k >= 9, (6) holds for all n >= 1.

Problem details · Reading note

Refutation of Zhang's character-sum constant restriction

Refuted. Lean theorem result, frozen in this repository 2026-09-19.

Wenpeng Zhang (2025). Some interesting number theory problems. DOI: 10.48550/arXiv.2506.17235. URL: https://arxiv.org/abs/2506.17235v1.

Claim. Question (D) asks whether the fixed constant difference c in identity (2), for fundamentally different integer-coefficient polynomials, can only be 0 or 2.

Problem details · Reading note

Refutation of Zhao's maximum preimage conjecture

Refuted. Lean theorem resultMaximum, frozen in this repository 2026-09-19.

William Zhao (2024). Stack-sorting with Stacks Avoiding Vincular Patterns. DOI: 10.1016/j.disc.2025.114834. URL: https://arxiv.org/abs/2410.17057v1.

Problem details · Reading note

Refutation of Zhao's second-largest preimage conjecture

Refuted. Lean theorem resultSecondLargest, frozen in this repository 2026-09-19.

William Zhao (2024). Stack-sorting with Stacks Avoiding Vincular Patterns. DOI: 10.1016/j.disc.2025.114834. URL: https://arxiv.org/abs/2410.17057v1.

Problem details · Reading note

Bastien–Khormali link-irregular tournament existence

Proved. Lean theorem result, frozen in this repository 2026-09-18.

Alexander Bastien and Omid Khormali (2025). On Link-irregular Digraphs. URL: https://arxiv.org/abs/2512.20494v1.

Claim. Conjecture 6 states that a link-irregular tournament exists on n vertices if and only if n is at least 6; the paper proves the assertion through order 8 and reports computational verification through order 100.

Problem details · Reading note

Refutation of Bhagat, Kulkarni, Larsson and Murali Conjecture 4

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Anjali Bhagat; Tanmay Kulkarni; Urban Larsson; Divya Murali (2026). Tie-breaking in self interest cumulative subtraction games. URL: https://arxiv.org/abs/2510.24280v2.

Claim. Conjecture 4 predicts that friendly tie-breaking for both players is never worse for either player than antagonistic tie-breaking for both.

Problem details · Reading note

Cao-Chen-Miller Conjecture 1: the multidimensional egg-drop bound

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Xiangwen Cao and Zongyun Chen and Steven J. Miller (2025). Egg Drop Problems: They Are All They Are Cracked Up To Be!. DOI: 10.48550/arXiv.2511.18330. URL: https://arxiv.org/abs/2511.18330.

Claim. Conjecture 1 proposes a real-power ceiling upper bound for multidimensional egg-drop strategies when the egg count is at least the dimension.

Problem details · Reading note

Even-hypergraph inverse-positivity refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Chaithra P; Shushma Rani; R. Venkatesh (2025). Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements. DOI: 10.48550/arXiv.2507.20847. URL: https://arxiv.org/html/2507.20847v1.

Claim. Section 8.1: for a simple hypergraph on a positive finite number of vertices, the reciprocal of its signed ordinary independence polynomial has nonnegative rational coefficients at every multiindex if and only if every edge has even cardinality.

Problem details · Reading note

Refutation of the equality conclusion in Fazekas et al. Conjecture 4

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Szilárd Zsolt Fazekas, Adam Mammoliti, Robert Mercaş, Jamie Simpson (2026). Binary Words Containing Few Abelian Squares. DOI: 10.48550/arXiv.2604.23188. URL: https://arxiv.org/abs/2604.23188v1.

Claim. Conjecture 4 asserts that equality in the floor n over 4 lower bound forces every abelian square to be trivial.

Problem details · Reading note

Refutation of Gonzalez D'Leon-Wachs Conjecture 4.13(2)

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Rafael S. Gonzalez D'Leon; Michelle L. Wachs (2026). Weighted bond posets and a new chromatic symmetric function. DOI: 10.48550/arXiv.2608.08692. URL: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13.

Claim. Conjecture 4.13(2): for a graph G on n vertices with k connected components and a spanning subgraph H with the same k components, (-1)^(n-k)(mu_G-mu_H) is real-rooted.

Problem details · Reading note

Refutation of the PNim heavy-interval conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Eric Gottlieb; Matjaž Krnc; Peter Muršič (2025). Nim on Integer Partitions and Hyperrectangles. DOI: 10.48550/arXiv.2506.04991. URL: https://arxiv.org/abs/2506.04991.

Claim. Section 1.2 defines PNim row and column subset deletion; section 2 defines Young's order; section 3 and Proposition 2 define heaviness; printed Conjecture 2 predicts a heavy interval below any heavy rectangle.

Problem details · Reading note

The all-dimensional Hypercube Inequality

Proved. Lean theorem result, frozen in this repository 2026-09-18.

Gennadiy Averkov; Katherina von Dichter; Ivan Soprunov (2026). On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube. URL: https://arxiv.org/html/2608.14909v1#S4.E16.

Claim. Equation (16) is the Hypercube Inequality for nonnegative weights on the vertices of the d-dimensional cube; Section 7.5 states that it remains open for d >= 4.

Problem details · Reading note

Refutation of Kok's dom-path diameter conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Johan Kok (2025). Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs. DOI: 10.48550/arXiv.2507.16500.

Problem details · Reading note

Refutation of the panchromatic pairing conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Mohammed Lalou; Nader Mbarek; Abdallah Skender; Olivier Togni (2026). Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity. DOI: 10.48550/arXiv.2512.15486. URL: https://arxiv.org/abs/2512.15486v2.

Claim. Conjecture 1 asserts that every hypergraph satisfies chi_p^2(H) = chi_p(H) - ceil(alpha_{chi_p(H)}(H)/2).

Problem details · Reading note

The ordinary Loopy polynomial does not determine the generalized degree polynomial

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Anatol Kirillov, Gleb Nenashev, Boris Shapiro, and Arkady Vaintrob (2026). The Loopy Polynomial: from Tutte's Universal V-Function to Bizonotopal Geometry. URL: https://arxiv.org/abs/2609.07728v1.

Claim. Definition 1.1 and Proposition 2.1 give the ordinary Loopy recursion and its choice independence; Problem 11.4 asks whether it determines the degree sequence for arbitrary graphs with loops included.

Problem details · Reading note

Misawa--Nishimura harmonic-strength pair refutation

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Ryutaro Misawa, Yusaku Nishimura (2025). Spherical Designs on S¹ of Finite Harmonic Strength. DOI: 10.48550/arXiv.2505.06893. URL: https://arxiv.org/abs/2505.06893v2.

Claim. Conjecture 3.3 asserts N({p,q},2) = 5 for all distinct integers p,q greater than one.

Problem details · Reading note

Refutation of Nathanson's additive h-basis strict inequality

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Melvyn B. Nathanson (2026). Problems in additive number theory, VII: The structure of additive h-bases for n. URL: https://arxiv.org/abs/2605.26425v3.

Claim. Problem 12 asks whether negative elements force ell_h(A) <= n-flat_h(k), or even the strict inequality ell_h(A) < n-flat_h(k).

Problem details · Reading note

Krizek's phitorial divisor-sum parity characterization

Proved. Lean theorem result, frozen in this repository 2026-09-18.

Jaroslav Krizek (2017). OEIS A280258, sum of phitorials over divisors. URL: https://oeis.org/A280258.

Claim. A280258 %N: a(n) = Sum_{d|n} pxi(d), where pxi(m) is the product of totatives of m (A001783). A280258 %C: Conjecture: a(n) is odd for numbers in A183300; a(n) is even for numbers in A001105 (2*n^2). A280258 %A: _Jaroslav Krizek_, Jan 01 2017

Problem details · Reading note

Marcus's A344083 nonpolygonal Pythagorean triple conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-18.

Michel Marcus (2021). OEIS A344083, a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k. URL: https://oeis.org/A344083.

Claim. A090467 consists of numbers not of the form 1 + k*m*(m-1)/2 - (m-1)^2 for k > 2 and m > 2. A344083 %C, May 09 2021: Conjecture: there are no other Pythagorean triples that give this minimum. In other words, it is the only triple with 3 A090467 terms. - _Michel Marcus_

Problem details · Reading note

Pandey's Parity Conjecture 4.1

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Rohan Pandey (2026). Parity-Dependent Real-Rootedness in Independence Polynomials of Generalized Petersen Graphs. URL: https://arxiv.org/abs/2601.03293v1.

Claim. Conjecture 4.1 asserts that for all integers n >= 2k+1 the independence polynomial of GP(n,k) has only real roots if and only if k is even, on the graph domain n >= 3 and 1 <= k < n/2.

Problem details · Reading note

Refutation of Pudelko Conjecture 8 at position zero

Refuted. Lean theorem result, frozen in this repository 2026-09-18.

Marc T. Pudelko (2025). Modular Periodicity of Random Initialized Recurrences. URL: https://arxiv.org/abs/2510.24882v5.

Claim. Conjecture 8 and equation (14) predict that the bounded-initialization probability of a Fibonacci absolute minimum at position zero converges to one quarter.

Problem details · Reading note

Refutation of Crux Problem 2623

Refuted. Lean theorem result, frozen in this repository 2026-09-17.

Faruk Zejnulahi; Sefket Arslanagic (2001). Problem 2623. URL: https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv27n2.pdf.

Claim. For positive cyclic variables, define S_k as the sum over j=1 through n of the quotient of the consecutive block sum x_j through x_(j+k) by the shifted block sum x_(j+1) through x_(j+k+1). Prove or disprove that S_k >= S_(k+1).

Problem details · Reading note

Fibonacci-word MAP maximum

Proved. Lean theorem result, frozen in this repository 2026-09-17.

Problem details · Reading note

First longest progressions at Fibonacci differences

Proved. Lean theorem result, frozen in this repository 2026-09-17.

Problem details · Reading note

Rotondo's A006972 Lucas-Carmichael sufficient condition

Proved. Lean theorem result, frozen in this repository 2026-09-17.

Davide Rotondo (2020). OEIS A006972, Lucas-Carmichael numbers, with Rotondo's sufficient condition for three distinct odd prime factors. URL: https://oeis.org/A006972.

Claim. %N Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1. %C Conjecture: if k = p*q*r, p = a*d - 1, q = b*d - 1, r = c*d - 1 are distinct odd primes, with d = gcd(p + 1, q + 1, r + 1) and a*b*c*d divides k + 1, then k is a Lucas-Carmichael number. - _Davide Rotondo_, Dec 23 2020

Problem details · Reading note

Prime square perfect-power representation

Proved. Lean theorem result, frozen in this repository 2026-09-17.

Cino Hilliard; Robert Israel (2006). OEIS A115039 and A115038, prime square perfect-power representation. URL: https://oeis.org/A115039.

Claim. Conjecture: There will always be an x,y,n such that x^2 + p = y^n for all primes p.

Problem details · Reading note

Refutation of Barker's A181278 recurrence

Refuted. Lean theorem result, frozen in this repository 2026-09-17.

Colin Barker (2018). OEIS A181278, binary matrices with ordered rows and decreasing mod-2 Gram rows. URL: https://oeis.org/A181278.

Claim. a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) for n>3.

Problem details · Reading note

Oudra's divisor-count formula for the Stechkin function

Proved. Lean theorem result, frozen in this repository 2026-09-17.

Robert Israel; Ridouane Oudra (2020). OEIS A274010: the Boris Stechkin function and an adjacent divisor-count formula. URL: https://oeis.org/A274010.

Claim. Conjecture: a(n) = tau(n) + tau(n-1) - 2, for n>=2.

Problem details · Reading note

Wu-Lou permutation-invariant Niven digit-sum bound

Proved. Lean theorem result, frozen in this repository 2026-09-17.

Huiling Wu; Sen-Yue Lou (2025). Permutation-Invariant Niven Numbers. DOI: 10.3390/sym18010186. URL: https://arxiv.org/abs/2508.01611.

Claim. Section 8.4 conjectures that, after excluding the stated cases, no permutation-invariant Niven numbers have digit sums outside 3 through 81.

Problem details · Reading note

Refutation of Zelinsky and Zhang's Conjecture 22

Refuted. Lean theorem result, frozen in this repository 2026-09-17.

Joshua Zelinsky; Kyle Zhang (2025). Kullback-Leibler divergence and primitive non-deficient numbers. DOI: 10.48550/arXiv.2501.04209. URL: https://arxiv.org/abs/2501.04209v2.

Claim. Conjecture 22: for every positive integer n outside {1, 12, 24, 30, 36, 48, 60, 72, 120, 180, 240, 360}, if p is the smallest prime factor of n, then v(n) >= 1/p^2, where v(n) is the divisor-weighted logarithmic sum defined in the paper.

Problem details · Reading note

Degree recovery from the ordinary Loopy polynomial with loops

Proved. Lean theorem loopy_determines_degree_multiset, frozen in this repository 2026-09-16.

Anatol Kirillov, Gleb Nenashev, Boris Shapiro, and Arkady Vaintrob (2026). The Loopy Polynomial: from Tutte's Universal V-Function to Bizonotopal Geometry. URL: https://arxiv.org/abs/2609.07728v1.

Problem details · Reading note

Schulte's A000680 half-factorial primality criterion

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2025). OEIS A000680, (2n)!/2^n, with the primality criterion (2n+1) | a(n) + 2^n iff 2n+1 is prime. URL: https://oeis.org/A000680.

Claim. %N a(n) = (2n)!/2^n. %C Conjecture: For n > 0 (a(n) + 2^n) is a multiple of (2*n + 1) if and only if (2*n + 1) is a prime number. - _Werner Schulte_, Oct 05 2025

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Schulte's Ω/ω binomial convolution

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2018). OEIS A001222, the Ω/ω binomial convolution conjecture (x+y)^Ω(n) = Σ_{d|n} x^Ω(d) (x+y)^{Ω(n/d)−ω(n/d)} y^{ω(n/d)}. URL: https://oeis.org/A001222.

Claim. Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - _Werner Schulte_, Feb 10 2018

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Schulte's factorial-square primality criterion

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2020). OEIS A006472, n!(n-1)!/2^(n-1), with the primality criterion n | 2 a(n-1) + 4 iff n is prime. URL: https://oeis.org/A006472.

Claim. %N a(n) = n!*(n-1)!/2^(n-1). %C Conjecture: For n > 1, n divides 2*a(n-1) + 4 if and only if n is prime. - _Werner Schulte_, Oct 04 2020

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Schulte's A008287 alternating-binomial expansion

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2015). OEIS A008287, quadrinomial coefficients, with the conjectured expansion T(n,k) = Sum_{j=0..k} (-2)^j binomial(n,j) binomial(3n-2j,k-j). URL: https://oeis.org/A008287.

Claim. %N Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1 + x + x^2 + x^3)^n. %F T(n,k) = Sum_{j=0..k} (-2)^j*binomial(n,j)*binomial(3*n-2*j,k-j) for n >= 0 and 0 <= k <= 3*n (conjectured). - _Werner Schulte_, Sep 09 2015

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Yanev's position identity for binary reversal

Proved. Lean theorem result, frozen in this repository 2026-09-16.

David W. Wilson; Ralf Stephan; Velin Yanev; Alois P. Heinz; Henry Bottomley (2017). OEIS A030101, binary digit reversal, with Yanev's position identity. URL: https://oeis.org/A030101.

Claim. A030101 %N: a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a decimal number. A030101 %F: Conjecture: a(n) = 2*w(n) - 2*w(A053645(n)) - 1 for n > 0, where w = A264596. - _Velin Yanev_, Sep 12 2017

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Quet's A079278 reduced-denominator recurrence

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Leroy Quet; N. J. A. Sloane (2003). OEIS A079278, denominators of the rational iteration b(n) = b(n-1) + 1/(1 + 1/b(n-1)). URL: https://oeis.org/A079278.

Claim. %N A079278 Define a rational sequence {b(n)} as b(1) = 1, b(n) = b(n-1) + 1/(1 + 1/b(n-1)) for n > 1; a(n) is the denominator of b(n). %F A079278 Conjecture (Quet): a(m+1) = a(m)^2 + a(m)^3 / a(m-1)^2 - a(m)*a(m-1)^2 for m >= 2.

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Kurkov's reversed q-Bell row polynomial

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Paul D. Hanna; Mikhail Kurkov (2025). OEIS A126347 q-Bell coefficient triangle and Kurkov's reversed-row conjecture. URL: https://oeis.org/A126347.

Claim. A126347 %N: Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n). A126347 %F: Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - _Mikhail Kurkov_, Jul 06 2025

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Bala's Lucas recurrence and its growing five-adic congruence

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Peter Bala (2022). OEIS A144837, Lucas(5^n), with the 5-adic congruence conjecture a(n+1) ≡ a(n) (mod 5^(n+r+1)). URL: https://oeis.org/A144837.

Claim. %N a(n) = Lucas(5^n). %F Conjecture: a(n+1) == a(n) (mod 5^(n+r+1)) for n >= r.

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A165719 integrality and the sole prime

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Vladimir Joseph Stephan Orlovsky; Bill McEachen (2009). OEIS A165719, Integers of the form k*(k+9)/8. URL: https://oeis.org/A165719.

Claim. %C A165719 Only one term is a prime number (17). Are all others composite? %F A165719 Conjecture: Valid k values are pairs of the form (8*m-1,8*m) for m=1,2,3.... If true, 17 is the only prime term. - _Bill McEachen_, May 06 2026

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Krizek's Fermat-Mersenne classification for A249759

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Jaroslav Krizek (2014). OEIS A249759, primes p for which sigma(p-1) is prime. URL: https://oeis.org/A249759.

Claim. Primes p such that sigma(p-1) is a prime q. Conjectures: 1) sequence is finite; 2) sequence is a subsequence of A019434 (Fermat primes); 3) sequence consists of Fermat primes p such that sigma(p-1) is a Mersenne prime; 4) a(n) = (A249761(n)+3)/2.

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Ordowski's A306270 semiprime classification

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Thomas Ordowski (2020). OEIS A306270, composite k with b^(k(k-1)) = 1 mod k^2 for all b coprime to k, with the conjecture that its semiprimes > 4 are the p(p^2-p+1) of A190275. URL: https://oeis.org/A306270.

Claim. %N Composite numbers k such that b^(k(k-1)) == 1 (mod k^2) for every b coprime to k. %C Conjecture: all semiprimes > 4 in this sequence are in A190275. - _Thomas Ordowski_, Jul 19 2020

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Caceres' A308090 gcd-factorial-power primality criterion

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Pedro Caceres (2019). OEIS A308090, gcd(2^n + n!, 3^n + n!, n+1), with the conjectured primality criterion gcd = n+1 implies n+1 prime. URL: https://oeis.org/A308090.

Claim. %N a(n) = gcd(2^n + n!, 3^n + n!, n+1). %C Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.

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Schulte's exponent-product multiplicativity conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2018). OEIS A322327, A005361(n)·A034444(n), with the conjecture that A005361(n)·k^ω(n) is multiplicative with prime-power values k·e. URL: https://oeis.org/A322327.

Claim. %N a(n) = A005361(n) * A034444(n). %C Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.

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OEIS A347293 shifted-product gcd row-sum convolution

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2022). OEIS A347293, shifted-product gcd triangle and row-sum convolution. URL: https://oeis.org/A347293.

Claim. %N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n. %F Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.

Problem details · Reading note

Schulte's gcd-quotient row-sum convolution

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Werner Schulte (2022). OEIS A350900, T(n,k) = Sum_{i=1..n} gcd(i,n)/gcd(gcd(i,k),n), with the conjectured row sums (n phi(n)) * (Sum_{d|n} d phi(d)). URL: https://oeis.org/A350900.

Claim. %N Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n. %F Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.

Problem details · Reading note

The A392095 column and A088713 source series

Proved. Lean theorem result, frozen in this repository 2026-09-16.

Mikhail Kurkov (2025). OEIS A392095, antidiagonal array and the shifted A088713 column. URL: https://oeis.org/A392095.

Claim. Conjecture: A(n,0) = A088713(n+1).

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An explicit membership graph outside G₂

Proved. Lean theorem u24_not_in_g2, frozen in this repository 2026-09-15.

Duncan Adamson, Amanita Dietz, Pamela Fleischmann, Annika Huch, and Silas Cato Sacher (2026). 2-word-π-representable Graphs. URL: https://arxiv.org/abs/2605.27183v1.

Claim. Definitions 1 and 14 use adjacency iff equality of two-letter projections in two k-uniform words; Conjecture 31 asks for proper adjacent inclusion at every positive k.

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Bartoli--Stănică reduced critical-point conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-15.

Daniele Bartoli; Pantelimon Stănică (2026). Reduced polynomial lifts of APN permutations over Galois rings and effective non-APN bounds. URL: https://arxiv.org/abs/2608.30808.

Claim. Conjecture 2 states that the reduced representative of every APN permutation over F_(2^m) has a critical point in F_(2^m).

Problem details · Reading note

Refutation of Detlefs's A000040 Fibonacci-Fermat prime characterization

Refuted. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Gary Detlefs (2014). OEIS A000040, The prime numbers. URL: https://oeis.org/A000040.

Claim. A000040 %N: The prime numbers. A000040 %F: Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(n-1) mod n = 1}. - _Gary Detlefs_, May 25 2014 A000040 %F disclosed but not claimed: Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(3*n) mod 3*n = 8}. - _Gary Detlefs_, May 28 2014

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The A001108 triangular-square indices have two odd divisor sums

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Jaroslav Krizek (2016). OEIS A001108, a(n)-th triangular number is a square. URL: https://oeis.org/A001108.

Claim. %N a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1. %C The squares of NSW numbers (A008843) interleaved with twice squares from A084703, where A008843(n) = A002315(n)^2 and A084703(n) = A001542(n)^2. Conjecture: Also numbers n such that sigma(n) = A000203(n) and sigma(n-th triangular number) = A074285(n) are both odd numbers. - _Jaroslav Krizek_, Aug 05 2016

Problem details · Reading note

Bala's cubic recurrence and its growing three-adic congruence

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Peter Bala (2022). OEIS A002000, the cubic recurrence a(n+1) = a(n)(a(n)^2 - 3) from 7, with Bala's 3-adic congruence conjecture. URL: https://oeis.org/A002000.

Claim. %N a(n+1) = a(n)*(a(n)^2 - 3) with a(0) = 7. %F Conjecture: a(n+1) == a(n) (mod 3^(n+r+2)) for n >= r.

Problem details · Reading note

Stern values invert the next Farey-tree numerator

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Gary W. Adamson (2023). OEIS A002487, Stern's diatomic series, with the Adamson Farey-tree inverse conjecture. URL: https://oeis.org/A002487.

Claim. %N Stern's diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1). %C It appears that a(n) is equal to the multiplicative inverse of A007305(n+1) mod A007306(n+1). For example, a(12) is 2, the multiplicative inverse of A007305(13) mod A007306(13), where A007305(13) is 4 and A007306(13) is 7.

Problem details · Reading note

Marcus's tent-map cycle length and the signed suborder of two

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Michel Marcus (2025). OEIS A003558, least m with 2^m = ±1 mod 2n+1, with Marcus's tent-map cycle-length conjecture. URL: https://oeis.org/A003558.

Claim. %N Least number m > 0 such that 2^m == +-1 (mod 2n + 1). %C It appears that a(n) is the length of the cycle starting at 2/(2*n+1) for the map x->1 - abs(2*x-1). - _Michel Marcus_, Jul 16 2025

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Kurkov's highest-bit recurrence for inverse Gray code

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Paul D. Hanna; Mikhail Kurkov (2023). OEIS A006068, inverse Gray code, with Kurkov's recurrence. URL: https://oeis.org/A006068.

Claim. A006068 %N: a(n) is Gray-coded into n. A006068 %F: Conjecture: a(n) = a(A053645(A063946(n))) + A053644(n) for n > 0 with a(0) = 0. - _Mikhail Kurkov_, Sep 09 2023

Problem details · Reading note

Refutation of Detlefs's A008365 four-residue characterization

Refuted. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Gary Detlefs (2011). OEIS A008365, 13-rough numbers: positive integers that have no prime factors less than 13. URL: https://oeis.org/A008365.

Claim. A008365 %N: 13-rough numbers: positive integers that have no prime factors less than 13. A008365 %C: Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - _Gary Detlefs_, Dec 30 2011

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Yanev's general sigma-radical identity

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Olivier Gérard; Velin Yanev (2017). OEIS A023887, sigma_n(n), with Yanev's sigma_m identity. URL: https://oeis.org/A023887.

Claim. %N a(n) = sigma_n(n): sum of n-th powers of divisors of n. %F Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - _Velin Yanev_, Aug 24 2017

Problem details · Reading note

The differences of A049473 occupy two shifted Beatty sequences

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Clark Kimberling (2014). OEIS A049473, Nearest integer to n/sqrt(2). URL: https://oeis.org/A049473.

Claim. %N Nearest integer to n/sqrt(2). %C Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - _Clark Kimberling_, Oct 05 2014 %N A001953 a(n) = floor((n + 1/2) * sqrt(2)). %N A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.

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Yanev's maximum prime exponent recurrence

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Labos Elemer; Velin Yanev (2017). OEIS A051903, maximum exponent in the prime factorization of n. URL: https://oeis.org/A051903.

Claim. %N Maximum exponent in the prime factorization of n. %F Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - _Velin Yanev_, Sep 02 2017

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Stephan's A054096 second-column conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Clark Kimberling; Ralf Stephan (2004). OEIS A054096, T(n,2), array T as in A054090. URL: https://oeis.org/A054096.

Claim. %N T(n,2), array T as in A054090. %C Conjecture: sequence is A006183 shifted right. - _Ralf Stephan_, Jan 15 2004. %N A054090: Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, ..., n, n >= 2, r(h) = sum of the numbers in row h of T. %F A054090: T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j). %N A006183: a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).

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Stephan's A059894 complement-reversal conjectures

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Marc LeBrun; Ralf Stephan (2003). OEIS A059894, complement and reverse all but the most significant bit. URL: https://oeis.org/A059894.

Claim. A059894 %N: Complement and reverse the order of all but the most significant bit in binary expansion of n. n = 1ab..yz -> 1ZY..BA = a(n), where A = 1-a, B = 1-b, ... . A059894 %F: a(1) = 1, a(2n) = a(n) + 2^(floor(log_2(n))+1), a(2n+1) = a(n) + 2^floor(log_2(n)) (conjectured). - _Ralf Stephan_, Aug 21 2003 A059894 %F: A000120(a(n)) = A000120(A054429(n)) = A023416(n) + 1 (conjectured). - _Ralf Stephan_, Oct 05 2003

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The Marcus and Bala formulas for the power of three in F(4n)

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Benoit Cloitre; Michel Marcus; Peter Bala (2022). OEIS A074724, highest power of 3 dividing F(4n), with the Marcus and Bala conjectures. URL: https://oeis.org/A074724.

Claim. %N Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number. %F a(n) = 3^A051064(n) (conjectured). - _Michel Marcus_, May 17 2022 %F Conjecture: a(n) = (sigma(3*n) - sigma(n))/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). Equivalently, a(n) = A088838(n) - A074724(n). - _Peter Bala_, Jun 10 2022

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Yanev's closed form for binary run compression

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Reinhard Zumkeller; Velin Yanev (2016). OEIS A090079, binary run compression. URL: https://oeis.org/A090079.

Claim. In binary expansion of n: reduce contiguous blocks of 0's to 0 and contiguous blocks of 1's to 1. Conjecture: a(n) = (2^(A005811(n)+1) + (1-(-1)^n)/2 - 2)/3. - _Velin Yanev_, Dec 12 2016

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Stephan's A091915 maximum even partition-product conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Jon Perry; Ralf Stephan (2004). OEIS A091915, Maximum of even products of partitions of n. URL: https://oeis.org/A091915.

Claim. %N A091915 Maximum of even products of partitions of n. %F A091915 For n>6, a(n+3) = 3a(n) (conjectured). - _Ralf Stephan_, Dec 02 2004

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Perry's A094802 least factorial index conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Jon Perry (2004). OEIS A094802, a(n) = smallest k such that all of 1 through n divides k!. URL: https://oeis.org/A094802.

Claim. %N A094802 a(n) = smallest k such that all of 1 through n divides k!. %C A094802 It is conjectured that after n=4 the sequence is prime for n prime or the previous prime for n not prime.

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Every row sum of the A105594 triangle is odd

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul Barry (2005). OEIS A105595, Row sums of number triangle A105594. URL: https://oeis.org/A105595.

Claim. %N Row sums of number triangle A105594. %C Conjecture : all terms are odd. %F a(n)=sum{k=0..n, mod(sum{j=0..n, abs(mu(binomial(n, j)))*mod(binomial(j, k), 2)}, 2)} %F A105594 T(n, k) = mod(Sum_{j=0..n}(abs(mu(binomial(n,j)))*mod(binomial(j,k),2)), 2).

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The A112041 prime scalings force the index to be four or divisible by ten

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Pierre Cami; Harvey P. Dale (2015). OEIS A112041, Numbers k such that 1*k + 1, 3*k + 1, 9*k + 1, 27*k + 1 are all primes. URL: https://oeis.org/A112041.

Claim. A112041 %N: Numbers k such that 1*k + 1, 3*k + 1, 9*k + 1, 27*k + 1 are all primes. A112041 %C: Conjecture: all terms except the first are multiples of 10. - _Harvey P. Dale_, Mar 26 2015

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OEIS A112302: Meijer's Lerch recurrence limit

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Johannes W. Meijer (2016). OEIS A112302, Lerch recurrence limit conjecture. URL: https://oeis.org/A112302.

Claim. With Phi the Lerch transcendent, LP(0)=1 and LP(n)=(1/n)*sum(Phi(1/2,n-k,1)*LP(k),k=0..n-1) for n>0; the comment conjectures that LP(n) tends to the real Somos constant represented by A112302.

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Layman's odd-power factorial residue classification

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paolo P. Lava; Giorgio Balzarotti; John W. Layman (2010). OEIS A119690, n! mod n*(n+1)/2. URL: https://oeis.org/A119690.

Claim. %N n! mod n*(n+1)/2. Target %C: It appears that f(n)=(n!)^(2k+1) modulo n(n+1)/2 is n if n is one less than an odd prime, else f(n) is 0, for any integer k. See A175567 for related results involving an even power of n!. - _John W. Layman_, Jul 12 2010. %F a(n) = n if n+1 is an odd prime, a(n) = 0 otherwise.

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Ordowski's binomial and power-sum least-prime conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Thomas Ordowski (2018). OEIS A133907, least prime p with binomial(n+p,p) ≡ 1 (mod p), with the power-sum least-prime conjecture. URL: https://oeis.org/A133907.

Claim. %N Least prime number p such that binomial(n+p, p) mod p = 1. %C Conjecture: a(n) is the smallest prime p such that Sum_{k=1..n} k^(p-1) == n (mod p). Thus a(n) >= A317358(n). - _Thomas Ordowski_, Jul 29 2018

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The A136162 next-prime product has four prime neighbors only at q=3

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Harry J. Smith; Jaroslav Krizek (2017). OEIS A136162, List of prime quadruplets {p, p+2, p+6, p+8}. URL: https://oeis.org/A136162.

Claim. %N List of prime quadruplets {p, p+2, p+6, p+8}. %C {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q*(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - _Jaroslav Krizek_, Jul 07 2017

Problem details · Reading note

Laguerre(n, 4) has odd numerator and denominator

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Robert Israel (2018). OEIS A160627, Numerator of Laguerre(n, 4). URL: https://oeis.org/A160627.

Claim. %N Numerator of Laguerre(n, 4). %C Conjecture: all terms are odd. - _Robert Israel_, Mar 29 2018 %N Denominator of Laguerre(n, 4). %C Conjecture: all terms are odd. - _Robert Israel_, Mar 29 2018

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Hanna's A204061 coefficients modulo five

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2012). OEIS A204061, G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ). URL: https://oeis.org/A204061.

Claim. G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers. a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.

Problem details · Reading note

Price's A228558 prime-exponent conjecture

Proved. Lean theorem result_a228558, frozen in this repository 2026-09-15.

Robert Price (2013). OEIS A228558, Numbers k such that (17^k + 4^k)/21 is prime. URL: https://oeis.org/A228558.

Claim. A228558 %N: Numbers k such that (17^k + 4^k)/21 is prime. %C: Conjecture: All terms are prime. A231329 %N: Numbers n such that (19^n + 4^n)/23 is prime. %C: Conjecture: All terms are prime.

Problem details · Reading note

Refutation of Stephan's A229140 least-coordinate distinctness conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-15.

Ralf Stephan (2013). OEIS A229140, Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares. URL: https://oeis.org/A229140.

Claim. A229140 %N: Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares (A001481). A229140 %C: Conjecture: the values between two zeros are always distinct from each other.

Problem details · Reading note

Price's A231329 prime-exponent conjecture

Proved. Lean theorem result_a231329, frozen in this repository 2026-09-15.

Robert Price (2013). OEIS A231329, Numbers n such that (19^n + 4^n)/23 is prime. URL: https://oeis.org/A231329.

Claim. %N Numbers n such that (19^n + 4^n)/23 is prime. %C Conjecture: All terms are prime.

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The immediate successor is the least weak pseudoprime exactly when it is an odd composite

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Robert Ferreol; Thomas Ordowski (2018). OEIS A239293, Smallest composite c > n such that n^c == n (mod c). URL: https://oeis.org/A239293.

Claim. %N Smallest composite c > n such that n^c == n (mod c). Context %C: a(n) is the smallest weak pseudoprime to base n that is > n. Context %C: If n is even and n+1 is composite, then a(n) = n+1. [Corrected by _Thomas Ordowski_, Aug 03 2018] Target %C: Conjecture: a(n) = n+1 if and only if n+1 is an odd composite number. - _Thomas Ordowski_, Aug 03 2018

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Kurkov's alternating binary run count

Proved. Lean theorem result, frozen in this repository 2026-09-15.

N. J. A. Sloane; Mikhail Kurkov (2021). OEIS A239907, alternating count of runs of ones, with Kurkov's A329320 identity. URL: https://oeis.org/A239907.

Claim. Let cn(n,k) denote the number of times 11..1 (k 1's) appears in the binary representation of n; a(n) = n - cn(n,1) + cn(n,2) - cn(n,3) + cn(n,4) - ... . Conjecture: a(n) = n - A329320(n) for n >= 0 (noticed by Sequence Machine). - _Mikhail Kurkov_, Oct 13 2021

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No A254074 term is a multiple of five

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Felix Fröhlich; Chai Wah Wu (2017). OEIS A254074, Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value. URL: https://oeis.org/A254074.

Claim. A254074 %N: Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value. A254074 %C: The first digit of a term is either 7, 8 or 9 and the last digit is either 5, 6 or 7. Conjecture: no term is a multiple of 5. - _Chai Wah Wu_, Sep 10 2017

Problem details · Reading note

The A260407 condition holds at every positive multiple of the order

Proved. Lean theorem result, frozen in this repository 2026-09-15.

M. F. Hasler; Jaroslav Krizek (2016). OEIS A260407, Numbers n such that (n-1)^2+1 divides 2^(n-1)-1. URL: https://oeis.org/A260407.

Claim. A260407 %N: Numbers n such that (n-1)^2+1 divides 2^(n-1)-1. A260407 %C: Conjecture: also numbers n such that ((2^k)^(n-1)-1) == 0 mod ((n-1)^2+1) for all k >= 1. - _Jaroslav Krizek_, Jun 02 2016

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Kurkov's binomial identity for binary-weight products

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Antti Karttunen; Mikhail Kurkov (2023). OEIS A284005, product of (1 + binary weight) over the binary shifts of n, with Kurkov's binomial conjecture. URL: https://oeis.org/A284005.

Claim. A284005 %N: a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)). A284005 %F: Conjecture: a(2^m*(2n+1)) = Sum_{k=0..m+1} binomial(m+1, k)*a(2^k*n) for m >= 0, n >= 0 with a(0) = 1. - _Mikhail Kurkov_, Apr 24 2023

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Schulte's Liouville and cube-Möbius product formula

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Werner Schulte (2018). OEIS A299406, coefficients of zeta(s) zeta(6s) / (zeta(2s) zeta(3s)), with the Liouville–A210826 product conjecture. URL: https://oeis.org/A299406.

Claim. %N Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s)*zeta(6*s))/(zeta(2*s)*zeta(3*s)). %F Conjecture: a(n) = A008836(n) * A210826(n).

Problem details · Reading note

Dyadic divisibility at Lang's Fibonacci indices

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Wolfdieter Lang (2018). OEIS A319197, factors of Fibonacci(2^(n-2)*3*m), with the dyadic divisibility conjecture. URL: https://oeis.org/A319197.

Claim. %N All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)*3*m) for n >= 3 and all m >= 0. %C It appears that Fibonacci(2^(n-2)*3*m)/(2^n) is a nonnegative integer for n >= 3 and all m >= 0.

Problem details · Reading note

A396596 divisor-power union: a counterexample

Refuted. Lean theorem result, frozen in this repository 2026-09-15.

Hal M. Switkay (2026). OEIS A396596: divisor-power records and the highly/deeply composite union conjecture. URL: https://oeis.org/search?q=id:A396596&fmt=text.

Claim. We conjecture that the present sequence can be constructed simply as a union of highly composite numbers and deeply composite numbers.

Problem details · Reading note

The eighth compositional iterate of A396798 modulo eight

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Claim. The fourth and eighth iterates have vanishing tails modulo eight; the third and fifth have periods 3,1,7,5 and 5,1,1,5 from degree two. The seventh has period 7,1,3,5 from degree two.

Problem details · Reading note

The fifth compositional iterate of A396798 modulo eight

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Claim. The fourth and eighth iterates have vanishing tails modulo eight; the third and fifth have periods 3,1,7,5 and 5,1,1,5 from degree two. The seventh has period 7,1,3,5 from degree two.

Problem details · Reading note

The fourth compositional iterate of A396798 modulo eight

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Claim. The fourth and eighth iterates have vanishing tails modulo eight; the third and fifth have periods 3,1,7,5 and 5,1,1,5 from degree two. The seventh has period 7,1,3,5 from degree two.

Problem details · Reading note

The seventh compositional iterate of A396798 modulo eight

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Claim. The fourth and eighth iterates have vanishing tails modulo eight; the third and fifth have periods 3,1,7,5 and 5,1,1,5 from degree two. The seventh has period 7,1,3,5 from degree two.

Problem details · Reading note

The third compositional iterate of A396798 modulo eight

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Claim. The fourth and eighth iterates have vanishing tails modulo eight; the third and fifth have periods 3,1,7,5 and 5,1,1,5 from degree two. The seventh has period 7,1,3,5 from degree two.

Problem details · Reading note

A396803 coefficients modulo three

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396803, A396805 and A396806: iterated-exponential congruences. URL: https://oeis.org/A396803.

Claim. For each of A396803, A396805 and A396806, a(n) is odd iff n is odd for n >= 1; for A396803, a(n) is congruent to n modulo 3; for A396806, a(n) is congruent to n modulo 6; for A396805, residues modulo 3 repeat 0, 1, 0 from n = 3.

Problem details · Reading note

A396805 coefficients modulo three from index three

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396805: iterated-exponential congruences at iterate count 5. URL: https://oeis.org/A396805.

Claim. For A396805, a(n) is odd iff n is odd for n >= 1; from n = 3 its residues modulo 3 repeat 0, 1, 0.

Problem details · Reading note

A396806 coefficients modulo six

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A396806: iterated-exponential congruences. URL: https://oeis.org/A396806.

Claim. For A396806 and every n >= 1, a(n) is odd iff n is odd, and a(n) is congruent to n modulo 3 and modulo 6.

Problem details · Reading note

OEIS A397315: the Three-Row blackout conjecture

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Arjun Pemmasani (2026). Blackouts Preserving Rectangle Identifiability on Grid Points: A partial solution to Problem 001 of Kagey's Open Problem Collection. URL: https://github.com/apemm/Kagey-Problems/blob/b9567e4dc1c7cf1e5034569571a9479b1b83dc57/kagey-problems/problem001/paper/main.tex.

Claim. The Three-Row conjecture asserts that the maximum valid blackout in a three-row grid with m columns is m+2 for every m>=3, where every lattice rectangle, including tilted rectangles, must remain identifiable from its visible corner set.

Problem details · Reading note

The integer source and dyadic parity of A397592

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Paul D. Hanna (2026). OEIS A397592: doubled linear rows and dyadic parity. URL: https://oeis.org/A397592.

Claim. Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).

Problem details · Reading note

OEIS A398726: the distinct cycle size limit

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Vaclav Kotesovec and Alois P. Heinz (2026). OEIS A398726, distinct cycle size asymptotic conjecture. URL: https://oeis.org/A398726.

Claim. The sum of distinct cycle sizes over all permutations of [n] is conjectured to be asymptotic to n*n!.

Problem details · Reading note

A399755: generalized Pell classes and the least square partner

Proved. Lean theorem result, frozen in this repository 2026-09-15.

Chai Wah Wu (2026). OEIS A399755: Numbers k such that the generalized Pell equation x^2-(k^2+1)*(y^2+1) = 0 has more than one fundamental solution in (x, y).. URL: https://oeis.org/A399755.

Claim. Conjecture: A399491(k) < 4*k^3+3*k if and only if k is a term of this sequence.

Problem details · Reading note

Vatter's largest derangement limit below one

Refuted. Lean theorem exists_derangement_limit_between, frozen in this repository 2026-09-15.

Vincent Vatter (2026). An Assortment of Problems in Permutation Patterns: Unimodality, Equivalence, Derangements, and Sorting. URL: https://arxiv.org/abs/2602.16355.

Claim. Section 4 records Conjecture 4.1 (123-avoidance limit 9/16), Question 4.2 (231-avoidance limit 0), and Question 4.3 (universal convergence of derangement ratios).

Problem details · Reading note

Starting value 47 refutes Erdős's proposed unique counterexample

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Paul Erdős (1985). Problems and results on consecutive integers and prime factors of binomial coefficients. DOI: 10.1216/RMJ-1985-15-2-353.

Claim. Put (21) ∏_{i=1}^{k} (x + i) = u_k(x) v_k(x), (u_k(x), v_k(x)) = 1 where v_k(x) is squarefree and all prime factors of u_k(x) occur with an exponent greater than 1. The representation in (21) is clearly unique. Clearly for k > k_0(x), u_k(x) > v_k(x). Perhaps one can estimate the smallest k_0(x) so that u_k(x) > v_k(x) for all k > k_0(x) quite well. I have not done this. For small values of k usually v_k(x) > u_k(x). I thought that for every x there is a k for which v_k(x) > u_k(x). x = 7 seemed a likely counterexample but if k = 7, u_7(7) = 2^7 3^3 < v_7(7) = 5·7·11·13. On the other hand a simple computation shows that n = 23 is a counterexample, i.e., for every k, v_k(23) < u_k(23). The reason for this is the existence of 24, 25, 27 and 32. I would not be surprised if 23 is the only counterexample. Perhaps in fact there is a k_0 so that for every k > k_0 and all n > n_0(k) (22) v_k(n) > u_k(n). (22) is perhaps too optimistic.

Problem details · Reading note

Refutation of Erdős's report of Mahler's binary-digit square uniqueness

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Paul Erdős (1989). Some personal and mathematical reminiscences of Kurt Mahler, Australian Mathematical Society Gazette 16(1) (1989), 1–2. URL: https://users.renyi.hu/~p_erdos/1989-34.pdf.

Claim. He told me about his last paper. Let e_i = 0 or 1. Then Σ_{i=1}^{n} e_i k^i = x^2 has infinitely many solutions for k = 2, 3 and 4. He conjectured that it has only a finite number of solutions for k > 4. In fact, the only nontrivial solution he found was 7^3 + 7^2 + 7 + 1 = 20^2, and perhaps there are no others (1 + (k^2 − 1) = k^2 counts as trivial).

Problem details · Reading note

Refutation of the divergence sentence in Kreh's Conjecture 18

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Martin Kreh (2015). Minimal Sets, Journal of Integer Sequences 18 (2015), Article 15.5.3. URL: https://cs.uwaterloo.ca/journals/JIS/VOL18/Kreh/kreh2.pdf.

Claim. Definition 16. For a given set M ⊂ N define a sequence δⁿ(M) of sets recursively by δ⁰(M) := M, δ(M) := δ¹(M) := M \ S(M), δⁿ⁺¹(M) := δ(δⁿ(M)) and let ηⁿ(M) := |S(δⁿ(M))|, η(M) := η¹(M). Conjecture 18. There are only countably many infinite sets M ⊂ N with η(M) ≤ η⁰(M). For all other sets we have ηⁿ(M) → ∞.

Problem details · Reading note

The A005590 zero set at multiples of three is characterized

Proved. Lean theorem result, frozen in this repository 2026-09-14.

N. J. A. Sloane; Ralf Stephan (2003). OEIS A005590, a(0)=0, a(1)=1, a(2n)=a(n), a(2n+1)=a(n+1)-a(n). URL: https://oeis.org/A005590.

Claim. %N a(0) = 0, a(1) = 1, a(2n) = a(n), a(2n+1) = a(n+1) - a(n). %F Conjecture: a(3n)=0 iff n in A003714. - _Ralf Stephan_, May 02 2003 _N. J. A. Sloane_

Problem details · Reading note

The A008474 prime-exponent sum iteration reaches five

Proved. Lean theorem ianakiev_a008474, frozen in this repository 2026-09-14.

Ivan N. Ianakiev; Olivier Gérard (2014). OEIS A008474, If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j). URL: https://oeis.org/A008474.

Claim. A008474 %N: If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j). A008474 %C: Conjecture: for m > 4, by iterating the map m -> A008474(m) one always reaches 5 [tested up to m = 320000]. - _Ivan N. Ianakiev_, Nov 10 2014 A008474 %F: Additive with a(p^e) = p + e. A008474 %A: _Olivier Gérard_

Problem details · Reading note

The A008590 gcd-filtered sums characterize the positive multiples of eight

Proved. Lean theorem result, frozen in this repository 2026-09-14.

N. J. A. Sloane; Lechoslaw Ratajczak (2017). OEIS A008590, Multiples of 8. URL: https://oeis.org/A008590.

Claim. %N Multiples of 8. %C From _Lechoslaw Ratajczak_, Sep 03 2017: (Start) Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers. a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even). a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even). a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even). The conjecture was checked for 5*10^4 consecutive integers. (End)

Problem details · Reading note

Refutation of the A018800 least-prime prefix suffix-bound conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Amarnath Murthy; David W. Wilson (2002). OEIS A018800, Smallest prime that begins with n. URL: https://oeis.org/A018800.

Claim. Smallest prime that begins with n. Conjecture: If a(n) = (n concatenated with k) then k < n. - _Amarnath Murthy_, May 01 2002 _David W. Wilson_

Problem details · Reading note

Products of distinct Mersenne primes satisfy exactly the sigma-tau rational-power relation

Proved. Lean theorem result, frozen in this repository 2026-09-14.

Labos Elemer; Jaroslav Krizek (2013). OEIS A046528, Numbers that are a product of distinct Mersenne primes. URL: https://oeis.org/A046528.

Claim. A046528 %N: Numbers that are a product of distinct Mersenne primes (elements of A000668). A046528 %C: n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit. A046528 %C: Supersequence of A051281 (numbers n such that sigma(n) is a power of tau(n)). Conjecture: numbers n such that sigma(n) = tau(n)^(a/b), where a, b are integers >= 1. Example: sigma(93) = 128 = tau(93)^(7/2) = 4^(7/2). - _Jaroslav Krizek_, May 04 2013

Problem details · Reading note

Refutation of the A049591 divisor-count prime-gap characterization

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Labos Elemer; Benoit Cloitre (2002). OEIS A049591, Odd primes p such that p+2 is composite. URL: https://oeis.org/A049591.

Claim. A049591 %N: Odd primes p such that p+2 is composite. A049591 %C: Sequence appears also to give all n > 1 such that there is no prime p satisfying the inequality n < p < n+tau(n)^2 where tau(n)=A000005(n). - _Benoit Cloitre_, Apr 13 2002 _Labos Elemer_

Problem details · Reading note

Refutation of the A089610 eventual strict-increase conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Cino Hilliard (2003). OEIS A089610, Number of primes between n^2 and (n+1/2)^2. URL: https://oeis.org/A089610.

Claim. Number of primes between n^2 and (n+1/2)^2. For small values of n, these numbers exhibit higher and lower values as n increases. Conjectures: After n=17 a(n) > 1. There exists an n_1 such that a(n) is < a(n+1) for all n >= n_1. Same as the number of primes between n^2 and n^2+n. Oppermann conjectured in 1882 that a(n)>0. - _T. D. Noe_, Sep 16 2008 _Cino Hilliard_, Dec 30 2003

Problem details · Reading note

Refutation of Cloitre's A090825 Bernoulli-integrality conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Benoit Cloitre (2004). OEIS A090825, nonprimes n such that (3/2)(1/n)(2n+1)(3^n+1)B(2n) is an integer. URL: https://oeis.org/A090825.

Claim. A090825 %N: Nonprimes n such that (3/2)*(1/n)*(2*n+1)*(3^n+1)*B(2*n) is an integer, where B(k) denotes the k-th Bernoulli number. A090825 %C: Conjecture: composite numbers with all prime factors in A053176 are in the sequence. - _Benoit Cloitre_, Feb 11 2004

Problem details · Reading note

Refutation of the A100952 log-bounded semiprime-partition conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Jonathan Vos Post; Reinhard Zumkeller (2004). OEIS A100952, Numbers that cannot be written as p*q+r with three distinct primes p, q and r. URL: https://oeis.org/A100952.

Claim. A100952 %N: Numbers that cannot be written as p*q+r with three distinct primes p, q and r. A100952 %C: A weaker conjecture: every integer greater than 60 (or some larger value based on further search) may be partitioned into a prime p and a semiprime qr, where the prime p is bounded by log(min(q,r)). Chen (1978) showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes. Zumkeller's conjecture effectively extends this from "even" to both even and odd integers. - _Jonathan Vos Post_, Nov 25 2004 A100952 %A: _Reinhard Zumkeller_, Nov 23 2004

Problem details · Reading note

Refutation of the A103585 period-43 comment

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Benoit Cloitre; Philippe Deléham; Ralf Stephan (2005). OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4. URL: https://oeis.org/A103585.

Claim. A103585 %N: Consider numbers k such that (A102370(k)-k)/2 = 1; read them mod 4 to get the sequence. A103585 %C: Sequence appears to have period 43. - _Ralf Stephan_, May 18 2007 _Benoit Cloitre_ and _Philippe Deléham_

Problem details · Reading note

Closed form of the A107928 harmonic-mean-numerator recurrence

Proved. Lean theorem stephan_a107928, frozen in this repository 2026-09-14.

Ralf Stephan (2010). OEIS A107928, a(n) is the numerator of harmonic mean of a(n-1) and a(n-2). URL: https://oeis.org/A107928.

Claim. %N a(n) is the numerator of harmonic mean of a(n-1) and a(n-2). %F a(1)=2; a(2)=3; n>=3: a(n) = numerator(2*a(n-1)*a(n-2)/(a(n-2)+a(n-1))). %F Conjecture: G.f.: x*(2+3*x+12*x^2+8*x^3+8*x^4)/[(1-2*x)*(1+2*x+4*x^2)]. a(3n) = (3/2)*8^n, a(3n+1) = 3*8^n, a(3n+2) = 2*8^n, for n>0. - _Ralf Stephan_, Dec 01 2010 %A _Zak Seidov_, Jun 10 2005

Problem details · Reading note

Cubic divisibility of generalized harmonic numerators along Adamchuk's progression

Proved. Lean theorem adamchuk_a116184, frozen in this repository 2026-09-14.

Alexander Adamchuk (2007). OEIS A116184, Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ]. URL: https://oeis.org/A116184.

Claim. Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ]. Conjecture: All terms of the arithmetic progression 3+36k belong to a(n). _Alexander Adamchuk_, Apr 08 2007

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Refutation of the A129598 differing-position characterization

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Antti Karttunen (2007). OEIS A129598, a(n) = n * A111089(n). URL: https://oeis.org/A129598.

Claim. A129598 %N: a(n) = n * A111089(n). A129598 %C: Conjecture: differs from A050399 at the positions given by A089966. E.g., a(15)=75, instead of A050399(15)=225, a(30)=150, instead of A050399(30)=450, a(33)=363, instead of A050399(33)=1089, a(45)=225, instead of A050399(45)=1225. - _Antti Karttunen_, May 01 2007

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Refutation of the A175406 floor formula conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Charles R Greathouse IV; Zak Seidov (2012). OEIS A175406, The greatest integer k such that (1+1/n)^k <= 2. URL: https://oeis.org/A175406.

Claim. A175406 %N: The greatest integer k such that (1+1/n)^k <= 2. A175406 %C: The sequence of first differences consists of zeros and ones, with no two consecutive zeros and no more than three consecutive ones. A175406 %F: a(n) = n log 2 + O(1). Conjecture: a(n) = floor((n + 1/2) log 2). - _Charles R Greathouse IV_, Apr 03 2012 A175406 %A: _Zak Seidov_, May 01 2010

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The A226857 Fibonacci sum-product recurrence is proved

Proved. Lean theorem barker_a226857, frozen in this repository 2026-09-14.

Colin Barker; Alonso del Arte (2014). OEIS A226857, Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers. URL: https://oeis.org/A226857.

Claim. %N Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers. %F Conjecture: a(n) = a(n-3)+a(n-6) for n>12. - _Colin Barker_, Nov 09 2014 %F Empirical g.f.: -x^2*(x^10 +x^9 +x^8 +2*x^7 +3*x^6 +3*x^5 +3*x^4 +3*x^3 +3*x^2 +2*x +1) / (x^6 +x^3 -1). - _Colin Barker_, Nov 09 2014 %C All Fibonacci numbers are in the sequence. The only prime numbers in this sequence are prime Fibonacci numbers. %A _Alonso del Arte_, Jun 19 2013

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The A254748 twin-prime mean has no superdivisor

Proved. Lean theorem gerasimov_a254748, frozen in this repository 2026-09-14.

Juri-Stepan Gerasimov (2015). OEIS A254748, Numbers without superdivisors. URL: https://oeis.org/A254748.

Claim. A254748 %N: Numbers without superdivisors: numbers n such that n/k + n fails to divide at least one of (n/k)^(n/k) + n, (n/k)^n + n/k or n^(n/k) + n/k for any divisor k of n. A254748 %C: Zerosuperdivisor numbers. Numbers n such that A247477(n) = 0. A254748 %C: Conjecture: Average of twin prime pairs (A014574) are zerosuperdivisor numbers. A254748 %A: _Juri-Stepan Gerasimov_, Feb 07 2015

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Refutation of the A260310 balanced-pair mod-six conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Robert G. Wilson v; Juri-Stepan Gerasimov (2015). OEIS A260310, Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members. URL: https://oeis.org/A260310.

Claim. Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members. Consider pairs (x,y) of numbers where sum(p|x) p + sum(q|y) q = x*sum(p|x) 1/p + y*sum(q|y) 1/q where p, q are primes and sum(p|x) p > sum(q|y) q. Conjecture: if a(2n-1) is prime then a(2n) is composite and vice versa. And when a(2n-1) is composite, it is congruent to 0 (mod 6). - _Robert G. Wilson v_, Jul 22 2015 _Juri-Stepan Gerasimov_, Jul 22 2015

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Refutation of the A319927 odd non-unitary power-sum conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-14.

Ivan N. Ianakiev (2018). OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. URL: https://oeis.org/A319927.

Claim. A319927 %N: Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k. A319927 %C: Conjecture: For any nonnegative integer power p the sum of the p-th powers of the odd non-unitary divisors of a(n) divides the sum of the p-th powers of the non-unitary divisors of a(n). A319927 %C: The start of this sequence, including the 58 terms currently shown in the data section, is consistent with a definition "nonsquarefree numbers not divisible by 4", but some larger terms are divisible by 4: for example, a(1305) = 9216 = 2^10 * 3^2. - _Peter Munn_, Sep 21 2020 A319927 %A: _Ivan N. Ianakiev_, Oct 02 2018

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Refutation of the one-divisor interval-density unimodality suggestion

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Gérald Tenenbaum (2013). Some of Erdős' Unconventional Problems in Number Theory, Thirty-four Years Later. DOI: 10.1007/978-3-642-39286-3_23.

Claim. Erdős (1979), restated as open by Tenenbaum (2013): Perhaps ε₁(n,m) is unimodular for m > n+1, but I know nothing about this.

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Refutation of Kok's A000149 domination clause

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Johan Kok (2025). Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs. DOI: 10.48550/arXiv.2507.16500.

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Schulte's alternating quadratic peak moment for Schröder paths

Proved. Lean theorem schulte_a060693, frozen in this repository 2026-09-13.

Werner Schulte (2017). OEIS A060693, Triangle read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks. URL: https://oeis.org/A060693.

Claim. Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks. T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *. Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - _Werner Schulte_, Jan 11 2017

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Refutation of the A072872 prime-index upper-bound conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Benoit Cloitre (2002). OEIS A072872, a(n) is the smallest positive number k such that n divides 2^k - k. URL: https://oeis.org/A072872.

Claim. a(n) is the smallest positive number k such that n divides 2^k - k. If n is a power of 2, a(n) = n. Conjecture : if n > 47, a(n) < prime(n). - _Benoit Cloitre_, Jul 28 2002

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Refutation of the A076502 nested-recurrence floor-offset conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Benoit Cloitre (2002). OEIS A076502, a(1)=1, a(n)=n-a(n-a(n-a(n-1))). URL: https://oeis.org/A076502.

Claim. a(1)=1, a(n)=n-a(n-a(n-a(n-1))); Conjecture : a(n) - c*n is bounded and a(n)=floor(c*n) + 0 + 1 or +2 - _Benoit Cloitre_, Nov 08 2002; a(n) is asymptotic to c*n with c=0.5698..is the positive root of x^3-x^2+2*x-1 - _Benoit Cloitre_, Nov 08 2002

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The trinomial odd diagonal of OEIS A077864

Proved. Lean theorem schulte_a077864, frozen in this repository 2026-09-13.

Werner Schulte (2015). OEIS A077864, Expansion of (1-x)^(-1)/(1-x-2*x^2-x^3). URL: https://oeis.org/A077864.

Claim. %N Expansion of (1-x)^(-1)/(1-x-2*x^2-x^3). %F a(0)=1, a(1)=2, a(2)=5, a(3)=11, a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - _Philippe Deléham_, Oct 25 2006 %F Conjecture: a(n) = Sum_{j=0..n/2} A027907(n+1-j,2*j+1), n >= 0. - _Werner Schulte_, Sep 29 2015 %N Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.

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Refutation of the A079063 square-root prime-gap lower-bound conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Benoit Cloitre (2003). OEIS A079063, Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1. URL: https://oeis.org/A079063.

Claim. Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1. Inspired by Andrica's conjecture. If it is true, a(n)>1 for all n. Conjecture: there is a constant c>0 such that for n large enough, a(n)>c*sqrt(n) and we can take c=0.4. More precisely, there are 2 constants A and B such that A=lim sup n ->infinity a(n)/sqrt(n) exists = 0.75....; B=lim inf n ->infinity a(n)/sqrt(n) exists =0.46.... - _Benoit Cloitre_, Feb 02 2003

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The least self-divisor exponent in OEIS A092028

Proved. Lean theorem firoozbakht_a092028, frozen in this repository 2026-09-13.

Farideh Firoozbakht (2004). OEIS A092028, a(n) is the smallest m > 1 such that m divides n^m-1. URL: https://oeis.org/A092028.

Claim. A092028 %N: a(n) is the smallest m > 1 such that m divides n^m-1. A092028 %C: Each prime factor of n-1 is a solution of the equation n^x-1 == 0 (mod x), so a(n) is not greater than the smallest prime factor of n-1. Conjecture 1: All terms of this sequence are primes. Conjecture 2: a(n) is the smallest prime factor of n-1 or for n>2, A092028(n) = A020639(n-1). A092028 %A: _Farideh Firoozbakht_, Mar 26 2004. A092028 %O: 3,1

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Exact factorial-square divisibility in A096127

Proved. Lean theorem factorial_square_exact_divisibility_iff_prime_power, frozen in this repository 2026-09-13.

Amarnath Murthy (2004). OEIS A096127, a(n) is the largest k such that (n^2)!/(n!)^k is an integer. URL: https://oeis.org/A096127.

Claim. a(n) is the largest k such that (n^2)!/(n!)^k is an integer. Conjecture: a(n)=n+1 only when n is prime or a power of a prime. [Verified for n=2..5000. - _Amiram Eldar_, Apr 06 2021]

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Square values in the A097602 square-counting recurrence

Proved. Lean theorem zumkeller_a097602, frozen in this repository 2026-09-13.

Reinhard Zumkeller; Vladeta Jovovic (2004). OEIS A097602, a(n+1) = a(n) + number of squares so far; a(1) = 1. URL: https://oeis.org/A097602.

Claim. A097602 %N: a(n+1) = a(n) + number of squares so far; a(1) = 1. A097602 %C: Conjecture: a(n) = m^2 iff m mod 3 > 0. A097602 %C: a(n) is a square iff n is congruent to {1, 4} mod 9. - _Vladeta Jovovic_, Aug 30 2004 A097602 %F: a(9*n+1) = (3*n+1)^2; a(9*n+4) = (3*n+2)^2. A097602 %A: _Reinhard Zumkeller_, Aug 30 2004

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Refutation of the A104863 floor-square-root recurrence conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Zak Seidov; Ralf Stephan (2005). OEIS A104863, a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=10, a(2)=30. URL: https://oeis.org/A104863.

Claim. a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=10, a(2)=30. For n>=17, a(n) = a(n-2) + a(n-4) + 1 (conjectured). If true then for m>5, a(2*m+1) = 4*F(m) + 25*F(m+1) + 1 and a(2*m+2) = 8*F(m) + 30*F(m+1) + 1 with F(n) = A000045(n). - _Ralf Stephan_, Nov 15 2010 _Zak Seidov_, Mar 28 2005

Problem details · Reading note

The zero indices of A110454

Proved. Lean theorem murthy_conjecture, frozen in this repository 2026-09-13.

Amarnath Murthy (2005). OEIS A110454, Largest composite number obtained by concatenation of parts of a distinct partition of n. URL: https://oeis.org/A110454.

Claim. Largest composite number obtained by concatenation of parts of a distinct partition of n, or 0 if no such number exist. Conjecture a(n) = 0 only for n = 1, 2 and 4. - _Amarnath Murthy_, Aug 04 2005

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Prime values of the divisor-pair product sum have exactly two preimages

Proved. Lean theorem seidov_conjecture, frozen in this repository 2026-09-13.

Zak Seidov (2006). OEIS A119623, Composite numbers for which the second elementary symmetric function of divisors is prime. URL: https://oeis.org/A119623.

Claim. A119623 %N (Zak Seidov, Jun 08 2006): Composite numbers for which the second elementary symmetric function of divisors (s2) is prime. A119623 %C (Zak Seidov, Jun 08 2006): Terms in A119616 are always prime if n is prime p and s2(p)=p, hence it is interesting to find composite numbers for which s2 is also prime. Relative values of s2 are: s2=47,97,163,457,733,2203,3733,7993,10723,11317,21313,22147,26557,33403,57283,61417,67153,79393,101467,149323,160453,162727,174337,272683,296827,318793,358273,432907,440383,486583,551767,639007,832687,843043,911917,961183,1152913,1202017,1277593,1322743,1375303,1462897,1567327,1824997,1878883. Otherwise the sequence s2 gives numbers which appear in A119616 at least twice (and conjecture is that exactly twice). A119616 %N (N. J. A. Sloane, based on email from Neven Juric, Jun 07 2006): Second elementary symmetric function of divisors of n.

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Hanna's mixed-parity block partition product in A124418

Proved. Lean theorem hanna_a124418, frozen in this repository 2026-09-13.

Paul D. Hanna (2006). OEIS A124418, Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries. URL: https://oeis.org/A124418.

Claim. Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)). Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(n,0)=A124419(n). Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind). - _Paul D. Hanna_, Nov 08 2006

Problem details · Reading note

Refutation of the A126762 least-witness exponent-shift conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Thomas Ordowski (2018). OEIS A126762, a(n) is the least k > n such that the remainder when n^k is divided by k is n. URL: https://oeis.org/A126762.

Claim. a(n) is the least k > n such that the remainder when n^k is divided by k is n. a(n) is the smallest number k > n such that n^k == n (mod k). Conjecture: a(n) is the smallest number k > n such that n^(k-1) == 1 (mod k). Thus a(n) is coprime to n. - _Thomas Ordowski_, Aug 03 2018

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Zeros of the A131853 binary Gaussian evaluation

Proved. Lean theorem zumkeller_a131853, frozen in this repository 2026-09-13.

Reinhard Zumkeller (2007). OEIS A131853, Numbers m such that z(m)=(0,0) with z as defined in A131851. URL: https://oeis.org/A131853.

Claim. A131853 %N: Numbers m such that z(m)=(0,0) with z as defined in A131851. A131853 %C: Intersection of A131854 and A131855: A131851(a(n))=0, A131852(a(n))=0; A131853 %C: conjecture: a(n) mod 5 = 0. A131853 %A: _Reinhard Zumkeller_, Jul 22 2007 A131851 %N: Real part of the function z(n)=Sum(d(k)*i^k: d as in n=Sum(d(k)*2^k), i=sqrt(-1)). A131851 %F: z(n) = if n=0 then (0, 0) else z(floor(n/2))*(0, 1) + (n mod 2, 0), complex multiplication.

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Palindrome-square reversal infinitude in A133901

Proved. Lean theorem brockhaus_seidov_a133901, frozen in this repository 2026-09-13.

Klaus Brockhaus; Zak Seidov (2007). OEIS A133901, Numbers in A128921 whose square is not a palindrome. URL: https://oeis.org/A133901.

Claim. A133901 %N: Numbers in A128921 whose square is not a palindrome. A128921 %N: Palindromes m such that reverse of m^2 is also a square. A133901 %C: Conjecture: Sequence is infinite. A133901 %e: A128921(41) = 1119111, 1119111^2 = 1252409430321 is not a palindrome, but its reverse is also a square: 1230349042521 = 1109211^2. A133901 %A: _Klaus Brockhaus_ and _Zak Seidov_, Sep 29 2007

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The A140869 Sundaram-type triangle complement

Proved. Lean theorem librandi_a140869, frozen in this repository 2026-09-13.

Vincenzo Librandi (2012). OEIS A140869, Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1. URL: https://oeis.org/A140869.

Claim. A140869 %N: T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1. A140869 %C: Conjecture: If h does not belong to the sequence, then 4*h+5 is prime. - _Vincenzo Librandi_, Nov 18 2012. A140869 %A: _Vincenzo Librandi_, Jan 16 2009

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The rational-denominator cube shift in A152020

Proved. Lean theorem cicuttin_a152020, frozen in this repository 2026-09-13.

Andres Cicuttin (2017). OEIS A152020, Denominator of 8/(9*n^2) divided by 9. URL: https://oeis.org/A152020.

Claim. Denominator of 8/(9*n^2) divided by 9. a(n) = n^2/gcd(n^2, 8). - _Andrew Howroyd_, Jul 25 2018 Conjecture: a(n) = denominator((n-2)^3/n^2). - _Andres Cicuttin_, Sep 19 2017

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A Gray-code closed form for the A163617 binary recurrence

Proved. Lean theorem yanev_a163617, frozen in this repository 2026-09-13.

Velin Yanev (2016). OEIS A163617, a(2*n) = 2*a(n), a(2*n + 1) = 2*a(n) + 2 + (-1)^n, for all n in Z. URL: https://oeis.org/A163617.

Claim. A163617 %N: a(2*n) = 2*a(n), a(2*n + 1) = 2*a(n) + 2 + (-1)^n, for all n in Z. A163617 %F: Conjecture: a(n) = A003188(n) + (6*n + 1 - (-1)^n)/4. - _Velin Yanev_, Dec 17 2016. A003188 %N: Decimal equivalent of Gray code for n. A003188 %F: a(n) = n XOR floor(n/2), where XOR is the binary exclusive OR operator.

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The A177680 twin-prime averages modulo five

Proved. Lean theorem dale_a177680, frozen in this repository 2026-09-13.

Juri-Stepan Gerasimov (2010). OEIS A177680, Numbers n such that 6n and 12n are both the average of twin prime pairs. URL: https://oeis.org/A177680.

Claim. A177680 %N: Numbers n such that 6n and 12n are both the average of twin prime pairs. A177680 %C: Conjecture: all terms but the first term are multiples of 5. - _Harvey P. Dale_, Sep 26 2017 A177680 %F: a(n)=A066388(n)/6. A177680 %A: _Juri-Stepan Gerasimov_, May 10 2010

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Collinear triples in the four-point grid

Proved. Lean theorem mathar_collinear_triples, frozen in this repository 2026-09-13.

Richard J. Mathar (2010). OEIS A178294, Number of collinear point triples in a 4 X 4 X 4 X... n-dimensional cubic grid. URL: https://oeis.org/A178294.

Claim. %N A178294 Number of collinear point triples in a 4 X 4 X 4 X... n-dimensional cubic grid. %A A178294 _R. H. Hardin_, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010 %F A178294 Conjecture: a(n) = 8^n/2-3*4^n/2+6^n. - _R. J. Mathar_, May 24 2010

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Refutation of the A192023 comb-Wiener determinant-count conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Clark Kimberling (2012). OEIS A192023, The Wiener index of the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes. URL: https://oeis.org/A192023.

Claim. The Wiener index of the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. Conjecture: for n>2, A192023(n-2) is the number of 2 X 2 matrices with all terms in {1,2,...,n} and determinant 2n. - _Clark Kimberling_, Mar 31 2012

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Prime alternating divisor sums in OEIS A193351

Proved. Lean theorem lagneau_a193351, frozen in this repository 2026-09-13.

Michel Lagneau (2012). OEIS A193351, Numbers k such that A071324(k) is prime. URL: https://oeis.org/A193351.

Claim. A193351 %N: Numbers k such that A071324(k) is prime. A193351 %C: Numbers k such that the alternating sum of all divisors of k (divisors nonincreasing, starting with k) is prime. A193351 %C: For k > 3, if the alternating sum of all divisors of k (divisors nonincreasing, starting with k) is a prime number, then k is either a square or twice a square. A193351 %C: This conjecture is verified up to k <= 10^8. - _Shreyansh Jaiswal_, Apr 25 2025 A193351 %A: _Michel Lagneau_, Dec 20 2012

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The A208342 coefficient formula for a jointly generated polynomial pair

Proved. Lean theorem schulte_a208342, frozen in this repository 2026-09-13.

Werner Schulte (2017). OEIS A208342, Triangle of coefficients of polynomials u(n,x) jointly generated with A208343. URL: https://oeis.org/A208342.

Claim. %N Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section. %F u(n,x) = u(n-1,x) + x*v(n-1,x), %F v(n,x) = x*u(n-1,x) + x*v(n-1,x), %F where u(1,x) = 1, v(1,x) = 1. %F T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - _Werner Schulte_, Mar 07 2017

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Refutation of the A231548 antisigma gap-three conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Jaroslav Krizek (2013). OEIS A231548, Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2). URL: https://oeis.org/A231548.

Claim. Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2). Also numbers n such that antisigma(n) < antisigma(n-2), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n. Conjecture: there are no numbers n such that antisigma(n) < antisigma(n-3). - _Jaroslav Krizek_, Nov 12 2013 Antisigma(n): Sum of the numbers less than n that do not divide n.

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Refutation of the A245786 integrality-implies-gcd-record conjecture

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Jaroslav Krizek (2014). OEIS A245786, Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer. URL: https://oeis.org/A245786.

Claim. Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer. Conjecture: Subsequence of A216793. Numbers n such that gcd(sigma(n), n) > gcd(sigma(m), m) for all m < n. - _Michel Marcus_, Sep 16 2012 _Jaroslav Krizek_, Aug 15 2014

Problem details · Reading note

OEIS A249621 terminal digits for square concatenations

Proved. Lean theorem wu_a249621, frozen in this repository 2026-09-13.

Chai Wah Wu (2014). OEIS A249621, Squares that remain squares when prepended with 1 and appended with 25 in base 10. URL: https://oeis.org/A249621.

Claim. A249621 %N: Squares that remain squares when prepended with 1 and appended with 25 in base 10. A249621 %C: Conjecture: all the numbers in the sequence end in either 00 or 56. - _Chai Wah Wu_, Nov 02 2014

Problem details · Reading note

The divisor-difference gcd and Heinz decoding identity in A258409

Proved. Lean theorem wiseman_a258409, frozen in this repository 2026-09-13.

Gus Wiseman (2019). OEIS A258409, Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n. URL: https://oeis.org/A258409.

Claim. Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n. Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164. - Gus Wiseman, Oct 16 2019. Heinz number of the multiset of differences between consecutive divisors of n. The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k). a(n) is the GCD of the indices j for which the j-th prime p_j divides n.

Problem details · Reading note

The antepenultimate-label count in OEIS A270236

Proved. Lean theorem mathar_f2, frozen in this repository 2026-09-13.

R. J. Mathar (2016). OEIS A270236, Triangle T(n,p): occurrences of p in the restricted growth functions of length n. URL: https://oeis.org/A270236.

Claim. NAME (R. J. Mathar, 2016-03-13): Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n. COMMENT (R. J. Mathar, 2016-03-13): The RG functions used here are defined by f(1)=1, f(j) <= 1+max_{i<j} f(i). FORMULA (R. J. Mathar, 2016-03-13): Conjecture: T(n,n-1) = 2+n*(n-1)/2 for n>1. FORMULA (R. J. Mathar, 2016-03-13; index correction Alois P. Heinz, 2016-03-18): Conjecture: T(n+1,n-1) = 2+n*(n+1)*(3*n^2-5*n+26)/24 for n>1.

Problem details · Reading note

The penultimate-label count in OEIS A270236

Proved. Lean theorem mathar_f1, frozen in this repository 2026-09-13.

R. J. Mathar (2016). OEIS A270236, Triangle T(n,p): occurrences of p in the restricted growth functions of length n. URL: https://oeis.org/A270236.

Problem details · Reading note

Kagey's modulo-ten congruence for no-leaf grid subgraphs

Proved. Lean theorem kagey_a301976_mod10, frozen in this repository 2026-09-13.

Peter Kagey; Colin Barker (2018). OEIS A301976, Number of no-leaf subgraphs of the 3 X n grid. URL: https://oeis.org/A301976.

Claim. A301976 %N: Number of no-leaf subgraphs of the 3 X n grid. A301976 %C: Also, the number of ways to lay unit-length matchsticks on a 3 X n grid of points in such a way that no end is "orphaned". A301976 %C: Conjecture: a(n) mod 10 = 3 for n > 2. A301976 %F: Conjectures from _Colin Barker_, Mar 30 2018: (Start) G.f.: x*(1 + x)*(1 - 8*x - 3*x^2) / (1 - 12*x + 6*x^2 + 20*x^3 + 5*x^4). a(n) = 12*a(n-1) - 6*a(n-2) - 20*a(n-3) - 5*a(n-4) for n>4. (End) A301976 %A: _Peter Kagey_, Mar 29 2018

Problem details · Reading note

Barker's recurrence for no-leaf subgraphs of the 3 X n grid

Proved. Lean theorem barker_a301976, frozen in this repository 2026-09-13.

Peter Kagey; Colin Barker (2018). OEIS A301976, Number of no-leaf subgraphs of the 3 X n grid. URL: https://oeis.org/A301976.

Problem details · Reading note

OEIS A302975 divisor-count power-ratio denominator square

Proved. Lean theorem krizek_a302975, frozen in this repository 2026-09-13.

Jaroslav Krizek (2018). OEIS A302975, a(n) = denominator of tau(n)^n / n^tau(n). URL: https://oeis.org/A302975.

Claim. A302975 %N: a(n) = denominator of tau(n)^n / n^tau(n). A302975 %C: tau(n) = the number of the divisors of n (A000005). Conjecture: all terms are squares. A302975 %A: _Jaroslav Krizek_, Apr 16 2018

Problem details · Reading note

Barker's recurrence for crossing- and nesting-free labeled graphs

Proved. Lean theorem barker_a326244, frozen in this repository 2026-09-13.

Colin Barker (2019). OEIS A326244, Number of labeled n-vertex simple graphs without crossing or nesting edges. URL: https://oeis.org/A326244.

Claim. Number of labeled n-vertex simple graphs without crossing or nesting edges. Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. Conjectures from _Colin Barker_, Jun 28 2019: (Start) G.f.: (1 - x)*(1 - 4*x) / (1 - 6*x + 8*x^2 - 4*x^3). a(n) = 6*a(n-1) - 8*a(n-2) + 4*a(n-3) for n>2. (End) Entry: _Gus Wiseman_, Jun 20 2019.

Problem details · Reading note

The divisor-count prime-factor bound in A328959

Proved. Lean theorem wiseman_a328959, frozen in this repository 2026-09-13.

Gus Wiseman (2019). OEIS A328959, a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n). URL: https://oeis.org/A328959.

Claim. a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222. Conjecture: All terms are nonnegative except for a(1) = -1. - _Gus Wiseman_, Nov 02 2019. The idea for this sequence came from _Mats Granvik_

Problem details · Reading note

The triangular-residue descent count in A329278

Proved. Lean theorem kagey_a329278, frozen in this repository 2026-09-13.

Peter Kagey (2019). OEIS A329278, Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n). URL: https://oeis.org/A329278.

Claim. A329278 %N: Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n). A329278 %C: Conjecture: for n > 0, the n-th row has 2^(n-1)-1 descents. A329278 %A: _Peter Kagey_, Nov 11 2019

Problem details · Reading note

First Occurrences in OEIS A335901

Proved. Lean theorem alkan_a335901, frozen in this repository 2026-09-13.

Altug Alkan (2020). OEIS A335901, a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1. URL: https://oeis.org/A335901.

Claim. A335901 %N: a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1. A335901 %C: Least k such that a(k) = 2^n are 1, 2, 5, 21, 169, 2705, ... (Conjecture: This sequence is A117261). - _Altug Alkan_, Jun 29 2020

Problem details · Reading note

First occurrences in the A335925 self-referential quotient recurrence

Proved. Lean theorem alkan_a335925, frozen in this repository 2026-09-13.

Altug Alkan (2020). OEIS A335925, a(n) = a(floor((n-1)/a(n-1))) + 1 with a(1) = 1. URL: https://oeis.org/A335925.

Claim. A335925 %N: a(n) = a(floor((n-1)/a(n-1))) + 1 with a(1) = 1. A335925 %C: Least k such that a(k) = n are 1, 2, 5, 16, 65, 326, 1957, ... (Conjecture: This sequence is A000522). A335925 %A: _Altug Alkan_, Jun 30 2020

Problem details · Reading note

Parity of the A397349 logarithmic-weight coefficients

Proved. Lean theorem parity_conjecture_holds, frozen in this repository 2026-09-13.

Paul D. Hanna (2026). OEIS A397349, logarithmic-weight coefficients: residue and parity. URL: https://oeis.org/A397349.

Claim. L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 3*n/(3*n^2 - 1) * a(n)*x^n ). Conjecture: a(n) is odd iff n is a power of 2. Conjecture: a(n) = [1,2] repeating (mod 3) for n >= 1.

Problem details · Reading note

Shin's four-point sumset compression question

Refuted. Lean theorem result, frozen in this repository 2026-09-13.

Henry Shin (2026). Iterated-sumset spectra: The complete exponent law and its rank geometry. DOI: 10.48550/arXiv.2609.01690.

Claim. The question following Corollary 10.5 asks whether every four-point sumset cardinality is realizable at the minimum diameter of a maximal-cardinality model.

Problem details · Reading note

Archer-Bourne cube-avoidance counting equality

Proved. Lean theorem card_avoids_312_321_cube_2143_eq_compositions, frozen in this repository 2026-09-12.

Kassie Archer, Noel Bourne (2026). Pattern avoidance in compositions and powers of permutations. DOI: 10.46298/dmtcs.17199. URL: https://arxiv.org/abs/2505.05218v3.

Problem details · Reading note

Necessary and sufficient conditions for the vertical slice result

Proved. Lean theorem verticalSlice_iff_two_traces, frozen in this repository 2026-09-12.

Morgan Fiebig, aBa Mbirika, Jürgen Spilker (2025). Period patterns, entry points, and orders in the Lucas sequences: theory and applications. URL: https://arxiv.org/abs/2408.14632v2.

Claim. Conjecture 5.2. Set $U_n:=U_n(p,q)$, and let $m \geq 1$ and $n \in \mathbb{Z}$. Assume that both $p$ and $m$ are even. If $U_{2n} \equiv 0 \pmod{m}$ and $U_{2n+1} \equiv q^n \pmod{m}$, then exactly one of the following two conclusions occur: (i) $U_n \equiv 0 \pmod{m}$, or (ii) $U_n \equiv \frac{m}{2} \pmod{m}$ where $n = c \cdot \frac{e_U(m)}{2}$, where $c$ is some odd integer and $e_U(m)$ is the entry point of $m$ in $(U_n)_{n \geq 0}$.

Problem details · Reading note

OEIS A395833: diagonal-exponent self-divisibility

Proved. Lean theorem exponent_self_divisibility, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A395833, g.f. A(x) satisfies [x^n] A(x/A(x)^(2*n-1)) = 0 for n > 1. URL: https://oeis.org/A395833.

Claim. G.f. A(x) satisfies [x^n] A( x/A(x)^(2*n-1) ) = 0 for n > 1. Conjecture: a(n) == 0 (mod 3) for n >= 2. Conjecture: (2*n-1) divides a(n) for n >= 1.

Problem details · Reading note

Parity of the A396803 iterated-exponential coefficients

Proved. Lean theorem parity_iterate_three, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A396803, A396805 and A396806: iterated-exponential congruences. URL: https://oeis.org/A396803.

Problem details · Reading note

Parity of the A396805 iterated-exponential coefficients

Proved. Lean theorem parity_iterate_five, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A396805: iterated-exponential congruences at iterate count 5. URL: https://oeis.org/A396805.

Claim. For A396805, a(n) is odd iff n is odd for n >= 1; from n = 3 its residues modulo 3 repeat 0, 1, 0.

Problem details · Reading note

Parity of the A396806 iterated-exponential coefficients

Proved. Lean theorem parity_iterate_six, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A396806: iterated-exponential congruences. URL: https://oeis.org/A396806.

Claim. For A396806 and every n >= 1, a(n) is odd iff n is odd, and a(n) is congruent to n modulo 3 and modulo 6.

Problem details · Reading note

Fibbinary parity of the A397244 coefficients

Proved. Lean theorem hanna_conjecture_a397244, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A397244, quadratic power-diagonal generating series. URL: https://oeis.org/A397244.

Claim. G.f. A(x) satisfies (2*n) * [x^n] A(x)^(2*n) = (2*n-1) * [x^n] A(x)^(2*n+1) for n > 1. Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for some k >= 0, where A003714 lists the Fibbinary numbers.

Problem details · Reading note

Alternation modulo three of the A397349 coefficients

Proved. Lean theorem hanna_conjecture_a397349_mod_three, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A397349, logarithmic-weight coefficients: residue and parity. URL: https://oeis.org/A397349.

Claim. L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 3*n/(3*n^2 - 1) * a(n)*x^n ). Conjecture: a(n) is odd iff n is a power of 2. Conjecture: a(n) = [1,2] repeating (mod 3) for n >= 1.

Problem details · Reading note

Parity of the A397356 reciprocal square-exponent diagonals

Proved. Lean theorem hanna_conjecture_a397356, frozen in this repository 2026-09-12.

Paul D. Hanna (2026). OEIS A397356, reciprocal square-exponent diagonal generating series. URL: https://oeis.org/A397356.

Claim. G.f. A(x) satisfies: [x^n] 1/A(x)^(n^2) = [x^n] 1/A(x)^(n^2-1) for n > 1 with A(0) = A'(0) = 1. Conjecture: for n >= 0, a(n) is odd iff n+1 is a power of 2. Conjecture: for n >= 0, a(n) is not divisible by 3 iff 2*(n+1) is a sum of two powers of 3 (A055235).

Problem details · Reading note

The even-parameter descent dichotomy for Lucas sequences

Proved. Lean theorem conjecture_five_two, frozen in this repository 2026-09-11.

Morgan Fiebig, aBa Mbirika, Jürgen Spilker (2025). Period patterns, entry points, and orders in the Lucas sequences: theory and applications. URL: https://arxiv.org/abs/2408.14632v2.

Problem details · Reading note

The single exceptional modulus for equality of even Lucas periods

Proved. Lean theorem even_lucas_periods, frozen in this repository 2026-09-11.

Morgan Fiebig, aBa Mbirika, Jürgen Spilker (2025). Period patterns, entry points, and orders in the Lucas sequences: theory and applications. URL: https://arxiv.org/abs/2408.14632v2.

Problem details · Reading note

A266489: clause C1, index divisibility on the frozen object

Proved. Lean theorem hanna_conjecture_a266489, frozen in this repository 2026-09-11.

Paul D. Hanna (2016). OEIS A266489, A300732, A300733, A292394, and A300734: vanishing-diagonal index divisibility. URL: https://oeis.org/A266489.

Claim. A266489: G.f. A(x) satisfies: [x^n] A( x/A(x)^n ) = 0 for n>1. (C1) n divides a(n) for n>=1: A268293(n) = a(n)/n. A300732: G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n) ) = 0 for n>=1. Conjecture: n divides a(n) for n>=1. A300733: G.f. A(x) satisfies: [x^n] A( x/A(x)^(3*n) ) = 0 for n>=1. Conjecture: n divides a(n) for n>=1. A292394: G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1. a(n) is divisible by n^2 for n>=1 (conjecture): A292395(n) = a(n)/n^2. A300734: G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n^2) ) = 0 for n>1. Conjecture: n^2 divides a(n) for n>=1.

Problem details · Reading note

A292394: square-index divisibility

Proved. Lean theorem hanna_conjecture_a292394, frozen in this repository 2026-09-11.

Paul D. Hanna (2017). OEIS A292394: vanishing-diagonal index divisibility. URL: https://oeis.org/A292394.

Claim. G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1. a(n) is divisible by n^2 for n>=1 (conjecture): A292395(n) = a(n)/n^2.

Problem details · Reading note

A300732: index divisibility

Proved. Lean theorem hanna_conjecture_a300732, frozen in this repository 2026-09-11.

Paul D. Hanna (2018). OEIS A300732: vanishing-diagonal index divisibility. URL: https://oeis.org/A300732.

Claim. G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n) ) = 0 for n>=1. Conjecture: n divides a(n) for n>=1.

Problem details · Reading note

A300733: index divisibility

Proved. Lean theorem hanna_conjecture_a300733, frozen in this repository 2026-09-11.

Paul D. Hanna (2018). OEIS A300733: vanishing-diagonal index divisibility. URL: https://oeis.org/A300733.

Claim. G.f. A(x) satisfies: [x^n] A( x/A(x)^(3*n) ) = 0 for n>=1. Conjecture: n divides a(n) for n>=1.

Problem details · Reading note

A300734: square-index divisibility

Proved. Lean theorem hanna_conjecture_a300734, frozen in this repository 2026-09-11.

Paul D. Hanna (2018). OEIS A300734: vanishing-diagonal index divisibility. URL: https://oeis.org/A300734.

Claim. G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n^2) ) = 0 for n>1. Conjecture: n^2 divides a(n) for n>=1.

Problem details · Reading note

Square divisibility of the A365095 perturbed diagonal

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-11.

Paul D. Hanna (2023). OEIS A365095, expansion of a generating function defined by a vanishing diagonal. URL: https://oeis.org/A365095.

Claim. Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0 for n > 1. Conjecture: [x^(n-1)] (1 + (k*n-1)*x*A(x)^2)^n / A(x)^n is divisible by n^2 for n > 0 and all integer k.

Problem details · Reading note

Modulo-four classification of the A372577 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-11.

Paul D. Hanna (2024). OEIS A372577, expansion of g.f. A(x) satisfying A(x)^2 = A(A(x*A(x) + x*A(x)^2)). URL: https://oeis.org/A372577.

Claim. Expansion of g.f. A(x) satisfying A(x)^2 = A(A( x*A(x) + x*A(x)^2 )). Conjecture: a(6*n - 3) == 3 (mod 4) and a(6*n - k) == 1 (mod 4) when k = {0,1,2,4,5} for n >= 1.

Problem details · Reading note

Coefficients of the A378580 quotient theta composition modulo four

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-11.

Paul D. Hanna (2025). OEIS A378580, g.f. A(x) satisfies A(x/A(x)) = theta_3(x). URL: https://oeis.org/A378580.

Claim. G.f. A(x) satisfies: A(x/A(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is square, else a(n) is divisible by 4 if n is nonsquare.

Problem details · Reading note

Dyadic denominators of the A381670 compositional-square series

Proved. Lean theorem scheuerle_conjecture, frozen in this repository 2026-09-11.

Thomas Scheuerle (2025). OEIS A381670, denominators of the compositional-square series. URL: https://oeis.org/A381670.

Claim. The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*A381669(k)/a(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)). Conjecture: All terms are powers of two.

Problem details · Reading note

Eventual periodicity of OEIS A383165

Proved. Lean theorem bala_conjecture_a383165, frozen in this repository 2026-09-11.

Seiichi Manyama; Peter Bala (2026). OEIS A383165, Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^2 / 2.. URL: https://oeis.org/A383165.

Claim. Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^2 / 2. Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic.

Problem details · Reading note

Eventual periodicity of OEIS A383166

Proved. Lean theorem bala_conjecture_a383166, frozen in this repository 2026-09-11.

Seiichi Manyama; Peter Bala (2026). OEIS A383166, Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^3 / 6.. URL: https://oeis.org/A383166.

Claim. Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^3 / 6. Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic.

Problem details · Reading note

A004123 totient-period conjecture

Proved. Lean theorem bala_conjecture_a004123, frozen in this repository 2026-09-10.

Peter Bala (2022). OEIS A004123, number of generalized weak orders on n points. URL: https://oeis.org/A004123.

Claim. A004123 NAME: Number of generalized weak orders on n points. Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, ...] with an apparent period of 6 = phi(7) starting at a(2). Cf. A000670.

Problem details · Reading note

A064618 totient-period conjecture

Proved. Lean theorem bala_conjecture_a064618, frozen in this repository 2026-09-10.

Peter Bala (2018). OEIS A064618, Stirling transform of (n!)^2. URL: https://oeis.org/A064618.

Claim. A064618 NAME: Stirling transform of (n!)^2. Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, ...), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - _Peter Bala_, Jan 15 2018

Problem details · Reading note

Parity of the A222014 factorial square-exponent sum

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-10.

Paul D. Hanna (2024). OEIS A222014, factorial square-exponent product-sum generating function and parity conjecture. URL: https://oeis.org/A222014.

Claim. Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

Problem details · Reading note

A266489: the mod-two congruence clause

Proved. Lean theorem hanna_conjecture_a266489, frozen in this repository 2026-09-10.

Paul D. Hanna (2016). OEIS A266489, g.f. A(x) satisfies: [x^n] A(x/A(x)^n) = 0 for n>1. URL: https://oeis.org/A266489.

Claim. G.f. A(x) satisfies: [x^n] A( x/A(x)^n ) = 0 for n>1. (C2) a(n) == 0 (mod 2) for n>=2.

Problem details · Reading note

Bala's OEIS A305550 totient-period conjecture

Proved. Lean theorem bala_conjecture_a305550, frozen in this repository 2026-09-10.

Peter Bala (2022). OEIS A305550, Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k). URL: https://oeis.org/A305550.

Claim. Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k). Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k).

Problem details · Reading note

Bala's general integral e.g.f. substitution conjecture

Proved. Lean theorem bala_conjecture_egf_general, frozen in this repository 2026-09-10.

Peter Bala (2022). Bala's general totient-period conjecture for integral G(exp(x) - 1). URL: https://oeis.org/A305550.

Claim. More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. The property is eventual periodicity modulo every positive k with period dividing phi(k).

Problem details · Reading note

The EGF coefficients of A338193 and Kurkov's recurrence

Proved. Lean theorem egf_coeff_eq_f, frozen in this repository 2026-09-10.

Vaclav Kotesovec; Mikhail Kurkov (2024). OEIS A338193, an integral-defined exponential generating function and Kurkov's recurrence. URL: https://oeis.org/A338193.

Claim. Conjecture: a(n) = f(0,n-1) for n > 0, where f is Kurkov's two-index recurrence.

Problem details · Reading note

Divisibility of the A338636 coefficients by eight

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-10.

Paul D. Hanna (2020). OEIS A338636, continued-fraction g.f. with odd-square numerators. URL: https://oeis.org/A338636.

Claim. G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2*x/(A(x) - 5^2*x/(A(x) - 7^2*x/(A(x) - 9^2*x/(A(x) - ...))))), a continued fraction relation. a(n) = 0 (mod 8) for n > 1 (conjecture).

Problem details · Reading note

The permutation question for A359482

Refuted. Lean theorem not_positive_permutation, frozen in this repository 2026-09-10.

Eric Angelini, Hans Havermann, and M. F. Hasler (2023). OEIS A359482 and the single-digit conjecture. URL: https://oeis.org/A359482.

Claim. A359482 is not a permutation of the positive integers; its only single-digit term is its initial 1.

Problem details · Reading note

The permutation conjecture for A363956

Proved. Lean theorem a363956_surjective, frozen in this repository 2026-09-10.

Scott R. Shannon (2023). OEIS A363956 — least unused multiples indexed by distinct prime factors. URL: https://oeis.org/A363956.

Claim. The entry conjectures that its greedy sequence is a permutation of the positive integers.

Problem details · Reading note

Parity of q-Catalan row square sums

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-10.

Paul D. Hanna (2024). OEIS A376527, square sums of Carlitz-Riordan q-Catalan rows and parity conjecture. URL: https://oeis.org/A376527.

Claim. Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

Problem details · Reading note

The A381362 bilateral product coefficient congruence

Proved. Lean theorem hanna_conjecture_a381362, frozen in this repository 2026-09-10.

Paul D. Hanna (2025). OEIS A381362, bilateral product generating function. URL: https://oeis.org/A381362.

Claim. G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(2*n-1) * (x^n + A(x))^(2*n-1). Conjecture: for n > 0, a(n) == 2 (mod 4).

Problem details · Reading note

The A381363 bilateral product coefficient congruence

Proved. Lean theorem hanna_conjecture_a381363, frozen in this repository 2026-09-10.

Paul D. Hanna (2025). OEIS A381363, bilateral product generating function. URL: https://oeis.org/A381363.

Claim. G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(3*n-1) * (x^n + A(x))^(3*n-1). Conjecture: for n > 0, a(n) == 2 (mod 4).

Problem details · Reading note

The A381364 scaled bilateral product congruence

Proved. Lean theorem hanna_conjecture_a381364, frozen in this repository 2026-09-10.

Paul D. Hanna (2025). OEIS A381364, scaled bilateral product generating function. URL: https://oeis.org/A381364.

Claim. G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(n-1) * (x^n + 2*A(x))^(n-1). Conjecture: for n > 0, a(n) == 6 (mod 9).

Problem details · Reading note

The A381365 scaled bilateral product congruence

Proved. Lean theorem hanna_conjecture_a381365, frozen in this repository 2026-09-10.

Paul D. Hanna (2025). OEIS A381365, scaled bilateral product generating function. URL: https://oeis.org/A381365.

Claim. G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(2*n-1) * (x^n + 2*A(x))^(2*n-1). Conjecture: for n > 0, a(n) == 6 (mod 9).

Problem details · Reading note

Constant Blocks and Distinct Run Sums

Proved. Lean theorem card_constantBlocks_eq_distinctRunSums, frozen in this repository 2026-09-10.

OEIS Foundation Inc. (2025). OEIS A382427. URL: https://oeis.org/A382427.

Claim. Constant-block partitions with distinct sums exist exactly when the parts have an ordering with distinct maximal run sums.

Problem details · Reading note

A393867: every logarithmic-derivative coefficient is odd

Proved. Lean theorem a393867_odd, frozen in this repository 2026-09-10.

Paul D. Hanna (2026). OEIS A393867, logarithmic derivative of A393866 (g.f. with [x^n] A^prime(n) = prime(n) [x^(n-1)] A^prime(n)). URL: https://oeis.org/A393867.

Claim. G.f. A(x) satisfies [x^n] A(x)^prime(n) = prime(n) * [x^(n-1)] A(x)^prime(n) for n >= 1. Logarithmic derivative of A393866. Conjecture: a(n) is divisible by prime(n) for n > 1. Conjecture: all terms are odd.

Problem details · Reading note

A395833: the mod-three congruence clause

Proved. Lean theorem hanna_conjecture_a395833, frozen in this repository 2026-09-10.

Paul D. Hanna (2026). OEIS A395833, g.f. A(x) satisfies [x^n] A(x/A(x)^(2*n-1)) = 0 for n > 1. URL: https://oeis.org/A395833.

Claim. G.f. A(x) satisfies [x^n] A( x/A(x)^(2*n-1) ) = 0 for n > 1. Conjecture: a(n) == 0 (mod 3) for n >= 2. Conjecture: (2*n-1) divides a(n) for n >= 1.

Problem details · Reading note

OEIS A396843: all coefficients are odd

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-10.

Paul D. Hanna (2026). OEIS A396843, g.f. satisfying A(x*A(x) - 3*x*A(x)^2) = x^2. URL: https://oeis.org/A396843.

Claim. Conjecture: all terms are odd.

Problem details · Reading note

Refutation of the A000364 pure-periodicity conjecture

Refuted. Lean theorem bala_conjecture_false, frozen in this repository 2026-09-09.

Peter Bala (2023). OEIS A000364, Euler (secant) numbers. URL: https://oeis.org/A000364.

Claim. Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x). Conjecture: taking the sequence [a(n) : n >= 1] modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, ...] with an apparent period of 6 = phi(21)/2. - _Peter Bala_, May 08 2023

Problem details · Reading note

The A038867 no-square conjecture

Proved. Lean theorem cube_gap_five_ne_sq, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2026). OEIS A038867. URL: https://oeis.org/A038867.

Claim. A comment on OEIS A038867 conjectures that no term of the sequence of cube differences at gap five is a perfect square.

Problem details · Reading note

The A091259 congruence

Proved. Lean theorem a091259_mod_three, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2024). OEIS A091259. URL: https://oeis.org/A091259.

Claim. A comment on OEIS A091259 conjectures that the reduced numerator of the ratio of the cubic to the linear divisor sum is congruent modulo three to a prime-exponent indicator.

Problem details · Reading note

Prime periods of the A122399 Stirling power-factorial sum

Proved. Lean theorem bala_conjecture, frozen in this repository 2026-09-09.

Peter Bala (2022). OEIS A122399 (Jovovic 2006), conjecture: a(n) mod p is purely periodic with period p - 1. URL: https://oeis.org/A122399.

Claim. a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k). Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1.

Problem details · Reading note

Prime periods of the A220181 alternating-weight Stirling sum

Proved. Lean theorem bala_conjecture, frozen in this repository 2026-09-09.

Peter Bala (2022). OEIS A220181, e.g.f. Sum_{n>=0} (1 - exp(-n*x))^n. URL: https://oeis.org/A220181.

Claim. E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n. a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k). Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1 ...], with an apparent period of 6. Cf. A122399. - _Peter Bala_, Jun 01 2022

Problem details · Reading note

Refutation of the all-prime A221077 period conjecture

Refuted. Lean theorem bala_conjecture_two_false, frozen in this repository 2026-09-09.

Peter Bala (2022). OEIS A221077, e.g.f. Sum_{n>=0} tanh(n*x)^n. URL: https://oeis.org/A221077.

Claim. E.g.f.: Sum_{n>=0} tanh(n*x)^n. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6. - _Peter Bala_, Jun 01 2022

Problem details · Reading note

Parity of the A222013 factorial product sum coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A222013, g.f. A(x) = Sum n! x^n A(x)^(n(n+1)/2) / Product (1 + k x A(x)^k), parity conjecture (Dec 2024). URL: https://oeis.org/A222013.

Claim. G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2) / Product_{k=1..n} (1 + k*x*A(x)^k). Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0. - _Paul D. Hanna_, Dec 06 2024

Problem details · Reading note

The prime-period conjecture for OEIS A224899

Refuted. Lean theorem bala_conjecture_two_false, frozen in this repository 2026-09-09.

Peter Bala (2022). OEIS A224899, e.g.f. Sum_{n>=0} sinh(n*x)^n. URL: https://oeis.org/A224899.

Claim. E.g.f.: Sum_{n>=0} sinh(n*x)^n. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322. - _Peter Bala_, May 29 2022

Problem details · Reading note

The A290322 admissible residue sum conjecture

Proved. Lean theorem residue_sum_ne_zero, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2024). OEIS A290322. URL: https://oeis.org/A290322.

Claim. A comment on OEIS A290322 conjectures that the sum of admissible units modulo n is nonzero whenever five divides n.

Problem details · Reading note

Prime periods of the A338040 weighted Stirling power sum

Proved. Lean theorem bala_conjecture, frozen in this repository 2026-09-09.

Peter Bala (2022). OEIS A338040 (Kotesovec 2020), conjecture: a(n) mod p purely periodic with period p − 1. URL: https://oeis.org/A338040.

Claim. E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1.

Problem details · Reading note

Bala's alternating gcd-sum formula for A344598

Proved. Lean theorem bala_conjecture, frozen in this repository 2026-09-09.

Peter Bala (2024). OEIS A344598, conjectured alternating gcd-sum formula. URL: https://oeis.org/A344598.

Claim. a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^2 - floor((n-1)/k)^2). Conjecture: a(n) = Sum_{k = 1..2*n} (-1)^k * gcd(k, 4*n). Cf. A344372. - _Peter Bala_, Jan 01 2024

Problem details · Reading note

Coefficient congruences for the A355872 bilateral theta reversion

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2022). OEIS A355872, g.f. satisfying x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2). URL: https://oeis.org/A355872.

Claim. G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2). Conjecture: a(n) == 2 (mod 4) for n >= 1. Conjecture: a(2*n-1) == 2 (mod 8) for n >= 1. Conjecture: a(2*n) == 6 (mod 8) for n >= 1.

Problem details · Reading note

The A363104 bilateral generating function modulo four

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2023). OEIS A363104, g.f. satisfying 4 = Sum_{n in Z} (-x)^n (4 A(x) + x^(n-1))^(n+1). URL: https://oeis.org/A363104.

Claim. Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1). Conjecture: g.f. A(x) == theta_3(x) (mod 4); a(n) == 2 (mod 4) iff n is a nonzero square and a(n) == 0 (mod 4) iff n is nonsquare.

Problem details · Reading note

The A363184 bilateral quarter-double-shift theta congruence

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2023). OEIS A363184, g.f. satisfying 4 = Sum_{n in Z} (-1)^n x^n (4 A(x) + x^(2n-1))^(n+1). URL: https://oeis.org/A363184.

Claim. Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^(n+1). Conjecture: g.f. A(x) == theta_3(x^2) (mod 4); a(n) == 2 (mod 4) if n = 2*k^2 for integer k > 0, and a(n) == 0 (mod 4) if floor(n/2) is nonsquare.

Problem details · Reading note

Parity of the cubic Catalan composition A363308

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2023). OEIS A363308, expansion of C(x*C(x)^3) with C the Catalan g.f.. URL: https://oeis.org/A363308.

Claim. Expansion of g.f. C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108). Conjecture: a(n) is odd iff n is a power of 2 or n = 0.

Problem details · Reading note

Divisibility of the A363560 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2023). OEIS A363560, g.f. satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9). URL: https://oeis.org/A363560.

Claim. Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9). Conjecture: a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).

Problem details · Reading note

Binomial parity of the A369083 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A369083, g.f. satisfying A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4. URL: https://oeis.org/A369083.

Claim. Expansion of g.f. A(x) satisfying A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4. Conjecture: a(n) == binomial(4*n+3,n) (mod 2) for n >= 0 (cf. A263133).

Problem details · Reading note

Divisibility modulo three of the A373308 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A373308, expansion of Product_{n>=0} (1 - x^(2^n))^3. URL: https://oeis.org/A373308.

Claim. Expansion of Product_{n>=0} (1 - x^(2^n))^3. Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0.

Problem details · Reading note

Square parity of the unit-reversion series A373312

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A373312, g.f. satisfying A(x)^2 = A(x*A(x)/(1-A(x))^2). URL: https://oeis.org/A373312.

Claim. Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^2 ). Conjecture: a(n) == 1 (mod 2) iff n = 2^k - 1 for k >= 1.

Problem details · Reading note

The A373561 divisor-bucketed quadruple sum

Proved. Lean theorem a373561, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2024). OEIS A373561. URL: https://oeis.org/A373561.

Claim. A comment on OEIS A373561 conjectures a closed form for a quadruple sum in which a greatest-common-divisor condition sorts the summands.

Problem details · Reading note

Powers-of-four parity of the A374568 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A374568, g.f. satisfying x = A(x-x^2) - A(x+x^2)^2. URL: https://oeis.org/A374568.

Claim. Expansion of g.f. A(x) satisfying x = A(x - x^2) - A(x + x^2)^2. Conjecture: a(n) is odd iff n = 4^k for k >= 0.

Problem details · Reading note

Parity of the Catalan-composition series A374570

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A374570, g.f. satisfying A(x)^2 = A(A(x)*C(x)) with C the Catalan function. URL: https://oeis.org/A374570.

Claim. Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*C(x) ), where C(x) = x + C(x)^2 is the Catalan function (A000108). Conjecture: for n > 1, a(n) is odd iff n = 2^k + 1 for k >= 0.

Problem details · Reading note

Fibbinary parity of the A374571 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A374571, g.f. satisfying A(x) = A(x^2) - x*A(x^2)^2. URL: https://oeis.org/A374571.

Claim. Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2. Conjecture: for n > 0, a(n) is odd iff n = A003714(k) for some k > 0, where A003714 lists Fibbinary numbers whose binary representation contains no two adjacent 1's.

Problem details · Reading note

Parity of the A375439 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A375439, expansion of A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3. URL: https://oeis.org/A375439.

Claim. Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3. Conjecture: a(n) is odd iff n is in A038754, which consists of numbers of the form 3^k and 2*3^k.

Problem details · Reading note

The A378252 running-sum residue conjecture

Proved. Lean theorem exists_zero_iff_modulus_pow_two, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2025). OEIS A378252. URL: https://oeis.org/A378252.

Claim. A comment on OEIS A378252 conjectures that the running-sum residue sequence built from a coprime pair reaches zero exactly when the modulus is a power of two.

Problem details · Reading note

The A378575 shifted-iterate fixed-point congruence

Proved. Lean theorem hanna_conjecture_five, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A378575, g.f. satisfying A(x) = x + x*A^5(x) (5-fold compositional iterate). URL: https://oeis.org/A378575.

Claim. G.f. A(x) satisfies A(x) = x + x*A(A(A(A(A(x))))), so that this sequence shifts left under the 5th self-COMPOSE. Conjecture: a(n) == 1 (mod 4) for n >= 1.

Problem details · Reading note

The A378576 shifted-iterate fixed-point congruence

Proved. Lean theorem hanna_conjecture_six, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A378576, g.f. satisfying A(x) = x + x*A^6(x) (6-fold compositional iterate). URL: https://oeis.org/A378576.

Claim. G.f. A(x) satisfies A(x) = x + x*A(A(A(A(A(A(x)))))), so that this sequence shifts left under the 6th self-COMPOSE. Conjecture: a(n) == 1 (mod 5) for n >= 1.

Problem details · Reading note

Strip-three ternary Catalan parity for A378578

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A378578, g.f. obtained by removing all factors of 3 from the coefficients of 1 + x*A(x)^3. URL: https://oeis.org/A378578.

Claim. G.f. A(x) equals the series obtained by removing all factors of 3 from the coefficients in 1 + x*A(x)^3. Conjecture: a(n) == binomial(3*n,n)/(2*n+1) (mod 2) for n >= 0.

Problem details · Reading note

The theta self-composition coefficients modulo four

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A378581, g.f. satisfying A(x*A(x)) = theta_3(x). URL: https://oeis.org/A378581.

Claim. G.f. A(x) satisfies: A(x*A(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is square, else a(n) is divisible by 4 if n is nonsquare.

Problem details · Reading note

Hanna's A379204 evenness conjecture

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2024). OEIS A379204, g.f. satisfying 1/x = Sum_{n in Z} A(x)^n (A(x)^n + 4)^(n+1). URL: https://oeis.org/A379204.

Claim. G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1). Conjecture: a(n) is even for n > 1. Conjecture: a(n) == 2 (mod 4) iff n = (k-1)^2 + 1 for some k > 1.

Problem details · Reading note

The A380056 divisibility conjecture at multiples of four

Proved. Lean theorem a380056_div_five, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2025). OEIS A380056. URL: https://oeis.org/A380056.

Claim. A comment on OEIS A380056 conjectures that every term whose index is a multiple of four is divisible by five.

Problem details · Reading note

Dyadic support of the odd coefficients of A380678

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A380678, g.f. satisfying A(x - A(x)^2/(1 - A(x)^2)) = x. URL: https://oeis.org/A380678.

Claim. G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x. Conjecture: for n>= 1, a(n) is odd iff n is a term of A027383, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2.

Problem details · Reading note

Fibbinary parity of the A380708 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A380708, g.f. satisfying A(x) = 1 + x*abs(1/A(x))^2. URL: https://oeis.org/A380708.

Claim. G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^2. Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 are the fibbinary numbers.

Problem details · Reading note

Binomial parity of the A380709 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A380709, g.f. satisfying A(x) = 1 + x*abs(1/A(x))^3. URL: https://oeis.org/A380709.

Claim. G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^3. Conjecture: a(n) == binomial(4*n-1, n) (mod 2) for n >= 0 (cf. A263132).

Problem details · Reading note

Absolute reciprocal squares and binary parity in A380710

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A380710, g.f. satisfying A(x) = 1 + x*A(x)*abs(1/A(x)^2). URL: https://oeis.org/A380710.

Claim. G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ). Conjecture: for n > 0, a(n) is odd iff n is a power of 2.

Problem details · Reading note

Parity of the A383377 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A383377, g.f. satisfying A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n). URL: https://oeis.org/A383377.

Claim. G.f. satisfies A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x). Conjecture: a(n) is even for n > 1.

Problem details · Reading note

Parity conjecture C.1 of OEIS A384267

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A384267, a(n) = abs([x^n] 1 + x/A(x)^2), conjecture C.1. URL: https://oeis.org/A384267.

Claim. a(n) = abs([x^n] 1 + x/A(x)^2), where A(x) is the g.f. (C.1) a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) (cf. A006013).

Problem details · Reading note

Parity of the A384270 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A384270, g.f. satisfying A(x) = A(x^4 + 4*x*A(x)^4)/A(x^3 + 3*x*A(x)^3). URL: https://oeis.org/A384270.

Claim. G.f. satisfies A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3). Conjecture: a(n) is odd iff n = 2^k for k >= 0.

Problem details · Reading note

Signed Catalan cubic substitution modulo three

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A386666, g.f. satisfying A(x)^2 = A(x^2 + 4*A(x)^3). URL: https://oeis.org/A386666.

Claim. G.f. A(x) satisfies: A(x)^2 = A( x^2 + 4*A(x)^3 ). Conjecture: a(3*n) (mod 3) == 2*a(3*n+1) (mod 3) == a(3*n+2) (mod 3) == 2*A039966(n) for n > 0, where A039966(n) equals the number of partitions of n into distinct powers of 3. A039966(n) is 1 if every base-3 digit of n is 0 or 1, and is 0 otherwise.

Problem details · Reading note

Multiple-of-Three Residue Coincidences

Proved. Lean theorem classify, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2025). OEIS A387319. URL: https://oeis.org/A387319.

Claim. A multiple-of-three residue coincidence characterizes composites except 4, 8, 10, and 25.

Problem details · Reading note

Choosable Initial Intervals and Reversed Partitions

Proved. Lean theorem card_choosable_eq_superdiagonal, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2025). OEIS A388711. URL: https://oeis.org/A388711.

Claim. Gus Wiseman conjectures that partitions with choosable initial intervals are counted by superdiagonal reversed partitions with the same sum and number of parts.

Problem details · Reading note

The A388986 Euler-form inclusion question

Proved. Lean theorem euler_form_lt, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2026). OEIS A388986. URL: https://oeis.org/A388986.

Claim. Every Euler-form number in A228058 is conjectured to satisfy the unitary-plus-squarefree divisor-sum inequality.

Problem details · Reading note

Odd-index parity of the A389472 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A389472, g.f. satisfying A(x) = A(x^2 + x^3)/x^2 - 1. URL: https://oeis.org/A389472.

Claim. G.f. satisfies A(x) = A(x^2 + x^3)/x^2 - 1. Conjecture: a(2*n-1) = 0 (mod 2) for n > 1. The separate conjecture modulo 3 is not claimed here.

Problem details · Reading note

Dyadic parity support of the A389476 quadratic reversion

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A389476, g.f. satisfying A(x - A(x)^2/(1 - A(x))^2) = x. URL: https://oeis.org/A389476.

Claim. G.f. A(x) satisfies: A( x - A(x)^2/(1 - A(x))^2 ) = x. Conjecture: a(n) is odd iff n is a term of A027383, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2 for m >= 0.

Problem details · Reading note

Alternating dyadic reversion modulo four for A389537

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A389537, g.f. satisfying A(A(x) - x) = A(x)^2. URL: https://oeis.org/A389537.

Claim. G.f. A(x) satisfies A( A(x) - x ) = A(x)^2. Conjecture: for n > 2, a(n) == 2 (mod 4) when n = 3*2^k for k >= 0 otherwise a(n) is divisible by 4.

Problem details · Reading note

G.f. A(x) satisfies A(x)^2 = A(2x - 2A(x)) / 2.

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A389540, g.f. satisfying A(x)^2 = A(2*x - 2*A(x))/2. URL: https://oeis.org/A389540.

Claim. G.f. A(x) satisfies A(x)^2 = A(2*x - 2*A(x)) / 2. Conjecture: a(n) is odd iff n = 2^k for k >= 0.

Problem details · Reading note

Quadratic square reversion modulo four for A389542

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2025). OEIS A389542, g.f. satisfying A(A(x)^2 - x^2) = 4*A(x)^3. URL: https://oeis.org/A389542.

Claim. G.f. A(x) satisfies A( A(x)^2 - x^2 ) = 4*A(x)^3. Conjecture: for n > 1, a(n) == 2 (mod 4) if n = 2^k + 1 (for some k >= 0) otherwise a(n) is divisible by 4.

Problem details · Reading note

The A390088 exception set

Proved. Lean theorem a390088, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2025). OEIS A390088. URL: https://oeis.org/A390088.

Claim. A comment on OEIS A390088 conjectures that exactly five numbers admit no base in which all their digits are prime.

Problem details · Reading note

Parity of the A392525 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A392525, g.f. satisfying A(x)^3 = A(x^3 + 15*x*A(x)^3). URL: https://oeis.org/A392525.

Claim. G.f. satisfies: A(x) = A( x^3 + 15*x*A(x)^3 )^(1/3), with A(0)=0, A''(0)=1. (1) A(x)^3 = A( x^3 + 15*x*A(x)^3 ). Conjecture: a(n) is odd iff n = 2^k for k >= 0.

Problem details · Reading note

Zero-Prepended First Sums and Odd Parts

Proved. Lean theorem card_zeroPrependedFirstSums_eq_distincts, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2026). OEIS A392694. URL: https://oeis.org/A392694.

Claim. Reversed partitions with weakly increasing zero-based partial alternating sums are conjectured to be counted by A000009.

Problem details · Reading note

Trimmed Alternating Sums of Ordinary Partitions

Proved. Lean theorem card_trimmedSums_eq_distincts, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2026). OEIS A392698. URL: https://oeis.org/A392698.

Claim. The partition-count conjecture on A392698 requires an index shift; the target is the number of distinct-part partitions of n+1.

Problem details · Reading note

Strict First Sums and Gapfree Odd Partitions

Proved. Lean theorem card_strictFirstSums_eq_gapfreeOdd, frozen in this repository 2026-09-09.

OEIS Foundation Inc. (2026). OEIS A392707. URL: https://oeis.org/A392707.

Claim. Strictly increasing zero-based partial alternating sums are conjectured to be counted by A053251.

Problem details · Reading note

The mod-four coefficients of the Catalan-shift square series

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A393172, g.f. satisfying A(x-x^2) = x + A(x)^2. URL: https://oeis.org/A393172.

Claim. G.f. A(x) satisfies A(x-x^2) = x + A(x)^2. Conjecture: for n > 1, a(n) == 2 (mod 4) iff n is a power of 2, otherwise a(n) is divisible by 4.

Problem details · Reading note

Prime divisibility of the A393867 logarithmic-derivative terms

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A393867, logarithmic derivative of A393866 (g.f. with [x^n] A^prime(n) = prime(n) [x^(n-1)] A^prime(n)). URL: https://oeis.org/A393867.

Problem details · Reading note

The A396099 period-four coefficient pattern

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396099, g.f. satisfying A(x) = x + A^2(x)*A^3(x) (compositional iterates). URL: https://oeis.org/A396099.

Claim. G.f. satisfies A(x) = x + A^2(x)*A^3(x). Conjecture: all terms are odd. Conjecture: a(n) = [1,3,3,1] repeating (mod 4) for n > 2. Conjecture: [x^n] A(A(x)) == 0 (mod 4) for n > 2. Conjecture: [x^n] ( A(x) - x*A(A(A(x))) ) == 2 (mod 4) for n > 2.

Problem details · Reading note

G.f. A(x) satisfies A(A(A(x))) = (1+x) * A(A(x))

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396102, g.f. satisfying A(A(A(x))) = (1+x)*A(A(x)). URL: https://oeis.org/A396102.

Claim. G.f. A(x) satisfies A(A(A(x))) = (1+x) * A(A(x)). Conjecture: a(n) == 1 (mod 3) for n >= 1.

Problem details · Reading note

Divisibility of the A396793 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396793, g.f. satisfying A(x) A(A(x)) = x^2 + 9 x^3. URL: https://oeis.org/A396793.

Claim. G.f. A(x) satisfies A(x) * A(A(x)) = x^2 + 9*x^3. Conjecture: a(n) == 0 (mod 3) for n > 1.

Problem details · Reading note

Divisibility of the A396795 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396795, g.f. satisfying A^2(x) A^3(x) = x^2 + 25 x^3. URL: https://oeis.org/A396795.

Claim. G.f. A(x) satisfies A^2(x) * A^3(x) = x^2 + 25*x^3, where A^n(x) is the n-th iteration of A(x). Conjecture: a(n) == 0 (mod 5) for n > 1.

Problem details · Reading note

OEIS A396804: the fourth-iterate EGF modulo four

Proved. Lean theorem mod_four, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396804 — E.g.f. satisfies A(x) = x exp(A^4(x)). URL: https://oeis.org/A396804.

Claim. The entry conjectures a(n) congruent to n modulo four for positive n; superscripts denote composition.

Problem details · Reading note

Refutation of the A396844 mod-sixteen conjecture

Refuted. Lean theorem hanna_conjecture_false, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396844, g.f. satisfying A(x*A(x) - 4*x*A(x)^2) = x^2. URL: https://oeis.org/A396844.

Claim. G.f. A(x) satisfies A( x*A(x) - 4*x*A(x)^2 ) = x^2. Conjecture: for n > 8, a(n) == 12 (mod 16) iff n is of the form 2*4^k+1 otherwise a(n) is divisible by 16.

Problem details · Reading note

The A396846 logarithmic coefficients modulo eight

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A396846, l.g.f. log(1 + x + Sum 4n/(4n^2-1) a(n) x^n), mod-8 conjecture. URL: https://oeis.org/A396846.

Claim. L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 4*n/(4*n^2-1) * a(n)*x^n ). Conjecture: a(n) == [1,7,5,7] repeating (mod 8) for n >= 1.

Problem details · Reading note

Oddness of all A397241 terms

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397241, g.f. with n [x^n] A^n = (n-1) [x^n] A^(n+1). URL: https://oeis.org/A397241.

Claim. G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1. Conjecture: all terms are odd. Conjecture: a(n) == 2 (mod 3) for n > 2. Conjecture: a(n) == 3 (mod 4) for n > 2.

Problem details · Reading note

OEIS A397241 coefficients modulo 4

Proved. Lean theorem hanna_conjecture_mod_four, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397241, g.f. with n [x^n] A^n = (n-1) [x^n] A^(n+1). URL: https://oeis.org/A397241.

Claim. G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1. Conjecture: all terms are odd. Conjecture: a(n) == 2 (mod 3) for n > 2. Conjecture: a(n) == 3 (mod 4) for n > 2.

Problem details · Reading note

OEIS A397241 coefficients modulo 3

Proved. Lean theorem hanna_conjecture_mod_three, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397241, g.f. with n [x^n] A^n = (n-1) [x^n] A^(n+1). URL: https://oeis.org/A397241.

Claim. G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1. Conjecture: all terms are odd. Conjecture: a(n) == 2 (mod 3) for n > 2. Conjecture: a(n) == 3 (mod 4) for n > 2.

Problem details · Reading note

Parity of the A397242 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397242, o.g.f. A = exp(x + Sum (n^2-1) a(n) x^n / n^2). URL: https://oeis.org/A397242.

Claim. O.g.f. satisfies A(x) = exp( x + Sum_{n>=2} (n^2-1) * a(n)*x^n / n^2 ). Conjecture: a(n) is odd iff n+1 is a power of 2. Conjecture: for n > 0, a(n) == 1 (mod 3) iff n+2 is of the form 3^k or 2*3^k. Conjecture: for n > 0, a(n) == 2 (mod 3) iff n+2 is a sum of 2 distinct powers of 3 (A038464).

Problem details · Reading note

The residue-one conjecture of OEIS A397242

Proved. Lean theorem hanna_conjecture_one, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397242, o.g.f. A = exp(x + Sum (n^2-1) a(n) x^n / n^2). URL: https://oeis.org/A397242.

Problem details · Reading note

The residue-two conjecture of OEIS A397242

Proved. Lean theorem hanna_conjecture_two, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397242, o.g.f. A = exp(x + Sum (n^2-1) a(n) x^n / n^2). URL: https://oeis.org/A397242.

Problem details · Reading note

A397345: square-weight congruence

Proved. Lean theorem parity_q5, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397345, A397348 and A397346: exponential square-weight congruences. URL: https://oeis.org/A397345.

Claim. For q=5 and q=3, a(n) is odd iff n+1 is a power of two; for q=2, a(n) modulo eight repeats 4,2,0,2 from n=2.

Problem details · Reading note

A397346: square-weight congruence

Proved. Lean theorem residues_q2, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397346: exponential square-weight congruence. URL: https://oeis.org/A397346.

Claim. Conjecture: a(n) == [4,2,0,2] repeating (mod 8) for n >= 2.

Problem details · Reading note

A397348: square-weight congruence

Proved. Lean theorem parity_q3, frozen in this repository 2026-09-09.

Paul D. Hanna (2026). OEIS A397348: exponential square-weight congruence. URL: https://oeis.org/A397348.

Claim. Conjecture: a(n) is odd iff n+1 is a power of 2.

Problem details · Reading note

Equation (11) for the reduced abelian complexity of the Thue-Morse word

Proved. Lean theorem reducedAbelianComplexity_even_difference, frozen in this repository 2026-09-08.

John M. Campbell, James Currie, Narad Rampersad (2025). Reduced complexities for sequences over finite alphabets. DOI: 10.48550/arXiv.2509.16034.

Claim. Section 3 leaves the odd-index recurrence and equation (11) for the reduced abelian complexity of the Thue-Morse word unproved.

Problem details · Reading note

Corneth's second recurrence for OEIS A068012

Proved. Lean theorem corneth_step, frozen in this repository 2026-09-08.

OEIS Foundation Inc.; Antti Karttunen; David A. Corneth (2025). OEIS A068012 subset sum doubling conjecture. URL: https://oeis.org/A068012.

Claim. For n greater than two with three not dividing n minus one, A068012(n) doubles.

Problem details · Reading note

The A068012 subset sum doubling conjecture

Proved. Lean theorem subset_sum_mod_six_doubling, frozen in this repository 2026-09-08.

OEIS Foundation Inc.; Antti Karttunen; David A. Corneth (2025). OEIS A068012 subset sum doubling conjecture. URL: https://oeis.org/A068012.

Claim. For n greater than two with three not dividing n minus one, A068012(n) doubles.

Problem details · Reading note

The A084068 Pell gcd conjecture

Proved. Lean theorem pell_companion_gcd, frozen in this repository 2026-09-08.

OEIS Foundation Inc.; Joseph A. Stocke (2025). OEIS A084068 gcd conjecture. URL: https://oeis.org/A084068.

Claim. The July 28, 2025 formula conjectures that A084068(n) equals gcd(A001108(n), A001109(n)).

Problem details · Reading note

The A146557 collinear triple divisibility conjecture

Proved. Lean theorem square_dvd_card_collinear_triples, frozen in this repository 2026-09-08.

OEIS Foundation Inc.; Peter Bala; Max Alekseyev (2025). OEIS A146557 collinear triple divisibility conjecture. URL: https://oeis.org/A146557.

Claim. Peter Bala conjectures that n squared divides A146557(n) when three does not divide n.

Problem details · Reading note

The printed index range of the sixth A329369 sum

Refuted. Lean theorem not_kurkovRowRecurrence, frozen in this repository 2026-09-08.

OEIS Foundation Inc.; Mikhail Kurkov (2024). OEIS A329369 sixth conjectured sum over the A373183 rows. URL: https://oeis.org/A329369.

Claim. The sixth conjectured sum is printed with the index q unrestricted, and that reading is false.

Problem details · Reading note

Mathar's linear recurrence for the A372991 factorial-quotient sequence

Proved. Lean theorem mathar_recurrence, frozen in this repository 2026-09-08.

Clark Kimberling; R. J. Mathar (2024). OEIS A372991, a(n) = (2n)!/(a(n-1)*a(n-2)). URL: https://oeis.org/A372991.

Claim. Conjecture D-finite with recurrence a(n) -2*n*(2*n-1)*a(n-3)=0. - R. J. Mathar, Jul 16 2024

Problem details · Reading note

Alternating row sums of the Fibonacci harmonic triangle

Proved. Lean theorem schulte_conjecture, frozen in this repository 2026-09-08.

Werner Schulte (2024). OEIS A378277, denominators in a harmonic triangle based on products of Fibonacci numbers. URL: https://oeis.org/A378277.

Claim. Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.

Problem details · Reading note

Palindromic composition of the A384222 rows

Proved. Lean theorem row_palindrome, frozen in this repository 2026-09-08.

Omar E. Pol (2025). OEIS A384222, lengths of the 2-dense sublists of divisors of n. URL: https://oeis.org/A384222.

Claim. Conjecture 1: row n is a palindromic composition of A000005(n). In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n. The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.

Problem details · Reading note

oeis-a385590-alternating-binomial

Proved. Lean theorem alternating_binomial_sum, frozen in this repository 2026-09-08.

Werner Schulte (2025). OEIS A385590, a triangle based on Fibonacci numbers. URL: https://oeis.org/A385590.

Claim. The alternating binomial sum of row n is zero for n greater than two and is (-1)^n for n less than three.

Problem details · Reading note

The least simultaneous digit-sum witness in A389000

Proved. Lean theorem wu_conjecture, frozen in this repository 2026-09-08.

Chai Wah Wu (2025). OEIS A389000, least k with n dividing the digit sums of k and k^2. URL: https://oeis.org/A389000.

Claim. Conjecture: a(9m+5) = 210^(2*m+1)-1 for m >= 0.

Problem details · Reading note

The A389536 square-cube substitution conjecture

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-08.

Paul D. Hanna (2025). OEIS A389536, G.f. A(x) = A(x^2 + 2*x^3)/x. URL: https://oeis.org/A389536.

Claim. G.f. A(x) satisfies: A(x) = A(x^2 + 2*x^3) / x. Conjecture: for n > 1, a(n) == 2 (mod 4) when n = 2^k + 1 (for some k >= 0) otherwise a(n) is divisible by 4.

Problem details · Reading note

First sums of decreasing partitions

Proved. Lean theorem wiseman_conjecture, frozen in this repository 2026-09-08.

Gus Wiseman (2025). OEIS A391620, partitions that are not first sums of a composition with all parts > 1. URL: https://oeis.org/A391620.

Claim. Conjecture: These are integer partitions of n whose least part is < 4. The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Problem details · Reading note

Coefficients of A393856 are one modulo five

Proved. Lean theorem hanna_a393856, frozen in this repository 2026-09-08.

Paul D. Hanna (2026). OEIS A393856, G.f. satisfies A(x - x*A(4*x)/4) = x. URL: https://oeis.org/A393856.

Claim. G.f. A(x) satisfies: A( x - x*A(4*x)/4 ) = x. Conjecture: a(n) == 1 (mod 5) for n >= 1.

Problem details · Reading note

Coefficients of A393857 are one modulo six

Proved. Lean theorem hanna_a393857, frozen in this repository 2026-09-08.

Paul D. Hanna (2026). OEIS A393857, G.f. satisfies A(x - x*A(5*x)/5) = x. URL: https://oeis.org/A393857.

Claim. G.f. A(x) satisfies A( x - x*A(5*x)/5 ) = x. Conjecture: a(n) == 1 (mod 6) for n >= 1.

Problem details · Reading note

Divisibility of the A394757 terms

Proved. Lean theorem sigma_gcd_divisibility, frozen in this repository 2026-09-08.

OEIS Foundation Inc. (2026). OEIS A394757. URL: https://oeis.org/A394757.

Claim. OEIS A394757 records the twin-prime center sequence whose terms satisfy the prime sigma-gcd condition and states that all terms are divisible by 18.

Problem details · Reading note

Divisibility of the A396794 coefficients

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-08.

Paul D. Hanna (2026). OEIS A396794, G.f. A(x) satisfies A(x)*A(A(A(x))) = x^2 + 16*x^3. URL: https://oeis.org/A396794.

Claim. G.f. A(x) satisfies A(x) * A(A(A(x))) = x^2 + 16*x^3. Conjecture: a(n) == 0 (mod 4) for n > 1.

Problem details · Reading note

Coefficients of the A396797 three-four iterate product series modulo six

Proved. Lean theorem hanna_conjecture, frozen in this repository 2026-09-08.

Paul D. Hanna (2026). OEIS A396797, G.f. satisfies A(x) = x + A^3(x)*A^4(x). URL: https://oeis.org/A396797.

Claim. G.f. satisfies A(x) = x + A^3(x)*A^4(x). Conjecture: a(n) = 1 (mod 6) for n >= 1.

Problem details · Reading note

Coefficient congruence for OEIS A396807

Proved. Lean theorem coefficient_congruence, frozen in this repository 2026-09-08.

Paul D. Hanna (2026). OEIS A396807, G.f. satisfies A(x) = x + A^5(x)*A^6(x). URL: https://oeis.org/A396807.

Claim. Conjecture: a(n) = 1 (mod 10) for n >= 1.

Problem details · Reading note

A negative answer to Vatter's Question 4.3

Refuted. Lean theorem vatter_question_4_3_answer_no, frozen in this repository 2026-09-08.

Vincent Vatter (2026). An Assortment of Problems in Permutation Patterns: Unimodality, Equivalence, Derangements, and Sorting. URL: https://arxiv.org/abs/2602.16355.

Claim. Section 4 records Conjecture 4.1 (123-avoidance limit 9/16), Question 4.2 (231-avoidance limit 0), and Question 4.3 (universal convergence of derangement ratios).

Problem details · Reading note

Two-parameter greedy three-sumfree membership

Proved. Lean theorem conjecture17, frozen in this repository 2026-09-06.

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, Thian Tromp (2025). Using Walnut to solve problems from the OEIS. DOI: 10.48550/arXiv.2503.04122.

Claim. Conjecture 17 gives a two-parameter periodic characterization of the greedy three-sumfree sequence with seeds 1, g, g+d.

Problem details · Reading note

Odd-index reduced abelian complexity of the Thue-Morse word

Proved. Lean theorem reducedAbelianComplexity_odd, frozen in this repository 2026-09-06.

John M. Campbell, James Currie, Narad Rampersad (2025). Reduced complexities for sequences over finite alphabets. DOI: 10.48550/arXiv.2509.16034.

Claim. Section 3 leaves the odd-index recurrence and equation (11) for the reduced abelian complexity of the Thue-Morse word unproved.

Problem details · Reading note

Not solved here (20)

Ball–Galvin–Hyry–Weingartner: unimodal coefficient permutations

T. Ball, D. Galvin, C. Hyry, and K. Weingartner (2022). Independent set and matching permutations. DOI: 10.1002/jgt.22724. URL: https://doi.org/10.1002/jgt.22724.

Claim. Section 4, Question 4.6 asks which unimodal permutations arise from coefficients of products of nonnegative real linear factors.

Problem details · Reading note

Classify negative base-phi prefix occurrence sequences

F. Michel Dekking (2023). The structure of base phi expansions. DOI: 10.48550/arXiv.2305.08349.

Claim. Occurrence sequences of negative-position digit prefixes in base-phi expansions are conjecturally Lucas-parameterized sequences drawn from three Sturmian families.

Problem details · Reading note

Chen-Sun's full double-character sum modulo sixteen

Deyi Chen and Zhi-Wei Sun (2024). On determinants involving (j+k/p) plus or minus (j-k/p). DOI: 10.48550/arXiv.2409.08213. URL: https://arxiv.org/html/2409.08213v4#S1.E11.

Claim. Conjecture 1.1(1.11) gives all four mod8 branches modulo16 for the inclusive double Legendre sum d_p, including the actual imaginary quadratic ideal class number in the p=3 mod4 branches.

Problem details · Reading note

CKZ Conjecture 2: the four actual record distributions

Joanna N. Chen, Sergey Kitaev, and Philip B. Zhang (2024). Distributions of statistics on separable permutations. DOI: 10.1016/j.dam.2024.05.004. URL: https://arxiv.org/abs/2404.18517v1.

Claim. Strict record distributions on literal 2413/3142 avoiders; Section 3 Conjecture 2 asks for all-length class-conditioned unimodality with a maximum at record count three.

Problem details · Reading note

Face counts for coordinate cross-polytope Minkowski sums

Ziyi Dai, Qilin Hou, Zhiyuan Liu, Warut Thawinrak and Hongyu Wang (2026). Counting Lattice Points in Minkowski Sums of Cross Polytopes. URL: https://arxiv.org/abs/2608.16037v2.

Claim. Equation (3) defines coordinate cross-polytope Minkowski sums; Problem 5.4 asks for their f-vectors and the ordinary h-vectors of the simple members.

Problem details · Reading note

Erdős #7: odd covering systems

T. F. Bloom (2026). Erdős Problem #7. URL: https://www.erdosproblems.com/7.

Claim. Is there a distinct covering system all of whose moduli are odd? The unrestricted question remains open on the problem page.

Problem details · Reading note

Equality of Lucas periods when p and m are both even, at q = -1

Morgan Fiebig, aBa Mbirika, Jürgen Spilker (2025). Period patterns, entry points, and orders in the Lucas sequences: theory and applications. URL: https://arxiv.org/abs/2408.14632v2.

Problem details · Reading note

Minimality of the base-4 golden-ratio DFAO

Aaron Barnoff and Curtis Bright and Jeffrey Shallit (2024). Using finite automata to compute the base-b representation of the golden ratio and other quadratic irrationals. DOI: 10.48550/arXiv.2405.02727.

Claim. A deterministic finite automaton with output reads the Zeckendorf representation of a power of b and emits the corresponding base-b digit of a quadratic irrational.

Problem details · Reading note

The Hemmer–Straub–Westrem weighted Ev character average

Claim. Conjecture 2.3 equates the alternating Ev sum over even three-row characters with the Ev sum of the two-row character; Lemma 3.1 supplies the Frobenius character interpretation.

Problem details · Reading note

Mamede--Santos--Soares Conjecture 5.1

Ricardo Mamede, Jose Luis Santos, Diogo Soares (2026). Maximum number of one-element commutation classes of a permutation. DOI: 10.48550/arXiv.2601.09395.

Claim. Lemma 3.6 gives a first-orientation source-word shape, and Proposition 3.8 constructs an injection by deleting part of its excursion.

Problem details · Reading note

Negative roots of the original Delannoy matrix square

Jianxi Mao and Lijie Wang (2026). The Narayana transformation. URL: https://arxiv.org/abs/2607.01572v1.

Claim. Conjecture 4.1 asks for real nonpositive roots of the row polynomials of the ordinary squares of the Eulerian and Delannoy triangles.

Problem details · Reading note

A fourth mutually unbiased basis in dimension six

Daniel McNulty, Stefan Weigert (2024). Mutually Unbiased Bases in Composite Dimensions -- A Review. DOI: 10.48550/arXiv.2410.23997.

Claim. Problem 10.2 states the nonexistence question for four mutually unbiased bases in dimension six; separately, the review's fourteen equivalent formulations concern complete MUB sets (Fin 7 in dimension six), not the Fin 4 problem.

Problem details · Reading note

Fibonacci divisor sums at square-plus-one arguments have even index

Michel Lagneau (2020). OEIS A339621: Sum of Fibonacci divisors of n^2 + 1. URL: https://oeis.org/A339621.

Claim. %N Sum of Fibonacci divisors of n^2 + 1. %C Conjecture: If the sum of the Fibonacci divisors of m^2 + 1 is a Fibonacci number, then this number belongs to the sequence A001906(n) = F(2n) where F(n) is the Fibonacci sequence.

Problem details · Reading note

Fibonacci Divisors of an Odd-Index Lucas Square Plus One

Michel Lagneau (2020). OEIS A339669, Number of Fibonacci divisors of Lucas(n)^2 + 1. URL: https://oeis.org/A339669.

Claim. A339669 %N: Number of Fibonacci divisors of Lucas(n)^2 + 1. A339669 %C: Particular attention must be paid to the regularity properties of the number of divisors of Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two conjectures verified for n < 156. A339669 %C: Conjecture 1: a(6*n+3) = 1. A339669 %C: Conjecture 2: a(6*n+1) = a(6*n+5) = 2. A339669 %A: _Michel Lagneau_, Dec 12 2020

Problem details · Reading note

Ternary support of the A397356 reciprocal square-exponent diagonals

Paul D. Hanna (2026). OEIS A397356, reciprocal square-exponent diagonal generating series. URL: https://oeis.org/A397356.

Problem details · Reading note

Zero-dimensional shadowing extensions

Dekui Peng (2026). Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits. URL: https://arxiv.org/abs/2606.14435v3.

Claim. Question 5.1 asks whether every compact Hausdorff shadowing system is a factor of a compact zero-dimensional shadowing system.

Problem details · Reading note

Gaussianity of random Zeckendorf game lengths via mixing

Justin Cheigh and Guilherme Zeus Dantas e Moura and Ryan Jeong and Jacob Lehmann Duke and Wyatt Milgrim and Steven J. Miller and Prakod Ngamlamai (2022). Towards the Gaussianity of Random Zeckendorf Games. DOI: 10.48550/arXiv.2210.11038.

Claim. A uniform summable strong-mixing bound on the split indicators of the random Zeckendorf game would yield a Gaussian limit for the game length.

Problem details · Reading note

Actual separable occupied-diagonal lower bound

Robert Dougherty-Bliss, Alejandro B. Galvan, Michaela Polley, and David Shuster (2026). Enumerating separable derangements. URL: https://arxiv.org/abs/2608.27583v1.

Claim. The paper gives an algorithm and asymptotic bounds for separable derangements, while Conjecture 15 leaves the limiting derangement proportion open.

Problem details · Reading note

Wall-Sun-Sun primes as a golden-unit lift problem

Minjia Shi and Xuan Wang and Bouazzaoui Zakariae and Jon-Lark Kim and Patrick Sole (2026). Second order Recurrences, quadratic number fields and cyclic codes. DOI: 10.48550/arXiv.2603.25343.

Claim. Equality of the Pisano period modulo p and modulo p squared corresponds to a failure of p-rationality of the associated real quadratic field.

Problem details · Reading note

Maximum order complexity along polynomial Zeckendorf subsequences

Damien Jamet and Pierre Popoli and Thomas Stoll (2021). Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences. DOI: 10.48550/arXiv.2106.09959.

Claim. The parity sequence of the Zeckendorf digit sum along a polynomial subsequence of degree d has maximum order complexity conjecturally of order N raised to one over two d.

Problem details · Reading note

How this list is made

Source revision: 6268c3f4.

Snapshot built at 2026-11-05T00:36:25.925532Z (UTC).

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The list is generated from problem files, reading notes, and resolution records in theorem pages at this source revision. The citation and claim of each entry are copied from the reading note’s front matter (authors, year, title, DOI or URL, claim). The claim is the note’s transcription of the source statement; whether it quotes the source exactly is not checked here. An entry whose reading note is not written yet cites the problem file’s DOI or URL. The records are read as text, so ordinary prose can produce one; this page does not check that a record came from the repository’s own verified claim. This page does not run the repository’s checks or verify the named theorems or their Lean proofs.

The “frozen in this repository” date is the date of the first commit that added each named theorem’s module to the frozen record. It is not the date the problem was solved in the world or the resolution was recorded. Solved entries are listed by their latest member freeze date, newest first, and by problem slug within a day; unsolved entries are listed by slug.

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