slug: kulhandjian-hanzo-2026-q2-distance-ceiling
bibkey: kulhandjian2026singer
doi: 10.48550/arXiv.2610.02392
url: https://arxiv.org/abs/2610.02392v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.result

The q = 2 Singer-quadric qubit codes have minimum distance 2 for every d ≥ 2

Problem

M. Kulhandjian and L. Hanzo, Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q), arXiv:2610.02392v1, construct for a prime power $q$ and $d\ge 2$ a stabilizer code $\mathcal{Q}(q,d)$ with $H=(A\mid M_{Q}A)$: $A=\mathrm{circ}({\mathbf{\tau }}_H)$ is the Singer incidence matrix of $\mathrm{PG}(d,q)$ and $M_Q=\mathrm{circ}({\mathbf{\tau }}_Q)$ the quadric circulant. Proposition 23 (Section VII.E) states for $q=2$ and $d\in \{2,\dots ,6\}$ that $u=M_Q{\mathbf{\tau }}_H\equiv 1$, that every $Z_i{Z}_j$ lies in the centralizer, and, by exhaustive search for $d\le 4$, that $Z_0{Z}_1$ is not a stabilizer, so $d_{\min }=2$. Conjecture 24 asserts the proposition for every $d\ge 2$, and the paper’s list of open problems asks to settle it analytically. The verbatim statements are in the literature note.

Issue #13470 fixes the reading. At $q=2$: $K=\mathrm{GF}(2^{d+1})$, $n=2^{d+1}-1$, $({\mathbf{\tau }}_H{)}_i=1$ iff $\mathrm{Tr}({\alpha }^i)=0$, $({\mathbf{\tau }}_Q{)}_i=1$ iff $\mathrm{Tr}({\alpha }^{3i})=0$, circulants have entries $v_{(i-j)\mathrm{mod}n}$, the centralizer is $\{(\mathbf{a},\mathbf{b}):\mathbf{a}{H}_x^{\top }=\mathbf{b}{H}_z^{\top }\}$ and the stabilizers are the row space of $H$. claim asserts, for every $d\ge 2$ and every primitive $\alpha$ (the paper fixes one): $u=1$; for $i\ne j$, $(e_i+e_j,0)$ is in the centralizer and not a stabilizer; every centralizer element outside the stabilizers has symplectic weight $\ge 2$. The witness has weight 2 also as a Hamming weight, so $d_{\min }=2$ under either reading of the paper’s weight.

Motivation

The construction is offered as a family of qudit stabilizer codes, and the paper’s discussion of the useful parameter regime rests on the $q=2$ case being limited to distance 2 for every $d$; the numerical evidence covered $d\le 6$ for $u$ and $d\le 4$ for the distance. D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.result proves the conjecture for every $d\ge 2$.

Gap

Issue #13470 records the literature check before any Lean. The paper has a single version (2026-10-01, accepted for IEEE Open Journal of the Communications Society); its remark gives a structural argument only for $d$ even and calls $d$ odd open, and its conclusions list the conjecture as an open problem. No settlement was found in the repository or in the search seat’s bounded checks; the Semantic Scholar citation API was rate limited. not-found-in-searched-scope.

Route

Let $m=d+1\ge 3$ and $n=2^m-1\ge 7$.

1. Power sums. ${\sum }_{x\in K^{\ast }}{x}^e$ is $-1$ if $n\mid e$ and $0$ otherwise. No exponent $2^r$, $3\cdot 2^s$ or $3\cdot 2^s-2^r$ with $0\le r,s<m$ is divisible by $n$: the last would give $2^t\equiv 3(\mathrm{mod}n)$ with $t<m$, impossible as $2^t,3<n$.
2. $u=1$. Over ${\mathbb{F}}_2$, $[\mathrm{Tr}y=0]=1+\mathrm{Tr}y$. Writing $\mathrm{Tr}x={\sum }_{i<m}{x}^{2^i}$, $u_k={\sum }_{x\in K^{\ast }}(1+\mathrm{Tr}x)(1+\mathrm{Tr}({\alpha }^{3k}{x}^{-3}))$ expands into $n\equiv 1$ plus multiples of vanishing power sums. Hence $H_x=M_{Q}A$ is the all-ones matrix and every $(e_i+e_j,0)$ is in the centralizer.
3. Trace fibers. For $b\ne 0$, $x\mapsto \mathrm{Tr}(bx)$ is a nonzero linear functional, so each fiber has $2^{m-1}$ elements. A row combination of $A$ has entries $c+\mathrm{Tr}(b{\alpha }^{-j})$, hence weight $0$, $n$, $2^{m-1}$ or $2^{m-1}-1$, never 2: $(e_i+e_j,0)$ is not a stabilizer.
4. No weight 1. A weight-1 vector in the centralizer would make the all-ones vector, a column of $A$ or its complement zero; every column of $A$ has weight $2^{m-1}-1\notin \{0,n\}$.

Falsifier

The proof would fail if $3$ were congruent to a power of $2$ modulo $n$, or if some trace functional $x\mapsto \mathrm{Tr}(bx)$ with $b\ne 0$ vanished identically.

Evidence

The canonical source is D5/S3/Quantum/Information/SingerQuadricQubitDistanceCeiling.lean. Its public declarations are K, n, tr, tauH, tauQ, Hz, Hx, centralizer, stabilizers, wt, claim and result; circulants are Mathlib’s Matrix.circulant. The proof applies FiniteField.sum_pow_units, FiniteField.algebraMap_trace_eq_sum_pow, traceForm_nondegenerate, AddMonoidHom.card_fiber_eq_of_mem_range, GaloisField.card, GaloisField.finrank, Matrix.circulant_mul and, for the dichotomy of ZMod 2 = Fin 2, Fin.eq_one_of_ne_zero. It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom. The frozen module state has statement identity sha256:30cb69ce3559f85841ba3baa45721da21f4c03b6fa6600a77293dc39be759a1e. The result declaration has statement identity sha256:cb2e7d43c3fa24263af085336d911223ae6568ea0c88ab654232d0dc36e3b6b6. The Freeze event is sha256:8a2d61eedbfaccef90ab155edfc2ac65e8e198ded3277efefd12d2e084becbdf; it has no project-level frozen prerequisite.

Numerical check (exact $\mathrm{GF}(2^m)$ arithmetic): $u\equiv 1$ for $d=2,\dots ,10$; the row-space weights of $A$ are exactly $\{0,3,4,7\}$, $\{0,7,8,15\}$, $\{0,15,16,31\}$ at $d=2,3,4$. Negative controls: with $\xi =2$ in place of $3$, or with a random ${\mathbf{\tau }}_Q$, $u$ is not all-ones at $d=2,\dots ,5$.

Triage

Tier 1 explicit conjecture of a 2026 paper, preregistered in issue #13470 before any Lean. theorem; resolution proved.

The escape witness exponent_exclusions is the arithmetic fact of Route step 1; it is on the proof path of result through correlation_units and cross_correlation. The private theorems whose proof path uses it (correlation_units, cross_correlation, hx_ones, pair_centralizer, small_centralizer_zero, distance_lower_bound) are content; every other private theorem, including inverse_power_sum_zero (which uses only the first conjunct $n\nmid 2^r$), is bind-only and is used on the proof path of result (CLAUDE.md §3.2 「有消费的辅助声明」). The public instance NeZero (n d) (instNeZeroNatN) is bind-only: it unfolds $n$ and applies Nat.one_lt_pow. hx_ones consumes it through the AddGroup (Fin (n d)) instance that Matrix.circulant_mul requires, and primitive_bijective, primitive_pow_sub, inverse_power_sum_zero and unit_inverse through NeZero.pos and NeZero.one_le. Utility is none: the module proves a universal statement and contains no finite enumeration, checker, numeric reduction or certified instance. There is no digestion atom.

What the settlement shows

Proved by result: for every $d\ge 2$ and every primitive element, the cross-correlation is all-ones, every $Z_i{Z}_j$ is a non-stabilizer element of the centralizer, and the minimum distance is 2.

Established inside the proof. The decisive fact is arithmetic: $3$ is not congruent to a power of 2 modulo $2^m-1$ (exponent_exclusions), so the exponents of the expanded correlation avoid $0$ modulo $n$. The row space of $A$ consists of the vectors $c+\mathrm{Tr}(b{\alpha }^{-j})$ (row_formula), whose weights are $0$, $n$, $2^{m-1}$ and $2^{m-1}-1$ (row_weights).

Argued, not formalized.

• The same argument gives $u\equiv 1$ for every exponent $\xi$ in place of 3 with $\xi \not\equiv 0$ and $\xi \not\equiv 2^t(\mathrm{mod}n)$, for both parities of $d$; the paper’s structural remark needs $\gcd (3,n)=1$ and covers only $d$ even. Both conditions are needed: for $\xi \equiv 0$ the quadric indicator is constant (at $d=2$, $\xi =7$, $\mathrm{Tr}(1)=1$ makes it zero and $u\ne 1$), and for $\xi \equiv 2^t$, $\mathrm{Tr}(x^{\xi })=\mathrm{Tr}(x)$ (with $\xi =2$, $u$ is not all-ones for $d\le 5$). Computed: for $\xi =5$ and $\xi =7$, every $d=2,\dots ,10$ satisfying both conditions gives $u\equiv 1$, including the cases where $\xi$ divides $n$ ($d=3,7$ for $\xi =5$; $d=5,8$ for $\xi =7$); the only excluded pair, $d=2$ with $\xi =7$, gives $u\ne 1$.
• The row space of $A$ is the image of the linear map $(c,b)\mapsto (c+\mathrm{Tr}(b{\alpha }^{-j}){)}_j$ on ${\mathbb{F}}_2\times K$. Row $i$ is the image of $(1,{\alpha }^i)$, so a combination with coefficients ${\epsilon }_i$ is the image of $({\sum }_i{\epsilon }_i,{\sum }_i{\epsilon }_i{\alpha }^i)$, and the pairs $(1,x)$ with $x\in K^{\ast }$ span ${\mathbb{F}}_2\times K$: for distinct nonzero $x,y$ the pairs at $x$, $y$ and $x+y$ sum to $(1,0)$, and $(0,x)=(1,x)+(1,0)$. The map is injective: for $b\ne 0$ the image is nonconstant, and $(1,0)$ maps to the all-ones vector. Hence the row space of $A$ has dimension $m+1=d+2$; so has that of $H=(A\mid J)$, whose right half is ${\sum }_i{\epsilon }_i$ times the all-ones vector, i.e. $c1$. The code is therefore $[[2^{d+1}-1,2^{d+1}-d-3,2]{]}_2$ for every $d\ge 2$, matching the paper’s computed $k=3,10,25$ at $d=2,3,4$.

Open. The paper’s distance conjecture for odd $q$ (its flagship family) and the Hamada-type rank formula for $q=p^t$ with $t>1$ are not addressed.

Effect on the paper. Conjecture 24 holds. The paper’s conclusion that the $q=2$ family is limited to distance 2 (error detection only) holds for all $d$, not only for the computed range; its other results and conjectures are unaffected.

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The search seat’s database readings are seat-reported.