slug: skalski-stroinski-2025-ising-uniqueness-sets
bibkey: skalski2025level
doi: 10.48550/arXiv.2511.20925
url: https://arxiv.org/abs/2511.20925v1
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/IsingUniquenessSets.result

Skalski and Stroiński’s conjecture on the smallest set of uniqueness for the Ising cone

Problem

Skalski and Stroiński work on X = {-1,+1}^k with the Rademacher functions r_j(x) = x_j and the Walsh functions w_L(x) = ∏_{j∈L} r_j(x); for 1 ≤ q ≤ k they set

B^k_q = Lin{w_L : L ⊂ {1,…,k} and |L| ≤ q}.

Following Bogdan, Bosy and Skalski, a subset U ⊂ X is a set of uniqueness for (B^k_q)_+, the nonnegative functions of B^k_q, if φ = 0 is the only such function vanishing on U, and u(k,q) is the size of the smallest one. The section on extremal sizes closes with the conjecture

u(k,2) = k+1, i.e., for every k there are no sets of uniqueness having at most k elements.

Issue #10022 fixes the readings: points of X are functions Fin k → Bool with true read as +1; B^k_2 is the real span of the w_L with |L| ≤ 2; the equality is read for k ≥ 3, and the edge k = 2 is recorded separately.

Motivation

The frozen declaration D5/S3/Combinatorics/IsingUniquenessSets.result proves that no set of uniqueness for (B^k_2)_+ has at most k points, for every k; that u(k,2) = k + 1 for every k ≥ 3; and that u(2,2) = 4. Sets of uniqueness control the existence of maximum likelihood estimators in the Ising exponential family, and u(k,2) is the sample size below which no sample can guarantee it.

Gap

Issue #10022 preregisters this published conjecture and its literature check. Semantic Scholar reports zero citations of arXiv:2511.20925; Crossref finds no journal version; MathDB /p/371692 has status open with zero solutions; an arXiv query for “set of uniqueness” returns only the source in this sense. The source proves u(k,2) ≤ k + 1 and a lower bound of order log k from a covering argument.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent proof.

Route

Lower bound, every k: if |U| ≤ k, the linear map sending v ∈ ℝ^{k+1} to the values v_0 + Σ_i v_i x_i at the points x ∈ U has a nonzero kernel vector. Since every coordinate squares to one, the square φ(x) = (v_0 + Σ_i v_i x_i)^2 is a combination of w_∅, the w_{i} and the w_{i,j}, so φ ∈ B^k_2; it is nonnegative and vanishes on U, and it is not identically zero, because comparing the all-plus point with the point where coordinate i is flipped forces v_i = 0 and then v_0 = 0. So U is not a set of uniqueness.

Upper bound, k ≥ 3: let e_m (f_m) be the point whose only +1 (-1) coordinate is m. Every φ ∈ B^k_2 satisfies

• Σ_m φ(f_m) − Σ_m φ(e_m) + (k−2)(φ(−𝟙) − φ(𝟙)) = 0,
• Σ_m φ(e_m) + Σ_m φ(f_m) − (k−4)(φ(𝟙) + φ(−𝟙)) = 8·2^{−k} Σ_x φ(x),

as each w_L with |L| ≤ 2 checks directly. If φ ≥ 0 vanishes at every e_m and at 𝟙, the first identity forces φ(−𝟙) = 0 and φ(f_m) = 0, the second then gives Σ_x φ(x) = 0, and nonnegativity gives φ = 0. So these k + 1 points form a set of uniqueness.

Edge k = 2: B^2_2 contains the indicator of every point, so only X itself is a set of uniqueness and u(2,2) = 4.

Falsifier

A refutation of the lower bound would be a set of at most k points on which every nonnegative function of B^k_2 vanishing there is zero; the square of an affine function through a kernel vector rules this out. A refutation of the equality for some k ≥ 3 would need the k + 1 points above to fail, which the two linear identities exclude.

Evidence

For k = 3..8, fifty random members of B^k_2 each satisfy both identities and Σ_x φ(x) = 2^k c_0 to within 10^{-8}; for twenty random sets of k points per k, the kernel vector yields a nonzero square vanishing on the set. At k = 2, the indicator of (−1,−1) is nonnegative and vanishes on the three points of the source’s Lemma rem:three. The Lean proof has only the standard axiom closure propext, Classical.choice, and Quot.sound. These finite checks support the reading but do not establish the universal theorem.

Triage

First-tier external named open problem: Skalski and Stroiński, arXiv:2511.20925v1 (2025), closing conjecture of the section on extremal sizes of sets of uniqueness, preregistered in issue #10022. Resolution: proved, with the equality read for k ≥ 3; at k = 2 the source’s own listing gives u(2,2) = 2^2 = 4, which the delivered statement records.

The public surface is exactly sgn, walsh, walshSpace, IsSetOfUniqueness, minUniqueness, claim, and result. This is a uniform theorem, not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance, so utility: none applies.

ASSUMED-UNVERIFIED

The linear-programming values u(3,2) = 4, u(4,2) = 5, u(5,2) = 6 reported by the screening search are not recomputed here. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The Lean kernel does not authenticate the external source, its version history, or the literature-check coverage. The finite checks do not establish the universal theorem.