Singularity of symmetric random sign matrices

Singularity of symmetric random sign matrices concerns the probability that a symmetric n × n matrix whose entries on and above the diagonal are independent random signs is singular. Two rows can coincide with probability of order 2⁻ⁿ, which gives a lower bound; Costello, Tao and Vu showed that the matrix is nonsingular with probability tending to one, and later work improved the upper bound to an exponential one with an unspecified constant. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), listed as its result 239 and dated 3 and 4 October 2026, claim the sharp exponential rate for fair and for biased signs. This page summarizes the published claims and records public follow-up work.

Background

• As recounted in the introduction of the fair-sign manuscript:
  • Independent entries. Komlós proved that a matrix with independent uniform 0/1 entries is nonsingular with probability tending to one, and Tikhomirov proved the sharp rate (1/2 + o(1))ⁿ for independent uniform signs. Symmetry changes how the matrix is revealed, since a row and its matching column are dependent.
  • Symmetric matrices. Costello, Tao and Vu proved that a random symmetric Bernoulli matrix is nonsingular with probability tending to one, with the bound O(n^−1/8+δ), developing quadratic Littlewood–Offord estimates. Nguyen improved this to O_C(n^−C) and Vershynin obtained a bound decaying like e to the power −n^c; further bounds, with exponents of order n^1/4√log n, √n, √n (log n)^1/4 and √(n log n), followed by Ferber and Jain, by Campos, Mattos, Morris and Morrison, by Jain, Sah and Sawhney, and by Campos, Jenssen, Michelen and Sahasrabudhe.
  • The exponential bound. Campos, Jenssen, Michelen and Sahasrabudhe proved an exponential bound e^−cn, using inverse Littlewood–Offord estimates for conditioned random walks; related work controls the least singular value and larger rank deficiencies.

The original results

• Fair signs (3 October 2026). The sharp exponential rate of singularity for symmetric Bernoulli matrices states that for a symmetric n × n matrix whose entries on and above the diagonal are independent uniform signs, the probability that the determinant vanishes is (1/2 + o(1))ⁿ.
• Biased signs (4 October 2026). The sharp singularity rate for biased symmetric sign matrices states that when each entry on and above the diagonal equals 1 with a fixed probability p in (0, 1) other than 1/2, the probability that the determinant vanishes is (p² + (1 − p)² + o(1))ⁿ, and that the event that two rows agree attains this rate.
• Method, as described: a Schur complement reduces the sign matrix to a symmetric matrix of bits of one dimension less, and a reduction to corank one turns singularity into a question about appending a new column to a nonsingular matrix; the expected number of such columns is controlled through the discriminant group of the matrix with its symmetric pairing, lattices of controlled covolume and the Reverse Minkowski Theorem of Regev and Stephens-Davidowitz. The biased manuscript reuses several deterministic lemmas of the fair-sign manuscript, which do not depend on the entry distribution.
• Sources. Commit 7ef5c0c of the openai/math repository:
  • The sharp exponential rate of singularity for symmetric Bernoulli matrices;
  • The sharp singularity rate for biased symmetric sign matrices.
• Formalization. The repository's contents list gives no Lean scope note or Comparator challenge for this result.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-06	Sharp singularity rates for symmetric random sign matrices, an article of the Latent Seminar blog at lowrank.github.io	Reading (exposition)	Explains family 239: a symmetric random matrix, the two claimed rates, the exact two-row lower bound, quadratic concentration as the obstruction for the upper bound, the structure of the fair-sign proof and the additional issues for biased signs, describing the claim that two coinciding rows give the sharp exponential singularity rate for fair and for biased symmetric sign matrices.	Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
2026-10-16	J. Jardine, Lattice24, referee reviews rank_405 (fair signs) and rank_489 (biased signs) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Both reviews report no gap. The review of the fair-sign paper states that it read all sections, about 5,000 lines of LaTeX, and checked the steps, and lists no issues. The review of the biased-sign paper states that it read sections 1 to 6 and checked the main-line argument. It records one minor issue: several load-bearing deterministic inputs are imported from the fair-sign paper, which it describes as a dependency rather than a circular argument, noting that the review of that paper found no gap in them.	Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0.
2026-10-17	AI Proved What?, episode 239 When is a random matrix singular?, remade version of 22 October 2026	Reading (exposition)	Video explainer of two OpenAI preprints, 101 pages, dated 3 and 4 October 2026: for a symmetric matrix of fair random signs the probability of singularity is 1/2 + o of 1, to the power n, and for signs equal to +1 with a fixed probability p other than 1/2 the rate is p² + the square of 1 − p, the chance that two signs agree, attained by two agreeing rows. The episode explains the two-row lower bound, recalls the conjecture of Costello, Tao and Vu that it gives the true exponential rate, the earlier exponential bound with an unspecified constant by Campos, Jenssen, Michelen and Sahasrabudhe, and the sharp rates of Tikhomirov and of Jain, Sah and Sawhney for matrices with all entries independent. It states that the sharper expected form of order n² over 2 to the n, any explicit error term, uniformity in p and other entry distributions are not claimed.	Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, 2 papers, 101 pages, dated October 3 and 4, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that the preprints have not been revised since release, that the second paper draws on 23 numbered results of the first, that there is no Lean formalization, that the result is not among those OSVerify reviewed, that one community site lists it as an unreviewed partial result because it does not reach the form with n², and that it found no expert reaction.

Notes

• The background summarizes the fair-sign manuscript's own account of earlier work.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.