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The sharp exponential singularity rate of symmetric random sign matrices: P(det A_n = 0) = (1/2 + o(1))^n

Let AnA_n be a symmetric n×nn\times n matrix whose entries on and above the diagonal are independent uniform signs. Two rows agree with probability about 2−n2^{-n}, so Pr⁡(det⁡An=0)≥2−n+o(n)\Pr(\det A_n=0)\ge2^{-n+o(n)}, and it is generally believed that Pr⁡(det⁡An=0)=Θ(n22−n)\Pr(\det A_n=0)=\Theta(n^22^{-n}), mirroring the non-symmetric case where Tikhomirov proved the rate (1/2+o(1))n(1/2+o(1))^n. Symmetry destroys row independence: Costello, Tao and Vu proved Pr⁡(det⁡An=0)→0\Pr(\det A_n=0)\to0, followed by polynomial and stretched-exponential bounds and the exponential bound e−cne^{-cn} of Campos, Jenssen, Michelen and Sahasrabudhe. Is the singularity probability of a symmetric random sign matrix (1/2+o(1))n(1/2+o(1))^n, i.e. governed by two equal rows at exponential scale?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Random matrix theory; discrete random matrices
Posed by
Kevin Costello, Terence Tao and Van Vu, and Van Vu in later surveys (as the belief that the probability is of order n^2 2^{-n})
Year posed
2006
Years open
20y
Solved
2026-10-03
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal theorem: for symmetric AnA_n with independent uniform signs on and above the diagonal, Pr⁡(det⁡An=0)=(1/2+o(1))n=2−n+o(n)\Pr(\det A_n=0)=(1/2+o(1))^n=2^{-n+o(n)}, via a Schur-complement reduction to symmetric 0/1 matrices, a corank-one reduction and a counting argument for admissible appended columns. Companion: for fixed bias p∈(0,1)∖{1/2}p\in(0,1)\setminus\{1/2\}, the rate is p2+(1−p)2p^2+(1-p)^2, attained by two agreeing rows. Not shown: the believed Θ(n22−n)\Theta(n^22^{-n}) asymptotic, uniformity in pp, or general entry distributions.

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts: the uniform-sign rate (October 3, 2026) and the fixed-bias analogue (October 4, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript (Pr⁡(det⁡An=0)=(1/2+o(1))n\Pr(\det A_n=0)=(1/2+o(1))^n) and Theorem 1.1 of the biased companion (lim⁡Pr⁡(det⁡An(p)=0)1/n=p2+(1−p)2\lim\Pr(\det A_n^{(p)}=0)^{1/n}=p^2+(1-p)^2 for fixed p≠1/2p\ne1/2) were read against the stated belief. Classified partial because the believed asymptotic Θ(n22−n)\Theta(n^22^{-n}) is sharper than the exponential rate proved. The posing attribution comes from Campos-Jenssen-Michelen-Sahasrabudhe, not from the manuscripts. Not refereed; no Lean formalization.

Sources

Changelog1 change

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