Sharp singularity rates for symmetric random sign matrices

What OpenAI claims

Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give Pr(det A_n=0)=(1/2+o(1))ⁿ; for fixed bias p ∈ (0,1)∖{1/2}, the rate is (p²+(1-p)²+o(1))ⁿ. In the biased case, agreeing rows attain this rate.

Papers (2)

1. The sharp exponential rate of singularity for symmetric Bernoulli matrices

   3 October 2026 · Preprint, not peer reviewed · PDF

   Let An be a symmetric n× n matrix whose entries on and above the diagonal are independent uniform signs. We prove

2. The sharp singularity rate for biased symmetric sign matrices

   4 October 2026 · Preprint, not peer reviewed · PDF

   Let An be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability p ∈ (0,1)∖{1/2}. We prove that ℙ(det A_n=0)=(p²+(1-p)²+o(1))ⁿ. The rate is attained by the event that two rows agree.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.