OpenAI · 239-02 · 2026-10-04 · Probability and statistical mechanics

The sharp singularity rate for biased symmetric sign matrices

preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026

Family 239. Sharp singularity rates for symmetric sign matrices.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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 $p \in (0, 1) ∖ {1/2}$ , the rate is $(p^{2} + (1 - p)^{2} + o (1))^{n}$ . In the biased case, agreeing rows attain this rate.

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

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.