# Random symmetric matrix singularity reference

Family 239: Sharp singularity rates for symmetric random sign matrices. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Numerical probability researchers can compare uniform or biased symmetric sign-matrix sampling against the claimed exponential singularity rates.

## Applicability and commercial boundary

The o(1) term is asymptotic, determinant zero is an exact arithmetic event, and floating-point near-singularity is different. The claim does not certify numerical conditioning of arbitrary matrices.

## Initial business decision

Conditional research. Buyer budget, commercial novelty and profitability are unvalidated.

## Next verification action

Use exact integer rank on small fixtures and inspect effective error bounds before estimating finite failure probability.

## Evidence scope

The catalog statement was individually reviewed. This record does not imply a manuscript proof review or formal-scope comparison. No independent Lean check was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Main source

[The sharp exponential rate of singularity for symmetric Bernoulli matrices](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf).