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

Principal theorem: for symmetric $A_n$ with independent uniform signs on and above the diagonal, $\Pr(\det A_n=0)=(1/2+o(1))^n=2^{-n+o(n)}$ , 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 $p\in(0,1)\setminus\{1/2\}$ , the rate is $p^2+(1-p)^2$ , attained by two agreeing rows. Not shown: the believed $\Theta(n^22^{-n})$ asymptotic, uniformity in $p$ , or general entry distributions.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Oct 3, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

Confidence: Not scored

Registry verification: unreviewed · announcement · partial

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

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript ( $\Pr(\det A_n=0)=(1/2+o(1))^n$ ) and Theorem 1.1 of the biased companion ( $\lim\Pr(\det A_n^{(p)}=0)^{1/n}=p^2+(1-p)^2$ for fixed $p\ne1/2$ ) were read against the stated belief. Classified partial because the believed asymptotic $\Theta(n^22^{-n})$ 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.

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

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Companion: The sharp singularity rate for biased symmetric sign matrices

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paper link supplied by the VibeMathed registry.

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Costello, Tao and Vu, Random symmetric matrices are almost surely nonsingular (2006)

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Companion: The sharp singularity rate for biased symmetric sign matrices evidence for this event

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

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

Costello, Tao and Vu, Random symmetric matrices are almost surely nonsingular (2006) evidence for this event

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

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