The sharp singularity exponent of biased symmetric sign matrices
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Definitions and mathematical statement
Statement recorded in the supplied source.
For each fixed $p\in (0,1)$ excluding 1/2, let $A_n$ be symmetric with independent signs on and above the diagonal, each +1 with probability p. Is $lim_{n\to \infty }$ P(det $A_n=0)^{1/n}=p^2+(1-p)^2$?Short English statement
For each fixed $p\in (0,1)$ excluding 1/2, let $A_n$ be symmetric with independent signs on and above the diagonal, each +1 with probability p. Is $lim_{n\to \infty }$ P(det $A_n=0)^{1/n}=p^2+(1-p)^2$?Sources
- S1 Original question source or mathematical context. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/build/sections/01-introduction.tex.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2865
With fixed sign bias $p\ne1/2$, the claimed symmetric-matrix singularity rate is $(p^2+(1-p)^2+o(1))^n$, attained by equal rows. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.
The sharp singularity exponent of biased symmetric sign matrices
OpenAI's claim
OpenAI presents a proof in the selected manuscript. With fixed sign bias $p\ne1/2$, the claimed symmetric-matrix singularity rate is $(p^2+(1-p)^2+o(1))^n$, attained by equal rows.Mathematical statement of the claimed result
Result 1.
Fix $p\in(0,1)\setminus\{1/2\}$. In a symmetric matrix $A_n^{(p)}$, let the upper-triangular entries, including the diagonal, be independent, equal to $1$ with probability $p$ and $-1$ otherwise. The source claims \[ \lim_{n\to\infty}\mathbb P(\det A_n^{(p)}=0)^{1/n}=p^2+(1-p)^2. \] This is a fixed-bias assertion; the unbiased case is supplied by a separate manuscript.Outline of the argument
Kernel vectors determine possible exposed columns, with sparse-tail cases separated from the conditioned bit model. Lattice tools and prime-field corank bounds control the structured alternatives; the biased column estimate matches the repeated-row lower rate.Formal verification
Family 239 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, The sharp singularity rate for biased symmetric sign matrices, 2026-10-04; citation key
OAI:The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.