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Math-2865-A1 · revision r1 · on Math-2865

OpenAI claim summary for Math-2865

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. The sharp singularity exponent of biased symmetric sign matrices
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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