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Math-2322 · mathematics · Probability

Singularity Probability of Random Symmetric Sign Matrices

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $Q_n$ be an $n\times n$ random symmetric matrix whose upper-triangular entries are independent Bernoulli variables taking values $\pm1$. Determine or estimate the probability that $Q_n$ is singular.

Formulation references: [S1], [S2].

Short English statement

Let $Q_n$ be an $n\times n$ random symmetric matrix whose upper-triangular entries are independent Bernoulli variables taking values $\pm1$. Determine or estimate the probability that $Q_n$ is singular.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2322-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2322

The manuscript uses a symmetric matrix with independent uniform signs on and above the diagonal and proves singularity probability 2^(-n+o(n)). This gives the sharp exponential estimate for exactly the problem statement symmetric Bernoulli sign model; independence of the full n^2 entries is never assumed. The main theorem determines the rate p^2+(1-p)^2 for each fixed nonfair bias p. The problem statement does not specify the Bernoulli parameter explicitly but its referenced sharp-rate problem convention is the fair case, which this manuscript excludes. Within these assumptions, this is a claimed affirmative answer to the stated question.

Singularity Probability of Random Symmetric Sign Matrices

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: The symmetric unbiased sign-matrix singularity probability is claimed to be $(1/2+o(1))^n$; 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. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Let $A_n$ be real symmetric, with independent uniform $\{-1,1\}$ entries on and above its diagonal. The claimed singularity asymptotic is \[ \mathbb P(\det A_n=0)=\left(\frac12+o(1)\right)^n=2^{-n+o(n)}. \] The diagonal entries are randomized as well as the off-diagonal entries.
Result 2.
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

Lattice covering separates robust kernel vectors from structured subspaces. Discriminant pairings and principal-minor algebra restrict admissible new columns; typical rank estimates and small-ball bounds then exclude other singular mechanisms at a larger exponential rate.

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

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