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Math-2862-A1 · revision r1 · on Math-2862

OpenAI claim summary for Math-2862

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.

For each fixed degree and interior bulk interval, the source allows sufficiently small fixed iid uniform diagonal disorder, proves positive continuous limiting density there, and gives the GOE point-process limit at every fixed energy after unfolding by that density. These are the precise model and limit conventions in the target.

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

Manuscript

Contents
  1. GOE bulk statistics for weakly disordered regular-graph operators
  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

GOE bulk statistics for weakly disordered regular-graph operators

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. The same GOE limit is claimed under small fixed uniform diagonal disorder, unfolded by the corresponding random-tree density of states.

Mathematical statement of the claimed result

Result 1.
Fix $d\ge3$, and take uniform simple labeled $d$-regular graphs, independent iid $\omega_i\sim\operatorname{Unif}[-1,1]$, and $H_{n,d,w}=A_{n,d}+w\operatorname{diag}(\omega_i)$. Let $\nu_{d,w}$ be the expected root spectral measure of the same operator on the infinite $d$-regular tree. For each $0<\kappa<2\sqrt{d-1}$, the source gives $w_0(d,\kappa)>0$. At every fixed $0<w<w_0$, $\nu_{d,w}$ has a continuous positive density $\rho_{d,w}$ near $I=[-2\sqrt{d-1}+\kappa,2\sqrt{d-1}-\kappa]$. For $E\in I$ and every nonnegative $f\in C_c(\mathbb R)$, \[ \left|\mathbb E e^{-\int f\,d\Xi_{n,E}}- \mathbb E e^{-\int f\,d\Xi_n^{\mathrm{GOE}}}\right|\to0, \quad\Xi_{n,E}=\sum_i\delta_{n\rho_{d,w}(E)(\lambda_i-E)}. \] GOE uses variances $1/n$ off diagonal and $2/n$ on diagonal, unfolded at zero by $n/\pi$. Convergence holds through $n>d$ with $nd$ even.

Outline of the argument

Cavity resolvent recursion, edge switches, and spectral averaging yield a distributional local law retaining the tree environment. Conditional entropy identifies Gaussian eigenvector marks without losing second moments; measure-preserving potential changes are compared with Dyson perturbation to obtain fixed-energy universality.

Formal verification

Family 219 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