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Math-2412-A1 · revision r1 · on Math-2412

OpenAI claim summary for Math-2412

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 at unsolvedmath.com: partially solved. 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.

The question asks for improved quantitative bounds on interval eigenvalue counts when the degree is at most logarithmic to the fourth power. The selected theorem describes the limiting bulk point process for each fixed degree and fixed interior energy. It consequently identifies limiting microscopic count distributions, but gives neither a convergence rate nor a uniform estimate in the requested degree range. Its auxiliary local counting estimates are imported inputs. This is a related universality result rather than a quantitative solution of the recorded counting problem.

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

Manuscript

Contents
  1. Eigenvalue Counts for Low-Degree Random Regular Graphs
  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

Eigenvalue Counts for Low-Degree Random Regular Graphs

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Every fixed regular degree $d\geq3$ is claimed to give GOE eigenvalue statistics at fixed interior bulk energies for uniformly sampled simple regular graphs.

Mathematical statement of the claimed result

Result 1.
For fixed $d\ge3$, let $G_{n,d}$ be uniform among simple labeled $d$-regular graphs, at admissible sizes, and $\lambda_i$ its adjacency eigenvalues. Fix $|E|<2\sqrt{d-1}$ and define \[ \rho_d(E)=\frac{d\sqrt{4(d-1)-E^2}}{2\pi(d^2-E^2)},\qquad \Xi_{n,d,E}=\sum_i\delta_{n\rho_d(E)(\lambda_i-E)}. \] For GOE eigenvalues $\mu_i$, with off-diagonal variance $1/n$ and diagonal variance $2/n$, set $\Xi_n^{\mathrm{GOE}}=\sum_i\delta_{(n/\pi)\mu_i}$. The claim is that each nonnegative continuous compactly supported $h$ satisfies \[ \left|\mathbb E e^{-\int h\,d\Xi_{n,d,E}} -\mathbb E e^{-\int h\,d\Xi_n^{\mathrm{GOE}}}\right|\to0. \] The random graph has no further conditioning.

Outline of the argument

An entropy method gives quantitatively accurate Gaussian eigenvector marks and high moments at sampled edges. Reversible edge switches promote those marks to microscopic orthogonal invariance; a Gaussian perturbation and fixed-energy Dyson relaxation identify the point process, with compact-count control completing convergence.

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