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Math-2795-A1 · revision r1 · on Math-2795

OpenAI claim summary for Math-2795

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.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 011; 2026-09-24).

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

Manuscript

Contents
  1. Poisson-Dirichlet factorization of prime predecessors
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Poisson-Dirichlet factorization of prime predecessors

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 011; 2026-09-24), the asserted result is that the logarithmically normalized ordered prime factors of $p-1$, for uniformly sampled primes, converge jointly to the parameter-one Poisson--Dirichlet partition.

Mathematical statement of the claimed result

Manuscript 1.
For each prime $p\ge3$, order prime factors $q_j(p)$ of $p-1$ decreasingly with multiplicity, pad by ones, and set $V_j(p)=\log q_j(p)/\log(p-1)$. Take independent uniform $U_j\in(0,1)$, form stick fragments $B_j=(\prod_{i<j}U_i)(1-U_j)$, and let $(L_j)$ be their decreasing rearrangement, the Poisson--Dirichlet law with parameter one. For every fixed $k\ge1$ and bounded continuous $F:[0,1]^k\to\mathbb R$, the claimed ordinary prime-average limit is \[ \lim_{x\to\infty}\frac1{\pi(x)-1}\sum_{3\le p\le x}F(V_1(p),\ldots,V_k(p)) =\mathbb EF(L_1,\ldots,L_k). \] The sum ranges over primes with equal weights, and the limit is through all real $x$.

Outline of the argument

For manuscript 1 (family 011; 2026-09-24), marked divisor moments are expressed using a determinant expansion and paired endpoint weights. Independent residue lines and a symmetric memory space control the signed graph error, allowing the finite-dimensional factor statistics to match the ordinary-integer model.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 011; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources