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Math-2786-A1 · revision r1 · on Math-2786

OpenAI claim summary for Math-2786

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 012; 2026-09-24).

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

Manuscript

Contents
  1. The joint Dickman law for consecutive integers
  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

The joint Dickman law for consecutive integers

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 012; 2026-09-24), the asserted result is that consecutive integers have independent limiting largest-prime-factor statistics with Dickman marginals, and either ordering of their largest factors has density one half.

Mathematical statement of the claimed result

Manuscript 1.
Let $P^+(n)$ be the largest prime factor and $\rho$ the Dickman function, equal to one on $[0,1]$, zero on negative arguments and satisfying $u\rho'(u)+\rho(u-1)=0$ for $u>1$. For every fixed $a,b\in(0,1)$, the claimed joint law is \[ \lim_{X\to\infty}\frac1X|\{2\le n\le X:P^+(n)\le n^a,\ P^+(n+1)\le n^b\}| =\rho(1/a)\rho(1/b). \] The limit uses ordinary unweighted counting through all real $X$, not logarithmic averaging.

Outline of the argument

For manuscript 1 (family 012; 2026-09-24), a positive divisor weight amplifies a putative mixed correlation into quadratic energy. An independent split model controls its first two moments, while residue smoothing suppresses nonconstant modes and forces the amplified energy to vanish.

Formal verification

For manuscript 1 (family 012; 2026-09-24), OpenAI lists Lean statements for this assertion: consecutive integers have independent limiting largest-prime-factor statistics with Dickman marginals, and either ordering of their largest factors has density one half.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are JointDickman.lean.

Their named declarations include OAI.JointDickmanPaper.decreasing_order, OAI.JointDickmanPaper.increasing_order, OAI.JointDickmanPaper.joint_law.

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