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Math-2792-A1 · revision r1 · on Math-2792

OpenAI claim summary for Math-2792

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 026; 2026-09-25).

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

Manuscript

Contents
  1. Positive lower density of arbitrarily large normalized prime gaps
  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

Positive lower density of arbitrarily large normalized prime gaps

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 026; 2026-09-25), the asserted result is that each fixed threshold $C\log p$ is exceeded by a proportion of consecutive prime gaps bounded away from zero, which also gives a positive lower proportion of increases in $p_n/n$.

Mathematical statement of the claimed result

Manuscript 1.
Writing $p_n$ for the $n$th prime, the claimed positive lower density is \[ \forall C>0\ \exists c(C)>0,N_0(C)\ \forall N\ge N_0(C): \quad|\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}|\ge c(C)N. \] The density constant and threshold may depend on the fixed gap multiplier, which does not grow with $N$.

Outline of the argument

For manuscript 1 (family 026; 2026-09-25), a square of signed divisor sums gives appreciable weight to a prime in one short interval and little weight to a prime in the adjacent interval. Two second-moment bounds convert weighted mass to ordinary starts; bounded multiplicity then counts distinct consecutive gaps.

Formal verification

For manuscript 1 (family 026; 2026-09-25), the documented Lean scope is positive lower density of increases in the sequence $p_n/n$; the full large-gap theorem is not part of that selected result.

The selected formal statement has narrower scope than the full manuscript claim.

The linked family Comparator checks are PrimeGaps.lean.

Their named declarations include OAI.Problem344.large_gaps_and_ratio_density.

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