What was already known
Polymath 8b proved the long-standing published bound H1 ≤ 246. Stadlmann independently and concurrently proved H1 ≤ 240. OpenAI’s manuscript combines Polymath 8a and Stadlmann equidistribution estimates with new support conditions and numerical optimization to claim DHL[40,2], yielding 186 (manuscript §1.1, pp. 1–2).
AI involvement
The primary manuscript names OpenAI as author and says the proof is due to GPT 6 Astra. The repository formalization metadata also names OpenAI as author; the exact amount of human input during theorem discovery is not specified.
OpenAI is named as author and the abstract explicitly credits GPT 6 Astra with the proof. The exact amount of human direction beyond problem selection is not documented.
Importance
If correct, 186 is a further numerical improvement to the bounded-prime-gap upper bound, while the twin-prime conjecture H1=2 remains open.
Limitations
The Lean code proves the deduction from three explicit inputs, not the unconditional bound: two Kloosterman estimates are cited from established literature, and 104 outer, 45 inner and three cap numerical inequalities rely on a Python/FLINT certificate. The reported certificate was not rerun here, and no complete outside check of the 39-page manuscript was found.
Technical verification
The source reports a Lean kernel check of DHL[40,2] and H1≤186 conditional on three explicit analytic/numerical inputs. Two Kloosterman estimates come from cited literature; the finite integral and cap inequalities rely on a Python/FLINT certificate not discharged in Lean or rerun here. No completed full manuscript review was located.
- Supporting source (github.com) — Conditional theorem declarations
Formal checking
Source-reported Lean 4 kernel/Comparator check covers the implication from three explicit external assumptions to DHL[40,2] and H1≤186. It does not check the assumptions themselves.
- Checker
- OpenAI-reported Lean 4 kernel and Comparator
- Scope
- Conditional deduction of DHL[40,2] and H1≤186 from two Kloosterman estimates and the finite numerical-integral/cap package.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The manuscript provides an outline, but no named human is documented as independently explaining the new complementary factorization and sieve-weight argument.
Correction and publication history
Claim, AI attribution, prior bounds, and conditional checking scope reviewed from primary sources for non-notifying local inclusion.
Read this published revision