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Benefit Ledger · provisional · Mathematics

GPT-6 Astra improves short and large prime-gap bounds

On 3 September 2026 OpenAI released GPT-6 Astra with two analytic number-theory results: a proof that infinitely many consecutive primes differ by at most 186, and an improvement to a large-gap bound that had been stuck for more than 80 years.

3 Sep 2026Tier 2 NotableMethodology 0.1

Current score

+0.42

3 base · Notable (tier 2 of 5, 3 pts)
× 0.8500 attribution · Primary causal contribution
× 0.4000 evidence · External expert evaluation
× 0.5500 realization · Experimentally validated
× 0.7500 durability
Event-level product before credit split: 0.42

Improved classical prime-gap bounds are notable mathematics (tier 2), not a Millennium problem. OpenAI attributes the short-gap proof to GPT-6 Astra (0.85). Evidence is a developer launch plus posted papers and a conditional Lean repo (0.40). Realization is a demonstrated theorem with incomplete Lean axioms (0.55). The bound itself is durable if the analysis holds.

What happened

OpenAI’s GPT-6 Astra launch materials present two prime-gap theorems as realized mathematical work by the named release, not as benchmark scores. The short-gap paper proves lim inf (p_{n+1} − p_n) ≤ 186 by establishing DHL[40,2] for an admissible 40-tuple of diameter 186, combining Polymath8a and Stadlmann equidistribution estimates with a larger Selberg-sieve support. Stadlmann independently had 240. A Lean 4 development (openai/PrimeGaps186) formalizes the main theorems conditionally on explicit exponential-sum and numerical-integral axioms, with a Python-FLINT certificate for the numerics. The large-gap result is described in the same announcement as improving a term unchanged for more than 80 years, with posted proofs. These are bounded improvements in analytic number theory, not Millennium Prize problems. Credit stays with Astra because OpenAI attributed the proofs to that named release.

Model attribution

+0.42

GPT-6

Named 3 September 2026 release credited with the short-gap theorem and the large-gap improvement.

The short-gap paper states the proof is due to GPT-6 Astra; the launch post presents both gap results as Astra work.

Attribution 0.8500 · Credit share 100% · OpenAI

Claims

  • The posted argument establishes lim inf of consecutive prime gaps ≤ 186 via DHL[40,2] on a diameter-186 admissible 40-tuple.

    outcome · supported

  • The Lean formalization is unconditional and replaces human review of the exponential-sum inputs.

    significance · disputed

  • These results solve a Clay Millennium Prize problem.

    significance · disputed

Sources

primary sources

other sources

Secondary domains: Computer Science

Revision history

  • 13 Sep 2026 · 0.00 → 0.42

    Imported September 2026 mathematics events under methodology 0.1 credit-unit rules.