{"slug": "gpt-6-astra-infinitely-pairs-of-consecutive-primes-with-distance-at-most-186", "title": "GPT-6-Astra: infinitely pairs of consecutive primes with distance at most 186", "summary": "OpenAI released a Lean 4 formalization, GPT-6-Astra, proving that infinitely many pairs of consecutive primes are separated by at most 186, but the proof is conditional on three explicit axioms not yet verified in Lean. The project, named PrimeGaps186, includes a Python certificate that recomputes the numerical bounds, and the Lean build passed without errors, though it does not discharge the axioms.", "body_md": "This repository contains a Lean 4 formalization of a prime-gap bound and a\nPython numerical certificate. The Lean results remain **conditional on three explicit input axioms**;\nthe cited mathematical estimates and numerical computations have not been\nturned into Lean proofs of those inputs.\n\nFor the sequence of primes\n\nThe development derives\n\nThe main declarations in [PrimeGaps186.lean](/openai/PrimeGaps186/blob/main/PrimeGaps186.lean), in namespace\n`PrimeGap186`\n\n, are:\n\n| Declaration | Result |\n|---|---|\n`dhl_40_2` |\n|\n`infinite_two_prime_translates_admissibleTuple` |\nInfinitely many two-prime translates of the explicit tuple. |\n`primeGapLiminf_le_186` |\nThe consecutive-prime gap bound. |\n\nFor a prime\n\nThe axiom `PrimeGap186.kloosterman3_bound`\n\nassumes the following bound for\n**every prime**\n\nThis follows from Deligne's theorem as stated in Nicholas M. Katz,\n[ Gauss Sums, Kloosterman Sums, and Monodromy Groups, Annals of Mathematics\nStudies 116, Princeton University Press (1988), Theorem 4.1.1(1)–(2), p. 49](https://web.math.princeton.edu/~nmk/Katz-GKM.pdf#page=29).\nWith\n\nThe axiom `PrimeGap186.kloosterman2_correlation_bound`\n\nassumes the following\nbound for **every prime**\n\nThis is Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel,\n[ The Friedlander–Iwaniec character sum, 14 June 2013, Proposition 2, p. 1](https://people.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf#page=1).\nTheir normalized\n\nThese estimates are established in the cited literature, but remain unproved inputs in this Lean development.\n\n`PrimeGap186.physical_integral_bounds`\n\nassumes 104 outer and 45 inner\nphysical-integral upper bounds, plus three cap bounds.\n\nThe [Python certificate](/openai/PrimeGaps186/blob/main/prime_gap_186_certificate.py) recomputes the trial\nfrom scratch. The tested environment used Python 3.12.13, NumPy 2.2.6,\npython-flint 0.9.0, and a custom FLINT 3.6.0 build with corrected signed\npolynomial convolution (not bundled).\n\n```\npython3 -B prime_gap_186_certificate.py --workers 4 --output prime_gap_186_fresh.json\n```\n\nUse a new output path. Keep `PYTHONOPTIMIZE`\n\nunset and do not use `-O`\n\nor `-OO`\n\n.\nMandatory floating-point and signed-convolution checks must pass. A successful\nrun produces a receipt with `passed: true`\n\n; it does not discharge any Lean\naxiom.\n\nThe project pins Lean 4.34.0-rc2 and its Mathlib dependencies. With\n[elan](https://github.com/leanprover/elan) installed, run:\n\n```\nlake exe cache get\nlake build PrimeGaps186\n```\n\nThe registered Lean build passed without errors or warnings. Comparator matched\nall three results to `Challenge.lean`\n\n, and Nanoda and Lean’s kernel accepted their\nproofs in a local Colima Linux VM. The [configuration](/openai/PrimeGaps186/blob/main/comparator/main.json) permits\nthe three documented project axioms plus `propext`\n\n, `Quot.sound`\n\n, and\n`Classical.choice`\n\n(six total); this verifies conditional proofs, not the inputs\nthemselves. The numerical certificate is unchanged from its earlier passing run.\n\n[Challenge.lean](/openai/PrimeGaps186/blob/main/Challenge.lean) specifies the statements and input assumptions,\nwith three intentional theorem placeholders. See the\n[Comparator instructions](/openai/PrimeGaps186/blob/main/comparator/README.md) and\n[formalization metadata](/openai/PrimeGaps186/blob/main/formalization.yaml) for the checking setup and status.\n\nProject contributions use [Apache 2.0](/openai/PrimeGaps186/blob/main/LICENSE); existing third-party notices\nremain applicable.", "url": "https://wpnews.pro/news/gpt-6-astra-infinitely-pairs-of-consecutive-primes-with-distance-at-most-186", "canonical_source": "https://github.com/openai/PrimeGaps186", "published_at": "2026-09-03 19:18:56+00:00", "updated_at": "2026-09-03 19:53:26.709547+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["OpenAI", "Lean 4", "PrimeGaps186", "Python", "NumPy", "FLINT", "Mathlib", "Nicholas M. Katz"], "alternates": {"html": "https://wpnews.pro/news/gpt-6-astra-infinitely-pairs-of-consecutive-primes-with-distance-at-most-186", "markdown": "https://wpnews.pro/news/gpt-6-astra-infinitely-pairs-of-consecutive-primes-with-distance-at-most-186.md", "text": "https://wpnews.pro/news/gpt-6-astra-infinitely-pairs-of-consecutive-primes-with-distance-at-most-186.txt", "jsonld": "https://wpnews.pro/news/gpt-6-astra-infinitely-pairs-of-consecutive-primes-with-distance-at-most-186.jsonld"}}