GoKawiilLean 4 formalization confirms prime gaps of at most 186, but relies on unproven axioms
A new repository presents a Lean 4 formal proof deriving the bound liminf(p_{n+1}-p_n) ≤ 186 for consecutive primes, building on the Maynard-Tao style DHL[40,2] admissible-tuple framework. The proof chain is conditional on three explicit input axioms covering Kloosterman sum bounds and related estimates, which have not themselves been formalized as Lean proofs despite being backed by established results like Deligne's theorem via Katz's monograph.
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Formal verification tools like Lean are increasingly used to check intricate number-theoretic arguments, and prime gap bounds sit at the heart of major analytic number theory advances following Zhang, Maynard, and the Polymath project. This work shows both the promise and current limits of formalization: the logical scaffolding is machine-checked, but the deepest analytic estimates still rest on trusted, unverified mathematical inputs rather than fully machine-certified proofs.
- The Lean development proves DHL[40,2] and a 186 prime-gap bound conditionally on three axioms.
- The unproven axioms concern Kloosterman sum bounds tied to Deligne's theorem, not yet formalized in Lean.
- The work illustrates the current boundary between formally verified logic and trusted classical analytic estimates in number theory.
Prime Obsession by John Derbyshire — If bounded prime gaps and admissible tuples pique your curiosity, this classic book walks through the Riemann Hypothesis and the deep structure of the primes in readable prose. It's a great companion for anyone following formalization projects like this one but wanting the story behind the mathematics.
See Prime Obsession by John Derbyshire on Amazon → Affiliate link — we may earn a commission on purchases, at no extra cost to you. Product picked by AI based on this article; it is not a tested recommendation.Source: github.com, 2026-09-03
Published there as: “Prime Gaps at Most 186”
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