GoKawiilPrime Gaps at Most 186
This repository contains a Lean 4 formalization of a prime-gap bound and a Python numerical certificate. The Lean results remain conditional on three explicit input axioms; the cited mathematical estimates and numerical computations have not been turned into Lean proofs of those inputs.
The result
For the sequence of primes $p_n$ , the target bound is
$$\liminf_{n\to\infty}(p_{n+1}-p_n)\le 186.$$
The development derives $\mathrm{DHL}[40,2]$ from the inputs below: every admissible set of forty integer shifts has infinitely many translates containing at least two primes. Admissibility means omitting a residue class modulo every prime. Applying this to the included tuple of diameter 186 gives the gap bound.
The main declarations in PrimeGaps186.lean, in namespace PrimeGap186 , are:
Declaration Result dhl_40_2 $\mathrm{DHL}[40,2]$ for every admissible integer tuple. infinite_two_prime_translates_admissibleTuple Infinitely many two-prime translates of the explicit tuple. primeGapLiminf_le_186 The consecutive-prime gap bound.
Assumed Deligne-type estimates
For a prime $p$ , write $e_p(x)=\exp(2\pi i\widetilde{x}/p)$ , where $\widetilde{x}$ is any integer representative of $x\in\mathbb{F}_p$ . Define
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