Prime 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

$$\mathrm{Kl}_3(c;p) =\frac1p\sum_{\substack{x_1,x_2,x_3\in\mathbb{F}_p\\x_1x_2x_3=c}} e_p(x_1+x_2+x_3),$$ $$K_2(c;p)=\sum_{u\in\mathbb{F}_p^\times}e_p(u+c/u).$$

The axiom PrimeGap186.kloosterman3_bound assumes the following bound for every prime $p$ and all $c\in\mathbb{F}_p^\times$:

$$\left|\mathrm{Kl}_3(c;p)\right|\le 3.$$

This follows from Deligne's theorem as stated in Nicholas M. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Annals of Mathematics Studies 116, Princeton University Press (1988), Theorem 4.1.1(1)–(2), p. 49. With $n=3$, trivial multiplicative characters, and $b_1=b_2=b_3=1$, rank three and weight two give the raw bound $3p$; our normalization divides by $p$.

The axiom PrimeGap186.kloosterman2_correlation_bound assumes the following bound for every prime $p$ and all $A,B\in\mathbb{F}_p^\times$:

$$\left|\sum_{t\in\mathbb{F}_p\setminus\{0,-1\}} K_2(A/t;p)\,K_2(B/(t+1);p)\right|\le 8p\sqrt p.$$

This is Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel, The Friedlander–Iwaniec character sum, 14 June 2013, Proposition 2, p. 1. Their normalized $\mathrm{Kl}_2(c)$ equals $K_2(c;p)/\sqrt p$ after inverting the summation variable, so their $8\sqrt p$ bound becomes $8p\sqrt p$ here. No condition $A\ne B$ is imposed; the two poles are excluded even when $A=B$.

These estimates are established in the cited literature, but remain unproved inputs in this Lean development.

Numerical input and certificate

PrimeGap186.physical_integral_bounds assumes 104 outer and 45 inner physical-integral upper bounds, plus three cap bounds.

The Python certificate recomputes the trial from scratch. The tested environment used Python 3.12.13, NumPy 2.2.6, python-flint 0.9.0, and a custom FLINT 3.6.0 build with corrected signed polynomial convolution (not bundled).

python3 -B prime_gap_186_certificate.py --workers 4 --output prime_gap_186_fresh.json

Use a new output path. Keep PYTHONOPTIMIZE unset and do not use -O or -OO. Mandatory floating-point and signed-convolution checks must pass. A successful run produces a receipt with passed: true; it does not discharge any Lean axiom.

Building and verification

The project pins Lean 4.34.0-rc2 and its Mathlib dependencies. With elan installed, run:

lake exe cache get
lake build PrimeGaps186

The registered Lean build passed without errors or warnings. Comparator matched all three results to Challenge.lean, and Nanoda and Lean’s kernel accepted their proofs in a local Colima Linux VM. The configuration permits the three documented project axioms plus propext, Quot.sound, and Classical.choice (six total); this verifies conditional proofs, not the inputs themselves. The numerical certificate is unchanged from its earlier passing run.

Challenge.lean specifies the statements and input assumptions, with three intentional theorem placeholders. See the Comparator instructions and formalization metadata for the checking setup and status.

Project contributions use Apache 2.0; existing third-party notices remain applicable.