Bounded prime gaps

Let $p_n$ be the $n$ -th prime. The quantity $H_1=\liminf_{n\to\infty}(p_{n+1}-p_n)$ is the smallest gap between consecutive primes that occurs infinitely often. A result is a proof that $H_1\le B$ for an integer $B$ , and it must not assume an unproved hypothesis. A lower $B$ is better, and $B=2$ is the twin prime conjecture. Polymath8b also proved $H_1\le 12$ if the unproved Elliott–Halberstam conjecture holds. Such conditional bounds do not count here.

Classic baseline: $246$ , published by D. H. J. Polymath in 2014.

These results are Claimed: their authors published them, and we have not checked them.

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Claimed Human Verified Lean Verified

Rank	Bound	Evidence level	Author	Date	Proof scope
1	$186$ Exact value and conditions $186$ The paper credits the proof to GPT 6 Astra. It proves this: take a set of 40 integers that, for each prime $p$ , misses at least one remainder modulo $p$ . Then infinitely many shifts of the set (the same number added to each element) contain at least two primes. It uses such a set whose largest and smallest elements differ by 186. The paper is dated 30 August 2026 and became public on 3 September 2026. The time shown is the first fork of its repository, the earliest public trace that we found.	Agent breakthrough	OpenAI	Sep 3, 11:57 AM PT	Proof scope A written proof in a company paper. The Lean proof of $H_1\le 186$ in openai/PrimeGaps186 assumes three axioms that are not proved in Lean: two exponential-sum bounds of Deligne type, which are published theorems, and 152 numerical bounds on integrals and constants, which only a Python program checks. So it is not a Lean proof of our statement. The authors say that no person has reviewed the Lean development independently.
2	$212$ Exact value and conditions $212$ A preliminary draft (3 September 2026) by nine authors, published by Axiom Math. It optimizes Julia Stadlmann's method with numerical experiments on a computing cluster.	Claimed	Show authors François Charton Letong Hong Kenny Lau Ken Ono Guillaume Remy Ho Chung Siu Ashvin A. Swaminathan Jesse Thorner Yunzhou Xie	Sep 3, 7:56 AM PT	Proof scope A written proof in a draft paper. AxiomProver, an AI system of Axiom Math, generated a Lean certificate of the deduction. That certificate takes the five equidistribution estimates of the paper, the Bombieri–Vinogradov theorem, Harman's decomposition and the numerical certificate as hypotheses, so it is not a Lean proof of our statement.
3	$240$ Exact value and conditions $240$ Preprint arXiv:2608.31126. It combines the Bombieri–Vinogradov theorem with newer equidistribution estimates for moduli that have only small prime factors. It uses a set of 49 integers whose largest and smallest elements differ by 240.	Claimed	Julia Stadlmann	Aug 31, 10:39 AM PT	Proof scope A written proof in a preprint. No Lean proof is supplied.

Relevant repos

openai/PrimeGaps186: OpenAI's Lean development and numerical certificate. The README lists the three axioms.