Proved
The preliminary draft proves H₁ ≤ 212 ↗
The paper’s unconditional assertion; formalization assumes analytic and variational inputs.
Observed Oct 13, 2026
Bound · version 1
Gaps at most 212, infinitely often
Earlier version 1. View current version 2 →
Statement
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 212; in other words H₁ ≤ 212.
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
Scope
An unconditional upper bound on the recurring gap, attributed to the 2026 preliminary draft; it does not imply every gap is this small.
Evidence
Source reports
Connections
In this problem · publication v1
How small can prime gaps be? →
Bound · Proved
Scope: The cited mathematical result bounds recurring consecutive-prime gaps by 212.
Limit: The target H₁ = 2 is not established.
Nested results concern the same prime-gap limit; they are not separate portions of a solved problem.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Bounded gaps between primes: the 212 bound →
- Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver → · Original passage ↗
Recorded wording
The preliminary draft states H₁ ≤ 212 and explains the analytic and exact rational calculations. Its appendix reports a formal deduction with analytic and variational hypotheses.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
François Charton · Letong Hong · Kenny Lau · Ken Ono · Guillaume Remy · Ho Chung Siu · Ashvin A. Swaminathan · Jesse Thorner · Yunzhou Xie · Preprint · Publication date Sep 3, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-09-03. Observed Oct 13, 2026.
The nine authors supply the mathematics, building on Julia Stadlmann and earlier sieve work. AxiomProver is credited with generating a formal deduction; this is not a claim that the model independently discovered the mathematical bound.
Source updated Sep 3, 2026.
Variants of the Selberg sieve, and bounded intervals containing many primes →
- Main bounded-gap results and introduction → · Original passage ↗
Recorded wording
The published Polymath8b argument establishes an unconditional recurring prime gap at most 246; the twin-prime target remains out of reach.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
D. H. J. Polymath · Published · Publication date 2014
Peer-reviewed historical result.
Revision & attribution
Source revision: 2014. Observed Oct 13, 2026.
No AI contribution is asserted by this source record.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
The preliminary draft proves H₁ ≤ 212 ↗
The paper’s unconditional assertion; formalization assumes analytic and variational inputs.
Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-prime-gaps-bound-212
- Statement version
- 1
- SHA-256
- b250b0ade3b47544df6e556b808ff38edd6177e4ff22155b24470c245b770e6b