Source authenticated

A tilted residue-class construction for long prime-free intervals

Claims $G(T)\gg\log T\log_2 T/\log_4 T$ against FGKMT's $\log T\log_2 T\log_4 T/\log_3 T$ , a gain of $\log_3 T/(\log_4 T)^2$ , together with $Y(X)\gg X\log X/\log_3 X$ for the covering problem behind it. Ben Green resists calling it incremental: "The sieving procedure is different to the Erdos-Rankin one which underpinned all bounds on the problem since 1938, and it wins $\log_3 N$ over that procedure. The [FGKMT] paper also wins a $\log_3 N$ . These wins are essentially independent of one another so one now wins basically $(\log_3 N)^2$ over Rankin's 1938 bound." He adds that the new sieve "by itself could have claimed the Erdos 10000 dollars for this question". What is new is narrow: only the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt; the hypergraph covering theorem and Maynard weight come from FGKMT. Readers on the thread note that several later sections reproduce FGKMT with no new content, and Green calls the exposition "horrific".

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: expert-verified. Publication: announcement. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: A tilted residue-class construction for long prime-free intervals · Large prime gaps record

Confidence: Not scored

Registry verification: expert verified · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

GPT 5.6 Sol
model · ai model contributor · OpenAI

DottedCalculator (prompting and submission)
human · human collaborator

Boris Alexeev (Lean formalisation)
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: A tilted residue-class construction for long prime-free intervals

specialist registry · passed

Expert-verified, by an unusually well-placed expert: Ben Green, a co-author of the FGKMT bound this beats, so a reader with every incentive to be sceptical rather than generous. On 26 August he called it "an important claim" that "survives an initial plausibility check", while warning that "the exposition is truly horrible" and that verifying it would take time. On 27 August, after "a few hours thought and some conversations with Terry Tao and James Maynard", he wrote that he "became more or less convinced this is correct", adding that "a lean formalisation exists so we know it's correct, and experts closest to the area basically understand informally what the key ideas are and why the proof works". Boris Alexeev, who formalised it, states that the result is "formalized unconditionally in Lean". That matches the audit done here on 30 August: 403 Lean files, 56,614 lines, no $\texttt{sorry}$ , no $\texttt{admit}$ , no $\texttt{native\_decide}$ and not one declared axiom, with the terminal theorem stated in primitive Mathlib terms and carrying no hypotheses at all. Short of Lean-verified on the anchoring half. erdosproblems.com hosts this as a proof claim under its standing disclaimer that listing is "no guarantee of proof correctness" - it has not accepted the claim - and nobody without a stake has audited the informal-to-formal correspondence. The Lean was not built here - no toolchain. No referee has read it, and Green expects a human-written exposition to take months.

Source ↗

Proof claim and expert discussion: Green, Bloom, Alexeev

independent reproduction · passed

independent link supplied by the VibeMathed registry.

Source ↗

Lean formalisation of the two main statements

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Ford, Green, Konyagin, Maynard, Tao - the record this improves

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Manuscript source (TeX) in the Lean repository

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Erdős problem #4, with the prior record and prize history

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

A tilted residue-class construction for long prime-free intervals

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Artifacts and verifiers

Lean formalisation of the two main statements

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

A tilted residue-class construction for long prime-free intervals parent of this event

How large can the gap between consecutive primes be, infinitely often? Writing $\log_k$ for the $k$ -fold iterated logarithm, Erdős asked (problem #4, a $10,000 prize) whether $p_{n+1}-p_n \gg C\log n\log_2 n\log_4 n/(\log_3 n)^2$ for every $C$ ; that was settled in 2016, and the record bound since has been Ford-Green-Konyagin-Maynard-Tao's $p_{n+1}-p_n \gg \log n\log_2 n\log_4 n/\log_3 n$ . This work claims a stronger bound, $G(T)\gg \log T\log_2 T/\log_4 T$ , an improvement by a factor of $\log_3 T/(\log_4 T)^2$ , together with $Y(X)\gg X\log X/\log_3 X$ for the covering problem behind it. parent of this event

Erdős problem #4, with the prior record and prize history evidence for this event

A tilted residue-class construction for long prime-free intervals evidence for this event

This event attributed to DottedCalculator (prompting and submission)

Proof claim and expert discussion: Green, Bloom, Alexeev evidence for this event

Lean formalisation of the two main statements evidence for this event

Manuscript source (TeX) in the Lean repository evidence for this event

Ford, Green, Konyagin, Maynard, Tao - the record this improves evidence for this event

This event attributed to GPT 5.6 Sol

This event attributed to Boris Alexeev (Lean formalisation)

VibeMathed record: A tilted residue-class construction for long prime-free intervals evidence for this event

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