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Precise statement
Let $G(X)$ denote the largest gap between consecutive primes not exceeding $X$, and let $\log_j$ denote the $j$-fold iterated logarithm. The paper proves that, for all sufficiently large $X$,
$$
G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2}.
$$
Equivalently, there is an absolute constant $c>0$ such that $G(X)$ is at least $c$ times the quantity above for all sufficiently large $X$. This improves Rankin's classical lower bound by a factor of $\log_2 X$.
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fidelity, correctness, priority, or novelty.
What AI did
GPT 6 Astra
The paper explicitly attributes the proof to GPT 6 Astra. The new argument introduces a short-translates proposition that allows a sparse residual set of positions to be made composite simultaneously. Its main construction uses specially chosen divisor-sum factors, a shared truncation of their product, and a nonnegative squared weight. The resulting proposition is then inserted into an Erdős--Rankin construction to obtain the improved maximal prime-gap bound.
Site-confirmed on 4 September 2026: this site built openai/LongGapsBetweenPrimes at commit 03a1190d from a clean checkout on GitHub Actions (run 33843996072). lake build completed all 8707 jobs, the build's own #print axioms line reads 'LongGapsBetweenPrimes.long_prime_gaps' depends on axioms: [propext, Classical.choice, Quot.sound], and leanchecker replayed the library through the kernel. The statement was read by hand: Challenge.lean asserts, for some c > 0 and all sufficiently large X, a consecutive prime gap below X exceeding c times log X (log_2 X)^2 log_4 X / (log_3 X)^2, which is the claimed bound and not a weaker cousin of it. What remains is what a kernel cannot see: there is no human author, and no named mathematician has commented yet.
The paper proves
$$
G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2}
$$
for all sufficiently large $X$. Its main new ingredient is a short-translates theorem: for any sufficiently small set $S\subseteq[1,H]$ with $|S|\le\delta x$, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the claimed length. This directly and asymptotically improves the August 2026 GPT-5.6 Sol bound
$$
G(X)\gg\frac{\log X\log_2 X}{\log_4 X}
$$
by the unbounded factor
$$
\frac{\log_2 X(\log_4 X)^2}{(\log_3 X)^2}.
$$
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.