{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:improved-maximal-prime-gap-lower-bound",
  "title": "Improved maximal prime-gap lower bound",
  "canonical_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 $X$,\n$$\nG(X)\\gg \\frac{\\log X\\,(\\log_2 X)^2\\,\\log_4 X}{(\\log_3 X)^2}.\n$$\nEquivalently, 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 $\\log_2 X$.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:improved-maximal-prime-gap-lower-bound:event",
      "problem_id": "vibemathed:improved-maximal-prime-gap-lower-bound",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-03T00:00:00.000Z",
      "title": "OpenAI self-publication",
      "summary": "The paper proves\n$$\nG(X)\\gg \\frac{\\log X\\,(\\log_2 X)^2\\,\\log_4 X}{(\\log_3 X)^2}\n$$\nfor 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\n$$\nG(X)\\gg\\frac{\\log X\\log_2 X}{\\log_4 X}\n$$\nby the unbounded factor\n$$\n\\frac{\\log_2 X(\\log_4 X)^2}{(\\log_3 X)^2}.\n$$",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:improved-maximal-prime-gap-lower-bound:attempt"
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      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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      ]
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  "attempts": [
    {
      "id": "vibemathed:improved-maximal-prime-gap-lower-bound:attempt",
      "problem_id": "vibemathed:improved-maximal-prime-gap-lower-bound",
      "model": "GPT 6 Astra",
      "provider": "OpenAI",
      "attempted_at": "2026-09-03T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "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.",
      "failed_routes": [],
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        },
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          "kind": "transcript",
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          "kind": "problem-record",
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        },
        {
          "label": "The tilted residue-class construction this improves",
          "url": "https://vibemathed.com/problem/tilted-residue-class-construction-for-long-prime-free-intervals",
          "kind": "other",
          "license": null
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        {
          "label": "OpenAI self-publication",
          "url": "https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/long_gaps.pdf",
          "kind": "primary-mathematical-source",
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        {
          "label": "VibeMathed: Improved maximal prime-gap lower bound",
          "url": "https://vibemathed.com/problem/improved-maximal-prime-gap-lower-bound",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:improved-maximal-prime-gap-lower-bound:verification",
      "problem_id": "vibemathed:improved-maximal-prime-gap-lower-bound",
      "solution_event_id": "vibemathed:improved-maximal-prime-gap-lower-bound:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "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.",
      "sources": [
        {
          "label": "VibeMathed: Improved maximal prime-gap lower bound",
          "url": "https://vibemathed.com/problem/improved-maximal-prime-gap-lower-bound",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "kind": "primary-mathematical-source",
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    {
      "label": "VibeMathed: Improved maximal prime-gap lower bound",
      "url": "https://vibemathed.com/problem/improved-maximal-prime-gap-lower-bound",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
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      "id": "vibemathed:improved-maximal-prime-gap-lower-bound:signal:recent_partial_result",
      "problem_id": "vibemathed:improved-maximal-prime-gap-lower-bound",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}