{
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  "source": "vibemath",
  "problem_id": "vibemathed:bounded-prime-gaps-at-most-212",
  "title": "A new bound for small gaps between primes: $H_1 \\le 212$",
  "canonical_statement": "Write $H_1 = \\liminf_{n\\to\\infty}(p_{n+1}-p_n)$. Stadlmann had recently proved $H_1 \\le 240$, improving the bound $246$ of Polymath8b. Building on her work, this paper proves $H_1 \\le 212$: infinitely many pairs of consecutive primes are at most $212$ apart.",
  "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",
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      "id": "vibemathed:bounded-prime-gaps-at-most-212:event",
      "problem_id": "vibemathed:bounded-prime-gaps-at-most-212",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-03T00:00:00.000Z",
      "title": "A new bound for small gaps between primes",
      "summary": "$H_1 \\le 212$, improving Stadlmann's $240$ of three days earlier and the $246$ of Polymath8b that had stood since 2014. The twin prime conjecture, $H_1 = 2$, is untouched. Held the record for hours at most: OpenAI's paper claiming $186$ is dated 30 August, four days before this one, though its Lean development appeared on 2 September.",
      "ai_contribution": "ai_assisted",
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      "id": "vibemathed:bounded-prime-gaps-at-most-212:attempt",
      "problem_id": "vibemathed:bounded-prime-gaps-at-most-212",
      "model": "AxiomProver",
      "provider": "Axiom Math",
      "attempted_at": "2026-09-03T00:00:00.000Z",
      "human_collaborators": [
        "François Charton",
        "Letong Hong",
        "Kenny Lau",
        "Ken Ono",
        "Guillaume Remy",
        "Ho Chung Siu",
        "Ashvin A. Swaminathan",
        "Jesse Thorner",
        "Yunzhou Xie"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The mathematics is the authors'. The AI contribution is the formal certificate, and the paper is precise about it in Appendix A: \"AxiomProver, an AI system under development by AxiomMath, autonomously generated from natural-language specifications a Lean certificate of the deduction of Theorem 1.1.\" The certificate takes as hypotheses the five Type I, Type II and Type III equidistribution estimates of Section 5, the bilinear Bombieri-Vinogradov theorem below the half-level, the Harman decomposition, and the variational certificate. Nothing in the paper claims the model found the argument, and the abstract does not mention AI at all.\n\nThe same group's AxiomProver had formalised the twelve-year-old 246 bound in Lean a few weeks earlier, which is the work this builds its tooling on.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "PrimeGapsLib, the Lean library",
          "url": "https://github.com/AxiomMath/PrimeGapsLib",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Blueprint of the 246 formalisation this builds on",
          "url": "https://primegaps.axiommath.ai/",
          "kind": "independent",
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        },
        {
          "label": "Stadlmann's 240, the bound this improves on",
          "url": "https://arxiv.org/abs/2608.31126",
          "kind": "independent",
          "license": null
        }
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          "label": "A new bound for small gaps between primes",
          "url": "https://primegaps.axiommath.ai/bgp212.pdf",
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        {
          "label": "VibeMathed: A new bound for small gaps between primes: $H_1 \\le 212$",
          "url": "https://vibemathed.com/problem/bounded-prime-gaps-at-most-212",
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      "solution_event_id": "vibemathed:bounded-prime-gaps-at-most-212:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint one day old, not peer reviewed. Its Lean certificate was produced by AxiomProver and is conditional on the equidistribution estimates and the Bombieri-Vinogradov theorem stated in the paper, so it certifies the deduction rather than the analytic inputs. Nine authors, several of whom work on exactly this, take responsibility for the mathematics. No independent expert has read it on the record, and this site has not rebuilt the certificate.",
      "sources": [
        {
          "label": "VibeMathed: A new bound for small gaps between primes: $H_1 \\le 212$",
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      "url": "https://vibemathed.com/problem/bounded-prime-gaps-at-most-212",
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      "problem_id": "vibemathed:bounded-prime-gaps-at-most-212",
      "kind": "recent_partial_result",
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      "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."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}