{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:prime-gaps-at-most-186",
  "title": "Prime Gaps at Most 186",
  "canonical_statement": "For the sequence of primes $p_n$, the project derives\n$$\\liminf_{n\\to\\infty}(p_{n+1}-p_n)\\le186.$$\nMore precisely, assuming three explicit analytic/numerical inputs, it proves $\\mathrm{DHL}[40,2]$: every admissible set of forty integer shifts has infinitely many translates containing at least two primes. Applying this to an explicit admissible $40$-tuple of diameter $186$ yields infinitely many consecutive prime gaps of size at most $186$. The Lean development verifies the deduction from the stated inputs; the two Kloosterman-type estimates and the finite physical-integral/cap bounds remain external assumptions.",
  "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:prime-gaps-at-most-186:event",
      "problem_id": "vibemathed:prime-gaps-at-most-186",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-03T00:00:00.000Z",
      "title": "Github",
      "summary": "Assuming three explicit input statements, the project proves $\\mathrm{DHL}[40,2]$: every admissible $40$-tuple contains at least two primes infinitely often after translation. An explicit admissible $40$-tuple of diameter $186$ then gives\n$$\\liminf_{n\\to\\infty}(p_{n+1}-p_n)\\le186.$$\nThe Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimates cited to Katz/Deligne and Fouvry--Kowalski--Michel; the third consists of finitely many numerical integral and cap inequalities backed by a Python/FLINT certificate.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:prime-gaps-at-most-186:attempt"
      ],
      "method_family_ids": [
        "vibemathed:prime-gaps-at-most-186:method"
      ],
      "source_assertion_ids": [
        "vibemathed:prime-gaps-at-most-186:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:prime-gaps-at-most-186:attempt",
      "problem_id": "vibemathed:prime-gaps-at-most-186",
      "model": "GPT 6 Astra",
      "provider": "OpenAI",
      "attempted_at": "2026-09-03T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The project metadata identifies GPT 6 Astra, operating through Codex, as the agent used for the formalization workflow. Its stated task was to formalize the source statements faithfully, simplify the proofs, and retain unresolved finite-field and numerical inputs as explicit axioms. The resulting development was refined through human-guided edits and automated proof checks. The repository metadata attributes both the project and its source article “Improved Gaps Between Primes” to OpenAI.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Lean proof",
          "url": "https://github.com/openai/PrimeGaps186/blob/main/PrimeGaps186.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Challenge.lean, the statements the comparator checks",
          "url": "https://github.com/openai/PrimeGaps186/blob/main/Challenge.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Formalization metadata",
          "url": "https://github.com/openai/PrimeGaps186/blob/main/formalization.yaml",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Numerical certificate",
          "url": "https://github.com/openai/PrimeGaps186/blob/main/prime_gap_186_certificate.py",
          "kind": "code",
          "license": null
        },
        {
          "label": "Preprint",
          "url": "https://github.com/openai/PrimeGaps186/blob/main/short_gaps_numerics.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Improved short gaps between primes (OpenAI, 30 August 2026)",
          "url": "https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Stadlmann's 240, the bound this improves on",
          "url": "https://arxiv.org/abs/2608.31126",
          "kind": "independent",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Github",
          "url": "https://github.com/openai/PrimeGaps186",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Prime Gaps at Most 186",
          "url": "https://vibemathed.com/problem/prime-gaps-at-most-186",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:prime-gaps-at-most-186:verification",
      "problem_id": "vibemathed:prime-gaps-at-most-186",
      "solution_event_id": "vibemathed:prime-gaps-at-most-186:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-checked, not Lean-verified, and the distinction is the whole of it. The Lean development in openai/PrimeGaps186 reports zero sorry in its three main declarations, and Comparator and Lean's kernel accept them - but all three depend on three project-specific axioms: a rank-three hyper-Kloosterman bound, a rank-two Kloosterman correlation bound, and a package of 104 outer, 45 inner and 3 cap numerical inequalities. So the kernel has checked that those three statements imply DHL[40,2] and the bound; it has not checked them. The first two are tied to established literature and the third is recomputed by a Python and FLINT certificate that does not discharge its Lean axiom. No independent expert has read the paper on the record.",
      "sources": [
        {
          "label": "VibeMathed: Prime Gaps at Most 186",
          "url": "https://vibemathed.com/problem/prime-gaps-at-most-186",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/openai/PrimeGaps186",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/openai/PrimeGaps186",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Prime Gaps at Most 186",
      "url": "https://vibemathed.com/problem/prime-gaps-at-most-186",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:prime-gaps-at-most-186:signal:recent_partial_result",
      "problem_id": "vibemathed:prime-gaps-at-most-186",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "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"
}