{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture",
  "title": "Amdeberhan-Medina-Moll Arctangent Sum Conjecture",
  "canonical_statement": "Let $x_n = \\tan\\left(\\sum_{k=1}^{n} \\arctan k\\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \\notin \\mathbb{Z}$ for every $n \\ge 5$. Any integer value $x_n = m$ must satisfy $|m| \\ge e^{(1/2+o(1)) n \\log n}$, which forces $\\#\\{1 \\le n \\le N : x_n \\in \\mathbb{Z}\\} = O(\\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \\approx 0.147$.",
  "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:amdeberhan-medina-moll-arctan-conjecture:event",
      "problem_id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-07T00:00:00.000Z",
      "title": "arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare",
      "summary": "density-one set of n; the conjecture itself remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:amdeberhan-medina-moll-arctan-conjecture:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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      ]
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  "attempts": [
    {
      "id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture:attempt",
      "problem_id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture",
      "model": "AxiomProver",
      "provider": "Axiom Math",
      "attempted_at": "2026-07-07T00:00:00.000Z",
      "human_collaborators": [
        "Ken Ono"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The system was given only a natural-language statement of the definitions and the three target results, plus an instruction to formalize and prove them with no sorry. From that input AxiomProver autonomously produced both the Lean formalization of the problem and a complete Lean proof. The human author then wrote the paper's exposition using the formal development as his reference, which reverses the usual order: the Lean came first and the prose was derived from it.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "AxiomProver Lean development",
          "url": "https://github.com/AxiomMath/TanArctan",
          "kind": "lean-proof",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare",
          "url": "https://arxiv.org/abs/2607.05739",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Amdeberhan-Medina-Moll Arctangent Sum Conjecture",
          "url": "https://vibemathed.com/problem/amdeberhan-medina-moll-arctan-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture:verification",
      "problem_id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture",
      "solution_event_id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A public Lean 4.28.0 development accompanies the paper, containing a formalization of the problem and a proof the author states is sorry-free and adds no axioms. We attempted to compile it and did not complete the build, so the axiom claim here rests on the author's statement rather than on our own check.",
      "sources": [
        {
          "label": "VibeMathed: Amdeberhan-Medina-Moll Arctangent Sum Conjecture",
          "url": "https://vibemathed.com/problem/amdeberhan-medina-moll-arctan-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare",
          "url": "https://arxiv.org/abs/2607.05739",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare",
      "url": "https://arxiv.org/abs/2607.05739",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Amdeberhan-Medina-Moll Arctangent Sum Conjecture",
      "url": "https://vibemathed.com/problem/amdeberhan-medina-moll-arctan-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture:signal:recent_partial_result",
      "problem_id": "vibemathed:amdeberhan-medina-moll-arctan-conjecture",
      "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"
}