{
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  "source": "vibemath",
  "problem_id": "vibemathed:han-xiong-integer-trace",
  "title": "The Han-Xiong Integer Trace Conjecture",
  "canonical_statement": "Han and Xiong extended the Gaussian binomial coefficient to positive rational index and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at the integer point. Ono's paper proves a support-dominance theorem settling the conjecture for a large family of rational parameters and reduces the full conjecture to unit fractions, with a finite computer verification covering every remaining case up to a fixed bound.",
  "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:han-xiong-integer-trace:event",
      "problem_id": "vibemathed:han-xiong-integer-trace",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-31T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Settles the conjecture for a large family and reduces the rest to unit fractions; the general unit-fraction case remains open.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:han-xiong-integer-trace:attempt"
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      "method_family_ids": [
        "vibemathed:han-xiong-integer-trace:method"
      ],
      "source_assertion_ids": [
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      ]
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      "problem_id": "vibemathed:han-xiong-integer-trace",
      "model": "AxiomProver",
      "provider": "Axiom Math",
      "attempted_at": "2026-07-31T00:00:00.000Z",
      "human_collaborators": [
        "Ken Ono"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The theoretical results were autonomously produced and verified in Lean by AxiomProver: the formal statements and proofs of Theorem 1.3, Corollary 1.4 and Theorem 1.5 were generated from a natural-language statement of the problem containing no proofs, then checked by the Lean proof assistant. An appendix records precisely what was and was not supplied to the system.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.00323",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Han-Xiong Integer Trace Conjecture",
          "url": "https://vibemathed.com/problem/han-xiong-integer-trace",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "cost_usd": null,
      "independence": "unknown"
    }
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      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The main theorems were formalized and kernel-checked in Lean by the same system that produced them; the human author wrote the paper from that formal development. Tier: AxiomProver generated the formal statements and proofs from a natural-language prompt; nobody independent has audited the statement fidelity.",
      "sources": [
        {
          "label": "VibeMathed: The Han-Xiong Integer Trace Conjecture",
          "url": "https://vibemathed.com/problem/han-xiong-integer-trace",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.00323",
          "kind": "primary-mathematical-source",
          "license": null
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      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.00323",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Han-Xiong Integer Trace Conjecture",
      "url": "https://vibemathed.com/problem/han-xiong-integer-trace",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:han-xiong-integer-trace:signal:recent_partial_result",
      "problem_id": "vibemathed:han-xiong-integer-trace",
      "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"
}