{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:ballantine-beck-feigon-maurischat-subsum",
  "title": "The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials",
  "canonical_statement": "Ballantine, Beck, Feigon and Maurischat introduced the subsum polynomial $\\mathrm{sp}(\\lambda,x) := \\prod_i (1+x^{\\lambda_i})$ attached to an integer partition $\\lambda$, studied rational functions built by summing reciprocals of these polynomials over natural classes of partitions, and posed ten conjectures. Six are now proved: the ordinary and binary coprimality and divisibility conjectures, and the odd and ternary special-value and recurrence conjectures.",
  "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:ballantine-beck-feigon-maurischat-subsum:event",
      "problem_id": "vibemathed:ballantine-beck-feigon-maurischat-subsum",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-05-20T00:00:00.000Z",
      "title": "arXiv:2605.21718 - Reciprocals of Partition Polynomials",
      "summary": "six of the ten conjectures proved; one was found false as printed and its corrected form remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:ballantine-beck-feigon-maurischat-subsum:attempt"
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      ]
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  "attempts": [
    {
      "id": "vibemathed:ballantine-beck-feigon-maurischat-subsum:attempt",
      "problem_id": "vibemathed:ballantine-beck-feigon-maurischat-subsum",
      "model": "AxiomProver",
      "provider": null,
      "attempted_at": "2026-05-20T00:00:00.000Z",
      "human_collaborators": [
        "Evan Chen",
        "Ken Ono",
        "Jujian Zhang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "AxiomProver autonomously produced Lean and mathlib formalizations and machine-checkable proofs of all six conjectures. It also discovered a counterexample to one of the conjectures as printed, so the same system both proved and refuted statements drawn from one list, which is a useful demonstration that it was reading the statements rather than pattern-matching toward the expected answer.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2605.21718 - Reciprocals of Partition Polynomials",
          "url": "https://arxiv.org/abs/2605.21718",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials",
          "url": "https://vibemathed.com/problem/ballantine-beck-feigon-maurischat-subsum",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
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      "id": "vibemathed:ballantine-beck-feigon-maurischat-subsum:verification",
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      "solution_event_id": "vibemathed:ballantine-beck-feigon-maurischat-subsum:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The six proofs are Lean and mathlib formalizations, so they are machine-checkable rather than dependent on refereeing. Note that the catalog has not compiled the artifact itself, so this records the authors' claim of formalization, not an independent build.",
      "sources": [
        {
          "label": "VibeMathed: The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials",
          "url": "https://vibemathed.com/problem/ballantine-beck-feigon-maurischat-subsum",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.21718 - Reciprocals of Partition Polynomials",
          "url": "https://arxiv.org/abs/2605.21718",
          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "arXiv:2605.21718 - Reciprocals of Partition Polynomials",
      "url": "https://arxiv.org/abs/2605.21718",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Ballantine-Beck-Feigon-Maurischat Conjectures on Subsum Polynomials",
      "url": "https://vibemathed.com/problem/ballantine-beck-feigon-maurischat-subsum",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:ballantine-beck-feigon-maurischat-subsum:signal:recent_partial_result",
      "problem_id": "vibemathed:ballantine-beck-feigon-maurischat-subsum",
      "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"
}