{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations",
  "title": "Balanced EF1 and fPO Allocations",
  "canonical_statement": "Does every instance of indivisible goods with additive valuations admit a balanced allocation (any two bundles differing in size by at most one) that is simultaneously envy-free up to one good (EF1) and fractionally Pareto optimal (fPO)? Kawase et al. established existence only for personalized bivalued valuations or at most two valuation types. Proved in general, via the Knaster-Kuratowski-Mazurkiewicz lemma applied to a weighted-welfare duality framework plus a new price interlacing lemma.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations:event",
      "problem_id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-06T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The paper records how the collaboration actually went: the authors first aimed at a counterexample showing EF1 and PO incompatible under matroid constraints, and when the model surfaced fundamental difficulties with that plan they redirected toward proving the positive result instead. The paper also extends the technique to category constraints and leaves a pseudopolynomial-time algorithm open.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations:attempt"
      ],
      "method_family_ids": [
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      "source_assertion_ids": [
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      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations:attempt",
      "problem_id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations",
      "model": "GPT-5.6-Sol (via OpenAI Codex), Claude Fable 5",
      "provider": "OpenAI, Anthropic",
      "attempted_at": "2026-08-06T00:00:00.000Z",
      "human_collaborators": [
        "Benjamin Cookson",
        "Nisarg Shah",
        "Paritosh Verma"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The acknowledgement is unusually specific: \"All the mathematical proofs and counterexamples were derived by OpenAI Codex (GPT-5.6-Sol at Max effort) based on research directions, literature connections, proof and search strategies, and inspirations supplied by the authors.\" The authors verified every detail and simplified the exposition with both models, and retain responsibility.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.06325",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Balanced EF1 and fPO Allocations",
          "url": "https://vibemathed.com/problem/fair-and-efficient-balanced-allocations-for-additive-valuations",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
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  "verifications": [
    {
      "id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations:verification",
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      "solution_event_id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "An arXiv preprint days old. The authors state they verified all the mathematics themselves, which is not independent review, and no formalization is reported.",
      "sources": [
        {
          "label": "VibeMathed: Balanced EF1 and fPO Allocations",
          "url": "https://vibemathed.com/problem/fair-and-efficient-balanced-allocations-for-additive-valuations",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.06325",
          "kind": "primary-mathematical-source",
          "license": null
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      ]
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  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.06325",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Balanced EF1 and fPO Allocations",
      "url": "https://vibemathed.com/problem/fair-and-efficient-balanced-allocations-for-additive-valuations",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations:signal:watch_only",
      "problem_id": "vibemathed:fair-and-efficient-balanced-allocations-for-additive-valuations",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}