{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:zonoid-volume-log-submodularity",
  "title": "Log-Submodularity of Zonoid Volume",
  "canonical_statement": "The conjecture that volume is log-submodular under Minkowski addition on zonoids, that is $|A||A+B+C| \\leq |A+B||A+C|$. Disproved by a four-dimensional zonotope generated by a 2-modular matrix together with two segments. Several related local mixed-volume, local Loomis-Whitney, projection-volume-ratio and volume-to-surface-area conjectures fall with it.",
  "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:zonoid-volume-log-submodularity:event",
      "problem_id": "vibemathed:zonoid-volume-log-submodularity",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-07T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:zonoid-volume-log-submodularity:attempt"
      ],
      "method_family_ids": [
        "vibemathed:zonoid-volume-log-submodularity:method"
      ],
      "source_assertion_ids": [
        "vibemathed:zonoid-volume-log-submodularity:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:zonoid-volume-log-submodularity:attempt",
      "problem_id": "vibemathed:zonoid-volume-log-submodularity",
      "model": "GPT-5.6 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-08-07T00:00:00.000Z",
      "human_collaborators": [
        "Ruben Skorupinski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper has a section called How the counterexample was found. The author proved the unimodular case and identified 2-modular matrices as the place to look; then \"Chat-GPT was then used to search for a counterexample within the space of the 2-modular matrices which led to the discovery of the counterexample within a specific class of 2-modular matrices\". The acknowledgment is more conservative, crediting GPT-5.6 Pro with \"literature searches and exploratory volume computations of 2-modular zonotopes\", all independently verified by the author. Classified on the lower of the two readings.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.07702",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Log-Submodularity of Zonoid Volume",
          "url": "https://vibemathed.com/problem/zonoid-volume-log-submodularity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:zonoid-volume-log-submodularity:verification",
      "problem_id": "vibemathed:zonoid-volume-log-submodularity",
      "solution_event_id": "vibemathed:zonoid-volume-log-submodularity:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old. The counterexample is an explicit finite object and the volumes are exactly computable, so it is checkable by anyone who wants to, but nobody independent has done so.",
      "sources": [
        {
          "label": "VibeMathed: Log-Submodularity of Zonoid Volume",
          "url": "https://vibemathed.com/problem/zonoid-volume-log-submodularity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.07702",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.07702",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Log-Submodularity of Zonoid Volume",
      "url": "https://vibemathed.com/problem/zonoid-volume-log-submodularity",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:zonoid-volume-log-submodularity:signal:watch_only",
      "problem_id": "vibemathed:zonoid-volume-log-submodularity",
      "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"
}