{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:levit-mandrescu-unimodality-conjecture",
  "title": "Levit–Mandrescu Unimodality Conjecture",
  "canonical_statement": "A graph on $n$ vertices is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial $i(G,x)$ of every very well-covered graph is unimodal, i.e. its coefficient sequence is nondecreasing and then nonincreasing.",
  "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:levit-mandrescu-unimodality-conjecture:event",
      "problem_id": "vibemathed:levit-mandrescu-unimodality-conjecture",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-22T00:00:00.000Z",
      "title": "Lucas B. (Jump Trading), LinkedIn announcement",
      "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.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:levit-mandrescu-unimodality-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:levit-mandrescu-unimodality-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:levit-mandrescu-unimodality-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:levit-mandrescu-unimodality-conjecture:attempt",
      "problem_id": "vibemathed:levit-mandrescu-unimodality-conjecture",
      "model": "GPT-5.6 Sol, Claude Fable 5",
      "provider": null,
      "attempted_at": "2026-07-22T00:00:00.000Z",
      "human_collaborators": [
        "Lucas B."
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The models produced an explicit counterexample: the whiskering of $E_{588} \\vee (39K_{11} \\sqcup 85K_{12})$, a very well-covered graph on 4,074 vertices, whose independence polynomial $(1+x)^{1449}(1+2x)^{588} + (1+x)^{1913}(1+12x)^{39}(1+13x)^{85} - (1+x)^{2037}$ has a strict local valley at $a_{1095}$. Per the announcement, the search took a few hours once the question was posed.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Lucas B. (Jump Trading), LinkedIn announcement",
          "url": "https://www.linkedin.com/posts/tesuji_levit-mandrescu-counterexample-activity-7487528446403768320-b1pc",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Levit–Mandrescu Unimodality Conjecture",
          "url": "https://vibemathed.com/problem/levit-mandrescu-unimodality-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:levit-mandrescu-unimodality-conjecture:verification",
      "problem_id": "vibemathed:levit-mandrescu-unimodality-conjecture",
      "solution_event_id": "vibemathed:levit-mandrescu-unimodality-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Announced on LinkedIn by Lucas B. (Head of AI Research, Jump Trading); no preprint yet, and the reviewers credited are internal to the team rather than independent. The stated polynomial was recomputed with exact integer arithmetic: $a_{1094} > a_{1095} < a_{1096}$ holds, so the polynomial given is genuinely not unimodal, and its low-order coefficients ($a_0 = 1$, $a_1 = 4074$) are consistent with a graph on 4,074 vertices. What remains unchecked is that the whiskered graph's independence polynomial equals the polynomial stated.",
      "sources": [
        {
          "label": "VibeMathed: Levit–Mandrescu Unimodality Conjecture",
          "url": "https://vibemathed.com/problem/levit-mandrescu-unimodality-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Lucas B. (Jump Trading), LinkedIn announcement",
          "url": "https://www.linkedin.com/posts/tesuji_levit-mandrescu-counterexample-activity-7487528446403768320-b1pc",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Lucas B. (Jump Trading), LinkedIn announcement",
      "url": "https://www.linkedin.com/posts/tesuji_levit-mandrescu-counterexample-activity-7487528446403768320-b1pc",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Levit–Mandrescu Unimodality Conjecture",
      "url": "https://vibemathed.com/problem/levit-mandrescu-unimodality-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:levit-mandrescu-unimodality-conjecture:signal:watch_only",
      "problem_id": "vibemathed:levit-mandrescu-unimodality-conjecture",
      "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"
}