{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:sombor-energy-conjecture",
  "title": "Sombor-Energy Conjecture",
  "canonical_statement": "Does every nontrivial finite simple graph have noninteger Sombor energy? If $\\rho_1,\\ldots,\\rho_n$ are the eigenvalues of the Sombor matrix of a graph $G$, its Sombor energy is\n\n$$E_{\\mathrm{SO}}(G)=\\sum_{i=1}^{n}|\\rho_i|.$$\n\nThe conjecture asserted that $E_{\\mathrm{SO}}(G)\\notin\\mathbb Z$ for every nontrivial graph. A connected graph on nine vertices is exhibited with $E_{\\mathrm{SO}}(G)=64$, disproving the conjecture.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:sombor-energy-conjecture:event",
      "problem_id": "vibemathed:sombor-energy-conjecture",
      "type": "disproved",
      "status": "candidate",
      "occurred_at": "2026-07-30T00:00:00.000Z",
      "title": "Sombor Energy Conjecture Counterexample",
      "summary": "VibeMathed reports this item as candidate. 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:sombor-energy-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:sombor-energy-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:sombor-energy-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:sombor-energy-conjecture:attempt",
      "problem_id": "vibemathed:sombor-energy-conjecture",
      "model": "GPT-5.6 Thinking",
      "provider": "OpenAI",
      "attempted_at": "2026-07-30T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The AI constructed a connected bipartite graph on nine vertices and calculated its Sombor spectrum exactly. Writing its Sombor matrix in the block form\n\n$$S(G)=\\begin{pmatrix}0&B\\\\B^{T}&0\\end{pmatrix},$$\n\nthe singular values of $B$ were found to be\n\n$$5,\\quad 5,\\quad 11+\\sqrt{31},\\quad 11-\\sqrt{31}.$$\n\nTherefore,\n\n$$E_{\\mathrm{SO}}(G)\n=2\\left(5+5+(11+\\sqrt{31})+(11-\\sqrt{31})\\right)\n=64.$$\n\nThe AI also audited the edge list, degrees, connectivity, bipartition, matrix multiplication, characteristic polynomial, singular values and final energy calculation, and produced a self-contained proof.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Sombor Energy Conjecture Counterexample",
          "url": "https://github.com/Lamptypeshi/Sombor-Energy-Conjecture-Counterexample/blob/main/somborconnectedintegerenergyFALSE.pdf",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Sombor-Energy Conjecture",
          "url": "https://vibemathed.com/problem/sombor-energy-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:sombor-energy-conjecture:verification",
      "problem_id": "vibemathed:sombor-energy-conjecture",
      "solution_event_id": "vibemathed:sombor-energy-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The proof has been internally audited using exact calculations. The graph has nine vertices, fifteen distinct edges and degree sequence $(4,4,4,3,3,3,3,3,3)$. Its connectivity, bipartition, Sombor matrix, product $B^{T}B$, singular values, complete spectrum and energy $64$ were independently recomputed within the AI conversation.\n\nThe result has not yet been checked by an independent graph-theory expert, peer reviewer or formal proof assistant. A literature search located the original conjecture, the 2023 partial-results paper and its 2024 corrigendum, but did not locate an equivalent connected counterexample. This search does not establish absolute priority, and no claim is made that this is the first or a new counterexample.",
      "sources": [
        {
          "label": "VibeMathed: Sombor-Energy Conjecture",
          "url": "https://vibemathed.com/problem/sombor-energy-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Sombor Energy Conjecture Counterexample",
          "url": "https://github.com/Lamptypeshi/Sombor-Energy-Conjecture-Counterexample/blob/main/somborconnectedintegerenergyFALSE.pdf",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Sombor Energy Conjecture Counterexample",
      "url": "https://github.com/Lamptypeshi/Sombor-Energy-Conjecture-Counterexample/blob/main/somborconnectedintegerenergyFALSE.pdf",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Sombor-Energy Conjecture",
      "url": "https://vibemathed.com/problem/sombor-energy-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:sombor-energy-conjecture:signal:verify_now",
      "problem_id": "vibemathed:sombor-energy-conjecture",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "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"
}