{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:graffiti-conjecture-806",
  "title": "Graffiti Conjecture 806",
  "canonical_statement": "Let $S$ be the set of square-free integers in $[2, n]$ and $G = PR[S]$ the graph on $S$ in which two integers are adjacent when they are not coprime. From the cases $n \\le 100$ and about twenty further values $n \\le 200$, Graffiti conjectured that the largest adjacency eigenvalue $\\lambda_1(G)$ is at most the number of distinct vertex degrees.\n\nFalse. At $n = 51$ the graph has $31$ vertices, $11$ distinct degrees and $\\lambda_1 > 11.846$; the conjecture fails again for every $n$ from $786$ to $5000$, and the deficit $\\lambda_1 - D$ grows roughly linearly in $n$, so no additive correction $\\lambda_1 \\le D + C$ survives either.",
  "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:graffiti-conjecture-806:event",
      "problem_id": "vibemathed:graffiti-conjecture-806",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-26T00:00:00.000Z",
      "title": "AI Village graffiti-verification (GitLab)",
      "summary": "The repository supplies an executable verifier in exact integer arithmetic; for $n = 51$ an integer vector $x$ satisfies $x^{T}Ax > 11\\,x^{T}x$, so $\\lambda_1 > 11 = D$. The deficit $\\lambda_1(n) - D(n)$ grows through $n = 5000$ in the repository's computations, but no asymptotic theorem proving divergence is claimed, so \"false for every constant $C$\" is a computed pattern, not a proved one.\n\nOne anomaly, which the repository records itself: $n = 51$ lies inside the range Graffiti is said to have tested when it made the conjecture.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:graffiti-conjecture-806:attempt"
      ],
      "method_family_ids": [
        "vibemathed:graffiti-conjecture-806:method"
      ],
      "source_assertion_ids": [
        "vibemathed:graffiti-conjecture-806:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:graffiti-conjecture-806:attempt",
      "problem_id": "vibemathed:graffiti-conjecture-806",
      "model": "Claude Opus 5 (AI Village)",
      "provider": "Anthropic",
      "attempted_at": "2026-08-26T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Claude Opus 5 identified and verified counterexamples to WOW conjecture 806 in the square-free non-coprimality graph family. It reconstructed the intended graph and degree-count invariant from the original Written on the Wall source, searched the family, found the smallest counterexample at $n=51$, and produced exact integer Rayleigh-quotient certificates showing that the largest adjacency eigenvalue exceeds the number of distinct degree values.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Repository: 197 refutations, each with a verifier script",
          "url": "https://gitlab.com/ai-village-agents/village/graffiti-verification",
          "kind": "code",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "AI Village graffiti-verification (GitLab)",
          "url": "https://gitlab.com/ai-village-agents/village/graffiti-verification/-/blob/main/verify/verify_wow1_806.py",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Graffiti Conjecture 806",
          "url": "https://vibemathed.com/problem/graffiti-conjecture-806",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:graffiti-conjecture-806:verification",
      "problem_id": "vibemathed:graffiti-conjecture-806",
      "solution_event_id": "vibemathed:graffiti-conjecture-806:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Site-confirmed: the repository's verifier, verify_wow1_806.py, was re-run here on 2 September 2026. Fast mode passed 160 assertions and full mode 175, in 21 seconds, with every counterexample certified without floating point - an explicit integer vector $x$ with $x^{T}Ax > D\\,x^{T}x$, which forces $\\lambda_1 > D$ by the Rayleigh principle. For $n = 51$ the certificate is $x^{T}Ax = 1117310362790 > 11 \\cdot 94319113125$.\n\nThe verifier also confirms the graph construction against direct gcd tests, scans every $n \\le 300$ for the least counterexample (it is $51$), and checks the neighbouring conjectures 802, 805, 807 and 808 of the same block as controls, which hold on Graffiti's stated range. Its check against the original Written on the Wall text was skipped here, since the text file is not in the repository; statement fidelity rests on the transcription in the script header and on those controls.\n\nNo independent specialist review.",
      "sources": [
        {
          "label": "VibeMathed: Graffiti Conjecture 806",
          "url": "https://vibemathed.com/problem/graffiti-conjecture-806",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "AI Village graffiti-verification (GitLab)",
          "url": "https://gitlab.com/ai-village-agents/village/graffiti-verification/-/blob/main/verify/verify_wow1_806.py",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "AI Village graffiti-verification (GitLab)",
      "url": "https://gitlab.com/ai-village-agents/village/graffiti-verification/-/blob/main/verify/verify_wow1_806.py",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Graffiti Conjecture 806",
      "url": "https://vibemathed.com/problem/graffiti-conjecture-806",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:graffiti-conjecture-806:signal:watch_only",
      "problem_id": "vibemathed:graffiti-conjecture-806",
      "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"
}