{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:crouzeix-s-conjecture",
  "title": "Crouzeix's Conjecture",
  "canonical_statement": "Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$, $\\lVert p(A)\\rVert \\leq 2 \\max_{z \\in W(A)} |p(z)|$, where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.",
  "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:crouzeix-s-conjecture:event",
      "problem_id": "vibemathed:crouzeix-s-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-27T00:00:00.000Z",
      "title": "Jin, The Numerical Range Is a 2-Spectral Set",
      "summary": "Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:crouzeix-s-conjecture:attempt"
      ],
      "method_family_ids": [
        "vibemathed:crouzeix-s-conjecture:method"
      ],
      "source_assertion_ids": [
        "vibemathed:crouzeix-s-conjecture:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:crouzeix-s-conjecture:attempt",
      "problem_id": "vibemathed:crouzeix-s-conjecture",
      "model": "GPT-5.6 Sol; ChatGPT 5.6 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-07-27T00:00:00.000Z",
      "human_collaborators": [
        "Shanmu Jin",
        "Emiel Lorist",
        "Felix L. Schwenninger"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "For the first proof: Jin, a neurosurgery resident with no specialized mathematical training, reports that the key result (Theorem 2) emerged during an approximately sixteen-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode - a public prompt adapted from the Cycle Double Cover run, web access denied, a branching portfolio of subagent strategies under adversarial audit, and no human intervention once started. Jin then simplified and verified the output; the repository publishes the prompt, successive manuscripts, a Lean formalization and an axiom audit. For the independent second proof, Lorist and Schwenninger disclose that ChatGPT 5.6 Pro was used to review previous approaches to the weaker spectral constant $1 + \\sqrt{2}$ and to identify a possible source of improvement in estimates involving iterates $f^n$ of extremal or approximately extremal functions.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Townsend and Greenbaum, The Neurosurgery Resident Who Proved Crouzeix's Conjecture (14 Aug 2026)",
          "url": "https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf",
          "kind": "independent",
          "license": null
        },
        {
          "label": "Jin's repository: prompt, successive manuscripts, Lean formalization, axiom audit",
          "url": "https://github.com/jinshanmu/CrouzeixConjecture",
          "kind": "code",
          "license": null
        },
        {
          "label": "Lorist and Schwenninger, A solution to Crouzeix's conjecture (arXiv, 4 Aug 2026) - the independent second proof",
          "url": "https://arxiv.org/abs/2608.03841",
          "kind": "independent",
          "license": null
        },
        {
          "label": "AIM workshop on Crouzeix's conjecture (2017)",
          "url": "https://aimath.org/pastworkshops/crouzeix.html",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Wikipedia: Crouzeix's conjecture",
          "url": "https://en.wikipedia.org/wiki/Crouzeix%27s_conjecture",
          "kind": "wikipedia",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Jin, The Numerical Range Is a 2-Spectral Set",
          "url": "https://www.preprints.org/manuscript/202607.1919",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Crouzeix's Conjecture",
          "url": "https://vibemathed.com/problem/crouzeix-s-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:crouzeix-s-conjecture:verification",
      "problem_id": "vibemathed:crouzeix-s-conjecture",
      "solution_event_id": "vibemathed:crouzeix-s-conjecture:event",
      "level": "independent_expert_verified",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Independently expert-verified, publicly on record: Townsend and Greenbaum's essay of 14 August 2026 states that both authors and Michel Crouzeix himself \"have checked the proof thoroughly and believe that Dr. Jin's manuscript is correct\" - the conjecture's own poser among the verifiers, and Greenbaum co-organized the 2017 AIM workshop on the problem. This site read that essay in full and audited Jin's repository (commit 9df0783): 82 Lean files with zero sorry, zero axiom declarations and zero native_decide with comments stripped, on toolchain v4.28.0, alongside an Annals-formatted manuscript and the complete autonomous-run prompt - though the Lean was not compiled here and its statement-to-conjecture correspondence not audited, so the tier rests on the expert endorsement, not the formalization. The independent second proof by Lorist and Schwenninger (arXiv:2608.03841) has no comparable public endorsement yet and the essay stops short of vouching for it. Neither manuscript is refereed.",
      "sources": [
        {
          "label": "VibeMathed: Crouzeix's Conjecture",
          "url": "https://vibemathed.com/problem/crouzeix-s-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Jin, The Numerical Range Is a 2-Spectral Set",
          "url": "https://www.preprints.org/manuscript/202607.1919",
          "kind": "primary-mathematical-source",
          "license": null
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      ]
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  ],
  "sources": [
    {
      "label": "Jin, The Numerical Range Is a 2-Spectral Set",
      "url": "https://www.preprints.org/manuscript/202607.1919",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Crouzeix's Conjecture",
      "url": "https://vibemathed.com/problem/crouzeix-s-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:crouzeix-s-conjecture:signal:replay_ready",
      "problem_id": "vibemathed:crouzeix-s-conjecture",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "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"
}