{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:lorist-schwenninger-remark-2-positivity-question", "title": "Lorist-Schwenninger Remark 2 positivity question", "canonical_statement": "Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives $\\kappa \\le 1 + \\sqrt{1 - \\Re\\langle E_1 Tx,x\\rangle}$, so positivity of $\\Re\\langle E_1 Tx,x\\rangle$ would prove the lemma outright. They write: \"it is unclear whether $\\Re\\langle E_1 Tx,x\\rangle \\ge 0$ in general.\"", "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:lorist-schwenninger-remark-2-positivity-question:event", "problem_id": "vibemathed:lorist-schwenninger-remark-2-positivity-question", "type": "disproved", "status": "resolved", "occurred_at": "2026-08-14T00:00:00.000Z", "title": "Lorist-Schwenninger, \"A solution to Crouzeix's conjecture\"", "summary": "Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:lorist-schwenninger-remark-2-positivity-question:attempt" ], "method_family_ids": [ "vibemathed:lorist-schwenninger-remark-2-positivity-question:method" ], "source_assertion_ids": [ "vibemathed:lorist-schwenninger-remark-2-positivity-question:assertion" ] } ], "attempts": [ { "id": "vibemathed:lorist-schwenninger-remark-2-positivity-question:attempt", "problem_id": "vibemathed:lorist-schwenninger-remark-2-positivity-question", "model": "Claude Opus 5.0", "provider": "Anthropic", "attempted_at": "2026-08-14T00:00:00.000Z", "human_collaborators": [], "exposure": "unknown", "prompt_public": null, "outcome": "The counterexample was found and verified computationally by Claude Opus 5.0 in a single session. The search identified the mechanism (the centres of the domain and the numerical range must be separated to make the quantity negative), found a 2x2 witness, reduced it to a closed-form algebraic expression, and verified the sign in exact arithmetic. The sharp consequence — that inequality (5) alone recovers exactly the Crouzeix-Palencia constant 1+sqrt(2) — was derived in the same session. The human operator directed the investigation to Remark 2 and approved the final write-up; all mathematical content was produced by the model.", "failed_routes": [], "artifacts": [ { "label": "GitHub", "url": "https://github.com/ZestyWombat854/crouzeix-dilation-positivity", "kind": "code", "license": null } ], "sources": [ { "label": "Lorist-Schwenninger, \"A solution to Crouzeix's conjecture\"", "url": "https://arxiv.org/abs/2608.03841", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Lorist-Schwenninger Remark 2 positivity question", "url": "https://vibemathed.com/problem/lorist-schwenninger-remark-2-positivity-question", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:lorist-schwenninger-remark-2-positivity-question:verification", "problem_id": "vibemathed:lorist-schwenninger-remark-2-positivity-question", "solution_event_id": "vibemathed:lorist-schwenninger-remark-2-positivity-question:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Reproduced by this site on 17 August 2026. Remark 2 was confirmed verbatim in the Lorist-Schwenninger LaTeX source, including the exact sentence \"it is unclear whether Re >= 0 in general\". The counterexample repository pins the claim in a statement file and was run here: the exact sympy certificate gives m = -4/95 - 1188*sqrt(90709)/8617355 = -0.0836..., with the sign certified through its minimal polynomial rather than floating point, and the independent mpmath implementation (no shared code) agrees; both also confirm the paper's own bounds still hold at the witness, so the counterexample refutes the remark's hope without touching Lemma 1. The mechanism note - the domain centre and numerical-range centre must separate - matches what the certificate shows. No human peer review; the tier records this site's own reproduction.", "sources": [ { "label": "VibeMathed: Lorist-Schwenninger Remark 2 positivity question", "url": "https://vibemathed.com/problem/lorist-schwenninger-remark-2-positivity-question", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Lorist-Schwenninger, \"A solution to Crouzeix's conjecture\"", "url": "https://arxiv.org/abs/2608.03841", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Lorist-Schwenninger, \"A solution to Crouzeix's conjecture\"", "url": "https://arxiv.org/abs/2608.03841", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Lorist-Schwenninger Remark 2 positivity question", "url": "https://vibemathed.com/problem/lorist-schwenninger-remark-2-positivity-question", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:lorist-schwenninger-remark-2-positivity-question:signal:watch_only", "problem_id": "vibemathed:lorist-schwenninger-remark-2-positivity-question", "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" }