{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:mathbb-c-infty-caratheodory-conjecture", "title": "The $C^\\infty$ Carathéodory Conjecture on Umbilic Points", "canonical_statement": "Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in $\\mathbb{R}^3$ has at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and it stands. The $C^\\infty$ case is false: an explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point. The same family disproves the smooth Loewner conjecture, whose member at $k=1$ has an isolated trace-free Hessian zero of winding number three.", "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:mathbb-c-infty-caratheodory-conjecture:event", "problem_id": "vibemathed:mathbb-c-infty-caratheodory-conjecture", "type": "disproved", "status": "candidate", "occurred_at": "2026-08-19T00:00:00.000Z", "title": "Levent Alpöge, X announcement of the smooth counterexample", "summary": "Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.", "ai_contribution": "ai_assisted", "attempt_ids": [ "vibemathed:mathbb-c-infty-caratheodory-conjecture:attempt" ], "method_family_ids": [ "vibemathed:mathbb-c-infty-caratheodory-conjecture:method" ], "source_assertion_ids": [ "vibemathed:mathbb-c-infty-caratheodory-conjecture:assertion" ] } ], "attempts": [ { "id": "vibemathed:mathbb-c-infty-caratheodory-conjecture:attempt", "problem_id": "vibemathed:mathbb-c-infty-caratheodory-conjecture", "model": "Claude, Codex", "provider": "Anthropic, OpenAI", "attempted_at": "2026-08-19T00:00:00.000Z", "human_collaborators": [ "Levent Alpöge", "John-Paul Smith" ], "exposure": "unknown", "prompt_public": null, "outcome": "Two distinct roles, neither of them the discovery. The formalization's module docstring records that Alpöge's announcement \"credits John-Paul Smith and Claude with checking the construction\", so the model's part was verifying a human construction. Separately, the Lean development was, in its author's words, \"developed with Codex and parallel proof-review agents\" - a formalization of a human result, which the methodology does not count as the contribution. Recorded as assisted rather than co-developed for that reason; the submission proposed co-developed.", "failed_routes": [], "artifacts": [ { "label": "Lean proof, sorry-free at the pinned commit", "url": "https://github.com/google-deepmind/formal-conjectures/blob/7aa855bb344450777d9b19fe1cf11f2f5f9fae09/FormalConjectures/Other/CaratheodoryLoewnerCounterexample.lean", "kind": "lean-proof", "license": null }, { "label": "The informal proof accompanying the formalization", "url": "https://github.com/google-deepmind/formal-conjectures/blob/7aa855bb344450777d9b19fe1cf11f2f5f9fae09/FormalConjectures/Other/CaratheodoryLoewnerCounterexample.md", "kind": "paper", "license": null }, { "label": "Lean statement of the smooth and analytic conjectures", "url": "https://github.com/google-deepmind/formal-conjectures/pull/5066", "kind": "lean-statement", "license": null }, { "label": "Ghomi, Open Problems in Geometry of Curves and Surfaces (Problem 8.1)", "url": "https://ghomi.math.gatech.edu/Papers/op.pdf", "kind": "problem-record", "license": null } ], "sources": [ { "label": "Levent Alpöge, X announcement of the smooth counterexample", "url": "https://x.com/__alpoge__/status/2089971359921156203", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The $C^\\infty$ Carathéodory Conjecture on Umbilic Points", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:mathbb-c-infty-caratheodory-conjecture:verification", "problem_id": "vibemathed:mathbb-c-infty-caratheodory-conjecture", "solution_event_id": "vibemathed:mathbb-c-infty-caratheodory-conjecture:event", "level": "lean_checked_statement_unaudited", "mathematical_correctness": "supported", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Audited by this site on 21 August 2026 at the commit the formal_proof attribute pins (7aa855b, google-deepmind/formal-conjectures). The formalized hypothesis is the classical statement and not a weakened one: IsConvexSphereOfClass requires Topology.IsEmbedding together with range F = frontier K for a compact convex K of nonempty interior, so \"parametrized\" names the Gauss parametrization rather than admitting mere immersions. The pinned line is not_caratheodoryConjectureOfClass_infty, the smooth statement, and the proof tree is 8013 lines carrying zero sorry, zero declared axioms and no native_decide. Not lean-verified, because that commit is NOT merged - it is diverged from main by 11 commits and behind by 26, and both upstream pull requests are drafts, #5070 saying \"I'm currently checking this ... please ignore\". The Lean was read here, not compiled, and the announcement itself is an X post.", "sources": [ { "label": "VibeMathed: The $C^\\infty$ Carathéodory Conjecture on Umbilic Points", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Levent Alpöge, X announcement of the smooth counterexample", "url": "https://x.com/__alpoge__/status/2089971359921156203", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Levent Alpöge, X announcement of the smooth counterexample", "url": "https://x.com/__alpoge__/status/2089971359921156203", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The $C^\\infty$ Carathéodory Conjecture on Umbilic Points", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:mathbb-c-infty-caratheodory-conjecture:signal:verify_now", "problem_id": "vibemathed:mathbb-c-infty-caratheodory-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" }