{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture", "title": "A Smooth Counterexample to the Trautman Conjecture", "canonical_statement": "The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\\mathbb{C}^2$. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.", "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:a-smooth-counterexample-to-the-trautman-conjecture:event", "problem_id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture", "type": "disproved", "status": "resolved", "occurred_at": "2026-09-02T00:00:00.000Z", "title": "arXiv", "summary": "For every neighborhood $U$ of the origin in $\\mathbb{C}\\times\\mathbb{R}$ and every $\\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\\phi$, supported in $U$ with $\\|\\phi\\|_{C^2}<\\varepsilon$, such that\n$$\nT^{0,1}_\\phi=\\operatorname{span}_{\\mathbb C}\\{L_\\phi\\}\n$$\nis strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:attempt" ], "method_family_ids": [ "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:method" ], "source_assertion_ids": [ "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:assertion" ] } ], "attempts": [ { "id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:attempt", "problem_id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture", "model": "ChatGPT (model not named; the paper says ChatGPT Plus)", "provider": "OpenAI", "attempted_at": "2026-09-02T00:00:00.000Z", "human_collaborators": [ "Sean N. Curry" ], "exposure": "unknown", "prompt_public": null, "outcome": "Sean Curry states that the counterexample construction itself was discovered through experimentation with ChatGPT Plus on August 18, 2026. ChatGPT was also used to produce an initial rough draft and later for proofreading. Curry independently checked all calculations and arguments and takes responsibility for the paper.", "failed_routes": [], "artifacts": [], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2609.03198", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: A Smooth Counterexample to the Trautman Conjecture", "url": "https://vibemathed.com/problem/a-smooth-counterexample-to-the-trautman-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:verification", "problem_id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture", "solution_event_id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Unreviewed. A single-author arXiv preprint two days old, labelled by its author \"Preliminary version. Comments welcome.\" The author states he independently checked all calculations and arguments. The construction is a modification of a standard nonembeddable strongly pseudoconvex CR 3-manifold, arranged so that the canonical bundle keeps a nowhere-zero closed section, so it is checkable by anyone who knows the Rosay-type examples; three CR specialists are thanked in the paper but none is on the record as having checked it. No peer review, no formalisation.", "sources": [ { "label": "VibeMathed: A Smooth Counterexample to the Trautman Conjecture", "url": "https://vibemathed.com/problem/a-smooth-counterexample-to-the-trautman-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2609.03198", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2609.03198", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: A Smooth Counterexample to the Trautman Conjecture", "url": "https://vibemathed.com/problem/a-smooth-counterexample-to-the-trautman-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture:signal:watch_only", "problem_id": "vibemathed:a-smooth-counterexample-to-the-trautman-conjecture", "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" }