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A Smooth Counterexample to the Trautman Conjecture

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 $T^{0,1}_\phi=\operatorname{span}_{\mathbb C}\{L_\phi\}$ is 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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 2, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A Smooth Counterexample to the Trautman Conjecture · Trautman Conjecture

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

ChatGPT (model not named; the paper says ChatGPT Plus)
model · ai model contributor · OpenAI

Sean N. Curry
human · human collaborator

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Named evidence

VibeMathed record: A Smooth Counterexample to the Trautman Conjecture

specialist registry · passed

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.

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A Smooth Counterexample to the Trautman Conjecture

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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This event attributed to Sean N. Curry

A Smooth Counterexample to the Trautman Conjecture parent of this event

This event attributed to ChatGPT (model not named; the paper says ChatGPT Plus)

VibeMathed record: A Smooth Counterexample to the Trautman Conjecture evidence for this event

A Smooth Counterexample to the Trautman Conjecture evidence for this event

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