A smooth counterexample to the Trautman conjecture
A strongly pseudoconvex CR three-manifold is constructed that carries a nowhere-zero closed section of its canonical bundle yet is not locally embeddable, disproving the Trautman conjecture.
- Model
- ChatGPT Plus, version not named
- Field
- Mathematics
- Date
- 2026-09-02
- Human collaborators
- Sean N. Curry
- Problem posed
- 1998 · open 28 yrs
Sources
Original work
Independent commentary
What was found
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 complex two-space. The paper modifies a standard construction of nonembeddable smooth strongly pseudoconvex CR three-manifolds so that the closed-section condition is preserved, producing a strongly pseudoconvex counterexample. The construction proceeds through Hamiltonian perturbations of the Heisenberg CR structure, bump functions on shrinking solid tori and weighted moment identities, and is then globalized to the three-sphere.
Novelty check
The conjecture is attributed to Andrzej Trautman and dated 1998, and the paper's introduction sets out its origins and the standard nonembeddability constructions it modifies. No prior counterexample was found in the four registries or in the CR geometry literature located; the point of the paper is that the existing nonembeddable examples do not preserve the closed canonical section, which is exactly what the modification supplies. The result is new.
Caveats and known objections
A single-author unrefereed preprint days old at entry, labelled by its author a preliminary version, with no independent check and no formalization. The AI disclosure sits in the acknowledgements rather than the abstract, which is the pattern worth noting: a reader of the abstract alone would not know a model was involved. Autonomy is graded ai-assisted on the strictest defensible reading. The acknowledgements say 'The counterexample construction presented below was discovered through experimentation using ChatGPT Plus on August 18, 2026' and that ChatGPT also drafted the manuscript and helped proofread, but the framing is the author's experimentation, the problem was posed to him by Pawel Nurowski and Rod Gover, and he states he 'independently checked all calculations and arguments presented here and takes full responsibility'. A case for ai-led could be made from the word discovered; the weaker grade wins.
Also recorded at
vibematheda-smooth-counterexample-to-the-trautman-conjecture
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
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Graded author verified for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). A smooth counterexample to the Trautman conjecture. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-09-02-trautman-conjectureBibTeX
@misc{whataifound-oklahomastateuniversity-2026-conjecture,
title = {A smooth counterexample to the Trautman conjecture},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Oklahoma State University. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-09-02-trautman-conjecture}
}