Counterexample to the smooth Carathéodory conjecture on umbilic points
An explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point, so the C-infinity form of Carathéodory's 1922 conjecture is false.
- Lab
- Independent
- Model
- Claude; Codex
- Field
- Mathematics
- Date
- 2026-08-19
- Human collaborators
- Levent Alpöge, John-Paul Smith
- Problem posed
- 1922 · open 104 yrs
Sources
Announcement
Independent commentary
What was found
Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in R^3 carries at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and that theorem is untouched. The counterexample is explicitly a C-infinity object: an explicit support function produces 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 the wrong index. The gap between the smooth and the real-analytic case is the whole content of the result.
Novelty check
Carathéodory's conjecture is a named problem with a long literature and a standing partial theorem, Hamburger's real-analytic case from 1940-41, and it appears as Problem 8.1 in Ghomi's published problem list. The claim is a counterexample in the smooth category only, which is consistent with rather than contradicting Hamburger. No prior smooth counterexample appears in the problem list or in the vibemathed record. The construction is new work, not a retrieval of something already in the literature.
Caveats and known objections
Announced in an X post rather than a preprint, and unrefereed. The Lean formalization is sorry-free at a commit but not merged, and vibemathed records its statement as read rather than compiled and audited, so nothing here is machine-checked end to end for this registry's purposes; the grade stays at claimed. Autonomy is graded ai-assisted rather than higher on a deliberately strict reading of two separate model roles, neither of which is the discovery: Alpöge's announcement credits John-Paul Smith and Claude with checking the construction, which makes the model a verifier of a human construction, and the Lean development was, in its author's words, developed with Codex and parallel proof-review agents, which is formalization of a human result. Only the smooth case falls; the classical real-analytic form of the conjecture remains true.
Also recorded at
vibemathedmathbb-c-infty-caratheodory-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 Claimed and AI-assisted.
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Graded claimed for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexample to the smooth Carathéodory conjecture on umbilic points. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-19-caratheodory-umbilicBibTeX
@misc{whataifound-independent-2026-umbilic,
title = {Counterexample to the smooth Carathéodory conjecture on umbilic points},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Claimed. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-19-caratheodory-umbilic}
}