Carathéodory's conjecture asserts that every closed strictly convex smooth surface in 
has at least two umbilic points (Gutiérrez
and Sánchez-Bringas 1997). It is related to Loewner's
conjecture, a local index bound for fields of principal
directions.
Ando and Umehara (2026) reported a smooth strictly convex closed surface with exactly one umbilic point, of index 2, following an earlier announced counterexample. The construction is smooth and non-real-analytic. Independent verification of the full construction has not been established, so the claimed refutation remains qualified. Ando and Umehara (2026) report AI assistance with searching for candidate functions in the mathematical construction.
