The Trautman conjecture (Trautman 1999) asserts that a smooth three-dimensional CR manifold with a nowhere-zero closed section of its
canonical bundle is locally CR embeddable in 
. Here the canonical bundle consists of complex 2-forms annihilated
by contraction with the CR line 
.
The hypothesis is necessary for an embedding: the form 
pulls back to such a section. The conjecture
asks whether this single closed form also supplies enough local CR functions to build
an embedding.
Curry (2026) reported a smooth counterexample, preserving the closed canonical form while modifying a standard nonembeddable CR structure. The example is strongly pseudoconvex, so even this positivity condition does not rescue the proposed implication. ChatGPT helped construct the argument, which Curry developed and checked. Independent external verification had not been reported as of Sep. 7, 2026.
