Remark: Family 334. A smooth surface metric with no local immersion in R 3

Constructs a smooth positive-definite metric on $(-1,1{)}^2$ for which no neighborhood of the origin admits a smooth isometric immersion into ${\mathbb{R}}^3$. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.