# Surface realization limitation audit Family 334: A smooth surface metric with no local isometric immersion in ℝ3. First application triage, 9 October 2026 Australia/Brisbane. ## Problem and potential new use Geometry and CAD researchers can flag a universal promise to realize every smooth local surface metric isometrically in three dimensions. ## Applicability and commercial boundary The special smooth counterexample does not rule out approximate realization, finite meshes or useful realizations of restricted metrics. ## Initial business decision Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product. ## Next verification action Extract an explicit computable metric and a numerical illustration with approximation limits labeled. ## Evidence scope The catalog statement was individually reviewed. The manuscript proof and formal-scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json). [A Smooth Metric with No Local Isometric Immersion into Three-Space](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf).