# Metric surface realization research library Family 333: Smooth isometric immersions of surfaces into ℝ4. First application triage, 9 October 2026 Australia/Brisbane. ## Problem and potential new use Geometry researchers can organize four-dimensional smooth isometric immersions for arbitrary closed surface metrics. ## Applicability and commercial boundary An immersion may self-intersect and is different from an embedding. The existence statement supplies no practical 3D fabrication or 4D numerical construction. ## 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 Inspect the constructive proof and separately certify injectivity if an application needs an embedding. ## Evidence scope The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scope notes were also inspected; family 354’s explicit main profile formula was read. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json). [Smooth isometric immersions of closed surfaces into Euclidean four-space](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf).