OpenAI · 333-01 · 2026-09-23 · Differential geometry
Smooth isometric immersions of closed surfaces into Euclidean four-space
preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026
Family 333. Smooth isometric immersions of surfaces into .
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Every closed smooth Riemannian surface admits a smooth isometric immersion into , resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.
Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 333: Smooth isometric immersions of surfaces into ℝ<sup>4</sup>. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.