OpenAI · 357-01 · 2026-09-25 · Differential geometry

Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026

Family 357. Bi-Lipschitz coordinates at every regular RCD point.

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

Proves that every regular point of a noncollapsed $RCD (K, n)$ space, for $K \in R$ and integer $n \geq 2$ , has an open neighborhood bi-Lipschitz to an open subset of $R^{n}$ . Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly $H^{n}$ , and the chart compares ambient distances with a point-dependent finite constant.

We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed $RCD (K, n)$ space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝn. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.

Preprint source and Lean scope

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