Bi-Lipschitz coordinates at every regular RCD point
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves that every regular point of a noncollapsed \mathop{RCD}\nolimits (K,n) space, for K ∈ ℝ and integer n ≥ 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 \mathcal H^n, and the chart compares ambient distances with a point-dependent finite constant.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces
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 ℝ^(n). The bi-Lipschitz constant depends only on n, and the neighborhood uses…
Full source abstract
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 ℝ^(n). The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.