Bi-Lipschitz charts at regular points of RCD spaces

Bi-Lipschitz charts at regular points of RCD spaces concern the local structure of metric measure spaces with Ricci curvature bounded below in a synthetic sense. A point of such a space is regular if every tangent cone there is Euclidean. The question is whether every regular point then has a neighbourhood bi-Lipschitz equivalent to an open subset of Euclidean space. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 25 September 2026, claims an affirmative answer for noncollapsed RCD(K, n) spaces. This page summarizes the published claim and lists publicly available follow-up work.

Background

• As recounted in the manuscript's introduction:
  • Ricci limit spaces. Cheeger and Colding obtained manifold neighbourhoods with bi-Hölder parametrizations near quantitatively regular points of noncollapsed Ricci limit spaces, and asked about a bi-Lipschitz upgrade in Theorem 5.14 and Remark 5.15 of their 1997 paper in the Journal of Differential Geometry.
  • Synthetic lower Ricci bounds. Lott–Villani and Sturm introduced curvature-dimension conditions through optimal transport; Ambrosio, Gigli and Savaré added the Riemannian condition of a quadratic Cheeger energy. In the noncollapsed setting, where the reference measure is n-dimensional Hausdorff measure, De Philippis and Gigli proved volume convergence and rigidity in a 2018 paper in the Journal de l'École polytechnique.
  • Known structure. Mondino and Naber proved countable bi-Lipschitz rectifiability by measurable pieces in a 2019 paper in the Journal of the European Mathematical Society, and Kapovitch and Mondino placed the regular set inside an open manifold region with bi-Hölder charts in a 2021 paper in Geometry & Topology. Colding and Naber gave bi-Lipschitz embeddings of uniform Reifenberg regions into possibly higher-dimensional Euclidean space, and examples without Hölder-continuous metric tensors in a 2013 paper in Advances in Mathematics.
  • The conjecture. The manuscript states that the conjecture that every regular point in the noncollapsed setting has a bi-Lipschitz chart is recalled in Section 1.2 of Honda and Zhang, arXiv:2608.16021v1, whose regularity theorems assume additionally a positive lower bound on the injectivity radius.

The original result

• Bi-Lipschitz coordinates (25 September 2026). Theorem 1.1 of the manuscript Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces states: for every integer n ≥ 2 there is a constant L_n ≥ 1 with the following property. Let K be real and let X be a noncollapsed RCD space with lower Ricci bound K, dimension n and reference measure Hⁿ. Then every point of X at which all pointed Gromov–Hausdorff tangents are isometric to Euclidean Rⁿ based at the origin has an open neighbourhood U and a homeomorphism from U onto an open subset of Rⁿ distorting the restricted distance by at most the factor L_n.
  • The manuscript's standing hypotheses: the full CD(K, n) condition with quadratic Cheeger energy; complete, separable, geodesic spaces whose reference measure is exactly Hⁿ, of full support and finite on bounded sets.
  • Corollary 1.2 gives an open region of full Hⁿ-measure containing every regular point, with a countable atlas of L_n-bi-Lipschitz charts.
  • Method, as described: harmonic approximations of Euclidean coordinates, accumulated Bochner mass to compensate for the degeneration of their Gram matrices across scales, and a directional pinching estimate keeping the metric error summable.
• Source. Commit 1581d4c of the openai/math repository: the manuscript Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces.
• Formalization. The collection has no Lean scope note (lean/docs/) or Comparator challenge for this manuscript.
  • The repository's Lean library contains a theorem every_regular_point_has_bilipschitz_coordinates in lean/OAI/Geometry/RCD/Main.lean. Its hypotheses are n ≥ 2, a structure NoncollapsedRCD K n X (defined in Foundations.lean by the standing hypotheses above), and regularity of the point. Its conclusion is an open bi-Lipschitz chart into Rⁿ whose constant L ≥ 1 may depend on the space and the point. The dimension-only constant L_n and Corollary 1.2 are not part of that statement.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
No public follow-up work has been found.

Notes

• The library theorem above is not among the Comparator challenges of the collection, so the collection-wide kernel checks of the Comparator challenges do not cover it.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.