View source for 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 ([https://github.com/openai/math 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, [https://arxiv.org/abs/2608.16021v1 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''<sub>''n''</sub> ≥ 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 ''H''<sup>''n''</sup>. Then every point of ''X'' at which all pointed Gromov–Hausdorff tangents are isometric to Euclidean '''R'''<sup>''n''</sup> based at the origin has an open neighbourhood ''U'' and a homeomorphism from ''U'' onto an open subset of '''R'''<sup>''n''</sup> distorting the restricted distance by at most the factor ''L''<sub>''n''</sub>.
** The manuscript's standing hypotheses: the full CD(''K'', ''n'') condition with quadratic Cheeger energy; complete, separable, geodesic spaces whose reference measure is exactly ''H''<sup>''n''</sup>, of full support and finite on bounded sets.
** Corollary 1.2 gives an open region of full ''H''<sup>''n''</sup>-measure containing every regular point, with a countable atlas of ''L''<sub>''n''</sub>-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 <code>1581d4c</code> of the openai/math repository: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026 Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces].
* '''Formalization.''' The collection has no Lean scope note (<code>lean/docs/</code>) or Comparator challenge for this manuscript.
** The repository's Lean library contains a theorem <code>every_regular_point_has_bilipschitz_coordinates</code> in [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/OAI/Geometry/RCD/Main.lean lean/OAI/Geometry/RCD/Main.lean]. Its hypotheses are ''n'' ≥ 2, a structure <code>NoncollapsedRCD K n X</code> (defined in [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/OAI/Analysis/RCDHeat/Foundations.lean Foundations.lean] by the standing hypotheses above), and regularity of the point. Its conclusion is an open bi-Lipschitz chart into '''R'''<sup>''n''</sup> whose constant ''L'' ≥ 1 may depend on the space and the point. The dimension-only constant ''L''<sub>''n''</sub> and Corollary 1.2 are not part of that statement.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 17 October 2026)
|-
| colspan="5" | 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.

[[Category:OpenAI results]]