---
title: "Bi-Lipschitz coordinates at every regular RCD point"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 357; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "differential-geometry"
  - "unformalized"
seed_rank: 2212
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces"
    url: "https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Bi-Lipschitz coordinates at every regular RCD point

## One-sentence takeaway

OpenAI's result family 357 (Differential geometry) claims: Proves that every regular point of a noncollapsed $\mathop{\mathrm{RCD}}\nolimits (K,n)$ space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn.

## Why it matters here

Differential geometry is the deep background for geometry processing, curvature flows and mesh work in graphics. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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.
- *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 $\mathrm{RCD}(K,n)$ 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces](https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math