Bi-Lipschitz charts at every regular point of a noncollapsed RCD(K,n) space
Let be a noncollapsed space, the synthetic setting containing noncollapsed Ricci limit spaces. A point is regular if every pointed tangent there is . Mondino-Naber proved bi-Lipschitz rectifiability on measurable pieces and Kapovitch-Mondino gave an open manifold region with bi-Holder charts; Cheeger and Colding (1997) had asked whether bi-Holder parametrizations of quantitatively regular Ricci-limit neighborhoods can be upgraded to bi-Lipschitz. The regular-point conjecture, recalled by Honda and Zhang (2026), asks: does every regular point of a noncollapsed space have an open neighborhood, with the restricted distance, bi-Lipschitz homeomorphic to an open subset of ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Metric measure spaces; synthetic lower Ricci bounds; RCD spaces
- Posed by
- Conjecture recalled by Shouhei Honda and Ruobing Zhang (arXiv:2608.16021, 2026), Section 1.2; antecedent question of Jeff Cheeger and Tobias Colding, J. Differential Geom. 46 (1997), Remark 5.15
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 24 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims that for every integer there is such that every regular point of every noncollapsed space, , has an open neighborhood -bi-Lipschitz homeomorphic, with the ambient distance, to an open subset of ; the neighborhood depends on the space and the point. Corollary: an open set of full measure containing all regular points carries a countable bi-Lipschitz atlas with transition maps. It does NOT claim that the regular set is open, any collapsed or non-RCD statement, or regularity of the metric tensor in the charts.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. No Lean formalization of this result is in the release.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as the manuscript states it; Honda-Zhang was not opened here. Scope limits the paper states: noncollapsed means the reference measure is exactly ; regularity means every tangent at the point is Euclidean; the chart is an open neighborhood with the restricted ambient distance; no Holder control of the metric tensor and no bounded-Laplacian coordinates are claimed (Colding-Naber examples show the former can fail). Note the release's family summary says the constant is point dependent, but the manuscript's Theorem 1.1 and abstract say the constant depends only on . No Lean formalization exists for this family.