The Phillips–Toms conjecture
From OpenAI Aftermath
The Phillips–Toms conjecture predicts that the radius of comparison of the crossed product of a minimal dynamical system equals one half of its mean dimension, linking an invariant of C*-algebra regularity with a dimension invariant of topological dynamics. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), both dated 25 September 2026, claim the integer-action case for every minimal homeomorphism of an infinite compact metrizable space, including infinite values, and the topological compression theorem in complex cobordism that the proof uses. The family has no Lean formalization. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the main manuscript:
- Mean dimension was introduced by Gromov and developed by Lindenstrauss and Weiss, and Toms introduced the radius of comparison. Giol and Kerr constructed minimal systems of positive mean dimension whose crossed products have arbitrarily large radius of comparison. The integer-action conjecture is recorded by Elliott and Niu and as Problem XXXVI of Schafhauser, Tikuisis and White; Niu states a broader formulation for free minimal actions of amenable groups.
- For upper bounds, Phillips proved a bound of 1 + 36 times the mean dimension, Elliott and Niu settled the case of mean dimension zero, and Niu proved the conjectured upper bound of one half of the mean dimension. Hirshberg and Phillips obtained lower bounds under a cohomological independence condition, and Phillips bounded the radius of comparison of C(Y) below in terms of the covering dimension of Y. The manuscript describes the lower bound as the remaining problem.
The original results
- Claim. Theorem 1.1 of Radius of comparison equals half the mean dimension states that for every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of C(X) ⋊h Z equals one half of the mean dimension of (X, h), in [0, ∞]. The manuscript also states that zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension, with nuclear dimension at most one in this case. Theorem 1.1 of Filtered products and boundary-preserving compression in complex cobordism states that if a based map from a finite pointed CW complex to a sphere is null in complex cobordism on a subcomplex, then for every large ℓ its ℓ-th smash power is stably null on the union of products in which at least half of the factors lie in the subcomplex; the manuscript derives boundary-preserving compression of maps into products of cubes.
- Method, as described: the main manuscript proves the lower bound by converting comparison of positive elements in the crossed product into bundle information and then into a reduction in the covering order of orbit covers, applying to every finite open cover without a cohomological or finite-dimensionality hypothesis; the cobordism manuscript supplies the compression step, and Niu's open tower theorem and upper bound are inputs.
- Sources. The manuscripts Radius of comparison equals half the mean dimension and Filtered products and boundary-preserving compression in complex cobordism at commit
1581d4c. - Formalization. The repository has no Lean scope note for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | G. A. Elliott and Z. Niu, Mean dimension and radius of comparison, arXiv:2610.11143v1 | Extension | For a minimal free action of a discrete amenable group with the uniform Rokhlin property and Cuntz comparison of open sets, proves that the radius of comparison of the crossed product equals one half of the mean dimension, and the same for unital simple AH algebras with diagonal maps. The abstract states that the arguments are based on the OpenAI preprint on integer actions and its topological ideas. | arXiv preprint, version 1 of 8 October 2026. The arXiv record states that the work was carried out using ChatGPT. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews of the main manuscript and the cobordism manuscript in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Both reviews report no gap and state that all sections were read. The review of the main manuscript has no minor notes and describes the lower bound as new and the upper bound as Niu's theorem. The review of the cobordism manuscript has one minor note: everything depends on a cited quantitative nilpotence input, a uniform sublinear vanishing curve for the complex-cobordism Adams filtration from Mathew and from Burklund, Hahn and Senger, which the reviewer did not check line by line, while finding its reformulation and use internally correct. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
