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The Phillips-Toms conjecture for minimal homeomorphisms: radius of comparison equals half the mean dimension

For a minimal homeomorphism hh of an infinite compact metrizable space XX, the crossed product C(X)⋊hZC(X)\rtimes_h\mathbb Z is a simple unital nuclear C*-algebra. Toms's radius of comparison rc(A)\mathrm{rc}(A) is the least rr such that dτ(a)+r<dτ(b)d_\tau(a)+r<d_\tau(b) for every trace τ\tau forces Cuntz subequivalence a≾ba\precsim b; the Lindenstrauss-Weiss mean dimension mdim(X,h)\mathrm{mdim}(X,h) measures dimension per orbit coordinate. Giol and Kerr built minimal systems of positive mean dimension with large radius of comparison; Phillips proved rc≤1+36 mdim\mathrm{rc}\le1+36\,\mathrm{mdim}, Elliott and Niu settled zero mean dimension, and Niu proved rc≤12mdim\mathrm{rc}\le\frac12\mathrm{mdim}. The Phillips-Toms conjecture, recorded for integer actions by Elliott-Niu and as Problem XXXVI of Schafhauser-Tikuisis-White, predicts equality. Is rc(C(X)⋊hZ)=12mdim(X,h)\mathrm{rc}(C(X)\rtimes_h\mathbb Z)=\frac12\mathrm{mdim}(X,h) for every such system, including the value ∞\infty?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Operator algebras; C*-dynamics and mean dimension
Posed by
N. Christopher Phillips and Andrew Toms; integer-action form recorded by Elliott and Niu (Duke 2017) and as Problem XXXVI of Schafhauser-Tikuisis-White
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
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every minimal homeomorphism hh of an infinite compact metrizable XX, rc(C(X)⋊hZ)=12mdim(X,h)\mathrm{rc}(C(X)\rtimes_h\mathbb Z)=\frac12\mathrm{mdim}(X,h) in [0,∞][0,\infty]. The new part is mdim≤2 rc\mathrm{mdim}\le2\,\mathrm{rc}, obtained by turning comparison witnesses into bundle data and compressing cube-valued maps via the companion's cobordism theorem, for every finite open cover and with no finite-dimensionality hypothesis on XX. Corollary 1.2: zero mean dimension, the small boundary property, Jiang-Su stability, finite nuclear dimension and nuclear dimension at most one are equivalent for these crossed products. It does not treat Zd\mathbb Z^d or general amenable group actions (Niu's broader formulation).

What the AI did

The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript proves the missing lower bound; a companion of the same date (filtered products and boundary-preserving compression in complex cobordism) supplies the compression theorem it uses.

Verification

No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the conjecture as recorded by Elliott-Niu and Schafhauser-Tikuisis-White. No Lean formalization exists for this family. The upper bound rc <= mdim/2 is Niu's published theorem, and Niu's open tower theorem is a second input; the new part is the lower bound, which depends on the companion's MU-based compression theorem. The regularity corollary also uses published Elliott-Niu, Rordam and Castillejos-Evington-Tikuisis-White-Winter theorems. The paper limits itself to integer actions. Not refereed here.

Sources

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