Schafhauser-Tikuisis-White Problem XXI: tracial real rank zero implies uniform property Gamma
Let be a unital separable simple infinite-dimensional nuclear stably finite C*-algebra with traces, its uniform tracial completion, and the uniform tracial ultrapower along a free ultrafilter. Uniform property asks for central projections that halve every limit trace on every coefficient. Schafhauser, Tikuisis and White asked (Problem XXI, in their completion-and-halving formulation) whether real rank zero of forces uniform property . Does real rank zero of the uniform tracial ultrapower imply uniform property ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Operator algebras; uniform tracial completions
- Posed by
- Christopher Schafhauser, Aaron Tikuisis and Stuart White
- Year posed
- 2025
- Years open
- 1y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for unital, separable, simple, infinite-dimensional, nuclear and stably finite with , if has real rank zero then there is a projection with for all and all limit traces , so has uniform property ; it also identifies all traces on . It does not show that real rank zero of holds in general, and Toms (2026) has shown nuclearity alone does not give uniform property .
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal is 'Tracial projection methods and uniform property Gamma' (September 23, 2026), one of four manuscripts in the family; the same manuscript also gives the unital Toms-Winter route recorded in the Toms-Winter entry.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Problem XXI and answers it positively in the STW formulation. lean/formalization.yaml lists comparator UniformGamma, declaration OAI.ComparatorModel.CurrentMain.main. The statement was read here: for every separable, topologically simple, infinite-dimensional, nuclear (unique C*-norm on tensor products), stably finite unital C*-algebra with a tracial state, and every free ultrafilter at which the family ultrapower of the uniform tracial completion has real rank zero, there is a central projection halving every limit trace on every element of the completion. The claim is wrapped in existential quantifiers over five auxiliary construction classes (norms on quotients, null ideals, Cauchy algebras, limit traces), which the proof must supply. It states the headline of Problem XXI; the strict-comparison results of the same paper are not in this statement. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. Listed as Unreviewed rather than Lean-checked because its formal statement takes the key constructions as supplied hypotheses.