Szabo's Conjecture A on equivariant Jiang-Su stability, in the unital stably finite case
Let be a separable simple nuclear C*-algebra with and let be an action of a countable discrete amenable group on . Szabo's Conjecture A predicts automatic equivariant Jiang-Su stability: should be cocycle conjugate to . Szabo proved it for finite algebras with finitely many rays of extremal traces and very weak comparison; the general case was open, in particular when the action moves traces. Is every amenable action on a separable simple nuclear -stable C*-algebra equivariantly -stable?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Operator algebras; C*-dynamics of amenable group actions
- Posed by
- Gabor Szabo
- Year posed
- 2021
- Years open
- 5y
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for simple, separable, unital, infinite-dimensional, nuclear, stably finite and -stable, every action of a countable discrete amenable group is cocycle conjugate to , with no restriction on how acts on the trace simplex, and including finite groups and trivial actions. It does not treat nonunital algebras, and stably infinite (purely infinite) algebras are not part of this theorem.
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 'Equivariant Jiang-Su stability for amenable actions in the unital stably finite case' (October 5, 2026), one of four manuscripts in the family; it uses Szabo-Wouters' characterisation by equivariant uniform property Gamma for the final absorption step.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Conjecture A. It proves the conclusion for every action of a countable discrete amenable group on a simple separable unital infinite-dimensional nuclear stably finite -stable algebra, with no condition on traces or outerness. The manuscript itself says nonunital algebras, which Conjecture A includes, are beyond its scope; purely infinite algebras are also not covered by this theorem. The proof was not refereed and is not formalised.