The prescribed Hanf threshold in Shelah's categoricity conjecture is not provable in ZFC
For an abstract elementary class put , the Hanf number for existence of arbitrarily large models. Beside the qualitative eventual categoricity conjecture, Shelah's program states a prescribed-threshold form: if is categorical in some , then is categorical in every . Under amalgamation and tameness this transfer is known (Vasey), and a transfer at this bound for arbitrary AECs has been claimed (Espindola 2023). Grossberg distinguishes the prescribed and unspecified-threshold forms, and Saroch-Trlifaj (2024) give the exact formulation. Does categoricity above always transfer to every cardinal at least , for arbitrary AECs?
- Result
- Disproved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Model theory; abstract elementary classes
- Posed by
- Saharon Shelah (threshold form; formulated by Grossberg and by Saroch and Trlifaj)
- Year posed
- 2009
- Years open
- 17y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Contested
- 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 (assuming CH): there is an AEC in a countable finitary relational language with , at least two nonisomorphic models of cardinality , and categorical in every cardinal . So categoricity at does not transfer down to , and the prescribed-threshold form is not a theorem of ZFC (if ZFC is consistent). The same class is eventually categorical, so this is compatible with the qualitative conjecture. It does not decide whether the prescribed form can hold under other hypotheses (for example failure of CH), and it does not concern the threshold for .
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. The family has two manuscripts dated September 24, 2026: this CH construction and the ZFC eventual categoricity theorem (its own entry). This manuscript is the formalised one.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the threshold form. Under CH it builds an AEC in a countable relational language with , two nonisomorphic models of size , and categoricity in every ; with Godel's consistency of CH the prescribed form is unprovable in ZFC if ZFC is consistent. It does not show the prescribed form false in ZFC. lean/formalization.yaml lists comparator CHObstruction, declaration OAI.CHObstruction.main; the statement was read here and asserts exactly Theorem 1.1, with the AEC axioms (coherence, chain unions over ordinals, LS bound) spelled out over ZFSet carriers. The nonprovability corollary is not formalised. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript says its construction contradicts Espindola's published claim (Theorem 4.1) of transfer at this bound. Listed as Contested because of the conflicting claim described in the claim issue.
Claim issue
This result conflicts with a published theorem. The release's manuscript states that its construction contradicts Espindola's published Theorem 4.1, which claims categoricity transfer at this bound. Until the conflict is settled in public, the entry is Contested.