Shelah's eventual categoricity conjecture for abstract elementary classes

Theorem 1.2: for every infinite cardinal $\lambda$ there is $\mu(\lambda)\ge\lambda$ such that every AEC $K$ with $LS(K)\le\lambda$ categorical in some $\kappa\ge\mu(\lambda)$ is categorical in every $\kappa'\ge\mu(\lambda)$ ; equivalently (Theorem 1.3) an AEC categorical in unboundedly many cardinals is categorical on a tail. It applies to $L_{\omega_1,\omega}$ sentences in countable languages. It gives no explicit value for $\mu(\lambda)$ ; the prescribed $\beth_{\omega_1}$ threshold for $L_{\omega_1,\omega}$ and the Hanf threshold for AECs are left separate (the companion refutes the latter under CH).

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Shelah's eventual categoricity conjecture for abstract elementary classes · Shelah's eventual categoricity conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Shelah's eventual categoricity conjecture for abstract elementary classes

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.2 was read against the conjecture. It gives, for every infinite $\lambda$ , a cardinal $\mu(\lambda)$ such that every AEC with $LS(K)\le\lambda$ categorical in one cardinal $\ge\mu(\lambda)$ is categorical in all cardinals $\ge\mu(\lambda)$ , in ZFC, with no amalgamation, joint embedding, tameness or no-maximal-models hypothesis and any starting cardinal. This is the qualitative conjecture. It does not give a numerical threshold. The long proof (order presentations, working schemes, geometry built from unbounded categoricity) was not refereed and is not formalised; the family's Lean formalization covers only the CH companion.

Source ↗

Companion: Eventual categoricity for abstract elementary classes

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

The prescribed Hanf threshold in Shelah's categoricity conjecture is not provable in ZFC

witness · pending

paper link supplied by the VibeMathed registry.

Source ↗

Boney and Vasey, Categoricity and infinitary logics (2015)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

VibeMathed record: Shelah's eventual categoricity conjecture for abstract elementary classes evidence for this event

The prescribed Hanf threshold in Shelah's categoricity conjecture is not provable in ZFC evidence for this event

Boney and Vasey, Categoricity and infinitary logics (2015) evidence for this event

Shelah's eventual categoricity conjecture for abstract elementary classes parent of this event

Companion: Eventual categoricity for abstract elementary classes evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph