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Math-2242-A1 · revision r1 · on Math-2242

OpenAI claim summary for Math-2242

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The eventual-categoricity paper supplies a uniform sufficiently-large threshold but does not identify it with the particular Hanf or beth threshold in this statement. Transfer at cardinals between that specified bound and the larger threshold is consequently not supplied.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Shelah's categoricity conjecture for \$L_{\textbackslash{}omega_1,\textbackslash{}omega}\$
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Shelah's categoricity conjecture for \$L_{
\textbackslash{}
omega_1,
\textbackslash{}
omega}\$

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 240; 2026-09-24), the asserted result is that in ZFC, each bound on Löwenheim--Skolem number has a uniform threshold above which categoricity in one cardinal implies categoricity throughout the same tail.

Mathematical statement of the claimed result

Manuscript 1.
For every infinite cardinal $\lambda$, the claim gives a uniform cardinal threshold $\mu(\lambda)\ge\lambda$ for all abstract elementary classes $K$ of Löwenheim--Skolem number at most $\lambda$: \[ (\exists\kappa\ge\mu(\lambda):I(K,\kappa)=1) \Longrightarrow(\forall\kappa'\ge\mu(\lambda):I(K,\kappa')=1). \] Here $I(K,\kappa)$ is the number of model isomorphism classes of cardinality $\kappa$. The same threshold occurs in the premise and conclusion. A second formulation asserts that categoricity in unboundedly many cardinals forces categoricity on a tail, without adding amalgamation or tameness assumptions.

Outline of the argument

For manuscript 1 (family 240; 2026-09-24), fixed order-indexed presentations encode short model orbits. A direct reversal theorem for strong maps preserves bounded tuple records; elimination and stabilization of strong-map classes then create the stable diagrams and closed-level amalgamation needed for tail transfer.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 240; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources