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

Theorem 1.1 (assuming CH): there is an AEC $K$ in a countable finitary relational language with $LS(K)=\aleph_0$ , at least two nonisomorphic models of cardinality $\beth_{\omega_2}=H(K)$ , and categorical in every cardinal $\ge\Lambda=\beth_{(2^{\aleph_1})^+}$ . So categoricity at $\Lambda^+$ does not transfer down to $H(K)$ , 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 $\beth_{\omega_1}$ threshold for $L_{\omega_1,\omega}$ .

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: contested. 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: The prescribed Hanf threshold in Shelah's categoricity conjecture is not provable in ZFC · Prescribed categoricity threshold fails under CH

Confidence: Not scored

Registry verification: contested · announcement · partial

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

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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 $LS(K)=\aleph_0$ , two nonisomorphic models of size $\beth_{\omega_2}=H(K)$ , and categoricity in every $\mu\ge\beth_{(2^{\aleph_1})^+}$ ; 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.