Equivalence of generic stability notions for Keisler measures
resolvedconfidence 70%
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Precise statement
Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $\mu$ in $T$, we show that the following conditions are equivalent:
$(i)$ $\mu$ is a frequency interpretation measure;
$(ii)$ $\mu$ is definable and its canonical “random extension” $r_{\mu}$ is generically stable in the randomization theory $T^{R}$;
$(iii)$ $\mu$ is “self-averaging”.
This result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\Rightarrow(ii)\Rightarrow(iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\Rightarrow(ii)\Rightarrow(i)$, which we obtain through the use of AI models.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
ChatGPT 5.5; Kimi K3; Claude Fable 5; ChatGPT 5.6 Sol
The disclosure is in the abstract itself, not only in an acknowledgment: "The primary focus of this paper is the reverse implications $(iii)\Rightarrow(ii)\Rightarrow(i)$, which we obtain through the use of AI models." The dedicated AI Acknowledgment gives the detail, verbatim: "A proof of Theorem 1.1[(iii) $\Rightarrow$ (ii) $\Rightarrow$ (i)] was initially obtained from a ChatGPT 5.5 query focusing on the case when $T$ is discrete. We were also able to independently find proofs using Kimi K3 and Claude Fable 5. These arguments were heavily reorganized and rewritten by the authors with further assistance from ChatGPT 5.6 Sol. Theorem 5.1 was obtained by the authors by modifying a different result found by ChatGPT while attempting Question 5.3." Worth noting what that describes: the proof came out of a model, and was then independently reproduced by two further models from different vendors. The authors' own labour is reorganizing and rewriting.
An arXiv preprint (v1, 25 August 2026, math.LO), unrefereed and with no independent endorsement. No mathematics was checked here and there is nothing mechanical to check it against - no formalization, no certificate. Verified here on 26 August 2026: the paper exists at arXiv:2608.24605 with the title and all three authors this entry lists; the statement above is its abstract near verbatim; the AI disclosure appears both in the abstract and in a dedicated AI Acknowledgment, quoted in full above; and the prior work it builds on is real and correctly characterised - Conant and Gannon, Ann. Pure Appl. Logic 171 (2020) for the originating observation, and Conant, Gannon and Hanson, J. Math. Log. (2025) for the chain of implications this paper reverses.
Let $T$ be a complete first-order theory in discrete or continuous logic, let $M\prec\mathcal{U}$, and let $\mu\in\mathfrak{M}_{x}(\mathcal{U})$ be Borel-definable over $M$. The paper proves that the following three conditions are equivalent:
$(i)$ $\mu$ is a frequency interpretation measure (fim) over $M$;
$(ii)$ $\mu$ is definable over $M$ and its canonical random extension $r_{\mu}$ is generically stable over $M^{\Omega}$;
$(iii)$ $\mu$ is self-averaging over $M$.
The new work proves the reverse implications $(iii)\Rightarrow(ii)\Rightarrow(i)$ and extends the characterization to continuous logic. The authors therefore make the equivalent conditions into a definitive definition of generic stability for Keisler measures.
The paper also proves a further characterization in terms of an order-property condition and derives consequences for closure under Morley products.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.