Equivalence of averaging and stability properties of Keisler measures
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For Borel-definable global Keisler measures over a model of a discrete first-order theory, are self-averaging, generic stability of the canonical random extension and the frequency interpretation property equivalent?
Why it matters and what counts as progress
Why it matters: It seeks a complete characterization of generic stability by linking model-theoretic averaging and frequency conditions in the discrete setting.
What counts as progress: Prove the missing reverse implications for Borel-definable measures in the discrete setting; extensions to other settings need their own hypotheses.
How the problem developed
- 2025 · Partial result [1]
Gabriel Conant, Kyle Gannon and James E. Hanson Established forward implications among frequency interpretation, random-extension generic stability and self-averaging for discrete theories.
- 25 Aug 2026 · Theorem [1]
Gabriel Conant, Kyle Gannon and James E. Hanson, with ChatGPT 5.5, Kimi K3, Claude Fable 5 and ChatGPT 5.6 Sol Completed the reverse implications and proved the three conditions equivalent for Borel-definable global measures in discrete or continuous logic. The authors say the reverse-direction proof first came from ChatGPT 5.5, was independently found with Kimi K3 and Claude Fable 5, then reorganized with ChatGPT 5.6 Sol.
AI contribution
Generically stable Keisler measures
For Borel-definable global Keisler measures over a model of a discrete first-order theory, the authors prove the reverse implications: self-averaging gives a definable measure whose canonical random extension is generically stable, which in turn gives a frequency interpretation measure. Together with earlier forward implications, these yield equivalence.
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