Equivalence of generic stability notions for Keisler measures
Three candidate definitions of generic stability for a Borel-definable global Keisler measure are shown to be equivalent, closing a research objective left open by earlier work.
- Lab
- Independent
- Model
- ChatGPT 5.5; ChatGPT 5.6 Sol; Kimi K3; Claude Fable 5
- Field
- Mathematics
- Date
- 2026-08-25
- Human collaborators
- Gabriel Conant, Kyle Gannon, James E. Hanson
- Problem posed
- 2020 · open 6 yrs
Sources
Original work
Independent commentary
What was found
For a first-order theory in discrete or continuous logic and a Borel-definable global Keisler measure, the paper shows three conditions equivalent: that the measure is a frequency interpretation measure, that it is definable and its canonical random extension is generically stable in the randomization theory, and that it is self-averaging. The forward implications were already established by the same authors for discrete theories; the paper's contribution is the two reverse implications, which is what makes the notion of generic stability robust and settles a long-term objective from that earlier work.
Novelty check
The prior state is unusually clean here because it is the authors' own: the implications from frequency interpretation to generic stability of the random extension to self-averaging were established in their previous work for discrete theories, and the reverse implications are what remained. The paper states that division explicitly, so what is new is not in doubt. No competing proof of the reverse implications appears.
Caveats and known objections
A preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-led on the paper's AI acknowledgment, which reports both discovery and cross-checking by several models followed by substantial human rewriting: a proof of the main theorem was initially obtained from a ChatGPT 5.5 query focused on the discrete case, the authors were also able to find proofs independently using Kimi K3 and Claude Fable 5, and the arguments were then heavily reorganized and rewritten by the authors with further assistance from ChatGPT 5.6 Sol.
Also recorded at
vibemathedequivalence-of-generic-stability-notions-for-keisler-measures
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
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Cite this entry
whataifound.org. (2026). Equivalence of generic stability notions for Keisler measures. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-25-generically-stable-keislerBibTeX
@misc{whataifound-independent-2026-keisler,
title = {Equivalence of generic stability notions for Keisler measures},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-25-generically-stable-keisler}
}