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arXiv · 2604.14020

Continuous first-order logic of abstract harmonic spaces

Haoming Wang
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Abstract

This paper studies Brelot harmonic spaces within an unbounded many-sorted continuous first-order language and its expansions. 1. In the finite evaluation reduct of the harmonic language, the Brelot theory together with the finite interpolation scheme eliminates bounded harmonic quantifiers while leaving point quantifiers untouched. 2. The Martin compactification is homeomorphic to a compact subset of the local type space, with the Martin boundary corresponding to non-principal types. 3. If the compact convex space of normalized positive harmonic functions is a Choquet simplex, then each Martin representing measure on the minimal boundary corresponds to a unique Keisler measure supported on minimal types and admits a complete lift. 4. Harnack compactness also implies stability of normalized evaluation formulas, yielding canonical bases and a harmonic barycenter factoring through Keisler measures. These constructions are further applied to fine potential theory, where boundary poles of potentials are Keisler null. First-jet expansions encode the Radó--Kneser--Choquet and Lewy planar rigidity phenomena, while higher-dimensional failures are analyzed through thorn-forking chains in o-minimal expansions.

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Haoming Wang. 2026-08-02. Continuous first-order logic of abstract harmonic spaces. https://arxiv.org/abs/2604.14020

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