Abstract

We posit that persisting and transforming similarity relations form the structural basis of any comprehensible dynamic system. This paper introduces Similarity Field Theory, a mathematical framework that formalizes the principles governing similarity values among entities and their evolution. We define: (1) a similarity field S:U×U→[0,1] over a universe of entities U, satisfying reflexivity S(E,E)=1 and treated as a directed relational field (asymmetry and non-transitivity are allowed); (2) the evolution of a system through a sequence Zp=(Xp,S(p)) indexed by p=0,1,2,…; (3) concepts K as entities that induce fibers Fα(K)=E∈U∣S(E,K)≥α, i.e., superlevel sets of the unary map SK(E):=S(E,K); and (4) a generative operator G that produces new entities. Within this framework, we formalize a generative definition of intelligence: an operator G is intelligent with respect to a concept K if, given a system containing entities belonging to the fiber of K, it generates new entities that also belong to that fiber. Similarity Field Theory thus offers a foundational language for characterizing, comparing, and constructing intelligent systems. At a high level, this framework reframes intelligence and interpretability as geometric problems on similarity fields -- preserving and composing level-set fibers -- rather than purely statistical ones. We prove two theorems: (i) asymmetry blocks mutual inclusion; and (ii) stability requires either an anchor coordinate or eventual confinement within a level set. These results ensure that the evolution of similarity fields is both constrained and interpretable, culminating in a framework that not only interprets large language models but also introduces a novel way of using them as experimental probes of societal cognition, supported by preliminary evidence across diverse consumer categories.