Similarity Field Theory: A General Mathematical Framework for Intelligence

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 $S: U \times U \to [0,1]$ 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 $Z_p = (X_p, S<sup>{(p)})$ indexed by $p=0,1,2,\ldots$; (3) concepts $K$ as entities that induce fibers $F_{\alpha}(K) = { E \in U \mid S(E,K) \ge \alpha }$, i.e., superlevel sets of the unary map $S_K(E) := S(E,K)$; 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. 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 an exploration of how the framework allows us to interpret LLMs and present empirical results using LLMs as experimental probes of societal cognition.