Similarity Field Theory: A Mathematical Framework for Intelligence

arXiv:2509.18218v4 Announce Type: replace
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 $Z_p=(X_p,S^(p))$ indexed by $p=0,1,2,ldots$; (3) concepts $K$ as entities that induce fibers $F_alpha(K)=Ein U mid S(E,K)ge alpha$, 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. 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 implies either an anchor coordinate or asymptotic confinement to the target level (up to arbitrarily small tolerance). Together, these results constrain similarity-field evolution and motivate an interpretive lens that can be applied to large language models.