Similarity Field Theory: A Mathematical Framework for Intelligence

We posit that transforming similarity relations form the structural basis of comprehensible dynamic systems. 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,ldots; (3) concepts K as entities that induce fibers Fα(K)=E∈ U mid 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 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 applicable to large language models. AI systems may be aligned less to safety as such than to human-observable and human-interpretable conceptions of safety, which may not fully determine the underlying safety concept.