Similarity Field Theory: A Mathematical Framework for Intelligence

TL;DR

Similarity Field Theory (SFT) reframes cognition around a primitive, directed similarity field $S: U × U → [0,1]$ and concept fibers $Fα(K)=\{E ∈ U | S(E,K) ≥ α\}$. It introduces a sequence-based evolution $Z_p=(X_p,S^{(p)})$ and a generative operator $G$, defining intelligence as generating new entities that remain within a concept's fiber; two foundational results—the Incompatibility Theorem for asymmetry and the Stability Theorem linking long-term behavior to anchors or confinement—constrain how similarity fields evolve. The framework provides an operational, fiber-preserving definition of intelligence and recasts interpretability as a geometric problem, enabling networks to be viewed as compositions of calibrated similarity fields. By applying this lens to neural networks and language models, SFT offers a principled path toward understanding, generalization, and interpretability beyond purely statistical accounts.

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 $F_α(K)={E\in U \mid S(E,K)\ge α}$, 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.

Figures (3)

• Figure 1: The Incompatibility Theorem visualized through a negotiation. Each party's "acceptance fiber" is defined by the other's perceived value. Party B's offer ($S(B,A)=0.3$) is too low to enter Party A's acceptance fiber (threshold 0.8). Conversely, Party A's offer ($S(A,B)=0.8$) easily clears Party B's threshold (0.3). Because the condition is not met for both, mutual agreement is impossible.
• Figure 2: A single neuron as a primitive similarity field.
• Figure 3: A neural network composes simple similarity fields into a complex one.

Theorems & Definitions (28)

• Definition 1: Entity
• Definition 2: Universe
• Definition 3: Concept
• Definition 4: Similarity Field
• Definition 5: Fiber
• Proposition 1: Fiber monotonicity
• proof
• Proposition 2: Intersection identity
• proof
• Proposition 3