Similarity Field Theory in Dynamic Systems

• Similarity Field Theory is a mathematical framework that defines directed, graded similarity between entities in dynamic systems.
• It employs asymmetry and non-transitivity to model evolving concepts, enabling graded membership and flexible categorization.
• The theory underpins generative intelligence by mapping concept fibers and stabilizing similarity fields across dynamic learning epochs.

Similarity Field Theory is a mathematical framework designed to formalize and analyze how similarity relations among entities persist and transform within dynamic systems. Its core objective is to provide foundational language and rigorous tools for characterizing, comparing, and understanding the structure and evolution of similarity—especially as it relates to concepts, generative processes, and intelligent system behavior. The framework generalizes similarity to a directed, real-valued field over a universe of entities, allows for asymmetry and non-transitivity, and organizes system evolution, conceptual fibers, and generative intelligence definitions in a precise way (Ng, 21 Sep 2025).

1. Definition and Structure of the Similarity Field

The defining object of Similarity Field Theory is the similarity field: $S : U \times U \to [0, 1]$ where $U$ is a universe of entities and $S(E_1, E_2)$ gives the degree of similarity from entity $E_1$ to entity $E_2$.

Fundamental properties:

• Reflexivity: $\forall E \in U,\, S(E, E) = 1$
• Asymmetry: $S(E_1, E_2) \ne S(E_2, E_1)$ is generally permitted
• Non-transitivity: No assumption that $S(E_1, E_2)$ and $S(E_2, E_3)$ high implies $S(E_1, E_3)$ high

These choices enable highly nuanced, context-sensitive and directional modeling of similarity relations. In technical terms, $U$0 functions as a directed relational field rather than a metric or inner product.

2. Evolution of Similarity Fields

System dynamics are described via a sequence

$U$1

where $U$2 is a progression index (e.g., time, interaction rounds, training epochs). Here, $U$3 represents the current set of entities and $U$4 the prevailing similarity field at index $U$5.

The evolution mechanism models both changes in the population of entities (elements can be added, removed, transformed) and the evolution of similarity relations among them. This formalism is suitable for tracking processes such as learning, adaptation, cultural or conceptual change, and system-wide reconfigurations.

3. Concepts, Fibers, and Graded Membership

A concept $U$6 is treated as an entity that induces a unary similarity map: $U$7 Given $U$8, the fiber of $U$9 at threshold $S(E_1, E_2)$0 is the superlevel set

$S(E_1, E_2)$1

This construction provides a graded, thresholded notion of conceptual membership. Unlike crisp, classically binary category assignments, fibers support degrees of association with concepts, enabling flexible modeling of similarity-based categorization, prototype theory, and progressive conceptual generalization.

4. Generative Operators and Intelligence

The framework introduces a generative operator

$S(E_1, E_2)$2

A rigorous generative definition of intelligence is formalized: $S(E_1, E_2)$3 is intelligent with respect to concept $S(E_1, E_2)$4 (and level $S(E_1, E_2)$5) if, given some existing entities in $S(E_1, E_2)$6, it generates new entities $S(E_1, E_2)$7 such that

$S(E_1, E_2)$8

This operationalizes intelligence as the continued (and possibly extended) generation of entities preserving membership in desired concept fibers. The definition encompasses many natural processes in artificial intelligence, cognitive science, and evolutionary dynamics, where adaptive generativity is judged by the ability to maintain or expand structured similarity.

5. Theoretical Results and Constraints

Two key theorems ground the evolution and structure of similarity fields:

Incompatibility Theorem (Asymmetry Blocks Mutual Inclusion):

Given asymmetric similarity between two entities ($S(E_1, E_2)$9, $E_1$0, $E_1$1), it is not possible for both $E_1$2 and $E_1$3 to hold simultaneously. That is, reciprocal inclusion at the level of the other’s similarity threshold is forbidden by directional asymmetry.

Stability Theorem:

Suppose a sequence of similarity fields (or values under derived functionals $E_1$4) stabilizes to a limiting value $E_1$5. Then, after a finite progression, all subsequent similarity vectors are confined within neighborhoods of $E_1$6, meaning long-term stability requires either:

• The existence of a convergent anchor coordinate (a persistent similarity with respect to some entity),
• Or eventual confinement of all evolution within a level set (a tube) corresponding to $E_1$7.

These results ensure that system evolution is constrained, interpretable, and explainable within the field-theoretic framework.

6. Interpretation and Applications: LLMs and Empirical Probing

Similarity Field Theory is directly applicable as a lens for interpreting large, compositional AI systems such as LLMs. In such systems:

• Tokens, phrases, and even neuron activations are treated as entities in $E_1$8.
• Concepts (e.g., "animal," "code correctness," "brand typicality") are modeled as target entities $E_1$9.
• The internal similarity field $E_2$0 provides graded membership and enables ranking of outputs.
• The generative operator $E_2$1 corresponds to the model's output mechanism; intelligence is measured by whether outputs are confined within fibers corresponding to the desired concept.

Empirically, the framework has been used to probe LLMs with pairwise typicality judgments to reconstruct global rankings that reflect collective cognition. Aggregating these preference data via models such as Bradley–Terry–Luce yields distributions that can be compared with extrinsic societal metrics, validating the presence of an internalized, structured similarity map.

A schematic diagram from the paper illustrates how neurons compute primitive similarity fields and how neural network layers compose these via weighted averages or pointwise products:

• A neuron evaluates $E_2$2.
• Concealed layers aggregate and transform these primitives.
• Training maximizes expected similarity over inputs in the fiber $E_2$3.

7. Implications for Foundations of Intelligence and Systems Design

By grounding intelligence in the language of directed similarity fields, the framework offers a unifying mathematical foundation for the study and construction of intelligent systems:

• It supports flexible, non-symmetric, non-transitive similarity relations, capturing subtleties of expertise, context, and directed influence.
• Evolution, learning, and generative design are rigorously defined via systemic transitions respecting conceptual fibers.
• Theorems ensure that structural stability and exclusion mechanisms are transparent and interpretable.
• The methodology enables explainable AI and the empirical analysis of emergent cognition and similarity relations within artificial and biological systems.

In summary, Similarity Field Theory provides technical, operational, and interpretive machinery for the representation, evolution, and application of similarity in dynamic, intelligent systems. By formalizing concepts, fibers, generative intelligence, and constraining results, it deepens our capacity to characterize and harness similarity as an organizing principle in mathematics, artificial intelligence, and the cognitive sciences (Ng, 21 Sep 2025).