PatternSense - A Computational Theory of Structural Identity

We present PatternSense, a formal mathematical framework for identifying structural

invariants across arbitrary transformations of context, scale, medium, and substrate.

In contrast to taxonomic approaches that classify patterns by surface features,

PatternSense isolates the generative relations that remain constant while their observable

realizations differ. This resolves a longstanding problem in structural realism:

how to define what persists when a system changes.

We show that the core components of a structural theory of identity—transformation

groups, equivalence relations, and structure-preserving morphisms—are not free parameters but are forced by minimum description length (MDL) minimization and computational isomorphism. PatternSense yields two complementary notions of identity:

a discrete, kernel-equivalent identity and a continuous, deformational identity. We introduce

Isomorphism Transfer Efficacy (ITE) as an empirical validation criterion: if two systems share the same generative kernel, interventions effective in one should map (via isomorphism) to effective interventions in the other.

The result is an objective, substrate-independent theory of identity with applications

to cryptographic identity, AI model continuity, cross-domain reasoning, and

evolutionary dynamics. 1 Introduction

Pattern recognition underlies scientific inference, cognitive practice, and computational modeling. Yet most existing approaches classify patterns through content similarity rather than generative structure. For example, a Lotka–Volterra predator–prey oscillation and a TCP congestion-control sawtooth share the same relational dynamics but appear in different domains. Taxonomic methods therefore treat them as unrelated, obscuring the underlying invariance.

This paper introduces PatternSense, a framework for identifying patterns by the invariants

of their generative mechanisms. A pattern is defined as an equivalence class of systems

under structure-preserving transformations, where the equivalence relation is determined by

minimum description length (MDL) minimization and computational isomorphism. Formal

definitions appear in Appendix A.

We contribute:

1. A proof that transformation groups, equivalence relations, and morphisms are uniquely

determined by the generative model class.

2. A computable identity criterion based on Kolmogorov complexity bounds.

3. A dual identity semantics: binary identity (kernel isomorphism) and continuous identity

(deformation distance).

4. An empirical validation method—Isomorphism Transfer Efficacy (ITE)—for testing

identity claims via cross-domain intervention transfer.

PatternSense thus yields a substrate-independent, mathematically grounded theory of

identity applicable across computational, biological, social, and ecological systems.

The synthesis presented here does not correct or supersede prior approaches. Each relevant field—causal modeling, algorithmic information theory, structural realism, dynamical systems, and modern generative modeling—developed powerful but partial frameworks suited to its own aims and available tools. PatternSense emerges only because these traditions have now matured to the point where their conjunction is possible.

The central concepts of the framework—minimal generative structure, causal invariance

under intervention, and stability under admissible transformations—require ingredients that

historically belonged to different intellectual lineages. None alone could have supplied the

whole picture. What is new is not the individual components, but the conditions under which

they can be combined and applied to contemporary systems that make cross-representation identity a practical concern.

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