Semantic Substrate Theory: An Operator-Theoretic Framework for Geometric Semantic Drift

TL;DR

This paper introduces a formal operator-theoretic framework called Semantic Substrate Theory to unify various signals of semantic drift, such as embedding changes and distributional shifts, within a single mathematical model.

Contribution

It formalizes semantic drift signals in a unified substrate model, introduces new measures like bridge mass, and provides theoretical tests to validate the model.

Findings

Bridge mass predicts future neighborhood rewiring.

The formal model relates multiple drift signals within a shared framework.

The paper establishes theoretical foundations and testing methods for semantic drift analysis.

Abstract

Most semantic drift studies report multiple signals e.g., embedding displacement, neighbor changes, distributional divergence, and recursive trajectory instability, without a shared explanatory theory that relates them. This paper proposes a formalization of these signals in one time-indexed substrate, $S_t=(X,d_t,P_t)$, combining embedding geometry with local diffusion. Within this substrate, node-level neighborhood drift measures changes in local conditional distributions, coarse Ricci curvature measures local contractivity of semantic diffusion, and recursive drift probes stability of iterated semantic operators. This manuscript specifies the formal model, assumptions, and tests that can refute the model. Herein, the paper introduces bridge mass, a node-level aggregate of incident negative curvature, as a predictor of future neighborhood rewiring. This paper provides the theory and test contracts; empirical performance is deferred to subsequent studies.