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

Stephen Russell

cs.CLFeb 21, 2026

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

Authors: Stephen Russell

Organizations: Intelligent Systems and Robotics Department University of West Florida

Abstract

Studies of semantic drift report heterogeneous signals, including embedding displacement, neighbor change, distributional divergence, and recursive trajectory instability, without a shared account that relates them. Semantic Substrate Dynamics Theory (SSDT) treats these signals as observables of one time-indexed substrate, St = (X, dt, Pt), that couples embedding geometry to a local diffusion kernel. The contribution is commensurability with a mechanism layer: the substrate separates within-basin churn from basin crossing, recursion-induced instability, and intervention-order effects, distinctions that a single detection score does not recover. Coarse Ricci curvature functions as a dense structural descriptor of basin and bridge geometry across the graph, and bridge mass, a node-level aggregate of incident negative curvature, functions as a sparse descriptor of the genuine bridge structure that is typically uncommon in embedding graphs. For recursive generation, node displacement relative to an origin decomposes into a radial component and a tangential component, which separates bounded departure from continuing reinterpretation. The predictions are stated in falsifiable form with a pre-declared rejection rule, and the predicted leading indicator of future rewiring is a local density statistic rather than the curvature aggregate. This manuscript provides the formal model, the assumptions, the observable roles, and the test contracts; empirical performance is deferred.

Figures & tables

Mode	Observable	What it measures
Translational	$D_{\mathrm{tr}}(x;t_{0},t_{1})$	displacement of a node’s own embedding
Rewiring	$D_{\mathrm{rw}}(x;t_{0},t_{1})=\operatorname{JS}(m_{x}^{(t_{0})}\\|m_{x}^{(t_{1})})$	redistribution of a node’s neighborhood
Dynamical	$x_{k+1}=F(x_{k})$	trajectory behavior under an iterated operator
Process	$H_{t}=G_{t}\circ F_{t}$	change from composing evolution with intervention

Table 1: The four drift modes and the substrate observable that defines each.

Quantity	Role	Scope
Coarse Ricci curvature $\kappa_{\mathrm{OR}}$	dense structural descriptor	basin versus bridge geometry across the whole graph
Bridge mass $B_{t}$	sparse structural descriptor	genuine bridge nodes, a thin tail of the distribution
Self-neighbor similarity $C_{k}$	predicted leading indicator of rewiring	population level
Radial component $r_{k}$	bounded-departure descriptor	recursive generation
Tangential component $\varepsilon_{k}^{\perp}$	persistent-reinterpretation descriptor	recursive generation
Rewiring drift $D_{\mathrm{rw}}$	outcome and criterion	the quantity predictions target

Table 2: Role assigned to each substrate quantity, separating description from prediction.

Figure 1: Semantic basins and bridges in substrate geometry. Blue wells depict positively curved basin regions ( $\kappa>0$ ) where local diffusion tends to be contractive and trajectories are pulled toward stable neighborhoods. The green ridge indicates a negatively curved bridge regime ( $\kappa\leq 0$ ), where diffusion is locally expansive and connectivity is fragile. Nodes near or traversing bridge structure are more susceptible to neighborhood rewiring under perturbation or intervention.

Semantic Drift in Language Models

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Comment-level Topic Drift Analysis in the Reddit Corpus

Steven Morse, Daniel Runfola, Trenton W. Ford

William & Mary

Semantic Change Detection Topic Modeling Social Media Analysis

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We present a novel application of embedding-based dynamic topic modeling techniques to detect and quantify topic drift at the comment level in a massive corpus. By leveraging pretrained language models to generate contextualized semantic embeddings for short text, we analyzed 12.7 billion Reddit comments spanning 2006 to 2022. Using unsupervised methods on these embeddings, we identify dynamically evolving topic clusters over time. Our primary contribution is a methodology for analysis of semantic drift and discourse evolution in the embedding space itself. We also demonstrate modifications to existing methods that enable this analysis at scale, and we propose and demonstrate a null model comparison test to filter spurious dynamics. Key findings suggest that politically and socially contentious topics exhibit significant directional drift in embedding space, with inter-topic distances changing systematically over time beyond what the null model can explain, whereas domains such as music and sports remain comparatively stable.

Abstract

Semantic Drift in Language Models

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