How to Tame Your LLM: Semantic Collapse in Continuous Systems

TL;DR

This paper introduces a theoretical framework modeling large language models as continuous dynamical systems, revealing how discrete symbolic semantics can emerge from continuous processes through spectral analysis.

Contribution

It formalizes semantic dynamics in LLMs using Continuous State Machines and proves the Semantic Characterization Theorem linking spectral properties to symbolic semantics.

Findings

Spectral analysis reveals finite invariant semantic basins.

Discrete symbolic semantics emerge from continuous models.

Slowly varying dynamics preserve semantic structure.

Abstract

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator $P: L^2(M,\mu) \to L^2(M,\mu)$ encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), $P$ is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of $P$ induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over $\mathbb{R}$. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.