How to Tame Your LLM: Semantic Collapse in Continuous Systems
This work provides a foundational theory for interpreting semantic dynamics in LLMs, which could impact all of ML/AI by offering a framework to tame and analyze these models.
The authors tackled the problem of understanding how discrete symbolic semantics emerge from continuous computation in large language models by formalizing them as Continuous State Machines and proving the Semantic Characterization Theorem, which shows that the continuous activation manifold collapses into a finite, logically interpretable ontology under mild regularity assumptions.
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 $P: L^2(M,μ) \to L^2(M,μ)$ 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.