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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.05162 |
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| _version_ | 1866914180369481728 |
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| author | Wyss, C. M. |
| author_facet | Wyss, C. M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05162 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How to Tame Your LLM: Semantic Collapse in Continuous Systems Wyss, C. M. Machine Learning Artificial Intelligence Dynamical Systems Probability 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. |
| title | How to Tame Your LLM: Semantic Collapse in Continuous Systems |
| topic | Machine Learning Artificial Intelligence Dynamical Systems Probability |
| url | https://arxiv.org/abs/2512.05162 |