{"id":8932,"date":"2025-12-08T07:02:54","date_gmt":"2025-12-08T07:02:54","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/12\/08\/2512-05162\/"},"modified":"2025-12-08T07:02:54","modified_gmt":"2025-12-08T07:02:54","slug":"2512-05162","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/12\/08\/2512-05162\/","title":{"rendered":"How to Tame Your LLM: Semantic Collapse in Continuous Systems"},"content":{"rendered":"<p>    How to Tame Your LLM: Semantic Collapse in Continuous Systems<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n    <!-- no image --><br \/>\n \t<BR><br \/>\n<BR><\/BR><\/p>\n<div>arXiv:2512.05162v1 Announce Type: new<br \/>\nAbstract: 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.<\/div>\n<p> \t<BR><br \/>\n <BR><\/BR><br \/>\n    C. M. Wyss<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n<a href=\"https:\/\/arxiv.org\/abs\/2512.05162\">Go to original source<\/a><br \/>\n \t<BR><br \/>\n <BR><\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How to Tame Your LLM: Semantic Collapse in Continuous Systems arXiv:2512.05162v1 Announce Type: new 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 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,187,113,1153,420,112],"tags":[1044,7,2950],"class_list":["post-8932","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-ai","category-cs-lg","category-math-ds","category-math-pr","category-stat-ml","tag-continuous","tag-how","tag-semantic"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/8932"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=8932"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/8932\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=8932"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=8932"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=8932"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}