Latent Semantic Manifolds in Large Language Models

• URL: http://arxiv.org/abs/2603.22301v1
• Date: Tue, 17 Mar 2026 13:05:56 GMT
• Title: Latent Semantic Manifolds in Large Language Models
• Abstract summary: We develop a mathematical framework that interprets Large Language Models hidden states as points on a latent semantic manifold.<n>We prove two theorems: a rate-distortion lower bound on distortion for any finite vocabulary, and a linear volume scaling law for the expressibility gap.<n>We discuss implications for architecture design, model compression, decoding strategies, and scaling laws.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Large Language Models (LLMs) perform internal computations in continuous vector spaces yet produce discrete tokens -- a fundamental mismatch whose geometric consequences remain poorly understood. We develop a mathematical framework that interprets LLM hidden states as points on a latent semantic manifold: a Riemannian submanifold equipped with the Fisher information metric, where tokens correspond to Voronoi regions partitioning the manifold. We define the expressibility gap, a geometric measure of the semantic distortion from vocabulary discretization, and prove two theorems: a rate-distortion lower bound on distortion for any finite vocabulary, and a linear volume scaling law for the expressibility gap via the coarea formula. We validate these predictions across six transformer architectures (124M-1.5B parameters), confirming universal hourglass intrinsic dimension profiles, smooth curvature structure, and linear gap scaling with slopes 0.87-1.12 (R^2 > 0.985). The margin distribution across models reveals a persistent hard core of boundary-proximal representations invariant to scale, providing a geometric decomposition of perplexity. We discuss implications for architecture design, model compression, decoding strategies, and scaling laws

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