Geometric Properties of the Voronoi Tessellation in Latent Semantic Manifolds of Large Language Models

TL;DR

This paper empirically analyzes the Voronoi tessellation in language model vector spaces, confirming a linear scaling law and demonstrating margin refinement procedures to reshape model geometry without retraining.

Contribution

It validates Mabrok's linear scaling law with high precision and introduces margin refinement procedures to reshape the tessellation, preserving model performance.

Findings

Confirmed Mabrok's linear scaling law with R^2 = 0.9997.

Demonstrated that margin refinement can reshape the tessellation without retraining.

Fisher information distance maximization achieves stable corrections with invariant downstream performance.

Abstract

Language models operate on discrete tokens but compute in continuous vector spaces, inducing a Voronoi tessellation over the representation manifold. We study this tessellation empirically on Qwen3.5-4B-Base, making two contributions. First, using float32 margin recomputation to resolve bfloat16 quantization artifacts, we validate Mabrok's (2026) linear scaling law of the expressibility gap with $R^2$ = 0.9997 - the strongest confirmation to date - and identify a mid-layer geometric ambiguity regime where margin geometry is anti-correlated with cross-entropy (layers 24-28, $\rho$ = -0.29) before crystallizing into alignment at the final layer ($\rho$ = 0.836). Second, we show that the Voronoi tessellation of a converged model is reshapable through margin refinement procedures (MRP): short post-hoc optimization runs that widen token-decision margins without retraining. We compare direct margin maximization against Fisher information distance maximization across a dose-response sweep. Both methods find the same ceiling of ~16,300 correctable positions per 256K evaluated, but differ critically in collateral damage. Margin maximization damage escalates with intervention strength until corrections are overwhelmed. Fisher damage remains constant at ~5,300 positions across the validated range ($\lambda$ = 0.15-0.6), achieving +28% median margin improvement at $\lambda$ = 0.6 with invariant downstream benchmarks - a geometric reorganization that compresses the expressibility gap while preserving its scaling law. However, frequency and token-class audits reveal that gains concentrate in high-frequency structural tokens (84% of net corrections at $\lambda$ = 0.6), with content and entity-like contributions shrinking at higher $\lambda$. Fisher MRP is therefore a viable geometric polishing tool whose practical ceiling is set not by aggregate damage but by the uniformity of token-level benefit.