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This paper empirically studies the Voronoi tessellation induced by language models in their latent semantic manifolds using Qwen3.5-4B-Base. It validates the linear scaling law of the expressibility gap with high accuracy by resolving quantization artifacts and identifies a geometric ambiguity regime in mid-layers. The paper also introduces and compares two margin refinement procedures (MRP) – direct margin maximization and Fisher information distance maximization – demonstrating that the latter can reshape the Voronoi tessellation to improve margins while preserving downstream performance, albeit with a bias towards high-frequency tokens.
Post-training "geometric polishing" can reshape LLM token decision boundaries, but the benefits disproportionately accrue to frequent structural tokens, suggesting a ceiling on uniform performance gains.
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.