The Geometry of Semantic Space: A Continuous Geometric Framework for the Transformer Architecture

2026-07-19

AI Summary

This work addresses the lack of a unified geometric interpretation for Transformers, which hinders understanding of their stability, context limitations, and optimization dynamics. Building upon the geometric axiom that token sequences form a discrete 1-manifold equipped with a measure-theoretic lattice, the paper introduces a continuous stochastic differential geometric framework that unifies core components—RMSNorm, RoPE, and Softmax attention—as integro-differential equations on a semantic fiber bundle. It reveals that attention corresponds to a Schrödinger bridge, while SGD manifests as an Itô diffusion violating detailed balance, and establishes a duality principle governing topological stability. Six geometric predictions—including ε⁻¹/² Lipschitz scaling, Poincaré recurrence suppression on the RoPE torus, and phase transitions at context limits—are validated across five mainstream large language models (124M–8B parameters), achieving R² = 1.000 consistently across optimizers.

Abstract

We present a continuous geometric framework that models the discrete algebraic operations of the Transformer architecture as an integro-differential equation (IDE) on a semantic fiber bundle $\calE = \calM \times \R^d$. Beginning from a single geometric axiom -- that the token sequence forms a discrete $1$-manifold equipped with a canonical measure lattice -- we translate every core component of the modern Transformer (RMSNorm, RoPE, Softmax Attention, FFN, Residual Stream, SGD, Weight Decay) into a cohesive vocabulary of differential geometry, measure theory, and stochastic calculus. The resulting framework yields quantitative predictions spanning entropic optimal transport (Attention as a Schrödinger bridge) and non-equilibrium thermodynamics (SGD as Itô diffusion violating detailed balance). We conduct a six-part experimental campaign across five architectures (Qwen3, LLaMA\nobreakdash-3.1, Gemma\nobreakdash-3, GPT-2, Mistral) spanning $124$M to $8$B parameters. The empirical observables are quantitatively consistent with the geometric predictions: the $ε^{-1/2}$ Lipschitz scaling calibration at machine precision ($R^2 = 1.000$), the Lie--Trotter operator-splitting torsion, the symmetric ablation instability confirming the Dual-Law of Topological Stability, the $\calO(1/\sqrt{k})$ thermodynamic suppression of Poincaré recurrence on the RoPE torus, the thermodynamic context-limit phase transition, and the Non-Equilibrium Steady State parameter vortex -- verified across two optimizers (AdamW and Pure SGD) to exclude momentum artifacts. The results demonstrate that analyzing Transformers through the lens of continuous stochastic differential geometry provides a predictive descriptive vocabulary for the stability limits, context bounds, and optimization dynamics of Large Language Models.

Problem

• Transformer architecture
• semantic space
• continuous geometric framework
• stochastic differential geometry
• large language models

Innovation

• semantic fiber bundle
• integro-differential equation
• stochastic differential geometry
• non-equilibrium thermodynamics
• Schrödinger bridge