Geometric Identity and Piecewise Constrained Manifolds: A Unified Framework for Logical Reasoning in Large Language Models

This study establishes a unified geometric framework for logical reasoning in large language models (LLMs) through empirical validation of Vector Geometry Theory (VGT). We demonstrate that digits 0–9 form a piecewise constrained manifold (PCM) in Layer 0 of Qwen3-4B, characterized by non-monotonic step distribution (coefficient of variation CV=0.664) with a 54.4% drop precisely at the decimal carry boundary (4→5). Logical connectives ("because"/"therefore") exhibit 2.65× anomalous clustering on this manifold, confirming geometric unification of arithmetic and causal reasoning. Crucially, our operator mirroring experiment—swapping the embedding vectors of "+" and "−"—reveals that arithmetic behavior is entirely determined by geometric position: after swapping, "5+1" outputs "4" while "5−1" outputs "6", with rule consistency fully preserved. This provides definitive evidence for the Geometric Identity Theory (GIT): a token's identity is uniquely determined by its geometric coordinates on the logical manifold; coordinate exchange equals identity exchange equals algorithmic rule exchange. We further propose the Geometric Phase Alignment (GPA) framework to explain why general-purpose LLMs (e.g., Qwen3-4B) possess manifold precursors yet fail at reliable extrapolation due to phase misalignment, while VGT training achieves 99.0% accuracy on 20-digit addition through enforced collapse and phase alignment. This work delivers the first quantifiable geometric diagnostic framework for LLM logical capabilities, with profound implications for AI safety and interpretability.