Geometric Parsimony: Inducing Logical Skeletons via Manifold Alignment in Tiny Language Models

Recent studies on Qwen3-4B suggest that logical capabilities are concentrated in a "logical skeleton" comprising specific layers and singularities. This paper investigates whether such structures can be induced in tiny models (8 layers) under extreme data scarcity (500MB). By introducing Piecewise Constrained Manifolds (PCM), we demonstrate the emergence of a robust geometric skeleton. Our results show an intra-segment cosine similarity of 1.0000 and a deliberate structural collapse at rule boundaries (Sim: -0.9086). These findings validate the theory that logic is an emergent property of geometric parsimony rather than a mere byproduct of massive scaling.