Data-Efficient Logical Learning via Piecewise Constrained Geometric Manifolds

While large language models (LLMs) are widely thought to acquire logical and arithmetic competence through massive scaling of parameters and datasets , this research demonstrates that logical generalization can be achieved in extremely small data regimes. By building on Vector Geometry Theory (VGT), we introduce Piecewise Constrained Manifolds (PCM)—a minimal inductive bias that encodes algorithmic rules through segmented geometric consistency rather than global linearity. Using a compact recurrent architecture, we show that logical rules can be represented as irreducible geometric features. Our findings suggest that reasoning is not merely a function of scale, but of geometric parsimony: embedding the correct invariants into the representation space allows logic to emerge reliably even under severe data constraints.