The Geometric Origin of Logic: Weight Polarization and Manifold Collapse

This study explores the emergence of logical extrapolation in neural networks. By introducing a Variable Geometric Tension (VGT) framework, we demonstrate that forcing hidden representations into a low-dimensional manifold via L2-norm pressure (Geometric Collapse) enables a Transformer model to transcend statistical pattern matching. Our VGT-Pro model achieves 99.0% accuracy on 20-digit addition while only being trained on 1-6 digits, suggesting that logic is a geometric necessity squeezed out of representational redundancy.