Approximating Discrete Logical Boundaries in Neural Networks

This paper studies how neural networks model discrete logical boundaries. We compare continuous manifold approximation and discrete rule execution. Microscopic analysis with a Vectorized Structured Logic Network (V-SLN) shows that standard networks fit functional manifolds well but fail to satisfy discrete topological constraints at logical boundaries. We then evaluate the Qwen2.5 model family (3B to 72B) on logical inference chains. The results show that: (1) pre-trained base models have low sensitivity to truth-value transitions; (2) alignment compresses representations toward sigmoid-like distributions; and (3) at 72B scale, output confidence approaches a step-like function while a large Kullback-Leibler (KL) Divergence spike indicates distributional shift rather than structured state transition. Finally, we propose the Heterogeneous Logic Neural Network (H-LNN). Using temperature annealing and parallel activation pathways, H-LNN improves boundary modeling precision under comparable parameter budgets