A Mechanistic Explanation of CHSH Violations in Classical Neural Networks

Non-classical correlations, certified by violations of Bell–CHSH-type inequalities entailed by classical probability theory, are generally assumed to lie beyond the expressive capacity of classical neural networks. In this work, we show that a CHSH-style statistic $S$ in standard feedforward multi-task networks can exceed the classical bound of $2$ inside a specific “emergence window”, thereby exhibiting non-classical correlations. We further provide a mechanistic explanation of this phenomenon: under the shared-capacity bottleneck of multi-task learning, gradient competition couples hidden neurons; under ordinary feedforward evaluation, easy and hard task pairs become unequally learnable, which unbalances the four correctness correlations and drives $S>2$. We observe this pattern in numerical simulation, text classification, and image recognition, and further verify that the window is governed by the alignment between network capacity and task complexity. Using CHSH-type inequalities as a probe, this work shows that the learning dynamics of classical networks can also give rise to non-classical statistical correlations, and provides a theoretical basis for understanding non-classical statistical modeling in quantum-inspired methods.