A Trained Iterative Classifier Reduces to Energy Gradient Descent: Recovering Closed-Form Dynamics from a Learned NLI System

We show that a trained neural update in an iterative natural language inference (NLI) classifier can be removed and replaced with a closed-form energy gradient, with no retraining and no loss in performance. A BERT-based iterative classification head is trained on the SNLI dataset, where a hidden state evolves over multiple steps using a learned MLP update. After training, the MLP is removed and replaced with the analytical gradient of an energy function defined over cosine similarities to class anchors: V(h) = -logsumexp(beta * cos(h, A_k)) The parameter-free update achieves identical performance on the full SNLI development set (82.21% vs 82.05%, n = 9,842), demonstrating that the trained system had converged to performing gradient descent on this energy function. We formalize the recovered behavior as three empirical patterns: relational initialization, an energy landscape over anchor similarities, and gradient-based collapse dynamics. Additional experiments show improved neutral recall under joint retraining and identify a universal fixed-point collapse as the primary limitation. This work demonstrates that interpretable, closed-form dynamics can emerge from trained neural systems and can replace learned components without degrading performance. Code and models: https://github.com/chetanxpatil/livnium