Learning Dynamics of LISP: A Gradient-Free Constraint-Satisfaction Family Containing Backpropagation

We characterize the learning dynamics of Layerwise Inequality-Sign Propagation (LISP), a gradient-free credit-assignment rule that never computes activation derivatives---forward and backward passes execute without autograd, and the only signal moved between layers is a constraint-violation vector $\mathbf{d}^{(l)}$ modulated by a binary monotonicity indicator $\Pi_\sigma$. We prove that (i) the raw LISP output update is gradient descent on a smooth squared hinge loss with $O(1/T)$ convergence, and (ii) for ReLU networks without batch normalization, LISP with cross-entropy is value-equivalent to backpropagation almost everywhere---placing standard BP as one point inside the gradient-free LISP family. Across $>$3,000 runs on CIFAR-10/100, LISP trails tuned BP by 0.4--2.8pp on clean data but outperforms BP+CE by 2.8pp under 40% label noise; a controlled ablation isolates per-sample RMS normalization as the operative mechanism, and transferring it to standard backprop reproduces the effect.