Conservative-field gradient envelope

Integrate this at the parameter-gradient stage of a network whose forward pass contains continuous piecewise-smooth operations such as ReLU, clamp, top-k routing, mixture-of-experts gates, or custom branches. The scalar minibatch loss is $L(\theta)$. Run the normal forward pass and record branch predicates and near-interface tensors. Define a near-interface event using a margin threshold $\tau$: for ReLU use $|z|<\tau$, and for routing use the top-two logit gap $|a_{(1)}-a_{(2)}|<\tau$. Pseudocode: loss = forward(theta, mode='base'); G = [backward(loss)]; for m=1,...,M, perturb only near-interface preactivations with $\epsilon\sim\mathrm{Uniform}[-\tau,\tau]$, replay the branch decisions, backpropagate the perturbed computation, and append the resulting gradient g_m to G; clip each gradient to a common norm; compute gbar = sum(w_m*g_m) with uniform weights or replay-frequency weights; update theta = theta - eta_k*gbar. Use $\eta_k=\eta_0/(k+1)^\alpha$ with $0.5<\alpha\le1$ for the conservative-field experiment, and compare against a constant-step AdamW control. The paper supplies $D_L$, its convex hull, and the pathwise chain rule; the implementation estimates $D_L$ through finite branch replay and estimates interface frequency empirically. Log gradient radius $R_k=\max_m\|g_m-\bar g\|$, pairwise cosine similarity, near-interface fraction, loss spikes, and validation accuracy. First test a 2-layer ReLU MLP on MNIST or CIFAR-10, then a small continuous top-2 MoE. Compare ordinary autodiff, standard subgradient SGD, and the envelope update at equal total backward FLOPs. Success is lower interface-associated gradient variance, fewer loss spikes, and improved validation accuracy or stability at equal compute; failure is no gain after accounting for the extra backward passes.