Noise-crossing band-pass neuron

Integrate the activation after a linear or convolutional preactivation d = W x + b, initially in one hidden layer of a small MLP or CNN. During training, retain the binary forward signal z, but use the analytic expected response as a surrogate for the backward pass: compute F(d), set m = 2F(d)(1-F(d)), and backpropagate with dm/dd = 2p(d)(1-2F(d)). For a uniform noise law, sample two independent tensors u_1 and u_2 from Uniform[-a,a], form z = XOR(d >= u_1, d >= u_2), and use m = 2q(1-q) for the surrogate path, where q = clip((d+a)/(2a), 0, 1). A practical autograd implementation is z_forward + m - stop_gradient(m), so the forward value is binary while the backward value follows m. Estimate event density directly as the batch mean of z, and optionally learn or schedule a; do not estimate F empirically for the uniform case. Start with a two-layer MLP on MNIST or Fashion-MNIST, replacing only the first hidden activation, and compare against ReLU and sigmoid-gated binary units at matched parameter count. Measure test accuracy, activation event rate, gradient norm variance, and loss at equal training FLOPs. Success would mean comparable accuracy with fewer active events, fewer saturated units, or lower gradient variance. Also test for collapse to nearly all-zero events when a is poorly scaled.