Coherent-Fluctuating Amplitude Units

Integrate this mechanism into a complex-valued GRU, recurrent state-space model, or stochastic latent block. Let the ordinary neural update produce u_(t+1) and v_(t+1). Set rho_(t+1) = softplus(u_(t+1)) and delta_theta_(t+1) = v_(t+1), retain theta_t as a phase state, and sample N_(t+1) from the geometric law with mean rho_(t+1). For a reparameterized continuous approximation, sample U uniformly from (0,1) and use N_(t+1) = -rho_(t+1) log(U). Construct z_(t+1) = sqrt(N_(t+1)) exp(i(theta_t + delta_theta_(t+1))) and pass its real and imaginary parts onward. For a minibatch, phase-align each sample by z_tilde_b = z_b exp(-i arg(sum over b of z_b)); compute z_bar = average of z_tilde_b and the empirical amplitude moments. Add alpha times L_coh to the task loss, warm alpha from zero, and clip rho or normalize amplitudes to avoid rare numerical explosions. The paper contributes the geometric occupation law, the pi/4 ratio, and the factorial moment hierarchy; phase alignment, recurrence, and minibatch estimation are neural adaptations. First run a one-layer complex GRU on sequential MNIST or permuted MNIST, comparing deterministic complex states, additive Gaussian noise, and this module. Measure R = |z_bar|^2 / mean(|z|^2) as a function of learned rho. The quantitative prediction is that channels with rho at least 50 should have R close to pi/4 = 0.785, within approximately 10 percent, and moment_q divided by mean(|z|^2)^q should approach q! for q from 2 to 4. If these signatures do not appear, the transfer mechanism is falsified even if accuracy changes.