Demographic Synchronizing Expert Layer

Integrate the mechanism into a mixture-of-experts or recurrent block. Instantiate M experts f_j(x), each with a fixed or trainable phase theta_j and natural frequency omega_j, and maintain positive abundances a_j as state variables in log coordinates u_j = log(a_j). For every minibatch, compute the expert outputs, form Z = sum_j exp(u_j) exp(i theta_j) divided by sum_j exp(u_j), and use normalized abundances p_j = exp(u_j) divided by the sum of all exp(u_k) as mixture weights. The block output is y = sum_j p_j f_j(x). After each optimizer update, compute r = absolute value of Z and Phi = argument of Z. Set G_j = K r cos(theta_j - Phi) - gamma (u_j - log(a_star,j)), then update u_j by u_j plus delta_t times G_j. Update phases by theta_j = theta_j + delta_t omega_j, reducing phases modulo 2 pi. Use RK2 if Euler updates become unstable. The paper supplies the growth law and fourth-root onset; K_c and finite-size corrections must be estimated empirically. First test an eight-expert MLP on CIFAR-10 or a synthetic mixture-of-Gaussians classification problem, comparing against static softmax routing and Switch top-k routing at equal parameter count. Sweep K with gamma fixed and measure r, load entropy, loss, and gradient variance after burn-in. The falsifiable prediction is a sharp transition from finite-size incoherence to macroscopic coherence, with a log-log slope near one-fourth for r versus K minus K_c, rather than the conventional one-half slope. Also test whether gamma prevents expert collapse: above a measurable gamma range, the minimum abundance should remain positive, while removing the logarithmic term should cause concentration onto a few experts.