Rayleigh-Jeans Condensing Router

1. Integration point: insert this router immediately before expert dispatch in a sparse MoE layer. For a batch of B tokens, compute ordinary router logits a_bi, convert them to soft assignments r_bi = softmax_i(a_b), and define batch occupations n_hat_i = sum_b r_bi. Define each expert energy as epsilon_i = c_latency times normalized measured latency_i plus c_loss times EMA(expert loss_i) minus c_utility times EMA(expert utility_i); normalize all components across experts. 2. Pseudocode: initialize T and mu below min(epsilon). On each batch, estimate n_hat, solve sum_i T/(epsilon_i - mu) = N and sum_i epsilon_i T/(epsilon_i - mu) = E_star using bisection or Newton iterations, and form n_RJ. Add the KL, energy, and mass penalties to the task loss. For dispatch, use adjusted logits a_prime_bi = a_bi + log(n_RJ_i + small_constant) - log(n_hat_i + small_constant), then apply top-1 or top-2 routing. Anneal T only after experts have learned distinct functions. 3. Computed directly from the mechanism: the Rayleigh-Jeans target, chemical-potential solve, and critical excited capacity. Estimated empirically: expert energies, occupations, and task-loss response. 4. First cheap experiment: train a 4- or 8-expert top-2 MLP-MoE on CIFAR-100 or a small language-model corpus, comparing standard softmax routing, Switch load balancing, and this router. Sweep T and E_star. Fit observed occupations to T/(epsilon_i - mu) and measure condensate fraction. The quantitative prediction is that when N crosses N_ex(T), the minimum-energy expert's load exhibits a sharp increase while the remaining loads follow approximately 1/(epsilon_i - epsilon_0). The measured transition should be within 20 percent of the numerically solved threshold. Also measure whether controlled condensation reduces communication and routing entropy without increasing validation loss.