Valuation-Threshold Congestion Router

Integrate this at the router logits of a sparse MoE Transformer. For each token representation h_b, compute a valuation vector v_b = W_v h_b in R^E, where E is the number of experts. Maintain one nonnegative congestion price tau_j per expert, shared across the minibatch and updated with an exponential moving average of recent loads. Use this pseudocode: v = Wv @ h; score[b,j] = v[b,j] - tau[j]; route[b] = argmax_j score[b,j]; dispatch tokens to selected experts up to capacity C_j; compute load[j] = number of assigned tokens; update load_ema = beta * load_ema + (1-beta) * load; update tau[j] = relu(tau[j] + eta_tau * (load_ema[j] / C_j - 1)). In the forward pass use the one-hot argmax route. During training use a straight-through estimator whose backward pass substitutes the gradient of softmax(score divided by temperature T), or a softmax over the top two experts. Use U(v_j,tau_j) = v_j - lambda * tau_j, with lambda either fixed to one or tuned over a small grid. Estimate loads directly from each batch; the first implementation does not require a valuation-space density estimate or an optimal-transport solver. Compare against top-1 routing with Switch-style auxiliary load balancing on a 12-layer, 8-expert Transformer trained on a small language-model corpus. Measure validation perplexity, overflowed or dropped-token fraction, expert-load coefficient of variation, router assignment churn, and training throughput at equal FLOPs. The main falsifiable target is fewer overflowed tokens and lower load variance at the same auxiliary-loss weight, or equal load variance with less routing regularization and improved perplexity.