Reflection-Equation Expert Reset

Integrate the operation immediately after the router softmax and before top-k expert selection. Partition the N experts into fixed groups \(G_1,\ldots,G_m\), for example experts serving different modalities, parameter shards, or capacity pools. For each group \(G_g\), maintain positive logits \(\theta_g\), define \(a_g=\mathrm{softmax}(\theta_g)\), and compute the group mass \(c_g=\sum_{i\in G_g}p_i\). Apply the exact block reset \(p'_i=(1-\eta)p_i+\eta c_g(a_g)_i\) for \(i\in G_g\). The unconstrained version uses one global a and becomes \(p'=(1-\eta)p+\eta a\). Set \(\eta=\mathrm{sigmoid}(h)\) or activate the reset when router entropy \(H(p)=-\sum_i p_i\log p_i\) falls below a threshold \(H_{\min}\); stop gradients through the trigger but not through a. Compute the update in O(N) time and memory, never forming the dense rank-one matrix \(a\mathbf1^{\mathsf T}\). If using the Yang–Baxter pair router, apply the reset at sequence boundaries or after each routing block as the analogue of an integrable boundary. The reflection equation is an exact compatibility check for the chosen discrete pair permutation and grouped kernel, not an estimated quantity. Start with a 6-layer, 16-expert Transformer on WikiText-103 or CIFAR-100, comparing standard routing, always-on reset, and entropy-triggered reset. Track expert-utilization variance, entropy collapse, overflowed tokens, validation loss, and wall-clock overhead. The desired signal is lower utilization variance and fewer overflow events with no accuracy loss; the method fails if it suppresses expert specialization or adds measurable latency.