Confluence: WHEN DOES CONSERVED-FLOW ROUTING HELP? A COUPLING CONDITION FOR MIXTURE-OF-EXPERTS, WITH A LEGALITY GUARANTEE

Mixture-of-Experts (MoE) models route each input through a few experts cho- sen by a learned gate, one layer at a time and independently. We ask when it is useful to instead use a coupled router, which commits to a single, globally con- sistent path of experts. We study a conserved-flow router: a differentiable Tero– Nakagaki (Physarum) dynamic that relaxes to one min-cost path through a graph whose edges are exactly the legal expert transitions, so illegal routing is impossi- ble by construction rather than merely penalized. Our central finding is a boundary condition. When a task’s output marginalizes over its intermediate steps, mean- ing each step can be scored independently, the coupling provides no benefit: on MetaQA 2-hop knowledge-graph QA the flow router ties a plain per-step softmax over the same legal graph. As the steps become mutually constraining, the flow’s advantage over independent gating grows monotonically, a crossover we chart on a controlled constraint-density sweep. In a small MoE transformer on a coupled compositional task, the flow router cuts the illegal-transition rate from 0.34 to 0.01 at one dominant expert per layer; its accuracy reaches 0.99 on some seeds but is unstable across seeds and below a top-1 gate on average, which we report seed by seed. To make the approach practical, we use implicit differentiation through the flow fixed point and a matrix-free solver, and verify the resulting gradients against finite differences. The contribution is the routing mechanism, a specific condition for when it helps, and a hard legality guarantee, rather than a new accuracy record.