When Does Composition Preserve Coherence? A Product-Structure Criterion and a Magnitude Law for Multi-Agent Belief Assembly

An agentic system that splits a question among specialists and reassembles their answers can produce a belief no specialist holds. Each component passes its own audit; the assembly is incoherent, and no component-level check can see it. We show when this happens and what it costs. Componentwise auditing is sound exactly when the joint feasible set is a product of the local ones, and otherwise the expected squared incoherence is bounded below by how much the roster disagrees along the coupling. The roster determines the available variation; the router determines how it is spliced across coupled coordinates. The residual is the price of splicing one question’s answer from several members: exactly zero under one-owner assignment, and higher with fragmentation on every clique we swept. Assigning each constraint-coupled clique to one member, rather than routing its coordinates independently, costs the same elicitations and pays no incoherence, while which member is chosen decides accuracy—assuming a common-agenda regime in which one member can answer and repair the whole clique. On the evaluated partition panel, a leave-one-clique-out selector using members’ historical Brier alone outperforms coordinatewise averaging.