The Reachability Quotient: When Network Realization Can Be Forgotten

AbstractThis paper asks when the path and factorization structure of a network of bounded evaluators may be quotiented away without changing a specified observable. Earlier fixed-carrier results showed that, under full disclosure, terminal capability depends only on reachability. Here that positive case is embedded in a more general transport model in which edges act by pullback on kernels. The first result shows that licensed disclosure itself is kernel-equivalent to transport: any sender-computable disclosure channel can be represented by a self-transport whose pullback induces the same kernel. Disclosure and carrier-changing transport therefore belong to one operator model at the set-level semantics used here. For this model, the terminal closure under any fair firing schedule is given exactly by a walk-meet law: each node receives the meet of the source kernels transported along all finite walks into it. Literal walk structure is nevertheless too fine. Closure always factors through the effective action obtained by identifying walks that induce the same pullback operator, and every family of transports between an ordered pair admits a joint transport normal form consisting of a single tupled operator. The resulting joint operator is canonical although its decomposition into component transports need not be. The paper then establishes a hierarchy of further quotient collapses. Under transport coherence, parallel walks collapse to one effective contribution per reachable ancestor; under identity transport, the structure collapses all the way to bare reachability. A meet-redundancy analysis proves an exact fixed-profile criterion and a rigidity theorem showing that nontrivial uniform domination among partition-pullback operators essentially collapses to equality. A second observable, revelation, does not generally factor through the same quotients. The paper proves an exact zero-excess criterion for any finite number of senders and shows that transport coherence and zero-excess attainability are logically independent. The common object is therefore the induced operator action, while quotient exactness remains observable-relative. The result is a theory of the interval between concrete network realization and coarse reachability: which distinctions can safely be forgotten, which cannot, and why. Scope is finite-carrier, deterministic, static, set-level transport under fair schedules.