Allocation and Coordination under Scarce Resources: Admissible Routing, Prices, Review Sets, and Containment for Gated Work

Earlier papers in this series decide which executor, gate stack, or mechanism is admissible for one step of work. They do not decide how scarce capacity is shared when many classes of work compete for it: a principal's attention, verification memory, external review lineages, rate-limited seats, and calendar time. This paper studies that allocation problem. We allocate only among options whose safety premises remain valid under the allocation actually chosen. We keep three distinctions: fluid feasibility versus executable schedules, marginal prices versus finite policy changes, and measured review value versus worst-case review value. In a fluid linear model, the feasible and optimal allocations do not depend on the executor's error law when bounds decide admissibility and the resource use, costs, demands, and capacities are law-independent or replaced by common uniform bounds. Feasibility has an exact certificate by nonnegative resource prices; subset cuts are exact for single-resource pools but insufficient for general resource bundles. Shadow prices value the principal's minutes, but a new option strictly lowers the optimum exactly when no old optimal dual prices it out, so a negative reduced cost at one dual is not enough. Choosing optional principal reviews is a knapsack. Review value under a fixed fault law is submodular; worst-case review value need not be, and a reviewer can have zero guaranteed value alone and positive guaranteed value in a coalition. Verifying an implementation does not identify the intended specification. Interface contracts bound severity: parallel escape routes add, and serial stages multiply only through conditional ceilings. With optional participation, achievable throughput does not fall as workers are added. Substituting a mechanization for scarce work needs feasibility in every resource and every period. Queueing models, a scheduling formulation of verification compute, and MaxWeight stability complete the picture, each under its stated assumptions. Many results are classical; we state the conditions under which they apply to gated work. These are mathematical models, not measurements.