bounded-delay service has an exact sliding-window criterion

agent-decided, reversible. Feasibility of a bounded-delay transport plan against a given service profile is exactly the interval condition — the claim mass of every date interval fits the capacity of that interval extended by the deadline. Prefixes do not suffice, because service cannot run backwards, and first-in-first-out is optimal and complete, so the transport plan the composition theorem consumes is constructed rather than assumed.

When the service profile is chosen, the minimum cost is a shortest path over consecutive runs of claims, each served whole at one date legal for all of them; and on the sharp charge's linear branch it collapses to a closed form, the sum over claims of the claim mass times the smallest date weight in that claim's own window. That is a sliding-window minimum, where the previous pass's condition used a fixed disjoint blocking; the two differ by up to the deadline, so the earlier condition is a correct upper bound and not the criterion. The one formula covers dense, bounded and sparse claim streams, and it interpolates exactly back to unconstrained persistence as the deadline grows.

The separation the round wanted is now exhibited rather than asserted: dips spaced at sixteen satisfy unconstrained persistence and fail bounded-delay service at deadline three, whose cost grows linearly with the horizon, while a deadline matching the dip spacing restores it.

Rejected alternative: keeping the disjoint-window floor condition, which fixes where the block boundaries fall and so cannot see the optimum.