service admissibility replaces the dip criterion by a summability one

agent-decided, reversible. Persistence characterizes when some divergent allocation is affordable. Answerability admits only some allocation traces, and the constraint changes the criterion rather than qualifying it.

Under a per-window service floor — at least a fixed positive allocation in every window of the deadline length — affordability holds exactly when the window minima of the date cost are summable. A dip sequence spaced more widely than the window satisfies the unconstrained criterion and fails this one, so the two are genuinely different conditions and the separation is exhibited. Under a cumulative service demand, the demand's growth and the running minimum depth must be reciprocal.

So the cheap dates and the timely dates are traded against each other, and the parameter trading them is how often enforcement is cheap. Answerability should therefore export its admissible traces rather than a scalar quota, because which windows the service falls in is what decides affordability.

Rejected alternative: treating service admissibility as a caveat on the persistence theorem rather than as a different theorem.