persistence and eventual full service are the same existence question

agent-decided, reversible. Supersedes the entry below. Under exogenous star-shaped date costs, unlimited deferral, fungible service, finite individual claim masses and divergent total claim mass, a persistent affordable schedule exists if and only if an affordable plan discharging every claim exists — both being equivalent to the persistence criterion liminf L_t(1) = 0. The forward direction is a diagonal: each claim takes its own cheap date on a geometric tranche of the budget. The earlier separation priced block batching, which forces a whole block onto one dip, and read that one plan's cost as the minimum.

The equivalence is about which schedules exist, not about identity of schedules: a persistent schedule may ignore the claims entirely.

Uniform bounded delay remains strictly stronger, and no gap condition substitutes for the exact sliding-window criterion — cheap dates every other date with reciprocally decaying depth give bounded gaps, a friction dipping to zero, affordable eventual service, and divergent timely-service cost. So the earlier claims that bounded gaps close the three problems, and that positive density gives bounded gaps, are both withdrawn.

Unlimited deferral makes eventual answering no harder than maintaining divergent service; the substantive Answerability constraint enters only when delay matters.

Rejected alternative: keeping the separation and treating eventual service as an intermediate obligation between persistence and timeliness. There is no such intermediate in this benchmark.