the batching theorem needs concavity and equal claim masses

agent-decided, reversible. Two hypotheses were understated. The no-splitting lemma needs concavity, not star-shapedness: with other load already at the two dates the comparison is between increments rather than levels, and a star-shaped non-concave cost with unit load at each of two legal dates makes an even split of a claim of mass two strictly cheaper than any atomic assignment. The run structure needs, in addition, equal claim masses: the exchange that sorts service into arrival order swaps two claims between dates, which leaves the loads unchanged only when the masses are equal, and with unequal masses a crossed assignment can strictly win when a saturating date should take the larger claim.

So each result now sits at its exact strength: persistence and finite-horizon concentration need star-shapedness, no-splitting needs concavity, the run structure needs concavity and equal masses, and the closed form needs linearity and no assumption on masses at all.

Rejected alternative: keeping the exchange proof, which changed the loads it was comparing.