The characterization is what every algorithm of the paper runs on, so it is stated about the schedule itself rather than assumed. $Feasible$ unfolds to the existence of a $JITSchedule$, a piece of data with five conditions; the theorem is what licenses replacing it by the two conditions on the set.

The depth condition is used throughout the paper and proved nowhere in it. It is not an immediate consequence of the partition formulation: the content is that greedily colouring the jobs in order of their start times never needs more colours than the largest number of jobs alive at one instant. It is stated here because the algorithms of Sections 4, 5 and 6 index their tables by the jobs alive at an instant, and that index is only the right one because of this equivalence.

The depth statement carries the hypothesis that the processing times of $Z$ are positive. It is not cosmetic. A job with $q_j = 0$ has an empty interval, is therefore running at no instant at all, and conflicts with nothing; it is invisible to the left-hand side of the equivalence, while the right-hand side still has to give it a machine — which is possible only if there is one. The paper assumes positive processing times throughout.

Decidability of feasibility is recorded as a separate statement, in the form the paper's Observation 2 needs: it is enough to test the counting condition at the start times of the selected jobs, of which there are at most $n$.