Conditions under which persistence comparisons can be composed without ambiguity are identified. Discernibility, persistence under variation, and composability of comparison suffice to generate a consistency problem once accumulated loss becomes path-dependent. In that regime, local comparability alone does not ensure representative-independent composed comparison. Consistency under recomposition requires the introduction of at least one transport rule relating equivalence classes of local descriptions across admissible variation. This transport rule is representative-independent and does not presuppose time, dynamics, geometry, symmetry, or metric structure. It is imposed solely to stabilize persistence bookkeeping under composition. Formal definitions and propositions establish the failure of representative independence without transport, the restoration of consistency in its presence, and the minimality of transport in the sense that any consistent mechanism induces such a rule. The result establishes existence, not uniqueness, of transport and precedes any geometric, dynamical, or gauge-theoretic specialization.