Transition Certification with Composite Indices: The Acceptance Gap Between Compensatory and Noncompensatory Rules

Composite indices can certify transitions under weight-adaptive or common-plan rules. We formulate the comparison as a robust finite-horizon assessment with path-wise constraints and declared disturbances. The full-cone aggregate is lossless at a fixed trajectory, yet the plan/weight quantifiers need not commute. For finite menus in the stated margin-first class, the weight-adaptive accepted set is $\conv(\mathcal D)+\Rp$ and the common-plan set is the union of single-plan acceptance orthants; an abstract convexification of required margins closes that gap, while physical implementation requires a separate control assumption.An exact rational witness has a nonempty gap interior: at its canonical point both normalized margins are $6/5$, so the equal-weight margin sum $s_1+s_2=12/5$ exceeds the witness's licensing threshold $2$; each weighting licenses at least one primitive plan, though no single primitive plan meets both typed floors. A closed-form reserve threshold separates rescuable from impossible states. Under the declared linear-flow blend, fractional control closes the action-indexed gap, while the specified shared-shock blend does not restore feasibility at the canonical point; the complete-excursion result is conditional on its union-of-tubes assumption. On the witness-defined diagonal slice, every positive-elasticity member of the stated CES family falsely certifies an exact interval; the Leontief limit is typed-safe under the declared collapse convention. The shared-shock gap survives the exact $1/2$-cut, while a distinct replace reading with identical plan- and weight-independent dips collapses the distinction (SI S4). Northern-cod values supply a historical unit comparison only; the benchmark is not a fitted fishery model.