Proposition: Endpoint sufficiency and its missing-premise test

If a pure-preparation isometry has all non-source outputs included in accessible radiation and its final source space is one-dimensional, then the radiation preserves the entire input code exactly.

A final mass label, or vanishing final source energy, does not by itself imply this one-dimensional condition.

Proof. A one-dimensional source factor makes its complementary channel constant, so Theorem thm:exact applies. Conversely, an energy eigenspace can have dimension greater than one. A code can remain in that degeneracy without changing the energy label.