Proposition: Empty complete residual

If the complete endpoint residual space is one-dimensional and the endpoint evolution is an isometry $W:C\to A\otimes B$, exact recovery exists. The disappearance of a macroscopic hole alone does not imply this hypothesis.

Proof. For $\dim B=1$, the complementary channel is the unique trace replacement into $B$. Apply Theorem thm:constant. A macroscopic mass variable does not specify the dimensions or state dependence of other inaccessible registers.