Corollary: Projective residuals when the continuation record fits

Under the hypotheses of Theorem thm:boundary, a readout of $S$ on an $n$-qubit input has its destructive projective residual on every positive branch whenever $n+\left|S\right|\le N$. If no qubits survive, the residual ray is unique without this budget condition.

Proof. Write $\psi ={\sum }_{\kappa }|\kappa {⟩}_S|{\xi }_{\kappa }{⟩}_B$. For a fixed unitary $V$ on all of $B$, compare the direct continuation $\mathsf{Enter}(V(B))$ with its typed deferral. The copied input fits by hypothesis. All outcome laws involved are Born by Theorem thm:boundary. Record equality gives, for $b_{\kappa }={\Vert {\xi }_{\kappa }\Vert }^2>0$,

Choose $V$ sending ${\xi }_{\kappa }/\sqrt{b_{\kappa }}$ to a fixed basis vector. The equality makes the strategy residual have unit overlap with that ray. Thus the rays agree. The chosen continuation is fixed within each experiment and does not depend on the first reported outcome.