Theorem: Uniform microscopic recovery

For ${\eta }_{\mathrm{m}}={\sin }^2\Theta (T)$ and $0\le \Theta (T)\le \pi /2$,

Arbitrary correlated $QMK$ inputs are allowed.

At full transfer, with $L_T=J{e}^{-\mathrm{i}(\Omega T+\pi /2)N_{\mathrm{B}}}{V}_{\mathrm{st}}$,

Recovery is exact. For independent gauge preparation, the complete message complement is replacement by $|\varnothing ⟩⟨\varnothing {|}_B\otimes {\sigma }_G$.

Proof. The corrected no-loss branch is $J{{\eta }_{\mathrm{m}}}^{N_{\mathrm{B}}/2}$. For a purified encoded input, overlap with ideal transfer and boundary vacuum is $A=\mathrm{tr}({\rho }_B{{\eta }_{\mathrm{m}}}^{N_{\mathrm{B}}/2})\ge 0$. Since $1-r^k\le k(1-r)$, $1-A\le E_{\mathrm{code}}(1-\sqrt{{\eta }_{\mathrm{m}}})$. Pure-state distance is at most $\sqrt{2(1-A)}$. Decode and contract.

At full transfer only the no-loss branch remains, so the isometry factors as $L_T$ times boundary vacuum. The partial traces give the endpoint identities, and the independent gauge purification gives its fixed complementary factor.