Theorem: Complete endpoint preservation

The endpoint map is an energy-conserving isometry and transfers every residual key, transfer-time record, and pending logical system into radiation. Its complement is

Hence

For input $\rho$,

and for the Bell input,

The endpoint conditional information increment is $2s_N\log d$.

Proof. The maps send an orthonormal residual basis to an orthonormal endpoint radiation basis with one fixed source factor. Every term preserves residual-plus-radiation energy. There is no residual record or purifier outside that fixed factor.

On completed branches, the earlier occupied block is decoded using $r$. On pending branches, the endpoint occupied subspace contains the encrypted logical system and its key label; apply ${U_r}^{†}$ there. The branch labels are orthogonal. This is an explicit exact decoder on the emitted state.

Alternatively, (eq:endpointcomplement) and Theorem thm:exact establish exact recovery. The full radiation map is an isometric encoding, so it preserves the nonzero spectrum of $\rho$, proving (eq:endpointentropy). The reference statements follow from purity, or from exact decoding. Their increment over (eq:MIradiation) is $2s_N\log d$.