Theorem: Exact positive-residual certificate

For (eq:padV),

The controlled inverse

exactly recovers the whole diary, including references.

At every ciphertext prefix $0\le k\le \ell$, for every diary input,

Thus the retained coherent curve reaches a plateau of $\ell$ bits at the cutoff, while the complete residual stays positive dimensional. Late radiation alone has recovery error $a_d$; augmented coherent radiation has error zero.

Proof. Orthogonality of the key labels makes $V$ an isometry. Tracing $B$ gives the accessible block channel, and tracing $RE$ gives the constant complete complement. The Pauli twirl gives the late-only replacement channel. Controlled inversion returns $X$ in every block, so the complete decoder is exact.

The late ciphertext marginal is $I_R/d$ for every input. Its $k$-qubit prefix is $I/2^k$. Both displayed entropy formulas follow, and Lemma lem:approx gives the late-only benchmark.