Manuscript Hawk06-A@1 · Hawk06-A.tex · section · c59fe67c

Why the literal proposed target needs repair

Background and the necessary corrections › Why the literal proposed target needs repair · lines 159–249

Hawk06-A@1#tab:repairs

Several statements in the proposed plan would invalidate the proof if retained without qualification.

{}1.12

incorporated in Hawk06-A.
Proposed statement	Correct statement
Proposed statement	Correct statement
Early $E$ is maximally entangled with the unknown input $L$ and is available to the decoder.	The inaccessible test reference is $Q$ , entangled with $L$ . The accessible early register $E$ purifies an independent old-vault register $H$ . Fixing a Bell state of $L E$ is not a preparation of an arbitrary variable input.
The residual becomes reconstructible because its channel is constant.	A constant residual carries no unknown logical dependence. What becomes reconstructible is the specified encoded diary algebra, not an independently variable full algebra of the physical residual.
After the threshold the residual is no longer an independent factor.	It remains a physical tensor factor. The logical reconstruction assignment changes; the tensor factorization does not.
A sufficiently small residual works for every scrambling unitary.	A finite Haar calculation yields a probability or existence statement. Some fixed unitaries can keep the diary in the residual up to the chosen cutoff.
Apply the pure Page–Sen mean to the early-entangled preparation.	For a pure diary input and $h > 1$ , the vault state is a random rank- $h$ projector divided by $h$ . Its radiation entropy is not generally the pure-state Page–Sen mean.
Enter on the late qubits permanently blocks recovery even when $E$ remains quantum.	The bound $1 - 1/ d$ applies to a purely classical returned interface. A record plus a surviving quantum $E$ is not purely classical. Teleportation can recover exactly through such an interface.
A Petz formula automatically inherits the optimal decoupling bound.	A proposed Petz map needs its own error theorem or a decoded-channel certificate. We instead specify an optimal finite decoder program and certify it.
Diamond closeness transfers entropy but never a decoder.	On correctly identified common accessible interfaces, the same fixed decoder transfers by CPTP contraction and the triangle inequality. Closeness does not create a receiver or change its access permissions.
A turnover time is the recovery threshold.	Entropy turnover, complementary decoupling, and decoder execution have different hypotheses and may occur at different cuts.
The full diamond norm equals a half-diamond bound.	We use $d_{⋄} = \frac{1}{2} ∥ \cdot ∥_{⋄}$ . A displayed half-diamond allowance $e$ gives a full diamond allowance $2 e$ .

These repairs preserve the finite target while refusing an unsupported gravitational conclusion.