Revision audit and remaining verification work

The revision makes the following local improvements.

1. The complement is treated as an operator-valued channel on all logical matrix units, not merely as a pair of state marginals.

2. The pair-distance leakage $L_E$ is explicitly defined. Its recovery bound is proved with the half-diamond normalization used throughout.

3. Exact recovery is linked to the named constant-complement criterion, with a complete proof and a Kraus-condition check.

4. The initially proposed complementary lower bound is retained, but it is not presented as the strongest witness. Equatorial distinguishability and the exact Bell optimum give stronger bounds in this channel.

5. A complete balanced decoder supplies an improved intermediate upper bound. Bell optimality and diamond optimality remain distinct.

6. The decoder instrument preserves the conditional quantum-output specification. Its code label is not a postselection-based recovery certificate.

7. The noise complement retains $F$, and the entanglement-breaking floor is stated for every deterministic decoder.

8. The Choi state is normalized explicitly. The input-dimension factors in its diamond conversion and the matrix-unit conversion are proved.

9. The purity-stability allowance is sharpened to $2\xi$ using an effect test.

10. The continuum endpoint obstruction is proved separately, without a numerical transmission value or a gravitational identification.

Before reporting an executed verification package, one should:

• compile until all cross-references stabilize and inspect the resulting PDF;

• construct $V_{\eta }$ and ${\tilde{V}}_{\eta ,\lambda }$ independently and check their isometry identities;

• trace full referenced states independently of the displayed marginal formulas;

• check partial-transpose and spin-flip spectra, including singular endpoints;

• verify the complete Kraus families for all three decoders and the flagged instrument;

• compare the balanced decoded channel directly with its Pauli probabilities;

• preserve software versions, tolerances, raw outputs, and failures for every executed check.

These are proposed checks, not execution claims. Floating-point agreement alone is not a rigorous certificate.