Manuscript Oai276-A@1 · Oai276-A.tex · section · 215068af

Gate 1: the rectangular zero-block assertion

Stage 1: audit the five external proof interfaces › Gate 1: the rectangular zero-block assertion · lines 613–649

Oai276-A@1#eq:audit-block

For rectangular $A, B, C$ of the same shape, set

M = (A C B 0), x = Tr A A^{†}, y = Tr B B^{†}, z = Tr C C^{†}, x + y + z = 1.

The source's thm:block asserts

S (M M^{†}) \geq x S (A A^{†} / x) + y S (B B^{†} / y) + z S (C C^{†} / z) + g_{2} (y z),

with zero-weight terms omitted SourceBlock276.

The audit obligations are:

prove the inequality for noncommuting blocks;
retain the direction of the entropy inequality;
justify singular blocks without inverses of those blocks;
preserve the nonzero singular values under rectangular zero padding.

A scalar check, commuting special case, or numerical singular-value comparison does not discharge the first obligation. A matrix entropy inequality is not yet a coding theorem.

Displayed formulas in this passage

Oai276-A@1#eq:audit-block · line 629
$S (M M^{†}) \geq x S (A A^{†} / x) + y S (B B^{†} / y) + z S (C C^{†} / z) + g_{2} (y z),$
unlabeled · line 616
$M = (A C B 0), x = Tr A A^{†}, y = Tr B B^{†}, z = Tr C C^{†}, x + y + z = 1.$