Manuscript XIX-D@1 · XIX-D.tex · lemma · 1f47e33b

Lemma: Transport of a defined identity requires a matching map

Benchmark shapes and the scope of reported quantum checks › Why a scoped identity does not transfer to an insertion · lines 1419–1448 · proof included or linked

XIX-D@1#lem:transport

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Let $s_{1}, s_{2}$ be linear gauge operations on declared spaces $A_{1}, A_{2}$ , and let $I : A_{1} \to A_{2}$ . If

s_{2} I = I s_{1}, O_{2} = I O_{1}, s_{1} O_{1} = 0,

then $s_{2} O_{2} = 0$ .

For affine repair maps $B_{i} : C_{i} \to V_{i}$ , maps $J : V_{1} \to V_{2}$ and $L : C_{1} \to C_{2}$ transport a specified repair if

D_{2} = J D_{1}, J B_{1} = B_{2} L .

No completeness of the target correction class follows merely from that transport.

Proof. The first conclusion is $s_{2} O_{2} = s_{2} I O_{1} = I s_{1} O_{1} = 0$ . For a source repair $D_{1} + B_{1} c = 0$ ,

D_{2} + B_{2} L c = J (D_{1} + B_{1} c) = 0.

The hypotheses identify one transported family only. They do not assert that $L$ covers every target correction.

Displayed formulas in this passage

unlabeled · line 1424
$s_{2} I = I s_{1}, O_{2} = I O_{1}, s_{1} O_{1} = 0,$
unlabeled · line 1432
$D_{2} = J D_{1}, J B_{1} = B_{2} L .$
unlabeled · line 1443
$D_{2} + B_{2} L c = J (D_{1} + B_{1} c) = 0.$

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section: Why a scoped identity does not transfer to an insertion