Manuscript Oai280-B@1 · Oai280-B.tex · proposition · 6d92ab75

Proposition: Patch isometry and normalization

A normalized two-pentagon network › The two-input orientation · lines 2036–2060 · proof included or linked

Oai280-B@1#prop:patchisometry

The matrix $W$ in (eq:Wpatch) is an isometry $C_{L} \otimes C_{R} \to H_{L} \otimes H_{R}$ . It admits both sequential orientations:

encode C_{L} by V, then apply F_{R} to its internal output and C_{R},

and

encode C_{R} by V, then apply F_{L} to C_{L} and its internal output .

Proof. Expanding either composition gives exactly (eq:Wpatch). Each composition is a tensor permutation and a composition of isometries, by Proposition prop:encoding and Lemma lem:oriented. Thus $W^{†} W = I_{4}$ . This also fixes the factor $2$ ; an unnormalized tensor contraction would not by itself establish an encoding isometry.

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unlabeled · line 2041
$encode C_{L} by V, then apply F_{R} to its internal output and C_{R},$
unlabeled · line 2046
$encode C_{R} by V, then apply F_{L} to C_{L} and its internal output .$

Proposition: Patch isometry and normalization

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