Manuscript VII-A@1 · VII-A.tex · proposition · 11da3fe6

Proposition: Classical comparison map

Support and comparison of switches › Classical comparison with fixed Cauchy data · lines 1687–1722 · proof included or linked

VII-A@1#prop:classicalcomparison, VII-A@1#eq:r21, VII-A@1#eq:cocycle

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The map

r_{21} ψ = ψ - G_{2} (D_{2} - D_{1}) ψ (mod κ^{3})

sends $Sol_{1}$ to $Sol_{2}$ , preserves Cauchy data, and is the unique map with these properties. It satisfies

r_{32} r_{21} = r_{31}, r_{12} r_{21} = id (mod κ^{3}) .

Proof. For $ψ \in Sol_{1}$ ,

D_{2} r_{21} ψ = D_{2} ψ - (D_{2} - D_{1}) ψ = D_{1} ψ = 0 (mod κ^{3}) .

The correction has zero Cauchy data. Coefficientwise uniqueness therefore characterizes $r_{21} ψ$ as the $D_{2}$ solution with the same data as $ψ$ . Composition and inversion follow from the same characterization.

Displayed formulas in this passage

VII-A@1#eq:r21 · line 1690
$r_{21} ψ = ψ - G_{2} (D_{2} - D_{1}) ψ (mod κ^{3})$
VII-A@1#eq:cocycle · line 1699
$r_{32} r_{21} = r_{31}, r_{12} r_{21} = id (mod κ^{3}) .$
unlabeled · line 1710
$D_{2} r_{21} ψ = D_{2} ψ - (D_{2} - D_{1}) ψ = D_{1} ψ = 0 (mod κ^{3}) .$

Proposition: Classical comparison map

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