Manuscript VII-A-ST@1 · VII-A-ST.tex · proposition · c39a67a3

Proposition: Classical Cauchy comparison and cocycle

Causal support, switch equality, and classical comparison › A unique comparison, rather than false equality · lines 1368–1409 · proof included or linked

VII-A-ST@1#prop:classicalcomparison, VII-A-ST@1#eq:r21, VII-A-ST@1#eq:comparison-equivariance

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The map

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

takes $Sol_{1}$ to $Sol_{2}$ and preserves formal Cauchy data. For each source solution it is the unique target solution with those data. Hence it is the unique map between these solution spaces with those properties, and

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

If a gauge transformation is supported in a common plateau,

r_{21} [F_{κ}^{*} γ] (F_{κ}^{*} ψ) = F_{κ}^{*} (r_{21} [γ] ψ) (mod κ^{3}) .

Proof. For $D_{1} ψ = 0 (mod κ^{3})$ ,

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

The correction has zero data. Coefficientwise uniqueness follows successively from Lemma lem:Cauchy. The composites in the cocycle assertions solve the same final equation with the same initial data, so uniqueness proves them.

For (eq:comparison-equivariance), both sides solve the transformed target equation by plateau covariance and have the same Cauchy data, since the transformation is the identity near $Σ_{0}$ . Apply uniqueness again.

Displayed formulas in this passage

VII-A-ST@1#eq:r21 · line 1371
$r_{21} [γ] ψ = ψ - G_{2} (D_{2} - D_{1}) ψ (mod κ^{3})$
unlabeled · line 1381
$r_{32} r_{21} = r_{31}, r_{12} r_{21} = id (mod κ^{3}) .$
VII-A-ST@1#eq:comparison-equivariance · line 1386
$r_{21} [F_{κ}^{*} γ] (F_{κ}^{*} ψ) = F_{κ}^{*} (r_{21} [γ] ψ) (mod κ^{3}) .$
unlabeled · line 1395
$D_{2} r_{21} ψ = D_{2} ψ - (D_{2} - D_{1}) ψ = D_{1} ψ = 0 (mod κ^{3}) .$

Proposition: Classical Cauchy comparison and cocycle

Displayed formulas in this passage

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