Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · lemma · 8ca47088

Lemma: Pure-gauge response

Stage B: Cauchy solution and regulator preflight › The decisive preflight stop · lines 619–645 · proof included or linked

X-A-conv-ST@1#lem:preflight, X-A-conv-ST@1#eq:preflight

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For every $ξ \in g_{C}$ ,

w [K ξ] = E_{0} C T_{g, ε} (K ξ) = S_{ε} ξ .

Proof. Because $g = 1$ on a neighborhood of $supp ξ$ , $g K ξ = K ξ$ . Constant-coefficient derivatives commute with convolution, and direct contraction gives

C K b = □_{η} b .

Consequently,

C T_{g, ε} (K ξ) = C K (S_{ε} ξ) = □_{η} (S_{ε} ξ) .

The support of $S_{ε} ξ$ lies in the Euclidean $ε$ -neighborhood of $supp ξ$ , separated from $t = 0$ . Both Cauchy data therefore vanish. Equation (eq:leftinverse) proves the result.

Displayed formulas in this passage

X-A-conv-ST@1#eq:preflight · line 622
$w [K ξ] = E_{0} C T_{g, ε} (K ξ) = S_{ε} ξ .$
unlabeled · line 633
$C K b = □_{η} b .$
unlabeled · line 637
$C T_{g, ε} (K ξ) = C K (S_{ε} ξ) = □_{η} (S_{ε} ξ) .$

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section: The decisive preflight stop