Manuscript XV-F-patch@1 · XV-F-patch.tex · lemma · 6d6ef16d

Lemma: Support consequence of the displayed coordinate derivative

Assembled ledger, debugging, and the revised F0 verdict › A support diagnostic that must be removed from the old record · lines 2108–2141 · proof included or linked

XV-F-patch@1#lem:K8support, XV-F-patch@1#eq:K8support

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For the copied defined incidence operation in (eq:K8),

supp K_{8} \subseteq S_{\partial} := {(a, b, z) : a = z, b - a \in \partial J^{+}} .

This support consequence does not prove a conormal order or a characteristic composition theorem.

Proof. The function $B$ in (eq:Rsource) is, almost everywhere, the constant $1/ (8 π)$ on the interior of $J^{+}$ and zero on its exterior. Its distributional first derivatives are therefore supported on $\partial J^{+}$ .

The incidence $Q$ is supported on $v = a = z$ . In the copied pushforward, differentiation in $b$ acts on the factor $B (b - v)$ ; $Q$ is independent of $b$ . Hence

\partial_{b^{α}} \int B (b - v) Q (v; a, z) d^{4} v = \int (\partial_{α} B) (b - v) Q (v; a, z) d^{4} v .

The copied incidence operation is defined, and its support is contained in the image of $v = a = z$ , $b - v \in \partial J^{+}$ . The outer $b$ derivative and smooth factors do not enlarge support. This gives (eq:K8support).

Displayed formulas in this passage

XV-F-patch@1#eq:K8support · line 2111
$supp K_{8} \subseteq S_{\partial} := {(a, b, z) : a = z, b - a \in \partial J^{+}} .$
unlabeled · line 2131
$\partial_{b^{α}} \int B (b - v) Q (v; a, z) d^{4} v = \int (\partial_{α} B) (b - v) Q (v; a, z) d^{4} v .$

Lemma: Support consequence of the displayed coordinate derivative

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