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

The existing internal kernel

Patch I: the local amplitude module and reconstruction › The existing internal kernel · lines 1167–1210

XV-F-patch@1#eq:Rsource, XV-F-patch@1#eq:Rsupport, XV-F-patch@1#eq:Rpair

{Defines in the source. XIV-I.tex; input:convolution, eq:R; reported display lines 329–335; the preceding subsection identifies $R$ as the kernel of $E_{0}^{R}$ .}

Within the copied XIV-I namespace, $R$ is the kernel of $E_{0}^{R}$ , and the displayed normalization is

R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}), B := R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .

The source records $□ E_{0}^{R} = id$ . The convolution display is a premise concerning existing internal kernels, not a new product prescription for the quantum graph.

Consequently, the support of this copied kernel is not a missing sentence:

supp R = \partial J^{+}, J^{+} = {(t, x) : t \geq ∣ x ∣} .

Indeed its cone formula has the measure pairing

⟨ R, φ ⟩ = \frac{1}{4 π} \int_{0}^{\infty} t \int_{S^{2}} φ (t, t ω) d ω d t .

It vanishes off the forward cone boundary and is nonzero on every open neighborhood meeting that boundary. This elementary support reading introduces no estimate and no uniqueness theorem.

The identification is internal to the supplied XIV-I statement. It does not identify the same printed symbol in an excluded manuscript. Thus a request for the support of the copied kernel is resolved; a request for a separate cross-sheet identification can still be open.

Displayed formulas in this passage

XV-F-patch@1#eq:Rsource · line 1175
$R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}), B := R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .$
XV-F-patch@1#eq:Rsupport · line 1189
$supp R = \partial J^{+}, J^{+} = {(t, x) : t \geq ∣ x ∣} .$
XV-F-patch@1#eq:Rpair · line 1195
$⟨ R, φ ⟩ = \frac{1}{4 π} \int_{0}^{\infty} t \int_{S^{2}} φ (t, t ω) d ω d t .$