Manuscript XIV-I@1 · XIV-I.tex · section · 29df501d

Both internal placements of K 5

I0: copied inputs and the exact predecessor stop › Both internal placements of K 5 · lines 1179–1273

XIV-I@1#sec:apply, XIV-I@1#eq:coefficientform, XIV-I@1#eq:A5, XIV-I@1#prop:both

The adapted coordinate maps are diffeomorphisms. Pulling the estimates back therefore bounds arbitrary joint tests in the original variables, including $z$ . The chain rule expresses an original-variable derivative as smooth-coefficient derivatives in the adapted coordinates. Its normal derivatives add polynomial factors and its tangential derivatives differentiate coefficients. This proves the final statement.

Uniformity in $z$ means this joint distributional control. It does not mean that the kernel admits a pointwise smooth slice at every $z$ .

{I2c: application to both $K_{5}$ placements}

coefficient extraction from the exact formula

In either adapted chart, the outer line is

R (d_{\pm}) = \frac{1}{2 π} δ (q_{\pm}) .

The inner line remains $R (r_{\pm})$ , including its vertex.

The chain rule and the Leibniz rule applied to the complete formula (eq:K5) give coefficient arrays $c_{\pm, j β}^{I, J}$ such that

K_{5, \pm}^{I, J} = j + ∣ β ∣ \leq 4 \sum c_{\pm, j β}^{I, J} (w_{\pm}) δ^{(j)} (q_{\pm}) \partial_{r_{\pm}}^{β} R (r_{\pm}) .

Factors $1/ (2 π)$ and all coordinate density factors are included in the coefficients.

Equation (eq:coefficientform) is an exact coefficient extraction rule: expand the finite differential operator in (eq:K5) in the adapted coordinates, transfer its normal derivatives to the two indicated factors, and retain every derivative of every coefficient. It is not a tensor truncation.

Define

A_{5, \pm}^{I, J} = j + ∣ β ∣ \leq 4 \sum c_{\pm, j β}^{I, J} (w_{\pm}) A_{j β} (θ, κ) \in A_{R}^{2} (V_{\pm}) .

The amplitudes (eq:A5) reconstruct the two placements in (eq:K5) exactly:

K_{5} = K_{5, +} + K_{5, -} = Rec (A_{5, +}) + Rec (A_{5, -}),

after the two adapted coordinate maps are pulled back to $W$ .

They satisfy the uniform transition estimates and exact lower-degree remainder of the preceding section. Both metric-slot placements and every inherited coefficient are retained.

Proof. The copied formula differentiates the two retarded factors at most four times. In adapted coordinates, its tangential derivatives act on smooth coefficients, whereas its normal derivatives act on $δ (q_{\pm})$ or $R (r_{\pm})$ . The finite Leibniz expansion is therefore precisely (eq:coefficientform).

Theorem thm:reconstruction proves the equality. Theorem thm:dyadic and Corollary cor:remainder apply to its finite coefficient arrays. No metric slot has been tested against $G$ or assigned a value.

The amplitudes use the two original relative variables $r_{+} = a - z$ and $r_{-} = z - a$ . The internal time orientation is therefore handled separately for each placement; one is not substituted for the other.

Displayed formulas in this passage

XIV-I@1#eq:coefficientform · line 1211
$K_{5, \pm}^{I, J} = j + ∣ β ∣ \leq 4 \sum c_{\pm, j β}^{I, J} (w_{\pm}) δ^{(j)} (q_{\pm}) \partial_{r_{\pm}}^{β} R (r_{\pm}) .$
XIV-I@1#eq:A5 · line 1230
$A_{5, \pm}^{I, J} = j + ∣ β ∣ \leq 4 \sum c_{\pm, j β}^{I, J} (w_{\pm}) A_{j β} (θ, κ) \in A_{R}^{2} (V_{\pm}) .$
unlabeled · line 1203
$R (d_{\pm}) = \frac{1}{2 π} δ (q_{\pm}) .$
unlabeled · line 1243
$K_{5} = K_{5, +} + K_{5, -} = Rec (A_{5, +}) + Rec (A_{5, -}),$

Both internal placements of K 5

Displayed formulas in this passage

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