Both internal placements of K 5

In the finite retarded-pole module of Definition def:amodule, the I2 amplitude map for both $K_5$ placements is Constructed:

1. one explicit module contains their regular diagonal, separated flow-out, and internal transition presentations;

2. (eq:pole) fixes the relation between the noncharacteristic multiplier and the retarded cone amplitude;

3. the untested reconstruction has uniform compact-set control with $z$ retained;

4. every finite lower-degree term is included in the exact remainder (eq:split);

5. the reconstructed internal kernels have the wavefront bounds (eq:Gamma5).

This is not a single-Lagrangian principal-symbol theorem. It supplies no joint characteristic product with the quantum cross-lines. It opens I3 only.

Proof. Proposition prop:pole proves the transition identity. Theorems thm:dyadic and thm:reconstruction prove the uniform estimates and reconstruction. Corollary cor:remainder proves the complete finite lower-degree statement. Proposition prop:both applies these results to both exact placements, and Proposition prop:WF supplies the untested wavefront control.

3: the common precomposition sum and the missing joint map

I2 is now supplied in its named amplitude module. I3 is therefore entered, but its target is stronger than internal reconstruction.

other precomposition pieces

The regrouped $B_4+B_7$ kernel can be written directly with the existing coordinate-response line $L$:

This retains the complete $C_G$ term. It does not reintroduce the canceled explicit ${\partial }^2\psi$ term.

Let ${\mathsf{D}}^{\alpha \mu ,I,J}$ be the constant bilinear fiber map determined by

The local $B_6$ piece has kernel

On the retained patch,

has a defined finite operator presentation with $z$ retained.