Both internal placements of K 5

The pieces have the following internal descriptions:

1. $K_{47}$ is a finite derivative of the two independent separated retarded factors $R(b-a)R(b-z)$;

2. $K_5$ has the two constructed internal amplitude maps of Theorem thm:I2;

3. $K_6,K_8,K_9$ have finite conormal amplitudes on ${\Lambda }_{\mathrm{resp}}$.

On regular conormal charts their upper orders are, respectively, one for $K_{47}$, $1/2$ for $K_6$, and $3/2$ for $K_8,K_9$. These are internal orders only.

Proof. Formula (eq:K47) is obtained by inserting the kernel of $X$ into the copied coefficient $C_G$ and transferring the scalar derivative to the resulting kernel. Its two relative variables are independent under the diffeomorphism

Both endpoints are separated on this patch. The two cone differentials are independent, and the total derivative count is at most four. In dimension sixteen and codimension two the copied conormal convention gives upper order one.

For (eq:K6), transfer the divergence to the incoming retarded line and then transfer the scalar derivative. The two signs cancel. The actual bilinear local vertex forces $a=z$. Its two normal derivatives give upper order $2-(16-10)/4=1/2$.

The $K_5$ statement is Theorem thm:I2; the two response statements are Input statement input:Hresponse. The copied regrouping then gives (eq:presum).

This proposition constructs a common precomposition presentation of the observable Hessian. It does not construct the operator map required for (eq:target) at $N$.

grouped cross-line distribution is not the joint product

The two-line tensor

is a defined distribution in its two independent relative variables. Its pullback by

is defined, since this map is a diffeomorphism onto its image. Denote the resulting grouped distribution by ${\mathcal{C}}_{\mathrm{cross}}$.

Only this grouped pullback is used. It is not expanded into mixed $H$–${G_{\mathrm{act}}}^{\mathrm{A}}$ products, and it is not replaced by the numerical factor of XIV-G's audit.

The local mixed Hessian of ${\mathcal{V}}_g$ supplies its inherited finite fiber contraction and differential operators. Applying those operators to a defined distribution is legitimate. It does not define multiplication of that distribution by $K_{\mathrm{ret}}$ at a characteristic covector cancellation.

actual missing I3 map

Let ${\mathcal{K}}_{\mathrm{int}}$ denote the finite direct-sum coefficient spaces in Proposition prop:precomposition. Their reconstruction map is defined on the full retained patch.