Both internal placements of K 5

Partial fractions followed by the same distributional limit give (eq:partialfraction) when $\rho >0$.

The cone formula has physical homogeneity degree $-2$ in dimension four. Its Fourier transform consequently has degree $-2$ as well. Alternatively the same identity follows by positive dilation in (eq:MR).

The damping in this proof computes the Fourier transform of an existing kernel. It is not a regulator for a graph product, and no restoring limit of the quantum coefficient is used.

uniform pole chart on either characteristic sheet

For $s\in \{+1,-1\}$, define

Take a conic neighborhood of that nonzero sheet satisfying

The opposite factor $\omega +s\rho$ is then nonzero.

Let

On the stated conic neighborhood,

The coefficient $a_s$ is smooth and homogeneous of degree $-1$, and $b_s$ is smooth and homogeneous of degree $-2$ there.

Off ${\lambda }_s=0$, the complete expression equals $-1/p(\kappa )$. Its singular and regular pieces in (eq:pole) are both retained.

Temporal inverse Fourier transformation gives exactly

Thus the pole and noncharacteristic multiplier reconstruct the same retarded kernel, whose separated-cone coefficient is the one in (eq:R).

Proof. For $s=+1$, isolate the first term of (eq:partialfraction); for $s=-1$, isolate the second. The opposite denominator is smooth on the selected chart, giving (eq:pole). The homogeneities are immediate.

Away from the pole, both boundary values become their ordinary reciprocals. Combining the two fractions gives $-1/p$.

The elementary Fourier identity

follows by Fourier transforming the right-hand side with positive damping. Applying it to (eq:partialfraction) gives $\Theta (t)\sin (\rho t)/\rho$. Its spatial inverse transform is the fixed cone kernel (eq:R), by Lemma lem:multiplier and Fourier injectivity.

The $i\pi \delta ({\lambda }_s)$ part is fixed by the original retarded support. It is not an adjustable internal counterterm. Replacing it would change $R$ and hence the frozen pair.

This pole chart is not a characteristic prescription for ${\mathcal{T}}_{\mathrm{form}}$. Only one pre-existing internal line is being represented.

2b: a finite distribution-valued amplitude space

coordinates for the two placements