Diagnostics and the excluded edge argument

Let $\xi \in {\mathfrak{g}}_U$, $\zeta =0$, and

The backgrounds vanish on $0\le t\le \delta$. The rescaled layer therefore has zero source and zero solution, and both limiting data at $t=\delta$ are zero. The upstream pure-gauge laws give ${\mathsf{X}}_{1,\epsilon }=\xi$ and ${\mathsf{X}}_{2,\epsilon }=\frac{1}{2}\xi \cdot \partial \xi$ for every small $\epsilon$. This checks the complete two-jet and the zero-layer regime, not the general layer estimate.

Near a causal cylinder containing the observation dependence region, take $h=\psi (t)\eta$, $k=0$, with $\psi ={\psi }_{\ast }$ near zero. Spatial cutoffs outside the cylinder leave this evaluation unchanged. Then $a=\psi \eta$, $A=a_2[\psi \eta ]=0$, and

For sufficiently small $\epsilon$, the exit positions are

while the exit velocities are $-{\psi }_{\ast }$ and ${\psi }_{\ast }^2$. The profile moments vanish with the positions. The quadratic velocity is retained and agrees with (eq:velocities).

Let $m=0$, $h=\psi (t)\eta$, $k=0$, with smooth temporal data near the relevant causal cylinder and spatial cutoffs outside it. Let $g=g_0(t)$ there. Define

Since $a_2[\psi \eta ]=0$, the massless second scalar vertex vanishes. On the smooth free solution $u=t$, X-C's completed representatives give

and

If $\psi$ is supported in $t>\delta$, these numbers are already independent of $\epsilon <\delta$. For a genuine layer with $\psi (0)\ne 0$, put

Then

On $\mathrm{supp}f$, ${\sigma }_{\epsilon }=1$, so

with $O(\epsilon )$ error in the ramp evaluation. The first limiting number is $\int f({\Psi }_0-\Gamma ){\mathrm{d}}^{4}x$.

The limiting time-coordinate velocities are $-\psi (0)$ and $\psi (0{)}^2$. The multiplied-equation first velocity instead tends to $-\psi (t)+\psi (0)$, displaying the homogeneous difference. These are classical numbers, not quantum expectation values. They check the rate and signs but do not replace the proof.