Theorem: Calculated scalar seed transition coefficients

Status: Proved for displayed mathematical data.

Fix a compact characteristic localization as above. For amplitudes $H$ supported there, and sufficiently small $ϵ$,

Here $K$ is a fixed enlarged compact neighborhood, and the constants are uniform for the stated support class. For the response measure,

exactly.

These are localization asymptotics in $ϵ$. They are not smoothing asymptotics in $\delta$.

Proof. For $5a$, use (eq:radial-5a). On the $r=-T{n}_{\mathbf{m}}$ cone, $|r{|}_{\mathrm{E}}^2=2T^2$. Replace the amplitude by its value at $T=0$. The removed cone factor has total mass

The remaining integral is $⟨{\mu }_0,H⟩$. The amplitude difference is at most $CT\Vert H{\Vert }_{1,K}$, and

The remaining cone and passive measures have bounded mass on a fixed compact set. This proves (eq:tip-5a).

For $5z$, use (eq:radial-5z). At $T=0$, the ambient point is $(z,z+S{n}_{\mathbf{n}},z,y)$, again giving ${\mu }_0$ after renaming the base variable. The same radial mass and Lipschitz estimate prove (eq:tip-5z).

For $47$, a sufficiently small coincidence neighborhood satisfies $\beta -\gamma \ge c_0>0$. Use (eq:47-integral) without a smoothing amplitude. On its seed,

Set $\gamma =ϵG$, $\omega =ϵW$. Then

and the cutoff confines $(G,W)$ to a fixed bounded set. The density becomes

Its leading factor is ${ϵ}^3/(8{\pi }^2)$. The cutoff differs from $b_0(2G^2+|W{|}^2)$ by $O({ϵ}^2)$, uniformly on that bounded set. The amplitude differs from its incidence value by $O(ϵ)\Vert H{\Vert }_{1,K}$, and the density factor differs from $1/(8{\pi }^2)$ by $O(ϵ)$. Therefore the leading coefficient is

Changing $\sqrt{2}G$ to a Euclidean coordinate and then using three-dimensional polar coordinates identifies this constant as $c_{47}$ in (eq:transition-constants). All errors are $O({ϵ}^4)\Vert H{\Vert }_{1,K}$. Finally ${\theta }_{ϵ}=1$ at $r=0$, proving the response identity.