Lemma: Convergence and independence of the auxiliary cutoff

The limit (eq:T0) exists, is independent of the radial cutoff used, and is a continuous linear functional on compact tests satisfying $\psi (0)=0$.

Proof. Use Euclidean polar coordinates $z=\rho \omega$, with $\omega \in S^3$. Homogeneity and (eq:rational) give the polar form

The angular distribution is well defined: at its zeros the argument is a submersion on $S^3$.

The compact sphere has a finite-order distributional estimate for $A$. Since $\psi (0)=0$, the function $\psi (\rho \omega )$ and each fixed finite set of its angular derivatives are $O(\rho )$, uniformly on the sphere. The integral is therefore absolutely convergent in its radial distributional estimate near zero. It vanishes for sufficiently large $\rho$.

The omitted radial interval contributes $O(\delta )$, with constants controlled by finitely many test seminorms. Consequently all such cutoffs give the same limit, and the same estimates prove continuity.