Proof: Proof of Theorem thm:supported

Proof. Leibniz's rule gives (eq:exact-supported); crossed derivatives are transferred to profile derivatives using

Apply Theorem thm:scalar with $N=q_{\sigma }$. Moment cancellation removes inverse-power logarithms. Each scaled remainder is $O(\delta (1+|\log \delta |))$. Scalar amplitude order is at most $36$, and normal plus output test derivatives add at most four, giving (eq:supported-remainder).

All singular coefficients are finite jets at $v=\xi =0$; the supported normal form already imposes $r=u=0$. They are therefore supported on $N_{\mathrm{mark}}$. Uniform compact-stage estimates prove distributional continuity.

No bounded-core cutoff occurred. Relative-cutoff independence follows from Appendix app:positive. Uniqueness follows by subtracting two expansions with vanishing remainders and successively testing their most singular nonzero difference. This also identifies the finite coefficient and gives coordinate-overlap consistency.