Assumption: Verified complete expansion

For one fixed package and profile,

where

The following requirements hold.

1. ${\mathcal{E}}_{-}\subset (-\infty ,0)$ is finite, with distinct real exponents. The exponent and logarithmic-degree inventory is one fixed finite inventory on $U$, independent of the compact test support.

2. Every $J_{\lambda },J_0$ belongs to ${\mathbb{N}}_0$.

3. The coefficients and $T$ are actual distributions in ${\mathcal{T}}^{\prime }$, not unverified formal expressions.

4. For every $L⋐U$, there are $m_L\in {\mathbb{N}}_0$, $C_L>0$, and ${\omega }_L(\delta )\ge 0$, tending to zero, such that

   $$|R_{\delta }(\Phi )|\le C_L{\omega }_L(\delta )\Vert \Phi {\Vert }_{m_L,L},\mathrm{supp}\Phi \subset L,$$

   for all sufficiently small scales, uniformly over these tests.

5. The calculation proves equality for the complete positive-scale family: every retained row, every vertex type, both terms of (eq:crossed), and every required localization sector are included.

Equal scalar scale factors have been collected.