Lemma: Weighted Taylor verification on a compact support

Fix $L⋐U$. Suppose the exact localized complete family has a finite representation

where ${\beta }_{\ell }\in \mathbb{R}$ and $k_{\ell }\in {\mathbb{N}}_0$. Assume:

1. each $D_{\ell }$ is a fixed compact integration domain, $a_{\ell }$ is integrable, and the $x_{\ell ,i}$ are bounded;

2. every $w_{\ell ,i}>0$;

3. $F_{\ell }[\Phi ]$ is linear in $\Phi$, smooth in its last variables on neighborhoods containing the Taylor segments, and its derivatives through order $N_{\ell }+1$ are uniformly bounded by $C_L\Vert \Phi {\Vert }_{m_L,L}$;

4. $${\beta }_{\ell }+w_{\ell ,\min }(N_{\ell }+1)>0,w_{\ell ,\min }=\underset{i}{\min }{w}_{\ell ,i}.$$

Taylor expansion gives, on this fixed support, a finite power–logarithm expansion with continuous coefficient functionals and a remainder $C_L{\delta }^{a_L}\Vert \Phi {\Vert }_{m_L,L}$, for some $a_L>0$.

A global expansion additionally requires compatible local coefficient distributions and one common finite inventory.

Proof. The Taylor polynomial is

Its uniform remainder is bounded by

Integration against $a_{\ell }$ preserves the bound. The prefactor leaves a positive remainder power, with at most a fixed logarithm.

Collect negative exponents and degree-zero logarithms into $S_{\delta }$, degree-zero constants into $T$, and positive powers into the remainder. A smaller positive power absorbs every finite logarithmic factor. The test-jet bounds give coefficient continuity on the fixed-support test space.