Theorem: Integration of a nonsummable scale expansion

Suppose

where the real exponents ${\lambda }_j\le 0$ are distinct, each $P_j$ is a distribution-valued polynomial, and locally

For $M_j=\mathrm{deg}{P}_j$, set

Then

for a distribution $T$, with the stated finite-order remainder. The raw family converges if and only if every $P_j$ is zero. Otherwise a fixed compact test detects unboundedness.

Proof. The finite polynomials solve

The scale derivative of $t_{\delta }-{\sum }_j{\delta }^{{\lambda }_j}{Q}_j(\log \delta )$ is $R_{\delta }$. Theorem thm:Dini gives the constant distribution $T$ and the remainder.

If a polynomial remains, select the smallest exponent with a nonzero distributional polynomial and then its highest nonzero coefficient. A fixed test detects that coefficient. Its inverse-power term dominates larger exponents and lower polynomial powers. When the smallest exponent is zero, its primitive has positive logarithmic degree. The fixed scalar pairing is unbounded in either case.