Rigorous control of the non-perturbative corrections to the double expansion in $g$ and $g^2\ln (g)$ for the ${\varphi }_3^4$-trajectory in the hierarchical approximation

We study the renormalization invariant trajectory of the ${\varphi }^4$-perturbation of the free field fixed point in the hierarchical approximation. We parametrize it by a running ${\varphi }^4$-coupling $g$ with linear step $\beta$-function. We rigorously control the non-perturbative corrections to finite order approximants from double perturbation theory in $g$ and $g^2\ln (g)$. The construction uses a contraction mapping for the extended renormalization group composed of a hierarchical block spin transformation with a flow of $g$.