Full Wick ordering and joint cutoff removal

For $r+s=3$, use the kernel $k_{r,s}={K_f}^{+\cdots +-\cdots -}$ in the normally ordered $(r,s)$ block. For $n\ge s$ and $m_n=n+r-s$, define

Their finite occupied-sector sum defines ${{\mathcal{C}}_f}^{(3)}$ on ${\mathcal{D}}_{\varphi }\times {\mathcal{D}}_{\varphi }$.

For $n\ge 1$, set

This is a continuous norm on the algebraic direct-sum Schwartz domain.

In each ordered summand insert:

• bounded ultraviolet profiles equal to one for $|p|\le \Lambda$ and zero for $|p|\ge 2\Lambda$;

• a bounded transfer profile zero for $|q|\le \delta$ and one for $|q|\ge 2\delta$;

• an outer response-edge profile zero for $s\le \epsilon$ and one for $s\ge 2\epsilon$.

Take these profiles in $[0,1]$. The initial data of (eq:OptionA) are not changed.