H}'_{\text{aux}}\right)}_{I}}
 \left({\bf K}'\right) \,=\, \left[G^+_{{\cal H}_{I}\,\otimes\,
   {\left({\cal H}_{\text{aux}}\right)}_{I}}\right]\,\circ\,
   {{\cal O}^{\text{system}}_{{\cal H}_{I}\otimes
   {\left({\cal H}_{\text{aux}}\right)}_{I}} \!\left({\bf K}\right)}\,\circ\,
    \left[G_{{\cal H}'_{I}\,\otimes\,
     {\left({\cal H}'_{\text{aux}}\right)}_{I}}\right]\end{aligned}$$ together with a global functional dependence ${\cal \tilde O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)$ defined by the decomposition\
$$\begin{aligned}
{\cal \tilde O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)
 \,:=\,
 \sum_{I\in {\frak I}}{\cal O}_{{\cal H}'_{I}
  \otimes{\left({\cal H}'_{\text{aux}}\right)}_{I}}\left({\bf K}'\right)
\quad\;{\text {or}}\quad\;
{\cal \tilde O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}\left({\bf K}'\right)
 \,:=\,
 \prod_{I\in {\frak I}}{\cal O}_{{\cal H}'_{I}
  \otimes{\left({\cal H}'_{\text{aux}}\right)}_{I}}\left({\bf K}'\right)\end{aligned}$$ and [**no**]{} further local relation governing the renormalization of the correlation block part occurs.

The main advantage of a perfect RGT is a rigorous mathematical description for a local RGT. Although the structure of the local Operator ${\cal O}_{{\cal H}_{I}\otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}$ is conserved, a perfect RGT does not make use of the invariance relation (\[inveq\]).

A local RGT is said to be [*exact*]{} if it is perfect and\
$$\begin{aligned}
{\cal Z}\left[{\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}
  \left({\bf K}\right)\right]
 \,=\,{\cal Z}\left[{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}
  \left({\bf K}'\right)\right]
 \,=\,{\cal Z}\left[{\cal \tilde O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}
  \left({\bf K}'\right)\right]\;.\end{aligned}$$

If a RGT is exact it includes all needed constraints and therefore we can compare the RGT to the classical situation where non commutativity effects are absent.\
At this point we like to give some important remarks on perfect and exact 