I recently established among other things in [@MAXCAT] that, for an FOOO’s Kuranishi structure as defined in the [@FOOO] (see \[KURANISHISTRUCTURE\]), there exists a Kuranishi structure which refines it (see definition \[KURANISHIREFINEMENT\]) and has two conditions \[MAXMATCHING\] (1) and (3) I required (see theorem \[FOOOISDIR\]); and any two such refinements are R-equivalent \[REQUIV\] as Kuranishi structures with these two conditions. This essentially removes those two conditions \[MAXMATCHING\] (1) and (3) I imposed[^3], because the theory in [@DI] factors through R-equivalence (admitting common refinements). Then by using the theory in [@DI], one immediately has a (Hausdorff) good coordinate system \[GCS\] (in a way without doing induction[^4]and applicable for a map between two such structures) which is a good coordinate system as defined in [@FOOO] (charts have global group actions and coordinate changes are global group equivariant embeddings intertwining sections with global group actions) with extra properties. The perturbation theory in [@DI] is a more structured treatment and a special case of FOOO’s chartwise inductive abstract perturbation explained in [@FO] and [@FOOO], and with the help of a level-1 structure (always achievable up to a precompact shrinking with the same index set \[LEVEL1CATEXIST\]), this perturbation agrees on the nose with their CF-perturbation 