theory agree on the nose via a level-1 structure; and one can still do compact perturbation in polyfold theory even if the ambient/base space does not support sc-smooth cut-off functions, by transferring via the forgetful map to compact perturbation of a Kuranishi structure R-equivalence class). The polyfold–Kuranishi correspondence is also important if one wants to construct a relative symplectic field theory which direcly extends both theory of symplectic field theory (SFT) [@SFT] using Hofer-Wysocki-Zehnder’s polyfold theory [@PolyfoldI], [@PolyfoldII], [@PolyfoldIII] and [@Polyfoldinteg] together with Fish-Hofer-Wysocki-Zehnder’s polyfold construction of SFT moduli spaces [@FHWZ] (based on [@Polyfoldanalysis] and [@PolyfoldGW]) and Fukaya-Oh-Ohta-Ono’s construction of Lagrangian Floer theory [@FOOO] using their original version of Kuranishi structures.

To illustrate key ideas better, we will not focus on respecting fiber product structures on the boundaries of the abstract structures that are present for the moduli spaces considered up to certain finite energies and combinatorial types, but we will occasionally refer to this more general situation in passing in various places. We recall that the theory of moduli spaces in symplectic geometry can be roughly divided into three parts (possibly interwoven/combined in various approaches), an analysis part showing moduli space has an abstract perturbative structure, a perturbation theory 