called a **polyfold**. One observes that $B(Q,\omega)$ is canonically a polyfold after fixing merely a choice of appropriate gluing profile and a sequence of strictly increasing non-zero weights below $2\pi$ for the sc-structure (important for sc-Fredholmness later), see also \[PKAPPLICATION\] (3).

A first order differential operator maps as a (local) section from the base into the fiber with 1 functional regularity lower. If this is defined notationally as $U_m\to F_m$, then $F_{m+1}$ (from functional regularity perspective) still makes sense over a point in $U_m$. This structure is encoded in a local **strong bundle**, written as $U\triangleleft F$, and is respected by strong bundle maps (linear in the fiber direction). A strong bundle retract typically is denoted by $K:=R(U\triangleleft F)$. A section and respectively an $sc^+$ section, which mapping from the base to fibers preserves the regularity and respectively increases the regularity by one, make sense. We can lift a good uniformizer of $B(Q,\omega)$ to a good uniformizer of $E(Q,\omega, J)$ with the parameter space of a good uniformizer for $E(Q,\omega, J)$ as a strong bundle retract. Then $E(Q,\omega, J)$ becomes a strong bundle polyfold the same way as $B(Q,\omega)$ is a polyfold. The Cauchy-Riemann operator $\bar\partial_J$ is an sc-smooth section 