geometric) rather than a zig-zag of refinements (of a cobordism flavor, less geometric) connecting two ‘essentially the same’ Kuranishi structures; and then for R-equivalence to be an equivalence relation, it is important to have some extra structure for achieving transitivity. This structure is captured by the notion of **level-1 structure** and is very useful for other things, e.g. using a level-1 structure ten distinct virtual theories can be unified in this article, and level-1 structures also play a key role in defining a good notion of morphisms between Kuranishi structures (via localizing a class of maps generated by embeddings and submersions between Kuranishi structures), and in doing a single perturbation for the whole non-compact moduli spaces with compact filtrations by virtual structures with corners. I did not include a level-1 structure in my starting structure, because a choice of level-1 structure can always be summoned (and choices can be compared via common level-1 refinements): one can construct such global level-1 structures for $\mathcal{K}_X$ and for an embedding $\mathcal{K}_X\Rightarrow \mathcal{K}'_X$, and it is less minimal to be included in the basic definition.

To get the exact correspondence between HWZ’s polyfold Fredholm structure and FOOO’s Kuranishi structure version as explained in [@FOOO], 