The Geometry of L^k-Canonization I: Rosiness from Efficient Constructibility

We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} of model of $T$ in a certain “efficient” way, then $T$ is rosy – in fact, a certain natural ${\mathrm{\aleph }}_0$-categorical completion $T^{\lim }$ of $T$ is super-rosy of finite $U^{\text{thorn}}$-rank. In an appendix, we also show that any $k$-variable theory $T$ of a finite structure for which the Strong $L^k$-Canonization Problem is efficient soluble has the necessary amalgamation properties up to taking an appropriate reduct.