An infinite stratum of representability; some cylindric algebras are more representable than others

Let $2<n<mleq omega$. Let $CA_n$ denote the class of cylindric algebras of dimension $n$ and $RCA_n$ denote the class of representable $CA_n$s. We say that $Ain RCA_n$ is representable up to $m$ if $CmAtA$ has an $m$-square representation. An $m$ square represenation is locally relativized represenation that is classical locally only on so called $m$-squares. Roughly if we zoom in by a movable window to an $m$ square representation, there will become a point determinded and depending on $m$ where we mistake the $m$ square-representation for a genuine classical one. When we zoom out the non-representable part gets more exposed. For $2<n<m<lleq omega$, an $l$ square represenation is $m$-square; the converse however is not true. The variety $RCA_n$ is a limiting case coinciding with $CA_n$s having $omega$-square representations. Let $RCA_n^m$ be the class of algebras representable up to $m$. We show that $RCA_n^{m+1}subsetneq bold RCA_n^m$ for $mgeq n+2$.