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

Let $2<n<m\le \omega$. Let ${\text{CA}}_n$ denote the class of cylindric algebras of dimension $n$ and ${\text{RCA}}_n$ denote the class of representable ${\text{CA}}_n$s. We say that $\text{A}\in {\text{RCA}}_n$ is representable up to $m$ if $\text{Cm}\text{At}\text{A}$ 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 $2<n<m<l\le \omega$, an $l$ square represenation is $m$-square; the converse however is not true. The variety ${\text{RCA}}_n$ is a limiting case coinciding with ${\text{CA}}_n$s having $\omega$-square representations. Let ${\text{RCA}}_n^m$ be the class of algebras representable up to $m$. We show that ${\text{RCA}}_n^{m+1}⊊{\mathbf{\text{RCA}}}_n^m$ for $m\ge n+2$.