For finite n\geq 3, and k\geq 4, the variety SNr_n\CA_{n+k} is not atom canonical

Abstract

We show that there exists an atomic representable polyadic equality algebra of finite dimension n\geq 3, such that the cylindric reduct of its completion is not in SNr_n\CA_{n+4}, hence the result in the title. This solves an open problem in algebraic logic, though the values for k=n+1, n+2, n+3, is still, to the best of our knowlege unknown.