The possible values of critical points between varieties of lattices

We denote by the (∨,0)-semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor in its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A∈V with at most ℵ2 elements such that is not isomorphic to for any B∈W. The bound ℵ2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices.