Profiniteness and representability of spectra of Heyting algebras

We prove that there exist profinite Heyting algebras are not isomorphic to the completion of any algebra. This resolves an open problem from 2009. More generally, we characterize those varieties in which completions. It turns out exists largest such. give different characterizations this variety and show it is finitely axiomatizable locally finite. From follows decidable whether a all members In addition, introduce representable algebras, thus drawing connection classical representing posets as prime spectra.