  Polymodal provability logic GLP is incomplete w.r.t. Kripke frames. It is
known to be complete w.r.t. topological semantics, where the diamond modalities
correspond to topological derivative operations. However, the topologies needed
for the completeness proof are highly non-constructive. The question of
completeness of GLP w.r.t. natural scattered topologies on ordinals is
dependent on large cardinal axioms of set theory and is still open. So far, we
are lacking a useable class of models for which GLP is complete.
  In this paper we define a natural class of countable general topological
frames on ordinals for which GLP is sound and complete. The associated
topologies are the same as the ordinal topologies introduced by Thomas Icard.
However, the key point is to specify a suitable algebra of subsets of an
ordinal closed under the boolean and topological derivative operations. The
algebras we define are based on the notion of a periodic set of ordinals
generalizing that of an ultimately periodic binary omega-word.
