  We consider q-deformations of Witt rings, based on geometric operations on
zeta functions of motives over finite fields, and we use these deformations to
construct q-analogs of the Bost-Connes quantum statistical mechanical system.
We show that the q-deformations obtained in this way can be related to Habiro
ring constructions of analytic functions over the field with one element and to
categorifications of Bost-Connes systems.
