Explicit classes in Habiro cohomology

We propose a cycle description of the Habiro cohomology of a smooth variety X over the spectrum B of an \'etale Z[λ]-algebra and construct explicit nontrivial cycles using either the Picard-Fuchs equation on $X/B$ of a hypergeometric motive, or a push-forward of elements of the Habiro ring of $X/B$. In particular, we give explicit classes for 1-parameter Calabi--Yau families. The q-hypergeometric origin of our cycles imply that they generate q-holonomic modules that define q-deformations of the classical Picard-Fuchs equation. We illustrate our theorems with three examples: the Legendre family of elliptic curves, the A-polynomial curve of the figure eight knot, and for the quintic three-fold, whose q-Picard Fuchs equation appeared in its genus $0$-quantum K-theory. Our methods give a unified treatment of quantum K-theory and complex Chern-Simons theory around higher dimensional critical loci.