  For a K3 surface S, we study motivic invariants of stable pairs moduli spaces
associated to 3-fold thickenings of S. We conjecture suitable deformation and
divisibility invariances for the Betti realization. Our conjectures, together
with earlier calculations of Kawai-Yoshioka, imply a full determination of the
theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The
work may be viewed as the third in a sequence of formulas starting with
Yau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical data
suggest the motivic invariants are linked to the Mathieu M_24 moonshine
phenomena.
  The KKV formula and the Pairs/Noether-Lefschetz correspondence together
determine the BPS counts of K3-fibered Calabi-Yau 3-folds in fiber classes in
terms of modular forms. We propose a framework for a refined P/NL
correspondence for the motivic invariants of K3-fibered CY 3-folds. For the STU
model, a complete conjecture is presented.
