  We describe a theory of logarithmic Chow rings and tautological subrings for
logarithmically smooth algebraic stacks, via a generalisation of the notion of
piecewise-polynomial functions. Using this machinery we prove that the
double-double ramification cycle lies in the tautological subring of the
(classical) Chow ring of the moduli space of curves, and that the logarithmic
double ramification cycle is divisorial (as conjectured by Molcho,
Pandharipande, and Schmitt).
