  This paper utilizes the properties of transforms of currents under
equidimensional cycles, as introduced in \cite{MR4498559}, to establish the
multiplicative nature of the resulting regulator map, in the derived category.
The construction relies on a synthetic presentation of the fundamental triples
of currents from \cite{MR4498559}, which exhibits group-like behavior under an
extended Eilenberg-Zilber morphism. A key component of the analysis is a
character of the permutation Hopf algebra that takes values in the function
field of an infinite-dimensional affine space.
