  The values of the so-called {\em Dedekind--Rademacher cocycle} at certain
real quadratic arguments are shown to be global $p$-units in the narrow Hilbert
class field of the associated real quadratic field, as predicted by conjectures
of Darmon, Dasgupta, and Vonk. The strategy for proving this result combines an
approach of Darmon-Pozzi-Vonk with one crucial extra ingredient: the study of
infinitesimal deformations of irregular Hilbert Eisenstein series of weight one
in the anti-parallel direction, building on the techniques in earlier work of
Betina, Dimitrov, and Pozzi.
