  For the observational modeling of horizontal abundance distributions and of
magnetic geometries in chemically peculiar (CP) stars, Zeeman Doppler mapping
(ZDM) has become the method of choice. Comparisons between abundance maps
obtained for CP stars and predictions from numerical simulations of atomic
diffusion have always proved unsatisfactory, with the blame routinely put on
theory. Expanding a previous study aimed at clarifying the question of the
uniqueness of ZDM maps, this paper inverts the roles between observational
modeling and time-dependent diffusion results, casting a cold eye on essential
assumptions and algorithms underlying ZDM, in particular the Tikhonov-style
regularization functionals, from 1D to 3D. We show that these have been
established solely for mathematical convenience, but that they in no way
reflect the physical reality in the atmospheres of magnetic CP stars.
Recognizing that the observed strong magnetic fields in most well-mapped stars
require the field geometry to be force-free, we demonstrate that many published
maps do not meet this condition. There follows a discussion of the frequent
changes in magnetic and abundance maps of well observed stars and a caveat
concerning the use of least squares deconvolution in ZDM analyses. It emerges
that because of the complexity and non-linearity of the field-dependent
chemical stratifications, Tikhonov based ZDM inversions cannot recover the true
abundance and magnetic geometries. As our findings additionally show, there is
no way to define a physically meaningful 3D regularization functional instead.
ZDM remains dysfunctional and does not provide any observational constraints
for the modeling of atomic diffusion.
