  Accurate modeling of galaxy distributions is paramount for cosmological
analysis using galaxy redshift surveys. However, this endeavor is often
hindered by the computational complexity of resolving the dark matter halos
that host these galaxies. To address this challenge, we propose the development
of effective assembly bias models down to small scales, i.e., going beyond the
local density dependence capturing non-local cosmic evolution. We introduce a
hierarchical cosmic web classification that indirectly captures up to
third-order long- and short-range non-local bias terms. This classification
system also enables us to maintain positive definite parametric bias
expansions. Specifically, we subdivide the traditional cosmic web
classification, which is based on the eigenvalues of the tidal field tensor,
with an additional classification based on the Hessian matrix of the negative
density contrast. We obtain the large-scale dark matter field on a mesh with
$\sim3.9\,h^{-1}$ Mpc cell side resolution through Augmented Lagrangian
Perturbation Theory. To assess the effectiveness of our model, we conduct tests
using a reference halo catalogue extracted from the UNIT project simulation,
which was run within a cubical volume of 1 $h^{-1}$ Gpc side. The resulting
mock halo catalogs, generated through our approach, exhibit a high level of
accuracy in terms of the one-, two- and three-point statistics. They reproduce
the reference power-spectrum within better than 2 percent accuracy up to
wavenumbers $k\sim0.8\,h$ Mpc$^{-1}$ and provide accurate bispectra within the
scales that are crucial for cosmological analysis. This effective bias approach
provides a forward model appropriate for field-level cosmological inference and
holds significant potential for facilitating cosmological analysis of galaxy
redshift surveys, particularly in the context of projects such as DESI, EUCLID,
and LSST.
