  We study the relationship between dark-matter haloes and matter in the MIP
$N$-body simulation ensemble, which allows precision measurements of this
relationship, even deeply into voids. What enables this is a lack of
discreteness, stochasticity, and exclusion, achieved by averaging over hundreds
of possible sets of initial small-scale modes, while holding fixed large-scale
modes that give the cosmic web. We find (i) that dark-matter-halo formation is
greatly suppressed in voids; there is an exponential downturn at low densities
in the otherwise power-law matter-to-halo density bias function. Thus, the
rarity of haloes in voids is akin to the rarity of the largest clusters, and
their abundance is quite sensitive to cosmological parameters. The exponential
downturn appears both in an excursion-set model, and in a model in which
fluctuations evolve in voids as in an open universe with an effective
$\Omega_m$ proportional to a large-scale density. We also find that (ii) haloes
typically populate the average halo-density field in a super-Poisson way, i.e.
with a variance exceeding the mean; and (iii) the rank-order-Gaussianized halo
and dark-matter fields are impressively similar in Fourier space. We compare
both their power spectra and cross-correlation, supporting the conclusion that
one is roughly a strictly-increasing mapping of the other. The MIP ensemble
especially reveals how halo abundance varies with `environmental' quantities
beyond the local matter density; (iv) we find a visual suggestion that at fixed
matter density, filaments are more populated by haloes than clusters.
