  We implement the no-boundary proposal for the wave function of the universe
in an exactly solvable Bianchi IX minisuperspace model with two scale factors.
We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 /
arXiv:1804.01102) to include the contribution from the $\mathbb{C}\text{P}^2
\setminus B^4$ topology. The resulting wave function yields normalizable
probabilities and thus fits into a predictive framework for semiclassical
quantum cosmology. We find that the amplitude is low for large anisotropies. In
the isotropic limit the usual Hartle-Hawking wave function for the de Sitter
minisuperspace model is recovered. Inhomogeneous perturbations in an extended
minisuperspace are shown to be initially in their ground state. We also
demonstrate that the precise mathematical implementation of the no-boundary
proposal as a functional integral in minisuperspace depends on detailed aspects
of the model, including the choice of gauge-fixing. This shows in particular
that the choice of contour cannot be fundamental, adding weight to the recent
proposal that the semiclassical no-boundary wave function should be defined
solely in terms of a collection of saddle points. We adopt this approach in
most of this paper. Finally we show that the semiclassical tunneling wave
function of the universe is essentially equal to the no-boundary state in this
particular minisuperspace model, at least in the subset of the classical domain
where the former is known.
