  The criticality problem in nuclear engineering asks for the principal
eigen-pair of a Boltzmann operator describing neutron transport in a reactor
core. Being able to reliably design, and control such reactors requires
assessing these quantities within quantifiable accuracy tolerances. In this
paper we propose a paradigm that deviates from the common practice of
approximately solving the corresponding spectral problem with a fixed,
presumably sufficiently fine discretization. Instead, the present approach is
based on first contriving iterative schemes, formulated in function space, that
are shown to converge at a quantitative rate without assuming any a priori
excess regularity properties, and that exploit only properties of the optical
parameters in the underlying radiative transfer model. We develop the
analytical and numerical tools for approximately realizing each iteration step
withing judiciously chosen accuracy tolerances, verified by a posteriori
estimates, so as to still warrant quantifiable convergence to the exact
eigen-pair. This is carried out in full first for a Newton scheme. Since this
is only locally convergent we analyze in addition the convergence of a power
iteration in function space to produce sufficiently accurate initial guesses.
Here we have to deal with intrinsic difficulties posed by compact but
unsymmetric operators preventing standard arguments used in the finite
dimensional case. Our main point is that we can avoid any condition on an
initial guess to be already in a small neighborhood of the exact solution. We
close with a discussion of remaining intrinsic obstructions to a certifiable
numerical implementation, mainly related to not knowing the gap between the
principal eigenvalue and the next smaller one in modulus.
