In Hilbert space, a linear source-to-flux problem in the critical (zero
eigenvalue) limit is ill-posed, but regularized by a constraint on a linear
functional, fulfilled by tuning some control variable. For any exciting
perturbation, I obtain, by spectral decomposition and perturbation theory, the
regularized flux and the regularizing control variable non-linear responses.
May the exciting perturbation be obtained, inversely, from observable
responses? Yes, in some cases, from the existence of a weight scale, a
perturbation series, determined by recursion relations, involving well-posed
source problems, and the possibility of obtaining this weight scale from
observables of both the unconstrained and constrained systems.