  We generalize the theory of periodic homogenization for multidimensional SDEs
with additive Brownian and stable L\'evy noise for $\alpha\in (1,2)$ to the
setting of singular periodic Besov drifts of regularity $\beta\in
((2-2\alpha)/3,0)$ beyond the Young regime. For the martingale solution from
Kremp, Perkowski '22 projected onto the torus, we prove existence and
uniqueness of an invariant probability measure with strictly positive Lebesgue
density exploiting the theory of paracontrolled distributions and a strict
maximum principle for the singular Fokker-Planck equation. Furthermore, we
prove a spectral gap on the semigroup of the diffusion and solve the Poisson
equation with singular right-hand side equal to the drift itself. In the CLT
scaling, we prove that the diffusion converges in law to a Brownian motion with
constant diffusion matrix. In the pure stable noise case, we rescale in the
scaling that the stable process respects and show convergence to the stable
process itself. We conclude on the periodic homogenization result for the
singular parabolic PDE.
