Philip Anderson has argued in a sequence of highly influential papers that disordered lattice Hamiltonians in 1 and 2 dimensions have pure point spectrum with spatially localized eigenvectors, unless extraordinary conditions are in place. Such extraordinary conditions are met at the transition between a strong topological insulator and a normal one, where one can witness a spectacular delocalization of the eigenvectors, which often occurs at extreme levels of disorder. The first part of my talk will exemplify these phenomena. Random lattice Hamiltonians can be generated from the crossed product algebra induced by the action of $\mathbb Z^d$ on a disorder configuration space equipped with a probability measure $(\Omega, {\rm d} \mathbb P)$. In the second part of the talk, I will formalize the localized character of the spectra in terms of certain Sobolev norms associated with derivations coming from natural circle actions and the trace induced by ${\rm d} \mathbb P$. Then I will point out that the maximal domains of the generating cyclic cocycles consist of related Sobolev spaces. Lastly, I will present local index theorems for the pairings of the top cyclic cocycles, which hold on the maximal domains. These results explain why these pairings continue to have quantized ranges when pushed over the Sobolev domains and why the localization-delocalization transition must occur whenever a change in the quantized valued of a pairing is observed.