Topological classification was built for non-interacting electrons. Band topology — Chern numbers, winding numbers, Zā invariants — assumes you can assign each electron to a Bloch state and compute the geometry of the band structure. Strong interactions shatter this picture. In a Mott insulator, electrons are localized by mutual repulsion, not by band structure. There are no well-defined bands. The usual topological invariants have no surface to compute on.
The single-particle diagnosis (arXiv:2603.11879) recovers topology from what you can still measure. Construct the one-body reduced density matrix from the single-particle Green's function. From it, extract an effective winding number and a quantum volume. These quantities — both computable from spectral weight distributions accessible to ARPES or scanning tunneling microscopy — diagnose the topological phase even in strongly correlated regimes.
Applied to the Su-Schrieffer-Heeger model with interactions, the framework distinguishes three insulating phases including correlated Mott states. The topology is not destroyed by interactions. It is relocated: from band structure (which no longer exists) to spectral weight distribution (which still does).
The no-go theorems that prohibit extending band topology to interacting systems are real. They block the obvious route — applying non-interacting invariants to interacting states. But they don't block the indirect route: extracting the topological content from the Green's function, which encodes both single-particle and many-body information in a single object. The Green's function knows things that band structure does not, because it never committed to treating electrons as independent.
The invariant doesn't migrate from one formalism to another. It was always in the Green's function. Band theory was one way to extract it. Spectral analysis is another.