econsidered again, starting from \[[@CR27]\], which proved the quantization of the Hall conductance of an interacting electron system using quasi-adiabatic evolution of the groundstate around a flux-torus, under the *assumption* of a volume-independent spectral gap. In \[[@CR22]\] we followed a different approach, and proved the quantization of the interacting Hall conductivity by writing it as a convergent series, and by showing that all the interaction corrections cancel exactly, thanks to Ward Identities. Our result holds for interacting fermionic Hamiltonians of the form $\documentclass[12pt]{minimal}
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                \begin{document}$${\mathcal {H}}_0+U {\mathcal {V}}$$\end{document}$, where $\documentclass[12pt]{minimal}
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                \begin{document}$${\mathcal {H}}_0$$\end{document}$ is quadratic and gapped, with the chemical potential in the middle of a spectral gap of width $\documentclass[12pt]{minimal}
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                \begin{document}$$\Delta _0$$\end{document}$, $\documentclass[12pt]{minimal}
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