Mobility-Gap Robustness and Středa-Hall Separation in Disordered Axion Pumping

We show that the noncommutative second Chern number of a $(3+1)$D disordered axion pump remains quantized beyond global spectral gap closure at $W_g\approx 14.5$. Large-scale computation of the second Chern number and finite-size scaling demonstrate convergence of ${\mathrm{C}\mathrm{h}}_2$ toward $-1$ well into the mobility-gap regime ($W\lesssim 16.5$), while the Středa-Hall response ${\mathrm{\partial }}_{\phi }{\mathrm{C}\mathrm{h}}_{0,3}$ departs rapidly after global spectral gap closure, serving as an internal control that distinguishes bulk mobility-gap protection from finite-size artifacts. Level statistics identify a 3D unitary Anderson transition at each pump slice and locate the mobility-edge bottleneck at $\tau =0$, with critical disorder $W_c\approx 18$.