Cantor Spectrum via a Reducibility-Duality Bridge for the Mosaic Almost Mathieu Operator

We study the mosaic Almost Mathieu operator, a quasiperiodic model that naturally admits a singular strip-Jacobi representation. By establishing a duality framework and extending the correspondence between the integrated density of states and the fibered rotation number to this setting, we obtain an effective reduction to $SL(2,\mathbb{R})$ cocycles. As a consequence, combining Aubry duality, reducibility theory, and the Moser–Pöschel argument, we prove that the spectrum is a Cantor set for all noncritical parameters.