Positive-Measure Acceleration Strata for Prevalent Perturbations of N-Fold Pullbacks of Supercritical Almost Mathieu Potentials

Fix a strip width $\rho >0$, a Diophantine frequency $\alpha$, an integer $N\ge 2$, and $\lambda >1$. For the family $$V_{\epsilon ,\delta }(x)=2\lambda \cos (2\pi Nx)+\epsilon \delta (x),$$ we prove that there is an open prevalent set ${\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}}$ of real-analytic strip-$\rho$ perturbations such that every $\delta \in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}}$ admits ${\epsilon }_0(\delta )>0$ for which $$\begin{array}{c}{\displaystyle \mathrm{Leb}\{E\in \mathrm{\Sigma }(V_{\epsilon ,\delta },\alpha ):\omega (\alpha ,E;V_{\epsilon ,\delta })=s\}>0,0<\epsilon <{\epsilon }_0(\delta ),s=1,\dots ,N\mathrm{.}}\end{array}$$ The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves $s$ pairs of simple zeros on a positive-measure spectral set.