Multiple Fractional Cohomological Equations and Quantitative Mixing on Nilmanifolds

We develop a new analytic method for quantitative mixing of automorphisms on nilmanifolds. The method is based on the introduction and solvability of \emph{multiple fractional cohomological equations of Type~$I$} (sum type). We prove that these equations are solvable in a cohomology-free range governed by the spectral behavior at the edge $0$, with estimates in partial Sobolev/H\"older norms along (weak) stable/unstable subgroup directions only. As consequences, we obtain exponential decay of order-two correlations under partial regularity, without transverse derivatives, and quantitative mixing of all orders (a quantitative Rokhlin theorem) with rates explicit in the dynamical data. In particular, we show that irrational automorphisms exhibit super-exponential mixing of all orders for $C^\infty$ observables. To our knowledge, these are the first examples of super-exponential mixing beyond the torus, and the first examples of all-orders super-exponential mixing.