Inverse Theorems for the Gowers Norms on Finite Abelian Groups and Higher-Order Fourier Analysis: A Transfinite Approach

We present a rigorous derivation of the inverse theoremsfor the Gowers uniformity norms U k+1(G) over arbitraryfinite abelian groups G, extending the classical theory intothe realm of transfinite ergodic systems. By constructing acorrespondence between the combinatorial properties of mul-tiplicative derivatives and the cohomology of nilmanifolds,we establish that any bounded function f : G → C withnon-negligible U k+1 norm correlates with a phase polyno-mial of degree k on a structured nilmanifold. We introducethe Hyper-Abelian Ultrafilter technique to resolve the ob-struction of characteristic factors in the limit of high-rankabelian groups. This work unifies the arithmetic combina-torics of Green, Tao, and Ziegler with the spectral theoryof higher-order dynamical systems.