Augmentation Realization of N-Series and Finite-Quotient Descent

Let G = H1 >= H2 >= ... be a prescribed N-series and let A be its weighted augmentation filtration. In every degree we identify the kernel of the canonical map Hn/Hn+1 -> An/An+1 with the cokernel of an explicit map between normalized bar groups. When the series is finite and separated, the resulting criterion reduces to integer linear algebra. For a countable cofinal tower of finite quotients, descent to the discrete integral group ring is equivalent to a uniform bound on Losey-expression width. Tahara's class-three finite 2-groups and the Hartl-Mikhailov-Passi description of fourth dimension quotients then give a countable product P and an element h in gamma3(P) such that h is in D4fin(P) but not in D4(P).