The Riemann Zeta Function: Advancement, Retained Tail Hierarchies, and Stationary Geometry in the Infinite Number Framework

This article studies finite cutoff representations of the Riemann zeta function in a germ-based Infinite Number framework, distinguishing exact finite identities, retained corrections, and optional classical limit projection. First, a complex asymptotic accumulation theorem with a non-cancellation condition and an interpolation derivative criterion supports exact Advancement decompositions of complex power sums and zeta partial sums. These decompositions recover classical Euler–Maclaurin corrections and lead to a fixed-parameter scaling dichotomy conditional on whether the zeta value is zero or nonzero, together with complementary infinitesimal corrections, two-scale cancellation, and a finite sum nonzero certificate. Second, complementary fractional part tails are resolved into periodic Bernoulli boundary hierarchies with retained integral remainders. Under symmetric complementary scaling, the power exponent difference of corresponding individual unweighted boundary contributions is independent of hierarchy level and balances on the critical line. This levelwise balance concerns the individual boundary contributions and does not assert a corresponding balance of the complete retained tails. Third, Abel-regularized Fourier analysis resolves the oscillatory tails into individual modes, recovers the classical dual-cutoff relation from stationary geometry, and establishes uniform Fresnel approximations on growing neighboring mode windows. The central mode exhibits a nonconstant normalized Fresnel cluster profile despite zero classical limit projection of its unnormalized contribution. The results organize classical and germ-level information without establishing new restrictions on zeta zeros or a new approximate functional equation.