The Purified Zeta Function (ζM30)

The classical continuous model for the Riemann Zeta function evaluates the totality of natural numbers, introducing static basal factors that generate substantial arithmetic noise. The Topological Residue Theory (TRT-M30) proposes the Purified Zeta Function (ζM30), mathematically and axiomatically bound to a discrete cylindrical lattice quotient space C ∼=Z8 × N. By extracting 73.33% of the structural noise, the geometric density of survival is revealed through the 8/30 isomorphism. The Phase Rigidity Lemma and the absolute value inequalities applied to Abel’s partial summation annihilate the asymptotic fractional error, forcing complex zeros to anchor exclusively as deterministic topological collision nodes. This framework redefines primality as an inherently subtractive positional property, transitioning the computational complexity from iterative arithmetic division to an O(1) spatial mapping paradigm. Consequently, this determinism profoundly disrupts contemporary cryptographic paradigms, shifting the generation of secure primes from probabilistic algorithmic heuristics to exact memory-bound geographic mapping.