Discrete Cylinder Topology, Root-Free Square Arithmetic, and the Algebraic Derivation of the Prime Number Distribution in 5D Fiber Space

This paper develops a self-contained, strictly discrete mathematical framework for describing and constructively computing the distribution of prime numbers. Based on a rigorously pi-free and root-free two-dimensional unit grid (Z^2), periodic compactification yields discrete cylinders (Zn x Z) governed by the counter-rotating helix operators H_+(n) = n + 1 and H_-(n) = n - 1. We demonstrate that these operators act as elementary discrete integration and differentiation steps that span the entire number space via a five-stage dimensional cascade (D = 1 to 5). From the flanking product H_+(p) * H_-(p) = p^2 - 1, we derive the exact invariance for all primes p >= 5: p^2 = 24m + 1 where the resonance height m = k(3k +/- 1) / 2 is governed by the generalized pentagonal numbers. Composite numbers residing on the primary helix traces unfold via the commutative semigroup operation: m_c = m_1 oplus m_2 = 24 * m_1 * m_2 + m_1 + m_2 as multi-dimensional product tubes into a fifth fiber dimension. Finally, we prove that the strictly positive global net torsion of nested cylinder shells, combined with Lagrange's Four-Square Theorem, enforces a closed algebraic 5D hypersphere, guaranteeing lossless scaling of the system from 10^0 to 10^36 and beyond.