Arithmetic Resonance in 5D Crystalline Lattices: Unified Derivation of the Fine-Structure Constant \alpha and Riemann Zeta Critical Line Invariants

This manuscript establishes a rigorous theoretical and computational framework for Arithmetic Resonance within a five-dimensional (5D) discrete crystalline lattice (Λ_ij). By bridging number theory and quantum geometrodynamics, the paper derives the exact analytical decoupling of the fine-structure constant via the master identity: α = e / (π · 𝒯) where 𝒯 denotes the topological pentagon rotation coefficient and π is evaluated through discrete algebraic ratios (π = Ξ / Σ ⇒ π ∈ ℚ). Key contributions of this work include:1. The Pentagon Resonance Operator (R_pentagon): Formulated to couple 5D lattice matrices with non-trivial Riemann zeta zeros (ρ_n = 1/2 + iγ_n), enforcing strict vertical alignment along the critical line Re(s) = 1/2.2. Hardware-in-the-Loop (HIL) POSIX C Verification: Empirical validation executed under strict discrete parameters within a Termux environment, achieving a zero-dispersion steady-state (η = 0) across iterative loop executions up to 10^6 steps.3. Continuum Manifold Demolition: Demonstrating that physical spacetime and quantum fields operate via discrete lattice locks rather than continuous topological manifolds, eliminating divergences inherent in traditional continuous field theories.