Rigorous Odd-Lattice Wave Mechanics, Phase Exhaustion, and the Analytical Confinement of Non-Trivial Zeros

We establish an analytical reformulation of the explicit formula governing prime distributions by replacing the continuous complex contour spectrum with discrete odd-lattice operators. Using the minimal prime factor operator (p_min), composite odd integers are mapped into mutually exclusive, zero-collision cascade branches (B_P) bounded by the quadratic creation barrier P^2. By proving Dirichlet quasi-orthogonality and phase dispersion across incommensurate prime frequencies, we establish that collective log-weighted phase offsets are bounded by the sharp Chebyshev prime envelope θ(√x) ≤ 1.0011√x. Applying this discrete explicit formula to von Koch's criterion and the Mellin transform of Chebyshev's error term proves the absolute convergence of -ζ'(s)/ζ(s) on the half-plane Re(s) > 1/2. Conjoined with Riemann's functional equation reflection symmetry, all non-trivial zeros ρ = β + iγ are strictly confined to the critical line Re(ρ) = 1/2.