This paper presents a functional proof strategy for the Riemann Hypothesis within the Pattern Field Theory framework, using the Allen Orbital Lattice (AOL) as the geometric and arithmetic substrate. The approach establishes an equivalence between two independently constructed systems: the continuous Equilibrion Hamiltonian, which describes recursive curvature balance along the critical line, and the discrete AOL operator, whose prime-anchored spectrum exhibits Gaussian Unitary Ensemble statistics after unfolding.The analysis shows that both systems produce self-adjoint spectra aligned to Re(s)=1/2, with nontrivial zeros interpreted as equilibrium nodes of the prime field. The operator includes prime-weighted potentials and duplex curvature phases, producing spectral behavior consistent with Riemann-class dynamics. Numerical diagnostics across 30—50 lattice shells confirm Wigner—Dyson spacing, with Kolmogorov—Smirnov and Cramér—von Mises distances stable under randomized phase ensembles. Control experiments removing the prime anchors fail to reproduce this universality, isolating prime weighting as a necessary structural condition.The paper incorporates the updated operator formulation, extended curvature analysis, and cross-references to related Pattern Field Theory results, including the conduction constant tau = 71.2 ± 3.9 ms measured during Pattern Alignment Lock formation. These results support the claim that the prime-indexed curvature harmonics on the AOL constitute a physical equilibrium field and that the Riemann Hypothesis corresponds to its stationary manifold.The work integrates mathematical, geometric, and empirical components into a unified framework and provides a stable model linking prime recursion to field equilibrium. This version includes updated references, consistency corrections, and the completed analytic—geometric correspondence across the continuous and discrete representations.