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ZPIF (Zero Pair Interaction Functional) is introduced as a quadratic spectral operator framework extending the classical explicit formula of the Riemann zeta function. Unlike the standard linear spectral decomposition, ZPIF incorporates second-order interactions between spectral modes within a Hilbert space formulation. The framework includes a rigorous operator definition, spectral expansion, trace-class regularization, and conditional convergence under truncation. A computational scheme based on numerical zeta zeros is also proposed. The novelty of ZPIF lies in introducing a quadratic spectral energy functional consistent with classical spectral heuristics without assuming unresolved conjectures. Numerical experiments demonstrate nonlinear growth behavior and quadratic interaction effects that are absent in classical linear formulations.