We introduce ZPIF (Zero Pair Interaction Functional), 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.
Ebrahim Elsayed (Mon,) studied this question.