ZPIF (Zero Pair Interaction Functional): A Quadratic Spectral Variant (QSV) within the Scale Regulated Aggregation (SRA) Framework-Analytical and Computational Perspectives

We introduce ZPIF (Zero Pair Interaction Functional) as a quadratic spectral variant within the Scale Regulated Aggregation (SRA) framework, extending the classical explicit formula of the Riemann zeta function. In contrast to the traditional linear spectral decomposition, ZPIF incorporates regulated second order self interactions between spectral modes within a Hilbert space formulation. The framework is articulated through a rigorous operator definition, spectral expansion, trace class regularisation, and conditional convergence under truncation. A computational scheme based upon numerically determined zeta zeros is also advanced. The novelty of ZPIF resides in its presentation of a quadratic spectral energy functional that is structurally consistent with the SRA law, thereby uniting heuristic spectral insights with a foundational operator theoretic base. Numerical experiments substantiate nonlinear growth behaviour and quadratic interaction effects absent from classical linear formulations. This study constitutes the first collaborative articulation of ZPIF as an SRA variant, preserving Elsayed’s interpretive extension whilst simultaneously affirming Vogt’s foundational regulated aggregation framework, thereby inaugurating a joint mathematical discourse of international calibre.