ZPIF (Zero Pair Interaction Functional) A Proposed Quadratic Spectral Framework for Exploring Connections Between Prime Gaps, Dark Energy, Consciousness, and Spacetime Structure

Source abstract

We introduce ZPIF (Zero Pair Interaction Functional) as a proposed quadratic spectral framework that explores potential connections between four fundamental puzzles: prime gaps, dark energy, consciousness, and spacetime structure. Departing from classical linear spectral methods, ZPIF incorporates second-order self-interactions between spectral modes within a separable Hilbert space. We also introduce genuine cross-mode interactions $\sum_{n \neq m} \gamma_n \gamma_m \overline{c_n} c_m$ to capture true pair interactions between distinct spectral modes.The framework is built around a quadratic spectral law $\lambda \sum_n \gamma_n^2 |c_n|^2$, where $\gamma_n$ denote spectral frequencies, $c_n$ are spectral coefficients, and $\lambda$ is an interaction parameter. This structure suggests a common mathematical origin for phenomena across different scales, though this remains a heuristic proposal.ZPIF provides operator-theoretic foundations with convergence proofs and numerical simulations using the first 100 non-trivial zeta zeros. We explicitly define all framework elements: the spectral operator $\mathcal{D}$, test functions $f_x(t) = e^{-t^2} \cos(t \log x)$, spectral coefficients $c_n$, and the interaction parameter $\lambda$.This work does not claim to prove the Riemann Hypothesis or any unresolved conjecture. It offers a mathematical proposal for further investigation, with clear acknowledgments of its heuristic nature and limitations. The connections to prime gaps, dark energy, consciousness, and spacetime are intended as motivational analogies rather than proven physical laws.