ZPIF Self-Interactions: An Integrated Mathematical Schema for Prime Gaps, Dark Energy, Neural Dynamics, and Spacetime Geometry

Source abstract

This study introduces a unified mathematical schema based on spectral self-modulation that extends the classical Riemann explicit formulation through bilinear, trilinear, quadrilinear, and fractional interactions of spectral components. The key contribution is the Active Spectral Mode (ASM) Principle, which asserts that Riemann zeta zeros operate as dynamic mathematical agents possessing intrinsic characteristics: internal energy, magnetic moment, oscillatory behavior, cyclic recurrence, nonlocal signaling, quantum coupling, autonomous transformation, and unbounded internal complexity.The schema integrates polynomial self-modulations across multiple orders, along with fractional self-modulations including reciprocal, radical, and logarithmic forms. It also incorporates nonlocal communication, quantum state transitions, spectral evolution, auto-encoding, spectral interlinking, and emergent structural weaving.The mathematical framework is constructed on a separable Hilbert space H = L²(R), using a self-adjoint spectral operator D and test functions fx(t) = e-t² cos(t log x). All theoretical claims are validated through explicit constructions and computational tests using the first 100 non-trivial zeta zeros. Seven graphical representations generated by programming code illustrate the core ideas.This study opens a new path in spectral theory, number theory, and mathematical physics, showing that Riemann zeta zeros act as active mathematical entities whose dynamics may underlie prime distribution, dark energy evolution, neural oscillations, spacetime architecture, and the hidden coded language of the cosmos.