Universal Mapping between Prime-Prime Lattice (PPL) Space and Physical Observables: Mathematical Foundation of the PPL-Jacobi Core Transformation Layer (MOD-00)

Abstract

A central challenge in unifying multi-scale physical phenomena is bridging discrete topologicalstructures with continuous physical observables measured in SI units. This paper presentsthe rigorous mathematical foundation of the PPL-Jacobi Core Transformation Layer (MOD-00), which establishes a bijective mapping between the multiplicative topological manifoldof Prime-Prime Lattice (PPL) space and the additive, continuous fields of physical space.By constructing a scale-invariant Jacobi kernel governed by prime number distributions andmodified Bessel functions of the second kind K0, we formalize both the forward transformation(converting continuous physical observables into topological phase dynamics) andthe inverse transformation (mapping phase control forces back into physical source termsand actuators). We demonstrate the universal applicability of this foundational layer acrosstwo critical domains: flux pinning density in high-temperature superconductors and phasesynchronization control in nuclear fusion plasma.