The Projector–Synchronization Program: A Dependency-Mapped Research Framework for Emergent Geometry, Cosmology, and Horizon-Scale Dynamics

This paper presents the Projector–Synchronization Program (PSP), a mathematical research framework that explores the emergence of spacetime geometry, gravitational dynamics, and cosmological behavior from a nonlocal operator structure. The program is organized as a dependency graph beginning with a bilocal density operator, proceeding through Grassmannian projector geometry, non-Abelian transport structures, emergent metric construction, cosmological effective-field behavior, and horizon-scale perturbation theory. The framework is not presented as a completed physical theory. Instead, it is presented as an explicit research program in which each layer is assigned a structural status: Derived, Constrained, Observable, Candidate Mechanism, or Open Problem. The central organizing quantity is the frustration order parameter Φ² = Tr(FμνFμν), which links microscopic projector dynamics to macroscopic gravitational phenomena. The paper summarizes the current state of development, identifies mathematically established components, and isolates remaining theoretical and observational bottlenecks.