4D Projection-Limit Geometric Formulation

This paper develops a formal 4-dimensional projection-limit geometric formulation (PLG) in which relativistic and quantum-foundational phenomena emerge from the properties of a higher-dimensional Euclidean manifold. Physical configurations are represented as static geometries in ℝ⁴, while observable dynamics arise through projection and relational metric decomposition of these structures onto an observer-defined 3-dimensional Reality Plane. Within this framework, temporal ordering is identified with geometric displacement along the fourth spatial dimension rather than an independent coordinate. Under this construction, the Lorentz factor and the constancy of light speed follow as direct consequences of a conserved 4-dimensional expansion invariant.

The framework is extended to a scalar expansion field dependent on local energy density. Its weak-field limit recovers Newtonian gravitation, while its large-scale behaviour yields asymptotically flat galactic rotation profiles without requiring additional dark-matter components. Photon configurations correspond to fixed projection-limit boundaries of the expansion invariant, from which the observer-independent spatial–temporal metric ratio c emerges. A geometric interpretation of quantum probability is developed, in which Born-type weighting arises from dimensional reduction of 4-dimensional configurations. The formulation provides a geometric framework for analysing relativistic and gravitational behaviour within a unified expansion-invariant structure.