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 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. 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, whose weak-field limit recovers Newtonian gravitation and leading-order relativistic redshift behaviour. Photon configurations correspond to projection-limit boundary states of the expansion invariant, from which the observer-independent spatial–temporal metric ratio c emerges geometrically. A geometric interpretation of quantum probability is developed, in which Born-type weighting arises from dimensional reduction of 4-dimensional configurations, and measurement-induced localisation follows from geometric compatibility constraints between projection-limit and finite-metric systems. The formulation provides a unified geometric framework for relativistic behaviour, gravitational dynamics, and quantum probability structure.