The Geometric Matrix of Space

ABSTRACTTitle: The Unified Geometric Matrix of the Space-Time Network: Exact Invariant Derivations from the Electron to the Cosmic HorizonThis paper introduces a non-probabilistic, relational geometric framework for the space-time network that derives physical invariants without primary mass involvement. By establishing a master spatial macro-lever of 253.25 meters generated purely by the kinematic and geometric constraints of the hydrogen ground state, we demonstrate a direct causal linkage between micro-atomic metrics and macroscopic scale bounds.Using a strict vacuum rigidity frame of 6.732515 * 10^26 kilograms per meter and a baseline Planck equator marker of 1.616255 * 10^-35 meters, we present exact, one-step geometric derivations for the classical radius of the electron, the proton, and heavier atomic structures.Crucially, a rigorous mathematical verification across twenty-six orders of physical mass variants—spanning from the 0.1 eV neutrino up to the 172.5 GeV top quark and complex chemical nuclei like Uranium-238—uncovers a permanent structural node within the network geometry. We prove that the ratio between the macro-spatial Compton vortex shift and the sub-Planck Schwarzschild preform contraction is an absolute invariant locked at exactly 2.000129 for all matter. This invariant symmetry occurs because mass identically cancels out of the unified tensor equation, revealing that physical particles act as scale-dependent modulations of a deeply ordered, stationary vacuum metric.Finally, extending this network rigidity matrix to the total estimated mass of the observable metagalaxy outputs an initial preform radius corresponding precisely to a stationary, non-expanding cosmic boundary of 13.75 billion light-years. These findings suggest that the universe operates as a self-contained, mathematically locked geometric matrix where the limits of elementary particles and the cosmic horizon are fundamentally unified.