Geometric Unification of Physical Interactions

Abstract This work proposes a background-independent geometric framework where gravitation, electrostatics, quantum synchronization, and subatomic confinement emerge as complementary asymptotic regimes of a single discrete three-dimensional spatial metric. The fundamental resolution limit of this causal graph is defined by the geometric pixel ℓ = 2 · l_P. The framework removes the necessity of treating fundamental interactions as independent forces, replacing them with a unified intrinsic acceleration (a_i) governed by the finite causal resolution of space. This underlying metric response dictates two fundamental geometric scaling limits: a volumetric regime scaling as m^(1/3) corresponding to gravitation, and a structural surface regime scaling as m^(-5/3) corresponding to electrostatics. The continuous transition between these regimes is governed by the mesoscopic dynamic stability scale m_φ ≈ 4.157 × 10^(-9) kg, which represents the unique minimum of geometric coherence cost. The physical state of any localized topological defect is quantified by its geometric intensity (Ψ), a dimensionless variable measuring its effective structural occupancy within the lattice. The complete interaction dynamics are governed by a single unified equation of intrinsic acceleration, which smoothly yields the macroscopic gravitational limit, the Coulomb limit, and the Schwarzschild saturation limit as asymptotic extremes of the same continuous geometric curve. Ultimately, the model shows that independent physical forces are merely emergent properties of a single geometric topology. The framework reframes the fundamental mass of elementary fermions as the deterministic topological yield strength (Ψ_max) of this discrete spatial metric, reducing the origin of matter to an explicit boundary condition of geometric saturation. Furthermore, the framework derives the fundamental electron mass from the macroscopic radius of the universe, providing a geometric boundary for the inflationary horizon and establishing mass as a dynamic relational state that scales inversely with cosmological expansion. v47: Conceptual refinements, new sections, typographical corrections.