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 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) determined by the finite causal resolution of space. This underlying metric response determines two 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 determined 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 configuration is quantified by its geometric intensity (Ψ), a dimensionless variable measuring its effective structural occupancy within the lattice. The complete interaction dynamics are determined by a single unified equation of intrinsic acceleration, which 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 mass of elementary fermions as the deterministic topological strength limit (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 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. The Planck length is derived not as a primitive background constant, but as an emergent limit of the discrete spatial metric resolution. The model reveals that the structural mismatch between the volumetric and surface scaling exponents generates a persistent topological shear pressure, frame-shifting cosmological expansion from an exogenous dark energy phenomenon into the inevitable mechanical relaxation rate of the discrete spatial metric itself. This derivation culminates in a closed-form geometric expression for the gravitational constant (G), establishing it as a nemergent relational property of the global cosmological volume and the particle boundary limit. The mutual dependence between the mass of the electron and the cosmological radius formalizes Mach's principle, proving that absolute dimensionful scale is not a pre-existing background property, but a dynamic and emergent relational state. v56: Editorial corrections.