Geometric Unification of Physical Interactions

[Superseded version. Please refer to the latest one.] 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 absolute upper bound of geometric strain that the discrete metric can coordinately sustain is defined by the topological yield strength Ψ_max ≈ 2.04 × 10^21. This absolute limit deterministically anchors the electron mass and provides the foundation for the fermionic mass hierarchy. The complete interaction dynamics are governed by a single unified equation of intrinsic acceleration: a_i = (c^2 · U · Ψ) / ((m_z · c^2 · r^2) - (w · U · r · Ψ)) This formulation yields the macroscopic gravitational limit, the Coulomb limit, and the Schwarzschild saturation limit as asymptotic extremes of the same geometric curve. The model shows that independent physical forces are emergent properties of a single geometric topology. v44: Major changes.