Geometric Unification of Physical Interactions

[Superseded version. Please refer to the latest one.] Abstract This work proposes a geometric unification framework in which gravitation and electrostatic interaction emerge as complementary scaling regimes of a single underlying geometric operator. The model is constructed from a discrete three-dimensional metric constrained by a fundamental structural cost and a minimum spatial scale defined as ℓ = 2 * l_P. The unified interaction is expressed through the superposition of volumetric and structural terms, naturally recovering Newtonian gravitation, Coulomb interaction, relativistic saturation, and mesoscopic transition behavior within a single continuous framework. The model identifies the gravitational constant G not as a fundamental constant, but as a derived scale-dependent ratio between more elementary geometric quantities. The framework predicts a characteristic transition mass m_ϕ ≈ 4.157 × 10^–9 kg corresponding to the minimum of intrinsic acceleration. This scale emerges from the optimization between volumetric and structural geometric exponents and implies a potentially observable mesoscopic deviation from purely classical inertial behavior. The reduced Planck constant ħ is reinterpreted as an emergent consequence of discrete spatial geometry, derived from the fundamental pixel scale, the maximum propagation velocity c, and the saturation limit of the metric. Time itself is described not as a fundamental dimension, but as the accumulated geometric path required to resolve motion within a constrained three-dimensional structure. Physical motion is interpreted as an extremal process of geometric propagation within a finite and discretized metric substrate. More generally, the framework suggests that several apparently independent physical constants and interaction regimes may arise from geometric scaling constraints associated with finite spatial resolution and discrete metric saturation. v31: Conceptual refinements, new sections, typographical corrections, Markdown format.