Geometric Unification of Physical Interactions

[Superseded version. Please refer to the latest one.] Abstract This work proposes a unified geometric framework in which gravitation and electrostatic interaction emerge as complementary asymptotic regimes of a single underlying discrete geometric substrate. The framework therefore removes the necessity of treating gravitation and electrostatic interaction as fundamentally independent forces. Both emerge relationally from the same intrinsic geometric propagation structure under different saturation states of the metric substrate. The model is constructed from a unified intrinsic acceleration equation derived from volumetric scaling, geometric projection, and relativistic saturation constraints. The framework predicts two asymptotic interaction regimes: a_g ~ m^(1 / 3) and: a_e ~ m^(-5 / 3) which emerge continuously from a unified geometric attenuation structure. The transition between both regimes is governed by the mesoscopic equilibrium scale: m_φ ≈ 4.157 × 10^(-9) kg corresponding to the unique minimum of the intrinsic acceleration function. The resulting geometric response may be expressed as: T_3(r) = r + r^(-5) recovering both asymptotic limits from a single continuous formulation. The intrinsic acceleration equation may be written in compact form as: a_i = (a_z · T_3(r)) / (1 - (L_t / r)) where the numerator represents the redistributed geometric response of the substrate and the denominator represents the remaining unsaturated metric capacity available for interaction. This compact formulation emerges from a unified geometric propagation equation combining volumetric scaling, structural attenuation, and relativistic saturation constraints. Acceleration is not interpreted as an externally imposed interaction, but as an intrinsic relational response of the discrete metric to geometric propagation constraints. Gravitation corresponds to coherent volumetric redistribution across the substrate, while the electrostatic regime emerges from structural saturation as the interaction scale approaches the geometric penetration limit. Coulomb-like behavior is therefore recovered as the asymptotic manifestation of geometric saturation near the penetration regime, rather than as the interaction of independent electric charges. The primitive interaction prefactor reduces to: ħ · c / m_P^2 while the fine-structure constant emerges as the geometric saturation ratio separating volumetric and structural propagation regimes: α = m_z^2 / m_P^2 associated with the transition between volumetric and structural propagation regimes. Under this interpretation, α ceases to represent a primitive empirical coupling and instead becomes a derived geometric ratio associated with the transition between cooperative volumetric propagation and structural saturation. The framework further suggests that several conventional physical quantities may admit a geometric reinterpretation. Several observed physical phenomena emerge from transitions in the effective dimensionality of geometric propagation across the discrete substrate. In particular, gravitation, inertia, relativistic saturation, and subatomic confinement emerge as complementary asymptotic manifestations of the same underlying geometric propagation structure. The relativistic energy identity: E = m · c^2 emerges from the relation between finite metric tension and geometric propagation length. Physical mass corresponds to an effective geometric coherence cost associated with stable propagation across the discrete metric substrate. The observed proton-electron mass ratio is geometrically consistent with the asymptotic confinement structure generated by the dual volumetric and structural scaling regimes of a discrete three-dimensional substrate. Numerical evaluation of the geometric scaling relations yields agreement with empirical mass ratios at a precision exceeding 99.99% without introducing free fitting parameters. The framework derives the Casimir force and its corresponding spectral coefficient from the geometric cost of discrete confined lattice modes, reinterpreting the phenomenon as the structural consequence of volumetric shielding collapse within a confined metric region. Discrete quantum energy states and atomic orbital quantization may emerge deterministically from the topological synchronization of closed propagation paths upon a discrete spatial lattice. This suggests that the standard Hilbert-space formalism of quantum mechanics acts as an effective statistical description of an underlying topological synchronization process. The linear confinement potential of the strong interaction emerges as a deterministic consequence of dimensional propagation collapse, where unshielded topological defects project non-diluting 1D flux structures across the discrete spatial lattice. The model suggests that gluonic dynamics may emerge effectively from the underlying geometric flux structure, without requiring fundamental independent gauge fields as primitive postulates. Whether this geometric framework corresponds to physical reality remains ultimately contingent upon experimental verification near the proposed mesoscopic transition regime. v40: Conceptual refinements, new sections, typographical corrections.