Geometric Unification of Physical Interactions

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 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.

Within the framework, 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.

The primitive interaction prefactor reduces to:

ħ · c / m_P^2

while the fine-structure constant appears secondarily as the geometric saturation ratio:

α = m_z^2 / m_P^2

associated with the transition between volumetric and structural propagation regimes.

The framework further suggests that several conventional physical quantities may admit a geometric reinterpretation. 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.

Within the geometric interpretation, physical mass corresponds to an effective geometric coherence cost associated with stable propagation across the discrete metric substrate.

The framework further suggests that the observed proton-electron mass ratio is geometrically consistent with the asymptotic confinement structure of a discrete three-dimensional manifold. Numerical evaluation of the geometric scaling relations yields agreement with empirical mass ratios at a precision exceeding 99.99% without introducing free fitting parameters.

Whether this geometric framework corresponds to physical reality remains ultimately contingent upon experimental verification near the proposed mesoscopic transition regime.

v37: Conceptual refinements, new sections, typographical corrections.