Geometric Unification of Physical Interactions

[Superseded version. Please refer to the latest version.]

• Empirically falsifiable derivation of G through mesoscopic measurement.
• Proton radius derived as r_p = 4 · ƛ_p , with 577 ppm discrepancy.
• Neutron-proton mass difference derived geometrically, with 709 ppm discrepancy.
• Finite radius of the electron derived geometrically at r_e ≈ 5.636 fm.
• F_s / F_g 10^38 scale gap derived as R_s = 1 / α_G = (m_P / m_p)^2.
• Ω_Λ / Ω_m ratio identified as the dynamic constant √5, with < 1% discrepancy.

This phenomenological framework presents a geometric unification of physical interactions based on the topology of a three-dimensional space governed by structural vacuum tension. Newton’s and Coulomb’s laws are integrated into a single equation, revealing the gravitational constant (G) as a scale-dependent hybrid artifact. The structural constant w = 2 is identified as a necessary component to explain the displacement between static geometry and dynamics in a three-dimensional space.

The unification is validated through precise retro-predictions without free parameters or extra dimensions. The proton radius is derived as r_p = 4 · ƛ_p, matching experimental data with a discrepancy of 577 ppm. A finite structural radius for the electron (r_e ≈ 5.636 fm) is defined, eliminating QED singularities and suggesting that the electron can not resolve the measurement of the proton due to its own size. The neutron-proton mass difference is derived as a geometric cost, matching experimental data with a discrepancy of 709 ppm. The scaling symmetry between a particle and its orbital dynamics is defined, where α acts as the scaling factor and w determines the topological boundaries. The hierarchy ratio between F_s and F_g (10^38) is derived as R_s = 1 / α_G = (m_P / m_p)^2. Fundamental quantum phenomena, including the Rydberg formula and the uncertainty principle, are described in geometric terms. The “dark energy” to “dark matter” ratio (Ω_Λ / Ω_m) is identified as the dynamic constant δ = √5, matching experimental data with a discrepancy < 1%. Time is described not as a fundamental dimension, but as the accumulated geometric distance required to resolve structural tension in a three-dimensional topology.

The model predicts the existence of a resonance mass (m_ϕ ≈ 4.157 × 10^–9 kg) and a deviation from the universality of free fall due to structural interaction with the vacuum. While the identity between inertial and gravitational mass is preserved, non-linear dynamics become a mathematical requisite at the mesoscopic scale for objects which are not strictly neutral. Experimental confirmation of this anomaly would fix the value of G with quantum precision.