Particles are treated as points, but reality is 3D.
Electrostatics and gravity have historically been considered separate interactions with an apparently inexplicable gap. I propose this is entirely explained by the ratio between mass and volume and the ratio between mass and surface.
If we define m_z = √α ⋅ m_P ≈ 1.859 × 10^–9 kg, the ratio of forces becomes identical to the geometric scaling ratio:
F_e / F_g = (m_z / m)^2
This is not an approximation. It is an exact geometric identity derived from the scaling of acceleration:
a_e ∼ m^−5/3
a_g ∼ m^1/3
Below m_z, surface scaling dominates. This is the electrostatic regime.
Above m_z, volume scaling dominates. This is the gravitational regime.
There is no hierarchy problem, only a transition between different geometric ratios.
This same geometric model consistently resolves major tensions across all scales with zero free parameters:
A. Proton radius (r_p):
Theory: r_p = 4 · ƛ_p
Calc.: 0.84123 × 10^–15 m
Exp.: 0.84075 × 10^–15 m
∆: 577 ppm
Electrostatics and gravity have historically been considered separate interactions with an apparently inexplicable gap. I propose this is entirely explained by the ratio between mass and volume and the ratio between mass and surface.
If we define m_z = √α ⋅ m_P ≈ 1.859 × 10^–9 kg, the ratio of forces becomes identical to the geometric scaling ratio:
This is not an approximation. It is an exact geometric identity derived from the scaling of acceleration:
Below m_z, surface scaling dominates. This is the electrostatic regime. Above m_z, volume scaling dominates. This is the gravitational regime.
There is no hierarchy problem, only a transition between different geometric ratios.
This same geometric model consistently resolves major tensions across all scales with zero free parameters:
A. Proton radius (r_p):
B. Muon anomaly (g – 2):
C. Hubble tension (H_0):
D. Fine-structure constant (α):
E. Cosmic ratio:
Preprint: https://doi.org/10.5281/zenodo.17847770