The hierarchy problem is strictly a geometric scaling identity

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

B. Muon anomaly (g – 2):

  Theory: Lepton as an icosahedron (12 vertices).
  Formula: a_μ = (α / (2 · π)) + (α^2 / 12)
  Calc.: 0.001165847
  Exp.: 0.001165920
  ∆: 63 ppm

C. Hubble tension (H_0):

  Theory: Local parallax due to electron topology (1 / 12).
  Formula: H_0_local = H_0_CMB · (1 + 1 / 12)
  Calc.: 73.01 km/s/Mpc
  Exp.: 73.04 km/s/Mpc
  ∆: 328 ppm

D. Fine-structure constant (α):

  Theory: α^–1 = (4 · π^3 + π^2 + π) – (α / 24)
  Calc.: 137.0359996
  Exp.: 137.0359991
  ∆: 0.005 ppm

E. Cosmic ratio:

  Theory: Ω_Λ / Ω_m = √5
  Calc.: 2.2360
  Exp.: 2.2144
  ∆: < 1%

Preprint: https://doi.org/10.5281/zenodo.17847770