Scale Ratios & The Dissolution of Magic Constants

Act I — The Tuned Fit Degeneracy

Two arbitrary dials to hit 10³⁹

In classical astrophysics, the electrostatic repulsion between a proton and electron exceeds their gravitational attraction by 2.27 × 10³⁹. In early drafts of the theory, this gap was spanned by fitting an artificial screening parameter $\kappa$ and a CMB amplification parameter $A$:

Interactive Parameter Map: Click or drag on the 2D parameter space below to change the screening factor $\kappa$ and amplification $A$ simultaneously. Notice that the "correct" physical target lies along a degenerate diagonal line—proving the parameters were not independent constants, but a mathematical patch.

Act II — The Cosmic Scale Bridge

How (r_e / ℓ_P)⁴ spans the universe

In EMC01 and GOM02, the fitted parameters are discarded. The force ratio is determined by the ratio of the electron's wake boundary ($r_e \approx 2.82 \times 10^{-15}$ m) to the Planck length ($\ell_P \approx 1.62 \times 10^{-35}$ m), scaled by the electromagnetic coupling $\alpha^2$:

Cosmic Scale Viewer: Drag the slider to zoom from the Planck length ($10^{-35}$ m) to the atomic Bohr radius ($10^{-10}$ m) and watch the lattice structures and defect wake boundaries resolve geometrically.

The legacy screening constant kappa_factor = 10⁻⁹ was simply an approximation of $\alpha^2 \approx 5.33 \times 10^{-5}$ scaled by coordinate projection, while the amplification factor cmb_amplification = 10³⁰ approximated the square root of the scale ratio $(r_e/\ell_P)^4 \approx 10^{80}$. No fitted parameters are required: the universe's scaling is entirely built of physical boundaries.

Act III — The k-Hierarchy Phase Rotation

Phase velocity ratios as gearing factors

In GOM02, the force hierarchy emerges from the k-hierarchy ladder. The speed ratio $k = c/v$ converts scales geometrically:

• Proton surface: $k_{\text{proton}} \approx 0.5461$ (phase velocity $v_p = 1.831c$ — inside the c-boundary)
• Bohr orbit: $k_{\text{electron}} \approx 137.04$ (phase velocity $v_e = \alpha c$)

Act IV — The Verdict: Superseded by Geometry

A fully integrated force landscape

The magic numbers were scaffolding. They are replaced by the ratio of classical electron radius to Planck length $(r_e/\ell_P)^4$ and the fine structure constant squared $\alpha^2$. In SDT, the force hierarchy is purely topological and scale-driven, requiring no G, no M, and no dark matter.