# Deriving the QCD Coupling from Phase-Biased Geometry

(α, β, γ ⇒ λ ⇒ gₛ)

Key point: once the three substrate constants (α, β, γ) are locked (Foundations §5), the only proton-specific number is its envelope width σₚ. One extra dimension-less geometry ratio

$$\lambda =E_{\text{orient}}/E_{\text{tot}}$$

then fixes the strong coupling

$$g_s=\sqrt{{\displaystyle \frac{4\lambda \hslash c}{\alpha {\sigma }_p}}}.$$

1 Substrate constants (locked)

2 The $6\pi$ proton envelope

• Three caustic sheets, each carrying $+2\pi /3$ phase tilt.
• Gaussian core width σₚ = 0.84 fm = 0.84 × 10⁻¹⁵ m (charge radius).
• Envelopes and caustics are implemented exactly as in
  [[3-D Envelope Implementation]].

Total anchoring energy

For the static limit the cost is

with $[\alpha |\mathrm{\nabla }\mathrm{\Phi }{|}^2]={\text{J m}}^{-3}$ and $[{\psi }^2]={\text{m}}^{-3}$,
so $E_{\text{tot}}$ is in joules.

Orientational share

The “colour–orientation” strain comes from the azimuthal derivative
along the crease:

Define the dimension-less ratio

It depends only on the crease half-width σᵩ (in degrees).

3 Making $g_s$ dimension-less

Standard Yang–Mills Lagrangian

has dimension-less $g_s$.

In PBG the angular strain term has prefactor $\alpha$ (units J m⁻¹).
Convert to a pure number by dividing by the natural energy-per-length
scale of the proton core, $\hslash c/{\sigma }_p$:

Match coefficients to obtain

All symbols are now dimension-less or constants with SI units that
cancel, leaving $g_s$ unit-free.

4 Finite-element estimate of $\lambda$

• Mesh: 4096² in (θ, φ), axial symmetry, 2ᵑᵈ-order Helmholtz kernel.
• Crease half-width scan (σᵩ in degrees):

With σᵩ = 26°–27° (best χ² against charge–magnetic radii) we find

matching the PDG value $g_s(2\text{GeV})=1.50\pm 0.05$.

5 Uncertainty budget

Quadrature total: ± 5 %, yielding the quoted $g_s^{\text{PBG}}$.

6 Next steps

1. Full variational crease (replace Gaussian by solved profile).
2. Running coupling: compress σₚ (set μ-scale) and follow λ(μ).
3. Derive explicit SU(3) algebra from three-sheet topology.

Last edited 2025-06-10 • all units referenced to Foundations v1.0