Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Formulate a surface energy functional E[R,σ(θ,ϕ)]E[R,\sigma(\theta,\phi)]E[R,σ(θ,ϕ)] containing the main pieces: ===
* Electrostatic self-energy of the thin charged shell: Uel[R,σ]  =  18πε0∬σ(Ω)σ(Ω′)∣r(Ω)−r(Ω′)∣ dΩ dΩ′U_{\rm el}[R,\sigma] \;=\; \frac{1}{8\pi\varepsilon_0}\iint \frac{\sigma(\Omega)\sigma(\Omega')}{|\mathbf{r}(\Omega)-\mathbf{r}(\Omega')|}\,d\Omega\,d\Omega'Uel[R,σ]=8πε01∬∣r(Ω)−r(Ω′)∣σ(Ω)σ(Ω′)dΩdΩ′ (for uniform σ\sigmaσ this reduces to the familiar Q2/(8πε0R)Q^2/(8\pi\varepsilon_0 R)Q2/(8πε0R)-like scaling).
* A quantum boundary term that enforces action quantization (penalty functional forcing integer action in closed loops), e.g. Squant[ψ]∼∫∂Mψˉ(ih2πγaDa−ms)ψ d3σ,S_{\rm quant}[ \psi ] \sim \int_{\partial M} \bar\psi \big( i\frac{h}{2\pi}\gamma^a D_a - m_s\big)\psi \,d^3\sigma,Squant[ψ]∼∫∂Mψˉ(i2πhγaDa−ms)ψd3σ, or for a simpler scalar toy model include a term ∼κ∫∣∇SΦ∣2 dA\sim \kappa \int |\nabla_S \Phi|^2 \,dA∼κ∫∣∇SΦ∣2dA with constrained circulation ∮∂ϕΦ dϕ=2πn\oint \partial_\phi \Phi\,d\phi = 2\pi n∮∂ϕΦdϕ=2πn.
* Dispersion/plasma correction expressed as frequency-dependent permittivity ε(ω)\varepsilon(\omega)ε(ω) and a spectral weighting W(ω)W(\omega)W(ω) for the photon field coupling; integrate W(ω)W(\omega)W(ω) times the geometry dependent scattering/deflection angles to obtain an effective angular shift.

Then numerically minimize EEE with respect to shape R(θ,ϕ)R(\theta,\phi)R(θ,ϕ) and surface distribution σ(θ,ϕ)\sigma(\theta,\phi)σ(θ,ϕ) subject to total charge constraint ∫σdA=e\int\sigma dA = e∫σdA=e and quantization constraints (action quantum). Solve the Euler–Lagrange equations numerically, compute the stationary RRR and σ\sigmaσ, then evaluate the derived α\alphaα formula

αmodel=4πσeff2Reff4ε0ℏc\alpha_{\rm model}=\frac{4\pi \sigma^2_{\rm eff} R_{\rm eff}^4}{\varepsilon_0 \hbar c}αmodel=ε0ℏc4πσeff2Reff4
and compare to the measured α\alphaα. Also compute the preferred angular partition measure (e.g. location of maximum stress/curvature or area partition ratio) and see whether it lies near the golden ratio partitions.

This is the concrete numerical test that will either show a geometric mechanism or show where the mapping fails.