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

=== Goal: test if a simple QAT geometric model can produce the observed fine-structure constant ===

α  =  e24πε0 ℏc\alpha \;=\; \frac{e^2}{4\pi\varepsilon_0 \,\hbar c}α=4πε0ℏce2
by expressing e2e^2e2 and/or eee in geometric terms of a spherical surface (surface radius rrr, surface charge density σ\sigmaσ, plus the action scale h/2πh/2\pih/2π).

We’ll treat α\alphaα as a derived quantity:
# Model the total charge eee as charge density σ\sigmaσ over a sphere of area 4πr24\pi r^24πr2: e≈σ⋅4πr2⇒e2≈σ2(4πr2)2=16π2 σ2 r4.e \approx \sigma \cdot 4\pi r^2 \quad\Rightarrow\quad e^2 \approx \sigma^2 (4\pi r^2)^2 = 16\pi^2 \,\sigma^2 \, r^4 .e≈σ⋅4πr2⇒e2≈σ2(4πr2)2=16π2σ2r4. (This is a simple model — we'll discuss refinements afterwards.)
# Plug that into the usual expression for α: α(σ,r)  =  e24πε0 ℏc  =  16π2 σ2 r44πε0 ℏc  =  4π σ2 r4ε0 ℏc.\alpha(\sigma,r) \;=\; \frac{e^2}{4\pi\varepsilon_0 \,\hbar c}                 \;=\; \frac{16\pi^2 \,\sigma^2 \, r^4}{4\pi\varepsilon_0 \,\hbar c}                 \;=\; \frac{4\pi \,\sigma^2 \, r^4}{\varepsilon_0 \,\hbar c}.α(σ,r)=4πε0ℏce2=4πε0ℏc16π2σ2r4=ε0ℏc4πσ2r4. (I’ll leave ℏ=h/2π\hbar = h/2\piℏ=h/2π explicitly in the code and algebra if you prefer.)
# Ask: what (σ,r)(\sigma,r)(σ,r) produce α≈1/137.035999…\alpha \approx 1/137.035999\ldotsα≈1/137.035999… ? More specifically: - Fix a range of plausible radii rrr (from subatomic up to atomic scales) and see what surface charge density σ\sigmaσ is required to match α. - Alternatively fix σ\sigmaσ (e.g., one elementary charge distributed on the sphere, or surface charge density coming from plasma/temperature models) and see predicted α.
# If the simple surface-charge model is not close, try small refinements: - Allow for an “effective fraction” factor fff so e=f⋅σ⋅4πr2e = f \cdot \sigma \cdot 4\pi r^2e=f⋅σ⋅4πr2 (to model concave/convex partitioning or incomplete coverage). - Consider event-rate / quantization corrections (photon exchange rate, Planck-time scale) as a secondary parameter.