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

=== Start from the standard expression for the fine-structure constant ===

α  =  e24πε0ℏc.\alpha \;=\; \frac{e^2}{4\pi\varepsilon_0 \hbar c}.α=4πε0ℏce2.
QAT idea: treat the electron charge eee as a surface quantity on a light/electron sphere:

e  =  σ⋅4πr2e \;=\; \sigma \cdot 4\pi r^2e=σ⋅4πr2
where σ\sigmaσ is an effective surface charge density on a spherical shell of radius rrr. (This is exactly the QAT “charge on a 4πr² manifold” idea.)

Substitute into α:

α  =  (σ⋅4πr2)24πε0ℏc=4π σ2r4ε0ℏc.\alpha \;=\; \frac{(\sigma\cdot 4\pi r^2)^2}{4\pi\varepsilon_0 \hbar c}
= \frac{4\pi\,\sigma^2 r^4}{\varepsilon_0 \hbar c}.α=4πε0ℏc(σ⋅4πr2)2=ε0ℏc4πσ2r4.
That is a compact geometric form: α ∝ σ² r⁴ (with the known physical constants as scale factors).

A neat alternative, widely used in mainstream physics and compatible with the geometry view, is

α  =  reλc\alpha \;=\; \frac{r_e}{\lambda_c}α=λcre
where re=e24πε0mec2r_e = \dfrac{e^2}{4\pi\varepsilon_0 m_e c^2}re=4πε0mec2e2 is the classical electron radius and λc=ℏmec\lambda_c = \dfrac{\hbar}{m_e c}λc=mecℏ is the (reduced) Compton length. This shows α is the ratio of an electromagnetic length to a quantum length — very geometric and directly evocative of QAT’s sphere-vs-center idea.