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

=== Constants (we keep h/2πh/2\pih/2π explicit): ===
* h=6.62607015×10−34 J⋅sh = 6.62607015\times10^{-34}\,\mathrm{J\cdot s}h=6.62607015×10−34J⋅s
* ℏ=h/(2π)\hbar = h/(2\pi)ℏ=h/(2π)
* c=299792458 m/sc = 299792458\ \mathrm{m/s}c=299792458 m/s
* ε0=8.8541878128×10−12 F/m\varepsilon_0 = 8.8541878128\times10^{-12}\ \mathrm{F/m}ε0=8.8541878128×10−12 F/m
* target α0=1/137.035999084\alpha_0 = 1/137.035999084α0=1/137.035999084 (CODATA-ish)

Choice of radii rrr to sweep (log scale):
* from 10−18 m10^{-18}\,\mathrm{m}10−18m (very small, sub-proton scale) up to 10−9 m10^{-9}\,\mathrm{m}10−9m (large atomic/molecular), covering: r={10−18,10−16,10−14,10−12,10−10,10−9}r = \{10^{-18},10^{-16},10^{-14},10^{-12},10^{-10},10^{-9}\}r={10−18,10−16,10−14,10−12,10−10,10−9} m, and a fine log-grid around the Bohr radius a0≈5.29×10−11 ma_0 \approx 5.29\times10^{-11}\,\mathrm{m}a0≈5.29×10−11m.

Scans:
A. For each r, compute the σ\sigmaσ required to obtain α0\alpha_0α0 using the analytic inversion:

σ(α0,r)=ε0 ℏc α04π r4.\sigma(\alpha_0,r) = \sqrt{\frac{\varepsilon_0 \,\hbar c\,\alpha_0}{4\pi\, r^4}}.σ(α0,r)=4πr4ε0ℏcα0.
B. For a few physically-motivated σ choices:
# σ1 = e / (4πr^2) (one elementary charge uniformly distributed) (gives consistency check: is σ1 near σ(required)?)
# σ2 = some plasma-like densities converted to surface density (we can pick a few values) C. Produce plots:
* Required σ vs r (log–log)
* Ratio (σ1 / σ_required) vs r
* If you like: α predicted vs r for σ = e/(4πr^2) (to see how far that simple idea gets us)

Refinements to try afterwards (if initial scan is promising or instructive):
* Introduce fff (fraction coverage) or an effective radius reffr_\text{eff}reff to model concave/convex distribution.
* Include a thermal/plasma parameter to estimate σ from charge density × shell thickness.
* Test sensitivity: small changes in r or σ and corresponding Δα/α.