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

=== I ran a small numerical survey (code executed above) and produced: ===

• Baseline α (from CODATA-like constants used in the script): α ≈ 0.007297352569278033 (this is the familiar 1/137.035999…).

• A contour plot of Δα/α (in parts-per-million, ppm) as a function of Δr/r (ppm) and Δσ/σ (ppm) over the range ±5 ppm for each variable.

• A small table of representative cases (I showed it interactively). Representative results from the table (exact, shown as ppm):
* Δr/r = -5 ppm, Δσ/σ = -5 ppm → Δα/α ≈ -30 ppm
* Δr/r = -1 ppm, Δσ/σ = -1 ppm → Δα/α ≈ -6 ppm
* Δr/r = -0.1 ppm, Δσ/σ = -0.1 ppm → Δα/α ≈ -0.6 ppm
* Δr/r =   0 ppm, Δσ/σ =   0 ppm → Δα/α = 0 ppm
* Δr/r = +0.1 ppm, Δσ/σ = +0.1 ppm → Δα/α ≈ +0.6 ppm
* Δr/r = +1 ppm, Δσ/σ = +1 ppm → Δα/α ≈ +6 ppm
* Δr/r = +5 ppm, Δσ/σ = +5 ppm → Δα/α ≈ +30 ppm

(Those numbers follow directly from the exact formula; the linear approx 2Δσ/σ + 4Δr/r is effectively identical for these tiny changes — the max absolute error between exact and linear approx in the computed grid was ~3.7×10⁻¹⁰ (i.e. negligible at ppm scale).)

I also produced a contour plot (Δα/α in ppm) so you can see the linear stripes (slope consistent with 4× effect of Δr/r and 2× effect of Δσ/σ).

[If you want I can export the table or figure as files for you to download.]