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

=== - For reff=a0/2r_\text{eff} = a_0/2reff=a0/2 (half Bohr radius): Planck-weighted mean angle ≈ 0.097°  (very small) — That is: if the electron-sphere radius is atomic scale (Bohr radius), optical photons (visible / IR) produce only a tiny fractional phase around the circumference. So the mean angle is tiny, not near 137°. ===
* To get a Planck-weighted mean angle ≈ 137.5078° (the golden angle) using this same mapping, I found a required effective radius: reff≈3.838×10−8 mr_\text{eff} \approx 3.838 \times 10^{-8}\ \mathrm{m}reff≈3.838×10−8 m ≈ 38.4 nm — That is roughly 725 × the Bohr radius, and about 0.0765 × the Wien peak wavelength for 5778 K (Wien peak ≈ 501 nm). In other words, you need a nanometre-scale sphere (tens of nm) interacting with visible photons to get a mean angle near the golden angle under this mapping.
* The value needed to hit 137.035999…° (the inverse-α number expressed as degrees) is essentially the same (because the two target angles differ only by ~0.47°); the radii required differ negligibly for this coarse mapping.
* For comparison, single strong hydrogen/stellar lines evaluated with reff=a0/2r_\text{eff} = a_0/2reff=a0/2 give angles of order 0.09°–0.15° (Hα, Hβ, Na D, Ca K lines — all tiny).