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

=== 1. The mean angles we computed using a realistic optical geometry (raindrop refraction + dispersion) produce values that are near the golden-angle region (137.5°) — in this toy model we got ~137.69°. That’s in the same ballpark as the golden angle (137.5078°) and noticeably above the inverse fine-structure constant value 1/α ≈ 137.035999… (so the mean here is between the two numbers). ===
# The precise mean angle is sensitive to details that we approximated: the exact dispersion n(λ) for liquid water, how strong the emission lines are relative to continuum, the spectral range used, and how narrow the lines are. With small adjustments (different dispersion coefficients, stronger narrow lines, different temperature), the weighted mean moves by tenths to a few degrees.
# Earlier you saw results ~139–140° from a different approximate dispersion model. that difference (≈1–3°) is exactly the kind of sensitivity you get from using different n(λ) choices or a different spectral band. So nothing paradoxical — the weighted mean moves noticeably with dispersion model and weighting details.
# The small offset you’ve been intrigued by (~0.47° between golden angle and 1/α) is much smaller than typical dispersion-model uncertainties and the shifts caused by modest changes in spectrum. So if you want to argue for a meaningful numerical connection (137.036 ↔ 137.5078 via spectral geometry), you’ll need higher-precision inputs and an explanation why the remaining tiny offset (∼0.47°) is robust and not just a modeling artifact.