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

=== 1. The primary-rainbow geometry for a water drop gives a typical angle in the 138° region for visible wavelengths (this is the familiar ~42° from the anti-solar point — the complementary number is ~138°). That geometry is set primarily by the refractive index n(λ) of the medium. The numbers we computed (≈138.33°) are consistent with that. ===
# Narrow emission lines (even strong ones, as modeled here) only slightly change the spectrally-weighted mean angle. To move the weighted mean by a large amount (e.g. ≈0.47° or ≈0.8°) you would need unrealistically huge line-to-continuum power ratios or a radically different refractive-index dispersion. In other words: a realistic spectrum of light passing through water droplets will not naturally produce an angle close to 137.036… or 137.508… by spectral-weighting alone.
# That does not by itself rule out a deeper geometric link between α and golden-angle for QAT — it only shows that rainbow/refraction through water droplets plus realistic spectra does not produce the tiny offsets you were asking about. The atomic-scale sphere proposed in QAT is not a water droplet: its physical refractive/interaction properties could be very different (so the water-rainbow calculation is a model / analogy, not a proof).
# Conformal / spherical geometry ideas and qualitative similarities (spheres, right-angles, 4π, etc.) are still interesting and useful as intuition. But numerically matching fine-structure constants to macroscopic spherical optics will demand either: (a) a different medium (not water), (b) a different geometric mapping from atomic-scale QAT spheres to the optical deviation formula, or (c) a genuinely different mechanism than the classical rainbow-caustic.