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

=== Goal: find the refractive index n for which the primary-rainbow geometry yields a deviation equal to either ===

A) the golden-angle complement: 180∘−(golden angle)≈42.493…∘180^\circ - \text{(golden angle)} \approx 42.493\dots^\circ180∘−(golden angle)≈42.493…∘, and
B) the quantity 180∘−(1/α)180^\circ - (1/\alpha)180∘−(1/α) (which gives the angle complementary to the numerical value 1/α ≈ 137.035999…).

Procedure (numerically): for each candidate n evaluate D(i) on the interval 0<i<90∘0 < i < 90^\circ0<i<90∘ and take its minimum; then compute β = 180° − D_min. Solve for n that gives the required β.

Results (rounded):
* refractive index that gives β = 360°(1−1/φ) (i.e. golden-angle complement): ngold≈1.3301645601.n_{\text{gold}} \approx 1.3301645601.ngold≈1.3301645601.
* refractive index that gives β = 180^\circ - 1/\alpha (so that Dmin⁡=1/αD_{\min} = 1/\alphaDmin=1/α in degrees): nα≈1.3269643615.n_{\alpha} \approx 1.3269643615.nα≈1.3269643615.

Difference: ngold−nα≈0.0032002n_{\text{gold}} - n_{\alpha} \approx 0.0032002ngold−nα≈0.0032002 (i.e. a change of about 0.24% in n).

What that means numerically:
* golden-angle ↔ effective n ≈ 1.33016
* 1/α ↔ effective n ≈ 1.32696
* these n values are very close to real optical indices for water-like media (water near visible wavelengths: n ≈ 1.333 with dispersion across color). So the tiny change in n needed to shift 137.5077° → 137.036° is small and physically plausible.