Farey pairs applied to photon trajectory on the blackwhole disc and photon spheres connections

For photon spheres the natural dimensionless ratio is: > **ρ = Ω_ph / (Ω_ph + λ)** Both quantities in [M⁻¹], ratio is dimensionless, always in (0,1). ✅ --- ## The Beautiful Result | Spin a/M | ρ | Farey bracket | |---|---|---| | 0.000 | **0.500000** | (3/7, 1/2) — exactly at midpoint | | 0.300 | 0.498850 | (3/7, 1/2) | | 0.700 | 0.488492 | (3/7, 1/2) | | 0.900 | 0.467932 | (3/7, 1/2) — approaching boundary | | 0.998 | 0.423837 | **(2/5, 3/7)** — crossed into next pair | Schwarzschild sits **exactly at ρ = 1/2** — the Farey midpoint of (0,1). As spin increases, the photon sphere becomes more unstable (λ grows faster than Ω), and ρ drifts downward through the Farey sequence. At near-maximal spin a=0.998, it crosses into the next Farey pair — a **topological transition** in the resonance structure. This also connects directly to **quasinormal modes**: ω_QNM ~ Ω_ph − iλ/2, so ρ encodes the QNM damping ratio too.