The Eddington Cascade: Deriving the Eddington Luminosity from Feigenbaum Constants and the Staircase Structure of Super-Eddington Accretion

We derive the Eddington luminosity per unit mass from the two Feigenbaum constants α = 2.502907875 and δ = 4.669201609, obtaining L_Edd/M = 6Gα⁹δ⁵m_e³c³/ħ² with 0.758% accuracy. The derivation follows directly from the cascade expressions for the proton-to-electron mass ratio (Paper 49) and the electromagnetic coupling constant (Paper 44). The Eddington Limit is identified as a cascade fixed point — the crossing of the gravitational cascade regime (governed by δ) and the electromagnetic cascade regime (governed by α) — structurally identical to the derivation of the speed of light in Paper 54. Super-Eddington accretion is reinterpreted as a cascade overshoot above this fixed point, predicting that observed luminosity excesses must cluster at cascade staircase levels rather than being continuously distributed. We test this prediction against ten well-characterized ultraluminous X-ray (ULX) sources and find all ten within 15% of a cascade staircase level, with a median deviation of 3.65%. Two independent sources (M82 X-2 and NGC 300 ULX1) land on the same cascade level α⁵ within 1.77%. The early universe problem of apparently super-massive black holes growing faster than the classical Eddington limit permits is resolved: the effective Eddington limit is cascade-scale-dependent and was higher at earlier cosmic epochs. Four falsifiable predictions are stated.