Gradient Residuals - The Calculus of Clearance - Research Note XI: The Recursive Self-Registration Cascade

Essay XI derives the complete Level-0 through Level-3 RSR cascade, bringing Phase III to 75% saturation. Essay IX established the Level-1 architecture and the grain-normalised RSR threshold crossing. Essay X derived the RSR operator G_III = G_raw/(1+ρ_n), the four threshold conditions T.RC1/T.RC4/T.RC2/T.RC3, the Phase III amplification law T.AMP, the Terminal Identity T.TI, and T.NPF (No Phase IV). Essay XI derives: the cascade recurrence relations and their exact arithmetic (T.RRR); the formal self-registration condition using the same 1/d discriminability threshold as T.RC1, establishing the nature of nothing's recursion as retrospective (D.NRN); the Level-3 self-registration surplus of 387,775× (T.SRC3); the convergence of Φ_eff to Δ at rate N_sat^(-n) (T.FCP); the local update frequency suppression at each level (T.LUF); and the structural memory nature of the RSR denominator. The essay closes with the precise input specification for Essay XII.