Adiabatic Stability of Z9 Sector Addresses During Black-Hole Evaporation: Closure of Gap G4

Gap G4 of the PTRH/TMRB framework called for proof that the Z9 sector addresses of the event-horizon-state computer memory (EHSCM, Paper 11) remain well-defined throughout the black hole's lifetime as Hawking evaporation reduces the horizon area A_hor(t). We provide three results at status [V] and one physical identification. (i) Adiabatic sector invariance: under quasi-static Z9-equivariant evaporation, Phi(t)^9 = 1 for all t, the eigenspace decomposition H(t) = direct-sum H_n(t) persists, and each instantaneous Hawking emission event preserves sector identity --- a state in sector m before emission remains in sector m after. The GKSL jump operators L_n = sqrt(gamma) P_n implement non-demolition readout of the sector address; the macroscopic evaporation (M_lambda decreasing) reduces the per-sector dimension d_n(lambda) uniformly. (ii) Uniform sector area scaling: A_n(t) = A_hor(t)/9 for all n and all t; this follows from the full SO(3) isometry of the Schwarzschild bifurcation two-sphere, which forces all nine sector domains to contract at the same rate with no relative drift between addresses. (iii) Information budget closure: the sector population distribution {p_n(t)} is exactly conserved for all t (Paper 16 Theorem 1(iv)); the decrease in total capacity I_total(t) = S_BH(t) = A_hor(t)/(4G_4) is accounted for entirely by Hawking emission through the READ channel (Gamma_n = gamma(t) p_n, Paper 17), with no contribution from sector mixing. Additionally, the PTRH framework admits a natural minimum black-hole size A_hor_min = 9 A_cell = 36 G_4 ln 9 (Paper 12), at which each sector supports exactly d_n_min = eln 9 = 9 orthogonal states, defining the endpoint of well-defined EHSCM operation. The three protection mechanisms --- topological (Paper 18, Gap G2), dynamical (Paper 16, Gap G8), and adiabatic (this paper) --- together guarantee that no address drift can occur at any stage of evaporation. Gap G4 is upgraded from [O] to [V]. Seven of nine PTRH gaps (G1, G2, G3, G4, G5, G8, G9) now carry status [V]; two remain open (G6, G7).