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 Aₕor (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 Hₙ (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 Lₙ = sqrt (gamma) Pₙ implement non-demolition readout of the sector address; the macroscopic evaporation (Mₗambda decreasing) reduces the per-sector dimension dₙ (lambda) uniformly. (ii) Uniform sector area scaling: Aₙ (t) = Aₕor (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 pₙ (t) is exactly conserved for all t (Paper 16 Theorem 1 (iv) ) ; the decrease in total capacity Iₜotal (t) = SBH (t) = Aₕor (t) / (4G₄) is accounted for entirely by Hawking emission through the READ channel (Gammaₙ = gamma (t) pₙ, Paper 17), with no contribution from sector mixing. Additionally, the PTRH framework admits a natural minimum black-hole size Aₕorₘin = 9 Acell = 36 G₄ ln 9 (Paper 12), at which each sector supports exactly dₙₘin = e^ln 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).
George H. Bressler (Sun,) studied this question.