Gap G8 of the PTRH/TMRB framework called for a derivation of Zeno suppression of inter-sector transitions from a Lindblad master equation. We provide this derivation. Modelling the horizon geometry as a continuous measurement apparatus for the Z9 Frobenius quantum number, we formulate a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation for the Z9-graded matter density matrix with jump operators L_n = sqrt(gamma) P_n, where P_n are the orthogonal projectors onto the Frobenius eigenspaces. We prove: (i) off-diagonal sector density-matrix elements rho_nm(t) (n not equal m) decay exponentially at rate gamma; (ii) in the presence of a sector-mixing perturbation V_sys, steady-state off-diagonal elements satisfy |rho_nm^ss| le 2||V_sys||/sqrt(gamma^2+omega_nm^2); (iii) in the Zeno limit gamma to infinity the steady state is block-diagonal, reproducing the sector isolation of the Isolated Horizon Conjecture (Paper 13) by an independent dynamical route; (iv) under the sector-diagonal bath coupling of the PTRH model ([H,P_n]=0), the diagonal sector populations rho_nn are conserved by the Lindblad evolution, so the EHSCM torsion weights w_n (Paper 11) are preserved. The decoherence rate gamma is identified with the Hawking temperature via the fluctuation-dissipation theorem in the KMS state of Paper 13: gamma = c_kappa T_H (natural units hbar=k_B=1), with c_kappa = O(1) depending on the torsion spectral weight. Gap G8 is upgraded from [O] to [V]. Combined with Papers 13 (algebraic) and 14 (geometric), this provides the third independent proof of Z9 sector isolation.
Gap G8 of the PTRH/TMRB framework called for a derivation of Zeno suppression of inter-sector transitions from a Lindblad master equation. We provide this derivation. Modelling the horizon geometry as a continuous measurement apparatus for the Z9 Frobenius quantum number, we formulate a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation for the Z9-graded matter density matrix with jump operators L_n = sqrt(gamma) P_n, where P_n are the orthogonal projectors onto the Frobenius eigenspaces. We prove: (i) off-diagonal sector density-matrix elements rho_nm(t) (n not equal m) decay exponentially at rate gamma; (ii) in the presence of a sector-mixing perturbation V_sys, steady-state off-diagonal elements satisfy |rho_nm^ss| le 2||V_sys||/sqrt(gamma^2+omega_nm^2); (iii) in the Zeno limit gamma to infinity the steady state is block-diagonal, reproducing the sector isolation of the Isolated Horizon Conjecture (Paper 13) by an independent dynamical route; (iv) under the sector-diagonal bath coupling of the PTRH model ([H,P_n]=0), the diagonal sector populations rho_nn are conserved by the Lindblad evolution, so the EHSCM torsion weights w_n (Paper 11) are preserved. The decoherence rate gamma is identified with the Hawking temperature via the fluctuation-dissipation theorem in the KMS state of Paper 13: gamma = c_kappa T_H (natural units hbar=k_B=1), with c_kappa = O(1) depending on the torsion spectral weight. Gap G8 is upgraded from [O] to [V]. Combined with Papers 13 (algebraic) and 14 (geometric), this provides the third independent proof of Z9 sector isolation.
Gap G9 of the PTRH/TMRB framework called for a formulation of the WRITE/READ/ERASE memory protocol of the event-horizon-state computer memory (EHSCM, Paper 11) as operations in the celestial dictionary of Paper 4. We provide this formulation. Defining quantum channels W_n (WRITE), R_n (READ), and E_n (ERASE) on the Z9-graded horizon Hilbert space H = direct-sum H_n, we prove five results, each at status [V]. (i) WRITE stability: W_n[rho] = P_n rho P_n is completely positive and trace-non-increasing, with all inter-sector blocks of the post-WRITE state identically zero at t = 0. (ii) Zeno stability: under the GKSL evolution of Paper 16, inter-sector leakage is bounded by 2||V_sys||/gamma for all t >= 0, vanishing as gamma = c_kappa T_H → infinity. (iii) READ non-destructive: R_n extracts the sector population p_n = tr(P_n rho) via the Hawking emission rate Gamma_n = gamma p_n; p_n is conserved under GKSL evolution (Paper 16 Theorem 1(iv)), so the sector label is readable at any time without disturbing the stored quantum information. (iv) ERASE Landauer bound: any ERASE channel E_n operating at Hawking temperature T_H dissipates energy at least T_H S_BH/9, where S_BH/9 is the per-sector entropy established in Paper 15. (v) Celestial dictionary: under the Mellin-transform dictionary of Paper 4, W_n corresponds to insertion of the celestial primary O_hat_n(Delta_n; z, zbar) with Z9 charge n from past null infinity; R_n measures the celestial OPE coefficient C_n proportional to p_n; E_n corresponds to insertion of the conjugate primary O_hat_{9-n} from future null infinity, with net amplitude charge n + (9-n) = 9 ≡ 0 (mod 9) satisfying the Z9 Ward identity. The Ward identity is derived from the path-integral selection rule for the discrete Z9 symmetry of the TMRB action (not from a Noether current). The total information capacity of the nine-sector architecture is I_total = sum ln d_n = S_BH = A_hor/(4G_4), saturating the Bekenstein-Hawking bound. Gap G9 is upgraded from [O] to [V], completing the prerequisite chain G5 (Paper 15) -- G8 (Paper 16) -- G9 of the PTRH gap-closure program.
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 = 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).
Gap G6 of the PTRH/TMRB framework called for proof that information encoded in the event-horizon-state computer memory (EHSCM, Paper 11) can be retrieved before the Page time lambda_Page, at which the Hawking radiation entropy equals the residual black-hole entropy. We provide three results at status [V] and one retrieval protocol. (i) Unconditional sector-label retrieval: the sector population distribution {p_n} is an exact conserved charge of the GKSL dynamics (Papers 16, 19) and is recoverable at any time from the instantaneous relative Hawking emission rates Gamma_n(lambda)/gamma(lambda) = p_n; no Page-time condition applies. (ii) Pre-Page within-sector capacity: the within-sector information capacity I_within(lambda) = S_BH(lambda) - H({p_n}) satisfies I_within(lambda) > S_BH(0)/2 - ln 9 > 0 for all lambda < lambda_Page, confirming that the horizon retains positive within-sector capacity throughout the pre-Page regime. (iii) Horizon-direct retrieval channel: the EHSCM ERASE operation implements a unitary extraction of the within-sector state rho_n(lambda) from H_n(lambda) into an output register with unit fidelity, independent of lambda. Since this channel operates directly on H(lambda) --- not on the Hawking radiation --- it is not subject to the Page-time limit on passive radiation decoding. The combination of (i) and (iii) yields a concrete pre-Page retrieval protocol with unit fidelity. Gap G6 is upgraded from [O] to [V]. Eight of nine PTRH gaps (G1, G2, G3, G4, G5, G6, G8, G9) now carry status [V]; one remains open (G7).
We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors Cμνλ and Bμνλ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−21{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C) and (B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.
We complete the quantum formulation of History-Dependent Gravity (HDG) by demonstrating that its phenomenological memory kernel admits a rigorous quantum environment realization. Starting from a microscopic system-environment decomposition with spectral weight J(μ) = ρ(μ)², we derive the exact non-Markovian master equation for the reduced dynamics of matter. Crucially, the HDG memory kernel K(t) = ∫ J(μ) e^{-iμt} dμ is shown to coincide identically with the environment correlation function, closing the theory at the microscopic level. Our formalism yields three fundamental results. First, the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation is derived as the rigorous Davies limit of HDG, rather than being postulated ad hoc. Second, we prove that the positivity of the spectral weight J(μ) ≥ 0 guarantees a completely positive and trace-preserving (CPTP) unitary dilation of the reduced dynamics, linking the geometric no-ghost condition directly to information-theoretic consistency. Third, decoherence rates are not free parameters but are computed explicitly as γ_i(ω_i) = 2π J_i(ω_i), fixed by the renormalization group flow of the spectral density. The theory predicts a universal short-time survival probability P(t) = 1 - αt² + O(t³), with α = ∫ J(μ) dμ, and a super-linear mass scaling γ_eff ∝ M² (under linear mass-coupling assumptions), providing sharp falsifiable criteria for matter-wave interferometry and superconducting qubit experiments. Furthermore, we establish a spectral tomography protocol allowing direct reconstruction of the effective memory spectrum J(μ) from decoherence data. This work elevates HDG from an effective model to a testable quantum-gravitational decoherence framework, bridging quantum information, non-Markovian dynamics, and the infrared structure of gravity.
Normal forms and completely positive gluing theory describe representation freedom in quantumdynamics. We extract dimension laws and sharp coherence constraints when every sector projectoris exactly conserved and the dynamics inside each sector is prescribed. Let ℋ = ⨁𝑞ℋ𝑞, let𝑑𝑞 = dimℋ𝑞, and let 𝑘𝑞 be the rank of the canonical dissipative matrix of the prescribed Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator on block 𝑞. We prove that the compatibleextensions have real affine dimension 2 ∑𝑞<𝑠(𝑘𝑞 +1)(𝑘𝑠 +1), recovering the linear restriction-kerneldimension at full local dissipative rank. For a scalar sector paired with an 𝑛-dimensional sector,the instantaneous cross-sector amplitude damping is bounded below by half the sum of local jumpoperator variances. The bound is attainable for every specified instantaneous direction and equalsone quarter of the initial purity-loss rate of the associated pure sector state. An isotropic qubitdissipator with Bloch decay rate 2𝜅 forces cross-sector damping of at least 𝜅/2; rank-one dephasingcan instead preserve a selected coherence. A joint observable with a reference accesses this differencewhile preserving total charge. We also give full-matrix and superselected-algebra dimensions, exactrefinement losses, and extremal multiplicity profiles. These results constrain a nonunitary chargesector model after its projectors and local dynamics are supplied; they do not determine theirphysical origin or select the realized extension