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).
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).
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 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.
The non-isometric holographic model of the black hole interior stands out as a potential resolution of the long-standing black hole information puzzle since it remedies the friction between the effective calculation and the microscopic description. In this study, combining the final-state projection model, the non-isometric model of black hole interior and Hayden-Preskill thought experiment, we investigate the information recovery from decoding Hawking radiation and demonstrate the emergence of the Page time in this setup. We incorporate the effective modes into the scrambling inside the horizon, which are usually disregarded in Hayden-Preskill protocols, and show that the Page time can be identified as the transition of information transmission channels from the EPR projection to the local projections. This offers a new perspective on the Page time. We compute the decoupling condition under which retrieving information is feasible and show that this model computes the black hole entropy consistent with the quantum extremal surface calculation. Assuming the full knowledge of the dynamics of the black hole interior, we show how Yoshida-Kitaev decoding strategy can be employed in the modified Hayden-Preskill protocol. Furthermore, we perform experimental tests of both probabilistic and Grover's search decoding strategies on the 7-qubit IBM quantum processors to validate our analytical findings and confirm the feasibility of retrieving information in the non-isometric model. This study would stimulate more interests to explore black hole information problem on the quantum processors.
The effective field theory description of a radiating black hole introduces redundant degrees of freedom that necessitate annihilation of those modes at late stages to conserve entropy. The prevailing view is that such effective process can result in information loss unless the redundant states are annihilated in maximally entangled pairs, resembling quantum teleportation. In this Letter, we demonstrate that this view can be relaxed in a new postselection model. We investigate information recoverability in a radiating black hole through the non-unitary dynamics that projects the randomly-selected modes from a scrambling unitary. We show that the model has the merit of producing the von Neumann entropy of black holes consistent with the island formula calculation and that information in the black hole interior can be decoded from the Hawking radiation without loss after the Page time. Moreover, in this model the Page time gains a new interpretation as the transition point between two channels of information transmission when sufficient amounts of effective modes are annihilated inside the horizon. We present two decoding strategies along with their quantum circuit realizations. The experimental verification of the strategies employs 7-qubit IBM quantum processors, demonstrating the viability of these strategies and the potential for quantum processors to probe the black hole interior.
Abstract Black-hole evaporation can be unitary only if Hawking's thermal spectrum is a coarse view of an underlying pure quantum process. Island and replica-wormhole calculations recover the Page curve but do not generally provide the amplitudes mapping a specified black-hole microstate to outgoing quanta. Here we construct such a state-resolved channel for an area-law quantum memory in matrix black-hole quantum mechanics. An exact symmetry separates the fermionic state into a geometric factor that controls emission and a protected factor untouched by the full real- and imaginary-time matrix and link dynamics. We prove that this memory occurs in every representation sector, enumerate all sectors through rank ten, and exhibit a self-complementary sector whose entropy scales with horizon area. When the matrix rank decreases, an exact branching isometry transfers the memory into daughter-hole and radiation degrees of freedom. Combining this isometry with the model's monopole-tunnelling weights gives every encoded state the same Hawking-thermal branch probabilities while its conditional radiation remains pure and microstate dependent. Finite-rank simulations show Page-like transfer and near-unit decoding; dephasing leaves the branch probabilities unchanged but erases recovery. The result provides an explicit amplitude-level mechanism by which Hawking thermality and unitary information transfer coexist in one microscopic model.
The nonisometric holographic model of the black hole interior stands out as a potential resolution of the long-standing black hole information puzzle since it remedies the friction between the effective calculation and the microscopic description. In this study, combining the final-state projection model, the nonisometric model of black hole interior, and Hayden-Preskill thought experiment, we investigate the information recovery from decoding Hawking radiation and demonstrate the emergence of the Page time in this setup. We incorporate the effective modes into the scrambling inside the horizon, which are usually disregarded in Hayden-Preskill protocols, and show that the Page time can be identified as the transition of information transmission channels from the EPR projection to the local projections. This offers a new perspective on the Page time. We compute the decoupling condition under which retrieving information is feasible and show that this model computes the black hole entropy consistent with the quantum extremal surface calculation. Assuming the full knowledge of the dynamics of the black hole interior, we show how Yoshida-Kitaev decoding strategy can be employed in the modified Hayden-Preskill protocol. Furthermore, we perform experimental tests of both probabilistic and Grover’s search decoding strategies on the seven-qubit IBM quantum processors to validate our analytical findings and confirm the feasibility of retrieving information in the nonisometric model. This study would stimulate more interest to explore black hole information problem on the quantum processors. Published by the American Physical Society 2024