Black-Hole Horizons as Accessibility Boundaries: Hawking Recovery, Information Repartition, and Hidden Conditional Dependence

Hawking's derivation of black-hole radiation turned a classical collapse problem into a joint problem in gravity, quantum theory, and thermodynamics [citation]. The Bekenstein--Hawking entropy relation, near-horizon analyses, and later universality and trans-Planckian studies show that the leading temperature law is robust even when the interpretation of the outgoing modes remains disputed [citation].

The information problem sharpened once late-time purification, scrambling, islands, interior reconstruction, and algebraic factorization were brought into a common frame [citation]. Recent work has made the horizon split itself part of the problem by analyzing local observable algebras, generalized entropy for subregions, crossed-product constructions, factorization in gravitational settings, and observer-relative descriptions of black-hole physics [citation].

Jackiw--Teitelboim (JT) gravity and near-AdS$_2$TeX_2 throats provide a controlled setting in which Hawking emission, bath-coupled evaporation, Page turnover, and low-temperature quantum corrections can be compared within a common effective description [citation].

The global information state $\Iglob$TeX\Iglob is distinguished from the locally accessible information $\Aacc_{\Dobs}$TeX\Aacc_{\Dobs} on a fixed observation domain $\Dobs=(\Odom_{\rm geom},T,B,\Ggate)$TeX\Dobs=(\Odom_{\rm geom},T,B,\Ggate); commit denotes irreversible redistribution rather than ontic deletion, and the threshold $\Xith=0$TeX\Xith=0 is used as a horizon diagnostic [citation].

The horizon is treated as an accessibility boundary, Hawking radiation as the exterior recovery of thermal-like commit-available modes, and late-time purification on the fixed retarded-time split as the decay of hidden conditional dependence. The analysis is restricted to a strict-fence interpretation extension on stationary horizons or adiabatically evaporating backgrounds with fixed exterior observation domains. No full quantum-gravity completion, early-universe overwrite, or cosmological extension is claimed.

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Accessibility language

The notation distinguishes the global information state $\Iglob(t)$TeX\Iglob(t) from the locally accessible information $\Aacc_{\Dobs}(t)$TeX\Aacc_{\Dobs}(t) associated with a fixed observation domain $\Dobs$TeX\Dobs [citation]. The domain tuple is

\Dobs \equiv (\Odom_{\rm geom},\,T,\,B,\,\Ggate),

where $\Odom_{\rm geom}$TeX\Odom_{\rm geom} is the geometric reachability region, $T$TeXT the operational time window, $B$TeXB the operational bandwidth or sensitivity window, and $\Ggate$TeX\Ggate the irreversible recording gate. Commit is treated not as ontic destruction but as irreversible redistribution of correlations outside the accessible sector.

Exterior algebra and state restriction

The exterior state is the restriction of a global state functional to the fixed exterior algebra:

\wacc^{\Dext}(A)=\wglob(A),
\qquad
A\in \Aextalg(\Dext).

Whenever an auxiliary finite-dimensional or type-I representation is chosen, one may write

\wacc^{\Dext}(A)=\mathrm{Tr}\!\bigl(\rhoacc^{\Dext}A\bigr),
\qquad
A\in \Aextalg(\Dext),

but reference is then only a mnemonic for reference; it is not a claim of unique microscopic tensor factorization across a quantum-gravitational horizon. No unique microscopic horizon factorization, completed interior algebra, or full microscopic theory of evaporation is assumed or derived here. Recent algebraic treatments of gravitational entropy and black-hole information make precisely this distinction between local observable algebras, generalized entropy for subregions, crossed-product or type-II bookkeeping constructions, factorization questions, and naive density-matrix factorization [citation].

Horizon threshold diagnostic

A normalized threshold is used as a horizon diagnostic [citation]:

\Xith \equiv \frac{\chi^\mu \nabla_\mu \HH}{\kappa} - (\Xith)_\star = 0,

with $\chi^\mu$TeX\chi^\mu the horizon generator and $\kappa$TeX\kappa the surface gravity. In the stationary recovery class this threshold is taken to align with the apparent-horizon condition; outside that class it remains a horizon diagnostic to be checked. Equation reference is not a singularity criterion. It is an accessibility-boundary criterion.

Propagation, event statistics, and wavefunction semantics

The propagation sector uses a common self-adjoint operator class, boundary-response realizations of the same propagation core, and a positive-frequency wave-mechanical regime on a fixed coherent window [citation]. Event probabilities are attached at the accessible-event interface rather than at the level of a raw global phase field [citation]. Accessible wavefunctions are treated as phase-statistical representations on fixed domains rather than as literal material substances [citation]. Throughout, ``interior'' or ``hidden'' refers only to the complement of fixed exterior accessibility unless additional structure is stated explicitly.

The distinction between existence and accessibility changes how a black-hole horizon is to be read. A horizon may bound what is geometrically reachable inside $\Odom_{\rm geom}$TeX\Odom_{\rm geom} without implying that global information ceases to exist. The threshold reference is therefore interpreted as the boundary at which the exterior accessibility class changes discontinuously, not as an information annihilator.

definition: Exterior accessibility functional. Let $\wglob$TeX\wglob be a global state functional compatible with the propagation sector fixed above. For a fixed exterior observation domain $\Dext$TeX\Dext, the exterior accessibility functional is the restricted functional

\wacc^{\Dext}(A)=\wglob(A),
\qquad
A\in \Aextalg(\Dext).

Whenever a finite-dimensional or type-I auxiliary model is fixed, one may write

\wacc^{\Dext}(A)=\mathrm{Tr}\!\bigl(\rhoacc^{\Dext}A\bigr),
\qquad
A\in \Aextalg(\Dext),

as a mnemonic representation of reference.

The density-matrix notation, when available, is auxiliary only: it is not used to assert a unique microscopic tensor-product factorization across a quantum-gravitational horizon. Crossed-product, relative state-counting, and observer-dependent entropy constructions show that semiclassical black-hole entropy bookkeeping can be algebraically well defined without selecting a unique horizon factorization [citation].

Whenever the fixed exterior-hidden split is represented on an auxiliary finite-dimensional or type-I model, ordinary entropy bookkeeping takes the exact form

S_{\rm glob}
=
\Sacc+\Shid-I_{\acc:\hid},

where $I_{\acc:\hid}$TeXI_{\acc:\hid} is the mutual information between the accessible and hidden sectors. Equation reference is used only on explicitly fixed bin decompositions or reduced models. In this reading, $\Sacc$TeX\Sacc is what the fixed exterior domain can reconstruct, $\Shid$TeX\Shid is what remains outside that accessibility class, and any apparent loss is to be read through the redistribution of $I_{\acc:\hid}$TeXI_{\acc:\hid}, not as ontic deletion of $\Iglob$TeX\Iglob.

remark: Horizon versus singularity. The distinction between a horizon and a singularity is kept explicit. A horizon marks a change in accessibility structure through reference; it is not itself a statement about curvature blow-up. This distinction is already built into the threshold diagnostic and is preserved here without strengthening it into a singularity-resolution program.

Once a horizon is read as an accessibility boundary, the black-hole information problem changes form. The relevant questions are what is retained in $\Iglob$TeX\Iglob, what is hidden from a fixed exterior $\Dext$TeX\Dext, and under what conditions hidden correlations can re-enter the exteriorly accessible sector at late times.

Leading recovery layer

The standard Hawking temperature law is retained as the leading recovery statement,

\Tawk = \frac{\hbar\,\kappa}{2\pi c\,k_B},

for a stationary horizon with surface gravity $\kappa$TeX\kappa [citation]. Strong-field quantum and thermodynamic links are kept at the recovery or compatibility level rather than treated as an independently closed microscopic sector [citation]. The leading Hawking law is therefore taken as the recovery layer before any finite-domain refinement is discussed.

Mode repartition rather than literal pair creation

The pair-creation picture is retained only as a mnemonic and is not adopted as the fundamental microscopic description. Once a horizon is read through reference as an accessibility boundary, strong near-horizon curvature and phase response repartition the propagation sector into exteriorly recoverable and hidden mode classes. What an exterior observer recovers is a thermallike spectrum of commit-available modes on a fixed $\Dext$TeX\Dext.

This formulation leaves intact the main physical success of the Hawking calculation while avoiding the over-literal idea that the horizon itself is a local particle factory. It also aligns with the line of thought in which outgoing modes are not modeled as originating from a literal trans-Planckian reservoir localized at the horizon [citation].

Accessible temperature and spectral correction classes

The leading law reference is not the only quantity an exterior observer may report. The reported quantity is an accessible temperature parameter associated with a fixed $\Dext$TeX\Dext. The minimal admissible correction class is

\Tacc
=
\Tawk\Bigl[1+\delta_{\Xi_H}+\delta_{\Omega}+\delta_{\Ggate}+\mathcal O(\Xibh^2)\Bigr],

where $\Xibh$TeX\Xibh is the horizon-shell specialization of the root phase-gradient hierarchy parameter, $\delta_{\Xi_H}$TeX\delta_{\Xi_H} is a near-horizon phase-response correction, $\delta_{\Omega}$TeX\delta_{\Omega} an exterior-domain accessibility correction, and $\delta_{\Ggate}$TeX\delta_{\Ggate} a gate or commit correction associated with the fixed exterior readout class. Equation reference organizes admissible correction sources on the fixed exterior domain. It does not assume that the displayed terms are independently measurable, linearly separable beyond the stated order, or uniquely fixed without further microscopic input.

Likewise, for a bosonic mode label $\omega$TeX\omega, the accessible exterior occupation class is written as

n_{\omega}^{\acc}
=
\frac{1}{\exp\!\bigl[\hbar\omega/(k_B\Tawk)\bigr]-1}\,
\Facc(\omega;\Xibh,\Dext),
\qquad
\Facc \to 1
\quad
\text{in the recovery limit.}

Equation reference is likewise an exterior organizing class. Exact Planckian thermality is recovered when $\Facc\to 1$TeX\Facc\to 1, but the equation does not assign a unique microscopic origin to every deviation from that limit. Once $\Dext$TeX\Dext is finite and hidden-sector correlations are not silently set to zero, a weakly nonthermal exterior appearance is admissible without changing the leading Hawking law.

remark: Near-horizon response labels. A family-fixed smooth monotone response relabeling $q_H=g_H(p)$TeXq_H=g_H(p) may be introduced on an admitted near-horizon window to parameterize trans-Planckian sensitivity without postulating a literal infinite-frequency reservoir. Its detailed form is recorded in Appendix reference. No later late-time identity or reduced JT result depends on a specific choice of $g_H$TeXg_H.

The distinction between global information and local accessibility rephrases the information problem in terms of accessible and hidden sectors. Equation reference already shows that growth of exterior thermal entropy by itself does not determine the fate of global information. What matters is the balance among $\Sacc$TeX\Sacc, $\Shid$TeX\Shid, and $I_{\acc:\hid}$TeXI_{\acc:\hid}.

Retarded-time binning and the exact return identity

Fix an exterior retarded time $u$TeXu. Let $R_{<u}$TeXR_{<u} denote the radiation already accessible on the fixed exterior domain before $u$TeXu, let $r_u$TeXr_u denote the incremental radiation bin emitted over $[u,u+\Delta u)$TeX[u,u+\Delta u), and let $H_u$TeXH_u denote the corresponding hidden complement. The one-body Hawking entropy rate is introduced by

S(r_u)=\dot s_H^{\rm eff}(u)\,\Delta u+o(\Delta u).

The late-time return quantity is then defined exactly by

\Jh(u)
\equiv
\lim_{\Delta u\to 0}\frac{1}{\Delta u}I(r_u:R_{<u}).

proposition: Exact return identity on a fixed retarded-time split. Let $S_{\Rrad}(u)$TeXS_{\Rrad}(u) denote the fine-grained entropy of the collected exterior radiation. Then

\dot S_{\Rrad}(u)
=
\dot s_H^{\rm eff}(u)-\Jh(u).

Equation reference is an exact identity once the split $(R_{<u},r_u,H_u)$TeX(R_{<u},r_u,H_u) has been fixed. Its derivation is given in Appendix reference. If the evaporation history begins and ends with vanishing collected-radiation entropy,

S_{\Rrad}(u_0)=S_{\Rrad}(u_{\rm evap})=0,

then integrating reference gives

\int_{u_0}^{u_{\rm evap}} \Jh(u)\,du
=
\int_{u_0}^{u_{\rm evap}} \dot s_H^{\rm eff}(u)\,du.

Equation reference is therefore an exact consequence of reference under the remnant-free completion condition reference; it is not an additional phenomenological postulate. Explicit evaporating models that recover Page-compatible turnover remain consistent with this reading, but they do not replace the exact identity itself [citation].

Hidden conditional dependence and exterior recoverability

The quantity that controls late-time purification of the collected exterior radiation on the fixed retarded-time split is the hidden conditional dependence of the incremental bin. Define

\mathcal C_H(u)
\equiv
\lim_{\Delta u\to 0}\frac{1}{\Delta u}I(H_u:r_u\mid R_{<u}).

proposition: Hidden conditional-dependence identity on a pure fixed retarded-time split. If the tripartite state on $(R_{<u},r_u,H_u)$TeX(R_{<u},r_u,H_u) is pure at each retarded-time step, then

\mathcal C_H(u)
=
\dot s_H^{\rm eff}(u)+\dot S_{\Rrad}(u)
=
2\dot s_H^{\rm eff}(u)-\Jh(u).

Equation reference identifies late-time purification of the collected exterior radiation on the fixed retarded-time split with the collapse of hidden conditional dependence rather than with an unspecified nonthermal correction. The Page-turnover condition becomes

\dot S_{\Rrad}(u_P)=0
\iff
\mathcal C_H(u_P)=\dot s_H^{\rm eff}(u_P),

while the post-Page regime is characterized by

\dot S_{\Rrad}(u)<0
\iff
\mathcal C_H(u)<\dot s_H^{\rm eff}(u).

On any auxiliary finite-dimensional or type-I realization of the fixed retarded-time split, the Fawzi--Renner recoverability theorem gives the fidelity bound

F_u(\Delta u)
\ge
\exp\!\left[-\frac{1}{2}\mathcal C_H(u)\,\Delta u + o(\Delta u)\right],

so $\mathcal C_H(u)\to0$TeX\mathcal C_H(u)\to0 is precisely the high-fidelity recoverability regime for the incremental exterior bin on such auxiliary realizations. Appendix reference gives the derivation. Appendix reference evaluates $\Jh(u)$TeX\Jh(u) and $\mathcal C_H(u)$TeX\mathcal C_H(u) on a sharp-limit JT+bath background, while Appendix reference gives the stationary Gaussian JT stretched-horizon reduction and its fast-return limit.

Stationary JT hidden-current reduction

The information-theoretic identities established above are exact on the fixed accessibility split. The constructions below do not strengthen those identities into a generic admissibility theorem for black-hole evaporation. They provide reduced stationary realizations in which the return quantities acquire explicit kernel forms only after the additional model assumptions stated in this subsection.

For an explicit reduced model, consider a stationary JT black hole with a stretched horizon $\Sigma_\epsilon$TeX\Sigma_\epsilon. Split a near-horizon matter field as $\phi=\phi_A+\phi_H$TeX\phi=\phi_A+\phi_H and couple the slow exterior shell amplitude $q(u)$TeXq(u) linearly to a hidden operator by

S_{\rm int}
=
\lambda_H \int du\, q(u)\,O_H(u),
\qquad
O_H(u)=\partial_u \phi_H(u)\big|_{\Sigma_\epsilon},

with $\lambda_H\sim \zeta_\epsilon M_{\rm eff}^2(\HH_h)$TeX\lambda_H\sim \zeta_\epsilon M_{\rm eff}^2(\HH_h) the horizon-shell coupling inherited from the root action.

proposition: Reduced local kernel within a stationary Gaussian JT stretched-horizon model. For a stationary JT stretched-horizon reduction with the dimension-one hidden current $O_H=\partial_u\phi_H|_{\Sigma_\epsilon}$TeXO_H=\partial_u\phi_H|_{\Sigma_\epsilon}, a free Hartle--Hawking KMS hidden bath of Ohmic low-frequency spectral density $\rho_H(\omega)=\chi_O\omega$TeX\rho_H(\omega)=\chi_O\omega, and the first-order slow-bin projection used for the exterior shell amplitudes here, the induced projected response kernel is

\Gamma_{\rm H}(u,s)=\gamma_H\,\delta(u-s),
\qquad
\gamma_H:=\lambda_H^2\chi_O.

Appendix reference gives the derivation. Different operator dimensions, non-Ohmic hidden spectra, Schwarzian-dressed nonstationary sectors, and moving-island kernels lie outside this reduction. If one additionally imposes the minimal hidden-survival law of Appendix reference, the corresponding return kernel is

K_{\rm ret}(u,s)=\gamma_H e^{-\gamma_H(u-s-\tau_{\rm ret})}\,\Theta(u-s-\tau_{\rm ret}).

Equation reference is not an imported exact identity of the information-theoretic sector. It is exact only within the stationary Gaussian reduction after the hidden-survival law has been imposed. Convolving reference with the sharp-limit JT Hawking rate of Appendix reference yields an explicit $\mathcal J_\gamma(u)$TeX\mathcal J_\gamma(u) and $\mathcal C_H^\gamma(u)$TeX\mathcal C_H^\gamma(u), and the fast-return limit $\gamma_H\to\infty$TeX\gamma_H\to\infty with $\tau_{\rm ret}=u_P$TeX\tau_{\rm ret}=u_P reproduces the Page-delay delta kernel $K_{\rm JT}(u,s)=\delta(u-s-u_P)$TeXK_{\rm JT}(u,s)=\delta(u-s-u_P). The Gaussian reduction provides one stationary reduced realization of the return quantities used above [citation].

The compact-object companion records are used only as bounded diagnostic context for the accessibility-boundary reading. For the BH1 lane, BH--VP0 is read only through its EHT-facing morphology and polarization surrogate support; the GWOSC-facing remnant and response record belongs to BH2 and is not imported here. BH--VP1 contributes only the official GWTC-3 Tests-of-GR posterior comparison context with final label $\texttt{BH-VP1-TGR-DIAGNOSTIC-PARTIAL}$TeX\texttt{BH-VP1-TGR-DIAGNOSTIC-PARTIAL}. BH--VP2 contributes only a bounded public-table reconstruction and literature-summary surface-commit return suppression audit for the no-ordinary-material-surface lane in black-hole candidates relative to neutron-star X-ray transients [citation].

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These companion labels do not replace the EHT image pipeline, LVK/GWOSC waveform pipeline, LVK Tests-of-GR likelihood, or standard X-ray-binary accretion/population analyses. They do not establish a microscopic entropy theorem, black-hole interior closure, return-kernel theorem, compact-object theorem, object-level horizon proof, GR-violation claim, or quantum-gravity solution.

The analysis is restricted to a strict-fence black-hole interpretation extension on stationary or adiabatically evaporating horizons with fixed exterior observation domains. A black-hole horizon marks a change of accessibility class diagnosed by $\Xith=0$TeX\Xith=0 in the stationary recovery regime. The basic exterior state object is the restriction of the global state to the fixed exterior algebra $\Aextalg(\Dext)$TeX\Aextalg(\Dext); density-matrix notation, when used on auxiliary reduced models, is mnemonic only and does not assume a microscopic horizon factorization. The leading Hawking law remains the exterior recovery law, while finite exterior domains admit weak departures from exact thermality through the correction class reference and spectrum class reference.

Late-time purification of the collected exterior radiation is governed by exact information-theoretic identities on a fixed retarded-time split. The return rate $\Jh(u)$TeX\Jh(u) is the mutual-information rate between an incremental exterior bin and the previously emitted radiation, while $\mathcal C_H(u)$TeX\mathcal C_H(u) measures the hidden conditional dependence whose decay is necessary for purification and simultaneously controls exterior recoverability. On a stationary JT hidden-current reduction with a Gaussian Hartle--Hawking hidden bath, the projected response kernel is local. The delayed-exponential return kernel is exact only after the hidden-survival closure of Appendix reference, and the reduced Page-delay delta kernel belongs only to the sharp-limit JT+bath fast-return limit.

The BH--VP0, BH--VP1, and BH--VP2 records add only bounded EHT-facing, Tests-of-GR comparison, and public-table surface-return diagnostic context for the BH1 lane. BH--VP1 adds official GWTC-3 Tests-of-GR posterior comparison with IMR diagnostic pass and ringdown, parametrized-deviation, and spin-quadrupole diagnostic stress labels. BH--VP2 adds only a conditional public-table reconstruction and literature-summary audit of quiescent-luminosity suppression and Type-I burst upper-limit/non-detection contrast in the ordinary material surface-return lane. These labels remain diagnostic/bounded-support and do not alter the theorem status of the accessibility-boundary construction.

No full quantum-gravity completion, early-universe or cosmological extension, generic admissibility theorem for nonstationary Schwarzian-dressed sectors, or unrestricted microscopic evaporation law is claimed here. The paper fixes only the accessibility-side reinterpretation and the reduced realizations stated explicitly above.

On a fixed near-horizon comparison family one may introduce a smooth monotone response relabeling

q_H = g_H(p),
\qquad
g_H(p)=p+\mathcal O\!\left(\frac{p^3}{\Lambda_H^2}\right)
\quad (|p|\ll \Lambda_H),

with $\Lambda_H$TeX\Lambda_H a horizon-response scale. On the comparison families used here, the admissible class is fixed in advance and consists of smooth monotone relabelings on the admitted near-horizon window; whenever the response class preserves the reflection symmetry $p\mapsto -p$TeXp\mapsto -p, the relabeling is taken odd to the working order. Equation reference fixes only a family-fixed response label on that window. It neither supplies a unique ultraviolet completion nor feeds any later conclusion through a distinguished choice of $g_H$TeXg_H. The relabeling serves only as bookkeeping for trans-Planckian sensitivity, consistent with the robust-temperature but UV-sensitive-amplitude pattern emphasized in the dispersive literature [citation]. It is not a microscopic ultraviolet completion, does not define a horizon-local origin law for Hawking quanta, and does not enter any exact information-theoretic identity used in the main text.

Write $R\equiv R_{<u}$TeXR\equiv R_{<u}, $r\equiv r_u$TeXr\equiv r_u, and $H\equiv H_u$TeXH\equiv H_u for a single retarded-time step. Then

S_{\Rrad}(u+\Delta u)-S_{\Rrad}(u)=S(r\mid R),

while

I(r:R)=S(r)-S(r\mid R).

Combining reference and reference with reference and taking $\Delta u\to0$TeX\Delta u\to0 yields reference. Likewise,

I(H:r\mid R)=S(HR)+S(rR)-S(R)-S(HrR).

If $(H,R,r)$TeX(H,R,r) is pure, then

S(HR)=S(r),\qquad S(rR)=S(H)=S(Rr),\qquad S(HrR)=0,

which gives reference after division by $\Delta u$TeX\Delta u and passage to the continuum limit.

The same tripartite split admits an operational recoverability statement on any explicitly fixed auxiliary finite-dimensional or type-I realization. For a tripartite state $\rho_{HRr}$TeX\rho_{HRr} on such a realization, the Fawzi--Renner theorem guarantees a recovery map from $R$TeXR to $Rr$TeXRr whose fidelity is bounded by the conditional mutual information [citation]. Using the same logarithmic convention as the entropies above,

F_u(\Delta u)
\ge
\exp\!\left[-\frac{1}{2}I(H_u:r_u\mid R_{<u})\right]
=
\exp\!\left[-\frac{1}{2}\mathcal C_H(u)\,\Delta u + o(\Delta u)\right].

Thus $\mathcal C_H(u)\to0$TeX\mathcal C_H(u)\to0 is the statement that the incremental radiation bin becomes recoverable from the past radiation without further reference to the hidden complement.

For explicit evaluation of the late-time quantities and of the Page-delay fast-return limit, consider a two-dimensional Jackiw--Teitelboim black hole coupled to a non-gravitating bath. In the evaporation setup of Almheiri, Mahajan, Maldacena, and Zhao, the early Hawking branch of the radiation entropy grows as

S_R^{\rm early}(t)
\sim
2S_i^{\rm Bek}\bigl(1-e^{-\kappa t/2}\bigr),

while the Page turnover is produced by the competing island branch of the same radiation entropy [citation]. Motivated by that structure, retain only the leading exponentially decaying late branch and define the sharp-limit JT reduction by

S_i \equiv S_i^{\rm Bek},
\qquad
b\equiv \frac{\kappa}{2},
\qquad
a\equiv \kappa S_i,

S_H^{(1)}(u)
:=
2S_i\bigl(1-e^{-bu}\bigr),
\qquad
\dot s_H^{\rm eff}(u)=a e^{-bu},

and

S_R^{\rm JT}(u)
=
\min\!\Bigl[
2S_i\bigl(1-e^{-bu}\bigr),\,
S_i e^{-bu}
\Bigr].

Equation reference is a sharp-limit leading-order truncation rather than a full island computation. The Page turnover occurs where the two branches coincide,

e^{-bu_P}=\frac{2}{3},
\qquad
u_P=\frac{1}{b}\log\!\frac{3}{2}
=\frac{2}{\kappa}\log\!\frac{3}{2}.

proposition: Exact rates within the sharp-limit JT+bath reduction. For the sharp-limit JT+bath background reference--reference, the exact return and hidden conditional-dependence rates are

\Jh^{\rm JT}(u)=

0,  0<u<u_P,
[4pt]
\dfrac{3}{2}a e^{-bu},  u>u_P,

and

\mathcal C_H^{\rm JT}(u)=

2a e^{-bu},  0<u<u_P,
[4pt]
\dfrac{1}{2}a e^{-bu},  u>u_P.

proof. For $0<u<u_P$TeX0<u<u_P, equation reference selects the increasing Hawking branch, so $\dot S_R^{\rm JT}(u)=\dot s_H^{\rm eff}(u)=a e^{-bu}$TeX\dot S_R^{\rm JT}(u)=\dot s_H^{\rm eff}(u)=a e^{-bu}. Equations reference and reference then give $\Jh^{\rm JT}(u)=0$TeX\Jh^{\rm JT}(u)=0 and $\mathcal C_H^{\rm JT}(u)=2a e^{-bu}$TeX\mathcal C_H^{\rm JT}(u)=2a e^{-bu}. For $u>u_P$TeXu>u_P, equation reference selects the decaying island branch, $S_R^{\rm JT}(u)=S_i e^{-bu}$TeXS_R^{\rm JT}(u)=S_i e^{-bu}, so $\dot S_R^{\rm JT}(u)=-bS_i e^{-bu}=-(1/2)a e^{-bu}$TeX\dot S_R^{\rm JT}(u)=-bS_i e^{-bu}=-(1/2)a e^{-bu}. Substituting again into reference and reference yields reference and reference.

The integrated balance relation is exact in this sharp model:

\int_0^{\infty} \Jh^{\rm JT}(u)\,du
=
2S_i
=
\int_0^{\infty}\dot s_H^{\rm eff}(u)\,du.

Thus the one-body Hawking entropy budget is matched exactly by the returned mutual-information budget. Within this limiting background, the post-Page regime appears as an order-one activation of the return rate rather than as a perturbatively tiny correction.

To pass from the rate reference to a return kernel, impose only the minimal age-translation closure

\Jh(u)=\int_0^u ds\,K_{\rm ret}(u,s)\,\dot s_H^{\rm eff}(s),
\qquad
K_{\rm ret}(u,s)=k(u-s),
\qquad
k(\tau)=0\ \text{for}\ \tau<0.

Then, with $\dot s_H^{\rm eff}(s)=a e^{-bs}$TeX\dot s_H^{\rm eff}(s)=a e^{-bs}, one finds

\Jh(u)=a e^{-bu}\int_0^u d\tau\,k(\tau)e^{b\tau}.

Hence the age-profile is recovered exactly by

k(u)=e^{-bu}\frac{d}{du}\!\left(e^{bu}\frac{\Jh(u)}{a}\right),

with the derivative understood distributionally when necessary. Applying reference to reference gives the unique kernel on this sharp-limit background

k(u)=\delta(u-u_P),

or equivalently

K_{\rm JT}(u,s)=\delta(u-s-u_P).

The corresponding returned-correlation density is

G_{\rm ret}^{\rm JT}(u,s)
:=
K_{\rm JT}(u,s)\,\dot s_H^{\rm eff}(s)
=
a e^{-bs}\,\delta(u-s-u_P).

Equations reference--reference do not constitute a general microscopic kernel law. They state only that, on the sharp-Page JT+bath background and under the minimal age-translation closure reference, the return kernel is fixed uniquely to a Page-delay delta kernel. Appendix reference shows that this Page-delay delta kernel is the fast-return limit of a delayed-exponential kernel obtained from an exact Gaussian stationary JT stretched-horizon reduction. This Page delay characterizes the return of Hawking one-body entropy packets, not a diary-decoding delay of the Hayden--Preskill type [citation]. If the omitted endpoint and logarithmic corrections are restored, the sharp step in reference is smoothed and the delta kernel broadens into a narrow positive bump, but the sharp limit already isolates the leading Page-delay structure.

The construction below stays within a stationary JT black hole with a stretched horizon $\Sigma_\epsilon$TeX\Sigma_\epsilon, where a Gaussian reduced model is available. Split a near-horizon matter field as $\phi=\phi_A+\phi_H$TeX\phi=\phi_A+\phi_H into an exterior accessible sector and a hidden complement, and couple a slow exterior shell amplitude $q(u)$TeXq(u) linearly to a hidden operator $O_H(u)$TeXO_H(u) by

S_{\rm int}
=
\lambda_H\int du\,q(u)\,O_H(u),
\qquad
\lambda_H\sim \zeta_\epsilon M_{\rm eff}^2(\HH_h).

Take

O_H(u)=\partial_u\phi_H(u)\big|_{\Sigma_\epsilon}.

For a free Hartle--Hawking hidden sector the influence functional is exact Gaussian, and for this dimension-one operator on the thermal cylinder the hidden Wightman function is

G_H^{>}(u)
:=
\langle O_H(u)O_H(0)\rangle_{\beta_H}
=
c_O\left(\frac{\pi T_H}{\sinh[\pi T_H(u-i0)]}\right)^2.

Its Fourier transform has Ohmic spectral density,

\rho_H(\omega)
:=
G_H^{>}(\omega)-G_H^{<}(\omega)
=
\chi_O\,\omega,
\qquad
\chi_O>0.

The retarded self-energy is therefore

\Sigma_H^R(\omega)
=
\delta\Omega^2-i\lambda_H^2\chi_O\,\omega
\equiv
\delta\Omega^2-i\gamma_H\omega,
\qquad
\gamma_H:=\lambda_H^2\chi_O,

where the local real part is absorbed into a renormalized shell frequency. The KMS noise kernel is then

N_H(\omega)=\gamma_H\,\omega\coth\!\left(\frac{\beta_H\omega}{2}\right).

Within the first-order slow-bin projection used for the exterior amplitudes in the main text, the induced Volterra kernel is therefore local:

\Gamma_{\rm H}(u,s)=\gamma_H\,\delta(u-s).

The local kernel is the projected response of the stationary JT hidden-current reduction fixed above. It does not include Schwarzian dressing, different operator dimensions, or non-Ohmic hidden spectra. When the Schwarzian mode is restored, near-extremal JT boundary correlators develop late-time power-law tails and branch-cut structure, so the resulting memory law is expected to become non-Markovian [citation].

The return kernel does not follow from the local response kernel reference alone. To pass from the response kernel to a return kernel, impose only the minimal hidden-survival closure

P_H(u|s)=1,
\qquad
u-s<\tau_{\rm ret},

\partial_u P_H(u|s)=-\gamma_H P_H(u|s),
\qquad
u-s>\tau_{\rm ret}.

Hence

P_H(u|s)=

1,  u-s<\tau_{\rm ret},
[4pt]
e^{-\gamma_H(u-s-\tau_{\rm ret})},  u-s\ge \tau_{\rm ret},

and the corresponding hazard kernel is

K_{\rm ret}(u,s)
:=-\partial_u P_H(u|s)
=
\gamma_H e^{-\gamma_H(u-s-\tau_{\rm ret})}\,\Theta(u-s-\tau_{\rm ret}).

This yields the return kernel of the Gaussian model once the minimal hidden-survival law is imposed.

Now insert the sharp-limit JT one-body Hawking rate

\dot s_H^{\rm eff}(s)=a e^{-bs}

from Appendix reference. For $\gamma_H\neq b$TeX\gamma_H\neq b, the returned-correlation rate becomes

\mathcal J_\gamma(u)
=
\int_0^u ds\,K_{\rm ret}(u,s)\,\dot s_H^{\rm eff}(s)
=
\frac{a\gamma_H}{\gamma_H-b}
\Bigl[
 e^{-b(u-\tau_{\rm ret})}-e^{-\gamma_H(u-\tau_{\rm ret})}
\Bigr]\Theta(u-\tau_{\rm ret}),

and the hidden conditional-dependence rate is

\mathcal C_H^\gamma(u)=2a e^{-bu}-\mathcal J_\gamma(u).

The special case $\gamma_H=b$TeX\gamma_H=b is obtained by continuity and gives $\mathcal J_b(u)=ab(u-\tau_{\rm ret})e^{-b(u-\tau_{\rm ret})}\Theta(u-\tau_{\rm ret})$TeX\mathcal J_b(u)=ab(u-\tau_{\rm ret})e^{-b(u-\tau_{\rm ret})}\Theta(u-\tau_{\rm ret}).

Finally choose

\tau_{\rm ret}=u_P

and take the fast-return limit $\gamma_H\to\infty$TeX\gamma_H\to\infty. Then

K_{\rm ret}(u,s)\to \delta(u-s-u_P),

\mathcal J_\gamma(u)\to a e^{-b(u-u_P)}\Theta(u-u_P)
=\frac{3}{2}a e^{-bu}\Theta(u-u_P),

and

\mathcal C_H^\gamma(u)\to

2a e^{-bu},  u<u_P,
[4pt]
\dfrac{1}{2}a e^{-bu},  u>u_P,

which reproduces reference, reference, and reference. Thus the sharp-limit delta kernel is not only an inversion byproduct of the age-translation closure. It is the fast-return limit of the Gaussian stationary JT stretched-horizon reduction once the same hidden-survival law is imposed.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.