JT-Bath Evaporation, Schwarzian Memory, and Non-Markovian Return Kernels

The late-time return problem can be formulated in terms of the return rate $\Jh(u)$TeX\Jh(u) and the hidden conditional-dependence rate $\Ch(u)$TeX\Ch(u), with a stationary Jackiw--Teitelboim (JT) stretched-horizon Gaussian reduction providing an exact local response kernel within the corresponding reduced sector [citation]. The question addressed here is how that kernel changes once Schwarzian boundary fluctuations are retained. The analysis does not re-found the exterior-state construction, the late-time identities, or a general evaporation law. It adds admissible analytical closure only at the level of the reduced return-kernel family.

Near-extremal JT and nearly-AdS$_2$TeX_2 dynamics isolate boundary reparametrization as the low-temperature throat sector [citation], while exact Schwarzian correlators and Wilson-line constructions give closed-form control of the corresponding boundary observables [citation]. The island/quantum extremal surface (QES) program and its later time-dependent extensions identify the onset data relevant to evaporating JT+bath backgrounds [citation]. At the level of real-time response, the retarded Schwarzian correlator develops a branch cut and late-time power-law decay rather than a purely local dissipative form [citation]. Recent studies of near-extremal evaporation, scattering, and Hawking-emission corrections likewise show that Schwarzian-sector effects remain dynamically relevant beyond the strict local baseline [citation]. Open-system and Schwinger--Keldysh treatments provide a direct language for the reduced causal kernels used below [citation].

The discussion below keeps the late-time identities for $\Jh$TeX\Jh and $\Ch$TeX\Ch fixed, takes the stationary Gaussian stretched-horizon reduction as the local baseline, and represents the Schwarzian sector by a minimal one-branch nonanalytic low-frequency term consistent with the branch-cut evidence. The hidden/exterior split serves only as accessibility bookkeeping rather than as a claim of unique microscopic horizon factorization or a completed interior algebra, in line with recent work on nonperturbative Hilbert spaces, reconstruction, and operator-algebraic formulations of black-hole information [citation]. The reduced return law is used physically only on parameter sectors or post-onset windows for which the associated survival function is positive, monotone, and kernel-nonnegative. No field-level observable theorem or phenomenological fit is extracted from the reduced-kernel family constructed here.

The objects and notation used below are fixed here.

The working setup distinguishes global information from locally accessible information on an observation domain $\Dobs=(\Odom_{\rm geom},T,B,\Ggate)$TeX\Dobs=(\Odom_{\rm geom},T,B,\Ggate), interprets commit as irreversible redistribution rather than ontic deletion, and uses the compact-object horizon diagnostic

\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 [citation].

On a fixed tripartite split $(R_{<u},r_u,H_u)$TeX(R_{<u},r_u,H_u), define

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

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

with the exact identities

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

\Ch(u)=2\dot s_H^{\rm eff}(u)-\Jh(u)

whenever the split is fixed and the tripartite state is pure at each retarded-time step [citation]. An explicit kernel description for $\Jh$TeX\Jh beyond the local Gaussian model is then sought in the form

\Jh(u)=\int_0^u ds\,K_{\rm ret}(u,s)\,\dot s_H^{\rm eff}(s).

Equation reference is not an identity of quantum information theory. It is a reduced-dynamics representation whose kernel must be computed or constrained.

Every reduced-state notation involving the hidden complement is used only as an accessibility bookkeeping device. No unique microscopic tensor factorization across a quantum-gravitational horizon is assumed, and no completed interior algebra is constructed here [citation]. The return-kernel law is imposed only on that bookkeeping sector.

The stationary JT stretched-horizon Gaussian reduction provides the local baseline for Schwarzian dressing [citation]. Take a near-horizon field decomposition

\phi=\phi_A+\phi_H,

where $\phi_A$TeX\phi_A is the exterior accessible sector and $\phi_H$TeX\phi_H the hidden complement on a stretched shell $\Sigma_\epsilon$TeX\Sigma_\epsilon just outside the horizon diagnostic $\Xith=0$TeX\Xith=0. Let the slow exterior shell mode be $q(u)$TeXq(u), with $u$TeXu the horizon-adapted retarded time. The minimal horizon-shell interaction is

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

with

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

For a free Hartle--Hawking hidden sector the influence functional is exactly Gaussian. The thermal Wightman function of the dimension-one horizon current 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,

whose spectral density is Ohmic,

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

Consequently the retarded self-energy is

\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,

with the real local term absorbed into a renormalized shell frequency. The KMS noise kernel is

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

Within the first-order slow-bin projection used below, the induced response kernel is therefore exact and local:

\Gamma_0(u,s)=\gamma_H\,\delta(u-s).

This is the stationary Gaussian baseline. It is exact inside the stretched-horizon Gaussian model, but it does not yet contain Schwarzian dressing or time-dependent island motion.

The local baseline reference cannot be the whole story once the quantum Schwarzian sector is restored. The near-extremal JT or nearly-AdS$_2$TeX_2 literature shows that low-temperature throat dynamics is governed by boundary reparametrization fluctuations [citation]. For the retarded boundary correlator, recent primary work finds a late-time power-law decay together with a branch cut in the complex frequency plane emanating from the origin [citation]. This means that a purely local projected kernel is no longer compatible with the dressed boundary dynamics.

The branch cut does not by itself fix a unique microscopic low-frequency law for every probe sector. We therefore retain only the minimal causal representative used below: a local Gaussian part together with a single nonanalytic contribution. Let $p$TeXp denote the one-sided Laplace variable for a causal projected kernel. A minimal one-branch reduced representative of the Schwarzian low-frequency sector is

\widetilde\Gamma_{\rm Sch}(p)
=
\gamma_H+\eta_H p^{\nu_H-1},
\qquad 0<\nu_H<1,

with $\eta_H>0$TeX\eta_H>0 a dressed horizon-response coefficient and $\nu_H$TeX\nu_H the branch-cut index left explicit. The local part $\gamma_H$TeX\gamma_H is the exact Gaussian coefficient from reference; the power-law memory term represents a one-branch low-frequency class and is not asserted as a unique microscopic exponent law for every dressed probe sector.

proposition: Time-domain kernel of the minimal reduced Schwarzian class. If the one-sided Laplace transform of the projected kernel is given by reference, then the time-domain kernel is

\Gamma_{\rm Sch}(u,s)
=
\gamma_H\delta(u-s)
+
\frac{\eta_H}{\Gamma(1-\nu_H)}(u-s)^{-\nu_H}\Theta(u-s).

proof. For $0<\nu_H<1$TeX0<\nu_H<1, the inverse one-sided Laplace transform of $p^{\nu_H-1}$TeXp^{\nu_H-1} is $t^{-\nu_H}/\Gamma(1-\nu_H)$TeXt^{-\nu_H}/\Gamma(1-\nu_H) for $t>0$TeXt>0. Adding the local term gives reference.

Equation reference states that the projected horizon-shell response is not merely delayed or broadened; it is genuinely non-Markovian with a power-law tail whose exponent is controlled by the Schwarzian branch-cut sector. This is the reduced-dynamics form of the qualitative statement that Schwarzian-dressed retarded correlators have power-law late-time decay rather than a purely exponential quasinormal response [citation]. Recent primary work on Hawking emission corrected by JT gravity fluctuations also supports the use of Schwarzian two-point functions as dynamical input for radiation corrections [citation].

remark: Scope of the branch-cut index. A universal value of $\nu_H$TeX\nu_H is not fixed here. Determining the branch-cut index from a specific dressed operator sector requires a more microscopic input than is assumed below. Once the primary Schwarzian branch-cut structure is accepted, the projected kernel is represented within the one-branch reduced family by the non-Markovian class reference--reference.

remark: Reduced Schwarzian class. Equation reference records the minimal nonanalytic low-frequency class considered below. Additional analytic corrections, multiple branch points, or more detailed sector-dependent dressings are not excluded; they are simply outside the reduced family used below.

The response kernel reference does not by itself determine the return kernel in reference. Fixing $K_{\rm ret}$TeXK_{\rm ret} therefore requires an additional reduced closure for the decay of hidden dependence after return onset. We use the minimal stationary closure compatible with the same local-plus-branch-cut structure that appears in the projected response kernel. Let

\Delta:=u-s-\tauret.

For $\Delta<0$TeX\Delta<0, the fresh packet has not yet entered the return window and hidden dependence remains unsuppressed:

P_H(u|s)=1,
\qquad \Delta<0.

For $\Delta>0$TeX\Delta>0, the hidden-dependence survival obeys the Volterra equation

\partial_u P_H(u|s)
=
-\gamma_H P_H(u|s)
-
\frac{\eta_H}{\Gamma(1-\nu_H)}
\int_{s+\tauret}^{u} dv\,(u-v)^{-\nu_H}P_H(v|s),
\qquad \Delta>0,

with initial condition $P_H((s+\tauret)^+|s)=1$TeXP_H((s+\tauret)^+|s)=1. This is the minimal hidden-survival law generated by the same local-plus-branch-cut structure that appears in the projected response kernel.

proposition: Exact stationary return kernel under the minimal hidden-survival closure. For the closure reference--reference, the one-sided Laplace transforms with respect to $\Delta$TeX\Delta are

\LtP(p)=\frac{1}{p+\gamma_H+\eta_H p^{\nu_H-1}},

\LtK(p;s)=e^{-p\tauret}\frac{\gamma_H+\eta_H p^{\nu_H-1}}{p+\gamma_H+\eta_H p^{\nu_H-1}}.

proof. Taking the one-sided Laplace transform of reference over $\Delta>0$TeX\Delta>0 gives

p\LtP(p)-1=-(\gamma_H+\eta_H p^{\nu_H-1})\LtP(p),
\nonumber

which yields reference. Since $K_{\rm ret}(u,s)=-\partial_u P_H(u|s)$TeXK_{\rm ret}(u,s)=-\partial_u P_H(u|s), the transformed return kernel is $\LtK=e^{-p\tauret}[1-p\LtP]$TeX\LtK=e^{-p\tauret}[1-p\LtP], giving reference.

corollary: Return-weight normalization in the stationary reduced model. For the stationary reduced model, the delayed return kernel is normalized in the sense

\LtK(0;s)=1,

and hence, whenever the inverse transform exists as an integrable causal kernel away from the onset shift,

\int_{s+\tauret}^{\infty} du\,K_{\rm ret}(u,s)=1.

proof. Taking the limit $p\to 0^+$TeXp\to 0^+ in reference gives $\LtK(0;s)=1$TeX\LtK(0;s)=1. Equation reference is the corresponding causal inverse-transform statement.

Equation reference is explicit, causal, and non-Markovian. The return kernel is fixed by the stationary Gaussian baseline, the Schwarzian branch-cut dressing, and the minimal hidden-survival law.

The corresponding return-rate transform is immediate. For any admissible one-body organizer $\dot s_H^{\rm eff}(u)$TeX\dot s_H^{\rm eff}(u),

\widetilde{\Jh}(p)=\LtK(p)\,\widetilde{\dot s_H^{\rm eff}}(p),

while reference gives

\widetilde{\Ch}(p)=2\widetilde{\dot s_H^{\rm eff}}(p)-\widetilde{\Jh}(p).

For the exponential Hawking organizer used in the sharp JT toy branch,

\dot s_H^{\rm eff}(u)=a e^{-bu}\Theta(u),
\qquad
\widetilde{\dot s_H^{\rm eff}}(p)=\frac{a}{p+b},

one finds the exact transformed return rate

\widetilde{\Jh}(p)=
\frac{a e^{-p\tauret}}{p+b}
\frac{\gamma_H+\eta_H p^{\nu_H-1}}{p+\gamma_H+\eta_H p^{\nu_H-1}}.

Equation reference gives the exact transformed return rate for the exponential organizer.

corollary: Markovian limit. If $\eta_H\to 0$TeX\eta_H\to 0, then

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

and for reference

\mathcal J_\gamma(u)
=
\frac{a\gamma_H}{\gamma_H-b}
\left[e^{-b(u-\tauret)}-e^{-\gamma_H(u-\tauret)}\right]\Theta(u-\tauret),
\qquad \gamma_H\neq b,

while at the resonant point $\gamma_H=b$TeX\gamma_H=b,

\mathcal J_\gamma^{\rm res}(u)
=
a\gamma_H (u-\tauret)e^{-\gamma_H(u-\tauret)}\Theta(u-\tauret).

proof. For $\eta_H\to 0$TeX\eta_H\to 0, equation reference becomes

\widetilde{\Jh}(p)=\frac{a\gamma_H e^{-p\tauret}}{(p+b)(p+\gamma_H)}.
\nonumber

When $\gamma_H\neq b$TeX\gamma_H\neq b, partial fractions give reference. At $\gamma_H=b$TeX\gamma_H=b, the Laplace transform has a repeated pole and the inverse transform gives reference.

corollary: Pure branch-cut analytical limit. If $\gamma_H=0$TeX\gamma_H=0, then for $\Delta>0$TeX\Delta>0

P_H^{\rm tail}(\Delta)=E_{2-\nu_H}\!\left[-\eta_H\Delta^{2-\nu_H}\right],

K_{\rm ret}^{\rm tail}(u,s)
=
\eta_H\Delta^{1-\nu_H}
E_{2-\nu_H,\,2-\nu_H}\!\left[-\eta_H\Delta^{2-\nu_H}\right]\Theta(\Delta),

where $E_{\alpha}$TeXE_{\alpha} and $E_{\alpha,\beta}$TeXE_{\alpha,\beta} are the standard Mittag--Leffler functions.

proof. For $\gamma_H=0$TeX\gamma_H=0, equation reference becomes $\LtP(p)=1/(p+\eta_H p^{\nu_H-1})$TeX\LtP(p)=1/(p+\eta_H p^{\nu_H-1}). Using the standard Laplace identity $\mathcal L\{E_{\alpha}(-a t^{\alpha})\}=p^{\alpha-1}/(p^{\alpha}+a)$TeX\mathcal L\{E_{\alpha}(-a t^{\alpha})\}=p^{\alpha-1}/(p^{\alpha}+a) with $\alpha=2-\nu_H$TeX\alpha=2-\nu_H gives reference. Differentiating and inserting the onset shift yields reference.

Equations reference--reference isolate the branch-cut contribution in closed form and give an exact analytical limit of the stationary family.

definition: Admissibility gate on a chosen post-onset window. Fix a post-onset window $\mathcal W_s=(0,\Delta_{\max}(s)]$TeX\mathcal W_s=(0,\Delta_{\max}(s)]. A stationary or packetwise kernel is used as a survival law on $\mathcal W_s$TeX\mathcal W_s only if

0\le P_H(\Delta)\le 1,
\qquad
\partial_\Delta P_H(\Delta)\le 0,
\qquad
K_{\rm ret}(\Delta)\ge 0,
\qquad \Delta\in\mathcal W_s.

In the pure branch-cut limit reference--reference, these properties are not automatic, and they must be checked on every chosen mixed local-plus-tail sector as well.

remark: Positivity and monotonicity of the pure-tail sector. For $\alpha:=2-\nu_H\in(1,2)$TeX\alpha:=2-\nu_H\in(1,2), the classical complete-monotonicity result for $E_\alpha(-x)$TeXE_\alpha(-x) applies only to $0<\alpha\le 1$TeX0<\alpha\le 1 [citation], whereas for $1<\alpha<2$TeX1<\alpha<2 the fractional-oscillation solution $E_\alpha(-t^\alpha)$TeXE_\alpha(-t^\alpha) can have zeros [citation]. The formulas reference--reference therefore define an exact analytical limit of the stationary family, but their use as a survival law requires the windowed admissibility test reference.

remark: Use of the pure-tail sector. The Mittag--Leffler branch is retained as an analytical benchmark for the branch-cut contribution. Physical applications use either admissibility-certified sectors of that branch on a chosen window or mixed local-plus-tail kernels whose admissibility has been certified on the same window.

corollary: Fast-return limit and the Page-delay delta kernel. Let $\tauret=u_P$TeX\tauret=u_P be fixed. Then in the fast-return limit $\gamma_H\to\infty$TeX\gamma_H\to\infty with finite $\eta_H$TeX\eta_H,

\LtK(p;s)\to e^{-pu_P},

and hence

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

For the organizer reference this reproduces the sharp Page-delay delta kernel and the corresponding sharp-limit return rates [citation].

proof. In reference, divide numerator and denominator by $\gamma_H$TeX\gamma_H and send $\gamma_H\to\infty$TeX\gamma_H\to\infty. The rational factor tends to one, leaving only the delay factor $e^{-pu_P}$TeXe^{-pu_P}. Its inverse one-sided Laplace transform is $\delta(u-s-u_P)$TeX\delta(u-s-u_P).

remark: Sharp-return limit. The delta kernel reference is a sharp illustrative limit of the stationary reduced family. It is not claimed as a generic late-time black-hole return law outside that limit.

The stationary kernel reference is exact only inside the stationary reduced model. To connect it to evaporating JT+bath Page-curve backgrounds, the packetwise onset delay is identified, at the reduced semiclassical level, with the first retarded time at which the relevant island/QES branch dominates the no-island branch in the corresponding geometry [citation]. The analysis is restricted to a packetwise adiabatic lift. In the declared analysis, $u_I(s)$TeXu_I(s) is imported reduced semiclassical onset data from the corresponding JT+bath island/QES background; no independent QES-location theorem is derived here.

For a packet emitted at retarded time $s$TeXs, let $u_I(s)$TeXu_I(s) denote the first retarded time at which the island/QES branch relevant to that packet dominates the no-island branch in the corresponding JT+bath geometry. Define the moving-island onset delay by

\tauI(s):=u_I(s)-s.

This onset delay is used as the reduced repayment onset for Hawking one-body entropy packets and need not coincide with a message-decoding time. The adiabatic lift consists of promoting the stationary parameters to slow functions of the emission time:

\gamma_H\to \gamma_H(s),
\qquad
\eta_H\to \eta_H(s),
\qquad
\nu_H\to \nu_H(s),
\qquad
\tauret\to \tauI(s).

Define the mixed-sector inverse-time scale

\omega_{\rm ad}(s):=\gamma_H(s)+\eta_H(s)^{1/(2-\nu_H(s))}

and use the packetwise lift only on emission-time windows where

\epsad(s):=
\max\!\left\{
\left|\partial_s\tauI(s)\right|,
\frac{\left|\partial_s\gamma_H(s)\right|}{\omega_{\rm ad}(s)^2},
\frac{\left|\partial_s\eta_H(s)\right|}{\omega_{\rm ad}(s)^{3-\nu_H(s)}},
\frac{\left|\partial_s\nu_H(s)\right|}{\omega_{\rm ad}(s)}
\right\}\ll 1.

Then the emission-time-dependent return kernel is

\widetilde{K}_{\rm evap}(p;s)
=
\exp\!\big[-p\tauI(s)\big]
\frac{\gamma_H(s)+\eta_H(s)p^{\nu_H(s)-1}}
{p+\gamma_H(s)+\eta_H(s)p^{\nu_H(s)-1}}.

The resulting return law is

\Jh(u)=\int_0^u ds\,K_{\rm evap}(u,s)\,\dot s_H^{\rm eff}(s),

with

\Ch(u)=2\dot s_H^{\rm eff}(u)-\Jh(u).

Equation reference is the packetwise adiabatic lift. It is the leading kernel on windows where reference is small and receives $O(\epsad(s))$TeXO(\epsad(s)) corrections beyond that order. It is obtained by freezing the kernel data at emission time $s$TeXs, setting the onset delay by the corresponding QES branch, and allowing the local Gaussian coefficient, the branch-cut coefficient, and the branch-cut index to drift only at adiabatic order. No resummation of the full two-time nonstationary memory functional is attempted, so reference is restricted to the packetwise windows fixed by reference.

proposition: Recovery of the sharp Page-delay branch in the constant-onset fast-return limit. If $\tauI(s)\equiv u_P$TeX\tauI(s)\equiv u_P is constant, $\eta_H(s)\to 0$TeX\eta_H(s)\to 0, $\gamma_H(s)\to\infty$TeX\gamma_H(s)\to\infty, and the Hawking organizer takes the sharp JT form $\dot s_H^{\rm eff}(s)=a e^{-bs}$TeX\dot s_H^{\rm eff}(s)=a e^{-bs}, then reference reduces to

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

and reference--reference reproduce the corresponding sharp-limit rates [citation].

proof. This is the simultaneous constant-onset and fast-return limit of reference in the packetwise regime with fixed onset and dominant local return.

The reduced return-kernel analysis may import compact-object companion records only as bounded diagnostic context. For the BH2 reduced-response lane, BH--VP0 is read only through its GWOSC-facing remnant and response surface; the EHT-facing morphology and polarization surrogate record belongs to BH1 and is not imported here. BH--VP1 contributes only the official GWTC-3 Tests-of-GR posterior comparison context with final label $\BHVPoneLabel$TeX\BHVPoneLabel [citation].

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These labels do not replace the EHT image pipeline, LVK/GWOSC waveform pipeline, or LVK Tests-of-GR likelihood. They do not certify a field-level return-kernel observable, official ringdown-residual closure, compact-object theorem, microscopic entropy theorem, or quantum-gravity solution.

The analysis refines only the reduced return-kernel family on near-extremal JT backgrounds once Schwarzian memory is retained. The stationary projected Gaussian model supplies the exact local baseline reference, and the low-frequency Schwarzian contribution is represented by reference--reference. Within the reduced stationary family, the hidden-survival closure reference--reference makes the delayed return kernel explicit in Laplace space. The Markovian delayed-exponential sector, including the resonant case reference, and the fast-return delta sector are exact stationary limits of the same family. The fast-return delta kernel is retained only as a sharp reduced-family limit and is not elevated here to a generic late-time return law for evaporating black holes. The pure-tail Mittag--Leffler formulas reference--reference are exact analytical benchmarks and are used physically only after admissibility certification on the chosen post-onset window. The evaporating kernel reference is a leading packetwise adiabatic lift whose validity is limited to emission-time windows where the reduced frozen-data approximation is justified. The BH--VP0 and BH--VP1 records add only bounded GWOSC-facing and Tests-of-GR comparison context for the BH2 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, but these labels remain diagnostic-only and do not change the theorem status of the reduced return-kernel family.

The declared analysis does not derive a full quantum-gravity completion, a generic evaporating-background return-kernel theorem, a full two-time nonstationary resummation, a field-level observable theorem, a message-decoding or reconstruction law, or any CMB / early-universe consequence. Its established result is confined to the reduced stationary family, its exact stationary limits, admissibility-certified post-onset windows, and the leading packetwise adiabatic evaporating lift.

This appendix records the short derivations used in the main text.

From the Ohmic horizon bath to the local projected kernel

The thermal Wightman function reference implies the Ohmic spectral density reference and hence the self-energy reference. In frequency space, the dissipative contribution is therefore linear in $\omega$TeX\omega,

\Sigma_H^R(\omega)-\delta\Omega^2=-i\gamma_H\omega.

The corresponding generalized Langevin equation for the exterior shell mode is

\ddot q(u)+\Omega_r^2 q(u)+\gamma_H\dot q(u)=\xi_H(u),

with the KMS noise kernel fixed by reference. Under the slow-bin first-order projection used in the main text, reference reduces to

\dot q(u)+\int_{u_0}^{u} ds\,\Gamma_0(u,s)q(s)=\xi(u),

with the exact stationary projected kernel

\Gamma_0(u,s)=\gamma_H\delta(u-s).

From the branch cut to the projected memory kernel

The primary Schwarzian boundary-correlator result needed here is qualitative but strong: the retarded correlator has a late-time power-law tail and a branch cut emanating from the origin in frequency space [citation]. For a one-sided causal projected kernel, the minimal low-frequency image considered below is the nonanalytic term $\eta_H p^{\nu_H-1}$TeX\eta_H p^{\nu_H-1} with $0<\nu_H<1$TeX0<\nu_H<1. Within the one-branch reduced family, this term serves as a representative rather than a proof of a unique universal microscopic exponent law. The inverse transform then gives reference.

Laplace-space return law

For $\Delta>0$TeX\Delta>0, rewrite reference as

\frac{dP_H}{d\Delta}+\gamma_H P_H
+\frac{\eta_H}{\Gamma(1-\nu_H)}\int_0^{\Delta} d\ell\,(\Delta-\ell)^{-\nu_H}P_H(\ell)=0,
\qquad P_H(0)=1.

The one-sided Laplace transform gives reference, and then

\LtK(p;s)=e^{-p\tauret}\left[1-p\LtP(p)\right]

produces reference.

Pure-tail inverse transform

When $\gamma_H=0$TeX\gamma_H=0, set $\alpha=2-\nu_H\in(1,2)$TeX\alpha=2-\nu_H\in(1,2). Then

\LtP(p)=\frac{1}{p+\eta_H p^{\nu_H-1}}=\frac{p^{\alpha-1}}{p^{\alpha}+\eta_H}.

The standard Mittag--Leffler transform identity yields reference, and differentiation gives reference. These formulas define the exact analytical pure-tail limit of the stationary family; physical admissibility requires the additional conditions stated around reference.

Fast-return limit

If $\gamma_H\to\infty$TeX\gamma_H\to\infty while $\tauret$TeX\tauret is fixed, then

\LtK(p;s)=e^{-p\tauret}\frac{\gamma_H+\eta_H p^{\nu_H-1}}{p+\gamma_H+\eta_H p^{\nu_H-1}}
\longrightarrow e^{-p\tauret}.

The inverse transform gives the delayed delta kernel reference.

Packetwise adiabatic order

Equation reference is obtained by freezing the stationary kernel parameters at the emission time $s$TeXs and setting the onset delay by $\tauI(s)$TeX\tauI(s). The scale $\omega_{\rm ad}(s)$TeX\omega_{\rm ad}(s) in reference and the parameter $\epsad(s)$TeX\epsad(s) in reference record the requirement that those quantities drift slowly across one packetwise response window. No resummation of the full two-time nonstationary memory functional along the same causal interval is attempted. Accordingly, reference is used only as the leading packetwise kernel, with omitted corrections of order $O(\epsad(s))$TeXO(\epsad(s)), on those packetwise windows.

The reduced-kernel construction has a fixed mathematical standing. The late-time identities reference--reference are exact on a fixed tripartite split, the stationary JT stretched-horizon Gaussian sector provides the exact local baseline reference within the stationary projected Gaussian model, and the Schwarzian contribution is represented by the one-branch low-frequency family reference--reference. With the added hidden-survival Volterra law reference, this yields the exact stationary return kernel reference, together with the exact stationary limits reference--reference. The pure-tail formulas reference--reference are exact analytical limits of that stationary family.

Physical use is narrower than mathematical availability. The pure-tail sector is used only on post-onset windows that satisfy the admissibility gate reference. The evaporating kernel reference is the leading packetwise adiabatic lift obtained by freezing the stationary kernel data at emission time and setting the onset delay by $\tauI(s)$TeX\tauI(s); it is restricted to windows where reference is small. No full two-time nonstationary resummation, field-level observable theorem, phenomenological fit, or message-reconstruction law is extracted from this reduced-kernel family.

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Funding and competing interests..

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