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  pdftitle={Hawk01-A: Delayed-Query Information Retrieval from Astrophysical Hawking Radiation},
  pdfsubject={Fixed-background Hawking radiation, operational recovery, and observer-access bounds}
}

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\title{\textbf{Hawk01-A}\\[0.4em]
Delayed-Query Information Retrieval from Astrophysical Hawking Radiation:\\
Fixed-Background Baselines, Conserving Completions, and Observer-Access Bounds}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We formulate delayed-query retrieval from Hawking radiation as an
operational task on outgoing quantum-field modes in a four-dimensional
Schwarzschild exterior. Starting from the affine-vacuum horizon
boundary condition and the scalar scattering problem, we derive the
normalized greybody-modified emission law and finite-packet covariance.
For a trusted four-state qubit preparation, we prove the exact
optimal retrieval formula, a message-independent semiclassical null,
a sharp measure-before-query bound, and complete-attempt robustness
bounds. We construct a finite-energy emission isometry on degenerate
scalar occupation sectors whose late-radiation marginal agrees with a
truncated Hawking-mode thermal state for every message. With a declared
early key resource, its joint radiation permits partial quantum
recovery; without that resource, the late radiation is exactly
message-independent. A second conserving completion has the same
late state and energy distribution but no retrieval advantage even
with the early resource. Thus emission spectra, including the complete
late marginal in this example, do not determine the assisted recovery
channel. We calculate the exact coherent-access, destructive-number,
and referenced-recovery optima of the construction. Finite-packet
cutoffs, collection channels, certified model errors, and attempted-run
sampling are treated explicitly. The results are direct field-mode
statements and conditional completion theorems, not a derivation of
microscopic black-hole dynamics, an astrophysical decoding experiment,
or a resolution of the information paradox.
\end{abstract}

\section{Scope, conventions, and principal conclusions}
\label{sec:scope}

Hawking's particle-creation calculation determines a semiclassical
emission law once the spacetime, state, field, and asymptotic time
normalization have been specified \cite{Hawking1975,Unruh1976}.
It does not by itself determine the channel transferring a specified
infalling quantum message into accessible outgoing correlations.
Hayden--Preskill recovery introduces additional assumptions concerning
unitary dynamics, mixing, prior radiation, and decoding access
\cite{HaydenPreskill2007}. These assumptions are not consequences of
substituting an astrophysical mass into a finite radiation model.

This paper separates three mathematical objects:
\begin{enumerate}
\item the semiclassical outgoing field state;
\item a declared message-to-radiation channel;
\item an observer's collection, storage, and measurement channel.
\end{enumerate}
All recovery statements refer to the complete composition of the
second and third objects. An emission spectrum is not substituted for
that composition.

The source normalization follows the fixed-background interface
reported in Cheung's Em03-A \cite{CheungEm03A}. The operational
separations are informed by the delayed-query, recovery, extraction,
and record interfaces in
Psy04-A/B, En02-B--E, Em05-A/B, and En03-A
\cite{CheungPsy04A,CheungPsy04B,CheungEn02B,CheungEn02C,
CheungEn02D,CheungEn02E,CheungEm05A,CheungEm05B,CheungEn03A}.
The relevant results are proved below rather than transferred between
those manuscripts' different models.

Except where SI units are displayed, we use
\[
G=c=\hbar=k_B=1.
\]
The time \(u\) is retarded time normalized by the asymptotic
Schwarzschild Killing field. Frequencies \(\omega\) are positive
angular frequencies conjugate to that time. For normal states and
finite-dimensional channels, respectively, write
\begin{equation}
\Dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1,
\qquad
\Ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_{\diamond}.
\label{eq:distances}
\end{equation}
Diamond-norm statements are made on the explicitly regulated
interfaces; no finite-dimensional tensor-product factorization of
the exact gravitational theory is asserted.

The principal established conclusions are:
\begin{enumerate}
\item the fixed-background scalar source law,
Theorem~\ref{thm:source};
\item exact delayed-query scoring and observer monotonicity,
Theorem~\ref{thm:score};
\item a conditional semiclassical null and its complete-attempt
correction, Theorem~\ref{thm:null};
\item an explicit finite-energy, spectrum-constrained completion
and a nonrecovering completion with the same late marginal,
Theorem~\ref{thm:completion};
\item exact access-dependent scores and quantum recovery error,
Theorem~\ref{thm:completionoptima};
\item regulator, model-error, and sampling certificates.
\end{enumerate}

\section{The four-dimensional semiclassical source}
\label{sec:source}

\subsection{Geometry, field, and boundary state}

For \(M>0\), consider the exterior \(r>2M\) with metric
\begin{equation}
\dd s^2=-F(r)\dd t^2+F(r)^{-1}\dd r^2+r^2\dd\Omega^2,
\qquad
F(r)=1-\frac{2M}{r}.
\label{eq:metric}
\end{equation}
Set
\begin{equation}
r_*=r+2M\log\left(\frac{r}{2M}-1\right),
\qquad
u=t-r_*,
\qquad
\kappa=\frac{1}{4M},
\qquad
\betah=\frac{2\pi}{\kappa}=8\pi M.
\label{eq:coordinates}
\end{equation}
The field in the proved source calculation is a real, free, massless,
minimally coupled scalar satisfying \(\Box_g\phi=0\).

\begin{assumption}[Stationary Unruh boundary prescription]
\label{ass:unruh}
Incoming modes from past null infinity are in their vacuum.
The outgoing horizon sector is the affine-vacuum sector whose
current two-point distribution in affine coordinate \(U\) is
\[
-\frac{1}{4\pi(U-U'-\ii0)^2}.
\]
On that sector the affine/Killing relation is
\begin{equation}
U(u)=-\kappa^{-1}e^{-\kappa u}.
\label{eq:affine}
\end{equation}
The state is quasifree and has zero one-point function.
\end{assumption}

This is a stationary boundary-state prescription of the kind used to
isolate the late-time source term \cite{Unruh1976,CheungEm03A}.
Equation~\eqref{eq:affine} is exact in this prescribed stationary
model. It is not asserted to be an exact affine map throughout an
arbitrary collapse spacetime. A collapse application requires a
separate peeling estimate and transient error control.

After spherical-harmonic decomposition, the radial equation is
\begin{equation}
\left[
-\frac{\dd^2}{\dd r_*^2}+V_\ell(r)
\right]f_{\ell\omega}(r_*)
=\omega^2f_{\ell\omega}(r_*),
\qquad
V_\ell(r)
=F(r)\left[
\frac{\ell(\ell+1)}{r^2}+\frac{2M}{r^3}
\right].
\label{eq:radial}
\end{equation}
Let \(t_\ell(\omega)\) be the horizon-to-infinity transmission
amplitude with unit incident flux, and define
\begin{equation}
\Gamma_\ell(\omega;M)=\abs{t_\ell(\omega)}^2.
\label{eq:greybody}
\end{equation}

\begin{lemma}[Horizon current and occupation]
\label{lem:current}
Under Assumption~\ref{ass:unruh}, the Killing-coordinate current
covariance is
\begin{equation}
C_\kappa(s)
=-\frac{\kappa^2}{16\pi}
\csch^2\left(\frac{\kappa(s-\ii0)}{2}\right),
\qquad s=u-u'.
\label{eq:current}
\end{equation}
For the Fourier convention
\(\widehat C(\nu)=\int_{\mathbb R}e^{\ii\nu s}C(s)\dd s\),
\begin{equation}
\widehat C_\kappa(\nu)
=\frac{\nu}{2(1-e^{-\betah\nu})},
\qquad
\widehat C_\kappa(0)=\frac{1}{2\betah}.
\label{eq:currentfourier}
\end{equation}
The horizon occupation at \(\omega>0\) is
\begin{equation}
n_{\mathrm H}(\omega)
=\frac{1}{e^{\betah\omega}-1}.
\label{eq:horizonoccupation}
\end{equation}
\end{lemma}

\begin{proof}
Pull back the affine current distribution with the two derivative
factors \(U'(u)U'(u')\). Direct substitution of
\eqref{eq:affine} gives \eqref{eq:current}, with the boundary value
preserved because \(U'(u)>0\).

For \(\nu>0\), integrate the meromorphic boundary-value expression
against \(e^{\ii\nu s}\) in upper-half-plane rectangles whose
horizontal edges lie between successive poles. The vertical
contributions vanish as their real parts tend to infinity. The
upper contribution vanishes as its height tends to infinity.
The poles are
\[
s_k=\ii0+\ii k\betah,\qquad k=0,1,2,\ldots,
\]
and the principal part at each is
\(-[4\pi(s-s_k)^2]^{-1}\).
Consequently each residue contributes
\((\nu/2)e^{-k\betah\nu}\).
Summing the geometric series yields \eqref{eq:currentfourier}
for \(\nu>0\). Closing in the lower half-plane for \(\nu<0\)
gives the same expression. Its continuous value at zero is
\(1/(2\betah)\).

The positive-frequency vacuum contribution in this normalization is
\(\omega/2\). The negative-frequency contribution is therefore
\[
\frac{\omega}{2}n_{\mathrm H}(\omega)
=\widehat C_\kappa(-\omega)
=\frac{\omega}{2(e^{\betah\omega}-1)},
\]
which proves \eqref{eq:horizonoccupation}.
\end{proof}

\begin{theorem}[Normalized scalar Hawking source]
\label{thm:source}
For the geometry, field, and state above,
\(0\le\Gamma_\ell(\omega;M)\le1\), and the outgoing covariance is
\begin{equation}
\left\langle
a_{\ell m}^{\mathrm{out}\dagger}(\omega)
a_{\ell' m'}^{\mathrm{out}}(\omega')
\right\rangle
=
\delta_{\ell\ell'}\delta_{mm'}
\frac{\Gamma_\ell(\omega;M)}
     {e^{8\pi M\omega}-1}
\delta(\omega-\omega').
\label{eq:outcovariance}
\end{equation}
For a normalized packet \(h\) in one angular sector,
\begin{equation}
b_h=\int_0^\infty\overline{h(\omega)}
a_{\ell m}^{\mathrm{out}}(\omega)\dd\omega,
\qquad
\int_0^\infty\abs{h(\omega)}^2\dd\omega=1,
\label{eq:packetoperator}
\end{equation}
its mean occupation is
\begin{equation}
\left\langle b_h^\dagger b_h\right\rangle
=
\int_0^\infty\abs{h(\omega)}^2
\frac{\Gamma_\ell(\omega;M)}
     {e^{8\pi M\omega}-1}\dd\omega.
\label{eq:packetoccupation}
\end{equation}
With one temporal mode per phase-space cell \(2\pi\),
\begin{equation}
\boxed{
\frac{\dd^2N}{\dd u\,\dd\omega}
=
\frac{1}{2\pi}
\sum_{\ell=0}^{\infty}(2\ell+1)
\frac{\Gamma_\ell(\omega;M)}
     {e^{8\pi M\omega}-1}.
}
\label{eq:sourcerate}
\end{equation}
\end{theorem}

\begin{proof}
The real radial potential has conserved Wronskian flux.
The scattering matrix relating the two asymptotic channels is
unitary. In particular, transmission plus reflection probability
equals one, proving the bound on \(\Gamma_\ell\).
The usual positive-frequency scattering solutions exist for
\(\omega>0\): the potential is integrable in \(r_*\), and the
corresponding Jost integral equations define the asymptotic
solutions.

The outgoing annihilation operator is a linear combination of
the incoming infinity mode and the incoming horizon mode.
The infinity input contributes zero number occupation, the two
inputs have no cross covariance, and Lemma~\ref{lem:current}
gives the horizon occupation. Multiplying that occupation by
\(\abs{t_\ell}^2\) proves \eqref{eq:outcovariance}.
Integrating the covariance against the packet profile proves
\eqref{eq:packetoccupation}.

For completeness, partition positive frequency into bins of width
\(\Delta\omega\). In a bin, the functions
\[
h_k(\omega)
=(\Delta\omega)^{-1/2}
e^{-2\pi\ii k\omega/\Delta\omega}
\]
form an orthonormal frequency-window packet family. Consecutive
temporal centers are separated by \(2\pi/\Delta\omega\).
The expected occupation per packet is the bin average of the
occupation density. Dividing by that temporal separation gives
the bin rate
\[
\frac{1}{2\pi}
\int_{\mathrm{bin}}
\frac{\Gamma_\ell(\omega;M)}
     {e^{8\pi M\omega}-1}\dd\omega.
\]
Summing \(m\) and passing to the spectral density gives
\eqref{eq:sourcerate}. This counting convention is part of the
rate definition, not an additional thermal assumption.
\end{proof}

\begin{remark}[What the source theorem does not establish]
The theorem is the free-scalar fixed-background result.
It does not prove evaporation backreaction, a collapse solution,
an interacting Standard Model spectrum, or a message-recovery law.
For other species, the appropriately normalized source term has
the form
\begin{equation}
\frac{\dd^2N_s}{\dd u\,\dd\omega}
=
\frac{1}{2\pi}
\sum_{\ell,m,p}
\frac{\Gamma_{s\ell mp}(\omega;M)}
     {e^{\omega/\TH}-(-1)^{2s}},
\label{eq:speciesrate}
\end{equation}
with the field-specific scattering problem and multiplicities
specified separately \cite{Page1976,ArbeyAuffinger2019}.
Equation~\eqref{eq:speciesrate} is not a claim that those additional
scattering calculations have been performed here.
\end{remark}

\subsection{Quasistatic mass evolution and its limitations}
\label{subsec:mass}

The outgoing source power in natural units is
\begin{equation}
P_{\mathrm{em}}(M)
=
\int_0^\infty
\omega\frac{\dd^2N}{\dd u\,\dd\omega}\dd\omega.
\label{eq:power}
\end{equation}
For a selected species list, a quasistatic mass ledger is
\begin{equation}
\frac{\dd M}{\dd u}
=-P_{\mathrm{em}}(M)+P_{\mathrm{abs}}(u,M).
\label{eq:massledger}
\end{equation}
In SI units the right-hand side is divided by \(c^2\) when
the powers are in watts and \(M\) is in kilograms.

For the isolated massless scalar, the radial equation depends on
\(M\omega\). Define
\begin{align}
\alpha
&=
\frac{1}{2\pi}
\sum_{\ell=0}^{\infty}(2\ell+1)
\int_0^\infty
\frac{z\,\Gamma_\ell(z;1)}
     {e^{8\pi z}-1}\dd z,
\label{eq:alpha}\\
\nu
&=
\frac{1}{2\pi}
\sum_{\ell=0}^{\infty}(2\ell+1)
\int_0^\infty
\frac{\Gamma_\ell(z;1)}
     {e^{8\pi z}-1}\dd z.
\label{eq:nu}
\end{align}
Here \(\Gamma_\ell(z;1)\) denotes the dimensionless scattering
function at dimensionless frequency \(z\).

\begin{proposition}[Scalar quasistatic scaling]
\label{prop:mass}
If the sums in \eqref{eq:alpha}--\eqref{eq:nu} converge and the
isolated quasistatic ledger is adopted, then
\begin{equation}
P_{\mathrm{em}}(M)=\frac{\alpha}{M^2},
\qquad
\dot N(M)=\frac{\nu}{M},
\qquad
M(u)^3=M_0^3-3\alpha(u-u_0).
\label{eq:masssolution}
\end{equation}
For \(M(u)\ge M_{\mathrm{cut}}>0\), the cumulative expected
scalar count is
\begin{equation}
N(u_0,u)
=\frac{\nu}{2\alpha}\left[M_0^2-M(u)^2\right].
\label{eq:cumulativecount}
\end{equation}
The local surface-gravity adiabatic indicator is
\begin{equation}
\frac{\abs{\dot\kappa}}{\kappa^2}
=\frac{4\alpha}{M^2}.
\label{eq:adiabaticindicator}
\end{equation}
\end{proposition}

\begin{proof}
Substitute \(z=M\omega\) in the power and number integrals.
This gives the first two identities. Integrating
\(\dot M=-\alpha/M^2\) gives the third. Using
\(\dd u=-M^2\dd M/\alpha\) in the number integral gives
\eqref{eq:cumulativecount}. Finally,
\(\kappa=1/(4M)\) gives
\(\dot\kappa=-\dot M/(4M^2)\), and the displayed ratio follows.
\end{proof}

The ledger is not an exact evaporating metric. A small value of
\eqref{eq:adiabaticindicator} is a diagnostic, not a proved
channel-error bound. Every use of a sequence of fixed-background
channels below requires an independently stated approximation
allowance. The evolution stops at a declared
\(M_{\mathrm{cut}}\) well above the regime where the adopted
semiclassical treatment is uncontrolled.

\subsection{Astrophysical temperature scales}

Restoring SI units,
\begin{equation}
\TH(M)
=
\frac{\hbar c^3}{8\pi Gk_BM}
\simeq
\frac{1.227\times10^{23}\ {\rm K\,kg}}{M}.
\label{eq:temperatureSI}
\end{equation}
Using the displayed SI constants and the conventional numerical
mass \(M_\odot=1.9885\times10^{30}\ {\rm kg}\), the proposed
study points give Table~\ref{tab:masses}. The conversion should
not be confused with treating the solar mass itself as an exact
SI defining constant \cite{NISTConstants,Mamajek2015}.

\begin{table}[ht]
\centering
\caption{Source-temperature scales, not identified target objects
or completed detector forecasts.}
\label{tab:masses}
\begin{tabular}{@{}lll@{}}
\toprule
Adopted mass & \(\TH\) & \(k_B\TH\)\\
\midrule
\(10M_\odot\)
& \(6.17\times10^{-9}\ {\rm K}\)
& \(5.32\times10^{-13}\ {\rm eV}\)\\
\(10^{11}\ {\rm kg}\)
& \(1.227\times10^{12}\ {\rm K}\)
& \(105.7\ {\rm MeV}\)\\
\(10^{12}\ {\rm kg}\)
& \(1.227\times10^{11}\ {\rm K}\)
& \(10.57\ {\rm MeV}\)\\
\(10^{13}\ {\rm kg}\)
& \(1.227\times10^{10}\ {\rm K}\)
& \(1.057\ {\rm MeV}\)\\
\bottomrule
\end{tabular}
\end{table}

For these numerical conversions we used
\(G=6.67430\times10^{-11}\ {\rm m^3\,kg^{-1}\,s^{-2}}\),
\(c=299792458\ {\rm m\,s^{-1}}\),
\(k_B=1.380649\times10^{-23}\ {\rm J\,K^{-1}}\), and
\(\hbar\simeq1.054571817\times10^{-34}\ {\rm J\,s}\).
The table does not import a lifetime estimate, assert the
existence of a suitable primordial object, or replace the
species and background calculations needed for a count forecast.

\section{Physical message, complete attempts, and the access boundary}
\label{sec:message}

\subsection{Trusted preparation}

Let \(A\) be a two-dimensional internal code carried by a normalized
ingoing test-body or field preparation. The labels
\(x=(x_1,x_2)\in\{0,1\}^2\) are uniformly sampled, and
\begin{equation}
\rho_x^A
=
\frac12\left[
I+
\frac{(-1)^{x_1}\sigmax+(-1)^{x_2}\sigmaz}{\sqrt2}
\right].
\label{eq:squareensemble}
\end{equation}
These are pure states. The preparation is a trusted input interface:
the decoder does not receive a second copy, a preparation label,
or a message-dependent classical side channel.

\begin{assumption}[Degenerate physical code and leakage audit]
\label{ass:physicalcode}
The internal states \(\ket0_A,\ket1_A\) have equal energy,
charge, and declared angular-momentum labels. The common
center-of-mass wavepacket, injection trajectory, and arrival-time
distribution do not depend on \(x\).
On the code, the specified smeared macroscopic source operators
act as scalar multiples of the identity.

All other accessible preparation, prompt-scattering, capture,
and exterior-probe records are included in the declared
output channel, or excluded by an explicit access boundary
whose leakage is separately bounded.
\end{assumption}

The assumption is stronger than equality of mean energy.
Scalar macroscopic source operators do not prevent a separate
internal-state probe from learning \(x\). Consequently the
assumption includes an instrument-level leakage audit, not
only a stress-energy expectation test. The exterior source
models and packet-effect probes in Em03-D/F/G are useful
examples of why an outgoing, message-dependent field signal
need not be information escaping from a black-hole interior
\cite{CheungEm03D,CheungEm03F,CheungEm03G}.
Stimulated-emission and ordinary-scattering contributions also
require separate treatment \cite{Wald1976}.

The equal-energy code is an explicitly idealized physical
interface. This paper does not claim that a suitable
astrophysical injection apparatus has been constructed.

\subsection{Complete channel and late query}

Let \(B\) denote the initially inaccessible source degrees of
freedom, and let \(E\) be a declared stored early-radiation
resource. Initially \(A\) is independent of \(BE\), whose state
\(\omega^{BE}\) may be correlated. A supplied linear physical
evolution and access prescription defines a CPTP map
\begin{equation}
\NN_u:A\longrightarrow Z_u,
\qquad
\rho_x^{Z_u}=\NN_u(\rho_x^A).
\label{eq:completechannel}
\end{equation}
The complete output \(Z_u\) contains every accessible quantum
system and classical flag surviving to the decoding boundary:
\begin{equation}
Z_u=
E_{\mathrm{stored}}\,
R_{\mathrm{late,acc}}(u)\,
F_{\mathrm{capture}}\,
F_{\mathrm{apparatus}}\,
F_{\mathrm{other}}.
\label{eq:completeoutput}
\end{equation}
A factor is absent if and only if its exclusion is part of the
declared boundary. Initially correlated early radiation is not
silently traced out and later restored to the decoder.

Only after that boundary is fixed, a uniform
\(J\in\{1,2\}\) is generated independently of \(X\), all
pre-query systems, and all accessible random seeds. The decoder
then reports \(B_{\mathrm{out}}\in\{0,1\}\). The score is
\begin{equation}
Q=\Pr(B_{\mathrm{out}}=X_J).
\label{eq:scoredefinition}
\end{equation}

\begin{definition}[Complete attempt]
\label{def:attempt}
An attempt includes capture failures, loss, vacuum outcomes,
saturation, storage failure, and decoding failure. Every
attempt produces a declared binary answer, possibly by
independent random guessing on a failure branch. A heralded
score is a different task and must report both the heralding
instrument and its probability.
\end{definition}

This complete-output requirement is the channel analogue of the
record and residual-instrument distinctions in En03-A and
Psy02-B \cite{CheungEn03A,CheungPsy02B}. Equality of a selected
record does not establish equality of the surviving quantum
instrument.

\section{Exact delayed-query scoring and classical nulls}
\label{sec:score}

For any complete output ensemble, define the normalized states
\begin{equation}
\tau_{b\mid j}
=
\frac12\sum_{x:x_j=b}\rho_x^{Z},
\qquad b\in\{0,1\},\quad j\in\{1,2\}.
\label{eq:taudef}
\end{equation}

\begin{theorem}[Optimal score, input ceiling, and data processing]
\label{thm:score}
For unrestricted binary measurements after receiving \(J\),
\begin{equation}
\boxed{
\Qopt(\{\rho_x^Z\})
=
\frac12+
\frac18\sum_{j=1}^2
\norm{\tau_{0\mid j}-\tau_{1\mid j}}_1.
}
\label{eq:helstromscore}
\end{equation}
For any CPTP channel applied to the trusted ensemble
\eqref{eq:squareensemble},
\begin{equation}
\frac12\le\Qopt\le
\qQ:=\frac12+\frac{1}{2\sqrt2}.
\label{eq:qubitceiling}
\end{equation}
If \(Z'\) is obtained from \(Z\) by a message-independent
CPTP map, then
\begin{equation}
\Qopt(Z')\le\Qopt(Z).
\label{eq:scoreprocessing}
\end{equation}
\end{theorem}

\begin{proof}
For a fixed query \(j\), the two requested-bit hypotheses
have equal prior probability and states
\(\tau_{0\mid j}\), \(\tau_{1\mid j}\).
Let \(H_j=\tau_{0\mid j}-\tau_{1\mid j}\).
For an effect \(0\le M\le I\), the binary success probability is
\[
\frac12+\frac12\Tr(MH_j).
\]
Since \(\Tr H_j=0\), the supremum is
\(\frac12+\frac14\norm{H_j}_1\), attained by the positive
spectral projector of \(H_j\), with an arbitrary decision on
its kernel. Averaging over \(j\) proves
\eqref{eq:helstromscore}. This is the binary
Holevo--Helstrom argument \cite{Watrous2018}.

At the input,
\[
\tau_{0\mid1}^A-\tau_{1\mid1}^A
=\frac{\sigmax}{\sqrt2},
\qquad
\tau_{0\mid2}^A-\tau_{1\mid2}^A
=\frac{\sigmaz}{\sqrt2}.
\]
Each trace norm is \(\sqrt2\). Contractivity under a CPTP
channel proves the upper bound and
\eqref{eq:scoreprocessing}. Random guessing proves the lower
bound. An untouched qubit with query-dependent \(X\) or \(Z\)
measurement attains the upper bound.
\end{proof}

\begin{proposition}[Continuity with the correct normalization]
\label{prop:scorecontinuity}
For two output ensembles on a common space,
\begin{equation}
\abs{\Qopt(\{\rho_x\})-\Qopt(\{\widetilde\rho_x\})}
\le
\frac14\sum_x
\Dtr(\rho_x,\widetilde\rho_x).
\label{eq:scorecontinuity}
\end{equation}
In particular, for regulated channels,
\begin{equation}
\abs{\Qopt(\NN)-\Qopt(\widetilde\NN)}
\le \Ddia(\NN,\widetilde\NN).
\label{eq:scorediamond}
\end{equation}
\end{proposition}

\begin{proof}
For a fixed decoder, put
\[
F_x=\frac12\sum_{j=1}^2 M_{x_j\mid j}.
\]
Then \(0\le F_x\le I\), and its score is
\(\frac14\sum_x\Tr(F_x\rho_x)\).
The variational characterization of trace distance bounds the
score difference by the right-hand side of
\eqref{eq:scorecontinuity}. Taking suprema in both directions
proves the assertion. A half-diamond bound controls the trace
distance for every input state, proving
\eqref{eq:scorediamond}.
\end{proof}

\begin{theorem}[Sharp measure-before-query bound]
\label{thm:classical}
Suppose the complete channel from the trusted qubit is
measure-and-prepare:
\begin{equation}
\NN(\rho)=\sum_y\Tr(E_y\rho)\sigma_y,
\qquad
E_y\ge0,\quad \sum_y E_y=I,
\label{eq:EBform}
\end{equation}
with the measurement performed before the fresh query.
Then
\begin{equation}
\boxed{\Qopt(\NN)\le\frac34.}
\label{eq:EBbound}
\end{equation}
The bound is sharp, even with a two-outcome measurement and a
one-bit classical record. It also holds when the complete
pre-query output is an arbitrary classical record generated
by a POVM on the trusted qubit.
\end{theorem}

\begin{proof}
Giving the decoder the entire label \(y\) can only improve its
performance. Write
\[
E_y=w_y(I+\bm r_y\cdot\bm\sigma),
\qquad
w_y\ge0,\quad \abs{\bm r_y}\le1,\quad
\sum_yw_y=1.
\]
For the classical label channel, the requested-bit probability
difference is
\[
p(y\mid x_j=0)-p(y\mid x_j=1)
=\frac{\Tr(E_y\sigma_j)}{\sqrt2},
\]
where \(\sigma_1=\sigmax\) and \(\sigma_2=\sigmaz\).
Equation~\eqref{eq:helstromscore} therefore gives
\[
Q
=\frac12+\frac{1}{4\sqrt2}
\sum_yw_y\bigl(\abs{r_{y,x}}+\abs{r_{y,z}}\bigr)
\le\frac12+\frac14.
\]
The final inequality uses
\(\abs{r_x}+\abs{r_z}\le\sqrt2\abs{\bm r}\le\sqrt2\).
Measuring along \((\sigmax+\sigmaz)/\sqrt2\) attains equality.
For a continuous outcome space the same proof uses the trace
measure of the POVM and its Bloch-vector density.
\end{proof}

Measure-and-prepare channels are the entanglement-breaking
channel class \cite{HorodeckiShorRuskai2003}.
The bound above is a trusted-input channel statement.
It is distinct from an arbitrary classical encoder with a
two-state storage bottleneck, although that bottleneck also
has the sharp value \(3/4\) in the task considered in Psy04-A
\cite{CheungPsy04A}. Neither statement is a universal
``classical black-hole'' bound.

In particular, if an early quantum resource survives outside
the asserted classical cut, then the complete channel must
include that resource before Theorem~\ref{thm:classical} can be
applied. Entanglement breaking of one selected radiation
factor does not establish entanglement breaking of the
complete accessible output.

\begin{corollary}[Robust classical null]
\label{cor:robustEB}
If a regulated complete channel is within half-diamond distance
\(\epsilon\) of a measure-and-prepare channel, then
\begin{equation}
\Qopt\le
\min\left\{\qQ,\frac34+\epsilon\right\}.
\label{eq:robustEB}
\end{equation}
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:scorecontinuity} and
Theorems~\ref{thm:score} and \ref{thm:classical}.
\end{proof}

\section{The semiclassical null and capture leakage}
\label{sec:null}

\begin{assumption}[Identity-action semiclassical message baseline]
\label{ass:null}
For the selected fixed-background calculation:
\begin{enumerate}
\item the initial state of the message-independent field,
source auxiliaries, and stored early systems does not depend
on \(x\);
\item the evolution of all decoder-accessible field and
apparatus degrees of freedom acts as the identity on the
internal message code, or factors through a subsystem whose
complete accessible output is independent of that code;
\item no prompt reflected carrier, preparation record, or
other message-dependent branch survives outside this
factorization;
\item the discarded message and inaccessible auxiliaries
are traced out without postselection.
\end{enumerate}
\end{assumption}

This assumption defines an exact, calculable null for a
noninteracting internal test-code baseline. It is not a claim
that every semiclassical infalling-field preparation obeys it.

\begin{theorem}[Conditional semiclassical null and leakage allowance]
\label{thm:null}
Under Assumption~\ref{ass:null}, the complete accessible states
are identical:
\begin{equation}
\rho_x^{Z_u}=\rho_*^{Z_u}
\quad\text{for every }x,
\qquad
\boxed{\Qopt(u)=\frac12.}
\label{eq:exactnull}
\end{equation}
More generally, if a common state \(\rho_*\) satisfies
\begin{equation}
\frac14\sum_x\Dtr(\rho_x^{Z_u},\rho_*)\le\epsilon_{\mathrm{sc}},
\label{eq:leakageassumption}
\end{equation}
then
\begin{equation}
\Qopt(u)\le
\min\left\{\qQ,\frac12+\epsilon_{\mathrm{sc}}\right\}.
\label{eq:robustnull}
\end{equation}
\end{theorem}

\begin{proof}
In the identity-action case, tracing the message multiplies
the accessible state by \(\Tr\rho_x^A=1\).
The stated factorization has the same consequence.
Completeness excludes normalization by a message-dependent
selected outcome. Thus the output is independent of \(x\).
Both differences in \eqref{eq:helstromscore} vanish.
The approximate statement follows from
Proposition~\ref{prop:scorecontinuity}, compared with the
constant ensemble.
\end{proof}

\begin{proposition}[Flags, conditioning, and the reflected-carrier example]
\label{prop:flags}
Suppose the complete output has orthogonal flags \(f\), with
message-independent probabilities \(p_f\), and normalized
conditional ensembles \(\rho_{x,f}\). Then
\begin{equation}
\Qopt=\sum_f p_f\,\Qopt(\{\rho_{x,f}\}).
\label{eq:flagaverage}
\end{equation}
In particular, if a message is captured with probability
\(\eta_{\mathrm{cap}}\), its captured accessible branch is
constant, and an intact reflected qubit is available on the
other branch, then
\begin{equation}
\Qopt
=\frac12+
\frac{1-\eta_{\mathrm{cap}}}{2\sqrt2}.
\label{eq:reflectionfalsepositive}
\end{equation}
\end{proposition}

\begin{proof}
The requested-bit differences are block diagonal in \(f\).
Their trace norms are sums of the block trace norms, proving
\eqref{eq:flagaverage}. Apply the constant-branch score
\(1/2\) and the intact-qubit score \(\qQ\) to obtain
\eqref{eq:reflectionfalsepositive}.
\end{proof}

Equation~\eqref{eq:reflectionfalsepositive} is an explicit
ordinary-access false positive for an unqualified
``information recovery'' claim. If flag probabilities depend
on \(x\), the flags themselves carry information and
\eqref{eq:flagaverage} with fixed \(p_f\) is not applicable;
the complete ensemble must be used directly.

\begin{proposition}[Scheduled-cut null]
\label{prop:cut}
In a complete finite CPTP network, an output whose scheduled
ancestors exclude the message input is independent of that
input, including when a fixed initial environment is correlated
with an untouched reference.
\end{proposition}

\begin{proof}
Pull an output observable backward in topological order.
The adjoint of a trace-preserving map is unital, so an identity
on all outputs of a gate pulls back to an identity on its inputs.
The observable's support can expand only to declared predecessor
ports of gates on its scheduled ancestry. If the message is not
among those ports, every pulled-back observable is identity on
the message input. Its expectation is therefore independent of
the message state.
\end{proof}

This is the finite-interface support statement used in
Psy04-B and VII-J \cite{CheungPsy04B,CheungVIIJ}.
Scheduled ancestry can overestimate actual influence.
It is not a physical light-cone theorem, and the proposition
does not extend a linear channel calculation to unspecified
nonlinear, message-dependent metric evolution.

\section{Finite outgoing packets and ordinary detection}
\label{sec:packets}

\subsection{Normal finite-packet restrictions}

Choose \(k\) orthonormal outgoing one-particle functions
\(h_1,\ldots,h_k\), with all angular and species labels included.
Assume their occupation integrals are finite. Let \(b_a=b_{h_a}\)
and define
\begin{equation}
K_{ab}:=\langle b_b^\dagger b_a\rangle
=
\int_0^\infty
\overline{h_a(\omega)}h_b(\omega)
n_{\mathrm{out}}(\omega)\dd\omega,
\label{eq:packetcovariance}
\end{equation}
with the appropriate angular-sector Kronecker factors.

\begin{proposition}[Finite Gaussian packet state]
\label{prop:gaussian}
The fixed-background state restricted to these modes is a
normal, zero-mean, gauge-invariant Gaussian state determined
by the full positive matrix \(K\). Its displacement
characteristic function is
\begin{equation}
\left\langle
\exp\left[
\sum_a(z_ab_a^\dagger-\overline z_ab_a)
\right]
\right\rangle
=
\exp\left[-\frac12\bm z^\dagger(I+2K)\bm z\right].
\label{eq:weylcharacteristic}
\end{equation}
After a passive mode transformation diagonalizing \(K\), the
state is a product of thermal oscillator states with mean
occupations equal to the eigenvalues of \(K\).
\end{proposition}

\begin{proof}
The boundary state is quasifree, and linear scattering and
restriction to a finite CCR subspace preserve quasifreeness.
Stationarity and positive-frequency gauge invariance give
vanishing anomalous covariance. Wick's rule, or expansion of
the Weyl exponential, gives \eqref{eq:weylcharacteristic}.
Diagonalize \(K\) by a unitary transformation of the
annihilation operators. Each diagonal mode has characteristic
function \(\exp[-(1/2+\mu)\abs z^2]\), which is that of
\[
\sum_{n=0}^\infty
\frac{\mu^n}{(1+\mu)^{n+1}}\kb nn.
\]
These factors are trace class, and their finite tensor product
is normal. Their characteristic function equals
\eqref{eq:weylcharacteristic}, proving the assertion.
\end{proof}

Keeping only diagonal packet means generally discards information:
orthogonal frequency-window packets need not have diagonal
\(K\) when the occupation density varies within a window.
The finite covariance and occupation-cutoff interfaces in
En02-E provide the corresponding source-record precedent
\cite{CheungEn02E}; no recovery theorem is inferred from
Gaussianity.

\subsection{A flagged occupation regulator}

Let \(N_{\mathrm{tot}}=\sum_{a=1}^k b_a^\dagger b_a\), and let
\(P_L\) project onto total occupation at most \(L\). The retained
Fock subspace has dimension \(\binom{L+k}{k}\).
Introduce an orthogonal overflow state \(\ket{\perp}\).

\begin{definition}[Complete cutoff channel]
\label{def:cutoff}
On a packet state, define
\begin{equation}
\RR_L(\rho)
=
P_L\rho P_L+
\Tr((I-P_L)\rho)\kb{\perp}{\perp}.
\label{eq:cutoffchannel}
\end{equation}
With an external spectator or reference, the overflow branch
retains its corresponding unnormalized reference state.
The channel is trace preserving; it does not condition on
low occupation.
\end{definition}

\begin{proposition}[Cutoff error and a moment certificate]
\label{prop:cutoff}
Embed the original state in the Fock space enlarged by the
overflow flag, and put
\(\delta_L=\Tr((I-P_L)\rho)\). Then
\begin{equation}
\Dtr(\rho,\RR_L(\rho))
\le
\min\left\{1,\sqrt{\delta_L}+\frac{\delta_L}{2}\right\}.
\label{eq:cutofferror}
\end{equation}
This statement holds with an untouched reference.
If \([\rho,P_L]=0\), the distance equals \(\delta_L\).
Moreover,
\begin{equation}
\delta_L\le
\min\left\{1,
\frac{\langle N_{\mathrm{tot}}\rangle_\rho}{L+1}
\right\}.
\label{eq:cutoffmoment}
\end{equation}
\end{proposition}

\begin{proof}
For a purification \(\ket\psi\), write its projected and
discarded components with squared norms \(1-\delta_L\)
and \(\delta_L\). On their two-dimensional span, direct
diagonalization gives
\[
\norm{
\kb\psi\psi-
P_L\kb\psi\psi P_L
}_1
=\sqrt{4\delta_L-3\delta_L^2}
\le2\sqrt{\delta_L}.
\]
Partial trace is contractive, so the same upper bound holds
for mixed states and with a reference. The added overflow
block has trace norm \(\delta_L\). The triangle inequality
proves \eqref{eq:cutofferror}.

If the state commutes with \(P_L\), there are no off-diagonal
low/high blocks. The removed high block and the added
overflow block have orthogonal supports and each has trace
\(\delta_L\), so the trace distance is \(\delta_L\).
Finally,
\(N_{\mathrm{tot}}\ge(L+1)(I-P_L)\), proving
\eqref{eq:cutoffmoment}.
\end{proof}

For the Gaussian state of Proposition~\ref{prop:gaussian},
the total-number cutoff commutes with the state. For
message-dependent joint states it need not commute, and the
square-root allowance must not be replaced by the Gaussian
tail identity without proof.

An occupation-tail certificate does not certify discarded
frequency or angular sectors. Those require separate
output-state or channel bounds.

\subsection{Loss and complete number readout}

A single-mode pure-loss channel of transmissivity
\(\eta\in[0,1]\) has Kraus operators
\begin{equation}
L_r
=
\sum_{n=r}^{\infty}
\sqrt{\binom nr}\,
\eta^{(n-r)/2}(1-\eta)^{r/2}
\kb{n-r}{n},
\qquad r=0,1,\ldots .
\label{eq:losskraus}
\end{equation}
On a number input \(n\), the detected-number law is
\begin{equation}
\Pr(k\mid n)=\binom nk\eta^k(1-\eta)^{n-k}.
\label{eq:binomialloss}
\end{equation}
For a thermal independent environment of occupation
\(n_{\mathrm{env}}\), the beam-splitter relation gives
\begin{equation}
n_{\mathrm{det}}
=
\eta n_{\mathrm{in}}+(1-\eta)n_{\mathrm{env}}.
\label{eq:backgroundoccupation}
\end{equation}

The complete destructive number instrument on \(k\) modes,
with spectator \(S\), is
\begin{equation}
\mathcal M_{\mathrm{num}}(\rho^{RS})
=
\sum_{\bm n}
\kb{\bm n}{\bm n}_{F}
\otimes
\bra{\bm n}\rho^{RS}\ket{\bm n}.
\label{eq:numberinstrument}
\end{equation}
The positive block sum converges in trace norm. Clipping or
saturation must retain an explicit overflow outcome.

\begin{proposition}[Number records do not determine residual coherence]
\label{prop:records}
Equation~\eqref{eq:numberinstrument} is a complete
trace-preserving instrument. Different packet states can
produce the same number record while having different
delayed-query or quantum-recovery performance.
\end{proposition}

\begin{proof}
The traces of its positive diagonal blocks sum to one, which
proves convergence and completeness.
For the second assertion, the states
\[
\frac{\ket{10}+\ket{01}}{\sqrt2},
\qquad
\frac{\ket{10}-\ket{01}}{\sqrt2}
\]
have the same joint number distribution and are orthogonal.
A coherent measurement distinguishes them perfectly; the
destructive number record does not distinguish them.
\end{proof}

These are the complete-record distinctions made concretely
in Em03-B/C and the extraction interfaces in Em05-A/B
\cite{CheungEm03B,CheungEm03C,CheungEm05A,CheungEm05B}.
No independence of two packet records is assumed.
A shared passive collection port must obey a contraction
budget: if
\(\bm b_{\mathrm{det}}=T\bm b_{\mathrm{in}}+R\bm e\),
then
\[
TT^\dagger+RR^\dagger=I.
\]
It cannot be represented as two independent perfect copies
of the same incident mode.

\section{Conserving emission interfaces and a recovery certificate}
\label{sec:conserving}

\subsection{Energy-labelled emission}

For an energy-eigenmode regulator, let
\(\HH_B(M,\bm q)\) be an internal source sector with energy
\(M\) and declared conserved labels \(\bm q\).
A one-step emission interface has the form
\begin{equation}
V:
\HH_B(M,\bm q)
\longrightarrow
\bigoplus_\alpha
\HH_B(M-E_\alpha,\bm q-\bm q_\alpha)
\otimes\ket{\alpha}_R,
\label{eq:generalemission}
\end{equation}
where the radiation occupation configurations \(\alpha\)
have energies \(E_\alpha\ge0\). Vacuum and no-emission
configurations are included.

Writing
\[
V=\sum_\alpha K_\alpha\otimes\ket{\alpha},
\]
physicality requires
\begin{equation}
\sum_\alpha K_\alpha^\dagger K_\alpha=I.
\label{eq:emissioncompleteness}
\end{equation}
A specified occupation calibration on a code subspace is an
operator constraint of the form
\begin{equation}
\sum_\alpha n_\lambda(\alpha)
K_\alpha^\dagger K_\alpha
=
\overline n_\lambda I_{\mathrm{code}},
\label{eq:operatorcalibration}
\end{equation}
or a weaker expectation constraint on explicitly selected
source states. Which of these is imposed must be declared.

Neither \eqref{eq:operatorcalibration} nor a diagonal
probability distribution fixes the off-diagonal terms
\(K_\alpha\rho K_\beta^\dagger\).
Sector dimensions must also permit the isometry:
a conservation label is not a proof that sufficiently many
internal states exist.

A finite emission interface is used here as an effective
completion on a prepared subspace. It is not identified with
a microscopic black-hole Hilbert space or assigned
dimension \(e^{S_{\mathrm{BH}}}\) by assumption.
Successive steps must act on the actual residual output;
resetting that residual between steps would define a
different dynamics.

\subsection{Constant complement and exact recovery}

\begin{theorem}[Exact recovery criterion]
\label{thm:complement}
Let \(V:\HH_A\to\HH_Z\otimes\HH_F\) be a finite-dimensional
isometry with accessible channel \(\NN\) and complementary
channel \(\CC\). The following are equivalent:
\begin{enumerate}
\item a CPTP decoder satisfies \(\DD\NN=\id_A\);
\item there is a state \(\tau_F\) such that
\begin{equation}
\CC(X)=\Tr(X)\tau_F
\quad\text{for all }X\in\BB(\HH_A);
\label{eq:constantcomplement}
\end{equation}
\item for a Kraus family of \(\NN\),
\begin{equation}
A_a^\dagger A_b=c_{ab}I_A
\label{eq:KL}
\end{equation}
with scalar coefficients \(c_{ab}\).
\end{enumerate}
\end{theorem}

\begin{proof}
Suppose a decoder exists and let \(T\) be its Stinespring
isometry. The composite \((T\otimes I_F)V\) dilates the
identity channel. Its Kraus operators are all proportional
to the identity: the Choi matrix of the identity is rank
one, and a sum of positive rank-one Choi contributions can
have that range only when every contribution lies in it.
Thus
\[
(T\otimes I_F)V\ket\psi
=\ket\psi_A\otimes\ket\chi
\]
for a fixed ancillary vector \(\chi\).
Tracing all but \(F\) proves
\eqref{eq:constantcomplement}.

Conversely, diagonalize
\(\tau_F=\sum_j\lambda_j\kb jj\), omitting zero eigenvalues,
and write
\[
V\ket i
=\sum_j\sqrt{\lambda_j}\ket{\phi_{ij}}_Z\ket j_F.
\]
Applying \eqref{eq:constantcomplement} to every matrix unit
gives
\[
\langle\phi_{i'j'}\mid\phi_{ij}\rangle
=\delta_{ii'}\delta_{jj'}.
\]
On their span \(S\), define the isometry
\(U\ket{\phi_{ij}}=\ket i_A\ket j_J\).
For a fixed state \(\sigma_A\),
\[
\DD(Y)
=
\Tr_J(UP_SY P_SU^\dagger)
+\Tr((I-P_S)Y)\sigma_A
\]
is CPTP and returns \(X\) when applied to \(\NN(X)\).
Finally,
\[
\CC(X)_{ab}=\Tr(A_b^\dagger A_aX),
\]
so constancy on every matrix unit is equivalent to
\eqref{eq:KL}.
\end{proof}

This is the exact code-recovery condition in the
Knill--Laflamme framework and the constant-complement
interface reported in Em05-B
\cite{KnillLaflamme1997,CheungEm05B}.
Constancy of the complement on only the four preparations
in \eqref{eq:squareensemble} is insufficient: those states
do not span the \(\sigmay\) direction.

\section{An explicit Hawking-mode completion and its nonidentifiability}
\label{sec:completion}

\subsection{Physical regulator and thermal calibration}

The following construction supplies a concrete, conserving
completion rather than an unspecified perturbation
\(\rho_{\mathrm{thermal}}+\delta\rho_x\).

Consider two identical neutral scalar species in the same
s-wave energy-eigenmode regulator, both with frequency
\(\omega>0\). They are additional declared field degrees
of freedom in this phenomenological example, not a claim
that such particles have been observed.
There is no imposed conserved flavor charge.
Every radiation configuration below has zero angular
momentum and zero electric charge.

The stationary outgoing mean occupation of either mode is
\begin{equation}
\overline n
=
\frac{\Gamma_0(\omega;M)}
     {e^{8\pi M\omega}-1},
\qquad
q=\frac{\overline n}{1+\overline n}
=
\frac{\Gamma_0(\omega;M)}
     {e^{8\pi M\omega}-1+\Gamma_0(\omega;M)}.
\label{eq:calibratedq}
\end{equation}
For \(0<q<1\), the two-mode thermal state is
\begin{equation}
\rhoH(q)
=
(1-q)^2\sum_{n=0}^{\infty}q^n
\sum_{j=0}^{n}\kb{n,j}{n,j},
\qquad
\ket{n,j}:=\ket j_1\ket{n-j}_2.
\label{eq:thermalpair}
\end{equation}
Its total radiation energy in this regulator is \(n\omega\)
on the \(n\)-particle sector.

A normalized time-localized packet is not an exact energy
eigenmode. Exact conservation in this section refers to the
energy-eigenmode regulator. Transferring the construction
to finite-time packets requires an independently bounded
spectral-width and dynamics error; no sharp-frequency,
sharp-time identification is made.

Fix a finite integer \(L\ge0\) with
\begin{equation}
L\omega\le M-M_{\mathrm{cut}},
\label{eq:energycut}
\end{equation}
and define
\begin{align}
Z_L(q)
&=(1-q)^2\sum_{n=0}^{L}(n+1)q^n
=1-(L+2)q^{L+1}+(L+1)q^{L+2},
\label{eq:thermalZ}\\
d_n
&=\frac{(1-q)^2q^n}{Z_L(q)},
\label{eq:dn}\\
\sigma_L(q)
&=\sum_{n=0}^{L}d_n\sum_{j=0}^{n}
\kb{n,j}{n,j}.
\label{eq:truncatedthermal}
\end{align}
This is the source state of a finite-energy completion,
not a data-selection rule. All attempts in that model are
retained.

Because the discarded thermal sectors are orthogonal,
\begin{equation}
\Dtr(\rhoH(q),\sigma_L(q))
=
1-Z_L(q)
=
(L+2)q^{L+1}-(L+1)q^{L+2}.
\label{eq:thermaltruncation}
\end{equation}
Unbounded observables require their own moment audit; trace
distance alone does not control energy or number expectation.
Here that audit is explicit:
\begin{equation}
\langle N_{\mathrm{tot}}\rangle_{\sigma_L}
=
\frac{(1-q)^2}{Z_L(q)}
\sum_{n=0}^{L}n(n+1)q^n,
\qquad
\langle N_{\mathrm{tot}}\rangle_{\rhoH}
=\frac{2q}{1-q}.
\label{eq:thermalmoments}
\end{equation}

\subsection{Paired occupation sectors and the early resource}

For each \(n\), pair successive occupation vectors:
\begin{equation}
J_{n,k}\ket0=\ket{n,2k},
\qquad
J_{n,k}\ket1=\ket{n,2k+1},
\qquad
0\le k<\left\lfloor\frac{n+1}{2}\right\rfloor.
\label{eq:pairisometries}
\end{equation}
Both codewords have the same exact energy \(n\omega\).
For even \(n\), the single unpaired vector is \(\ket{n,n}\).
Set
\begin{equation}
p_{n,k}=2d_n,
\qquad
w_n=d_n\quad(n\ \text{even}),
\label{eq:pairweights}
\end{equation}
and define the transferable weight
\begin{align}
p_L(q)
&=\sum_{n=0}^{L}
2\left\lfloor\frac{n+1}{2}\right\rfloor d_n
\label{eq:pLsum}\\
&=
1-
\frac{(1-q)^2}{Z_L(q)}
\frac{1-q^{2(\lfloor L/2\rfloor+1)}}{1-q^2}.
\label{eq:pLclosed}
\end{align}
Thus
\begin{equation}
\sum_{n,k}p_{n,k}
+\sum_{\substack{0\le n\le L\\n\ {\rm even}}}w_n
=1.
\label{eq:weightscomplete}
\end{equation}

Let \(r=(r_1,r_2)\in\{0,1\}^2\), and write
\[
P_r=\sigmax^{r_1}\sigmaz^{r_2}.
\]
A stored early system \(E\) and an internal key \(K\) are initially
in the fixed state
\begin{equation}
\ket{\Omega}_{EK}
=
\frac12\sum_r\ket r_E\ket r_K.
\label{eq:earlykey}
\end{equation}
The key sectors are degenerate. A common early-system energy
may be added to every term and does not affect the construction.
The resource \eqref{eq:earlykey} is independent of \(X\) and
is explicitly priced as four orthogonal early states.
Its preparation is an additional assumption; it is not
derived from the Hawking spectrum. A shared classical key
would suffice for the reduced channels below, so this
construction is not offered as a theorem that specifically
requires Hayden--Preskill entanglement.

The residual source contains orthogonal states
\(\ket{b_{n,k}}\) of energy \(M-n\omega\), and orthogonal
two-dimensional residual subspaces with isometries \(I_n\)
for the unpaired even sectors, also at energy \(M-n\omega\).
Their existence is part of the declared source-sector
dimension ledger.

\begin{definition}[Transfer completion]
\label{def:transferV}
On the prepared input subspace define
\begin{equation}
\begin{split}
V_1\bigl(\ket\psi_A\ket r_K\ket M\bigr)
=
\ket r_K\Bigg[
&\sum_{n,k}\sqrt{p_{n,k}}\,
\ket{b_{n,k}}_B
\otimes J_{n,k}P_r\ket\psi_R\\
&+
\sum_{\substack{0\le n\le L\\n\ {\rm even}}}
\sqrt{w_n}\,
I_n\ket\psi_B\otimes\ket{n,n}_R
\Bigg].
\end{split}
\label{eq:transferV}
\end{equation}
The residual subspaces used by distinct terms are
orthogonal. \(E\) is untouched. The common internal
message energy has been included in the initial source
energy; its logical Hamiltonian is zero.
\end{definition}

\begin{theorem}[Conserving completions with identical late radiation]
\label{thm:completion}
The map \(V_1\) is an isometry on the prepared subspace.
Every term has total energy \(M\), and the residual mass
obeys the cutoff \eqref{eq:energycut}.
With input \eqref{eq:earlykey} and with \(BK\) inaccessible,
its accessible channel is
\begin{equation}
\begin{split}
\NN_1(X)
=
\frac14\sum_r\kb rr_E\otimes
\Bigg[
&\sum_{n,k}p_{n,k}
J_{n,k}P_rXP_r^\dagger J_{n,k}^\dagger\\
&+\Tr(X)
\sum_{\substack{0\le n\le L\\n\ {\rm even}}}
w_n\kb{n,n}{n,n}
\Bigg].
\end{split}
\label{eq:transferchannel}
\end{equation}
For every operator \(X\),
\begin{equation}
\Tr_E\NN_1(X)=\Tr(X)\sigma_L(q),
\qquad
\Tr_R\NN_1(X)=\Tr(X)\frac{I_E}{4}.
\label{eq:completionmarginals}
\end{equation}

There is a second energy-conserving isometry \(V_0\), on
a sufficiently large but otherwise identical declared
residual sector space, whose accessible channel is
\begin{equation}
\NN_0(X)
=
\Tr(X)\frac{I_E}{4}\otimes\sigma_L(q).
\label{eq:nullcompletion}
\end{equation}
The two completions have the same entire late-radiation
state, the same early marginal, and the same emitted-energy
distribution for every input. Their assisted delayed-query
scores differ whenever \(p_L(q)>0\).
\end{theorem}

\begin{proof}
Each paired map \(J_{n,k}P_r\) and each residual map \(I_n\)
is an isometry on the message. Orthogonality of the residual
labels and \eqref{eq:weightscomplete} give
\(V_1^\dagger V_1=I\). Distinct key inputs remain orthogonal.
Every term carries radiation energy \(n\omega\) and
residual energy \(M-n\omega\), so energy is conserved.
The radiation is neutral and s-wave, and the stipulated
residual labels conserve the declared charge and angular
momentum.

Apply \(V_1\) to the fixed \(EK\) resource and trace the
orthogonal residual sectors and \(K\). This gives
\eqref{eq:transferchannel}. The Pauli twirl satisfies
\[
\frac14\sum_rP_rXP_r^\dagger
=\frac{\Tr X}{2}I_A.
\]
Since \(p_{n,k}/2=d_n\), the twirled paired sectors and
unpaired terms assemble exactly into \(\sigma_L(q)\).
Tracing \(R\) gives the uniform early marginal.

For the second completion, choose mutually orthogonal
residual two-dimensional embeddings \(I_{n,j}^{(0)}\)
of energy \(M-n\omega\), and set
\[
V_0\bigl(\ket\psi_A\ket r_K\ket M\bigr)
=
\ket r_K
\sum_{n=0}^{L}\sum_{j=0}^{n}
\sqrt{d_n}\,
I_{n,j}^{(0)}\ket\psi_B\otimes\ket{n,j}_R.
\]
Normalization \(\sum_n(n+1)d_n=1\) proves that this is an
isometry. Each term conserves energy and the declared
charges. Tracing its residual and \(K\) gives
\eqref{eq:nullcompletion}.

Both isometries may be embedded in a common enlarged
residual Hilbert space. Within each finite total-energy
sector, an isometry on a prepared subspace extends to a
unitary after completing orthonormal bases of that sector.
This extension establishes finite-interface unitary
physicality, not a microscopic black-hole Hamiltonian.

Finally, the score difference follows from
Theorem~\ref{thm:completionoptima} below and the
message-independent channel \eqref{eq:nullcompletion}.
\end{proof}

\begin{remark}[Strength and limitation of the counterexample]
The ambiguity is stronger than equality of single-particle
spectra: the complete late density operator is identical
in the two completions. The difference lies in the joint
\(ER\) correlations. No measurement on the late radiation
alone can distinguish the messages.

The isometries nevertheless differ from the
identity-action semiclassical null. They expressly introduce
an internal-to-outgoing transfer interaction. Matching an
outgoing thermal marginal does not make that interaction
a consequence of local semiclassical evolution, nor does
it preserve a near-horizon pair state. Thus the example
does not evade small-correction theorems by merely changing
a thermal diagonal \cite{Mathur2009}.
\end{remark}

\subsection{Exact coherent, number-record, and referenced-recovery optima}

Put \(p=p_L(q)\).

\begin{theorem}[Exact access-dependent optima]
\label{thm:completionoptima}
For the transfer completion:
\begin{align}
\Qopt(R_{\mathrm{late}})
&=\frac12,
\label{eq:lateoptimum}\\
\Qopt(ER_{\mathrm{late}})
&=\frac12+\frac{p}{2\sqrt2},
\label{eq:jointoptimum}\\
\Qopt(EF_{\mathrm{num}})
&=\frac12+\frac{p}{4\sqrt2},
\label{eq:numberoptimum}
\end{align}
where \(F_{\mathrm{num}}\) is the fixed destructive number
record in the occupation basis of
\eqref{eq:thermalpair}, obtained before \(J\).
Without \(E\), that number record has score \(1/2\).

Over all deterministic CPTP decoders from \(ER\) to a
message qubit,
\begin{align}
\sup_{\DD}F_e(\DD\NN_1)
&=\frac{1+3p}{4},
\label{eq:optimalFe}\\
\inf_{\DD}\Ddia(\DD\NN_1,\id_A)
&=\frac34(1-p),
\label{eq:optimaldiamond}
\end{align}
where \(F_e\) is the overlap with a maximally entangled
reference--message state.
For \(\NN_0\), even with \(E\), the optimal delayed-query
score is \(1/2\), the optimal entanglement fidelity is
\(1/4\), and the optimal half-diamond recovery error is
\(3/4\).
\end{theorem}

\begin{proof}
The late marginal is constant by
\eqref{eq:completionmarginals}, proving
\eqref{eq:lateoptimum}. With \(E\), the key and pair-sector
projectors are mutually orthogonal and can be read without
measuring the logical qubit. On each paired sector the
decoder applies \(J_{n,k}^\dagger\) and \(P_r^\dagger\);
the recovered logical state is \(\rho_x^A\).
Unpaired sectors are constant. Equation~\eqref{eq:flagaverage}
therefore gives \eqref{eq:jointoptimum}, with attainability
by query-dependent \(X\) or \(Z\) measurement on the
recovered state.

For a paired sector, destructive number measurement is
logical \(Z\) measurement, with its outcome flip determined
by \(r_1\). After the key correction, its distribution has
no dependence on \(x_1\), so the first requested bit is
guessed with probability \(1/2\). For \(x_2\), the optimal
success probability is
\[
\frac12+\frac{1}{2\sqrt2}.
\]
Averaging over \(J\) gives
\(1/2+1/(4\sqrt2)\) on every paired branch.
Unpaired branches score \(1/2\), proving
\eqref{eq:numberoptimum}. Twirling proves the unassisted
number statement.

For referenced recovery, paired sectors permit perfect
recovery and contribute Bell overlap at most one.
Unpaired sectors are message-independent. After any local
decoder their reference--output state has the form
\(I_{\mathrm{ref}}/2\otimes\sigma_{\mathrm{out}}\), whose
overlap with a normalized Bell state is \(1/4\).
Hence every decoder satisfies
\[
F_e\le p+\frac{1-p}{4}.
\]
The coherent branchwise decoder, preparing \(I_A/2\) on
an unpaired sector, attains this bound. Its complete
decoded channel is
\begin{equation}
\Lambda_p=p\,\id_A+(1-p)\mathcal T,
\qquad
\mathcal T(X)=\Tr(X)\frac{I_A}{2}.
\label{eq:decodeddepolarizing}
\end{equation}

For any decoder, testing the Bell projector gives
\(\Ddia(\DD\NN_1,\id_A)\ge1-F_e\), establishing the lower
bound in \eqref{eq:optimaldiamond}.
The Pauli identity
\[
\mathcal T=\frac14
\left(\id+\Ad_{\sigmax}+\Ad_{\sigmay}+\Ad_{\sigmaz}\right)
\]
shows
\(\Ddia(\mathcal T,\id)\le3/4\) by the triangle inequality,
since each unitary-channel distance is at most one in
half-diamond normalization. A Bell input gives four
orthogonal Bell outputs and proves the matching lower
bound \(3/4\). Homogeneity now gives
\[
\Ddia(\Lambda_p,\id)=\frac34(1-p),
\]
which is attained by the stated decoder.

The constant channel \(\NN_0\) has no message-dependent
output. The same Bell argument gives fidelity \(1/4\)
and error at least \(3/4\); preparing \(I_A/2\) attains it.
\end{proof}

\begin{corollary}[Thermal-limit weight and conditional threshold]
\label{cor:threshold}
As \(L\to\infty\) with fixed \(0<q<1\),
\begin{equation}
p_L(q)\longrightarrow
p_\infty(q)=\frac{2q}{1+q}
=\frac{2\overline n}{1+2\overline n}.
\label{eq:thermalweightlimit}
\end{equation}
For a finite, energy-admissible \(L\), coherent access exceeds
the trusted measure-before-query bound if and only if
\begin{equation}
p_L(q)>\frac{1}{\sqrt2}.
\label{eq:finiteRACthreshold}
\end{equation}
In the mathematical untruncated limit this condition becomes
\begin{equation}
q>\frac{1}{2\sqrt2-1},
\qquad\text{equivalently}\qquad
\overline n>\frac{\sqrt2+1}{2}.
\label{eq:thermalRACthreshold}
\end{equation}
\end{corollary}

\begin{proof}
In \eqref{eq:pLclosed}, let \(Z_L\to1\) and the remaining
geometric tail tend to zero. Then
\[
1-p_\infty
=\frac{(1-q)^2}{1-q^2}
=\frac{1-q}{1+q}.
\]
The threshold follows by solving
\(1/2+p/(2\sqrt2)>3/4\).
Substitute \(q=\overline n/(1+\overline n)\) for the final
form.
\end{proof}

The limit in \eqref{eq:thermalweightlimit} is an analytic
comparison, not an admissible \(L=\infty\) emission from
finite mass \(M\). In particular, the actual
greybody-calibrated occupation in
\eqref{eq:calibratedq}, not temperature alone, determines
whether even the ideal conditional completion crosses the
trusted classical bound. A nonzero advantage over \(1/2\)
and a violation of \(3/4\) are different claims.

The recovery error in
\eqref{eq:optimaldiamond} is also a different quantity from
the delayed-query score. This distinction parallels the
separate recovery and executable-access optima in En02-C/D
and the key/transfer separation in Em03-E
\cite{CheungEn02C,CheungEn02D,CheungEm03E}.
Those finite models are not imported as Hawking dynamics.

\section{Observer access, collection, and robust certification}
\label{sec:observer}

\subsection{Three access classes}

For a supplied outgoing channel \(\NN_u^{\mathrm{out}}\),
the observer sees
\begin{equation}
\NN_u^{\mathrm{obs}}
=
\CC_u\circ\NN_u^{\mathrm{out}},
\label{eq:observercomposition}
\end{equation}
where \(\CC_u\) includes collection, propagation, storage,
backgrounds, and any pre-query measurement.

We distinguish:
\begin{enumerate}
\item \emph{ideal asymptotic coherent access}: all declared
outgoing modes and early resource are available;
\item \emph{finite-aperture coherent access}: a declared
collection channel retains the captured quantum state;
\item \emph{ordinary detection access}: a declared complete
instrument leaves only classical detector records before
the query.
\end{enumerate}
The third is not identified with the second.

For isotropic emission, a small far-field aperture has the
geometric estimate
\begin{equation}
f_{\mathrm{geom}}\simeq\frac{A_{\mathrm{det}}}{4\pi d^2}.
\label{eq:geometricfraction}
\end{equation}
This estimate requires the corresponding angular and
wavelength regime. Otherwise mode-overlap and diffraction
must be calculated. The expected source-derived detected
count is
\begin{equation}
N_{\mathrm{det}}
=
\int_{u_0}^{u_1}\dd u
\int_0^\infty\dd\omega
\sum_\lambda
\eta_\lambda(\omega,u)
\frac{\dd^2N_\lambda}{\dd u\,\dd\omega}.
\label{eq:detectedcounts}
\end{equation}
Background counts are a separate apparatus contribution.

Equation~\eqref{eq:detectedcounts} is a mean-count forecast,
not a coherent decoding channel. A scalar efficiency
cannot replace the multimode channel
\eqref{eq:observercomposition}.

\begin{proposition}[A declared all-or-nothing collection model]
\label{prop:erasureaccess}
For the completion of Section~\ref{sec:completion}, suppose:
\begin{enumerate}
\item capture succeeds with message-independent probability
\(\eta_{\mathrm{cap}}\);
\item on successful capture, the coherent logical branch is
collected without distortion with probability \(\chi\);
\item every other complete branch is message-independent,
including its surviving early state;
\item the corresponding flags are accessible.
\end{enumerate}
Then
\begin{equation}
\Qopt^{\mathrm{obs}}
=
\frac12+
\frac{\eta_{\mathrm{cap}}\chi p_L(q)}{2\sqrt2}.
\label{eq:erasurescore}
\end{equation}
\end{proposition}

\begin{proof}
The complete output is a flagged mixture of an intact
trusted-qubit branch with probability
\(\eta_{\mathrm{cap}}\chi p_L(q)\) and constant branches.
Apply Proposition~\ref{prop:flags}.
\end{proof}

This is an explicitly declared whole-code erasure model.
It is not the general bosonic loss channel
\eqref{eq:losskraus}. In particular, setting
\(\chi=f_{\mathrm{geom}}\) without a mode-level argument
would not be justified.

\subsection{Error budgets and sequential stability}

\begin{proposition}[Composition and state-error budgets]
\label{prop:stability}
For composable regulated CPTP channels \(\Phi_i\) and
\(\widetilde\Phi_i\),
\begin{equation}
\Ddia(
\Phi_n\circ\cdots\circ\Phi_1,
\widetilde\Phi_n\circ\cdots\circ\widetilde\Phi_1)
\le
\sum_{i=1}^{n}\Ddia(\Phi_i,\widetilde\Phi_i).
\label{eq:telescoping}
\end{equation}
Consequently, if the complete physical and modeled
ensembles have a certified total average trace-distance
allowance
\begin{equation}
\epsilon_{\mathrm{tot}}
=
\epsilon_{\mathrm{prep}}
+\epsilon_{\mathrm{source}}
+\epsilon_{\mathrm{adiabatic}}
+\epsilon_{\mathrm{modes}}
+\epsilon_{\mathrm{occupation}}
+\epsilon_{\mathrm{collection}}
+\epsilon_{\mathrm{storage}}
+\epsilon_{\mathrm{decoder}},
\label{eq:errorbudget}
\end{equation}
where each term is defined on the relevant common
state/channel interface, then the score changes by at
most \(\epsilon_{\mathrm{tot}}\).
\end{proposition}

\begin{proof}
Expand the difference of the channel compositions as a
telescoping sum with one factor difference in each term.
CPTP pre- and post-composition do not increase the diamond
norm. The triangle inequality proves
\eqref{eq:telescoping}.
For a mixture of state and channel allowances, insert
the intermediate output ensembles and use trace-distance
contractivity and the triangle inequality. Apply
Proposition~\ref{prop:scorecontinuity}.
\end{proof}

A classical norm estimate is not automatically a diamond
estimate. For classical intermediate maps with Lipschitz
constants \(L_i\), an error recursion
\[
e_i\le L_ie_{i-1}+\delta_i
\]
gives, by induction,
\begin{equation}
e_n\le
e_0\prod_{i=1}^{n}L_i+
\sum_{i=1}^{n}\delta_i
\prod_{j=i+1}^{n}L_j.
\label{eq:lipschitzrecursion}
\end{equation}
Converting this bound into
\(\epsilon_{\mathrm{adiabatic}}\) requires a theorem
relating that classical norm to the selected field state
or complete channel. The source-input stability estimates
in Em06-B and the interface allowances in En03-A are
relevant precedents, not substitutes for that theorem
\cite{CheungEm06B,CheungEn03A}.

\subsection{Certified astrophysical nulls}

Let \(\Theta_{\mathrm{sc}}\) be the declared uncertainty
class of semiclassical channels, including legitimate
preparation, background, capture, and access uncertainties.
Define
\begin{equation}
b_{\mathrm{sc}}(u)
=
\sup_{\theta\in\Theta_{\mathrm{sc}}}
\Qopt(\NN^{\mathrm{sc}}_{u,\theta}).
\label{eq:physicalnull}
\end{equation}
This is the astrophysical null. It equals \(1/2\) only
under a proved message-independence statement, and it is
not set to \(3/4\) unless the complete trusted-input
measure-before-query null is separately imposed.

\begin{proposition}[Finite-cover null certificate]
\label{prop:nullcover}
Suppose cells \(\Theta_k\) cover
\(\Theta_{\mathrm{sc}}\). For each cell there is a
regulated representative \(\widetilde\NN_k\) and a bound
\[
\sup_{\theta\in\Theta_k}
\Ddia(\NN^{\mathrm{sc}}_{u,\theta},\widetilde\NN_k)
\le r_k.
\]
If \(\widetilde Q_k\) is a computed score with certified
absolute numerical error at most \(e_k\), then
\begin{equation}
b_{\mathrm{sc}}(u)
\le
\min\left\{
\qQ,\,
\max_k(\widetilde Q_k+e_k+r_k)
\right\}.
\label{eq:nullcover}
\end{equation}
\end{proposition}

\begin{proof}
For \(\theta\in\Theta_k\),
Proposition~\ref{prop:scorecontinuity} gives
\[
\Qopt(\NN^{\mathrm{sc}}_{u,\theta})
\le\widetilde Q_k+e_k+r_k.
\]
Take the supremum over the covering cells and impose the
trusted-input ceiling.
\end{proof}

In finite dimension, \eqref{eq:helstromscore} is evaluated
by the positive and negative spectral parts of two
Hermitian matrices. A numerical plot without interval,
residual, or equivalent spectral-error control is not a
certificate for \eqref{eq:nullcover}.

\subsection{Recovery time and prefix consistency}

For a fixed null and access prescription, define
\begin{equation}
u_{\mathrm{RAC}}
=
\inf\{u:\Qacc(u)>b_{\mathrm{null}}(u)\},
\qquad \inf\varnothing=+\infty.
\label{eq:recoverytime}
\end{equation}
A robust crossing requires a certified lower bound for
the accessible score to exceed a certified upper bound
for the null.

If an earlier accessible radiation prefix can be obtained
by a CPTP restriction of a later prefix, and earlier
quantum storage has not degraded, data processing proves
that the ideal score is nondecreasing. A changing,
decohering observer need not have that monotonicity.

No equality between \eqref{eq:recoverytime} and a Page time
is claimed. Fine-grained entropy calculations, including
replica-wormhole calculations in their specified
gravitational settings, do not by themselves construct
the present operational decoder
\cite{AlmheiriEtAl2020}.

\section{Complete-attempt assessment and observability}
\label{sec:assessment}

\subsection{A finite-sample certificate}

Fix the preparation, collection interval, decoder, access
boundary, and null before using the assessment sample.
For attempt \(i\), sample independent uniform \(X_i,J_i\)
under the declared protocol, and set
\begin{equation}
W_i=\mathbf 1\{B_i=X_{i,J_i}\},
\qquad
\widehat Q_n=\frac1n\sum_{i=1}^{n}W_i.
\label{eq:attemptvariables}
\end{equation}
The sample size is the number of complete attempts, not
the number of photons or within-attempt detector bins.

\begin{theorem}[Independent-attempt rejection certificate]
\label{thm:sampling}
Suppose \(W_1,\ldots,W_n\) are independent and each
null mean is at most \(b\). Then for \(t>0\),
\begin{equation}
\Pr_{\mathrm{null}}(\widehat Q_n\ge b+t)
\le e^{-2nt^2}.
\label{eq:hoeffding}
\end{equation}
Thus at significance level \(\zeta\in(0,1)\), the null is
rejected whenever
\begin{equation}
\widehat Q_n>
b+\sqrt{\frac{\log(1/\zeta)}{2n}}.
\label{eq:rejection}
\end{equation}
If a model predicts a score lower bound \(q_-\) and
\(q_->b\), then under independent attempts with means
at least \(q_-\),
\begin{equation}
\Pr\!\left(
\widehat Q_n\le q_--t
\right)\le e^{-2nt^2}.
\label{eq:powerbound}
\end{equation}
\end{theorem}

\begin{proof}
For a bounded variable \(W\in[0,1]\),
\[
\mathbb E e^{s(W-\mathbb EW)}
\le e^{s^2/8}.
\]
For Bernoulli \(W\), this follows by differentiating the
log moment-generating function twice: its second
derivative is a Bernoulli variance and is at most \(1/4\);
the value and first derivative at zero vanish.
Independence and Chernoff's inequality give
\[
\Pr\left(\sum_i(W_i-\mathbb EW_i)\ge nt\right)
\le\inf_{s>0}e^{-snt+ns^2/8}
=e^{-2nt^2}.
\]
The null means satisfy
\(\sum_i\mathbb EW_i\le nb\), proving
\eqref{eq:hoeffding}. Apply the same argument to \(-W_i\)
for \eqref{eq:powerbound}.
\end{proof}

This is the attempted-run Hoeffding certificate
\cite{Hoeffding1963}; the separation between scalar scores
and complete records is also part of the En03-A
interface \cite{CheungEn03A}.
Using the same sample to select the best decoder or time
interval requires an additional selection correction or
a separate validation sample.

For a complete finite record alphabet of size \(K\),
coordinatewise Hoeffding and a union bound imply, with
probability at least \(1-\zeta\),
\begin{equation}
\TV(\widehat P,P)
\le
\min\left\{
1,\,
\frac{K}{2}
\sqrt{\frac{\log(2K/\zeta)}{2n}}
\right\}.
\label{eq:recordsampling}
\end{equation}
Dependence among records within an attempt does not
increase \(n\).

\begin{remark}[Repeated messages in one evolving source]
Repeated injections into one black hole are not automatically
independent attempts. Shared residual memory, changing mass,
and overlapping radiation can violate the assumptions of
Theorem~\ref{thm:sampling}. A history-conditional protocol
would need a fresh-label condition and a separately proved
conditional null after every admissible history.
No such certificate is inferred merely from counting
additional radiation packets.
\end{remark}

\subsection{Record equality and the passive-astronomy boundary}

\begin{proposition}[Testing obstruction]
\label{prop:testing}
For two complete observable record laws \(P_0,P_1\), every
randomized test with type-I and type-II errors
\(\alpha,\beta\) satisfies
\begin{equation}
\alpha+\beta\ge1-\TV(P_0,P_1).
\label{eq:testingobstruction}
\end{equation}
If the two laws are equal, then \(\alpha+\beta=1\).
\end{proposition}

\begin{proof}
Let \(\varphi\in[0,1]\) be the probability of declaring
model one. Then
\[
\alpha+\beta
=\mathbb E_0\varphi+\mathbb E_1(1-\varphi)
=1+\mathbb E_0\varphi-\mathbb E_1\varphi.
\]
The variational definition of total variation gives
\eqref{eq:testingobstruction}.
\end{proof}

Identical complete interval channels and instruments
produce identical adaptive record laws by induction on
their unnormalized branch states. This is the
complete-process comparison in Psy02-B
\cite{CheungPsy02B}. Equal selected number records alone
do not establish that stronger premise.

Passive astronomical observation may constrain source
spectra and backgrounds. It does not ordinarily provide
the independently known injected labels \(X_i\) required
to score \eqref{eq:attemptvariables}. Replacing a
classical carrier by a quantum one cannot create that
missing observation or intervention channel; the
corresponding observation-boundary distinction is
emphasized in Psy03-D \cite{CheungPsy03D}.
Its numerical uncued-rule bound is not imported into this
different astrophysical task.

The three assessment settings must therefore remain separate:
\begin{enumerate}
\item passive observation of an uncontrolled source;
\item a proposed controlled astrophysical intervention;
\item theoretical repeated preparations or simulations.
\end{enumerate}
Only the second and third can directly instantiate the
known-label task as defined here. The third is not an
observation of an astrophysical black hole.

\section{Mathematical release and physical conclusion}
\label{sec:release}

\begin{theorem}[Hawk01-A release]
\label{thm:release}
For the fixed-background scalar source, trusted message
ensemble, and explicitly declared access interfaces:
\begin{enumerate}
\item the horizon boundary prescription and radial scattering
give the normalized source law
\eqref{eq:sourcerate} and packet expectation
\eqref{eq:packetoccupation};
\item every complete message-to-output channel has exact
delayed-query score \eqref{eq:helstromscore}, with the
trusted-input ceiling \eqref{eq:qubitceiling};
\item the identity-action semiclassical baseline has score
\(1/2\), and deviations are bounded by
\eqref{eq:robustnull};
\item the finite-energy isometries \(V_0,V_1\) conserve the
declared energy and charges and reproduce the same
truncated late Hawking-mode state for every input;
\item their assisted retrieval performances differ, with
the exact access and recovery optima of
Theorem~\ref{thm:completionoptima};
\item occupation regulators, channel compositions, finite
uncertainty covers, and independent attempted runs have
the explicit certificates proved above.
\end{enumerate}
These statements do not determine a microscopic
astrophysical recovery law or an actual recovery time.
\end{theorem}

\begin{proof}
Item (i) is Theorem~\ref{thm:source}.
Item (ii) is Theorem~\ref{thm:score}.
Item (iii) is Theorem~\ref{thm:null}.
Item (iv) is Theorem~\ref{thm:completion}.
Item (v) is Theorem~\ref{thm:completionoptima}.
Item (vi) follows from
Propositions~\ref{prop:cutoff}, \ref{prop:stability},
\ref{prop:nullcover}, and Theorem~\ref{thm:sampling}.

A microscopic recovery law is not among the source
assumptions, and the conserving completions explicitly
supply different such effective interactions. The common
late state therefore cannot select one of them or fix a
time schedule. No numerical value of an astrophysical
recovery time follows.
\end{proof}

The direct astrophysical content is the four-dimensional
geometry, asymptotic time normalization, quantum-field
source, greybody transmission, radiation-mode access,
and energy ledger. The exact score and recovery results
are then evaluated on supplied channels on those modes.

The construction establishes a precise limitation:
even a complete late-radiation thermal marginal need not
determine retrieval from the joint accessible state.
Conversely, a message-dependent outgoing signal need not
originate in a post-horizon mechanism. Prompt carriers,
exterior couplings, early resources, and detector
instruments must be included before either interpretation
is justified.

A \(3/4\) violation, if certified for the complete trusted
measure-before-query null, excludes that null. It does
not by itself establish quantum non-Markovianity,
environmental quantum memory, black-hole interior escape,
or a solution of the information paradox. An identity
storage channel on an ordinary qubit already attains
\(\qQ\), as also emphasized in Psy04-A/B
\cite{CheungPsy04A,CheungPsy04B}.

\appendix

\section{Source interfaces and non-transfer boundaries}
\label{app:sources}

Table~\ref{tab:sourceledger} records the source roles used
in this paper. A manuscript identifier is not a declaration
that its theorem applies to a different Hilbert space,
field, geometry, code, or instrument.

\begin{table}[ht]
\small
\caption{Cheung source interfaces and their restricted use here.}
\label{tab:sourceledger}
\begin{tabularx}{\linewidth}{@{}p{0.22\linewidth}X@{}}
\toprule
Source record & Role and boundary\\
\midrule
Em03-A
& Fixed-background scalar emission and packet normalization.
Not an evaporating metric or recovery channel.\\
Em03-B/C
& Complete destructive records, loss, and joint apparatus
bookkeeping. Not coherent receiver construction.\\
Em03-D/F/G
& Exterior probes and effective exterior-source channels.
Not interior information escape or a gravitational
backreaction law.\\
Em03-E
& Separation of key availability and logical transfer.
A finite benchmark, not Hawking dynamics.\\
Psy04-A/B; VII-J
& Delayed query, complete access cut, and scheduled ancestry.
Not a physical causal-cone or memory-resource theorem.\\
En02-B--E
& Observable access, recovery, executable resource limits,
and finite-packet regulation.
Not a derivation of black-hole evaporation.\\
Em05-A/B
& Extraction versus coherence and exact
constant-complement recovery.
Spectrum agreement is not that criterion.\\
Em06-B; En03-A
& Input stability, record compatibility, and sampling
interfaces. Their norms must be connected to this
paper's states or channels before an error is imported.\\
Psy02-B; Psy03-D
& Complete-process testing and observation-boundary
distinctions. Their separate task-specific numerical
bounds are not astrophysical nulls.\\
\bottomrule
\end{tabularx}
\end{table}

The supplied excerpts of XIV-H/I/J describe a different
analytic programme. In particular, XIV-I constructs an
internal amplitude interface while leaving its full
joint characteristic map unresolved; XIV-J supplies a
supported-sector reference on a restricted domain and
does not admit the required full quantum coefficient
\cite{CheungXIVH,CheungXIVI,CheungXIVJ}.
Nothing in the finite-channel proofs of Hawk01-A constructs
that missing distributional operation.

For clarity, the elementary reconstruction-descent
criterion is:
\begin{equation}
\widetilde{\mathsf M}
\text{ factors through }\Rec
\quad\Longleftrightarrow\quad
\ker\Rec\subseteq\ker\widetilde{\mathsf M}.
\label{eq:descentappendix}
\end{equation}
Indeed, equal reconstructions differ by an element of
\(\ker\Rec\), so this condition is necessary and sufficient
for their images to agree. It defines a map on a quotient
only after the candidate map exists. It does not construct
a characteristic product.

Likewise \(q\,\delta'(q)=-\delta(q)\) shows that polynomial
degree in a presentation need not be intrinsic to the
reconstructed distribution. This identity does not select
an extension across an unresolved characteristic locus.
The relative-lifting interface reported in XV-B remains
a separate conditional linear problem
\cite{CheungXVB}.

Accordingly, Hawk01-A does not:
\begin{itemize}
\item construct the XIV full joint characteristic map,
its joint section, or its output wavefront theorem;
\item admit the coefficient denoted \(R_1^{(1)}\) in the
supplied analytic ledger;
\item open a downstream admission gate by citing a
finite-dimensional decoder;
\item infer a Ward identity from source spectra,
channel physicality, or record agreement.
\end{itemize}
No nonexistence theorem for the unresolved analytic map
is asserted either.

\section{Mathematical and physical corrections incorporated}
\label{app:debug}

The following corrections are part of the manuscript's
definitions and proofs, not outstanding claims.

\begin{enumerate}
\item \textbf{Normalized hypotheses.}
The states \(\tau_{b\mid j}\) in \eqref{eq:taudef}
contain a factor \(1/2\); the optimal-score prefactor
is consequently \(1/8\).

\item \textbf{Correct null hierarchy.}
Message independence gives \(1/2\).
A complete trusted-qubit measure-before-query channel
gives \(3/4\).
A general semiclassical null gives its certified
supremum \eqref{eq:physicalnull}.
These bounds are not interchanged.

\item \textbf{Complete attempts.}
The primary score retains failed capture and apparatus
branches. Input-dependent flags are included in the
state rather than treated as harmless heralds.

\item \textbf{Exact versus quasistatic geometry.}
An instantaneous Schwarzschild sequence is not an exact
evaporating spacetime. The adiabatic indicator does not
supply its own channel-error theorem.

\item \textbf{Energy eigenmodes versus packets.}
Exact energy-shell conservation is proved in the declared
energy-eigenmode regulator. Finite-time packet transfer
requires an additional approximation certificate.

\item \textbf{Finite source energy.}
An unbounded thermal occupation tail is not emitted from
a finite mass in the exact completion. The finite \(L\)
obeys \eqref{eq:energycut}; the thermal discrepancy and
number moments are displayed separately.

\item \textbf{Full covariance.}
Packet orthogonality does not justify dropping the
off-diagonal matrix elements of \(K\).

\item \textbf{Quantum versus classical cutoff errors.}
A commuting Gaussian cutoff has exact trace-distance
tail \(\delta_L\). A general joint quantum state uses
the gentle-projection allowance
\eqref{eq:cutofferror}.

\item \textbf{Spectra versus channels.}
The completions are given by isometries and complete
partial traces, not arbitrary additive density-matrix
corrections. Conservation, positivity, and normalization
are proved.

\item \textbf{Recovery certificate on all operators.}
Constant complement must hold on every matrix unit,
including the \(\sigmay\) direction, not only on the
four square preparations.

\item \textbf{Access and resource pricing.}
The early key and coherent radiation are declared
resources. Destructive number records have their own
strictly smaller optimum in
\eqref{eq:numberoptimum}.

\item \textbf{Statistical unit.}
The sample size counts complete independent attempts.
A simulated source and an actual astronomical source
are not identified.

\item \textbf{Unresolved dependencies.}
Partial source records and quotient criteria are not
substituted for missing microscopic dynamics or
missing characteristic distributional constructions.
\end{enumerate}

\section{Reference provenance and reproducibility}
\label{app:provenance}

Published references below use their original article,
publisher, or author-hosted bibliographic records.
Cheung's works are cited by the manuscript identifiers
and versions supplied with the brief. Their publication
dates, canonical full titles, and persistent article
identifiers were not supplied and are not invented.
They are therefore identified as unpublished manuscripts,
with the relevant theorem or interface labels where
available.

The references to those manuscripts are acknowledgments
and interface comparisons. The proofs of this paper's
operational, cutoff, conserving-completion, and sampling
claims are given in the text. A submission should provide
reviewers with the cited manuscript versions or an
appropriate archival or supplementary route.

No greybody numerical table, detector forecast, source
catalogue, or astrophysical intervention data are claimed.
A numerical realization should archive:
\begin{enumerate}
\item the source field/species list and greybody solver;
\item the asymptotic frequency and time conventions;
\item mass range and semiclassical termination rule;
\item packet profiles and the full covariance matrix;
\item all cutoff tails and unbounded-moment audits;
\item the complete message, capture, and observer channels;
\item the decoding program and quantum/classical resource
ledger;
\item the parameter cover and certified numerical errors;
\item the attempted-run sampling and validation procedure.
\end{enumerate}
Spectrum software such as BlackHawk can cross-check
emission and evolution inputs, but does not supply the
message-dependent recovery dynamics
\cite{ArbeyAuffinger2019}. The exact version and inputs
must be recorded if such a cross-check is performed.

\begin{thebibliography}{99}

\bibitem{Hawking1975}
S.~W.~Hawking,
``Particle creation by black holes,''
\emph{Communications in Mathematical Physics}
\textbf{43}, 199--220 (1975).
doi: \href{https://doi.org/10.1007/BF02345020}
{10.1007/BF02345020}.

\bibitem{Unruh1976}
W.~G.~Unruh,
``Notes on black-hole evaporation,''
\emph{Physical Review D}
\textbf{14}, 870--892 (1976).
doi: \href{https://doi.org/10.1103/PhysRevD.14.870}
{10.1103/PhysRevD.14.870}.

\bibitem{Page1976}
D.~N.~Page,
``Particle emission rates from a black hole:
Massless particles from an uncharged, nonrotating hole,''
\emph{Physical Review D}
\textbf{13}, 198--206 (1976).
doi: \href{https://doi.org/10.1103/PhysRevD.13.198}
{10.1103/PhysRevD.13.198}.

\bibitem{Wald1976}
R.~M.~Wald,
``Stimulated-emission effects in particle creation near
black holes,''
\emph{Physical Review D}
\textbf{13}, 3176--3182 (1976).
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{10.1103/PhysRevD.13.3176}.

\bibitem{HaydenPreskill2007}
P.~Hayden and J.~Preskill,
``Black holes as mirrors: quantum information in random
subsystems,''
\emph{Journal of High Energy Physics}
09 (2007), 120.
doi: \href{https://doi.org/10.1088/1126-6708/2007/09/120}
{10.1088/1126-6708/2007/09/120}.
arXiv: \href{https://arxiv.org/abs/0708.4025}{0708.4025}.

\bibitem{ArbeyAuffinger2019}
A.~Arbey and J.~Auffinger,
``BlackHawk: A public code for calculating the Hawking
evaporation spectra of any black hole distribution,''
\emph{European Physical Journal C}
\textbf{79}, 693 (2019).
doi: \href{https://doi.org/10.1140/epjc/s10052-019-7161-1}
{10.1140/epjc/s10052-019-7161-1}.
The revised v2.0 manual is
arXiv: \href{https://arxiv.org/abs/1905.04268v3}
{1905.04268v3}.

\bibitem{Watrous2018}
J.~Watrous,
\emph{The Theory of Quantum Information}
(Cambridge University Press, 2018),
especially the Holevo--Helstrom theorem and channel
distance results.
Author-hosted manuscript and errata:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{HorodeckiShorRuskai2003}
M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai,
``General entanglement breaking channels,''
\emph{Reviews in Mathematical Physics}
\textbf{15}, 629--641 (2003).
doi: \href{https://doi.org/10.1142/S0129055X03001709}
{10.1142/S0129055X03001709}.
arXiv: \href{https://arxiv.org/abs/quant-ph/0302031}
{quant-ph/0302031}.

\bibitem{KnillLaflamme1997}
E.~Knill and R.~Laflamme,
``Theory of quantum error-correcting codes,''
\emph{Physical Review A}
\textbf{55}, 900--911 (1997).
doi: \href{https://doi.org/10.1103/PhysRevA.55.900}
{10.1103/PhysRevA.55.900}.

\bibitem{Mathur2009}
S.~D.~Mathur,
``The information paradox: a pedagogical introduction,''
\emph{Classical and Quantum Gravity}
\textbf{26}, 224001 (2009).
doi: \href{https://doi.org/10.1088/0264-9381/26/22/224001}
{10.1088/0264-9381/26/22/224001}.
arXiv: \href{https://arxiv.org/abs/0909.1038}{0909.1038}.

\bibitem{AlmheiriEtAl2020}
A.~Almheiri, T.~Hartman, J.~Maldacena,
E.~Shaghoulian, and A.~Tajdini,
``Replica wormholes and the entropy of Hawking radiation,''
\emph{Journal of High Energy Physics}
05 (2020), 013.
doi: \href{https://doi.org/10.1007/JHEP05(2020)013}
{10.1007/JHEP05(2020)013}.
arXiv: \href{https://arxiv.org/abs/1911.12333}{1911.12333}.

\bibitem{Hoeffding1963}
W.~Hoeffding,
``Probability inequalities for sums of bounded random
variables,''
\emph{Journal of the American Statistical Association}
\textbf{58}, 13--30 (1963).
doi: \href{https://doi.org/10.1080/01621459.1963.10500830}
{10.1080/01621459.1963.10500830}.

\bibitem{NISTConstants}
National Institute of Standards and Technology,
\emph{Fundamental Physical Constants}.
Constant values and SI conversion records:
\url{https://physics.nist.gov/cuu/Constants/index.html}.
Numerical constants adopted for this manuscript are
displayed in Section~\ref{sec:source}.

\bibitem{Mamajek2015}
E.~E.~Mamajek et al.,
``IAU 2015 Resolution B3 on Recommended Nominal
Conversion Constants for Selected Solar and
Planetary Properties'' (2015).
arXiv: \href{https://arxiv.org/abs/1510.07674}{1510.07674}.

\bibitem{CheungEm03A}
N.~Cheung,
\emph{Em03-A}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: fixed-background scalar emission law,
\texttt{thm:main}; horizon occupation, packet expectation,
and rate interfaces.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03B}
N.~Cheung,
\emph{Em03-B}, version 2,
unpublished manuscript, n.d.
Supplied excerpt: complete destructive record,
\texttt{thm:record}, and apparatus/loss interfaces.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03C}
N.~Cheung,
\emph{Em03-C}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: joint referenced destructive record,
\texttt{thm:jointrecord}, and shared capture interface.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03D}
N.~Cheung,
\emph{Em03-D}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: quantitative local packet-effect
correspondence, \texttt{thm:main}, with packet and
transfer interfaces.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03E}
N.~Cheung,
\emph{Em03-E}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: exact optimal recovery,
\texttt{thm:recovery}, and key/transfer separation.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03F}
N.~Cheung,
\emph{Em03-F}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: exact deterministic packet recovery
and effective exterior-source channel,
\texttt{thm:recovery}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm03G}
N.~Cheung,
\emph{Em03-G}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: ordered exterior-source release,
\texttt{thm:release}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm05A}
N.~Cheung,
\emph{Em05-A}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: signed extraction physicality and
complementary channels, \texttt{thm:channels}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm05B}
N.~Cheung,
\emph{Em05-B}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: exact recovery if and only if the
complement is constant,
\texttt{thm:constantComplement}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEm06B}
N.~Cheung,
\emph{Em06-B}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: classical source-input stability,
\texttt{thm:input-stability}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02B}
N.~Cheung,
\emph{En02-B}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: resource-certified decoder soundness,
\texttt{thm:B1}, and separation of readout, subsystem,
and instrument tasks.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02C}
N.~Cheung,
\emph{En02-C}, version 1,
unpublished manuscript, n.d.
Supplied abstract: finite unitary radiation models,
information-theoretic recovery, and executable decoding.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02D}
N.~Cheung,
\emph{En02-D}, version 4,
unpublished manuscript, n.d.
Supplied excerpt: energy-labelled benchmark recovery
and memory optima, \texttt{thm:benchmark}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn02E}
N.~Cheung,
\emph{En02-E}, version 2,
unpublished manuscript, n.d.
Supplied source descriptions: finite Gaussian packet
calibration and occupation cutoff.
Supplied theorem excerpt: exact calibrated recovery
and memory optima, \texttt{thm:recovery}.
Source collection: Enter Calculus Companion.

\bibitem{CheungEn03A}
N.~Cheung,
\emph{En03-A}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: exact record compatibility,
\texttt{thm:pullback}.
Supplied source descriptions: stability, network cuts,
and scalar/record sampling interfaces.
Source collection: Enter Calculus Companion.

\bibitem{CheungPsy04A}
N.~Cheung,
\emph{Psy04-A}, version 1,
unpublished manuscript, n.d.
Supplied source descriptions: complete delayed-query
storage boundary and classical null.
Supplied excerpt: non-entailment of quantum
environmental memory, \texttt{thm:nonentailment}.
Source collection: Enter Calculus Companion.

\bibitem{CheungPsy04B}
N.~Cheung,
\emph{Psy04-B}, version 1,
unpublished manuscript, n.d.
Supplied source descriptions: query freshness,
scheduled ancestry, and complete joint access cut.
Supplied excerpt: \texttt{thm:nonentailment}.
Source collection: Enter Calculus Companion.

\bibitem{CheungPsy02B}
N.~Cheung,
\emph{Psy02-B}, version 2,
unpublished manuscript, n.d.
Supplied excerpt: complete-process equality and testing
obstruction, \texttt{thm:process}.
Source collection: Enter Calculus Companion.

\bibitem{CheungPsy03D}
N.~Cheung,
\emph{Psy03-D}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: carrier-independent uncued
observation obstruction, \texttt{cor:qobstruction}.
Source collection: Enter Calculus Companion.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{VII-J}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: scheduled support bounds actual
influence, \texttt{thm:schedule}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXIVH}
N.~Cheung,
\emph{XIV-H}, version 1,
unpublished manuscript, n.d.
Supplied excerpts: exact two-placement internal kernel
and unresolved internal transition,
\texttt{prop:K5exact}, \texttt{thm:H2},
\texttt{thm:release}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{XIV-I}, version 1,
unpublished manuscript, n.d.
Supplied excerpts: constructed internal amplitude
interface and unresolved joint characteristic map,
\texttt{thm:I2}, \texttt{thm:I3},
\texttt{thm:release}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{XIV-J}, version 1,
unpublished manuscript, n.d.
Supplied excerpts: reconstruction descent,
supported-sector reference, unresolved full map,
and no quantum admission,
\texttt{prop:descent}, \texttt{thm:reference},
\texttt{thm:J2}, \texttt{cor:noadmission}.
Source collection: Enter Calculus Companion.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B}, version 1,
unpublished manuscript, n.d.
Supplied excerpt: conditional relative-lifting
criterion, \texttt{thm:lifting}.
Source collection: Enter Calculus Companion.

\end{thebibliography}

\end{document}