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% External-source checks:
% Hawking's original fixed-background calculation: ([doi.org](https://doi.org/10.1007/BF02345020?utm_source=openai))
% Unruh's horizon-boundary-state prescription: ([journals.aps.org](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.870?utm_source=openai))
% Rigorous Schwarzschild Unruh-state construction: ([arxiv.org](https://arxiv.org/abs/0907.1034))
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\hypersetup{
  pdftitle={Em03-A: A fixed-background Hawking emission theorem},
  pdfsubject={Free scalar radiation, horizon state, Schwarzschild scattering,
              and documentary scope audit}
}

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\title{\textbf{Em03-A}\\[0.4em]
A Fixed-Background Hawking Emission Theorem\\[0.3em]
\large An explicit horizon-state calculation, Schwarzschild scattering,
and a non-transfer audit}
\author{}
\date{Version 1 --- October 8, 2026}

\begin{document}
\maketitle

\begin{abstract}
We give an audited formulation of the standard free-scalar emission law
on a fixed Schwarzschild background. The geometry, time normalization,
field, state, observable, and scattering problem are specified separately.
Surface gravity is calculated from the metric. A vacuum defined by an
affine horizon coordinate is pulled back to Killing time, and its
distributional Fourier transform is evaluated without inserting a
temperature or an occupation table. Schwarzschild transmission
probabilities are constructed from the radial wave equation. For every
normalized positive-frequency wavepacket supported in a compact band
away from zero, the resulting number expectation is the greybody-weighted
Bose occupation integral. A trace-class time--frequency window gives the
corresponding number-rate normalization.

The boundary calculation and its regulator removal are proved here.
The bulk existence and Hadamard property of the specified Unruh state
are a precisely declared external input, not a result attributed to an
earlier manuscript in the series. We also prove a compact-test
late-time estimate under explicit peeling and state-regularity
hypotheses. This estimate is not silently promoted to a collapse
wavepacket theorem. The supplied manuscript record is audited with its
retrieval limitations retained. No quantum-gravity correction,
backreaction, evaporation law, island construction, or reopening of an
inherited admission gate is claimed.
\end{abstract}

\tableofcontents

\section{Scope, evidence, and the declared model}
\label{sec:scope}

\subsection{What is completed}

Hawking's original result is a semiclassical field-theory calculation,
not a theorem of nonperturbative quantum gravity
\cite{Hawking1975}. The present paper is an explicit identification and
calculation of one fixed-background observable. It is not a novelty
claim for the Hawking effect.

The mathematical release is:
\[
\boxed{
 \text{declared Schwarzschild metric}
 \;+\;
 \text{declared free scalar and Unruh boundary state}
 \;\Longrightarrow\;
 \text{one proved asymptotic emission law}.
}
\]
The temperature is not supplied as calibration data. It emerges from
the affine-to-Killing coordinate relation after surface gravity has been
computed. The greybody factor is not supplied as a numerical table. It
is defined constructively by the same metric's radial scattering
equation.

The choice of state is nevertheless essential. This paper does not
claim that a metric and a classical wave equation uniquely determine a
quantum emission law.

\subsection{Documentary boundary}

The internal record supplied with the problem consists of excerpts and
summaries associated with corpus identifier
\texttt{ca99b12b}, retrieval identifier \texttt{retrieval-3}
\cite{SuppliedRecord}. It explicitly describes a local retrieval rather
than a complete scan of the series.

Accordingly, the documentary conclusion below is restricted to the
supplied record. Statements about Em02-E, En02-C, En02-D, and XVI-P that
are available only as summaries are identified as such. Their full
manuscripts have not been independently checked here. An internal label
is a source locator, not a theorem proved by this paper.

In particular, ``not constructed'' and ``not admitted'' are not replaced
by ``mathematically nonexistent.'' Similarly, ``XV-B remains closed''
means that its downstream admission condition remains undischarged; it
does not mean that a manuscript named XV-B is absent.

\subsection{Geometry and field}

We use units
\[
 c=\hbar=k_{\mathrm B}=1,
 \qquad G>0,
 \qquad M>0.
\]
The background is the right Schwarzschild exterior
\[
 \mathcal E=\R_t\times(\rh,\infty)_r\times S^2,
 \qquad \rh=2GM,
\]
with metric
\begin{equation}
 \mathrm g
 =-F(r)\dd t^2+F(r)^{-1}\dd r^2+r^2\dd\Omega^2,
 \qquad
 F(r)=1-\frac{\rh}{r}.
 \label{eq:metric}
\end{equation}
The Killing vector is \(\chi=\partial_t\), normalized by
\(\mathrm g(\chi,\chi)\to-1\) as \(r\to\infty\).

The field is a real, massless, minimally coupled scalar:
\begin{equation}
 S[\phi]
 =-\frac12\int \mathrm g^{ab}
          (\nabla_a\phi)(\nabla_b\phi)\dd\mathrm{vol}_{\mathrm g},
 \qquad
 \Box_{\mathrm g}\phi=0.
 \label{eq:action}
\end{equation}
Schwarzschild is Ricci flat, so the massless conformally coupled and
minimally coupled wave operators coincide on this background.

Let
\begin{equation}
 x=r+\rh\log\left(\frac r{\rh}-1\right),
 \qquad
 u=t-x,
 \qquad
 v=t+x.
 \label{eq:tortoise}
\end{equation}
The additive constant in \(x\) is immaterial. We use real orthonormal
spherical harmonics
\[
 Y_{\ell\alpha},
 \qquad
 \ell\in\N\cup\{0\},
 \qquad
 \alpha=1,\ldots,2\ell+1.
\]
This avoids imposing an independent reality convention on complex
\(m\)-modes. Every result below is for one fixed \((\ell,\alpha)\).

\subsection{State and the one imported existence theorem}
\label{subsec:external}

We prescribe independent incoming boundary states:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item on the complete past Kruskal horizon, the vacuum defined by its
 affine coordinate \(U\);
 \item on past null infinity \(\cI^-\), the vacuum defined by positive
 Killing frequency in \(v\).
\end{enumerate}
The product of these states is the Unruh boundary prescription
\cite{Unruh1976}. On the exterior part of the past horizon,
\begin{equation}
 U=-\kappa^{-1}e^{-\kappa u}.
 \label{eq:U}
\end{equation}
Here \(\kappa\) is calculated from \eqref{eq:metric} in
Lemma~\ref{lem:geometry}; its value is not an independent parameter.

\begin{externaltheorem}
For the massless scalar on Schwarzschild, the past-horizon and
past-null-infinity characteristic maps give a symplectic
bulk-to-boundary embedding. Pulling back the product of the affine
horizon vacuum and the Killing-frequency vacuum on \(\cI^-\) gives a
quasifree Unruh state. This state is Hadamard on the exterior and on its
stated extension through the future horizon.
\end{externaltheorem}

The external input is the construction and regularity result of
Dappiaggi, Moretti, and Pinamonti, specifically their
bulk-to-boundary results and Theorem~4.1
\cite{DMP2011}. We do not reproduce their global microlocal proof.
Thus the calculation below is self-contained at the radiation-boundary
level, while its identification with a regular bulk state uses this
declared theorem.

The normalized horizon field for a partial wave is \(\rh\phi\); the
normalized field at null infinity is \(r\phi\). Both have the
one-dimensional canonical normalization used below.

\begin{remark}[State selection is not rate calibration]
The input is positive affine frequency on the horizon and positive
Killing frequency on \(\cI^-\). No Bose occupation function is included
in this input. Its appearance is a conclusion of the Fourier
calculation in Section~\ref{sec:thermal}. Choosing a state is necessary
physical data, not a claim that thermality follows from the classical
metric alone.
\end{remark}

\section{The main emission theorem}
\label{sec:main}

For a compact frequency interval
\[
 B=[\omega_-,\omega_+]\subset(0,\infty),
\]
let \(h\in C^\infty_c((0,\infty))\) satisfy
\[
 \supp h\subset B,
 \qquad
 \int_0^\infty |h(\omega)|^2\dd\omega=1.
\]
Define the outgoing packet annihilator and number observable by
\begin{equation}
 \aout(h)
 =\int_0^\infty \overline{h(\omega)}
                         \aout(\omega)\dd\omega,
 \qquad
 N_h=\aout(h)^*\aout(h).
 \label{eq:packet}
\end{equation}
Their construction and finite expectation are proved below.

\begin{theorem}[Fixed-background scalar emission law]
\label{thm:main}
For the geometry, field, state, and normalization of
Section~\ref{sec:scope}, the following statements hold.

\begin{enumerate}[label=\textup{(\roman*)}]
 \item The normalized Schwarzschild Killing horizon has
 \begin{equation}
  \kappa=\frac1{4GM},
  \qquad
  \betah=\frac{2\pi}{\kappa}=8\pi GM.
  \label{eq:kappabeta}
 \end{equation}

 \item The incoming horizon current, expressed in Killing coordinate
 \(u\), has two-point distribution
 \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:thermalcurrent}
 \end{equation}
 whose Fourier transform is
 \begin{equation}
  \widehat C_\kappa(\nu)
  =\frac{\nu}{2(1-e^{-\betah\nu})},
  \qquad
  \widehat C_\kappa(0)=\frac1{2\betah}.
  \label{eq:thermalfourier}
 \end{equation}

 \item The radial scalar equation constructs a unitary two-channel
 scattering matrix. Its horizon-to-infinity transmission coefficient
 \(\tau_\ell(\omega)\) defines
 \begin{equation}
  \Gamma_\ell(\omega)=|\tau_\ell(\omega)|^2,
  \qquad
  0<\Gamma_\ell(\omega)\le1
  \quad(\omega>0).
  \label{eq:gamma}
 \end{equation}

 \item The outgoing number covariance is the distributional identity
 \begin{equation}
 \stateU\bigl(
     \aout(\omega)^*\aout(\omega')\bigr)
 =
 \frac{\Gamma_\ell(\omega)}
      {e^{\betah\omega}-1}\,
       \delta(\omega-\omega'),
 \label{eq:numberdensity}
 \end{equation}
 and consequently
 \begin{equation}
  \boxed{
  \stateU(N_h)
  =\int_0^\infty
       |h(\omega)|^2
       \frac{\Gamma_\ell(\omega)}
            {e^{8\pi GM\omega}-1}\dd\omega.
  }
  \label{eq:packetlaw}
 \end{equation}

 \item With the time--frequency counting convention of
 Proposition~\ref{prop:rate}, the number rate per angular mode is
 \begin{equation}
  \boxed{
   f_\ell(\omega;M)
   =\frac1{2\pi}
       \frac{\Gamma_\ell(\omega)}
            {e^{8\pi GM\omega}-1}.
  }
  \label{eq:rate}
 \end{equation}
 This formula is derived from the boundary state and the radial
 scattering equation, not assumed as calibration data.
\end{enumerate}
\end{theorem}

The proof is assembled at the end of Section~\ref{sec:observables}.
The result is exact in the declared stationary free-field model.
No late-time approximation is used in \eqref{eq:packetlaw}.
Section~\ref{sec:peeling} treats a different, explicitly conditional
late-time limit.

\begin{remark}[What a monochromatic formula means]
There is no normalized one-particle vector concentrated at one
continuum frequency, and \(\delta(0)\) is not a number expectation.
Equation~\eqref{eq:numberdensity} is a covariance density;
equation~\eqref{eq:packetlaw} is a finite observable expectation.
The frequently written expression
\[
 \mathbb E N_{\omega\ell m}
 =\frac{\Gamma_\ell(\omega)}
        {e^{8\pi GM\omega}-1}
\]
must be understood with a packet, box, or spectral-density
normalization specified.
\end{remark}

\section{Geometry and exact exterior scattering}
\label{sec:scattering}

\subsection{Surface gravity and affine horizon coordinate}

\begin{lemma}[Geometric normalization]
\label{lem:geometry}
For \eqref{eq:metric}, surface gravity is
\[
 \kappa=\frac12F'(\rh)=\frac1{2\rh}=\frac1{4GM}.
\]
The coordinate \(U\) in \eqref{eq:U} is affine on the past Kruskal
horizon, and its exterior restriction satisfies
\[
 \chi(U)=-\kappa U.
\]
\end{lemma}

\begin{proof}
In ingoing Eddington--Finkelstein coordinates,
\[
 \mathrm g=-F(r)\dd v^2+2\dd v\dd r+r^2\dd\Omega^2,
 \qquad
 \chi=\partial_v.
\]
The relevant connection coefficients are
\[
 \Gamma^v_{vv}=\frac12F'(r),
 \qquad
 \Gamma^r_{vv}=\frac12F(r)F'(r).
\]
At \(r=\rh\), therefore,
\[
 \nabla_\chi\chi=\frac12F'(\rh)\chi.
\]
Since \(F'(r)=\rh/r^2\), the stated value follows. The normalization of
\(\chi\) at infinity removes a rescaling ambiguity.

Set
\[
 U=-\kappa^{-1}e^{-\kappa u},
 \qquad
 V=\kappa^{-1}e^{\kappa v}.
\]
Their product is
\[
 UV=-\kappa^{-2}e^{2\kappa x},
\]
and \(2\kappa=1/\rh\). Hence the radial metric extends across the
horizons in the form
\[
 -A(UV)\dd U\dd V
\]
with \(A\) smooth and nonzero at \(UV=0\). On the past horizon \(V=0\),
the coefficient is constant along \(U\), so the horizon generators
have vanishing affine acceleration in coordinate \(U\). Finally,
\[
 \chi=\kappa(-U\partial_U+V\partial_V),
\]
which gives \(\chi(U)=-\kappa U\).
\end{proof}

\subsection{The radial equation}

\begin{lemma}[Partial-wave reduction]
\label{lem:radial}
Writing
\[
 \phi(t,r,\Omega)
 =\frac1r\sum_{\ell,\alpha}
       \psi_{\ell\alpha}(t,x)Y_{\ell\alpha}(\Omega)
\]
reduces \(\Box_{\mathrm g}\phi=0\) to
\begin{equation}
 \left(-\partial_t^2+\partial_x^2-V_\ell(x)\right)
                   \psi_{\ell\alpha}=0,
 \label{eq:radialwave}
\end{equation}
where
\begin{equation}
 V_\ell(x)
 =F(r(x))
       \left(\frac{\ell(\ell+1)}{r(x)^2}
                    +\frac{\rh}{r(x)^3}\right).
 \label{eq:potential}
\end{equation}
The potential is nonnegative, smooth, and integrable:
\begin{equation}
 \int_\R V_\ell(x)\dd x
 =\frac{\ell(\ell+1)}{\rh}+\frac1{2\rh}.
 \label{eq:potentialL1}
\end{equation}
\end{lemma}

\begin{proof}
In Schwarzschild coordinates,
\[
 \Box_{\mathrm g}\phi
 =-F^{-1}\partial_t^2\phi
   +r^{-2}\partial_r(r^2F\partial_r\phi)
   +r^{-2}\Delta_{S^2}\phi.
\]
Insert \(r^{-1}\psi Y_{\ell\alpha}\), use
\(\Delta_{S^2}Y_{\ell\alpha}=-\ell(\ell+1)Y_{\ell\alpha}\),
and use \(\partial_x=F\partial_r\). Multiplication by \(rF\)
gives \eqref{eq:radialwave}.

All terms in \eqref{eq:potential} are nonnegative. Since
\(\dd x=F^{-1}\dd r\),
\[
 \int_\R V_\ell(x)\dd x
 =\int_{\rh}^{\infty}
       \left(\frac{\ell(\ell+1)}{r^2}+\frac{\rh}{r^3}\right)\dd r,
\]
which evaluates to \eqref{eq:potentialL1}.
\end{proof}

\subsection{Constructive transmission probabilities}

\begin{lemma}[Positive-frequency scattering]
\label{lem:scattering}
For every fixed \(\ell\) and \(\omega>0\), the radial equation
\begin{equation}
 q''(x)+\bigl(\omega^2-V_\ell(x)\bigr)q(x)=0
 \label{eq:ODE}
\end{equation}
has uniquely normalized Jost solutions \(f_+\) and \(f_-\) with
\[
 f_+(x,\omega)\sim e^{\ii\omega x}\quad(x\to+\infty),
 \qquad
 f_-(x,\omega)\sim e^{-\ii\omega x}\quad(x\to-\infty),
\]
including the corresponding derivative asymptotics.

They construct a unitary scattering matrix \(S_\ell(\omega)\).
The horizon-to-infinity coefficient is
\[
 \tau_\ell(\omega)=a_\ell(\omega)^{-1},
\]
where
\[
 f_+(x,\omega)
 =a_\ell(\omega)e^{\ii\omega x}
   +b_\ell(\omega)e^{-\ii\omega x}+o(1)
 \quad(x\to-\infty).
\]
In particular,
\[
 |a_\ell(\omega)|^2-|b_\ell(\omega)|^2=1,
 \qquad
 0<|\tau_\ell(\omega)|^2\le1.
\]
All these coefficients are continuous on compact subsets of
\((0,\infty)\).
\end{lemma}

\begin{proof}
The outgoing right Jost solution is constructed by the Volterra
equation
\begin{equation}
 f_+(x,\omega)
 =e^{\ii\omega x}
   +\int_x^\infty
       \frac{\sin(\omega(y-x))}{\omega}
                  V_\ell(y)f_+(y,\omega)\dd y.
 \label{eq:Volterra}
\end{equation}
Successive iteration converges absolutely. Indeed, the \(n\)-th
iterated correction is bounded by
\[
 \frac1{n!}
 \left(\frac1\omega\int_x^\infty V_\ell(y)\dd y\right)^n,
\]
and consequently
\[
 |f_+(x,\omega)|
 \le
 \exp\left(\frac{\norm{V_\ell}_{L^1}}{\omega}\right).
\]
The same Volterra estimate applied to a difference proves uniqueness.
Differentiation of \eqref{eq:Volterra} gives \eqref{eq:ODE} and the
derivative asymptotics at \(+\infty\).

Expanding the sine in \eqref{eq:Volterra} gives, at the other end,
\[
 a_\ell
 =1-\frac1{2\ii\omega}
       \int_\R e^{-\ii\omega y}
                    V_\ell(y)f_+(y,\omega)\dd y,
 \qquad
 b_\ell
 =\frac1{2\ii\omega}
       \int_\R e^{\ii\omega y}
                    V_\ell(y)f_+(y,\omega)\dd y.
\]
The omitted tails tend to zero because \(V_\ell f_+\in L^1\).
Differentiating the same expressions gives the left derivative
asymptotics.

For every solution of \eqref{eq:ODE}, the current
\[
 J[q]=\operatorname{Im}(\overline q\,q')
\]
is constant. For \(f_+\), its right-end value is \(\omega\), whereas its
left-end value is
\[
 \omega\bigl(|a_\ell|^2-|b_\ell|^2\bigr).
\]
Thus \(a_\ell\ne0\), and the horizon-incident solution
\(q_{\mathrm H}=f_+/a_\ell\) has asymptotics
\[
 q_{\mathrm H}\sim
 \begin{cases}
  e^{\ii\omega x}
       +(b_\ell/a_\ell)e^{-\ii\omega x},&x\to-\infty,\\
  a_\ell^{-1}e^{\ii\omega x},&x\to+\infty.
 \end{cases}
\]
This proves the transmission formula and its bounds.

The left Jost solution is obtained from the analogous Volterra
equation integrated from \(-\infty\). Its right incoming coefficient
is likewise nonzero by current conservation, so it constructs the
solution \(q_{\mathrm I}\) incident from infinity.

For arbitrary incoming amplitudes \(A_{\mathrm H},A_{\mathrm I}\),
the corresponding combination of \(q_{\mathrm H}\) and
\(q_{\mathrm I}\) has outgoing amplitudes
\[
 \begin{pmatrix}B_{\mathrm H}\\B_{\mathrm I}\end{pmatrix}
 =
 S_\ell(\omega)
 \begin{pmatrix}A_{\mathrm H}\\A_{\mathrm I}\end{pmatrix}.
\]
Current conservation gives
\[
 |A_{\mathrm H}|^2+|A_{\mathrm I}|^2
 =
 |B_{\mathrm H}|^2+|B_{\mathrm I}|^2
\]
for every pair of incoming amplitudes. Therefore
\(S_\ell^*S_\ell=I\), and the square matrix \(S_\ell\) is unitary.

On a compact positive-frequency interval, the Volterra series have
uniform majorants. Their terms and the coefficient integrals depend
continuously on \(\omega\). Dominated convergence proves the last
assertion.
\end{proof}

\begin{remark}[No geometric-optics replacement]
The potential \eqref{eq:potential} has not been discarded.
The transmission probability in \eqref{eq:gamma} is exact for this
ODE. This is the separation between the horizon occupation factor and
exterior scattering. Numerical approximations to the ODE may be used
later, but they require their own error bound.
\end{remark}

\section{Quantum input, parametrix, and the thermal calculation}
\label{sec:thermal}

\subsection{Canonical current normalization}

For a normalized radiation field \(\psi(s)\), write \(j(s)=\partial_s
\psi(s)\). The positive-frequency convention is
\begin{equation}
 j(s)
 =-\ii\int_0^\infty\sqrt{\frac{\omega}{4\pi}}
   \left(a(\omega)e^{-\ii\omega s}
                -a(\omega)^*e^{\ii\omega s}\right)\dd\omega,
 \qquad
 [a(\omega),a(\omega')^*]=\delta(\omega-\omega').
 \label{eq:currentexpansion}
\end{equation}
This yields
\begin{equation}
 [j(s),j(s')]=\frac{\ii}{2}\delta'(s-s').
 \label{eq:currentCCR}
\end{equation}

The vacuum current covariance is the tempered distribution
\begin{equation}
 C_0(s-s')
 =-\frac1{4\pi(s-s'-\ii0)^2}
 =\lim_{\epsilon\downarrow0}
    \int_0^\infty\frac{\omega}{4\pi}
       e^{-\ii\omega(s-s')-\epsilon\omega}\dd\omega.
 \label{eq:vacuumcurrent}
\end{equation}
In particular, for any complex compact test \(g\),
\begin{equation}
 \int_{\R^2}\overline{g(s)}\,C_0(s-s')g(s')\dd s\dd s'
 =
 \int_0^\infty\frac{\omega}{4\pi}
       \left|\int_\R g(s)e^{\ii\omega s}\dd s\right|^2\dd\omega
 \ge0.
 \label{eq:vacuumpositive}
\end{equation}
This formula also proves finite covariance on compact smooth tests.

We work with currents rather than an unsmeared massless scalar
zero mode. The number observables used below have frequency support
bounded away from zero.

\subsection{Pullback of the affine vacuum}

\begin{lemma}[Horizon pullback and singularity subtraction]
\label{lem:pullback}
Let \(j_U\) have covariance \eqref{eq:vacuumcurrent} in the affine
coordinate \(U\). On \(U<0\), set
\[
 j_{\mathrm H}(u)=U'(u)j_U(U(u)).
\]
This defines a positive quasifree current state with the canonical
commutator \eqref{eq:currentCCR}. Its covariance is
\eqref{eq:thermalcurrent}. Moreover,
\begin{equation}
 C_\kappa(s)-C_0(s)
 \quad\text{is smooth on }\R,
 \qquad
 (C_\kappa-C_0)(0)=\frac{\kappa^2}{48\pi}.
 \label{eq:subtraction}
\end{equation}
\end{lemma}

\begin{proof}
For \(g\in C^\infty_c(\R)\), change variables from \(u\) to \(U\):
\[
 \int g(u)U'(u)j_U(U(u))\dd u
 =
 \int g(u(U))j_U(U)\dd U.
\]
The transformed test is smooth and compactly supported in \(U<0\);
its extension by zero is a valid test on the affine line. Positivity
therefore follows from \eqref{eq:vacuumpositive}. The antisymmetric
covariance is preserved by this change of variable, which proves the
current commutator.

The covariance is the pullback distribution
\begin{equation}
 -\frac1{4\pi}
   \frac{U'(u)U'(u')}
        {(U(u)-U(u')-\ii0)^2}.
 \label{eq:pullbackcovariance}
\end{equation}
The boundary-value prescription is well defined because \(U\) is a
smooth strictly increasing coordinate.

Away from \(u=u'\), elementary substitution gives
\[
 \frac{U'(u)U'(u')}{(U(u)-U(u'))^2}
 =
 \frac{\kappa^2}
      {4\sinh^2(\kappa(u-u')/2)}.
\]
At the diagonal, the pullback keeps the same positive-frequency
boundary orientation. Equivalently, factor
\(U(u)-U(u')=(u-u')D(u,u')\) with \(D>0\), and use the
boundary-value distribution in the variable \(u-u'\).
This gives exactly \eqref{eq:thermalcurrent}, including its
distributional extension.

Finally,
\[
 \csch^2 z=z^{-2}-\frac13+O(z^2)
\]
at \(z=0\). The only real singularity of \(C_\kappa\) is its diagonal
singularity, which coincides with that of \(C_0\). Their difference
therefore extends smoothly, with the value in \eqref{eq:subtraction}.
\end{proof}

\begin{remark}[Which parametrix is used]
For the boundary current calculation, the singular reference
distribution is explicitly \(C_0\), and
\eqref{eq:subtraction} proves the required local subtraction.
For a local bulk observable, the appropriate object would instead
be the four-dimensional Hadamard parametrix associated with
\(\mathrm g\). The imported Hadamard theorem guarantees its
applicability to the selected bulk state; this paper does not identify
the one-dimensional boundary subtraction with a bulk stress-tensor
renormalization prescription. These distinctions are standard in
curved-spacetime QFT \cite{HollandsWald2015}.
\end{remark}

\subsection{Fourier evaluation and regulator removal}

Our Fourier convention is
\[
 \widehat C(\nu)=\int_\R e^{\ii\nu s}C(s)\dd s.
\]

\begin{lemma}[Thermal spectral density]
\label{lem:fourier}
The distribution \eqref{eq:thermalcurrent} has Fourier transform
\eqref{eq:thermalfourier}. It has the spectral representation
\begin{equation}
 C_\kappa(s)
 =
 \int_0^\infty\frac{\omega}{4\pi}
 \left[
  (1+\nh(\omega))e^{-\ii\omega s}
       +\nh(\omega)e^{\ii\omega s}
 \right]\dd\omega,
 \qquad
 \nh(\omega)=\frac1{e^{\betah\omega}-1},
 \label{eq:thermalspectral}
\end{equation}
understood in \(\cS'(\R)\). In particular,
\begin{equation}
 \widehat C_\kappa(-\omega)
 =e^{-\betah\omega}\widehat C_\kappa(\omega),
 \qquad \omega>0.
 \label{eq:detailedbalance}
\end{equation}
\end{lemma}

\begin{proof}
Let
\[
 C(z)=-\frac{\kappa^2}{16\pi}
               \csch^2(\kappa z/2).
\]
It is analytic in the strip
\(-\betah<\operatorname{Im}z<0\).
Its poles are \(z=\ii n\betah\), \(n\in\mathbb Z\), and its
principal part at every pole is
\[
 -\frac1{4\pi(z-\ii n\betah)^2}.
\]
For \(0<\eta<\betah\), consider
\[
 I_\eta(\nu)
 =\int_{\R-\ii\eta} e^{\ii\nu z}C(z)\dd z.
\]
The integrand decays exponentially as
\(|\operatorname{Re}z|\to\infty\).

For \(\nu>0\), close the contour upwards. The double-pole residue at
\(z=\ii n\betah\) is
\[
 -\frac{\ii\nu}{4\pi}e^{-n\betah\nu}.
\]
The poles enclosed are \(n\ge0\). Taking rectangles whose horizontal
edges stay halfway between consecutive poles, and then taking their
limits, gives
\[
 I_\eta(\nu)
 =2\pi\ii\sum_{n=0}^\infty
        \left(-\frac{\ii\nu}{4\pi}e^{-n\betah\nu}\right)
 =\frac{\nu}{2(1-e^{-\betah\nu})}.
\]
For \(\nu<0\), close downwards with clockwise orientation.
The enclosed poles are \(n\le-1\), so
\[
 I_\eta(\nu)
 =-\frac{\nu}{2}\sum_{k=1}^\infty e^{k\betah\nu}
 =\frac{\nu}{2(1-e^{-\betah\nu})}.
\]
The horizontal edges vanish because of the sign of \(\nu\), and the
vertical edges vanish by the real-direction exponential decay.

Putting \(z=s-\ii\eta\) shows that
\[
 \int_\R e^{\ii\nu s}C(s-\ii\eta)\dd s
 =
 e^{-\eta\nu}\frac{\nu}{2(1-e^{-\betah\nu})}.
\]
For fixed \(\eta\), the left-hand Fourier transform is continuous,
so the value at \(\nu=0\) is the continuous limit \(1/(2\betah)\).
As \(\eta\downarrow0\), the right-hand side converges in
\(\cS'(\R)\): on positive frequencies it has at most linear growth;
on negative frequencies it decays exponentially, uniformly for
\(0<\eta\le\betah/2\). This proves the regulator removal and
\eqref{eq:thermalfourier}.

Separating positive and negative frequencies in the inverse Fourier
transform yields \eqref{eq:thermalspectral}. Formula
\eqref{eq:detailedbalance} follows directly.
\end{proof}

\begin{remark}[Thermal relation and its domain]
Equation~\eqref{eq:detailedbalance} is the two-point KMS
detailed-balance relation for the horizon current in Killing time.
The full state includes a second incoming channel in its vacuum, and
the greybody-weighted outgoing state is not generally a KMS state at
the same temperature. We therefore do not transfer horizon KMS
thermality to every detector or every exterior observable.
\end{remark}

\subsection{Number covariance without a continuum-Fock assumption}

\begin{lemma}[Horizon occupation covariance]
\label{lem:horizonoccupation}
For positive-frequency tests supported in a compact subset of
\((0,\infty)\), the horizon state has
\begin{align}
 \stateH\bigl(\ainH(\omega)^*\ainH(\omega')\bigr)
   &=\nh(\omega)\delta(\omega-\omega'),
   \label{eq:Hnumber}\\
 \stateH\bigl(\ainH(\omega)\ainH(\omega')\bigr)
   &=0.
   \label{eq:Hanomalous}
\end{align}
The infinity input is independent and has
\begin{equation}
 \stateI\bigl(\ainI(\omega)^*\ainI(\omega')\bigr)=0.
 \label{eq:Inumber}
\end{equation}
These statements define finite packet covariances without requiring
the stationary state to be a density matrix in the outgoing vacuum
Fock representation.
\end{lemma}

\begin{proof}
Comparing \eqref{eq:currentexpansion} with
\eqref{eq:thermalspectral} identifies the coefficients of the positive
and negative frequency terms. Stationarity excludes an anomalous
covariance supported at \(\omega+\omega'=0\) on the strictly
positive-frequency test domain. This gives
\eqref{eq:Hnumber}--\eqref{eq:Hanomalous}.

For an explicit realization on any compact positive-frequency band,
take two independent vacuum oscillator fields \(b_1,b_2\) and set
\[
 \ainH(\omega)
 =\sqrt{1+\nh(\omega)}\,b_1(\omega)
       +\sqrt{\nh(\omega)}\,b_2(\omega)^*.
\]
Its commutator is \(\delta(\omega-\omega')\), and its vacuum
covariances are precisely \eqref{eq:Hnumber} and
\eqref{eq:Hanomalous}. All multiplication factors are bounded on the
chosen band. A third independent vacuum oscillator field realizes
\(\ainI\) and \eqref{eq:Inumber}. The horizon occupation function in
this realization is the function already derived in
Lemma~\ref{lem:fourier}, not an additional input.
\end{proof}

\section{Outgoing observables, rate normalization, and error bounds}
\label{sec:observables}

\subsection{Scattering of the quantum channels}

Let the row of the scattering matrix corresponding to future null
infinity be
\[
 \bigl(\tau_\ell(\omega),\rho_{\mathrm I,\ell}(\omega)\bigr).
\]
Define
\begin{equation}
 \aout(\omega)
 =\tau_\ell(\omega)\ainH(\omega)
       +\rho_{\mathrm I,\ell}(\omega)\ainI(\omega).
 \label{eq:quantumscatter}
\end{equation}
Because the classical scattering is stationary, it preserves Killing
frequency and does not mix annihilation with creation in this
equation. The affine-to-Killing change has already been accounted for
in the horizon state.

\begin{proposition}[Outgoing packet expectation]
\label{prop:packet}
The definition \eqref{eq:quantumscatter} gives canonical outgoing
commutators. Its number covariance is \eqref{eq:numberdensity},
and \(N_h\) has the finite expectation \eqref{eq:packetlaw}.
\end{proposition}

\begin{proof}
The two incoming channels commute, and unitarity gives
\[
 |\tau_\ell(\omega)|^2
       +|\rho_{\mathrm I,\ell}(\omega)|^2=1.
\]
Hence the outgoing commutator is canonical. The input state is a
product state with vanishing means and no cross-channel covariance.
Lemma~\ref{lem:horizonoccupation} therefore gives
\[
 \stateU\bigl(
   \aout(\omega)^*\aout(\omega')\bigr)
 =
 \overline{\tau_\ell(\omega)}\tau_\ell(\omega')
       \nh(\omega)\delta(\omega-\omega').
\]
This is \eqref{eq:numberdensity}. Smearing proves
\eqref{eq:packetlaw}. The integral is finite because
\(\Gamma_\ell\le1\), \(\omega\ge\omega_->0\) on the support, and
\(h\in L^2\).
\end{proof}

\begin{remark}[Greybody weighting is not exact KMS thermality]
The full outgoing current has spectral covariance
\[
 C_{\mathrm{out}}(s)
 =\int_0^\infty\frac{\omega}{4\pi}
 \left[
 (1+\Gamma_\ell(\omega)\nh(\omega))e^{-\ii\omega s}
   +\Gamma_\ell(\omega)\nh(\omega)e^{\ii\omega s}
 \right]\dd\omega.
\]
Its negative-to-positive spectral ratio is
\[
 \frac{\Gamma_\ell\nh}{1+\Gamma_\ell\nh},
\]
not generally \(e^{-\betah\omega}\).
The appropriate statement is a greybody-weighted emission law, not
global thermal equilibrium.
\end{remark}

\subsection{A precise \(1/(2\pi)\) counting convention}

\begin{definition}[Time--frequency number window]
\label{def:window}
On the one-particle space \(L^2((0,\infty),\dd\omega)\), let \(P_B\)
denote multiplication by the indicator of a compact positive-frequency
interval \(B\). For \(T>0\), define
\[
 (A_{B,T}h)(t)
 =\frac1{\sqrt{2\pi}}\int_B
            e^{-\ii\omega t}h(\omega)\dd\omega,
 \qquad -T/2\le t\le T/2,
\]
and
\[
 K_{B,T}=A_{B,T}^*A_{B,T}.
\]
The expected number in this mode window is
\[
 \mathcal N_{B,T}
 :=\Tr\bigl(M_{\nout}K_{B,T}\bigr),
 \qquad
 \nout(\omega)=\Gamma_\ell(\omega)\nh(\omega),
\]
where \(M_{\nout}\) is the multiplication operator.
Equivalently, this is the monotone expectation of finite-rank positive
number-functionals approximating \(K_{B,T}\).
\end{definition}

\begin{proposition}[Mode-window number rate]
\label{prop:rate}
The operator \(K_{B,T}\) is positive and trace class, and
\begin{equation}
 \mathcal N_{B,T}
 =\frac{T}{2\pi}\int_B
       \frac{\Gamma_\ell(\omega)}
            {e^{\betah\omega}-1}\dd\omega.
 \label{eq:windowcount}
\end{equation}
Thus the rate density under Definition~\ref{def:window} is
\eqref{eq:rate}.
\end{proposition}

\begin{proof}
The integral kernel of \(A_{B,T}\) is
\[
 (2\pi)^{-1/2}e^{-\ii\omega t}\mathbf1_B(\omega).
\]
It is Hilbert--Schmidt, with
\[
 \norm{A_{B,T}}_{\mathrm{HS}}^2=\frac{T|B|}{2\pi}.
\]
Consequently \(K_{B,T}=A_{B,T}^*A_{B,T}\) is positive and trace
class.

The function \(\nout\) is nonnegative and bounded on \(B\). Therefore
\[
 \Tr(M_{\nout}K_{B,T})
 =
 \norm{A_{B,T}M_{\sqrt{\nout}}}_{\mathrm{HS}}^2
 =
 \frac{T}{2\pi}\int_B\nout(\omega)\dd\omega.
\]
Substitution of its proved value gives \eqref{eq:windowcount}.
Dividing by \(T\) identifies the density.
\end{proof}

\begin{remark}[A counting convention is not an instrument]
The positive mode-window functional is not asserted to be a sharply
localized detector count or a finite measurement instrument. A
detector implementation requires a coupling, switching, input class,
and error estimate. Those are separate from this asymptotic
number-rate normalization.

Equation~\eqref{eq:rate} is per angular mode. Summing over the
\(2\ell+1\) real harmonics at fixed \(\ell\) introduces that
degeneracy. No all-\(\ell\) luminosity theorem is asserted here.
\end{remark}

\subsection{Controlled approximations}

\begin{corollary}[Transmission and narrow-band errors]
\label{cor:error}
Let \(h\) be as in Theorem~\ref{thm:main}.

If an approximation \(\widetilde\Gamma_\ell\) satisfies
\[
 \sup_{\omega\in B}
       |\widetilde\Gamma_\ell(\omega)-\Gamma_\ell(\omega)|
 \le\epsilon_\Gamma,
\]
then
\begin{equation}
 \left|
 \stateU(N_h)
 -
 \int_B |h(\omega)|^2
     \frac{\widetilde\Gamma_\ell(\omega)}
          {e^{\betah\omega}-1}\dd\omega
 \right|
 \le
 \frac{\epsilon_\Gamma}{e^{\betah\omega_-}-1}.
 \label{eq:gammaerror}
\end{equation}

Writing \(q(\omega)=\Gamma_\ell(\omega)/(e^{\betah\omega}-1)\),
for every \(\omega_0\in B\),
\begin{equation}
 |\stateU(N_h)-q(\omega_0)|
 \le \sup_{\omega\in B}|q(\omega)-q(\omega_0)|.
 \label{eq:banderror}
\end{equation}
Thus normalized packets in bands shrinking to a fixed
\(\omega_0>0\) have expectation tending to \(q(\omega_0)\).
\end{corollary}

\begin{proof}
For \eqref{eq:gammaerror}, integrate the absolute error against
\(|h|^2\), use the stated uniform transmission error, and use
\[
 (e^{\betah\omega}-1)^{-1}
 \le(e^{\betah\omega_-}-1)^{-1}.
\]
For \eqref{eq:banderror}, subtract \(q(\omega_0)\) inside
\eqref{eq:packetlaw} and use \(\int |h|^2=1\).
Continuity follows from Lemma~\ref{lem:scattering}.
\end{proof}

\subsection{Completion of the main proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Lemma~\ref{lem:geometry} proves the surface gravity and its geometric
normalization. Lemmas~\ref{lem:pullback} and \ref{lem:fourier} prove the
horizon covariance, its Fourier transform, and the distributional
regulator removal. Lemmas~\ref{lem:radial} and
\ref{lem:scattering} construct the exact radial scattering
coefficients. Proposition~\ref{prop:packet} proves the outgoing
number covariance and packet formula.
Proposition~\ref{prop:rate} proves the counting normalization.

The declared external theorem identifies the boundary prescription
with the Hadamard bulk Unruh state on the stated Schwarzschild domain.
No earlier internal manuscript is used as a substitute for that
theorem. The exponent in the emission law is
\(\betah\omega=8\pi GM\omega\), calculated rather than calibrated.
\end{proof}

\section{A separate late-time limit with a proved remainder}
\label{sec:peeling}

The exact stationary theorem does not by itself prove that a
specified collapse state converges to the Unruh state. This section
gives a useful local transfer theorem with every additional assumption
displayed.

\begin{assumption}[Peeling map and regular affine remainder]
\label{ass:peeling}
Let \(p\) be a smooth strictly increasing function for \(u\ge u_0\).
For constants \(A,\kappa,\lambda>0\), suppose
\begin{equation}
 p(u)=U_{\mathrm H}-Ae^{-\kappa u}+b(u),
 \qquad
 |b^{(j)}(u)|\le B_j e^{-(\kappa+\lambda)u},
 \quad j=0,1,2,3.
 \label{eq:peeling}
\end{equation}
Suppose a positive current state has affine-coordinate covariance
\begin{equation}
 C_{\mathrm{aff}}(U,U')
 =-\frac1{4\pi(U-U'-\ii0)^2}+R(U,U'),
 \label{eq:affineremainder}
\end{equation}
where \(R\) is smooth and bounded in a neighborhood of
\((U_{\mathrm H},U_{\mathrm H})\).

The pulled-back current covariance is
\begin{equation}
 C_p(u,u')
 =p'(u)p'(u')C_{\mathrm{aff}}(p(u),p(u')).
 \label{eq:Cp}
\end{equation}
\end{assumption}

Positivity in this assumption is not inferred from an arbitrary
choice of \(R\). The statement concerns a state whose covariance
already has the displayed form.

\begin{theorem}[Compact-test thermal limit]
\label{thm:peeling}
Under Assumption~\ref{ass:peeling}, fix \(L>0\).
There are constants \(t_L,C_L>0\) such that, for \(t\ge t_L\),
\begin{equation}
 C_p(t+x,t+y)-C_\kappa(x-y)
\end{equation}
is represented on \([-L,L]^2\) by a continuous kernel \(E_t(x,y)\)
satisfying
\begin{equation}
 \sup_{|x|,|y|\le L}|E_t(x,y)|
 \le C_L\left(e^{-\lambda t}+e^{-2\kappa t}\right).
 \label{eq:localerror}
\end{equation}
Consequently, for \(f,g\in C^\infty_c((-L,L))\),
\begin{align}
 &\left|
  \int_{\R^2}\overline{f(x)}
     \bigl[C_p(t+x,t+y)-C_\kappa(x-y)\bigr]
                   g(y)\dd x\dd y
 \right|
 \nonumber\\
 &\hspace{2em}
 \le C_L\left(e^{-\lambda t}+e^{-2\kappa t}\right)
                       \norm f_{L^1}\norm g_{L^1}.
 \label{eq:smearederror}
\end{align}
This proves a local distributional thermal limit for one translated
current covariance.
\end{theorem}

\begin{proof}
Affine translation and positive affine rescaling leave the singular
ratio in \eqref{eq:Cp} unchanged. Define
\[
 q_t(x)
 =\frac{p(t+x)-U_{\mathrm H}}{Ae^{-\kappa t}},
 \qquad
 q_\infty(x)=-e^{-\kappa x}.
\]
Equation~\eqref{eq:peeling} gives
\begin{equation}
 \norm{q_t-q_\infty}_{C^3([-L,L])}
 \le C_L e^{-\lambda t}.
 \label{eq:qconvergence}
\end{equation}
For sufficiently large \(t\), both \(q_t'\) and \(q_\infty'\) are
bounded below by a positive constant on this fixed interval.

Set
\[
 D_q(x,y)
 =\int_0^1 q'(y+\theta(x-y))\dd\theta,
 \qquad
 A_q(x,y)
 =\frac{q'(x)q'(y)}{D_q(x,y)^2}.
\]
Then \(q(x)-q(y)=(x-y)D_q(x,y)\), and the singular pullback is
\[
 -\frac1{4\pi}A_q(x,y)(x-y-\ii0)^{-2}.
\]

Write \(m=(x+y)/2\) and \(s=x-y\). The function \(A_q\), expressed in
\((m,s)\), is even in \(s\), satisfies \(A_q(m,0)=1\), and has
\(\partial_sA_q(m,0)=0\). Taylor's formula gives
\[
 A_q(m,s)=1+s^2B_q(m,s),
 \qquad
 B_q(m,s)
 =\int_0^1(1-\theta)
                   \partial_s^2A_q(m,\theta s)\dd\theta.
\]
The map \(q\mapsto A_q\) is locally Lipschitz from \(C^3\) to
\(C^2\) when \(q'\) has a common positive lower bound.
Indeed, its derivatives through order two are finite sums of
products of derivatives of \(q\) through order three and inverse
powers of \(D_q\); the denominators are uniformly bounded away from
zero. Thus \eqref{eq:qconvergence} implies
\[
 \sup|B_{q_t}-B_{q_\infty}|
 \le C_Le^{-\lambda t}.
\]

The distributional identity
\[
 s^2(s-\ii0)^{-2}=1
\]
now shows that the difference of the two singular pullbacks is the
continuous kernel
\[
 -\frac1{4\pi}
             \bigl(B_{q_t}(m,s)-B_{q_\infty}(m,s)\bigr).
\]
The \(q_\infty\) pullback is \(C_\kappa\) by
Lemma~\ref{lem:pullback}. This proves the first error term.

The smooth remainder contributes
\[
 p'(t+x)p'(t+y)
       R(p(t+x),p(t+y)).
\]
On the fixed square, \(p'(t+x)=O_L(e^{-\kappa t})\), and \(R\) is
bounded near the limiting affine point. This contribution is
\(O_L(e^{-2\kappa t})\), proving \eqref{eq:localerror}.
Integrating the continuous kernel against
\(\overline f\otimes g\) proves \eqref{eq:smearederror}.
\end{proof}

\begin{remark}[The precise unpassed collapse gate]
Theorem~\ref{thm:peeling} assumes the peeling estimates and the
affine regularity statement. It does not derive them from a stellar
collapse metric. It also controls compact time tests, whereas a
strictly positive-frequency packet generally has noncompact time
support. Passing from \eqref{eq:smearederror} to a collapse packet
number limit requires additional tail estimates and a controlled
identification of the propagated observable.

Likewise, a generic statement that a bulk state is Hadamard does not,
without a characteristic restriction theorem, supply
\eqref{eq:affineremainder}. These are genuine hypotheses, not
consequences of terminology.
\end{remark}

The distinction between a local scaling calculation and the full
collapse observable is consistent with the role of scaling limits in
the Fredenhagen--Haag approach \cite{FredenhagenHaag1990}. Their
collapse analysis is an external comparison, not an unproved internal
premise of this paper.

\section{Why the supplied landmarks do not substitute for this proof}
\label{sec:audit}

\subsection{A documentary audit, not a universal impossibility theorem}

A complete certificate for the observable studied here needs:
\begin{enumerate}[label=\textup{(\arabic*)}]
 \item a named background and normalized time;
 \item a named quantum field;
 \item a specified state with an admissible two-point distribution;
 \item a defined number or detector observable;
 \item a scattering or propagation identification;
 \item an evaluated expectation and any required regulator or
 late-time limit.
\end{enumerate}
The supplied record does not exhibit this certificate in any of the
pre-existing landmarks discussed below. This is a restricted
documentary conclusion. It does not assert that no future
construction, or no unexamined manuscript, can provide it.

\renewcommand{\arraystretch}{1.14}
\begin{longtable}{@{}p{0.23\textwidth}p{0.70\textwidth}@{}}
\toprule
Landmark or cluster & Supplied result and the non-transfer boundary\\
\midrule
\endfirsthead
\toprule
Landmark or cluster & Supplied result and the non-transfer boundary\\
\midrule
\endhead
\bottomrule
\endfoot

I--V; I \texttt{thm:main}
&
The excerpts specify finite Born semantics and distinct
representation or instrument problems. They do not construct the
Schwarzschild wave equation, the affine horizon state, or its
number expectation. A measurement rule consumes an observable
and channel; it does not select the field state.\\

VI--VIII; VII-F \texttt{prop:spacelike};
VIII-B \texttt{thm:dictionary};
VIII-C \texttt{thm:C5}
&
The supplied theorems concern relational descent, a specified conical
family, spacelike slices, and support for prescribed block channels.
The cited constructions select neither the black-hole horizon nor
the state used in this paper. A backward channel-support set is not
identified with a spacetime causal past or an island.\\

IX--X
&
According to the supplied summary, these are sharp-product and
ramp-convolution models. Their remainder coefficients are not
identified with an asymptotic scalar number covariance.\\

XI-A \texttt{lem:compatibility}
&
The excerpt proves the classical necessary condition
\((\partial_\mu g_{\mathrm{XI}})T_0^{\mu\nu}=0\).
Here \(g_{\mathrm{XI}}\) denotes the inherited source coefficient,
not the metric \(\mathrm g\) of \eqref{eq:metric}.
The condition is not a state, a Bogoliubov calculation, or a
stress-tensor expectation.\\

XII--XV; XIV-I, XIV-J; XV-B
&
The excerpts distinguish defined internal kernels, partial
off-characteristic references, and the unresolved full characteristic
map. XIV-J stops at J2 without quantum admission. XV-B exists as a
technical interface manuscript but does not supply the inherited full
map or its admission. None of these statements constructs
\eqref{eq:numberdensity}.\\

XVI-N; XVI-O
&
The supplied obstruction in XVI-N is conditional on its declared
prepared-source class. XVI-O constructs a kernel subsector under a
specified prescription while retaining bulk and insertion limits.
A constructed kernel Ward subsector is not bulk symmetry restoration,
and a counterterm is not the present emission observable.
O.1 remains \texttt{NOT\_CLOSED} in the supplied record.\\

XVI-P
&
The supplied summary separates classical vertex and gauge-map
verification from later quantum evaluation. Its full text is not
verified here. The separation is a useful correction-program
discipline, not a free-field emission theorem.\\

Em02-D \texttt{thm:main}
&
The excerpt gives a fixed seven-row inverse-cubic distributional
asymptotic and a separated divergence witness. Its model is not
identified with the state and scattering problem of this paper.\\

Em02-E \texttt{thm:release}
&
The supplied summary identifies a classical quadratic bridge on a
power-law inflationary background and an incomplete quantum audit.
Its full theorem is not independently checked here. Neither a
relational lapse nor a classical quadratic kernel specifies the
Unruh state.\\

En02-C
&
The supplied summary distinguishes recovery from a restricted
streaming error and refuses a Page-curve interpretation. The full
manuscript is not independently checked here. Recovery and entropy
claims do not determine a frequency occupation density.\\

En02-D \texttt{thm:D1}, \texttt{thm:D2}
&
The supplied summary describes energy-sector isometries and a
quadrature estimate against an external rate. Those theorem texts are
not reproduced in the supplied excerpts. The rate is an input there;
in \eqref{eq:rate} it is the output of a field-state and scattering
calculation.\\
\end{longtable}

\subsection{Classical data do not select a flux}

\begin{proposition}[State underdetermination]
\label{prop:underdetermination}
The fixed geometry, free wave equation, and stationary scattering
matrix do not uniquely determine the outgoing occupation density.
\end{proposition}

\begin{proof}
Keep the metric, wave equation, scattering matrix, and infinity-input
vacuum fixed. On any compact positive-frequency band, one admissible
incoming horizon state is the Killing-frequency vacuum. For that
choice, both incoming number covariances vanish, and
\eqref{eq:quantumscatter} gives zero outgoing number covariance.

A second choice is the affine horizon vacuum restricted to the
exterior half-horizon. Lemmas~\ref{lem:fourier} and
\ref{lem:horizonoccupation} give its positive occupation
\(\nh(\omega)\), and Proposition~\ref{prop:packet} gives
\(\Gamma_\ell(\omega)\nh(\omega)>0\).

Thus the same classical scattering data admit different outgoing
occupations. The first choice is not asserted to have the same future
horizon regularity as the second. That difference is precisely why a
state and its regularity requirements must be part of a derivation.
\end{proof}

This proposition excludes the inference from ``a classical scalar
action exists'' to ``the Hawking flux has been proved.'' It does not
exclude a derivation after the missing quantum state data are supplied.

\subsection{Energy-preserving finite isometries do not select a spectrum}

\begin{proposition}[Finite-sector underdetermination]
\label{prop:finite}
Energy preservation and the isometry condition alone do not select a
Hawking occupation law or a dimensional emission rate.
\end{proposition}

\begin{proof}
Fix a source energy \(E\), a source degeneracy label
\(a=1,\ldots,d\), and permitted radiation energies
\(\omega_1,\ldots,\omega_k\le E\).
Choose orthonormal daughter--radiation states
\[
 |E-\omega_j,a\rangle_{\mathrm D}
                  \otimes|\omega_j,j\rangle_{\mathrm R}
\]
whose total energy is \(E\). The additional label \(j\) may distinguish
degenerate radiation sectors.

For every probability vector \(w=(w_1,\ldots,w_k)\), define
\[
 V_w|E,a\rangle
 =
 \sum_{j=1}^k\sqrt{w_j}\,
       |E-\omega_j,a\rangle_{\mathrm D}
                     \otimes|\omega_j,j\rangle_{\mathrm R}.
\]
Orthogonality gives
\[
 V_w^*V_w=I,
 \qquad
 (H_{\mathrm D}+H_{\mathrm R})V_w=V_wH_{\mathrm S}.
\]
The radiation-energy probabilities are exactly \(w_j\).

For \(k\ge2\), two different probability vectors give different
spectra while preserving both displayed identities. Nothing in these
identities singles out a Bose factor or a transmission probability.
Moreover, no duration or evolution schedule has been specified, so
they cannot determine a rate per unit time.
\end{proof}

The proposition explains the logical direction of calibration:
a finite operational model may consume an independently derived
emission law, but an energy-sector dilation does not produce that law
from its isometry condition.

\section{Debugging the proposed identifications}
\label{sec:debug}

The following corrections are part of the release.

\begin{enumerate}[label=\textbf{D\arabic*.},leftmargin=*]
 \item \textbf{A stationary exterior is not a collapse geometry.}
 The main theorem fixes Schwarzschild and a particular horizon
 boundary state. It does not replace the collapse initial-value
 problem by a metric name. The conditional local transfer theorem
 states its peeling hypotheses separately.

 \item \textbf{Hadamard is a regularity condition, not a complete state
 specification.}
 The selected state is the affine-horizon/Killing-infinity product
 state. Bulk Hadamard regularity is supplied by the declared external
 theorem. An arbitrary Hadamard state is not automatically this state.

 \item \textbf{An apparent horizon is not by itself sufficient
 thermal data.}
 A nonstationary foliation does not automatically provide the
 normalized Killing generator, affine-frequency state, and
 propagation estimates used here. A Vaidya extension needs its own
 precise hypotheses.

 \item \textbf{Horizon detailed balance is not global Unruh KMS
 equilibrium.}
 The incoming infinity channel is in its vacuum.
 Greybody scattering produces the occupation
 \(\Gamma_\ell\nh\), not a generally KMS outgoing state.

 \item \textbf{A continuum mode is not a finite number observable.}
 The distribution \eqref{eq:numberdensity}, the packet observable
 \eqref{eq:packet}, and the mode-window counting convention
 \eqref{eq:windowcount} are different objects with their
 normalizations made explicit.

 \item \textbf{A number observable does not require a bulk
 stress-tensor calculation.}
 The number covariance has been evaluated directly. The boundary
 subtraction is not represented as a four-dimensional
 \(\langle T_{ab}\rangle_{\mathrm{ren}}\) calculation. No bulk
 stress tensor is evaluated in this paper.

 \item \textbf{A sufficient product criterion failing is not universal
 nonexistence.}
 The supplied XIV--XV unresolved gates are retained as unresolved,
 not promoted to proofs that every characteristic construction is
 impossible.

 \item \textbf{The XVI obstruction is not automatically transported
 to another background.}
 Changing the metric neither proves a completion of the old
 prepared insertion nor proves that the old conditional obstruction
 applies unchanged to a new insertion. Either transfer requires a
 hypothesis-by-hypothesis identification.

 \item \textbf{A restored kernel subsector is not restored bulk
 symmetry.}
 The supplied XVI-O record distinguishes its constructed subsector,
 bulk matched set, and inherited insertion. Those verdicts are not
 collapsed.

 \item \textbf{A rate does not determine a microscopic channel.}
 Even after \eqref{eq:rate}, a finite consumer must specify time
 scheduling, multiple-emission rules, energy cutoffs, coherent
 amplitudes, and dilation data. Decoder performance and an entropy
 curve do not provide those choices.

 \item \textbf{Package checks are not proofs.}
 No numerical test, archive manifest, compilation flag, or
 machine-checking certificate is used as a premise here. The
 manuscript provides mathematical arguments; it does not claim
 formal verification.
\end{enumerate}

\section{Release and the remaining programmes}
\label{sec:release}

\begin{theorem}[Em03-A mathematical release]
\label{thm:release}
For the declared fixed-background free-scalar model:

\begin{enumerate}[label=\textup{(\roman*)}]
 \item surface gravity is computed from the normalized Schwarzschild
 metric;
 \item the affine horizon vacuum is specified before the thermal
 calculation;
 \item its Killing-time covariance and distributional Fourier limit
 are proved;
 \item the exact positive-frequency transmission probability is
 constructed from the Schwarzschild radial equation;
 \item the outgoing compact-band number expectation and its
 mode-window rate are proved;
 \item approximation errors are controlled by
 Corollary~\ref{cor:error};
 \item under the additional assumptions of
 Section~\ref{sec:peeling}, a compact-test late-time covariance
 limit has the proved remainder
 \eqref{eq:smearederror}.
\end{enumerate}

The bulk existence and Hadamard identification are the declared
external input. No collapse packet limit, all-mode luminosity,
backreaction, mass-loss equation, quantum-gravity correction, island,
Page curve, microscopic evaporation channel, or inherited
characteristic-coefficient admission is established by this release.
\end{theorem}

\begin{proof}
Assertions (i)--(v) are Theorem~\ref{thm:main}.
Assertion (vi) is Corollary~\ref{cor:error}.
Assertion (vii) is Theorem~\ref{thm:peeling}, with its hypotheses
retained.

The proof contains no construction of a dynamical metric, graviton
insertion, island extremization, entropy reconstruction, or inherited
characteristic graph. The compact-test transfer theorem explicitly
does not identify a noncompact collapse number packet. None of the
unasserted conclusions follows as an additional part of the proved
statements.
\end{proof}

\subsection{Programme A: a genuine collapse packet theorem}

A successor must fix one collapse spacetime, its initial state, and a
propagated positive-frequency packet observable. It must then prove:
\begin{enumerate}[label=\textup{(\arabic*)}]
 \item the affine or characteristic identification used by the packet;
 \item the relevant two-point regularity after characteristic
 restriction;
 \item the peeling and propagation estimates;
 \item tail control sufficient for the noncompact packet smearing;
 \item convergence to \eqref{eq:packetlaw}, with a stated error.
\end{enumerate}
Theorem~\ref{thm:peeling} supplies a local comparison tool only after
its assumptions have been verified. It is not the missing global
collapse proof.

\subsection{Programme B: one relational measurement}

A relational successor may replace or implement the asymptotic packet
by one declared smearing or detector. It must specify the rods or
coordinates, prove descent on the chosen classical and quantum
domains, and identify the represented observable and its coupling.
Compact smearing does not automatically prove compact support after
a nonlocal coordinate response.

The earlier relational sheets may inform the methodology. They are
not formal premises unless their domains, transformations, and
observables are explicitly matched. A free scalar on a fixed metric
does not inherit a graviton gauge-fixing obligation merely because a
later theory will include metric fluctuations.

\subsection{Programme C: one controlled effective correction}

Only after the base observable has been identified can a correction
to that observable be stated unambiguously. A correction sheet must
declare:
\begin{enumerate}[label=\textup{(\arabic*)}]
 \item the effective action, background, field content, and order;
 \item a verified vertex and dependency inventory;
 \item the gauge-fixing and state prescription where relevant;
 \item the admissible bulk and composite counterterms;
 \item the evaluated insertion, its regulator removal, and the
 applicable Ward identities;
 \item a common observable and error topology for comparing the
 corrected and free results.
\end{enumerate}
These are requirements for a new proof. The present scalar calculation
does not close XVI-O O.1, discharge the XVI-N prepared-insertion
obstruction, or admit the separate XIV--XV characteristic coefficient.

\subsection{Programme D: operational consumers}

A finite energy-sector model may now use the rate
\eqref{eq:rate} as a derived input rather than an external table,
provided its calibration procedure and all required errors are
specified. Its energy-sector isometry and implementation theorems must
still hold for the chosen weights and schedule.

An operational recovery theorem, a memory lower bound, or a matching
entropy shape remains a statement about that finite channel. None
reverses Proposition~\ref{prop:finite} or reconstructs the field-state
calculation that produced the rate.

\subsection{Final scope statement}

Relative to the supplied pre-existing record, no cited landmark
provides the completed certificate for the observable studied here.
This paper adds a new, explicitly specified fixed-background
free-field calculation. It derives the semiclassical emission law
within that model, using a declared external bulk state-construction
theorem and proofs of the boundary thermal and scattering
calculations.

It does not derive quantum-gravitational evaporation. It does not
rename a conical dictionary, a classical lapse, a prepared
counterterm, a decoder, or an entropy curve as Hawking radiation.

\begin{thebibliography}{9}

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

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

\bibitem{DMP2011}
C.~Dappiaggi, V.~Moretti, and N.~Pinamonti,
\emph{Rigorous construction and Hadamard property of the Unruh state
in Schwarzschild spacetime},
Advances in Theoretical and Mathematical Physics
\textbf{15} (2011), 355--447.
Preprint: \texttt{arXiv:0907.1034}.
The external theorem declared in
Section~\ref{subsec:external} uses the bulk-to-boundary construction
and the Hadamard theorem; it is not reproved in this manuscript.

\bibitem{FredenhagenHaag1990}
K.~Fredenhagen and R.~Haag,
\emph{On the derivation of Hawking radiation associated with the
formation of a black hole},
Communications in Mathematical Physics
\textbf{127} (1990), 273--284.
DOI: \texttt{10.1007/BF02096757}.

\bibitem{HollandsWald2015}
S.~Hollands and R.~M.~Wald,
\emph{Quantum fields in curved spacetime},
Physics Reports
\textbf{574} (2015), 1--35.
Preprint: \texttt{arXiv:1401.2026}.
DOI: \texttt{10.1016/j.physrep.2015.02.001}.

\bibitem{SuppliedRecord}
\emph{Supplied manuscript excerpts and companion summaries},
corpus identifier \texttt{ca99b12b},
retrieval identifier \texttt{retrieval-3},
as reproduced with the problem statement.
This is the documentary source for the internal audit, not an
independently verified complete manuscript registry.
Internal labels in Section~\ref{sec:audit} are retained source
locators. The summaries concerning Em02-E, En02-C, En02-D, and XVI-P
are not represented as full-text verification.

\end{thebibliography}

\end{document}