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% EVIDENCE BOUNDARY
% Analytical constructions and proofs are supplied.
% Compilation, machine verification, numerical execution, experimental
% calibration, and laboratory implementation are not asserted.
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% SCOPE REPAIR
% The proved packet correspondence is for one massless s-wave packet.
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% region grows with the spatial approximation parameter.
% It is not a pointlike detector or a fixed-zone arbitrary-accuracy
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% External inputs are declared explicitly:
%   - complete stationary scalar scattering;
%   - Green operators and local linear system--probe scattering;
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% packet-matching assumption.
%=====================================================================

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\hypersetup{
  pdftitle={Em03-D: A Local System--Probe Realization
            of a Hawking-Packet Observable},
  pdfauthor={Ngo Cheung},
  pdfsubject={Compact scalar coupling, outgoing packet matching,
              bounded Weyl effects, and quantitative errors}
}

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\title{\textbf{Em03-D}\\[1mm]
\textbf{A Local System--Probe Realization
of a Hawking-Packet Observable}\\[2mm]
\large Compact coupling, infrared-controlled bulk approximation,
and a bounded Weyl effect}

\author{Ngo Cheung\\
Independent Researcher\\
Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}}

\date{Version 1 --- 8 October 2026}

\begin{document}

\maketitle

\begin{abstract}
We construct a compactly coupled scalar probe scheme for one bounded
outgoing Hawking-packet quadrature effect on the Schwarzschild
exterior. The result is restricted to a massless s-wave packet and
uses full angular-shell access.

Complete stationary scattering identifies the packet with a
two-channel bulk coefficient. We construct real, compactly supported
bulk tests approaching that coefficient in the Unruh covariance
norm. A time-derivative filter permits a quantitative spatial
Cauchy-data truncation without assuming that the thermal
low-frequency weight is harmless. Separately, the explicit
zero-energy s-wave solutions give a local infrared bound for
arbitrary compact reduced tests.

A second massless scalar is coupled to the system by a smooth compact
bilinear interaction. We calculate the exact induced Weyl effect and
bound its deviation from the selected compact bulk test by a convergent
advanced-Green-operator construction. Residual probe noise is retained.
A finite-dimensional symplectic squeezing of the probe preparation
suppresses that noise while keeping the probe readout a bounded
effect; dividing an effect by its attenuation coefficient is not used.

The resulting expectation comparison is uniform on an explicitly
defined class of regular system states with dominated second moments.
It specializes to the inherited Unruh baseline and has a constructive
resource-dependent accuracy schedule. The coupling region expands
with the packet approximation, the readout test grows at weak coupling,
and a conjugate probe variance diverges along the noise-suppression
schedule.

This is a local field--probe observable correspondence, not a
pointlike detector, a packet-number instrument, coherent state
transfer, or channel equivalence with the effective apparatus of
Em03-C. No collapse geometry, microscopic evaporation dynamics,
or quantum-gravity correction is constructed.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Scope, target, and declared external inputs}
\label{sec:scope}
%=====================================================================

The construction follows
\[
\boxed{
\text{compact local coupling}
\longrightarrow
\text{exact induced Weyl effect}
\longrightarrow
\text{quantitative outgoing-packet comparison}.
}
\]

Its target is one effect, not the two-memory apparatus of Em03-C
\cite{Em03C}. For a normalized outgoing packet \(h\), set
\[
Q_h=\frac{a_{\mathrm{out}}(h)+a_{\mathrm{out}}(h)^\dagger}{\sqrt2},
\qquad
E_{h,\vartheta}
=\frac12\left(I+\Repart e^{\ii\vartheta Q_h}\right),
\qquad
\vartheta\in\R\setminus\{0\}.
\]
Since the real part of a unitary has spectrum in \([-1,1]\),
\[
0\le E_{h,\vartheta}\le I.
\]

We prove a state-class comparison of the form
\[
\left|
(\omega\otimes\sigma_{R,\epsilon})
 \left[\Theta_{R,\epsilon}
       (I\otimes O_{R,\epsilon})\right]
-\omega(E_{h,\vartheta})
\right|
\le \epsilon_{\mathrm{local}}.
\]
Here \(R\) is a spatial approximation parameter and
\(\epsilon>0\) is a coupling parameter. The probe state,
coupling profile, and readout test are specified below.

The scheme uses a full spherical shell. For each finite \(R\) its
coupling is compactly supported strictly outside the horizon.
No fixed shell is claimed to realize arbitrary packet accuracy.

\subsection{What is inherited}

The selected system state is the massless scalar Unruh prescription
of Em03-A \cite{Em03A}. Its horizon input is thermal in normalized
Killing frequency, its infinity input is vacuum, and the two inputs
are independent and centered quasifree. The outgoing packet mean is
\[
\nu_h=\int_0^\infty |h(\omega)|^2
 \frac{\Gamma_0(\omega)}{e^{\betah\omega}-1}\dd\omega,
\qquad
\betah=8\pi GM.
\]
The subscript \(0\) denotes the s-wave greybody probability.

The one-mode Gaussian characteristic function used here follows
from this prescription and the finite-mode construction in
Em03-C. It does not require a density operator for the entire
stationary continuum.

The effective Markov port, capture controls, and finite registers
of Em03-B and Em03-C are not premises of the local construction.

\subsection{External theorems actually used}

\begin{externaltheorem}[Stationary scalar scattering]
For the massless scalar on the Schwarzschild exterior, the past and
future horizon and infinity channels give complete stationary
scattering. In the s-wave reduction, the incoming generalized
eigenfunctions define a unitary spectral transform
\[
\mathcal U:L^2(\R,\dd x)
 \longrightarrow L^2((0,\infty),\dd\omega;\C^2)
\]
intertwining the radial operator with multiplication by
\(\omega^2\). The flux-normalized stationary scattering matrix is
unitary.
\end{externaltheorem}

We use the complete wave-scattering result of Dimock
\cite{Dimock}, together with the ordinary one-dimensional spectral
resolution for a real short-range potential \cite{DeiftTrubowitz}.
Baskin--Wang provides a complementary geometric account of the
two radiation ends and their energy normalization
\cite{BaskinWang}. The exact mode and test normalizations used below
are displayed explicitly.

\begin{externaltheorem}[Green operators and local linear scattering]
A formally self-adjoint normally hyperbolic operator on a globally
hyperbolic spacetime has the advanced and retarded Green operators
and Cauchy evolution used below. Adding a smooth compactly supported
symmetric zeroth-order matrix coupling gives a well-posed coupled
theory and a symplectic scattering map. It induces the corresponding
automorphism of the Weyl algebra.
\end{externaltheorem}

The PDE input is the standard Green-operator and Cauchy theory
\cite{BGP}. The local system--probe construction and its scattering
interpretation are those of Fewster--Verch \cite{FewsterVerch}.
Its Weyl-algebra version is described by Fewster--Jubb--Ruep
\cite{FewsterJubbRuep}. We give the particular test transformation
and estimates required here.

\begin{externaltheorem}[Bulk state identification]
The affine-horizon and Killing-infinity product prescription
defines a quasifree Hadamard Unruh state on the Schwarzschild
exterior.
\end{externaltheorem}

This is the Dappiaggi--Moretti--Pinamonti construction
\cite{DMP}, already declared in Em03-A. Its global microlocal proof
is not reproduced.

None of these external statements supplies the compact
bulk-to-packet approximation proved in Section~\ref{sec:packetbridge}.

\paragraph{Evidence boundary.}
The manuscript supplies analytical constructions. Compilation,
formal verification, numerical evaluation, calibration, and
experimental realization are not asserted.

%=====================================================================
\section{Geometry, signs, and canonical radiation coordinates}
\label{sec:geometry}
%=====================================================================

\subsection{The exterior and dimensionless estimates}

In physical units,
\[
\mathcal E=\R_t\times(\rh,\infty)_r\times S^2,
\qquad
\rh=2GM,
\]
\[
g=-F(r)\dd t^2+F(r)^{-1}\dd r^2+r^2\dd\Omega^2,
\qquad F(r)=1-\frac{\rh}{r}.
\]
The time generator is normalized at infinity.

For the PDE estimates we measure lengths in units of \(\rh\):
\[
\widetilde t=t/\rh,\qquad
\widetilde r=r/\rh,\qquad
\widetilde\omega=\rh\omega.
\]
The packet rescales as
\[
\widetilde h(\widetilde\omega)
=\rh^{-1/2}h(\widetilde\omega/\rh).
\]
This preserves both packet normalization and its annihilator.
After this paragraph, tildes are suppressed:
\[
\rh=1,\qquad
F(r)=1-r^{-1},\qquad
x=r+\log(r-1),\qquad
\beta=4\pi.
\]
The occupation integrals are unchanged by the rescaling.
All constants in the estimates are dimensionless in this convention.

The physical coupling associated with a dimensionless profile
\(\epsilon\rho\) is \(\rh^{-2}\epsilon\rho\).
A dimensionless scalar smearing is converted to its physical
smearing with the corresponding factor \(\rh^{-3}\).
Thus the profile choices below do not equate quantities of
different physical dimensions.

The exterior is globally hyperbolic, with \(t=\mathrm{constant}\)
Cauchy surfaces. Indeed, in the conformally related optical metric,
a causal curve has \(|\dd x/\dd t|\le1\). On every bounded
\(x\)-interval the angular optical geometry is regular and compact.
An inextendible causal curve cannot escape the spatial geometry at
finite \(t\), and therefore crosses every time slice once.

\subsection{Wave operator and s-wave reduction}

We use
\[
P=-\Box_g.
\]
For both the real system field \(\phi\) and real probe field \(\psi\),
the free action is
\[
S_0=-\frac12\int
 \left(\nabla_a\phi\nabla^a\phi+
       \nabla_a\psi\nabla^a\psi\right)\dd\mathrm{vol}_g.
\]
Both fields are massless.

Let \(Y_0=(4\pi)^{-1/2}\). Writing
\[
\phi(t,r,\Omega)=\frac{Y_0}{r}\Psi_\phi(t,x)
\]
in the selected angular sector gives
\[
P\left(\frac{Y_0}{r}u\right)
=\frac{Y_0}{rF}\Dop u,
\qquad
\Dop=\partial_t^2+\Lop,
\qquad
\Lop=-\partial_x^2+V(x),
\qquad
V(x)=\frac{F(r(x))}{r(x)^3}.
\]

For a real reduced test \(f(t,x)\), define its bulk representative
\begin{equation}
\mathfrak F[f](t,r,\Omega)
=\frac{Y_0}{rF}f(t,x).
\label{eq:bulkrepresentative}
\end{equation}
Then
\[
\phi(\mathfrak F[f])
=\int_{\R^2}f(t,x)\Psi_\phi(t,x)\dd t\,\dd x.
\]
This follows from
\(\dd\mathrm{vol}_g=r^2F\,\dd t\,\dd x\,\dd\Omega\).
Compact reduced support gives compact bulk support outside the horizon.

The causal propagator convention is
\[
E=G_{\mathrm A}-G_{\mathrm R}.
\]
It is consistent with
\[
[\phi(f),\phi(g)]
=\ii\int f\,Eg\,\dd\mathrm{vol}_g.
\]
For \(\Dop=\partial_t^2+\Lop\), the retarded solution has the
usual positive sine kernel. The advanced-minus-retarded
propagator has the corresponding negative sine kernel.

\subsection{Incoming modes and the weighted coefficient space}

Let \(f_+(x,\omega)\sim e^{\ii\omega x}\) at \(+\infty\) and
\(f_-(x,\omega)\sim e^{-\ii\omega x}\) at \(-\infty\) be Jost
solutions. Define
\[
W(\omega)=
f_-(x,\omega)f_+'(x,\omega)
-f_-'(x,\omega)f_+(x,\omega),
\]
\[
\tau(\omega)=\frac{2\ii\omega}{W(\omega)},\qquad
q_{\mathrm H}=\tau f_+,\qquad
q_{\mathrm I}=\tau f_-.
\]
These are the horizon-incident and infinity-incident modes.
The infinity-incident mode has an outgoing infinity reflection
amplitude denoted \(\rho_{\mathrm I}(\omega)\).

The incoming spectral transform is
\[
(\mathcal Uu)_j(\omega)
=\frac1{\sqrt{2\pi}}
  \int_\R\overline{q_j(x,\omega)}u(x)\dd x,
\qquad j\in\{\mathrm H,\mathrm I\}.
\]
It is understood in its unitary \(L^2\) extension.

The canonical field expansion is
\[
\Psi_\phi(t,x)
=\sum_j\int_0^\infty
 \frac{q_j(x,\omega)a_j(\omega)e^{-\ii\omega t}
       +\overline{q_j(x,\omega)}a_j(\omega)^\dagger
          e^{\ii\omega t}}
      {\sqrt{4\pi\omega}}\dd\omega.
\]
Consequently a compact real reduced test has coefficient
\begin{equation}
k_j(f)(\omega)
=\frac1{\sqrt{4\pi\omega}}
 \int_{\R^2}
 f(t,x)\overline{q_j(x,\omega)}e^{\ii\omega t}
                                         \dd t\,\dd x,
\label{eq:testcoefficient}
\end{equation}
so that
\[
\phi(\mathfrak F[f])=X(k(f)),\qquad
X(k)=a(k)+a(k)^\dagger.
\]

Annihilators are conjugate-linear in their tests:
\[
[a(k),a(l)^\dagger]=\langle k,l\rangle.
\]
Define the real symplectic form and Weyl convention by
\[
s(k,l)=2\Impart\langle k,l\rangle,\qquad
W(k)=e^{\ii X(k)},
\]
\begin{equation}
W(k)W(l)=e^{-\ii s(k,l)/2}W(k+l).
\label{eq:Weyl}
\end{equation}

The Unruh second-moment norm is
\begin{equation}
\|k\|_{\mathrm U}^2
=\int_0^\infty
 \left[(1+2n_{\mathrm H}(\omega))|k_{\mathrm H}(\omega)|^2
                         +|k_{\mathrm I}(\omega)|^2\right]\dd\omega,
\qquad
n_{\mathrm H}(\omega)=(e^{\beta\omega}-1)^{-1}.
\label{eq:Unruhnorm}
\end{equation}
For real linear observables,
\[
\stateU(X(k)^2)=\|k\|_{\mathrm U}^2,\qquad
\stateU(W(k))=e^{-\|k\|_{\mathrm U}^2/2}.
\]
The norm dominates the ordinary two-channel \(L^2\) norm.

%=====================================================================
\section{An explicit local infrared and covariance estimate}
\label{sec:infrared}
%=====================================================================

This section is specific to the s-wave potential. No higher-angular
threshold estimate is silently imported.

\begin{lemma}[Zero-energy normalization]
\label{lem:threshold}
The potential \(V\) is smooth, nonnegative, and satisfies
\[
\int_\R V(x)\dd x=\frac12,\qquad
\int_\R(1+|x|)V(x)\dd x<\infty.
\]
Its zero-energy Jost solutions are
\[
f_-(x,0)=r(x),\qquad
f_+(x,0)=-r(x)\log F(r(x)),
\]
and
\[
W(0)=-1.
\]
There is an \(\omega_c>0\) for which
\[
|W(\omega)|\ge\frac12
\quad (0\le\omega\le\omega_c).
\]
\end{lemma}

\begin{proof}
Since \(\dd x=F^{-1}\dd r\),
\[
\int V\dd x=\int_1^\infty r^{-3}\dd r=\frac12.
\]
Near the horizon, \(x\) grows logarithmically in \(r-1\);
at infinity, \(x=r+O(\log r)\). These facts prove the first-moment
bound.

Substitution in \(\Lop f=0\) verifies the two solutions.
They tend to one at their respective normalization ends.
Using \(\partial_xr=F\) and \(F'=r^{-2}\) gives their
Wronskian \(-1\).

The Jost Volterra equations have the zero-frequency kernel
\(y-x\) in the right equation and its left-end analogue.
The first-moment bound gives convergent series and continuity
at \(\omega=0\), including derivatives on compact \(x\)-sets.
The nonzero Wronskian therefore gives the final assertion.
\end{proof}

Choose one such \(\omega_c\). For \(X>0\), set
\[
m_X=\max\left\{
 \int_{-X}^\infty(y+X)V(y)\dd y,\,
 \int_{-\infty}^{X}(X-y)V(y)\dd y
\right\},
\]
\begin{equation}
C_X=\max\{4\omega_c e^{m_X},\,e^{1/(2\omega_c)}\}.
\label{eq:Jostconstant}
\end{equation}

\begin{lemma}[Compact-region mode bound]
\label{lem:modebound}
For \(j=\mathrm H,\mathrm I\), \(|x|\le X\), and \(\omega>0\),
\[
|q_j(x,\omega)|
\le C_X\min\{\omega/\omega_c,1\}.
\]
\end{lemma}

\begin{proof}
For \(0\le\omega\le\omega_c\), use
\(|\sin(\omega z)/\omega|\le|z|\) in the Jost Volterra series.
It gives \(|f_\pm(x,\omega)|\le e^{m_X}\) on the compact region.
Together with
\(|\tau|\le4\omega\), this proves the low-frequency bound.

For \(\omega\ge\omega_c\), the Volterra series gives
\[
|f_\pm(x,\omega)|
\le \exp(\|V\|_1/\omega_c)=e^{1/(2\omega_c)}.
\]
Current conservation gives \(|\tau|\le1\). This proves the
remaining bound.
\end{proof}

For a reduced test supported at \(|x|\le X\), define
\begin{equation}
\begin{split}
\mathcal B_X(f)^2
:=\frac{C_X^2}{2\pi}\Bigg[
&\left(\frac12+\frac{2}{\beta\omega_c}\right)\|f\|_1^2\\
&+\left(\frac1{2\omega_c^2}
             +\frac{2}{3\beta\omega_c^3}\right)
                                      \|\partial_tf\|_1^2
\Bigg].
\end{split}
\label{eq:compactcovariancebound}
\end{equation}
The \(L^1\) norms are over \(\R^2\).

\begin{proposition}[Explicit compact-test covariance bound]
\label{prop:compactcovariance}
Every compact smooth real reduced test satisfies
\[
\|k(f)\|_{\mathrm U}\le\mathcal B_X(f).
\]
In particular, its thermal infrared integral is finite.
\end{proposition}

\begin{proof}
At \(0<\omega\le\omega_c\), use
\eqref{eq:testcoefficient} and Lemma~\ref{lem:modebound}:
\[
|k_j(f)(\omega)|
\le\frac{C_X\sqrt\omega}{\sqrt{4\pi}\,\omega_c}\|f\|_1.
\]
At \(\omega\ge\omega_c\), integrate once by parts in \(t\):
\[
|k_j(f)(\omega)|
\le\frac{C_X}{\sqrt{4\pi}\,\omega^{3/2}}
                                      \|\partial_tf\|_1.
\]
Use
\(1+2n_{\mathrm H}(\omega)\le1+2/(\beta\omega)\)
for both channel bounds. Summing the two channel estimates and
performing the elementary integrals gives
\eqref{eq:compactcovariancebound}.
\end{proof}

This local threshold calculation is not a replacement of
unselected radiation by vacuum. Both incoming channels appear
in the estimate, with the actual horizon thermal weight.

%=====================================================================
\section{The outgoing target and a controlled state class}
\label{sec:target}
%=====================================================================

Fix a normalized
\[
h\in C_c^\infty((0,\infty)),\qquad
\supp h\subset[\omega_-,\omega_+],\qquad
0<\omega_-<\omega_+.
\]
Stationary scattering gives
\[
a_{\mathrm{out}}(h)
=a_{\mathrm H}(\overline\tau h)
 +a_{\mathrm I}(\overline{\rho_{\mathrm I}}h).
\]
Thus
\begin{equation}
k_*=\frac{\vartheta}{\sqrt2}
       (\overline\tau h,\overline{\rho_{\mathrm I}}h),
\qquad
\vartheta Q_h=X(k_*).
\label{eq:packetcoefficient}
\end{equation}
This is the coefficient of the outgoing scattering observable
in the bulk incoming coordinates. It is not a finite-radius
pointwise limit of an unsmeared quantum field.

Unitarity gives
\[
\|k_*\|_0^2=\vartheta^2/2,\qquad
\|k_*\|_{\mathrm U}^2
=\vartheta^2(\nu_h+1/2).
\]
Consequently
\begin{equation}
\boxed{
\stateU(E_{h,\vartheta})
=\frac12\left[1+
 e^{-\vartheta^2(2\nu_h+1)/4}\right].
}
\label{eq:Hawkingbaseline}
\end{equation}
There is no packet-count rate factor \(1/(2\pi)\) in this formula.

\begin{definition}[Second-moment-dominated system states]
\label{def:stateclass}
For \(C\ge1\), let \(\mathfrak S_C\) consist of normal states
in the regular Unruh representation of the completed
two-channel Weyl algebra for which
\[
\omega(X(k)^2)\le C\|k\|_{\mathrm U}^2
\]
on its real weighted coefficient domain. The second moment
includes any nonzero mean.
\end{definition}

The baseline belongs to \(\mathfrak S_1\).
The class is not a statement about all Hadamard states.

It is nontrivial beyond the baseline. A Weyl displacement by
\(l\) changes \(X(k)\) by a scalar of magnitude
\(|s(l,k)|\le2\|l\|_0\|k\|_{\mathrm U}\).
The displaced Unruh state therefore belongs to
\(\mathfrak S_{1+4\|l\|_0^2}\).

\begin{lemma}[Weyl expectation comparison]
\label{lem:Weylcomparison}
For \(\omega\in\mathfrak S_C\) and weighted coefficients \(k,l\),
\[
|\omega(W(k))-\omega(W(l))|
\le \sqrt C\,\|k-l\|_{\mathrm U}+\frac{|s(k,l)|}{2}.
\]
In particular,
\begin{equation}
\left|
\omega\left(\frac{I+\Repart W(k)}2\right)
-\omega\left(\frac{I+\Repart W(l)}2\right)
\right|
\le
\frac{\sqrt C+\|l\|_{\mathrm U}}2
                                      \|k-l\|_{\mathrm U}.
\label{eq:effectcomparison}
\end{equation}
\end{lemma}

\begin{proof}
Put \(d=k-l\). The Weyl relation expresses \(W(k)\) as
\(W(l)W(d)\) times a scalar phase of angle of magnitude
\(|s(k,l)|/2\).
Cauchy--Schwarz for the state and spectral calculus give
\[
|\omega(W(l)(W(d)-I))|
\le\sqrt{\omega((W(d)-I)^*(W(d)-I))}
\le\sqrt{\omega(X(d)^2)}.
\]
The phase costs at most \(|s(k,l)|/2\). This proves the first
inequality. Since
\[
|s(k,l)|=|s(d,l)|
\le2\|d\|_0\|l\|_0
\le2\|d\|_{\mathrm U}\|l\|_{\mathrm U},
\]
taking real parts and dividing by two proves the second.
\end{proof}

The estimate uses exact Weyl relations rather than a formal
unbounded-operator Baker--Campbell--Hausdorff manipulation.

%=====================================================================
\section{Bridge A: compact bulk tests approaching the packet}
\label{sec:packetbridge}
%=====================================================================

\subsection{Real Cauchy data for a chosen coefficient}

For real Cauchy data \((q,p)\), define the linear observable
\[
\mathcal L(q,p)
=\int_\R\bigl(p(x)\Psi_\phi(0,x)
                    -q(x)\partial_t\Psi_\phi(0,x)\bigr)\dd x.
\]
Its incoming coefficient is
\begin{equation}
k(q,p)(\omega)
=\frac{\mathcal Up(\omega)-\ii\omega\mathcal Uq(\omega)}
                              {\sqrt{2\omega}}.
\label{eq:Cauchycoefficient}
\end{equation}
For compact smooth data this follows directly from the canonical
field expansion. Its indicated spectral extensions follow by
continuity on their moment domains.

Let
\[
\mathcal C=\mathcal U\,(\text{complex conjugation})\,\mathcal U^{-1}.
\]
It is an antiunitary involution commuting with real functions of
\(\omega\). It incorporates the reality convention of the
incoming scattering modes; componentwise conjugation must not
be substituted for it.

Set
\[
k_-=\omega^{-2}k_*,
\qquad b=\sqrt{2\omega}\,k_-,
\]
and define
\begin{equation}
\mathcal Up=\frac{b+\mathcal Cb}{2},\qquad
\mathcal Uq=\frac{\mathcal Cb-b}{2\ii\omega}.
\label{eq:targetCauchydata}
\end{equation}
These give real \(q,p\), satisfy \(k(q,p)=k_-\), and have spectral
support in the compact positive band. They belong to every
integer Sobolev space, by spectral functional calculus and
elliptic regularity for \(\Lop\).

\subsection{Compact Cauchy truncation and time-slice test}

Choose a real \(\chi\in C_c^\infty((-2,2))\), equal to one on
\([-1,1]\). For \(R\ge1\), put
\[
q_R=\chi(x/R)q,\qquad p_R=\chi(x/R)p.
\]
Let \(s_R\) solve
\[
\Dop s_R=0,\qquad
s_R(0)=q_R,\qquad \partial_ts_R(0)=p_R.
\]
Choose a smooth time function \(\eta_T\), equal to zero for
\(t\le-T\) and one for \(t\ge T\), with \(T>0\). Define
\begin{equation}
f_{R,T}
=-\partial_t^2\Dop(\eta_Ts_R),\qquad
F_{R,T}=\mathfrak F[f_{R,T}].
\label{eq:compactpackettest}
\end{equation}
Because
\[
\Dop(\eta_Ts_R)=\eta_T''s_R+2\eta_T'\partial_ts_R,
\]
the reduced test is smooth and compact, with
\[
|t|\le T,\qquad |x|\le2R+T.
\]
The full support includes the angular sphere \(S^2\).

\begin{lemma}[Exact time-slice identity and derivative filter]
\label{lem:timeslice}
The reduced field observable of \(\Dop(\eta_Ts_R)\) is
\(\mathcal L(q_R,p_R)\). The coefficient of \(F_{R,T}\) is
\begin{equation}
k_R=\omega^2k(q_R,p_R).
\label{eq:kR}
\end{equation}
The time-slice identity is exact for every \(T>0\).
\end{lemma}

\begin{proof}
Pair the test with a homogeneous field solution and integrate
the \(\eta_T''\) term once by parts:
\[
\int \Dop(\eta_Ts_R)\Psi_\phi
=\int\eta_T'
  \bigl(\partial_ts_R\,\Psi_\phi
                     -s_R\,\partial_t\Psi_\phi\bigr).
\]
The integrated symplectic pairing is constant in time.
Since \(\int\eta_T'=1\), it equals \(\mathcal L(q_R,p_R)\).

Alternatively the advanced and retarded solutions of
\(\Dop(\eta_Ts_R)\) are \((\eta_T-1)s_R\) and \(\eta_Ts_R\),
respectively, confirming the sign convention.
In \eqref{eq:testcoefficient}, applying \(-\partial_t^2\)
to the test multiplies its coefficient by \(\omega^2\).
This proves \eqref{eq:kR}.
\end{proof}

The time filter is not an assertion that a compact time window
exactly selects a band-limited packet. It first acts on a
band-limited auxiliary coefficient \(k_-\), then controls the
low-frequency leakage caused by spatial Cauchy truncation.

\subsection{Explicit packet error}

Write
\[
\delta q_R=q_R-q,\qquad \delta p_R=p_R-p.
\]
Define
\begin{equation}
\begin{split}
\Delta_R^2={}&
 \|\Lop^{3/4}\delta p_R\|_2^2
 +\|\Lop^{5/4}\delta q_R\|_2^2\\
&+\frac2\beta\left(
 \|\Lop^{1/2}\delta p_R\|_2^2
 +\|\Lop\delta q_R\|_2^2\right).
\end{split}
\label{eq:packeterror}
\end{equation}
All fractional powers are those of the positive self-adjoint
radial operator.

For a bound using only ordinary spatial derivatives, put
\[
D_p(R)=\|\delta p_R\|_2^2+\|\Lop\delta p_R\|_2^2,
\]
\[
\begin{split}
D_q(R)={}&
 \|\delta q_R\|_2^2+\|\Lop\delta q_R\|_2^2
 +\|\partial_x(\Lop\delta q_R)\|_2^2\\
&+\int_\R V|\Lop\delta q_R|^2\dd x.
\end{split}
\]
Then
\begin{equation}
\Delta_R^2\le
(1+2/\beta)\bigl(D_p(R)+D_q(R)\bigr).
\label{eq:ordinarytailbound}
\end{equation}

\begin{theorem}[Quantitative bulk-to-packet bridge]
\label{thm:packetbridge}
The compact tests \eqref{eq:compactpackettest} satisfy
\[
\|k_R-k_*\|_{\mathrm U}\le\Delta_R,\qquad
\Delta_R\longrightarrow0.
\]
For every \(\omega\in\mathfrak S_C\),
\begin{equation}
\boxed{
\left|
\omega\left(\frac{I+\Repart\Wsys(F_{R,T})}{2}\right)
-\omega(E_{h,\vartheta})
\right|
\le H_C\Delta_R,
\qquad
H_C=\frac{\sqrt C+\|k_*\|_{\mathrm U}}2.
}
\label{eq:packetbridge}
\end{equation}
\end{theorem}

\begin{proof}
From \eqref{eq:Cauchycoefficient},
\[
k_R-k_*=
\omega^2
 \frac{\mathcal U\delta p_R-\ii\omega\mathcal U\delta q_R}
                                      {\sqrt{2\omega}}.
\]
Use
\[
|z-\ii\omega w|^2\le2|z|^2+2\omega^2|w|^2
\]
and \(1+2n_{\mathrm H}\le1+2/(\beta\omega)\).
The resulting integral is bounded by
\[
\int\left[
 (\omega^3+2\omega^2/\beta)|\mathcal U\delta p_R|^2
 +(\omega^5+2\omega^4/\beta)|\mathcal U\delta q_R|^2
\right]\dd\omega.
\]
Spectral unitarity gives precisely \eqref{eq:packeterror}.

For \(\lambda\ge0\),
\(\lambda^{3/2},\lambda\le1+\lambda^2\) and
\(\lambda^{5/2}\le1+\lambda^3\).
Also
\[
\|\Lop^{3/2}u\|_2^2
=\|\partial_x(\Lop u)\|_2^2+\int V|\Lop u|^2.
\]
These facts prove \eqref{eq:ordinarytailbound}.

Multiplication by \(\chi(x/R)\) tends to the identity in every
integer Sobolev norm on fixed Sobolev functions. This follows
from the \(L^2\) derivative tails and the bounds
\(\partial_x^j\chi(x/R)=O(R^{-j})\).
In particular,
\(\delta p_R\to0\) in \(H^2\) and
\(\delta q_R\to0\) in \(H^3\).
The potential and its required derivatives are bounded, so
\(D_p(R)+D_q(R)\to0\).

Finally,
\(\Wsys(F_{R,T})=W(k_R)\) in the common regular representation.
Apply Lemma~\ref{lem:Weylcomparison} with \(l=k_*\).
\end{proof}

This proves the packet bridge from bulk Cauchy evolution and
complete scattering. It does not assume the effective temporal
port of Em03-C.

The spatial truncation accounts for both radiation ends.
Its support moves toward spatial infinity and toward the horizon
as \(R\) increases. It is therefore not a fixed finite-radius,
outgoing-only detector construction.

%=====================================================================
\section{Bridge B: one compact local scalar coupling}
\label{sec:localcoupling}
%=====================================================================

Fix \(R,T\) and abbreviate
\[
f=f_{R,T},\qquad F=F_{R,T},\qquad
X=2R+T,\qquad K=\supp F.
\]

\subsection{Probe solution and future processing test}

Choose a spacelike-compact free probe solution \(v_{\mathrm{bulk}}\)
equal to the constant solution \(1\) on a neighborhood of \(K\).
Such a solution can be constructed explicitly.

At \(t_0=-T-1\), prescribe physical probe Cauchy position equal
to one on the spatial projection of \(J^-(K)\) at that time,
extend it smoothly with compact radial support, and prescribe
zero velocity. The constant function solves \(P1=0\).
Finite propagation and Cauchy uniqueness imply equality with
one on \(K\).

Write
\[
v_{\mathrm{bulk}}=\frac{Y_0}{r}v(t,x).
\]
Thus \(v=\sqrt{4\pi}\,r(x)\) on \(K\).

Choose \(\eta_+\) equal to zero for \(t\le T+1\) and one for
\(t\ge T+2\), and put
\[
g_{\mathrm{red}}=-\Dop(\eta_+v),\qquad
g=\mathfrak F[g_{\mathrm{red}}].
\]
This is a real compact test in the future processing slab.
Its advanced solution satisfies
\[
\Gadv g=(1-\eta_+)v_{\mathrm{bulk}},
\]
and therefore equals one on \(K\).

\begin{specification}[Local interaction and bounded probe effect]
\label{spec:local}
Let
\[
\rho=-F,\qquad \lambda=\epsilon\rho,\qquad \epsilon>0.
\]
The interaction is
\begin{equation}
S_{\mathrm{int}}
=-\epsilon\int_{\mathcal E}
                   \rho(x)\phi(x)\psi(x)\dd\mathrm{vol}_g.
\label{eq:localaction}
\end{equation}
The coupled equations are
\[
T_\epsilon
\begin{pmatrix}\phi\\\psi\end{pmatrix}=0,\qquad
T_\epsilon=
\begin{pmatrix}P&\epsilon\rho\\
                \epsilon\rho&P\end{pmatrix}.
\]
The postinteraction probe effect is
\begin{equation}
O_\epsilon=
\frac12\left(I+\Repart\Wprobe(g/\epsilon)\right).
\label{eq:probeeffect}
\end{equation}
The complementary outcome is \(I-O_\epsilon\).
Both outcomes are retained.
\end{specification}

The profile is smooth, compact, and spherically symmetric.
All angular sectors decouple in the coupled equations.
The selected probe test is an s-wave test, so it induces only
the selected system angular sector.

The readout effect itself is a specified final probe operation.
The manuscript does not construct its laboratory electronics.

\subsection{Exact scattering tests}

Let
\[
w_\epsilon=
G_{\mathrm A,T_\epsilon}
                         \begin{pmatrix}0\\g\end{pmatrix}.
\]
Define
\begin{equation}
F_\epsilon=-\rho(w_\epsilon)_\psi,\qquad
g_\epsilon=g/\epsilon-\rho(w_\epsilon)_\phi.
\label{eq:exactscatteredtests}
\end{equation}

\begin{theorem}[Exact induced Weyl effect]
\label{thm:induced}
The local scattering morphism obeys
\[
\Theta_\epsilon(I\otimes\Wprobe(g/\epsilon))
=\Wsys(F_\epsilon)\otimes\Wprobe(g_\epsilon).
\]
For a centered quasifree probe state with variance convention
\[
\sigma(\Wprobe(u))
=\exp[-V_\sigma(u,u)/2],
\]
the induced effect is
\begin{equation}
\boxed{
\varepsilon_{\sigma,\epsilon}(O_\epsilon)
=\frac12\left[
 I+c_\epsilon\Repart\Wsys(F_\epsilon)\right],
\qquad
c_\epsilon=e^{-V_\sigma(g_\epsilon,g_\epsilon)/2}.
}
\label{eq:inducedeffect}
\end{equation}
Its expectation in every system state is
\[
\omega(\varepsilon_{\sigma,\epsilon}(O_\epsilon))
=(\omega\otimes\sigma)
      \Theta_\epsilon(I\otimes O_\epsilon).
\]
The induced system test is compactly supported in \(K\).
\end{theorem}

\begin{proof}
For future-supported free test vectors \(H\), comparison of
advanced coupled and uncoupled solutions gives the test
transformation
\[
\theta_\epsilon H
=H-(T_\epsilon-P\oplus P)
                           G_{\mathrm A,T_\epsilon}H.
\]
Equivalently, pair the coupled solution with its advanced
test solution and use Green's identity in an early Cauchy
region, where both theories agree.
The test classes are those modulo the respective wave operators.

The transformation preserves the symplectic form: compare
the conserved coupled symplectic current on past and future
Cauchy surfaces outside \(K\).
It therefore gives the Weyl scattering morphism
\(\Theta_\epsilon(W(H))=W(\theta_\epsilon H)\).
This is the local linear scattering construction stated in
Section~\ref{sec:scope}.

Applying it to \(H=(0,g/\epsilon)\) yields
\eqref{eq:exactscatteredtests}.
System and probe Weyl operators belong to separate tensor
factors, so no additional mixed phase appears.
Apply \(\id\otimes\sigma\), then take the real part.
Since \(c_\epsilon\) is real and nonnegative, this gives
\eqref{eq:inducedeffect}. Multiplication by \(\rho\) gives
the support assertion.
\end{proof}

This is an equality of bounded effects in a Weyl-algebra
quantization. The unbounded polynomial field algebra alone
would not supply the effect \eqref{eq:probeeffect}.

%=====================================================================
\section{A quantitative transfer estimate from advanced evolution}
\label{sec:transfer}
%=====================================================================

\subsection{Reduced advanced operator and explicit constants}

The s-wave coupled operator is
\[
\begin{pmatrix}\Dop&\epsilon a\\
                \epsilon a&\Dop\end{pmatrix},
\qquad
a(t,x)=F(r(x))\rho(t,r(x)).
\]
In particular, because \(\rho=-F\) and
\(F=Y_0f/(rF(r))\),
\[
a=-Y_0f/r,\qquad
-av=f\quad\text{on }K.
\]
Here the first \(F\) in \(\rho=-F\) is the bulk test, while
\(F(r)=1-r^{-1}\) is the metric coefficient.

Work on the finite time interval
\[
I=[t_-,t_+]=[-T-1,T+3],\qquad \Delta=t_+-t_-.
\]
For a reduced function \(u\), define
\[
\|u\|_{\mathcal X}^2
=\sup_{t\in I}\sum_{j=0}^2
 \left(\|\partial_t\partial_x^ju\|_2^2
      +\|\partial_x^{j+1}u\|_2^2
      +\|\partial_x^ju\|_2^2\right).
\]
Let \(G_{\mathrm A,\Dop}\) be the reduced free advanced Green
operator, and define
\[
Au=G_{\mathrm A,\Dop}(au).
\]
Its source is compactly supported inside the coupling slab.

The following deliberately conservative constants suffice:
\[
c_2=64\left(1+\sum_{j=0}^2\|\partial_x^jV\|_\infty\right),
\]
\[
M_2(a)=8\sup_{t\in I}
                         \sum_{j=0}^2\|\partial_x^ja(t,\cdot)\|_\infty,
\qquad
\Lambda=\Delta e^{c_2\Delta}M_2(a).
\]

\begin{lemma}[Advanced energy estimate]
\label{lem:energy}
The operator \(A\) satisfies
\[
\|Au\|_{\mathcal X}\le\Lambda\|u\|_{\mathcal X}.
\]
\end{lemma}

\begin{proof}
For a free inhomogeneous solution with zero final data,
differentiate the sum of energies in the definition of
\(\mathcal X\). Spatial integration by parts cancels the
principal wave terms. The remaining potential and commutator
terms contain only \(V,V',V''\).
Cauchy--Schwarz and the product rule give
\[
\frac{\dd}{\dd t}\sqrt{\mathcal E_2(t)}
\le c_2\sqrt{\mathcal E_2(t)}+\|J(t)\|_{H^2},
\]
with the reversed-time version for advanced evolution.
The stated \(c_2\) dominates all the finite product-rule
coefficients.

The spatial product estimate
\[
\|a(t)u(t)\|_{H^2}\le
                       M_2(a)\sqrt{\mathcal E_2(u;t)}
\]
follows by expanding the derivatives through order two.
Backward Gronwall inequality on \(I\) gives
\[
\|G_{\mathrm A,\Dop}(au)\|_{\mathcal X}
\le\Delta e^{c_2\Delta}M_2(a)\|u\|_{\mathcal X}.
\]
\end{proof}

Let \(v_{\mathrm A}=G_{\mathrm A,\Dop}g_{\mathrm{red}}\).
It equals \(v\) on \(K\) and has finite \(\mathcal X\) norm.
Put
\[
z_\epsilon=\epsilon^2\Lambda^2.
\]
Assume \(z_\epsilon<1\).

\begin{proposition}[Exact convergent transfer formula]
\label{prop:Neumann}
The reduced advanced coupled solution is
\[
(w_\epsilon)_\psi=(I-\epsilon^2A^2)^{-1}v_{\mathrm A},
\qquad
(w_\epsilon)_\phi=-\epsilon A(w_\epsilon)_\psi.
\]
Hence the reduced system test is
\[
f_\epsilon=-a(w_\epsilon)_\psi=f+r_\epsilon,
\]
where
\[
r_\epsilon=-a\left[
 (I-\epsilon^2A^2)^{-1}v_{\mathrm A}-v_{\mathrm A}\right].
\]
The reduced residual probe test is
\begin{equation}
(g_\epsilon)_{\mathrm{red}}
=g_{\mathrm{red}}/\epsilon+\epsilon j_\epsilon,
\qquad
j_\epsilon=aA(I-\epsilon^2A^2)^{-1}v_{\mathrm A}.
\label{eq:residualprobe}
\end{equation}
Moreover,
\[
\left\|(w_\epsilon)_\psi-v_{\mathrm A}\right\|_{\mathcal X}
\le
\frac{\epsilon^2\Lambda^2}{1-z_\epsilon}
                                      \|v_{\mathrm A}\|_{\mathcal X}.
\]
\end{proposition}

\begin{proof}
The two advanced equations give
\[
(w_\epsilon)_\phi=-\epsilon A(w_\epsilon)_\psi,\qquad
(w_\epsilon)_\psi=v_{\mathrm A}-\epsilon A(w_\epsilon)_\phi.
\]
Substitution gives the displayed inverse.
Lemma~\ref{lem:energy} proves convergence in \(\mathcal X\).
The resulting solution has zero final data and satisfies the
coupled equation. Cauchy uniqueness identifies it with the
exact coupled advanced solution; smooth-source regularity
makes it smooth. The test formulas and remainder estimate
follow by substitution.
\end{proof}

\subsection{Conversion to the Unruh moment topology}

Abbreviate the two scalar coefficients in
\eqref{eq:compactcovariancebound} by
\[
A_0=\frac12+\frac{2}{\beta\omega_c},\qquad
A_1=\frac1{2\omega_c^2}+\frac{2}{3\beta\omega_c^3}.
\]
Choose a rectangular coupling support bound
\(I\times[-X,X]\), and let \(V_K=2X\Delta\).
Put
\[
a_0=\|a\|_\infty,\qquad a_t=\|\partial_ta\|_\infty,
\]
\begin{equation}
L_R=\frac{2V_KC_X}{\sqrt{2\pi}}
                 \sqrt{A_0a_0^2+A_1(a_t+a_0)^2}.
\label{eq:transferconstant}
\end{equation}

\begin{lemma}[Product covariance estimate]
\label{lem:productbound}
For a smooth \(u\) of finite \(\mathcal X\) norm,
\[
\|k(au)\|_{\mathrm U}\le L_R\|u\|_{\mathcal X}.
\]
\end{lemma}

\begin{proof}
The one-dimensional Sobolev inequality gives, with a
conservative constant,
\[
\sup_{t,x}|u(t,x)|,\ \sup_{t,x}|\partial_tu(t,x)|
\le2\|u\|_{\mathcal X}.
\]
Therefore
\[
\|au\|_1\le2V_Ka_0\|u\|_{\mathcal X},\qquad
\|\partial_t(au)\|_1
\le2V_K(a_t+a_0)\|u\|_{\mathcal X}.
\]
Apply Proposition~\ref{prop:compactcovariance}.
\end{proof}

\begin{theorem}[Quantitative compact-observable transfer]
\label{thm:transfer}
The exact induced system test satisfies
\begin{equation}
\boxed{
\|k(f_\epsilon)-k_R\|_{\mathrm U}
\le d_{\mathrm{trf}}(R,\epsilon)
:=
L_R\frac{\epsilon^2\Lambda^2}{1-z_\epsilon}
                                      \|v_{\mathrm A}\|_{\mathcal X}.
}
\label{eq:transfererror}
\end{equation}
The residual probe correction satisfies
\[
\|k(j_\epsilon)\|_{\mathrm U}
\le
L_R\frac{\Lambda}{1-z_\epsilon}
                                      \|v_{\mathrm A}\|_{\mathcal X}.
\]
\end{theorem}

\begin{proof}
Apply Lemma~\ref{lem:productbound} to the two functions in
Proposition~\ref{prop:Neumann}, and use
\(\|A\|\le\Lambda\).
\end{proof}

All constants here are fixed by the packet-derived compact test,
the chosen finite processing slab, and the radial Green problem.
The estimate is not an unspecified calibration error.

%=====================================================================
\section{Probe preparation, noise, and squeezing cost}
\label{sec:probe}
%=====================================================================

\subsection{Why fixed probe noise is not negligible}

Prepare a reference probe in an independent copy
\(\sigma_{\mathrm{ref}}\) of the bulk Unruh state.
This is a centered regular quasifree Hadamard state of the same
massless probe equation. For its real s-wave tests,
\[
V_{\mathrm{ref}}(u,u)=\|k(u_{\mathrm{red}})\|_{\mathrm U}^2.
\]

The probe reference could be replaced by another explicitly
controlled covariance. It must not be replaced by a formal
``zero-noise'' state.

The leading residual test in \eqref{eq:residualprobe} is
\(g/\epsilon\). Since \([g]\ne0\),
\(V_{\mathrm{ref}}(g,g)>0\), and
\[
V_{\mathrm{ref}}(g_\epsilon,g_\epsilon)
=\epsilon^{-2}V_{\mathrm{ref}}(g,g)+O(1)
\quad(\epsilon\downarrow0)
\]
for fixed \(R,T\).
Thus a fixed reference preparation gives
\(c_\epsilon\to0\), not one. Its induced effect tends toward
\(I/2\).

Dividing the measured Weyl observable by \(c_\epsilon\)
would in general destroy the effect bound.
The following construction suppresses noise through the
preparation instead.

\subsection{An exact finite-plane squeeze}

The free probe test quotient has symplectic form
\[
s_\psi(u,w)=\int u\,Ew\,\dd\mathrm{vol}_g.
\]
Choose a real compact s-wave test \(p_0\) for which
\[
c_0=s_\psi(g,p_0)\ne0.
\]
Such a test exists by nondegeneracy of the free scalar
symplectic space. It can also be constructed from compact
Cauchy data symplectically paired with the nonzero solution
\(Eg\), followed by the time-slice construction.

For sufficiently small \(\epsilon\), define
\[
p_\epsilon=
\frac{p_0}{s_\psi(g_\epsilon,p_0)}.
\]
Then \(s_\psi(g_\epsilon,p_\epsilon)=1\).

Let \(\Pi_\epsilon\) be the symplectic projection onto their
real plane:
\[
\Pi_\epsilon u
=s_\psi(u,p_\epsilon)g_\epsilon
-s_\psi(u,g_\epsilon)p_\epsilon.
\]
Define a symplectic map on the probe test quotient by
\begin{equation}
S_\epsilon u
=u-\Pi_\epsilon u
+\epsilon^2s_\psi(u,p_\epsilon)g_\epsilon
-\epsilon^{-2}s_\psi(u,g_\epsilon)p_\epsilon.
\label{eq:squeeze}
\end{equation}
It scales \(g_\epsilon\) by \(\epsilon^2\), scales
\(p_\epsilon\) by \(\epsilon^{-2}\), and fixes the symplectic
orthogonal complement.

Define the independently prepared probe state by
\[
\sigma_{R,\epsilon}(\Wprobe(u))
=\sigma_{\mathrm{ref}}(\Wprobe(S_\epsilon u)).
\]

\begin{proposition}[Admissible squeezed preparation]
\label{prop:squeezedstate}
The state \(\sigma_{R,\epsilon}\) is centered, regular,
quasifree, and Hadamard for every allowed finite \(R,\epsilon\).
Its residual variance is
\begin{equation}
V_{\sigma_{R,\epsilon}}(g_\epsilon,g_\epsilon)
=\epsilon^4V_{\mathrm{ref}}(g_\epsilon,g_\epsilon).
\label{eq:squeezedvariance}
\end{equation}
\end{proposition}

\begin{proof}
The scaling on a Darboux plane and identity on its symplectic
complement preserve the symplectic form. Hence
\(W(u)\mapsto W(S_\epsilon u)\) is a Weyl automorphism.
Pullback of the reference state proves positivity and
quasifreeness. Its covariance remains continuous on compact
test spaces, so regularity is preserved.

The difference between the reference and transformed
two-point distributions is a finite sum of products of smooth
free solutions associated with \(g_\epsilon,p_\epsilon\) and
their contractions against the reference covariance.
For a Hadamard bisolution, smearing one variable with a smooth
compact test gives a smooth solution in the other variable.
The covariance difference is therefore smooth.
The Hadamard singularity is unchanged.

Finally \(S_\epsilon g_\epsilon=\epsilon^2g_\epsilon\),
which proves \eqref{eq:squeezedvariance}.
\end{proof}

This is an explicitly declared preparation. A local cooling or
squeezing protocol that creates it is not constructed.

\subsection{A numerical form of the noise bound}

Choose \(X_g\) containing the spatial support of
\(g_{\mathrm{red}}\), and set
\[
b_g=\mathcal B_{X_g}(g_{\mathrm{red}}).
\]
For \(z_\epsilon\le1/2\), put
\[
b_J=2L_R\Lambda\|v_{\mathrm A}\|_{\mathcal X}.
\]
Then
\[
\sqrt{V_{\mathrm{ref}}(g_\epsilon,g_\epsilon)}
\le b_g/\epsilon+\epsilon b_J.
\]
For \(0<\epsilon\le1\), define
\[
N_R=(b_g+b_J)^2.
\]
It follows that
\begin{equation}
\boxed{
V_{\sigma_{R,\epsilon}}(g_\epsilon,g_\epsilon)
\le N_R\epsilon^2,\qquad
\frac{1-c_\epsilon}{2}\le\frac{N_R\epsilon^2}{4}.
}
\label{eq:noisebound}
\end{equation}
The second bound uses \(1-e^{-u}\le u\).

The normalization of \(p_\epsilon\) is also quantitatively
controlled. Put
\[
b_P=\sqrt{V_{\mathrm{ref}}(p_0,p_0)}.
\]
The denominator cannot vanish if
\[
\epsilon^2\,2b_Jb_P<|c_0|/2,
\]
because the symplectic form is bounded by twice the ordinary
coefficient norms, which are dominated by the reference norms.

\begin{proposition}[Required conjugate noise growth]
\label{prop:effort}
The conjugate probe quadrature obeys
\[
V_{\sigma_{R,\epsilon}}(p_\epsilon,p_\epsilon)
\ge\frac1{4N_R\epsilon^2}.
\]
The squeezing parameter has magnitude
\(2\log(1/\epsilon)\), and the processing test is \(g/\epsilon\).
\end{proposition}

\begin{proof}
The two quadratures have commutator \(\ii I\).
The uncertainty inequality gives the variance product at least
\(1/4\). Apply \eqref{eq:noisebound}.
The remaining assertions are the two scalings in
\eqref{eq:squeeze} and \eqref{eq:probeeffect}.
\end{proof}

Noise suppression therefore has a divergent preparation effort
along the weak-coupling schedule. No finite-energy or
thermodynamic optimum is asserted.

%=====================================================================
\section{The local Hawking-packet correspondence theorem}
\label{sec:main}
%=====================================================================

\begin{theorem}[Quantitative local packet-effect correspondence]
\label{thm:main}
Fix the normalized outgoing s-wave packet \(h\), the nonzero
quadrature parameter \(\vartheta\), and \(C\ge1\).
Use the compact tests of Section~\ref{sec:packetbridge}, the local
coupling and future probe effect of
Specification~\ref{spec:local}, and the independent preparation
of Section~\ref{sec:probe}.

For every \(\omega\in\mathfrak S_C\), and every allowed
\(R,T,\epsilon\),
\begin{equation}
\boxed{
\begin{split}
&\left|
(\omega\otimes\sigma_{R,\epsilon})
 \left[\Theta_\epsilon(I\otimes O_\epsilon)\right]
-\omega(E_{h,\vartheta})
\right|\\
&\qquad\le
\min\left\{1,\,
 \frac{1-c_\epsilon}{2}
 +H_C\bigl(\Delta_R+d_{\mathrm{trf}}(R,\epsilon)\bigr)
\right\}.
\end{split}
}
\label{eq:mainbound}
\end{equation}
Thus one may take
\[
\epsilon_{\mathrm{packet}}=H_C\Delta_R,\qquad
\epsilon_{\mathrm{transfer}}=H_Cd_{\mathrm{trf}},\qquad
\epsilon_{\mathrm{probe}}=(1-c_\epsilon)/2.
\]

For \(z_\epsilon\le1/2\) and \(\epsilon\le1\), define
\[
T_R=2L_R\Lambda^2\|v_{\mathrm A}\|_{\mathcal X}.
\]
Then the completely displayed upper bound is
\begin{equation}
\boxed{
\epsilon_{\mathrm{local}}
\le
\min\{1,\,
 H_C\Delta_R+(H_CT_R+N_R/4)\epsilon^2\}.
}
\label{eq:simplifiedmain}
\end{equation}
The additional smallness condition ensuring the definition of
\(p_\epsilon\) is retained.
\end{theorem}

\begin{proof}
The exact induced effect is \eqref{eq:inducedeffect}.
Replacing \(c_\epsilon\) by one changes its expectation by at
most \((1-c_\epsilon)/2\).

The coefficient of its system Weyl operator is \(k(f_\epsilon)\).
Theorems~\ref{thm:packetbridge} and \ref{thm:transfer} give
\[
\|k(f_\epsilon)-k_*\|_{\mathrm U}
\le\Delta_R+d_{\mathrm{trf}}.
\]
Apply Lemma~\ref{lem:Weylcomparison} directly with target
\(k_*\), and then cap the difference of two effect expectations
at one. This proves \eqref{eq:mainbound}.

When \(z_\epsilon\le1/2\),
\(d_{\mathrm{trf}}\le T_R\epsilon^2\).
Equation~\eqref{eq:noisebound} supplies the probe term, proving
\eqref{eq:simplifiedmain}.
\end{proof}

\begin{corollary}[Inherited Hawking prediction]
\label{cor:baseline}
In the Unruh system state,
\[
\left|
(\stateU\otimes\sigma_{R,\epsilon})
 \left[\Theta_\epsilon(I\otimes O_\epsilon)\right]
-\frac12\left[1+
 e^{-\vartheta^2(2\nu_h+1)/4}\right]
\right|
\le\epsilon_{\mathrm{local}}
\]
with \(C=1\).

The exact local prediction is
\[
\frac12\left[
1+c_\epsilon e^{-\|k(f_\epsilon)\|_{\mathrm U}^2/2}
\right].
\]
\end{corollary}

\begin{proof}
Use \eqref{eq:Hawkingbaseline} and
Theorem~\ref{thm:main}. The exact formula follows by applying
the Unruh characteristic function to
\eqref{eq:inducedeffect}.
\end{proof}

The main theorem is stronger than agreement in one calibrated
state: it compares a fixed induced effect with the target on
the named class \(\mathfrak S_C\).
It remains weaker than an operator-norm or unrestricted channel
comparison.

\subsection{Constructive accuracy schedule}

\begin{theorem}[Resource-dependent tolerance schedule]
\label{thm:tolerance}
Fix \(h,\vartheta,C,T\) and a desired expectation tolerance
\(\alpha>0\). Within the family permitting the stated growing
shells, compact couplings, future tests, and squeezed probe
preparations, finite parameters meeting
\(\epsilon_{\mathrm{local}}\le\alpha\) exist.

A sufficient choice is:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=*]
\item choose a finite integer \(R\) with
\(H_C\Delta_R\le\alpha/2\);

\item for that fixed \(R\), choose \(\epsilon>0\) satisfying
\[
\epsilon\le1,\qquad
\epsilon^2\Lambda^2\le1/2,
\]
\[
(H_CT_R+N_R/4)\epsilon^2\le\alpha/2,
\]
and the nonzero-denominator condition for \(p_\epsilon\).
\end{enumerate}
\end{theorem}

\begin{proof}
Theorem~\ref{thm:packetbridge} proves \(\Delta_R\to0\).
The ordinary derivative integrals in
\eqref{eq:ordinarytailbound} provide a terminating search
criterion on increasing integer \(R\).

For fixed \(R\), all displayed Green, support, and covariance
constants are finite. The inequalities in the second step are
therefore satisfied by sufficiently small positive \(\epsilon\).
Equation~\eqref{eq:simplifiedmain} proves the result.
\end{proof}

No polynomial rate in \(R\) is claimed. The spatial derivative
tails are actual packet-dependent quantities.
The theorem proves existence of finite parameters at every
fixed tolerance, while explicitly allowing resources to grow.

It does not prove arbitrary accuracy in a fixed coupling zone,
with bounded squeezing, bounded readout-test size, or fixed
laboratory hardware.

%=====================================================================
\section{A worked packet family and the apparatus resource ledger}
\label{sec:resources}
%=====================================================================

\subsection{One specified smooth packet}

Choose \(0<\omega_-<\omega_+\), and define
\[
h(\omega)=Z^{-1/2}
\begin{cases}
\displaystyle
\exp\left[-\frac1{(\omega-\omega_-)(\omega_+-\omega)}\right],
                         &\omega_-<\omega<\omega_+,\\[2mm]
0,                       &\text{otherwise},
\end{cases}
\]
where \(Z\) is its squared \(L^2\) integral before normalization.
This is a real normalized smooth compact-band packet.

For a chosen \(\vartheta\), the construction consists of:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item solve the same s-wave radial scattering equation used
for \(\tau,\rho_{\mathrm I}\);

\item form \(k_*\) and the real spectral Cauchy data
\eqref{eq:targetCauchydata};

\item calculate the spatial derivative-tail certificate
\eqref{eq:ordinarytailbound};

\item form \(f_{R,T}\), its bulk representative, the coupling
\(\rho=-F_{R,T}\), and the explicitly time-sliced probe test;

\item calculate the constants
\(C_X,L_R,\Lambda,b_g,b_J,N_R\);

\item choose \(\epsilon\) by
Theorem~\ref{thm:tolerance} and prepare
\(\sigma_{R,\epsilon}\).
\end{enumerate}
These are analytical specifications. Numerical execution of this
list is not asserted.

The inherited mean is
\[
\nu_h=\int_{\omega_-}^{\omega_+}
  |h(\omega)|^2
  \frac{|\tau(\omega)|^2}{e^{\beta\omega}-1}\dd\omega.
\]
It determines the target probability in
\eqref{eq:Hawkingbaseline}; no numerical greybody table is
inserted.

For \(\vartheta\ne0\), that target probability lies strictly
between \(1/2\) and \(1\). Choosing an error smaller than
half its distance from \(1\) gives a nontrivial probability
certificate rather than the vacuous upper bound one.

\subsection{Access and effort}

\begin{center}
\small
\renewcommand{\arraystretch}{1.13}
\begin{longtable}{@{}p{0.24\textwidth}p{0.66\textwidth}@{}}
\toprule
Resource & Declared quantity and boundary\\
\midrule
\endfirsthead
\toprule
Resource & Declared quantity and boundary\\
\midrule
\endhead
\bottomrule
\endfoot

Geometry and angular access &
The right Schwarzschild exterior; the full angular sphere.
The interaction is not pointlike and is not a restricted
angular-patch detector.\\

Coupling support &
A compact subset of
\(|t|\le T,\ |x|\le2R+T\), times \(S^2\).
Its inner radius is strictly greater than the horizon radius
for every finite \(R\).\\

Spatial accuracy &
The explicit spectral or ordinary derivative tails
\(\Delta_R,D_p(R),D_q(R)\).
Increasing \(R\) enlarges access toward both radiation ends.\\

Coupling amplitude and derivatives &
\(\lambda=\epsilon\rho\).
For every fixed derivative order,
\(\|\partial^\gamma\lambda\|_\infty
=\epsilon\|\partial^\gamma\rho\|_\infty\)
in the stated dimensionless coordinates.
Physical values require the displayed length conversion.\\

Probe species &
One additional real massless scalar on the same exterior.
The use of the constant free solution is specific to this
declared mass choice.\\

Probe initial data &
Compact Cauchy data equal to the constant solution on the
causal predecessor of the coupling zone.
The initial support and subsequent processing support grow
with the coupling zone.\\

Processing test &
The compact future test \(g/\epsilon\).
Its test seminorms grow as \(\epsilon^{-1}\) for fixed \(R,T\).\\

Probe covariance &
The independent reference Unruh covariance, followed by the
exact finite-plane squeeze \eqref{eq:squeeze}.
The residual variance is bounded by \(N_R\epsilon^2\).\\

Conjugate preparation effort &
The squeezing magnitude is \(2\log(1/\epsilon)\).
A canonically conjugate variance is at least
\(1/(4N_R\epsilon^2)\).
No bounded-effort limit is claimed.\\

Final measurement &
The two effects \(O_\epsilon,I-O_\epsilon\).
Their probabilities are well defined and complete.
A laboratory implementation of the final probe POVM is a
separate obligation.\\

Clock &
Normalized asymptotic Killing time.
A static proper-time implementation at radius \(r_0\) must
convert by
\(\dd\tau_{\mathrm{proper}}=\sqrt{F(r_0)}\,\dd t\).\\

Energy and preparation work &
No evaluated cooling work, squeezing energy, or optimal
reset cost is supplied.
Hadamard regularity is not a bound on preparation resources.\\

Repeated trials &
Fresh independent system and probe preparations are additional
assumptions if independent-trial statistics are used.\\

\end{longtable}
\end{center}

In dimensionless tortoise coordinates,
\(r(-X)-1\sim e^{-X-1}\) as \(X\to\infty\).
Thus the inner-shell access can approach the horizon exponentially
in the coordinate support parameter. The construction must not be
described as maintaining a fixed proper distance from the horizon.

%=====================================================================
\section{Complete binary records and relation to Em03-C}
\label{sec:records}
%=====================================================================

The two probe effects define a binary law
\[
p_{\mathrm{local}}(1)
=(\omega\otimes\sigma_{R,\epsilon})
                  \Theta_\epsilon(I\otimes O_\epsilon),
\qquad
p_{\mathrm{local}}(0)=1-p_{\mathrm{local}}(1).
\]
The ideal binary target has
\(p_h(1)=\omega(E_{h,\vartheta})\).
For binary laws,
\[
d_{\mathrm{TV}}(p_{\mathrm{local}},p_h)
=|p_{\mathrm{local}}(1)-p_h(1)|
\le\epsilon_{\mathrm{local}}.
\]

This is an effect and probability comparison. It does not
identify the conditional postmeasurement system state with the
conditional output of an Em03-C instrument.

If a final laboratory readout has a separately justified binary
complete-law error \(e_{\mathrm{read}}\), retaining all declared
failures in a common output alphabet, then classical contraction
and the triangle inequality add that error. Such a readout error
is not supplied by local field scattering alone.

\begin{corollary}[Local bounded-departure test]
\label{cor:departure}
Let two system states belong to the named controlled class, with
ideal packet-effect gap
\[
\Delta=
|\omega_1(E_{h,\vartheta})-\omega_0(E_{h,\vartheta})|.
\]
Using the same declared local scheme, with its corresponding
bounds \(\epsilon_1,\epsilon_0\), their local binary probability
gap is at least
\[
\max\{0,\Delta-\epsilon_1-\epsilon_0\}.
\]
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:main} to each state and use the reverse
triangle inequality.
\end{proof}

For independent repeated preparations, a binary sample mean has
\[
\Pr(|\widehat p-p_{\mathrm{local}}|>u)
\le2e^{-2nu^2}.
\]
This follows by the exponential-moment bound for independent
variables in \([0,1]\), followed by Markov inequality.
Thus the confidence interval for the ideal packet probability
enlarges by \(\epsilon_{\mathrm{local}}\).
Independence of repeated preparations is not inferred from the
one-shot construction.

The outcome says whether a specified bounded exterior observable
departs from its stated baseline. It does not identify the cause as
quantum gravity, microscopic information recovery, or evaporation.

%=====================================================================
\section{Repairs to the proposed proof architecture}
\label{sec:repairs}
%=====================================================================

\begin{enumerate}[label=\textbf{D\arabic*.},leftmargin=*]

\item \textbf{A bounded effect requires the Weyl algebra.}
The polynomial scalar field algebra is not used as though it
contained the requested nontrivial effects.

\item \textbf{The two signs are fixed together.}
With \(P=-\Box_g\), the interaction
\(-\epsilon\rho\phi\psi\) gives the positive off-diagonal
operator entries. The causal propagator is advanced minus
retarded, and the time-slice pairing is
\(p\Psi-q\partial_t\Psi\).

\item \textbf{The packet is identified in the bulk before coupling.}
The coefficient \(k_*\), the real Cauchy data, and
\(F_{R,T}\) are constructed using complete stationary
scattering. Em03-C's effective port is not used to prove the
bulk-to-radiation identification.

\item \textbf{A derivative filter is needed for the truncation estimate.}
Energy completeness alone would not control the
\(n_{\mathrm H}\sim1/\omega\) covariance weight.
The filtered Cauchy construction supplies the powers of
\(\omega\) used in \eqref{eq:packeterror}.

\item \textbf{Generic compact transfer errors need their own infrared bound.}
They are not assumed to retain the target's derivative filter.
The s-wave zero-energy calculation proves their finite
covariance through
Proposition~\ref{prop:compactcovariance}.

\item \textbf{The time-slice parameter is not a packet-time cutoff.}
Its identity is exact. The actual approximation is spatial
Cauchy truncation of a backward-continued outgoing packet.

\item \textbf{Residual probe noise remains in the induced effect.}
Weak coupling with a fixed probe state would suppress the
signal. The squeezed preparation is explicit and its resource
growth is proved.

\item \textbf{No effect is divided by an attenuation factor.}
The final probe observable stays between zero and one.
This differs from reconstructing a general Weyl element by
unbounded amplification of its measurement estimator.

\item \textbf{Full-shell access is a real restriction.}
The proof does not establish a pointlike or finite angular-patch
scheme. Its increasing coupling region includes access near
both spatial ends.

\item \textbf{State-class convergence is not operator-norm convergence.}
The error uses a common regular representation and controlled
second moments. It is not a uniform bound over all states.

\item \textbf{One observable is not a transfer channel.}
No number distribution, coherent oscillator extraction,
joint-packet tomography, or reference-stable channel equivalence
with Em03-C is proved.

\item \textbf{The probe measurement chain has a declared endpoint.}
A local bilinear interaction and induced effect are constructed.
An engineering realization of the final probe POVM and of its
squeezed preparation remains separate.

\end{enumerate}

%=====================================================================
\section{Remaining physical programmes and release}
\label{sec:release}
%=====================================================================

The present construction closes one narrow local-observable bridge.
The following remain outside it:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item a fixed-zone or restricted-angular-access approximation;
\item a probe-preparation protocol with bounded energy and
specified laboratory controls;
\item exact or approximate packet-number detection;
\item a local instrument comparison retaining quantum outputs
and external references;
\item a collapse geometry and a noncompact late-time packet limit;
\item backreaction or a dynamical mass-loss law;
\item an interacting matter or quantum-gravity correction to
the same observable;
\item microscopic input-dependent evaporation and recovery.
\end{enumerate}

The supplied internal documentary accounts do not alter these
proof obligations. This local free-field calculation does not
close O.1, discharge a separate XIV--XV characteristic-product
problem, or admit an inherited quantum-gravitational coefficient.
No manuscript registry or corpus coverage claim is used as a
mathematical premise.

\begin{theorem}[Em03-D analytical release]
\label{thm:release}
For one normalized positive-frequency massless s-wave packet
on the Schwarzschild exterior:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item its bounded quadrature effect and inherited Unruh
probability are identified;

\item real compact bulk tests converge to its scattering
coefficient with the explicit covariance error
\eqref{eq:packeterror};

\item a smooth compact local bilinear scalar coupling and
future probe effect have the exact induced form
\eqref{eq:inducedeffect};

\item the advanced evolution gives the explicit
compact-observable transfer error
\eqref{eq:transfererror};

\item a specified independent quasifree Hadamard probe
preparation suppresses residual noise with
\eqref{eq:noisebound}, while its conjugate variance and
readout-test effort grow as stated;

\item these constructions prove the state-class expectation
comparison \eqref{eq:mainbound} and the finite
resource-dependent schedule of
Theorem~\ref{thm:tolerance}.
\end{enumerate}

The construction uses full angular-shell access and a coupling
zone that may grow with accuracy. Complete stationary scattering,
standard local Green/scattering theory, and the bulk Unruh-state
identification are the declared external inputs.

The release is a local system--probe realization of one bounded
packet-observable comparison. It is not a pointlike detector,
a packet-number instrument, coherent state transfer, or
equivalence with Em03-C's effective Markov apparatus.
No collapse-packet theorem or quantum-gravitational evaporation
result is established.
\end{theorem}

\begin{proof}
Equation~\eqref{eq:packetcoefficient} and
Corollary~\ref{cor:baseline} establish (i).
Theorem~\ref{thm:packetbridge} proves (ii), using its displayed
Cauchy and spectral constructions.
Specification~\ref{spec:local} and
Theorem~\ref{thm:induced} prove (iii).
Proposition~\ref{prop:Neumann} and
Theorem~\ref{thm:transfer} prove (iv).
Propositions~\ref{prop:squeezedstate} and \ref{prop:effort},
together with \eqref{eq:noisebound}, prove (v).
Theorems~\ref{thm:main} and \ref{thm:tolerance} prove (vi).

The support, species, state-class, and external-theorem
restrictions are retained throughout. No excluded dynamical,
instrument-level, or laboratory construction appears in these
proofs.
\end{proof}

\paragraph{Final scope statement.}
Em03-D constructs a compactly coupled local scalar probe scheme
whose induced bounded effect approximates one specified outgoing
Hawking-packet quadrature effect, with an explicit state-class
error bound. The approximation uses growing full-shell access,
a scaled processing test, and a specified squeezed probe state.
It does not establish packet-number detection, coherent state
transfer, or channel equivalence with the apparatus of Em03-C.

%=====================================================================
\section*{Declarations}
%=====================================================================

\paragraph{Data and verification availability.}
The analytical specifications and estimates are contained in
this source. No numerical scattering table, executed resource
schedule, compilation record, formal proof certificate,
experimental calibration, or apparatus demonstration is supplied.

\paragraph{Attribution.}
The Unruh radiation prescription is inherited from the supplied
Em03-A source, and the one-mode target is consistent with the
supplied Em03-C source. Complete stationary scattering, local
linear system--probe scattering, and Hadamard state existence
are declared external inputs. The filtered compact bulk
approximation, particular constant-solution probe matching,
quantitative estimates, and effect-preserving noise-suppression
schedule are the constructions analyzed here.

%=====================================================================
\begin{thebibliography}{99}

\bibitem{Em03A}
N.~Cheung,
\emph{Em03-A: A Fixed-Background Hawking Emission Theorem},
version 1, 8 October 2026.
Supplied source \code{Em03-A-v1.tex}.
Used for the named scalar model, incoming Unruh prescription,
stationary scattering conventions, and packet occupation law.
Its separate compact-test peeling theorem is not a collapse
packet premise of this manuscript.

\bibitem{Em03C}
N.~Cheung,
\emph{Em03-C: Joint Hawking-Packet Correlations and Certified Tests
of Semiclassical Departures},
version 1, 8 October 2026.
Supplied source \code{Em03-C.tex}.
Finite-mode Gaussian and bounded-observable context.
The effective capture and register apparatus is not imported
as a local-field realization.

\bibitem{Dimock}
J.~Dimock,
\emph{Scattering for the wave equation on the Schwarzschild metric},
General Relativity and Gravitation
\textbf{17} (1985), 353--369.
doi:10.1007/BF00759679.
Complete classical wave scattering and its horizon/infinity
channels.

\bibitem{DeiftTrubowitz}
P.~Deift and E.~Trubowitz,
\emph{Inverse scattering on the line},
Communications on Pure and Applied Mathematics
\textbf{32} (1979), 121--251.
doi:10.1002/cpa.3160320202.
One-dimensional short-range Jost and spectral-scattering
framework used for the stated radial transform.

\bibitem{BaskinWang}
D.~Baskin and F.~Wang,
\emph{Radiation fields on Schwarzschild spacetime},
Communications in Mathematical Physics
\textbf{331} (2014), 477--506.
arXiv:1305.5273.
Geometric radiation-field regularity, the two radiation ends,
and energy normalization.
Its energy theorem is not substituted for the weighted
quantum covariance estimate proved here.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007.
arXiv:0806.1036.
Normally hyperbolic Cauchy problems, Green operators,
causal support, and scalar quantization.

\bibitem{FewsterVerch}
C.~J.~Fewster and R.~Verch,
\emph{Quantum fields and local measurements},
Communications in Mathematical Physics
\textbf{378} (2020), 851--899.
arXiv:1810.06512.
doi:10.1007/s00220-020-03800-6.
General induced-observable framework and the compact bilinear
scalar coupling; in particular the advanced scattering-test
transformation in the linear model.

\bibitem{FewsterJubbRuep}
C.~J.~Fewster, I.~Jubb, and M.~H.~Ruep,
\emph{Asymptotic measurement schemes for every observable
of a quantum field theory},
Annales Henri Poincar\'e
\textbf{24} (2023), 1137--1184.
arXiv:2203.09529.
doi:10.1007/s00023-022-01239-0.
Compact local measurement constructions and the Weyl-algebra
scattering and induced-observable formulas.
Its local asymptotic theorem is not used as an unproved
outgoing-packet approximation.

\bibitem{DMP}
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.
arXiv:0907.1034.
Bulk Unruh-state construction and Hadamard property.
The global microlocal proof remains an external input.

\end{thebibliography}

\end{document}