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% EVIDENCE BOUNDARY
% This source supplies analytical constructions and proofs.
% Compilation, formal verification, numerical execution, experimental
% calibration, and laboratory implementation are not asserted.
%
% MAIN RELEASE
% Two ordered, noncommuting effective exterior-source interactions have
% an exact scalar-field channel, including its interference terms.
% A compact-source family has a non-entanglement-breaking restriction
% to an identified outgoing Schwarzschild scalar packet.
% An explicit mathematical packet decoder has a referenced recovery
% certificate approaching zero in a growing-resource family.
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% LOCAL-ACCESS BOUNDARY
% The packet decoder assumes information-theoretic quantum access to
% the identified packet. It is not a local field-capture construction.
% Local bounded-observable access is established separately.
% No local coherent receiver, interior escape, gravitational source
% derivation, backreaction, or evaporation theorem is claimed.
%=====================================================================

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\hypersetup{
  pdftitle={Em03-G: Coherence Transfer and Recovery Limits
            in Physical Scalar Radiation},
  pdfauthor={Ngo Cheung},
  pdfsubject={Ordered exterior-source interactions, exact packet channels,
              referenced recovery, environmental leakage, and local access}
}

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\title{\textbf{Em03-G}\\[1mm]
\textbf{Coherence Transfer and Recovery Limits
in Physical Scalar Radiation}\\[2mm]
\large Noncommuting exterior-source interactions,
referenced packet channels, and local-access requirements}

\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 solve an effective exterior-source model consisting of two
temporally ordered couplings of a degenerate qubit to the massless
s-wave scalar field on Schwarzschild. The first coupling uses the
source observable \(Z\), and the second uses \(X\). Both source
profiles are smooth, compact, strictly exterior, and have full
angular-shell support. The initial scalar field is the independently
specified Unruh state.

The exact radiation channel is evaluated on every source matrix
unit. Its outgoing packet restrictions are normal channels with
explicit interference terms, rather than independently assigned
displaced Gaussian outputs. We also calculate the retained-source
channel and construct a complete complementary channel in a fixed
purified field representation.

A phase-matched one-packet comparison model admits an explicit
mathematical decoder. Its worst-case referenced recovery error
approaches zero when the first displacement grows and the second
decreases with their symplectic pairing fixed. Filtered compact
bulk approximations realize this comparison with ordered exterior
profiles and a source-input half-diamond error bound. Consequently,
finite compact parameters exist for which an identified outgoing
scalar packet has a provably entangled reference-output state.
This is a sufficient transmission regime, not a classification of
every parameter choice.

At every finite first-pulse coefficient, the retained source has
nonzero input distinguishability. We derive a corresponding
nonzero recovery lower bound, while keeping Bell fidelity,
worst-case entanglement fidelity, and half-diamond distance distinct.
Complete environmental leakage bounds retain unselected radiation,
future-horizon degrees of freedom, and background purification.

We prove a new common second-moment bound for the interference-bearing
outputs and use it for local bounded-Weyl probability certificates.
Coherent local packet access is not assumed or constructed.
Causal support, finite signed excess radiative energy, interference
energy, switching work, and resource growth are treated separately.

The interactions are effective exterior source laws, not a
microscopic gravitational derivation. No black-hole interior
information escape or evaporation theorem is established.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Question, predecessors, and the two release gates}
\label{sec:scope}
%=====================================================================

The principal chain is
\[
\boxed{
\text{qubit input}
\longrightarrow
\text{ordered compact source interactions}
\longrightarrow
\text{identified outgoing scalar packet channel}
\longrightarrow
\text{referenced transmission and recovery bounds}.
}
\]

The operational receiver is a separate chain:
\[
\text{physical scalar field}
\longrightarrow
\text{local receiver}
\longrightarrow
\text{quantum memory}.
\]
The first chain is constructed here. A coherent local realization
of the second chain is not asserted.

Em03-A--C \cite{Em03A,Em03B,Em03C} provide the selected stationary
scalar prescription and finite-packet context. Em03-D
\cite{Em03D} supplies a local bounded-observable architecture.
Em03-E \cite{Em03E} has additional finite radiation payloads.
Em03-F \cite{Em03F} instead identifies source-dependent information
in the scalar field itself, but its single-source-observable
radiation channel is measure-and-prepare.

Quantum transmission through scalar fields already has explicit
nonperturbative protocols in flat and curved spacetimes
\cite{Simidzija,KasprzakTjoa}. We do not claim novelty for its
possibility, for controlled displacement, or for the use of
noncommuting source observables. Simple-generated qubit--field
calculations also have established precedents \cite{Tjoa}.
The present contribution is the stated compact Schwarzschild
source realization, its exact interference-bearing packet channel,
and its source-input and recovery certificates.

\paragraph{Two release gates.}
We distinguish:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item \textbf{Information-theoretic scalar transmission:}
the identified outgoing packet channel and a mathematical CPTP
decoder on that packet;

\item \textbf{Local coherent reception:}
a specified spacetime receiver whose quantum output has a proved
referenced comparison with the packet decoder.
\end{enumerate}
Passing the first gate does not pass the second.

\paragraph{Evidence boundary.}
All constructions and estimates are analytical. Compilation,
formal verification, numerical execution, calibration, and physical
implementation are not asserted. A theorem about a specified
effective interaction is not a derivation of its gravitational origin.

\paragraph{Metrics.}
For normal states and channels,
\[
\dist(\rho,\tau)=\frac12\|\rho-\tau\|_1,\qquad
\ddiam(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond.
\]
The channel distance includes arbitrary external references.
We use squared pure-target fidelity. In particular,
\[
F_e(\Lambda)=
\bra\Phi(\id_R\otimes\Lambda)
                  (\ketbra\Phi\Phi)\ket\Phi,
\qquad
\ket\Phi=\frac{\ket{00}+\ket{11}}{\sqrt2}.
\]
Worst-case entanglement fidelity is defined separately in
Section~\ref{sec:recovery}.

\subsection{The obstruction that receivers cannot repair}

\begin{proposition}[Postprocessing the Em03-F channel]
\label{prop:Fobstruction}
Suppose
\[
\NN_F(\rho)=\sum_{z=0}^1\bra z\rho\ket z\,\rho_z.
\]
For every deterministic radiation-only access channel
\(\mathcal A\) and decoder \(\DD\),
\[
\DD\mathcal A\NN_F(\rho)
=\sum_z\bra z\rho\ket z\,\DD\mathcal A(\rho_z)
\]
is entanglement breaking. Its qubit Bell fidelity is at most
\(1/2\), and its recovery half-diamond error is at least \(1/2\).
\end{proposition}

\begin{proof}
The displayed channel remains measure-and-prepare. For a referenced
input its output is a sum of positive product operators.
Postprocessing on the receiver side preserves separability.
A separable two-qubit state has Bell overlap at most \(1/2\):
the pure-product bound follows by Cauchy--Schwarz and the mixed
bound by convexity. The Bell-projector test gives the distance
lower bound.
\end{proof}

This is the standard entanglement-breaking mechanism
\cite{EntanglementBreaking}, with its numerical consequence
proved directly. Better reception cannot recreate source coherence
already absent from F's radiation output. The modification below
is therefore made at the source.

%=====================================================================
\section{Field conventions and the independent background}
\label{sec:field}
%=====================================================================

\subsection{Geometry, reduction, and measures}

In physical units the background is the right Schwarzschild exterior,
\[
g=-\left(1-\frac{r_{\mathrm h}}r\right)\dd t^2
 +\left(1-\frac{r_{\mathrm h}}r\right)^{-1}\dd r^2
 +r^2\dd\Omega^2,\qquad r_{\mathrm h}=2GM.
\]
Killing time is normalized at infinity.

For the estimates we measure lengths in units of \(r_{\mathrm h}\).
Suppressing tildes,
\[
r_{\mathrm h}=1,\qquad
\fmet(r)=1-r^{-1},\qquad
x=r+\log(r-1),\qquad \beta=4\pi.
\]
Physical frequencies and energies are \(r_{\mathrm h}^{-1}\)
times the displayed values. A dimensionless bulk scalar smearing
corresponds to physical smearing with factor \(r_{\mathrm h}^{-3}\).
Packet normalization is preserved by
\[
\widetilde h(\widetilde\omega)
=r_{\mathrm h}^{-1/2}h(\widetilde\omega/r_{\mathrm h}).
\]

The free real scalar has
\[
P=-\Box_g,\qquad E=G_{\mathrm A}-G_{\mathrm R}.
\]
Let \(Y_0=(4\pi)^{-1/2}\). The selected s-wave field is
\[
\phi(t,r,\Omega)=Y_0\Psi(t,x)/r,
\]
with reduced operator
\[
\Dop=\partial_t^2+\Lop,\qquad
\Lop=-\partial_x^2+V(x),\qquad
V(x)=\fmet(r(x))/r(x)^3.
\]
For a real compact reduced test \(f\), define
\[
\Bulk[f]=\frac{Y_0}{r\fmet(r)}f(t,x).
\]
Because
\(\dd\mathrm{vol}_g=r^2\fmet(r)\dd t\,\dd x\,\dd\Omega\),
\[
\phi(\Bulk[f])=\int f(t,x)\Psi(t,x)\dd t\,\dd x.
\]
All finite compact reduced supports give compact supports
strictly outside the horizon.

\begin{externalinput}[Free scattering, Green operators, and state]
We use complete stationary massless-scalar scattering on Schwarzschild,
the free normally hyperbolic Cauchy and Green-operator theory, and
the bulk Hadamard identification of the Unruh boundary prescription.
The horizon input is centered quasifree with
\[
n_{\mathrm H}(\omega)=(e^{\beta\omega}-1)^{-1},
\]
the infinity input is vacuum, and the inputs are independent.
\end{externalinput}

The classical scattering and Green inputs are
\cite{Dimock,BGP}. The selected bulk state construction is
\cite{DMP}. They do not supply the new source dynamics or a
coherent receiver.

\subsection{Coefficients and Weyl normalization}

Use horizon-incident and infinity-incident modes \(q_{\mathrm H}\),
\(q_{\mathrm I}\), normalized as in Em03-D--F. Their spectral
transform is
\[
(\mathcal Uu)_j(\omega)
=\frac1{\sqrt{2\pi}}\int
           \overline{q_j(x,\omega)}u(x)\dd x.
\]
It is unitary from \(L^2(\mathbb R)\) onto
\(L^2((0,\infty);\mathbb C^2)\), and takes \(\Lop\) to
multiplication by \(\omega^2\).

The field expansion is
\[
\Psi(t,x)=\sum_j\int_0^\infty
 \frac{q_j a_j e^{-\ii\omega t}
       +\overline{q_j}a_j^\dagger e^{\ii\omega t}}
      {\sqrt{4\pi\omega}}\dd\omega.
\]
Consequently
\begin{equation}
k_j(f)(\omega)
=\frac1{\sqrt{4\pi\omega}}
 \int f(t,x)\overline{q_j(x,\omega)}
                           e^{\ii\omega t}\dd t\,\dd x.
\label{eq:coeff}
\end{equation}
Annihilation tests are conjugate-linear:
\[
a(k)=\sum_j\int\overline{k_j(\omega)}a_j(\omega)\dd\omega.
\]
Set
\[
X(k)=a(k)+a(k)^\dagger,\qquad W(k)=e^{\ii X(k)},
\]
\[
\symp(k,l)=2\Impart\langle k,l\rangle,\qquad
W(k)W(l)=e^{-\ii\symp(k,l)/2}W(k+l).
\]
Then \(\phi(\Bulk[f])=X(k(f))\).

Define
\[
\Covop=\diag(1+2n_{\mathrm H},1),\qquad
\|k\|_{\mathrm U}^2=\langle k,\Covop k\rangle,
\qquad
\Qchar(k)=e^{-\|k\|_{\mathrm U}^2/2}.
\]
The ordinary two-channel norm is \(\|\cdot\|_0\). We have
\[
\stateU(W(k))=\Qchar(k),\qquad
\stateU(X(k)^2)=\|k\|_{\mathrm U}^2,\qquad
\|k\|_0\le\|k\|_{\mathrm U}.
\]

\subsection{Compact-profile domains}

\begin{lemma}[Compact-profile regularity]
\label{lem:compactregularity}
For real \(f\in C_c^\infty(\mathbb R^2)\), its coefficient
\(k(f)\) has finite covariance norm and obeys
\[
\|\omega^{1/2}k(f)\|_0<\infty,\qquad
\|\omega^{1/2}\Covop k(f)\|_0<\infty.
\]
The instantaneous coefficient
\[
k_{t,j}(\omega)=
\frac{e^{\ii\omega t}}{\sqrt{4\pi\omega}}
       \int f(t,x)\overline{q_j(x,\omega)}\dd x
\]
is smooth in the covariance norm and has compact temporal support.
\end{lemma}

\begin{proof}
Here are the specific s-wave threshold facts needed for the proof.
The zero-energy Jost solutions are
\[
f_-(x,0)=r(x),\qquad
f_+(x,0)=-r(x)\log\fmet(r(x)),
\]
with Wronskian \(-1\). Also
\[
\int V\dd x=\frac12,\qquad
\int(1+|x|)V\dd x<\infty.
\]
The Jost Volterra equations are therefore continuous at zero.
On each compact spatial interval they give
\[
|q_j(x,\omega)|\le C_X\min\{\omega/\omega_c,1\}
\]
for suitable finite \(C_X\) and \(\omega_c>0\).
At zero use the Volterra kernel bound
\(|\sin(\omega z)/\omega|\le|z|\) and the nonzero
Wronskian; above \(\omega_c\) use
\(\exp(\|V\|_1/\omega_c)\) and flux conservation.

Thus a compact spacetime test has
\[
|k_j(f)(\omega)|=O(\sqrt\omega)\quad(\omega\downarrow0).
\]
At high frequency, one time integration by parts gives
\(O(\omega^{-3/2})\). Further time integrations are available.
Since \(n_{\mathrm H}\le1/(\beta\omega)\), these bounds prove
all three stated integrability claims, including the extra
\(\Covop^2\) in the last one.

For the instantaneous coefficient,
\[
k_t=e^{\ii\omega t}\mathcal U(f(t,\cdot))/\sqrt{2\omega}.
\]
The threshold estimate controls its low-frequency covariance.
Time derivatives introduce finite powers of \(\omega\);
compact spatial smoothness and spectral unitarity control those
powers by applying powers of \(\Lop\) to \(f(t,\cdot)\).
This proves smoothness of every order.
\end{proof}

This is the weighted domain relevant to the source. It does not
replace the thermal horizon sector by vacuum.

%=====================================================================
\section{Ordered noncommuting source dynamics}
\label{sec:dynamics}
%=====================================================================

The message is encoded by a fixed identification into the source
qubit \(S\). Its computational basis is the \(Z\) basis:
\[
Z\ket0=\ket0,\qquad Z\ket1=-\ket1,\qquad H_S=0.
\]
The initial state is
\[
\rho_{RS}\otimes\stateU.
\]
The source and receiver have no prior message-dependent field
preparation.

Choose real compact profiles \(f_1,f_2\) with disjoint time slabs
\(I_1<I_2\), and set \(F_j=\Bulk[f_j]\). The interaction is
\begin{equation}
H_I(t)=g_1Z\otimes\Phi_{F_1}(t)
                       +g_2X\otimes\Phi_{F_2}(t).
\label{eq:interaction}
\end{equation}
The exact measure convention is
\[
\Phi_F(t)=\int_{\Sigma_t}NF(t,\cdot)\phi(t,\cdot)\dd\Sigma_t
=\int f(t,x)\Psi(t,x)\dd x,
\]
where
\[
N=\sqrt{\fmet(r)},\qquad
\dd\Sigma_t=r^2\sqrt{\fmet(r)}\dd x\,\dd\Omega.
\]
Thus \(\int\Phi_F(t)\dd t=\phi(F)\).

Abbreviate
\[
a_1=g_1k(f_1),\qquad a_2=g_2k(f_2),\qquad
\theta=\symp(a_1,a_2).
\]

\subsection{Representation and common domain}

A useful fixed purified realization is the doubled Fock space
with real-linear map
\[
Tk=
\bigl((\sqrt{1+n_{\mathrm H}}k_{\mathrm H},k_{\mathrm I}),
                   \sqrt{n_{\mathrm H}}\,\overline{k_{\mathrm H}}\bigr).
\]
It satisfies
\[
\|Tk\|^2=\|k\|_{\mathrm U}^2,\qquad
2\Impart\langle Tk,Tl\rangle=\symp(k,l).
\]
Let \(\Omega\) be its vacuum. Its physical Weyl subalgebra has
exactly the prescribed characteristic function.
No global density in the undoubled stationary radiation Fock
space is assumed.

Write \(N_{\mathrm d}\) for doubled Fock particle number.
Segal fields are essentially self-adjoint on finite-particle
vectors, and
\(\mathcal D((N_{\mathrm d}+1)^{1/2})\) is their common form
domain. Weyl displacements preserve this domain.

\begin{theorem}[Exact ordered evolution]
\label{thm:unitary}
The interaction \eqref{eq:interaction} has a unitary propagator.
After both slabs have ended, its endpoint is
\begin{equation}
\boxed{
U_G=e^{\ii(\gamma_1+\gamma_2)}
       e^{-\ii X\otimes X(a_2)}
       e^{-\ii Z\otimes X(a_1)}.
}
\label{eq:UG}
\end{equation}
The individual phases are
\[
\gamma_j
=-\frac{g_j^2}{2}
 \int_{I_j}\dd t\int_{I_j\cap\{u<t\}}\dd u\,
                      \symp(k_{j,t},k_{j,u}).
\]
The two factors preserve the same number form domain.
On that domain their chronological composition solves the strong
Schr\"odinger equation into the Hilbert space.
\end{theorem}

\begin{proof}
For one slab and source observable \(O\in\{X,Z\}\), \(O^2=I\).
If \(K(t)=g_j\int_{I_j\cap\{u\le t\}}k_{j,u}\dd u\), its
propagator is
\[
e^{\ii\gamma_j(t)}
 \sum_{o=\pm1}P_o^O\otimes W(-oK(t)).
\]
Chronological Weyl products give exactly the displayed scalar
phase. Smooth covariance-norm continuity from
Lemma~\ref{lem:compactregularity} gives the strong product limit.

For a differentiable coefficient path \(r(t)\), the Weyl relation
and the Stone derivative give, on the number form domain,
\[
\partial_tW(r(t))
=\left[\ii X(\dot r(t))
       +\frac{\ii}{2}\symp(\dot r(t),r(t))\right]W(r(t)).
\]
The scalar derivative cancels \(\partial_t\gamma_j\), proving
the Schr\"odinger equation on the slab. Creation/annihilation
estimates and the displacement identity show domain invariance.

The slabs do not overlap. Their endpoint propagators therefore
compose in chronological order on the same domain, proving
\eqref{eq:UG}. Differentiating the inner product of two candidate
solutions gives uniqueness. No central-commutator claim is made
for overlapping \(X\)- and \(Z\)-couplings.
\end{proof}

Below we omit the irrelevant common scalar phase.

\subsection{Finite displacement branches}

For signs \(z,x\in\{+1,-1\}\), put
\[
P_z^Z=(I+zZ)/2,\qquad P_x^X=(I+xX)/2,
\]
\[
B_{xz}=W(-xa_2)W(-za_1)
=e^{\ii xz\theta/2}W(-za_1-xa_2).
\]
For \(z=+1\) identify \(\ket z=\ket0\), and for \(z=-1\)
identify \(\ket z=\ket1\). Let
\[
\epsilon_z=(1-z)/2,\qquad
m_{xz}=\bra x_X z\ket=x^{\epsilon_z}/\sqrt2.
\]
Then
\begin{equation}
U_G=\sum_{x,z}P_x^XP_z^Z\otimes B_{xz},\qquad
U_G(\ket z\otimes\Omega)
=\sum_xm_{xz}\ket x_X\otimes B_{xz}\Omega.
\label{eq:branches}
\end{equation}

These branch phases are part of the dynamics.
Dropping them before interference calculations is not valid.

%=====================================================================
\section{The complete scalar radiation channel}
\label{sec:channel}
%=====================================================================

First define the field-output channel as a map into state functionals:
\[
\NN_G(\rho)(A)
=(\tr_S\otimes\stateU)
                  [U_G^\dagger(I\otimes A)U_G(\rho\otimes I)].
\]
Equivalently, its values on source matrix units
\(E_{zw}=\ketbra z w\) are
\begin{equation}
\boxed{
\NN_G(E_{zw})(A)
=\sum_xm_{xz}m_{xw}\,
                     \stateU(B_{xw}^\dagger A B_{xz}).
}
\label{eq:matrixfunctional}
\end{equation}
This formulation does not assume a global field density.

For a weighted field coefficient \(l\), write
\[
A_l=\symp(a_1,l),\qquad B_l=\symp(a_2,l).
\]

\begin{theorem}[All four matrix-unit characteristic functions]
\label{thm:matrixunits}
The exact source-to-field channel obeys
\begin{equation}
\boxed{
\begin{aligned}
\NN_G(\ketbra00)(W(l))
 &=\Qchar(l)e^{-\ii A_l}\cos B_l,\\
\NN_G(\ketbra11)(W(l))
 &=\Qchar(l)e^{+\ii A_l}\cos B_l,\\
\NN_G(\ketbra01)(W(l))
 &=-\ii\sin B_l\,\Qchar(l-2a_1),\\
\NN_G(\ketbra10)(W(l))
 &=-\ii\sin B_l\,\Qchar(l+2a_1).
\end{aligned}
}
\label{eq:fourchars}
\end{equation}
The characteristic functional of every input follows by linearity.
In particular, off-diagonal source dependence is generally nonzero.

If \(a_2=0\), the result is the single-\(Z\)-observable channel
of Em03-F. If \(a_1=0\), it is a single-\(X\)-observable
measure-and-prepare channel.
\end{theorem}

\begin{proof}
Let \(r_{xz}=-za_1-xa_2\). Weyl multiplication gives
\[
\begin{split}
\stateU(B_{xw}^\dagger W(l)B_{xz})
={}&
\exp\left[-\frac{\ii}{2}(z+w)A_l-\ii xB_l\right]\\
&\times\Qchar(l+(w-z)a_1).
\end{split}
\]
The phase from
\(\symp(r_{xw},r_{xz})=x(w-z)\theta\)
cancels the explicit branch-phase difference.
For \(z=w\), the coefficient \(m_{xz}m_{xw}=1/2\).
For \(z\ne w\), it is \(x/2\).
Summing over \(x\) proves all four formulas.
The two reductions follow either from the endpoint unitary or
from these characteristic functions.
\end{proof}

A nonzero off-diagonal output is coherence dependence, not by
itself an entanglement-transmission theorem. For example, the
single-\(X\) reduction already depends on computational
off-diagonals and is nevertheless entanglement breaking.

\subsection{Outgoing packet coordinates}

Stationary scattering gives
\[
b_h=a_{\mathrm{out}}(h)=a(v_h),\qquad
v_h=(\overline\tau h,\overline{\rho_{\mathrm I}}h).
\]
For normalized \(h\),
\[
\|v_h\|_0=1,\qquad
\|v_h\|_{\mathrm U}^2=1+2\nu_h,
\]
\[
\nu_h=\int|h(\omega)|^2
           \frac{|\tau(\omega)|^2}{e^{\beta\omega}-1}\dd\omega.
\]
The complete outgoing coefficient of pulse \(j\) is
\[
d_j=\mathsf S a_j,
\]
where \(\mathsf S(\omega)\) is the unitary scattering matrix.
Its infinity component is
\[
d_{j,\infty}=\tau a_{j,\mathrm H}
                         +\rho_{\mathrm I}a_{j,\mathrm I}.
\]
Thus
\[
\langle v_h,a_j\rangle
=\int\overline h\,d_{j,\infty}\dd\omega.
\]
Every selected mode below is an outgoing mode of this scalar,
not an additional radiation payload.

\subsection{Normal packet restrictions and interference kernels}

Choose orthonormal compact-band outgoing packets
\(h_1,\ldots,h_m\). Put
\[
\mathfrak k=\Span_{\mathbb C}\{v_{h_1},\ldots,v_{h_m},a_1,a_2\}.
\]
Its dimension is at most \(m+2\), and its occupation matrix is
finite. Choose an ordinary orthonormal basis whose first \(m\)
vectors are the selected packets.

\begin{lemma}[Finite participating Gaussian density]
\label{lem:participating}
The baseline restriction to \(\FF(\mathfrak k)\) has a normal
centered gauge-invariant Gaussian density \(\rho_{\mathfrak k}\).
It need not factor between the selected packet space and its
orthogonal complement in \(\mathfrak k\).
\end{lemma}

\begin{proof}
In an orthonormal basis \(e_i\), its occupation matrix is
\[
A_{ij}=\stateU(a(e_j)^\dagger a(e_i)).
\]
It is positive and finite. Diagonalize \(A\) by a passive
finite-mode unitary. Each diagonal covariance is represented by
the geometric density of that mean. Their product, rotated back,
has the required Gaussian characteristic function and trace one.
Uniqueness of finite-mode Weyl characteristic functions identifies
the restricted state. Off-diagonal covariance is retained.
\end{proof}

For a branch pair define the trace-class operator
\[
T_{xz,yw}
=\Tr_{\mathfrak k\ominus\mathfrak h_{\mathrm{sel}}}
       (B_{xz}\rho_{\mathfrak k}B_{yw}^\dagger).
\]

\begin{theorem}[Normal outgoing-packet channel and Choi state]
\label{thm:normal}
The selected packet channel is the normal CPTP map
\begin{equation}
\boxed{
\NN_{\boldsymbol h}(E_{zw})
=\sum_xm_{xz}m_{xw}T_{xz,xw}.
}
\label{eq:normalchannel}
\end{equation}
Its normalized reference-output state is
\begin{equation}
\boxed{
J_G=\frac12\sum_{z,w}
       \ketbra z w_R\otimes\NN_{\boldsymbol h}(E_{zw}).
}
\label{eq:Choi}
\end{equation}
Equations~\eqref{eq:fourchars} determine its selected Weyl
characteristic functions exactly.

The diagonal outputs are equal-weight mixtures of the two
displaced baseline packet states corresponding to
\(-za_1-a_2\) and \(-za_1+a_2\).
The off-diagonal outputs are interference operators and are
not positive preparation states.
\end{theorem}

\begin{proof}
Both source endpoint factors act only on
\(S\otimes\FF(\mathfrak k)\).
Apply them to \(\rho\otimes\rho_{\mathfrak k}\), trace the
source and unselected participating modes, and use
\eqref{eq:branches}. This proves the trace-class formula and
complete positivity with trace preservation on every source
operator and reference extension.

For a diagonal source input there are two orthogonal final
source \(X\)-branches, each of weight \(1/2\).
Each full-field Weyl conjugation displaces the selected
Gaussian baseline. For off-diagonals the two branch coefficients
have opposite signs; their trace is zero by
\eqref{eq:fourchars} at \(l=0\). There is no
measure-and-prepare interpretation for those terms.
\end{proof}

The rest of the participating field has not been prepared in vacuum
or assumed independent. Only its actual restricted density was used.

%=====================================================================
\section{Retained source and a complete complementary channel}
\label{sec:complement}
%=====================================================================

Define
\[
q=e^{-2\|a_1\|_{\mathrm U}^2},\qquad
r=e^{-2\|a_2\|_{\mathrm U}^2},
\]
\[
q_\pm=e^{-2\|a_1\pm a_2\|_{\mathrm U}^2},\qquad
c_+=\frac{q_++q_-}{2},\qquad
c_-=\frac{q_--q_+}{2}.
\]

\begin{theorem}[Exact retained-source channel]
\label{thm:source}
For input Bloch vector \((u_x,u_y,u_z)\), the retained source has
Bloch vector
\begin{equation}
\boxed{
\bigl(q u_x,\quad r\sin(2\theta)+c_+u_y,\quad
      c_-u_x+r\cos(2\theta)u_z\bigr).
}
\label{eq:sourceBloch}
\end{equation}
Equivalently,
\[
\begin{aligned}
\BB_G(\ketbra00)
 &=\tfrac12[I+r\sin(2\theta)Y+r\cos(2\theta)Z],\\
\BB_G(\ketbra11)
 &=\tfrac12[I+r\sin(2\theta)Y-r\cos(2\theta)Z],\\
\BB_G(\ketbra01)
 &=\tfrac12(qX+c_-Z+\ii c_+Y),\\
\BB_G(\ketbra10)
 &=\BB_G(\ketbra01)^\dagger.
\end{aligned}
\]
For every finite \(a_1\), \(q>0\).
\end{theorem}

\begin{proof}
Write \(A=X(a_1)\), \(B=X(a_2)\).
The second pulse leaves \(X\) invariant, while the first gives
\[
e^{\ii ZA}Xe^{-\ii ZA}=X\cos(2A)-Y\sin(2A).
\]
Averaging in the centered Gaussian state gives \(qX\).

For \(Y\),
\[
e^{\ii XB}Ye^{-\ii XB}=Y\cos(2B)-Z\sin(2B).
\]
In a diagonal \(Z\)-block, the first pulse shifts \(B\) by
\(-z\theta\). The diagonal expectation is therefore
\(r\sin(2\theta)\), independent of \(z\).
In an off-diagonal block use
\[
W(a_1)W(\pm2a_2)W(a_1)=W(2a_1\pm2a_2).
\]
The cosine sandwich contributes \(c_+Y\).

Similarly,
\[
e^{\ii XB}Ze^{-\ii XB}=Z\cos(2B)+Y\sin(2B).
\]
Its diagonal average is \(r\cos(2\theta)Z\), and the
sine sandwich gives \(c_-X\).
This proves the adjoint identities and all displayed formulas.
\end{proof}

\subsection{Fixing the complete environment}

A complement cannot be defined by tracing selected modes from an
unpurified mixed packet marginal. We fix the following representation.

In the doubled realization of Section~\ref{sec:dynamics}, the regular
selected finite-mode CCR representation has a tensor factorization
\[
\HH_{\mathrm U}\cong
\FF(\mathbb C^m)\otimes\HH_{\mathrm{env}}.
\]
The selected packet algebra acts on the first factor.
One way to obtain this factorization is to complete its finitely
many canonical quadratures to a real symplectic basis in their
finite-dimensional doubled one-particle span, implement that finite
canonical transformation, and leave the orthogonal doubled Fock
space unchanged. This is the finite CCR multiplicity construction.

We fix this factorization once. The vector \(\Omega\) is its fixed
purification of the selected background. The environment contains:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item unselected participating radiation;
\item all other outgoing-infinity modes;
\item the future-horizon field algebra;
\item the purification degrees of freedom of the incoming thermal
horizon sector.
\end{enumerate}
These physical field subalgebras need not be independent tensor
factors in the chosen background. They are not discarded from the
environmental representation.

Define the Stinespring isometry
\begin{equation}
\mathcal V\ket z
=\sum_xm_{xz}\ket x_S\otimes B_{xz}\Omega.
\label{eq:dilation}
\end{equation}
Its accessible output is the selected packet factor, and its
environment is \(S\otimes\HH_{\mathrm{env}}\).

\begin{theorem}[Complete complement at the fixed interface]
\label{thm:complement}
A complete complementary channel is
\begin{equation}
\boxed{
\NN_{\boldsymbol h}^c(E_{zw})
=\sum_{x,y}m_{xz}m_{yw}\ketbra x y_S
 \otimes
 \Tr_{\mathrm{sel}}
       \ketbra{B_{xz}\Omega}{B_{yw}\Omega}.
}
\label{eq:complement}
\end{equation}
Tracing its environmental factor gives the retained-source channel
of Theorem~\ref{thm:source}.

The finite participating calculation of
Theorem~\ref{thm:normal} gives the same accessible channel.
Its canonical finite-background purification is embedded
isometrically into this full environmental interface.
\end{theorem}

\begin{proof}
The isometry follows from unitarity of \(U_G\) and normalization
of \(\Omega\). Taking the two respective partial traces gives
the accessible channel and \eqref{eq:complement}.
A further trace gives \(\BB_G\).

For the last assertion, factor the finite participating CCR
space from the full regular representation. Its vector
restriction is a purification of \(\rho_{\mathfrak k}\).
A Schmidt decomposition embeds the canonical purification
space into the remaining representation space.
Since \(U_G\) acts only on the participating physical modes
and the source, this embedding intertwines the dilations.
No source-dependent information is removed by that replacement
of purification coordinates.
\end{proof}

We do not identify retained-source leakage with complete leakage.
It is a lower bound on the latter because discarding environmental
systems is contractive.

%=====================================================================
\section{Recovery metrics and finite-coupling lower bounds}
\label{sec:recovery}
%=====================================================================

For a selected packet channel \(\NN\), define
\[
\varepsilon_{\mathrm{rec}}(\NN)
=\inf_{\DD}\ddiam(\DD\NN,\id_2).
\]
For a qubit density \(\rho\), choose a purification \(\ket{\psi_\rho}\)
and put
\[
F_{\mathrm{wc}}(\Lambda)
=\inf_\rho
 \bra{\psi_\rho}(\id\otimes\Lambda)
          (\ketbra{\psi_\rho}{\psi_\rho})\ket{\psi_\rho}.
\]
Its value is independent of the purification convention.
This is squared worst-case entanglement fidelity against identity.

The general recovery/environment literature treats metric-specific
duality and continuity \cite{BenyOreshkov,KSW}. The constants below
are instead derived directly for the present qubit input.

\begin{lemma}[Environmental distinguishability from a Bell test]
\label{lem:leakageBell}
Let \(\NN\) be a qubit-input channel with a fixed complete complement.
If a decoder has Bell fidelity \(F_e\), then the complementary
outputs for any orthogonal qubit pair have trace distance at most
\[
2\sqrt{1-F_e}.
\]
There is also a replacement state \(\tau\) on the original complete
environment with
\[
\ddiam(\NN^c,\RR_\tau)\le
                         \min\{1,2\sqrt{1-F_e}\}.
\]
\end{lemma}

\begin{proof}
Dilate the decoder and apply the complete evolution to a Bell input.
The result is a pure vector \(\ket\Psi\) on reference, decoded qubit,
and enlarged environment. Its projection onto the target Bell vector
has norm \(\sqrt{F_e}\).
For \(F_e>0\), normalize that projection to an environmental vector
\(\ket e\). Then
\[
\dist(\ketbra\Psi\Psi,\ketbra\Phi\Phi\otimes\ketbra e e)
=\sqrt{1-F_e}.
\]
The assertion for \(F_e=0\) is trivial.
Tracing the decoded qubit and decoder environment shows that the
normalized Choi state of the original complement is within
\(\sqrt{1-F_e}\) of \(I_R/2\otimes\tau\), for the corresponding
reduced \(\tau\).

For an orthogonal input pair with Bloch direction \(L\),
\[
\Tr_R[(L^{\mathsf T}\otimes I)
       (J_{\NN^c}-I_R/2\otimes\tau)]
=\NN^c(L)/2.
\]
The complete trace norm of the difference bounds this partial
contraction. Hence
\[
\frac12\|\NN^c(L)\|_1\le2\sqrt{1-F_e},
\]
which is the first assertion.

For a qubit-input Hermiticity-preserving map \(\Lambda\),
\[
\frac12\|\Lambda\|_\diamond
\le2\,\frac12\|J_\Lambda\|_1.
\]
Indeed every pure referenced qubit input is
\((A\otimes I)\ket\Phi\), with \(\|A\|^2\le2\);
apply the trace-norm product bound and then convexity.
Reference dimension two suffices by Schmidt decomposition.
The Choi estimate proves the replacement-channel bound.
\end{proof}

\begin{theorem}[Finite first-pulse recovery obstruction]
\label{thm:lower}
For every fixed compact two-pulse model and every selected packet
family,
\begin{equation}
\boxed{
\varepsilon_{\mathrm{rec}}(\NN_{\boldsymbol h})
\ge \frac{q^2}{4}
=\frac14e^{-4\|a_1\|_{\mathrm U}^2}>0.
}
\label{eq:lower}
\end{equation}
Every decoder also obeys
\[
F_e\le1-q^2/4,\qquad
F_{\mathrm{wc}}\le1-q^2/4.
\]
For every replacement channel on the complete complement,
\[
\ddiam(\NN_{\boldsymbol h}^c,\RR_\tau)\ge q/2.
\]
\end{theorem}

\begin{proof}
The retained-source outputs for the two \(X\)-eigenstate inputs
have an \(X\)-Bloch separation \(2q\). Their distance is at least
\(q\). The complete complementary distance for that pair is
therefore at least \(q\).

Lemma~\ref{lem:leakageBell} gives
\(q\le2\sqrt{1-F_e}\).
Thus \(1-F_e\ge q^2/4\).
The Bell projector gives
\(\ddiam(\DD\NN,\id)\ge1-F_e\).
Worst-case fidelity is no larger than Bell fidelity.

For any fixed replacement output, the triangle inequality for
the two complementary inputs gives maximal replacement error
at least half their distance, hence at least \(q/2\).
\end{proof}

The theorem excludes exact finite-coupling packet recovery.
It does not exclude a family with recovery error tending to zero
while \(q\) tends to zero and resources diverge.

%=====================================================================
\section{A phase-matched packet model and an explicit decoder}
\label{sec:matched}
%=====================================================================

Fix one normalized compact-band outgoing packet \(h\), and put
\[
v=v_h,\qquad \mathcal V_h=\|v\|_{\mathrm U}^2=1+2\nu_h.
\]
For \(\lambda>0\), choose comparison coefficients
\begin{equation}
a_1^{(0)}=\lambda v,\qquad
a_2^{(0)}=\ii\mu v,\qquad
\mu=\frac{\pi}{8\lambda}.
\label{eq:matched}
\end{equation}
Then
\[
\theta_0=2\lambda\mu=\pi/4,
\]
\[
q_0=e^{-2\mathcal V_h\lambda^2},\qquad
r_0=e^{-2\mathcal V_h\mu^2},\qquad
c_+=q_0r_0,\qquad c_-=0.
\]
The endpoint unitary \(U_0\) acts only on the source and this
one outgoing packet. The packet's initial density is the
geometric density of mean \(\nu_h\).

The coefficients in \eqref{eq:matched} are not compact bulk tests.
They are a comparison model; compact realizations are constructed
in Section~\ref{sec:compact}.

Let \(\ket{\chi}=(\ket0+\ii\ket1)/\sqrt2\), the \(Y=+1\)
source state. Define the deterministic packet decoder
\begin{equation}
\boxed{
\DD_\lambda(\tau)
=\Tr_h\!\left[
 U_0^\dagger(\ketbra\chi\chi\otimes\tau)U_0
\right].
}
\label{eq:decoder}
\end{equation}
Its output source basis is identified with the original message.
It uses a fresh independent qubit in \(\ket\chi\), the inverse
two-pulse unitary on that qubit and packet, and packet discard.
It is CPTP on every normal packet input.

\begin{lemma}[A product-reset comparison]
\label{lem:reset}
For a state \(\zeta_{SB}\), let
\(P=\ketbra\chi\chi\otimes I\), and
\(p=1-\tr(P\zeta)\). Then
\[
\dist(\zeta,\ketbra\chi\chi\otimes\zeta_B)
\le t(p):=\min\{1,\sqrt p+p/2\}.
\]
The bound retains arbitrary systems in \(B\).
\end{lemma}

\begin{proof}
Purify \(\zeta\) and apply \(P\). The rank-two pure-state
calculation, followed by partial trace, gives
\[
\frac12\|\zeta-P\zeta P\|_1\le\sqrt p.
\]
Moreover
\[
\ketbra\chi\chi\otimes\zeta_B
=P\zeta P+\ketbra\chi\chi\otimes\zeta_B^{\mathrm{bad}},
\]
where
\(\zeta_B^{\mathrm{bad}}
=\Tr_S((I-P)\zeta(I-P))\succeq0\)
has trace \(p\). This additional term costs \(p/2\).
\end{proof}

\begin{theorem}[Referenced packet recovery upper bound]
\label{thm:matchedrecovery}
For the comparison channel \(\NN_0\), define
\[
p_\lambda=\frac{1-r_0+q_0r_0}{2}.
\]
Then
\begin{equation}
\boxed{
\ddiam(\DD_\lambda\NN_0,\id_2)
\le E_\lambda:=t(p_\lambda),
\qquad E_\lambda\longrightarrow0
\quad(\lambda\to\infty).
}
\label{eq:matchedrecovery}
\end{equation}
Consequently
\[
F_{\mathrm{wc}}(\DD_\lambda\NN_0)\ge1-E_\lambda,
\qquad F_e(\DD_\lambda\NN_0)\ge1-E_\lambda.
\]
A useful explicit upper estimate is
\[
p_\lambda\le
\frac{\mathcal V_h\pi^2}{64\lambda^2}+\frac{q_0}{2}.
\]
\end{theorem}

\begin{proof}
Theorem~\ref{thm:source} gives the final source \(Y\)-mean
\[
r_0+q_0r_0u_y.
\]
For every input and every reference extension, the probability
outside \(\ket\chi\) is therefore at most \(p_\lambda\).

Let \(\zeta_{RSh}\) be the actual joint source-packet output.
Lemma~\ref{lem:reset} compares it with
\(\ketbra\chi\chi\otimes\zeta_{Rh}\).
Apply \(U_0^\dagger\) on \(Sh\), retaining \(R\), and trace the
packet. The first state returns the input exactly; the second
gives \((\id_R\otimes\DD_\lambda\NN_0)(\rho)\).
Contractivity proves the uniform referenced bound.

Since \(1-e^{-x}\le x\),
\[
(1-r_0)/2\le\mathcal V_h\mu^2,\qquad
q_0r_0/2\le q_0/2.
\]
This proves the explicit estimate and convergence.
An effect expectation changes by at most trace distance,
giving both fidelity bounds.
\end{proof}

No product assumption about this packet and unselected background
modes is needed: \(U_0\) acts only on \(Sh\), so its reduced
channel is determined by the packet marginal.

\subsection{A bounded referenced entanglement witness}

Define
\[
\mathcal W_\lambda
=(\id_R\otimes\DD_\lambda^*)
             \left(\frac12 I_{R\widehat M}-\ketbra\Phi\Phi\right).
\]
This is a bounded self-adjoint operator on reference and packet,
with \(\|\mathcal W_\lambda\|\le1/2\).

\begin{corollary}[A proved transmission regime]
\label{cor:witness}
For every separable reference-packet state \(\tau\),
\[
\tr(\mathcal W_\lambda\tau)\ge0.
\]
For the comparison channel,
\[
\tr(\mathcal W_\lambda J_0)
\le E_\lambda-\frac12.
\]
Thus \(E_\lambda<1/2\) proves a non-entanglement-breaking
packet channel.
\end{corollary}

\begin{proof}
Local decoding preserves separability, and a separable decoded
two-qubit state has Bell overlap at most \(1/2\).
For \(J_0\), its decoded Bell overlap is at least \(1-E_\lambda\).
\end{proof}

The witness is a mathematical reference-packet witness.
A local implementation of its decoder is not inferred from its
boundedness.

%=====================================================================
\section{Compact ordered realization and source-input stability}
\label{sec:compact}
%=====================================================================

\subsection{Filtered compact realization of a band coefficient}

We reproduce the needed bulk approximation rather than assuming
that a packet coefficient is already a compact source.

For a compact-band coefficient \(u\), choose a time center \(t_j\).
Let
\[
\mathcal C=\mathcal U(\text{complex conjugation})\mathcal U^{-1},
\qquad
b=e^{-\ii\omega t_j}\sqrt{2\omega}\,\omega^{-2}u.
\]
Set real Cauchy data by
\[
\mathcal Up=\frac{b+\mathcal Cb}{2},\qquad
\mathcal Uq=\frac{\mathcal Cb-b}{2\ii\omega}.
\]
They have coefficient \(\omega^{-2}u\) at \(t_j\) and belong
to every integer Sobolev space.

Choose a real \(\chi\in C_c^\infty((-2,2))\), equal to one
on \([-1,1]\). For \(L\ge1\), let
\[
q_L=\chi(x/L)q,\qquad p_L=\chi(x/L)p.
\]
Let \(s_L\) be their homogeneous Cauchy solution at \(t_j\).
With \(\eta_T\) zero before \(t_j-T\) and one after \(t_j+T\),
define
\[
f_{L,T}^{u,t_j}=-\partial_t^2\Dop(\eta_Ts_L).
\]
Its support is contained in
\[
|t-t_j|\le T,\qquad |x|\le2L+T.
\]

\begin{lemma}[Compact coefficient approximation]
\label{lem:bridge}
The real compact test just defined satisfies
\[
\|k(f_{L,T}^{u,t_j})-u\|_{\mathrm U}\le\Delta_L(u,t_j),
\qquad \Delta_L(u,t_j)\longrightarrow0,
\]
where, with \(\delta q=q_L-q,\ \delta p=p_L-p\),
\[
\begin{split}
\Delta_L^2={}&
\|\Lop^{3/4}\delta p\|_2^2+
\|\Lop^{5/4}\delta q\|_2^2\\
&+\frac2\beta\left(
\|\Lop^{1/2}\delta p\|_2^2+\|\Lop\delta q\|_2^2\right).
\end{split}
\]
\end{lemma}

\begin{proof}
The time-slice identity is
\[
\phi(\Bulk[\Dop(\eta_Ts_L)])
=\int(p_L\Psi(t_j)-q_L\partial_t\Psi(t_j))\dd x.
\]
Integration by parts proves it using the conserved Cauchy pairing
and \(\int\eta_T'=1\).
Applying \(-\partial_t^2\) multiplies its coefficient by
\(\omega^2\). Therefore the coefficient error is
\[
e^{\ii\omega t_j}\omega^2
 \frac{\mathcal U\delta p-\ii\omega\mathcal U\delta q}
                                      {\sqrt{2\omega}}.
\]
Use
\(|z-\ii\omega w|^2\le2|z|^2+2\omega^2|w|^2\)
and \(1+2n_{\mathrm H}\le1+2/(\beta\omega)\).
Spectral unitarity gives the displayed bound.

The cutoffs converge in \(H^2\) for \(p\) and \(H^3\) for \(q\).
Derivative tails and
\(\partial_x^r\chi(x/L)=O(L^{-r})\) prove this.
The potential and its derivatives are bounded.
The fractional norms are controlled by these integer graph norms,
using
\(\lambda^{3/2},\lambda\le1+\lambda^2\) and
\(\lambda^{5/2}\le1+\lambda^3\).
Hence \(\Delta_L\to0\).
\end{proof}

The derivative filter is needed for the weighted infrared topology.
The parameter \(T\) is an exact time-slice choice, not a packet
time-window approximation.

Choose \(t_2-t_1>2T\). The construction for \(u=v\) at \(t_1\)
and \(u=\ii v\) at \(t_2\) gives strictly ordered compact
source profiles. Set \(g_1=\lambda,\ g_2=\mu\).

\subsection{An error bound on the finite-dimensional source input}

Let comparison coefficients be \(a_1,a_2\), and let implemented
coefficients be \(\widehat a_1,\widehat a_2\).
Write
\[
\delta_j=\widehat a_j-a_j,\qquad
r_j=\|\delta_j\|_{\mathrm U}.
\]
Scalar endpoint phases are immaterial.

\begin{theorem}[Referenced two-pulse profile stability]
\label{thm:stability}
At any fixed selected packet interface, with the same independently
prepared background,
\begin{equation}
\boxed{
\ddiam(\widehat\NN,\NN)
\le e_{\mathrm{src}}
:=\min\{1,r_1+r_2+|\symp(a_1,\delta_2)|\}
\le\min\{1,r_1+(1+2\|a_1\|_0)r_2\}.
}
\label{eq:sourceerror}
\end{equation}
The same bound compares the chosen complete complements when
their full environmental identification is retained.

This is a source-input channel estimate. It is not unrestricted
bosonic diamond continuity of Weyl unitaries.
\end{theorem}

\begin{proof}
Align the scalar Weyl phases in each comparison.
For the first controlled factor on a product input
\(\ket{\psi}_{RS}\otimes\Omega\), the spectral inequality
\[
\|(W(\delta_1)-I)\Omega\|
\le\|X(\delta_1)\Omega\|=r_1
\]
gives a uniform vector bound.

Telescope the two endpoint factors, pricing the second-factor
difference on the ideal first-pulse output.
Because the second source projectors are orthogonal, its squared
norm is bounded by the squared norm of
\(X(\delta_2)\) on that joint output.
The first-pulse conjugation identity is
\[
U_1^\dagger(I\otimes X(\delta_2))U_1
=I\otimes X(\delta_2)-Z\otimes
                         \symp(a_1,\delta_2)I.
\]
Thus its norm on the initial vector is at most
\(r_2+|\symp(a_1,\delta_2)|\).
This argument is uniform in the source-reference vector.

Trace distance of pure output vectors is bounded by their
vector-norm difference after phase alignment. Partial traces
contract it, proving both channel comparisons.
Finally,
\(|\symp(a_1,\delta_2)|\le2\|a_1\|_0r_2\).
\end{proof}

For the compact construction,
\[
r_1\le\lambda\Delta_{L_1}(v,t_1),\qquad
r_2\le\mu\Delta_{L_2}(\ii v,t_2).
\]
Hence
\[
e_{\mathrm{src}}
\le\lambda\Delta_{L_1}
       +(1+2\lambda)\mu\Delta_{L_2}.
\]

\begin{theorem}[Compact scalar transmission and recovery]
\label{thm:compacttransmission}
For the compact ordered model and decoder \(\DD_\lambda\),
\begin{equation}
\boxed{
\ddiam(\DD_\lambda\widehat\NN_h,\id_2)
\le E:=\min\{1,E_\lambda+e_{\mathrm{src}}\}.
}
\label{eq:compactrecovery}
\end{equation}
The decoded Bell and worst-case entanglement fidelities are at least
\(1-E\), and
\[
\tr(\mathcal W_\lambda\widehat J_G)\le E-\frac12.
\]
For every desired positive recovery allowance, finite compact
ordered source parameters exist with \(E\) below that allowance,
within the stated growing-resource family.

In particular, finite compact parameters exist for which
the identified outgoing scalar packet channel is not
entanglement breaking.
\end{theorem}

\begin{proof}
Postcompose Theorem~\ref{thm:stability} with the same complete
decoder and use Theorem~\ref{thm:matchedrecovery}.
Effect contraction proves the fidelity and witness bounds.

First choose finite \(\lambda\) so that \(E_\lambda\) is small.
For that fixed \(\lambda\), \(\mu\) is finite and positive.
Lemma~\ref{lem:bridge} permits finite \(L_1,L_2\) making
\(e_{\mathrm{src}}\) arbitrarily small, while the time slabs
remain ordered. If \(E<1/2\), the witness is negative.
\end{proof}

\begin{example}[A nonvacuous analytical parameter certificate]
Choose \(\lambda\) so that
\[
\lambda\ge\pi\sqrt{\mathcal V_h},\qquad q_0\le1/32.
\]
Then \(p_\lambda\le1/32\), and
\[
E_\lambda\le1/\sqrt{32}+1/64<1/4.
\]
Choose finite compact approximation parameters with
\(e_{\mathrm{src}}\le1/8\).
Equation~\eqref{eq:compactrecovery} then gives
\(E<3/8\), decoded Bell fidelity greater than \(5/8\), and
a strictly negative bounded reference-packet witness.

This is an analytical specification, not a numerical or laboratory
demonstration.
\end{example}

The lower bound of Theorem~\ref{thm:lower} still applies at each
finite compact parameter choice. Arbitrarily accurate recovery in
a sequence is not exact recovery at one finite coupling.

\subsection{Complete leakage bounds in the transmission regime}

Let the compact model's
\(q_{\mathrm c}=e^{-2\|\widehat a_1\|_{\mathrm U}^2}\), and define
\[
L_{\mathrm{env}}
=\inf_\tau\ddiam(\widehat\NN_h^c,\RR_\tau).
\]
Then
\begin{equation}
\boxed{
q_{\mathrm c}/2\le L_{\mathrm{env}}
\le\min\{1,2\sqrt E\}.
}
\label{eq:leakagerange}
\end{equation}
The lower bound is Theorem~\ref{thm:lower}.
The upper bound follows from
Lemma~\ref{lem:leakageBell} and the decoder fidelity
\(F_e\ge1-E\).
It concerns the complete fixed complement, not just the source.

%=====================================================================
\section{Coherence witnesses, moments, and local observable access}
\label{sec:observables}
%=====================================================================

\subsection{A direct scalar coherence witness}

For any weighted \(l\), define the bounded effects
\[
E_{\sin,l}=\frac12(I+\Impart W(l)),\qquad
E_{\cos,l}=\frac12(I+\Repart W(l)).
\]
Consider source inputs \(\ket{\pm_X}\), which have identical
computational populations.

\begin{proposition}[Exact coherence-dependent scalar gap]
\label{prop:coherencegap}
The sine-effect probability gap for these two inputs is
\begin{equation}
\boxed{
\Delta_X(l)=
\frac{|\sin B_l|}{2}
       \left[\Qchar(l-2a_1)+\Qchar(l+2a_1)\right].
}
\label{eq:coherencegap}
\end{equation}
It is nonzero whenever \(\sin B_l\ne0\).
For the matched packet comparison, \(l=2\lambda v\) gives
\[
\Delta_X(l)
=\frac12\left(1+e^{-8\mathcal V_h\lambda^2}\right).
\]
\end{proposition}

\begin{proof}
The input difference is \(X=E_{01}+E_{10}\).
Equation~\eqref{eq:fourchars} gives
\[
\NN_G(X)(W(l))
=-\ii\sin B_l
       [\Qchar(l-2a_1)+\Qchar(l+2a_1)].
\]
Take the imaginary part and divide by two.
For the comparison, \(B_l=-2\theta_0=-\pi/2\);
one Gaussian argument is zero and the other is \(4\lambda v\).
\end{proof}

This diagnoses computational-basis coherence dependence.
Entanglement transmission is established by the separate referenced
witness in Theorem~\ref{thm:compacttransmission}, not by this
probability gap alone.

\subsection{A new moment bound for the two-pulse outputs}

\begin{definition}[Moment-dominated field class]
Let \(\mathfrak S_C\) consist of normal field states in the fixed
regular representation satisfying
\[
\omega(X(l)^2)\le C\|l\|_{\mathrm U}^2
\]
on the real weighted coefficient domain.
The moment includes any nonzero mean.
\end{definition}

\begin{theorem}[Uniform output moment domination]
\label{thm:moments}
For every source density, including every reference extension,
its two-pulse reduced field output belongs to
\(\mathfrak S_{C_G}\), where
\begin{equation}
\boxed{
C_G=\left[1+2(\|a_1\|_0+\|a_2\|_0)\right]^2.
}
\label{eq:CG}
\end{equation}
The compact-source output two-point functions retain the
Hadamard singularity of the baseline.
\end{theorem}

\begin{proof}
Let
\[
R=U_1^\dagger(X\otimes I)U_1.
\]
It is a self-adjoint unitary. Direct conjugation gives
\[
U_G^\dagger(I\otimes X(l))U_G
=I\otimes X(l)-Z\,\symp(a_1,l)-R\,\symp(a_2,l).
\]
On any initial purified source-reference vector times \(\Omega\),
the norm of this expression is at most
\[
\|l\|_{\mathrm U}
 +|\symp(a_1,l)|+|\symp(a_2,l)|
\le\sqrt{C_G}\|l\|_{\mathrm U}.
\]
Squaring proves the moment bound without assuming a Gaussian
output.

For compact source tests, the exact branch formulas are finite
sums of Weyl sandwiches of the initial Hadamard state.
Differentiating their characteristic functions twice shows that
the change of the two-point function consists of products of
smooth causal solutions and contractions of the original
two-point function against compact smooth source tests.
Such one-variable smearings of a Hadamard bisolution are smooth.
The two-point change is therefore smooth.
\end{proof}

The single-displacement constant of Em03-F is not reused for the
interference-bearing output.

\subsection{Local sine-effect access, not coherent access}

For \(\omega\in\mathfrak S_C\), exact Weyl relations and
Cauchy--Schwarz give
\[
|\omega(W(k))-\omega(W(l))|
\le\sqrt C\|k-l\|_{\mathrm U}
                         +|\symp(k,l)|/2.
\]
Hence either sine or cosine effect satisfies
\[
|\omega(E_{\bullet,k})-\omega(E_{\bullet,l})|
\le H_C(l)\|k-l\|_{\mathrm U},
\qquad
H_C(l)=\frac{\sqrt C+\|l\|_{\mathrm U}}2.
\]
For completeness, write \(k=l+d\), use the Weyl phase, and bound
\[
|\omega(W(l)(W(d)-I))|
\le\sqrt{\omega(X(d)^2)}.
\]
Also
\(|\symp(k,l)|\le2\|d\|_{\mathrm U}\|l\|_{\mathrm U}\).

Choose a measurement time slice strictly later than both source
slabs. The compact packet construction of
Lemma~\ref{lem:bridge} gives a real compact test \(B_L\) with
coefficient \(l_L\) and error \(\Delta_L\).

Use the independent massless scalar probe and compact bilinear
coupling described explicitly in Appendix~\ref{app:local}.
Its final effect is
\[
O_{\sin,\epsilon}
=\frac12(I+\Impart W_\psi(z/\epsilon)).
\]
Its exact induced system effect has the form
\[
\frac12\left[I+c_\epsilon\Impart W_\phi(B_\epsilon)\right],
\qquad
c_\epsilon=e^{-V_{\sigma_{L,\epsilon}}
                     (z_\epsilon,z_\epsilon)/2}.
\]

\begin{theorem}[Local probability comparison for G outputs]
\label{thm:local}
For every two-pulse source output, the local probe satisfies
\begin{equation}
\boxed{
\left|
(\omega\otimes\sigma_{L,\epsilon})
 \Theta_\epsilon(I\otimes O_{\sin,\epsilon})
-\omega(E_{\sin,l})
\right|
\le e_{\mathrm{loc}},
}
\label{eq:local}
\end{equation}
where
\[
e_{\mathrm{loc}}
=\min\left\{1,\,
(1-c_\epsilon)/2+
 H_{C_G}(l)(\Delta_L+d_{\mathrm{trf}}(L,\epsilon))\right\}.
\]
The transfer and noise quantities are explicitly specified in
Appendix~\ref{app:local}. For fixed source and target they can
meet every positive mathematical probability tolerance by first
enlarging the shell and then reducing \(\epsilon\), with the
stated preparation costs retained.

For the two source \(X\)-eigenstates, the local gap is at least
\[
\max\{0,\Delta_X(l)-2e_{\mathrm{loc}}\}.
\]
\end{theorem}

\begin{proof}
The source-output moment hypothesis is
Theorem~\ref{thm:moments}. Remove the induced attenuation at cost
\((1-c_\epsilon)/2\), then compare its coefficient with \(l\)
using the packet and transfer errors and the preceding Weyl
estimate. The gap follows by the reverse triangle inequality.
The finite-parameter schedule is proved in the appendix.
\end{proof}

This compares effect expectations. It does not approximate a
quantum-output receiver channel.

\subsection{Complete records and statistics}

Suppose final local readout has a separately justified complete-law
allowance \(e_{\mathrm{read},s}\), including all declared failure
records with fixed scores in \([0,1]\). For each preparation,
a sufficient observable allowance is
\[
e_s=e_{\mathrm{loc},s}+e_{\mathrm{read},s},
\]
capped at one. Any independently justified coefficient or source
error is added at its own comparison interface.

For independent attempted trials, with fresh preparations and
fixed parameters, define
\[
u_s=\sqrt{\frac{\log(4/\beta_{\mathrm{conf}})}{2n_s}},
\qquad 0<\beta_{\mathrm{conf}}<1.
\]
Then with probability at least \(1-\beta_{\mathrm{conf}}\),
\[
|\widehat p_+-\widehat p_-|
\ge\Delta_X(l)-e_+-e_--u_+-u_-.
\]
The proof is the exponential-moment bound
\(\Pr(|\widehat p-p|>u)\le2e^{-2nu^2}\) for independent
scores in \([0,1]\), followed by a union bound.
Every failure is scored, not removed.

These statistical and observable allowances are not inserted into
a coherent-recovery diamond-distance ledger.

%=====================================================================
\section{Coherent local reception remains a separate gate}
\label{sec:receiver}
%=====================================================================

The decoder \eqref{eq:decoder} acts on an identified outgoing
packet and a fresh qubit. Its CPTP definition is complete.
Its physical implementation has not been supplied.

A coherent local receiver would require a specified compact
field--probe interaction, independent receiver preparation,
a retained quantum memory output, and a comparison on the
source input including its reference. The general local
system--probe framework provides scattering and induced
observables \cite{FewsterVerch,FewsterJubbRuep};
it does not identify the above decoder with a local receiver.

Two valid future interfaces are:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item \textbf{Packet factorization:}
prove that the receiver acts through the named packets with
a fixed environmental channel or preparation;

\item \textbf{Direct source-to-memory calculation:}
derive the composed source--field--receiver channel without
assuming a packet/environment product state.
\end{enumerate}
The second option is especially relevant when unselected background
modes are correlated with the selected packet.

\begin{proposition}[Conditional coherent-receiver composition]
\label{prop:receiver}
Suppose a receiver and its implemented decoder have a justified
source-input referenced comparison
\[
\ddiam(\widehat\Lambda_{\mathrm{receiver}},
                         \DD_\lambda\widehat\NN_h)
\le e_{\mathrm{access}}+e_{\mathrm{dec}}.
\]
Then
\[
\ddiam(\widehat\Lambda_{\mathrm{receiver}},\id_2)
\le\min\{1,E_\lambda+e_{\mathrm{src}}
                         +e_{\mathrm{access}}+e_{\mathrm{dec}}\}.
\]
\end{proposition}

\begin{proof}
Use the triangle inequality and
Theorem~\ref{thm:compacttransmission}.
\end{proof}

The hypothesis is not supplied by the local sine-effect theorem.
It must include the actual receiver's unselected modes, noise,
initial preparations, and quantum outputs. In particular, a
protocol that first converts all message-bearing field influence
into classical records remains entanglement breaking.

No unrestricted bosonic diamond convergence follows merely from
coefficient convergence. The channel stability established here is
on the specified finite-dimensional source input and fixed background.

%=====================================================================
\section{Causality, interference energy, and switching work}
\label{sec:energy}
%=====================================================================

\subsection{Causal support}

\begin{proposition}[Spacelike observation invariance]
\label{prop:causality}
If a compact local field test \(G\) is spacelike separated from
\(\supp F_1\cup\supp F_2\), then
\[
U_G^\dagger(I\otimes W(k_G))U_G=I\otimes W(k_G).
\]
Before the first slab the propagator is identity; between slabs
only the first endpoint has acted.
\end{proposition}

\begin{proof}
Causal support gives
\(\symp(k(F_j),k_G)=\int F_jEG\,\dd\mathrm{vol}_g=0\).
Each controlled Weyl factor therefore commutes with the observable.
The chronological assertions follow from the disjoint switching
supports and Theorem~\ref{thm:unitary}.
\end{proof}

The source is exterior and spatially extended over a shell.
These field-observation support statements do not turn it into a
covariant microscopic matter theory. Asymptotic packets have
noncompact time profiles; their compact approximants have the
separate growing-support ledger already stated.

\subsection{The exact additional occupation kernel}

Let the initial source Bloch \(Y\)-component be \(u_y\), and retain
\(q=e^{-2\|a_1\|_{\mathrm U}^2}\).
For an incoming test \(v\), exact conjugation gives
\begin{equation}
U_G^\dagger a(v)U_G
=a(v)-\ii\langle v,a_1\rangle Z
                         -\ii\langle v,a_2\rangle R,
\qquad
R=U_1^\dagger XU_1.
\label{eq:annconjugation}
\end{equation}
The last source operator depends on the field; it must not be
replaced by an independent classical sign.

Let
\[
d_j=\mathsf S a_j,\qquad
e_1=\mathsf S\Covop a_1.
\]
The two radiation ends are indexed by
\(\alpha\in\{\infty,\mathrm{hor}\}\).

\begin{theorem}[Finite signed excess radiative energy]
\label{thm:energy}
The diagonal change of the outgoing occupation kernel is
\begin{equation}
\boxed{
\delta n_\alpha(\omega)
=|d_{1,\alpha}(\omega)|^2+|d_{2,\alpha}(\omega)|^2
-2q\,u_y\Impart\!
       \left(\overline{e_{1,\alpha}(\omega)}
                                d_{2,\alpha}(\omega)\right).
}
\label{eq:occupationchange}
\end{equation}
Its frequency-weighted integral is absolutely convergent:
\[
\delta E_\alpha=\int_0^\infty
                              \omega\delta n_\alpha(\omega)\dd\omega.
\]
The total excess is
\begin{equation}
\boxed{
\delta E_{\mathrm{tot}}
=\|\omega^{1/2}a_1\|_0^2+\|\omega^{1/2}a_2\|_0^2
-2q\,u_y\Impart
 \langle\omega^{1/2}\Covop a_1,\omega^{1/2}a_2\rangle.
}
\label{eq:totalenergy}
\end{equation}
It equals \(\delta E_\infty+\delta E_{\mathrm{hor}}\).

The change is not asserted nonnegative. The interference term can
change the background energy, and is not a rank-one mean-amplitude
addition of the Em03-F type.
\end{theorem}

\begin{proof}
Let \(N_{\mathrm{in}}=\diag(n_{\mathrm H},0)\).
Gaussian differentiation of the Weyl characteristic function gives
\[
\stateU(a(v)^\dagger\sin(2X(a_1)))
=2q\langle a_1,N_{\mathrm{in}}v\rangle,
\]
and the cosine contraction vanishes.
Also
\[
R=X\cos(2X(a_1))-Y\sin(2X(a_1)),\qquad
\langle ZR\rangle=\ii q u_y.
\]
Expand the occupation expectation using
\eqref{eq:annconjugation}. The squared displacement terms give
\(|d_1|^2+|d_2|^2\). Their source cross term is
\(-2q u_y\Impart(\overline{d_1}d_2)\).
The contractions with the background add
\(-4q u_y\Impart(
 \overline{(\mathsf SN_{\mathrm{in}}a_1)}d_2)\).
Since \(\Covop=I+2N_{\mathrm{in}}\), this is
\eqref{eq:occupationchange}. The computation can first be made
on compact-band test kernels and then extended by continuity.

Lemma~\ref{lem:compactregularity} and pointwise scattering
unitarity give
\[
\|\omega^{1/2}d_j\|_0<\infty,\qquad
\|\omega^{1/2}e_1\|_0<\infty.
\]
Cauchy--Schwarz proves absolute integrability of the interference
term. Summing the two ends and using scattering unitarity proves
\eqref{eq:totalenergy}.
\end{proof}

The energy is defined by this finite change of the two-point kernel,
not by subtracting two divergent stationary total energies.
A nonmonochromatic packet is not assigned the exact Hamiltonian
\(\omega_0N_h\).

\subsection{Switching work for both pulses}

The source has zero internal Hamiltonian. It does not contain an
internal-energy fuel for the process.

A precise bookkeeping realization is available in the doubled
representation. Its Killing generator is
\[
K_{\mathrm U}
=\dd\Gamma(\omega)_{\mathrm{first}}
                         -\dd\Gamma(\omega)_{\mathrm{purifying}},
\]
with \(K_{\mathrm U}\Omega=0\).
This signed generator implements physical Killing translations on
the Weyl algebra. Its change of expectation equals the
baseline-subtracted physical radiation energy. It is not a
positive total-energy operator for the stationary thermal
background.

\begin{proposition}[Two-pulse switching-work identity]
\label{prop:work}
For the prescribed source controls, the total switching work is
\begin{equation}
\boxed{
W_{\mathrm{ctrl}}
=\sum_{j=1}^2g_j
 \int \partial_tf_j(t,x)
       \langle O_j\Psi_{\mathrm{Sch}}(t,x)\rangle
                                             \dd t\,\dd x
=\delta E_{\mathrm{tot}},
}
\label{eq:work}
\end{equation}
where \(O_1=Z,\ O_2=X\), and the expectation is in the
chronologically evolved joint state. Both pulses are included.
The second expectation contains the correlations created by the first.
\end{proposition}

\begin{proof}
In the Schr\"odinger description the explicitly time-dependent
interaction is
\[
H_{\mathrm{int,Sch}}(t)
=\sum_jg_jO_j\int f_j(t,x)\Psi_{\mathrm{Sch}}(0,x)\dd x.
\]
On the common finite-energy form domain,
\[
\frac{\dd}{\dd t}
 \langle K_{\mathrm U}+H_{\mathrm{int,Sch}}(t)\rangle
=\langle\partial_tH_{\mathrm{int,Sch}}(t)\rangle.
\]
This follows by the Schr\"odinger equation; the commutator
terms cancel in the total generator.

The identity is legitimate for the present paths:
instantaneous compact spatial tests have arbitrarily many
high-frequency spectral moments, and the threshold estimates
control the required doubled energy moments at zero.
Weyl products preserve these form domains.
Alternatively one may first insert finite-particle and spectral
cutoffs, prove the identity there, and pass to the form limits
using these bounds.

The switched interaction vanishes at both endpoints, and the
initial generator expectation is zero. Integration gives the
first expression in \eqref{eq:work}.
The final generator expectation is the same Wick expansion as
the summed two-point change in Theorem~\ref{thm:energy},
with frequency inserted. It is exactly
\(\delta E_{\mathrm{tot}}\).
\end{proof}

There is no independent classical-mean energy formula for this
interference-bearing process. Negative excess energy, when it occurs,
means extraction of work from the specified initial field and source
preparation by the prescribed controls. No autonomous black-hole
energy balance, metric response, or mass-loss law is inferred.

%=====================================================================
\section{Resources, limitations, and the submission ledger}
\label{sec:resources}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.25\textwidth}p{0.65\textwidth}@{}}
\toprule
Resource or interface & Declared quantity and boundary\\
\midrule
\endfirsthead
\toprule
Resource or interface & Declared quantity and boundary\\
\midrule
\endhead
\bottomrule
\endfoot

Source &
One degenerate qubit, fixed encoding and Pauli axes.
Its exterior full-shell extension is an effective source law.\\

Switching &
Two real compact profiles with strictly ordered time slabs.
Overlapping noncommuting pulses are outside the exact
Hamiltonian theorem.\\

Physical radiation &
The massless Schwarzschild scalar s-wave.
The output packets are fixed scattering modes of that field,
with no added information-bearing payload.\\

Transmission scaling &
First comparison coefficient \(\lambda v\);
second \(\ii\pi v/(8\lambda)\).
The phase pairing is fixed while the first displacement and
its control effort grow.\\

Compact realization &
Filtered Cauchy-data cutoffs for both source profiles.
Their full shells may grow toward both radial ends.
For every finite cutoff their supports remain strictly exterior.\\

Approximation accuracy &
The source-input error
\(r_1+r_2+|\symp(a_1,\delta_2)|\), not an unrestricted
field-unitary operator-norm error.\\

Background &
The complete participating Gaussian covariance.
Selected and unselected modes need not be independent.
The full dilation retains background purification.\\

Packet decoder &
One fresh qubit in the \(Y=+1\) state, inverse two-pulse
packet unitary, and packet discard.
This is an explicit mathematical decoder, not a local receiver.\\

Local observable probe &
Independent massless probe, compact bilinear coupling,
scaled future readout test, and finite-plane squeezing.
It certifies one bounded probability, not quantum capture.\\

Probe effort &
Growing shell support; processing test proportional to
\(\epsilon^{-1}\); squeezing magnitude
\(2\log(1/\epsilon)\); diverging conjugate variance.\\

Energy and work &
Finite signed excess radiation energy, including interference.
The switching apparatus provides or extracts the computed work.
No gravitational backreaction is included.\\

Statistics &
Fresh independent attempted trials and complete visible failures.
One-shot channel bounds do not prove repeated-trial independence.\\

Physical identification &
No microscopic gravitational source law, interior algebra,
collapse construction, or evaporation endpoint dynamics.\\

\end{longtable}
\end{center}

\subsection{What has and has not been classified}

The manuscript proves:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item entanglement breaking for either single-observable reduction;
\item an exact channel for every ordered two-pulse parameter choice;
\item a nonzero finite-coupling recovery obstruction;
\item an explicit non-entanglement-breaking compact-source family
with arbitrarily small information-theoretic recovery certificates.
\end{enumerate}
It does not claim a necessary-and-sufficient entanglement-breaking
criterion for all two-pulse parameters, nor exact recovery optima
for the general interference channel.

\subsection{A physically justified source is a further programme}

Horizon response matching can constrain absorption and emission
correlators in a stated effective-theory regime
\cite{HorizonEFT}. It does not derive this degenerate exterior qubit,
its Pauli controls, or its unknown-input channel.

A successor must supply the source spectrum, state-dependent response,
transition matrix elements, higher correlation functions,
coupling motivation, and controlled error regime.
The local receiver also needs its own quantum-output proof.
No documentary corpus claim is a mathematical premise.

\subsection{Logical repairs retained in this release}

\begin{enumerate}[label=\textbf{D\arabic*.},leftmargin=*]
\item \textbf{Two-pulse time ordering is chronological composition.}
The single-observable central-phase calculation is used only
inside each nonoverlapping slab.

\item \textbf{All matrix units are evaluated.}
Nonzero off-diagonal output is shown explicitly, but is not
substituted for a referenced entanglement test.

\item \textbf{Interference terms are not Gaussian preparation states.}
Normality is obtained from the joint participating-mode density
and actual unitary evolution.

\item \textbf{The complete complement has a fixed purification.}
Retained-source distinguishability is only a lower bound on
complete environmental information.

\item \textbf{Metrics have different definitions.}
Bell fidelity, worst-case entanglement fidelity, and half-diamond
distance are linked only by the printed arguments and constants.

\item \textbf{The phase-matched packet interaction is first a comparison.}
Compact ordered source profiles are supplied with a proved
source-input error before physical scalar transmission is claimed.

\item \textbf{The decoder is not free local access.}
Its abstract packet operations do not identify a laboratory
receiver or a compact coherent field-capture map.

\item \textbf{The moment class is reproved for G.}
The field output may be non-Gaussian. Its uniform bound follows
from exact Heisenberg conjugation, not from F's displaced-state
formula.

\item \textbf{The energy contains interference.}
The excess is signed and absolutely integrable.
F's positive rank-one energy addition is not reused.

\item \textbf{Growing-resource convergence is not finite exact recovery.}
The retained source has \(q>0\) at every finite first coefficient.
All local shell and preparation costs remain in the ledger.

\item \textbf{The exterior model is not interior information escape.}
No horizon-crossing communication law or gravitational source
derivation is supplied.

\end{enumerate}

%=====================================================================
\section{Final analytical release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Em03-G ordered exterior-source release]
\label{thm:release}
Under the specified stationary scalar input, complete scattering,
and ordered effective exterior-source interactions:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item the two-pulse unitary evolution is defined on a common domain,
with the individual time-ordering phases retained;

\item the radiation channel is calculated on all source matrix units,
with exact Weyl characteristic functions and normal outgoing-packet
restrictions;

\item the retained-source channel and a complete fixed-interface
complementary channel are constructed;

\item finite first-pulse coefficients impose the recovery and
environmental lower bounds of Theorem~\ref{thm:lower};

\item a phase-matched packet comparison has the explicit decoder
and worst-case referenced recovery allowance of
Theorem~\ref{thm:matchedrecovery};

\item filtered compact ordered source profiles realize that comparison
with the source-input half-diamond error of
Theorem~\ref{thm:stability};

\item finite compact exterior models have provably entangled
reference-outgoing-packet states, and a growing-resource family
has recovery certificates tending to zero;

\item coherence-dependent scalar probability gaps and a common
output moment bound are proved, permitting separate local
bounded-observable certificates;

\item causal observation support, finite interference-bearing
radiative excess energy, and both-pulse switching work are calculated.
\end{enumerate}

The information-theoretic decoder assumes quantum access to the
identified outgoing packet. A coherent local receiver is not
constructed. Local observable approximation is not promoted to
quantum-output channel equivalence.

The source law is effective and exterior. No microscopic
gravitational derivation, black-hole interior information escape,
collapse theorem, Page curve, backreaction, mass-loss equation,
or complete evaporation result is established.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:unitary} proves (i).
Theorems~\ref{thm:matrixunits} and \ref{thm:normal} prove (ii).
Theorems~\ref{thm:source} and \ref{thm:complement} prove (iii).
Theorem~\ref{thm:lower} proves (iv).
Theorem~\ref{thm:matchedrecovery} proves (v).
Lemma~\ref{lem:bridge} and
Theorem~\ref{thm:stability} prove (vi).
Theorem~\ref{thm:compacttransmission} proves (vii).
Proposition~\ref{prop:coherencegap},
Theorem~\ref{thm:moments}, and
Theorem~\ref{thm:local} prove (viii).
Proposition~\ref{prop:causality},
Theorem~\ref{thm:energy}, and
Proposition~\ref{prop:work} prove (ix).

No local quantum-memory receiver or gravitational source law
is among the model inputs or constructions.
Their separate release gates are retained throughout.
\end{proof}

\paragraph{Final scope statement.}
Em03-G proves that ordered noncommuting effective exterior-source
interactions can transfer referenced quantum information into an
identified outgoing Schwarzschild scalar packet. It supplies exact
packet-channel formulas, finite-coupling obstructions, and a
compact-source family with explicit information-theoretic recovery
certificates. Coherent local access remains a separately identified
obligation; the local construction here certifies bounded
observable probabilities only. No microscopic gravitational
derivation, interior information escape, or evaporation theorem
is inferred.

%=====================================================================
\appendix
\section{Explicit local sine-probe parameters}
\label{app:local}
%=====================================================================

This appendix specifies the local quantities used in
Theorem~\ref{thm:local}. It specializes the compact linear
system--probe construction of Em03-D and proves the sine-effect
adaptation.

Fix a compact witness test \(B=\Bulk[b]\) constructed after the
source slabs. Choose a spacelike-compact free probe solution
\(v_{\mathrm{bulk}}\) equal to one near \(\supp B\).
Compact Cauchy data equal to the constant solution on the causal
predecessor of that set give it by finite propagation.
Write \(v_{\mathrm{bulk}}=Y_0v/r\).

Choose a switch \(\eta_+\) in a compact future slab, zero before
the slab and one after it. Define
\[
z_{\mathrm{red}}=-\Dop(\eta_+v),\qquad
z=\Bulk[z_{\mathrm{red}}].
\]
Then \(G_{\mathrm A}z=(1-\eta_+)v_{\mathrm{bulk}}=1\)
on the coupling support.

Use
\[
S_{\mathrm{probe,int}}
=-\epsilon\int\rho\phi\psi\,\dd\mathrm{vol}_g,\qquad
\rho=-B,
\]
\[
T_\epsilon=
\begin{pmatrix}P&\epsilon\rho\\
                \epsilon\rho&P\end{pmatrix}.
\]
Set
\[
w_\epsilon=G_{\mathrm A,T_\epsilon}(0,z)^{\mathsf T},
\quad
B_\epsilon=-\rho(w_\epsilon)_\psi,
\quad
z_\epsilon=z/\epsilon-\rho(w_\epsilon)_\phi.
\]
Advanced scattering gives
\[
\Theta_\epsilon(I\otimes W_\psi(z/\epsilon))
=W_\phi(B_\epsilon)\otimes W_\psi(z_\epsilon).
\]
It follows by applying
\[
\theta_\epsilon H
=H-(T_\epsilon-P\oplus P)G_{\mathrm A,T_\epsilon}H
\]
to \(H=(0,z/\epsilon)\). Green's identity and conserved coupled
symplectic current verify this test map and its Weyl morphism.

For any centered quasifree independent probe,
\[
\varepsilon_{\sigma,\epsilon}(O_{\sin,\epsilon})
=\frac12[I+c_\epsilon\Impart W_\phi(B_\epsilon)].
\]
The probe characteristic coefficient is real, so taking the
imaginary part commutes with the probe expectation. No mixed
tensor-factor Weyl phase is present.

Here are finite quantitative constants.
If the coupling is supported in \(I\times[-X,X]\), let
\(\Delta=|I|\), \(V_K=2X\Delta\), and
\[
a_{\mathrm p}=\fmet\rho=-Y_0b/r,\qquad
A_{\mathrm p}u=G_{\mathrm A,\Dop}(a_{\mathrm p}u),
\qquad
v_{\mathrm A}=G_{\mathrm A,\Dop}z_{\mathrm{red}}.
\]
On the coupling set, \(-a_{\mathrm p}v_{\mathrm A}=b\).

Use
\[
\|u\|_{\mathcal X}^2
=\sup_{t\in I}\sum_{j=0}^2
\left(
 \|\partial_t\partial_x^ju\|_2^2+
 \|\partial_x^{j+1}u\|_2^2+
 \|\partial_x^ju\|_2^2
\right).
\]
Choose
\[
c_2=64\left(1+\sum_{j=0}^2\|\partial_x^jV\|_\infty\right),
\]
\[
M_2=8\sup_{t\in I}
                     \sum_{j=0}^2\|\partial_x^ja_{\mathrm p}\|_\infty,
\qquad
\Lambda=\Delta e^{c_2\Delta}M_2.
\]
Spatial wave-energy differentiation through order two, the
product rule, and backward Gronwall inequality give
\[
\|A_{\mathrm p}u\|_{\mathcal X}\le\Lambda\|u\|_{\mathcal X}.
\]

For a test supported in \(|x|\le X\), the compact-mode estimate
in Lemma~\ref{lem:compactregularity} gives
\[
\|k(f)\|_{\mathrm U}\le\mathcal B_X(f),
\]
\[
\mathcal B_X(f)^2=
\frac{C_X^2}{2\pi}
  (A_0\|f\|_1^2+A_1\|\partial_tf\|_1^2),
\]
\[
A_0=\frac12+\frac{2}{\beta\omega_c},\qquad
A_1=\frac1{2\omega_c^2}+\frac{2}{3\beta\omega_c^3}.
\]
The constants are obtained by integrating the low-frequency
\(O(\sqrt\omega)\) and high-frequency \(O(\omega^{-3/2})\)
coefficient estimates, retaining the horizon weight.

With
\[
a_0=\|a_{\mathrm p}\|_\infty,\qquad
a_t=\|\partial_ta_{\mathrm p}\|_\infty,
\]
put
\[
L_{\mathrm p}
=\frac{2V_KC_X}{\sqrt{2\pi}}
                 \sqrt{A_0a_0^2+A_1(a_t+a_0)^2}.
\]
The one-dimensional Sobolev inequality gives
\[
\|k(a_{\mathrm p}u)\|_{\mathrm U}
\le L_{\mathrm p}\|u\|_{\mathcal X}.
\]

When \(\epsilon^2\Lambda^2<1\), the exact reduced solution is
\[
(w_\epsilon)_\psi
=(I-\epsilon^2A_{\mathrm p}^2)^{-1}v_{\mathrm A},\qquad
(w_\epsilon)_\phi=-\epsilon A_{\mathrm p}(w_\epsilon)_\psi.
\]
The Neumann series converges in \(\mathcal X\), has zero final
data, and solves the coupled equation; Cauchy uniqueness identifies
it with the exact advanced solution. Consequently
\[
d_{\mathrm{trf}}(L,\epsilon)
=L_{\mathrm p}
 \frac{\epsilon^2\Lambda^2}{1-\epsilon^2\Lambda^2}
                                   \|v_{\mathrm A}\|_{\mathcal X}
\]
bounds \(\|k(B_\epsilon)-l_L\|_{\mathrm U}\).

The residual probe test is
\[
(z_\epsilon)_{\mathrm{red}}
=z_{\mathrm{red}}/\epsilon+\epsilon j_\epsilon,
\quad
j_\epsilon=
a_{\mathrm p}A_{\mathrm p}
 (I-\epsilon^2A_{\mathrm p}^2)^{-1}v_{\mathrm A}.
\]
For \(\epsilon^2\Lambda^2\le1/2\),
\[
\|k(j_\epsilon)\|_{\mathrm U}
\le b_J:=2L_{\mathrm p}\Lambda\|v_{\mathrm A}\|_{\mathcal X}.
\]

Take an independent reference probe with Unruh covariance.
Choose a real compact test \(p_0\) with
\(c_0=\symp_\psi(z,p_0)\ne0\).
For small enough \(\epsilon\), set
\[
p_\epsilon=p_0/\symp_\psi(z_\epsilon,p_0).
\]
Define the symplectic transformation
\[
\begin{split}
S_\epsilon u={}&u-
 [\symp_\psi(u,p_\epsilon)z_\epsilon
   -\symp_\psi(u,z_\epsilon)p_\epsilon]\\
&+\epsilon^2\symp_\psi(u,p_\epsilon)z_\epsilon
 -\epsilon^{-2}\symp_\psi(u,z_\epsilon)p_\epsilon.
\end{split}
\]
It squeezes \(z_\epsilon\) by \(\epsilon^2\), squeezes its
conjugate by \(\epsilon^{-2}\), and fixes the symplectic
orthogonal complement. Prepare
\[
\sigma_{L,\epsilon}(W_\psi(u))
=\sigma_{\mathrm{ref}}(W_\psi(S_\epsilon u)).
\]
Symplecticity proves positivity and quasifreeness.
The finite-rank covariance change is smooth, by compact-test
smearing of the reference Hadamard bisolution; its Hadamard
singularity is unchanged.

Choose \(X_z\) containing the spatial support of \(z_{\mathrm{red}}\),
and set
\[
b_z=\mathcal B_{X_z}(z_{\mathrm{red}}),\qquad
N_L=(b_z+b_J)^2.
\]
For \(\epsilon\le1,\ \epsilon^2\Lambda^2\le1/2\),
\[
V_{\sigma_{L,\epsilon}}(z_\epsilon,z_\epsilon)
=\epsilon^4V_{\mathrm{ref}}(z_\epsilon,z_\epsilon)
\le N_L\epsilon^2,
\qquad
(1-c_\epsilon)/2\le N_L\epsilon^2/4.
\]
A sufficient normalization condition is
\[
2\epsilon^2b_J
 \sqrt{V_{\mathrm{ref}}(p_0,p_0)}<|c_0|/2.
\]
It follows by expanding the denominator as
\(c_0/\epsilon+\epsilon\symp_\psi(j_\epsilon,p_0)\).

Thus, with
\[
T_L=2L_{\mathrm p}\Lambda^2\|v_{\mathrm A}\|_{\mathcal X},
\]
the local theorem has the explicit sufficient allowance
\[
e_{\mathrm{loc}}
\le\min\left\{1,\,
H_{C_G}(l)\Delta_L+
       [H_{C_G}(l)T_L+N_L/4]\epsilon^2\right\}.
\]
First choose finite \(L\) to reduce the packet term, then finite
positive \(\epsilon\) to meet the remaining inequalities.
No rate in \(L\) is claimed.

Finally the uncertainty inequality gives
\[
V_{\sigma_{L,\epsilon}}(p_\epsilon,p_\epsilon)
\ge\frac1{4N_L\epsilon^2}.
\]
The processing test grows as \(\epsilon^{-1}\), and the squeeze
magnitude is \(2\log(1/\epsilon)\).
A fixed reference probe would instead have residual variance of
order \(\epsilon^{-2}\), suppressing the signal.
No effect is divided by its attenuation, and no bounded-energy
probe-preparation protocol is supplied.

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

\paragraph{Verification and data availability.}
The analytical specifications and proofs are contained in this source.
No numerical scattering calculation, executed tolerance schedule,
compiled archive, formal proof certificate, calibration record, or
apparatus demonstration is supplied or asserted.

\paragraph{Attribution.}
Controlled qubit--field displacements, nonperturbative scalar-field
communication, Gaussian restrictions, and general channel
recovery/complementarity methods are established constructions.
The manuscript claims the particular compact Schwarzschild
realization and its printed model-specific channel and error
calculations, not novelty for those underlying principles.

\paragraph{Predecessor status.}
The supplied Em03-A--F sources are used only through the identified
interfaces. The source, packet-channel, recovery, and stability
calculations required for this release are printed here.
Unrelated manuscript registry and documentary accounts are not
mathematical premises.

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

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N.~Cheung,
\emph{Em03-A: A Fixed-Background Hawking Emission Theorem},
version 1, 8 October 2026.
Supplied source \code{Em03-A-v1.tex}.
Stationary scalar prescription and outgoing occupation law;
not microscopic evaporation dynamics.

\bibitem{Em03B}
N.~Cheung,
\emph{Em03-B: A Certified Apparatus-Model Readout of One Hawking Packet},
version 2, 8 October 2026.
Supplied source \code{Em03-B-v2.tex}.
Effective capture and complete-record context;
not a local coherent receiver premise.

\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 Gaussian restrictions, phase-sensitive tests,
and reference-stable model comparisons.

\bibitem{Em03D}
N.~Cheung,
\emph{Em03-D: A Local System--Probe Realization
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version 1, 8 October 2026.
Supplied source \code{Em03-D-v1.tex}.
Filtered compact packet approximation and quantitative
full-shell local-observable construction.

\bibitem{Em03E}
N.~Cheung,
\emph{Em03-E: Microscopic Input Dependence and Certified Recovery
in a Dynamical Radiation Model},
version 1, 8 October 2026.
Supplied source \code{Em03-E.tex}.
Separate finite information-bearing payload model.

\bibitem{Em03F}
N.~Cheung,
\emph{Em03-F: Source-Dependent Information in Physical Scalar Radiation},
version 1, 8 October 2026.
Supplied source \code{Em03-F.tex}.
Single-observable exterior source, scalar packet identification,
and the entanglement-breaking benchmark.

\bibitem{Tjoa}
E.~Tjoa,
\emph{Non-perturbative simple-generated interactions with a quantum
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P.~Simidzija, A.~Ahmadzadegan, A.~Kempf, and
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arXiv:1908.07523.
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Nonperturbative scalar-field transmission precedent;
not the compact Schwarzschild packet certificate proved here.

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M.~Kasprzak and E.~Tjoa,
\emph{Transmission of quantum information through quantum fields
in curved spacetimes},
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\textbf{58} (2025), 095301.
arXiv:2408.00518.
doi:10.1088/1751-8121/adb66b.
Curved-spacetime scalar-field communication precedent.
Its receiver construction is not imported as the local receiver
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\bibitem{DMP}
C.~Dappiaggi, V.~Moretti, and N.~Pinamonti,
\emph{Rigorous construction and Hadamard property of the Unruh
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Advances in Theoretical and Mathematical Physics
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arXiv:0907.1034.
Declared bulk state-construction and Hadamard input.

\bibitem{Dimock}
J.~Dimock,
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General Relativity and Gravitation
\textbf{17} (1985), 353--369.
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Complete classical Schwarzschild wave-scattering input.

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C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
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Normally hyperbolic Cauchy theory and Green operators.

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M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai,
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Worst-case entanglement-fidelity recovery context;
its constants are not substituted for the direct qubit
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\bibitem{KSW}
D.~Kretschmann, D.~Schlingemann, and R.~F.~Werner,
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General channel-continuity and complementarity context.

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C.~J.~Fewster, I.~Jubb, and M.~H.~Ruep,
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Local-observable measurement context;
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\bibitem{HorizonEFT}
W.~D.~Goldberger and I.~Z.~Rothstein,
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\end{thebibliography}

\end{document}