%===================================================================== % Em03-F.tex % % Standalone source for pdfLaTeX. % % Build: % pdflatex -interaction=nonstopmode -halt-on-error Em03-F.tex % pdflatex -interaction=nonstopmode -halt-on-error Em03-F.tex % % No external bibliography, figures, generated data, or shell escape % are required. % % EVIDENCE BOUNDARY % Analytical constructions and proofs are supplied. % Compilation, formal verification, numerical execution, experimental % calibration, and laboratory realization are not asserted. % % RELEASE BOUNDARY % The source is an effective, spatially extended exterior qubit. % Its interaction is specified in normalized Schwarzschild Killing time. % Information is carried by identified modes of the massless scalar % field, not by an additional radiation payload. % % The single-source-observable radiation channel is entanglement % breaking. Classical source dependence is not quantum recovery. % % Local access concerns one bounded sine-Weyl effect. It uses growing % full-shell access and an explicitly squeezed independent probe. % No coherent field-to-memory channel or local instrument equivalence % is asserted. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Em03-F: Source-Dependent Information in Physical Scalar Radiation}, pdfauthor={Ngo Cheung}, pdfsubject={Exterior quantum source, controlled Weyl dynamics, outgoing scalar packets, local witnesses, and recovery obstruction} } \setcounter{secnumdepth}{3} \setcounter{tocdepth}{2} \numberwithin{equation}{section} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.2em} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{externalinput}{Declared external input} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{specification}[theorem]{Model specification} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\code}[1]{\texttt{\detokenize{#1}}} \newcommand{\HH}{\mathcal H} \newcommand{\FF}{\mathcal F_{\mathrm s}} \newcommand{\NN}{\mathcal N} \newcommand{\DD}{\mathcal D} \newcommand{\id}{\mathrm{id}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\,\mathrm d} \newcommand{\dist}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\tv}{d_{\mathrm{TV}}} \newcommand{\stateU}{\omega_{\mathrm U}} \newcommand{\sigm}{\sigma} \newcommand{\fmet}{\mathsf f} \newcommand{\Lop}{\mathsf L} \newcommand{\Dop}{\mathsf D} \newcommand{\Bulk}{\mathfrak F} \newcommand{\DeltaZ}{\Delta_Z} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Repart}{Re} \DeclareMathOperator{\Impart}{Im} \DeclareMathOperator{\diag}{diag} \title{\textbf{Em03-F}\\[1mm] \textbf{Source-Dependent Information in Physical Scalar Radiation}\\[2mm] \large Compact source--field dynamics, outgoing packet identification, bounded witnesses, and quantum-transfer limitations} \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 an exactly solvable effective exterior quantum source coupled to the massless scalar field on Schwarzschild. The benchmark is restricted to the s-wave sector and uses a spatially extended source with full angular-shell support. A degenerate qubit controls a smooth compact linear field interaction in normalized Killing time. We construct its unitary propagator, including the finite time-ordering phase, in a regular representation of the inherited Unruh state. Tracing the source gives a measure-and-prepare field channel. Its restriction to finitely many identified outgoing scalar packets is a normal channel whose two conditional outputs are oppositely displaced versions of the semiclassical Gaussian baseline. All source matrix units are evaluated. The channel transmits a classical source observable but is entanglement breaking. For a finite packet output, its optimal deterministic recovery error is expressed through the trace distance of the two conditional radiation states and is strictly greater than one half at every finite coupling. We prove a nonzero outgoing signal for a compact separable exterior profile. A bounded sine-Weyl effect distinguishes the source signs, whereas all selected joint number probabilities are identical for those signs. The phase-sensitive gap is linear at weak coupling, while the occupation-mean departure is quadratic. A compact local scalar probe realizes the sine-effect comparison. We verify a common second-moment bound for the source-displaced states and give separate packet, transfer, probe-noise, profile, readout, and statistical allowances. The local construction uses growing full-shell access, a scaled processing test, and a specified squeezed probe preparation. Causal support, retained-source dephasing, finite excess radiative energy, and switching work are calculated separately. The source law is a declared exterior model, not a derivation from microscopic quantum gravity. No information escape from a black-hole interior, coherent local field transfer, or evaporation theorem is established. \end{abstract} \tableofcontents %===================================================================== \section{Scope, dependencies, and the change from Em03-E} \label{sec:scope} %===================================================================== The construction follows \[ \boxed{ \text{source input} \longrightarrow \text{compact source--field dynamics} \longrightarrow \text{identified outgoing scalar packets} \longrightarrow \text{bounded source-dependent witness}. } \] Em03-E \cite{Em03E} uses an independently specified information-bearing payload beside occupation registers. Here there is no such payload. Every radiation output used for the information claim is a restriction of the massless scalar field already appearing in Em03-A--D. The interaction is a deliberately solvable benchmark. Its simple-generated structure and exact qubit--field calculations have established precedents \cite{Tjoa}; neither that interaction class nor controlled Weyl displacement is claimed as new. The present construction identifies the output with Schwarzschild scattering packets, proves a compact-profile signal, evaluates its witnesses and recovery limitations, and composes it with a quantitative local probe scheme. \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabular}{@{}p{0.37\textwidth}p{0.53\textwidth}@{}} \toprule Ingredient & Status\\ \midrule Schwarzschild background and normalized time & Specified as in Em03-A.\\ Stationary scalar scattering & Declared external input, with normalizations printed below.\\ Unruh covariance and bulk identification & Inherited prescription; bulk existence and Hadamard property remain external inputs.\\ Finite packet Gaussian restriction & Constructed below from the stated covariance.\\ Compact s-wave estimates & Reproduced in the needed specialization of Em03-D.\\ Source, encoding, and interaction & New effective-model specifications.\\ Source-to-scalar radiation channel & Calculated from the specified evolution.\\ Local sine-effect comparison & Constructed and proved below.\\ Gravitational origin of the source law & Not established.\\ \bottomrule \end{tabular} \end{center} Em03-B and Em03-C \cite{Em03B,Em03C} provide operational context, not a local source or an automatically available coherent scalar receiver. Em03-D \cite{Em03D} supplies the relevant full-shell local-observable architecture. Its cosine readout is not used as though it distinguished opposite displacements: the imaginary part is treated explicitly here. \paragraph{Evidence boundary.} The manuscript supplies analytical arguments. Compilation, formal verification, numerical execution, calibration, and physical implementation are not asserted. ``Certified'' refers to the printed mathematical estimates under their stated hypotheses. \paragraph{Distance conventions.} For normal states and channels, \[ \dist(\rho,\tau)=\frac12\|\rho-\tau\|_1,\qquad \ddiam(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond. \] The channel metric retains arbitrary external references. An expectation comparison is not silently promoted to a diamond-distance comparison. %===================================================================== \section{Exterior field, scattering coordinates, and infrared control} \label{sec:field} %===================================================================== \subsection{Geometry and units} In physical units, \[ g=-\left(1-\frac{r_{\mathrm h}}r\right)\dd t^2 +\left(1-\frac{r_{\mathrm h}}r\right)^{-1}\dd r^2 +r^2\dd\Omega^2, \qquad r_{\mathrm h}=2GM. \] The time generator is normalized at infinity. For the estimates, measure length in units of \(r_{\mathrm h}\). Thus, after rescaling, \[ r_{\mathrm h}=1,\qquad \fmet(r)=1-r^{-1},\qquad x=r+\log(r-1),\qquad \beta=4\pi. \] A physical frequency is \(r_{\mathrm h}^{-1}\omega\), and a physical energy is \(r_{\mathrm h}^{-1}\) times the displayed dimensionless energy. Packet normalization is preserved by \[ \widetilde h(\widetilde\omega) =r_{\mathrm h}^{-1/2} h(\widetilde\omega/r_{\mathrm h}). \] A dimensionless bulk smearing used below corresponds to a physical smearing with factor \(r_{\mathrm h}^{-3}\). The right exterior is globally hyperbolic. We use \[ P=-\Box_g,\qquad E=G_{\mathrm A}-G_{\mathrm R}. \] The free scalar action is \[ S_0=-\frac12\int\nabla_a\phi\nabla^a\phi\,\dd\mathrm{vol}_g. \] Let \(Y_0=(4\pi)^{-1/2}\). In the s-wave sector, \[ \phi(t,r,\Omega)=\frac{Y_0}{r}\Psi(t,x), \] \[ P\left(\frac{Y_0}{r}u\right) =\frac{Y_0}{r\fmet(r)}\Dop u,\qquad \Dop=\partial_t^2+\Lop,\qquad \Lop=-\partial_x^2+V(x),\qquad V(x)=\frac{\fmet(r(x))}{r(x)^3}. \] For a real reduced test \(f\in C_c^\infty(\mathbb R^2)\), put \begin{equation} \Bulk[f](t,r,\Omega) =\frac{Y_0}{r\fmet(r)}f(t,x). \label{eq:bulk} \end{equation} Then \[ \phi(\Bulk[f]) =\int_{\mathbb R^2}f(t,x)\Psi(t,x)\dd t\,\dd x. \] The factor follows from \(\dd\mathrm{vol}_g=r^2\fmet(r)\dd t\,\dd x\,\dd\Omega\). \subsection{External field inputs} \begin{externalinput}[Stationary scattering and free evolution] The incoming horizon and infinity modes give complete stationary scattering for the massless Schwarzschild scalar. In the s-wave sector there is a unitary spectral transform \[ \mathcal U:L^2(\mathbb R,\dd x) \longrightarrow L^2((0,\infty),\dd\omega;\mathbb C^2) \] intertwining \(\Lop\) with multiplication by \(\omega^2\). The flux-normalized two-channel scattering matrix is unitary. The free Cauchy problem and advanced/retarded Green operators have their usual causal support. \end{externalinput} The classical scattering input is the Schwarzschild wave-scattering construction \cite{Dimock}; the Green-operator input is standard normally hyperbolic theory \cite{BGP}. The specific weighted estimates needed here are proved below. \begin{externalinput}[Selected quantum state] The initial field is the centered, regular, gauge-invariant quasifree Unruh prescription. The incoming horizon sector has occupation \(n_{\mathrm H}(\omega)=(e^{\beta\omega}-1)^{-1}\); the incoming infinity sector is vacuum; the sectors are independent. This prescription has the bulk Hadamard identification of Dappiaggi--Moretti--Pinamonti \cite{DMP}. \end{externalinput} The global Hadamard proof is not reproduced. Hadamard regularity alone would not select this state. \subsection{Mode normalization} Let \(f_+(x,\omega)\sim e^{\ii\omega x}\) at \(+\infty\) and \(f_-(x,\omega)\sim e^{-\ii\omega x}\) at \(-\infty\). Define \[ \mathcal W(\omega)=f_-f_+'-f_-'f_+,\qquad \tau(\omega)=\frac{2\ii\omega}{\mathcal W(\omega)}, \] \[ q_{\mathrm H}=\tau f_+,\qquad q_{\mathrm I}=\tau f_-. \] Let \(\rho_{\mathrm I}\) be the outgoing-infinity reflection amplitude of the infinity-incident mode. The spectral convention is \[ (\mathcal Uu)_j(\omega) =\frac1{\sqrt{2\pi}}\int \overline{q_j(x,\omega)}u(x)\dd x. \] The canonical field expansion is \[ \Psi(t,x)=\sum_{j=\mathrm H,\mathrm I}\int_0^\infty \frac{q_j(x,\omega)a_j(\omega)e^{-\ii\omega t} +\overline{q_j(x,\omega)}a_j(\omega)^\dagger e^{\ii\omega t}} {\sqrt{4\pi\omega}}\dd\omega. \] Annihilation tests are conjugate-linear: \[ a(k)=\sum_j\int\overline{k_j(\omega)}a_j(\omega)\dd\omega. \] Write \[ X(k)=a(k)+a(k)^\dagger,\qquad W(k)=e^{\ii X(k)}, \] \[ \sigm(k,l)=2\Impart\langle k,l\rangle,\qquad W(k)W(l)=e^{-\ii\sigm(k,l)/2}W(k+l). \] For a compact reduced test, \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, \qquad \phi(\Bulk[f])=X(k(f)). \label{eq:coefficient} \end{equation} The common covariance norm is \begin{equation} \|k\|_{\mathrm U}^2 =\int_0^\infty \left[(1+2n_{\mathrm H})|k_{\mathrm H}|^2 +|k_{\mathrm I}|^2\right]\dd\omega. \label{eq:Unorm} \end{equation} The ordinary two-channel norm is denoted \(\|k\|_0\). It is dominated by \(\|k\|_{\mathrm U}\). In the baseline, \[ \stateU(X(k)^2)=\|k\|_{\mathrm U}^2,\qquad \stateU(W(k))=e^{-\|k\|_{\mathrm U}^2/2}. \] \subsection{A compact-test estimate} \begin{lemma}[S-wave threshold bound] \label{lem:threshold} The potential is smooth and nonnegative, \[ \int V\dd x=\frac12,\qquad \int(1+|x|)V(x)\dd x<\infty. \] The zero-energy Jost solutions are \[ f_-(x,0)=r(x),\qquad f_+(x,0)=-r(x)\log\fmet(r(x)), \qquad \mathcal W(0)=-1. \] Choose \(\omega_c>0\) with \(|\mathcal W(\omega)|\ge1/2\) on \([0,\omega_c]\). For every \(X>0\), a finite constant \(C_X\) exists with \[ |q_j(x,\omega)|\le C_X\min\{\omega/\omega_c,1\}, \qquad |x|\le X. \] One explicit choice is \[ \begin{split} m_X=\max\Bigl\{& \int_{-X}^{\infty}(y+X)V(y)\dd y,\, \int_{-\infty}^{X}(X-y)V(y)\dd y\Bigr\},\\ C_X&=\max\{4\omega_c e^{m_X},e^{1/(2\omega_c)}\}. \end{split} \] \end{lemma} \begin{proof} Use \(\dd x=\fmet^{-1}\dd r\) to obtain \(\int V\dd x=\int_1^\infty r^{-3}\dd r=1/2\). The first moment follows from logarithmic horizon behavior and \(x=r+O(\log r)\) at infinity. Direct substitution gives the zero-energy solutions and their Wronskian. The first-moment condition makes the Jost Volterra series continuous at zero, so its nonzero Wronskian supplies \(\omega_c\). In the low-frequency Volterra series use \(|\sin(\omega z)/\omega|\le |z|\). This gives \(|f_\pm|\le e^{m_X}\) on the compact region; \(|\tau|\le4\omega\) gives the threshold bound. For \(\omega\ge\omega_c\), the series is bounded by \(e^{\|V\|_1/\omega_c}\), and current conservation gives \(|\tau|\le1\). \end{proof} Define \[ A_0=\frac12+\frac{2}{\beta\omega_c},\qquad A_1=\frac1{2\omega_c^2}+\frac{2}{3\beta\omega_c^3}, \] \begin{equation} \mathcal B_X(f)^2 =\frac{C_X^2}{2\pi} \left(A_0\|f\|_1^2+A_1\|\partial_tf\|_1^2\right). \label{eq:Bbound} \end{equation} \begin{lemma}[Weighted compact-test continuity] \label{lem:compact} For a real compact smooth test supported in \(|x|\le X\), \[ \|k(f)\|_{\mathrm U}\le\mathcal B_X(f). \] Furthermore, \begin{equation} \int_0^\infty\omega\sum_j|k_j(f)(\omega)|^2\dd\omega \le\frac{C_X^2}{2\pi} \left(\frac{\omega_c}{3}\|f\|_1^2 +\frac1{\omega_c}\|\partial_tf\|_1^2\right). \label{eq:energybound} \end{equation} \end{lemma} \begin{proof} At low frequency, \[ |k_j(f)|\le \frac{C_X\sqrt\omega}{\sqrt{4\pi}\omega_c}\|f\|_1. \] At high frequency, one time integration by parts gives \[ |k_j(f)|\le \frac{C_X}{\sqrt{4\pi}\omega^{3/2}}\|\partial_tf\|_1. \] Use \(1+2n_{\mathrm H}\le1+2/(\beta\omega)\), sum both channels, and integrate the two intervals. This gives \eqref{eq:Bbound}. Integrating the same estimates with weight \(\omega\) gives \eqref{eq:energybound}. \end{proof} These estimates retain the thermal horizon sector. They do not replace unselected radiation by vacuum. %===================================================================== \section{The source and its exact compact interaction} \label{sec:interaction} %===================================================================== \subsection{Encoding and localization} The message and source are qubits. The encoding is the fixed unitary identification \[ V_{\mathrm{enc}}\ket0_M=\ket+_S,\qquad V_{\mathrm{enc}}\ket1_M=\ket-_S, \] where \[ Z_S\ket s=s\ket s,\qquad s\in\{+1,-1\},\qquad H_S=0. \] The symbols \(\ket+\), \(\ket-\) here denote \(Z_S\) eigenstates, not \(X\) eigenstates. The initial state is \[ \rho_{RS}\otimes\stateU. \] There is no message dependence in the initial field and no additional information-bearing radiation factor. Choose a real compact reduced profile \(f\), and set \(F=\Bulk[f]\). Its support is strictly exterior for every fixed compact spatial support. The same qubit controls the full shell. This is an effective extended source in a specified slicing, not a covariant microscopic matter field. \subsection{The actual Hamiltonian path} The lapse and spatial volume on a time slice are \[ N=\sqrt{\fmet(r)},\qquad \dd\Sigma_t=r^2\sqrt{\fmet(r)}\,\dd x\,\dd\Omega. \] Consequently \[ \Phi_F(t)=\int_{\Sigma_t}NF(t,\cdot)\phi(t,\cdot)\dd\Sigma_t =\int_{\mathbb R}f(t,x)\Psi(t,x)\dd x, \] and \[ \int\Phi_F(t)\dd t=\phi(F). \] The interaction-picture Hamiltonian is \begin{equation} H_I(t)=g\,Z_S\otimes\Phi_F(t),\qquad g\in\mathbb R. \label{eq:HI} \end{equation} The corresponding action term is \[ S_{\mathrm{int}}=-g\int F(x)Z_S\phi(x)\dd\mathrm{vol}_g. \] Define the instantaneous coefficient \[ k_{t,j}(\omega)= \frac{e^{\ii\omega t}}{\sqrt{4\pi\omega}} \int f(t,x)\overline{q_j(x,\omega)}\dd x. \] Then \(k_F=\int k_t\dd t=k(f)\). \begin{lemma}[Instantaneous coefficient regularity] \label{lem:instant} The map \(t\mapsto k_t\) is smooth in the weighted norm and has compact temporal support. Each \(\Phi_F(t)=X(k_t)\) is a well-defined essentially self-adjoint field operator on the standard finite-particle core of a quasifree Fock realization. \end{lemma} \begin{proof} The compact-region threshold bound controls the weighted low-frequency integral uniformly on compact time intervals. At high frequency, \[ k_t=e^{\ii\omega t}\mathcal U(f(t,\cdot))/\sqrt{2\omega}. \] Time derivatives introduce finite powers of \(\omega\) and derivatives of \(f\). Spectral unitarity and the compact spatial smoothness of \(f\) give the required bounds by applying sufficiently many powers of \(\Lop\) to those derivatives. Thus all time derivatives exist in the weighted norm. For clarity, a doubled Fock realization is available. On its one-particle space use the real-linear map \[ T k= \bigl((\sqrt{1+n_{\mathrm H}}\,k_{\mathrm H},k_{\mathrm I}), \sqrt{n_{\mathrm H}}\,\overline{k_{\mathrm H}}\bigr). \] It obeys \[ \|Tk\|^2=\|k\|_{\mathrm U}^2,\qquad 2\Impart\langle Tk,Tl\rangle=\sigm(k,l). \] The vacuum Weyl characteristic function is the required one. The Segal field \(a(Tk)+a(Tk)^\dagger\) is essentially self-adjoint on finite-particle vectors: the usual creation/annihilation estimates make those vectors analytic for each such field. This realizes \(X(k_t)\) and its closure. It does not posit a global stationary density in the physical, undoubled radiation Fock space. \end{proof} \subsection{Time ordering without an unbounded Duhamel shortcut} For \(t\ge t_0\), define \[ K(t,t_0)=\int_{t_0}^{t}k_u\dd u, \] \begin{equation} \gamma(t,t_0) =-\frac{g^2}{2}\int_{t_0}^{t}\dd u \int_{t_0}^{u}\dd v\,\sigm(k_u,k_v). \label{eq:phase} \end{equation} \begin{theorem}[Exact Hamiltonian evolution] \label{thm:unitary} The interaction \eqref{eq:HI} has the unitary propagator \begin{equation} \boxed{ U(t,t_0)=e^{\ii\gamma(t,t_0)} \sum_{s=\pm1}\ketbra{s}{s}\otimes W(-sgK(t,t_0)). } \label{eq:propagator} \end{equation} It satisfies the composition law and the Schr\"odinger equation on the common field-number form domain in the doubled Fock realization. After the source has switched off, \begin{equation} \boxed{ U_F=e^{\ii\gamma_F} \sum_{s=\pm1}\ketbra{s}{s}\otimes W(-sgk_F). } \label{eq:UF} \end{equation} Its adjoint action depends only on the free test class of \(F\); the time-ordering phase is common to both source branches. \end{theorem} \begin{proof} All integrals in \eqref{eq:phase} are finite by Lemma~\ref{lem:instant} and \(|\sigm(k,l)|\le2\|k\|_{\mathrm U}\|l\|_{\mathrm U}\). Weyl operators are unitary, so the displayed formula is unitary. A chronological product of short interactions has factors \(W(-sgk_{t_j}\Delta t_j)\), with later times on the left. The exact Weyl product gives the phase \[ -\frac{g^2}{2} \sum_{j>i}\sigm(k_{t_j},k_{t_i})\Delta t_j\Delta t_i. \] Smooth weighted continuity makes its limit \eqref{eq:phase}; the summed coefficients converge to \(K(t,t_0)\). Strong continuity of the Weyl representation gives \eqref{eq:propagator}. Splitting the same integrals proves the propagator composition law. The number form domain is invariant under Weyl operators. Indeed the displacement identity for annihilators and the creation/annihilation estimates bound its graph norm after a finite displacement. On this domain, a differentiable one-particle path \(r(t)\) obeys \[ \frac{\dd}{\dd t}W(r(t)) =\left[\ii X(\dot r(t)) +\frac{\ii}{2}\sigm(\dot r(t),r(t))\right]W(r(t)). \] This follows from the Weyl relation and the Stone derivative of \(W(\delta r)\); approximation by coherent finite-particle vectors extends it in the form-domain graph topology. For \(r=-sgK\), the scalar derivative term is \(\ii g^2\sigm(k_t,K)/2\). It cancels \(\ii\dot\gamma\), leaving \(\partial_tU=-\ii H_I(t)U\). This also establishes uniqueness on that domain by comparing two solutions and differentiating their unitary relative propagator. Finally \(\phi(Pu)=0\), so replacing \(F\) by \(F+Pu\) does not change \(k_F\). A different path phase is an overall phase and leaves the adjoint action unchanged. \end{proof} No operator-norm estimate for an unbounded field Hamiltonian has been inferred from the finite-generator estimate of Em03-E. The construction uses exact Weyl products and a stated domain. %===================================================================== \section{The source-to-field channel and retained-source back-action} \label{sec:channels} %===================================================================== For the rest of the manuscript abbreviate \[ a=gk_F,\qquad r_s=-sa. \] Define the conditional field states \[ \omega_s(A)=\stateU(W(r_s)^\dagger A W(r_s)). \] \begin{theorem}[All source matrix units] \label{thm:fieldchannel} The field-output map is the channel into field state functionals \begin{equation} \boxed{ \NN_F(\rho)= \sum_{s=\pm1}\bra s\rho\ket s\,\omega_s. } \label{eq:fieldchannel} \end{equation} Its Heisenberg adjoint is the unital CP map \[ \NN_F^*(A)=\sum_s\omega_s(A)\ketbra{s}{s}. \] In particular, \[ \NN_F(\ketbra{+}{+})=\omega_+,\qquad \NN_F(\ketbra{-}{-})=\omega_-, \] \begin{equation} \boxed{ \NN_F(\ketbra{+}{-}) =\NN_F(\ketbra{-}{+})=0. } \label{eq:offdiagonalzero} \end{equation} For every reference-source input, its referenced output is \[ \sum_s \bra s\rho_{RS}\ket s\otimes\omega_s. \] \end{theorem} \begin{proof} Expand the controlled unitary \eqref{eq:UF}. Tracing the source keeps exactly its diagonal blocks. The global phase cancels. This proves the channel formula and all matrix-unit identities. Positivity of each state functional proves complete positivity of the displayed adjoint; their normalization proves unitality. The same block calculation retains \(R\). \end{proof} \begin{theorem}[Exact retained-source channel] \label{thm:backaction} The retained-source channel preserves diagonal matrix units and multiplies each off-diagonal one by \begin{equation} \boxed{ \chi_F=\stateU(W(2a))=e^{-2\|a\|_{\mathrm U}^2}. } \label{eq:sourcecoherence} \end{equation} Equivalently, \[ \mathcal B_F=\chi_F\id+(1-\chi_F)\DeltaZ. \] Its half-diamond distance from the identity is \((1-\chi_F)/2\), and its Bell-input fidelity is \((1+\chi_F)/2\). \end{theorem} \begin{proof} The coefficient of \(\ketbra{s}{t}\) after tracing the field is \[ \stateU(W(r_t)^\dagger W(r_s)) =\stateU(W(-(s-t)a)). \] The branch directions are collinear, so there is no Weyl phase. For \(s\ne t\), \((s-t)^2=4\), giving \eqref{eq:sourcecoherence}. The resulting qubit channel is \[ \frac{1+\chi_F}{2}\id+ \frac{1-\chi_F}{2}\Ad_Z. \] On a Bell input its two Pauli outputs are orthogonal; the triangle inequality supplies the matching diamond upper bound. Their identity weight is the Bell fidelity. \end{proof} For superposed source inputs, the joint evolution creates source--field correlations. Removing the retained source removes the off-diagonal source blocks from the radiation channel; it is not a claim that the joint unitary itself destroys coherence. \paragraph{Complementary-channel boundary.} The retained-source channel is not the complete complement of a selected packet channel. That complement would also contain unselected scalar modes, the future-horizon output, and a specified purification of the mixed background. No complementary distance is asserted from \(\mathcal B_F\) alone. %===================================================================== \section{Outgoing scalar packet identification and normality} \label{sec:packets} %===================================================================== For a normalized compact-band packet \(h\), stationary scattering gives \[ b_h=a_{\mathrm{out}}(h)=a(v_h),\qquad v_h=(\overline\tau h,\overline{\rho_{\mathrm I}}h). \] Thus \[ \|v_h\|_0=1,\qquad Q_h=X(v_h)/\sqrt2,\qquad P_h=X(\ii v_h)/\sqrt2. \] Define the outgoing source coefficient \begin{equation} d_\infty(\omega) =\tau(\omega)k_{F,\mathrm H}(\omega) +\rho_{\mathrm I}(\omega)k_{F,\mathrm I}(\omega), \qquad c_h=\langle v_h,k_F\rangle =\int\overline h\,d_\infty\dd\omega. \label{eq:outcoefficient} \end{equation} If \(\mathsf S(\omega)\) is the complete scattering matrix, the other component of \(\mathsf S k_F\) is denoted \(d_{\mathrm{hor}}\). \begin{proposition}[Calculated packet displacement] \label{prop:displacement} The conditional outgoing packet amplitude is \begin{equation} \boxed{ \alpha_{s,h}=-\ii s g\,c_h. } \label{eq:alpha} \end{equation} Consequently, \[ \omega_s(Q_h)=\sqrt2\Repart\alpha_{s,h},\qquad \omega_s(P_h)=\sqrt2\Impart\alpha_{s,h}. \] These are calculated from the bulk profile and scattering map. \end{proposition} \begin{proof} The Weyl displacement identity gives \[ W(r)^\dagger a(v)W(r)=a(v)+\ii\langle v,r\rangle. \] Insert \(r=-sgk_F\). The initial state is centered, so its conditional mean is \eqref{eq:alpha}. Taking the quadrature combinations proves the result. \end{proof} \subsection{Finite packet densities} Choose orthonormal \(h_1,\ldots,h_m\) in a common compact positive-frequency band. Set \(b_i=b_{h_i}\) and \[ A_{ij}=\stateU(b_j^\dagger b_i) =\int\overline{h_i}h_j \frac{|\tau(\omega)|^2}{e^{\beta\omega}-1}\dd\omega. \] This convention retains the covariance transposition: \(\stateU(b_i^\dagger b_j)=A_{ji}\). Let \[ D(z)=\exp\left(\sum_i z_i b_i^\dagger-\overline{z_i}b_i\right). \] \begin{lemma}[Normal finite-mode baseline] \label{lem:gaussian} The selected baseline has a unique normal density \(\rho_0\) on \(\FF(\mathbb C^m)\) with \[ \tr(\rho_0D(z)) =\exp[-z^\dagger(A+I/2)z]. \] If \(A=W\diag(\lambda_1,\ldots,\lambda_m)W^\dagger\), it is the passive rotation of the product densities \[ \bigotimes_j \left[ \frac1{1+\lambda_j} \sum_{n=0}^\infty \left(\frac{\lambda_j}{1+\lambda_j}\right)^n\ketbra n n \right]. \] It commutes with total number. \end{lemma} \begin{proof} The restricted covariance is finite, positive, centered, and has zero anomalous part. Construct independent circular complex Gaussian amplitudes of variances \(\lambda_j\), rotate them by \(W\), and form their coherent-state mixture. It is positive trace class of trace one and has exactly the displayed characteristic function. For one variance \(\lambda>0\), angular integration kills off-diagonal number terms and radial integration gives \(\lambda^n/(1+\lambda)^{n+1}\). Zero variance means vacuum. Equality of finite-dimensional Weyl characteristic functions identifies the normal density and proves uniqueness. Passive rotation preserves total number. \end{proof} \begin{theorem}[Normal scalar-output channel] \label{thm:packetnormal} Define \[ \boldsymbol\alpha_s =(-\ii sg c_{h_1},\ldots,-\ii sg c_{h_m}). \] The conditional finite packet states are \begin{equation} \boxed{ \rho_s=D(\boldsymbol\alpha_s)\rho_0 D(\boldsymbol\alpha_s)^\dagger. } \label{eq:packetstates} \end{equation} The source-to-packet channel is the normal CPTP map \[ \NN_{\boldsymbol h}(X) =\sum_s\bra sX\ket s\,\rho_s. \] This includes all source matrix units and all references. \end{theorem} \begin{proof} Conjugating a selected Weyl operator by \(W(r_s)\) multiplies its baseline characteristic function by the displacement phase determined by \eqref{eq:alpha}. This is the characteristic function of \eqref{eq:packetstates}. The latter is a normal density by Lemma~\ref{lem:gaussian}, so characteristic-function uniqueness proves normality of the restricted output. The channel formula follows from Theorem~\ref{thm:fieldchannel} and is CPTP by its measure-and-prepare form. \end{proof} The selected packets do not exhaust the source signal. The components of \(d_\infty\) orthogonal to their span and the coefficient \(d_{\mathrm{hor}}\) remain unselected. The retained source also remains a physical factor of the model. %===================================================================== \section{Quantum-transfer obstruction and exact recovery optimum} \label{sec:obstruction} %===================================================================== The measure-and-prepare characterization of entanglement-breaking channels is standard \cite{EntanglementBreaking}. The following bounds are proved directly for the present normal packet channel. \begin{theorem}[Referenced quantum-transfer obstruction] \label{thm:EB} For every reference-source density, \[ (\id_R\otimes\NN_{\boldsymbol h})(\rho_{RS}) =\sum_s\bra s\rho_{RS}\ket s\otimes\rho_s \] is separable across reference and radiation. Every radiation-only deterministic CPTP decoder to a qubit has Bell-input fidelity at most \(1/2\) and recovery error at least \(1/2\): \[ F_e(\DD\NN_{\boldsymbol h})\le\frac12,\qquad \ddiam(\DD\NN_{\boldsymbol h},\id_M)\ge\frac12. \] \end{theorem} \begin{proof} The displayed positive product decomposition proves separability. Local decoding preserves it. A normalized product vector has overlap at most \(1/2\) with \[ \ket\Phi=\frac{\ket{0,+}+\ket{1,-}}{\sqrt2}, \] by Cauchy--Schwarz. Convexity extends this bound to separable states. Testing the Bell projector gives trace-distance error at least \(1-F_e\ge1/2\), hence the diamond lower bound. \end{proof} Let \[ d_{\boldsymbol h}=\dist(\rho_+,\rho_-). \] \begin{theorem}[Exact optimal deterministic packet recovery] \label{thm:recovery} For the declared finite packet output, \begin{equation} \boxed{ \sup_{\DD}F_e(\DD\NN_{\boldsymbol h}) =\frac{1+d_{\boldsymbol h}}4,\qquad \inf_{\DD}\ddiam(\DD\NN_{\boldsymbol h},\id_M) =\frac{3-d_{\boldsymbol h}}4. } \label{eq:recovery} \end{equation} These are information-theoretic decoder optima, not local decoder-implementation claims. At every finite \(g\) and compact smooth source profile, \[ d_{\boldsymbol h} \le\sqrt{1-e^{-4g^2\|k_F\|_{\mathrm U}^2}}<1. \] Thus the optimal recovery error is strictly greater than \(1/2\). \end{theorem} \begin{proof} For any decoder, let \(E=\DD^*(\ketbra+ +)\). Its Bell overlap is \[ F_e=\frac14\left[\tr(E\rho_+)+\tr((I-E)\rho_-)\right] \le\frac{1+d_{\boldsymbol h}}4, \] using the positive-part characterization of trace distance. Let \(\Pi=e^{\ii\pi N_{\mathrm{tot}}}\). Lemma~\ref{lem:gaussian} gives \(\Pi\rho_+\Pi^\dagger=\rho_-\). For \(T=\rho_+-\rho_-\), choose the Helstrom effect \(E_+=(I+\operatorname{sign}T)/2\), with value \(1/2\) on the kernel. Then \(\Pi E_+\Pi^\dagger=I-E_+\). Both conditional correct-decision probabilities equal \(q=(1+d_{\boldsymbol h})/2\). Measure this effect and prepare the corresponding source basis state. The decoded qubit channel has Pauli probabilities \[ \lambda_I=\lambda_Z=q/2,\qquad \lambda_X=\lambda_Y=(1-q)/2. \] For a Pauli channel its half-diamond distance from identity is \(1-\lambda_I\): the Bell outputs are orthogonal and the triangle inequality gives the matching upper bound. This decoder attains both values in \eqref{eq:recovery}. Every decoder has error at least \(1-F_e\), proving optimality. For the strict upper bound on \(d_{\boldsymbol h}\), compare the two vector states \(W(-a)\Omega_{\mathrm U}\) and \(W(a)\Omega_{\mathrm U}\) in the regular quasifree realization. Their overlap modulus is \(e^{-2\|a\|_{\mathrm U}^2}\). Their pure-state distance is \(\sqrt{1-e^{-4\|a\|_{\mathrm U}^2}}\). Restriction to packets is contractive. Lemma~\ref{lem:compact} makes the exponent finite. \end{proof} A nonzero classical signal is therefore compatible with a quantified impossibility of faithful unknown-qubit transfer. %===================================================================== \section{Phase-sensitive witnesses and what number counts miss} \label{sec:witness} %===================================================================== For a weighted coefficient \(l\), define \[ E_{\sin,l}=\frac12(I+\Impart W(l)),\qquad E_{\cos,l}=\frac12(I+\Repart W(l)), \] where \[ \Impart W(l)=\frac{W(l)-W(l)^\dagger}{2\ii}. \] Both are effects because the real and imaginary parts of a unitary are self-adjoint contractions. \begin{theorem}[Exact source-sign witness] \label{thm:witness} With the evolution sign fixed in \eqref{eq:UF}, \begin{equation} \boxed{ \omega_s(W(l)) =e^{-\|l\|_{\mathrm U}^2/2} e^{-\ii s g\sigm(k_F,l)}. } \label{eq:shiftedcharacteristic} \end{equation} Hence \[ \omega_s(E_{\sin,l}) =\frac12\left[ 1-s e^{-\|l\|_{\mathrm U}^2/2} \sin(g\sigm(k_F,l))\right], \] and the sign gap is \begin{equation} \boxed{ \Delta_{\sin} =e^{-\|l\|_{\mathrm U}^2/2} |\sin(g\sigm(k_F,l))|. } \label{eq:singap} \end{equation} The cosine expectations are identical for both signs. If \(l\) belongs to the selected outgoing packet algebra, \(d_{\boldsymbol h}\ge\Delta_{\sin}\). \end{theorem} \begin{proof} The exact Weyl identity is \[ W(r)^\dagger W(l)W(r) =e^{\ii\sigm(r,l)}W(l). \] Insert \(r=-sgk_F\), then the centered Gaussian characteristic function. Real and imaginary parts give the two effects. An effect probability difference is bounded by state trace distance, proving the last assertion. \end{proof} \subsection{Identical number statistics for opposite labels} \begin{theorem}[Number records conceal the source sign] \label{thm:counts} For the packet family in Theorem~\ref{thm:packetnormal}, every joint number-diagonal probability is the same in \(\rho_+\) and \(\rho_-\). Each marginal occupation mean is \begin{equation} \boxed{ \omega_s(N_i)=\nu_i+g^2|c_{h_i}|^2,\qquad \nu_i=A_{ii}. } \label{eq:occupation} \end{equation} For one packet, its generating function is \begin{equation} \boxed{ G_s(t)= \frac{1}{1+\nu_h(1-t)} \exp\left[ -\frac{g^2|c_h|^2(1-t)}{1+\nu_h(1-t)} \right], \qquad 0\le t\le1. } \label{eq:displacedPGF} \end{equation} \end{theorem} \begin{proof} The total parity \(\Pi\) maps \(\rho_+\) to \(\rho_-\) and acts on each joint number vector by a scalar sign. Its diagonal probabilities are therefore unchanged, even if the baseline packet covariance has off-diagonal correlations. Conjugation by displacement replaces \(b_i\) by \(b_i+\alpha_{s,i}\). The centered baseline gives \(\nu_i+|\alpha_{s,i}|^2\). For one packet the baseline is a circular coherent-state Gaussian mixture of variance \(\nu_h\). After displacement its coherent amplitude has mean \(\alpha_{s,h}\). Conditional number counts are Poisson. The circular Gaussian integral of \(e^{-(1-t)|z|^2}\) gives \eqref{eq:displacedPGF}; the vacuum case follows by its zero-variance limit. \end{proof} The two labels have identical count laws, but those laws need not equal the undisplaced Hawking baseline. If \(B=\sigm(k_F,l)\ne0\), then for fixed profiles and witness, \[ \omega_s(N_h)-\nu_h=g^2|c_h|^2,\qquad \Delta_{\sin} =e^{-\|l\|_{\mathrm U}^2/2}|B|\,|g|+O(|g|^3). \] For \(|gB|\le\pi/2\), \[ \Delta_{\sin}\ge \frac2\pi e^{-\|l\|_{\mathrm U}^2/2}|gB|. \] Thus a phase-sensitive bounded witness can be first order while the occupation-mean departure is second order. %===================================================================== \section{A compact exterior profile with a proved outgoing signal} \label{sec:benchmark} %===================================================================== This section supplies an analytical benchmark, not an evaluated experiment. Choose any \(\omega_0>0\). The two incoming generalized eigenfunctions are linearly independent, and \(\tau(\omega_0)\ne0\). Therefore \[ M(x,\omega_0) =\tau(\omega_0)\overline{q_{\mathrm H}(x,\omega_0)} +\rho_{\mathrm I}(\omega_0) \overline{q_{\mathrm I}(x,\omega_0)} \] is not the zero solution. Choose a finite \(x_0\) with \(M_0=M(x_0,\omega_0)\ne0\). Let \[ b(u)= \begin{cases} \exp[-1/(1-u^2)],&|u|<1,\\ 0,&|u|\ge1. \end{cases} \] Use normalized rescalings to choose real nonnegative bumps \[ \chi_\delta(t)=C_\delta b(t/\delta),\qquad \zeta_\delta(x)=D_\delta b((x-x_0)/\delta), \] with both integrals one. Choose \(\delta\) small enough that \[ \left|\int\zeta_\delta(x)M(x,\omega_0)\dd x-M_0\right| <|M_0|/4, \] \[ \left|\int\chi_\delta(t)e^{\ii\omega_0t}\dd t-1\right|<1/4. \] These finite choices exist by continuity. Set \begin{equation} f(t,x)=\chi_\delta(t)\zeta_\delta(x),\qquad F=\Bulk[f]. \label{eq:workedprofile} \end{equation} This has fixed compact exterior shell support. For this profile, \[ k_{F,j}(\omega) =\frac{\widehat\chi_{\mathrm{plain}}(\omega)} {\sqrt{4\pi\omega}} \int\zeta_\delta(x)\overline{q_j(x,\omega)}\dd x, \] where \(\widehat\chi_{\mathrm{plain}}(\omega)=\int\chi_\delta(t) e^{\ii\omega t}\dd t\). In particular \(d_0=d_\infty(\omega_0)\ne0\). Choose a compact positive-frequency band \(B_0\) around \(\omega_0\) on which \[ |d_\infty(\omega)-d_0|<|d_0|/2. \] Let \(H\) be a nonnegative normalized smooth bump inside \(B_0\), and define \[ h=e^{\ii\arg d_0}H. \] Then \[ \Repart c_h =\Repart\int e^{-\ii\arg d_0}H d_\infty\dd\omega \ge\frac{|d_0|}{2}\int H\dd\omega>0. \] \begin{theorem}[Nonzero compact-source packet coupling] \label{thm:nonzero} The explicitly specified compact profile and packet satisfy \(c_h\ne0\). For \(\vartheta>0\), choose \[ l=\frac{\vartheta}{\sqrt2} e^{\ii(\arg c_h+\pi/2)}v_h. \] Then \[ \sigm(k_F,l)=\sqrt2\,\vartheta|c_h|>0,\qquad \|l\|_{\mathrm U}^2=\vartheta^2(\nu_h+1/2). \] Consequently \begin{equation} \boxed{ \Delta_{\sin} =e^{-\vartheta^2(2\nu_h+1)/4} |\sin(\sqrt2\,g\vartheta|c_h|)|, \qquad \delta\nu_h=g^2|c_h|^2. } \label{eq:workedgap} \end{equation} For \(0<|g|\sqrt2\vartheta|c_h|\le\pi/2\), the gap is strictly positive. \end{theorem} \begin{proof} The preceding bump estimates prove \(d_0\ne0\), and the band estimate proves \(c_h\ne0\). Since \[ \langle k_F,l\rangle =\frac{\vartheta}{\sqrt2} e^{\ii(\arg c_h+\pi/2)}\overline{c_h}, \] its imaginary part is \(\vartheta|c_h|/\sqrt2\). Scattering unitarity and the baseline covariance give the stated norm. Apply Theorems~\ref{thm:witness} and \ref{thm:counts}. \end{proof} No independently chosen packet displacement is used in this construction. The phase of the packet and of the witness is a declared experimental reference. The fixed source shell is sufficient for nonzero coupling. Growing shells will enter the local approximation to its noncompact outgoing packet observable, not the source-existence proof. %===================================================================== \section{Controlled state classes and profile perturbations} \label{sec:stability} %===================================================================== \begin{definition}[Moment-dominated system states] For \(C\ge1\), let \(\mathfrak S_C\) be the normal states in the regular Unruh representation satisfying \[ \omega(X(u)^2)\le C\|u\|_{\mathrm U}^2 \] on the real weighted s-wave coefficient domain. The second moment includes any nonzero mean. \end{definition} \begin{proposition}[Common source-state domination] \label{prop:sourceclass} Both source-displaced states, and every mixture of them, belong to \(\mathfrak S_{C_{\mathrm{src}}}\), where \begin{equation} \boxed{ C_{\mathrm{src}}=1+4g^2\|k_F\|_0^2. } \label{eq:Csrc} \end{equation} Their two-point functions differ from the baseline by smooth terms, so the conditional states and their mixtures retain the Hadamard two-point singularity. \end{proposition} \begin{proof} Weyl conjugation shifts \(X(u)\) by \(-sg\sigm(k_F,u)\). The baseline is centered, so \[ \omega_s(X(u)^2) =\|u\|_{\mathrm U}^2+g^2\sigm(k_F,u)^2 \le(1+4g^2\|k_F\|_0^2)\|u\|_{\mathrm U}^2. \] Convexity gives the mixture statement. The displacement mean is the pairing with the smooth solution \(EF\). The two-point change is its smooth product. For mixtures the analogous two-point change is still smooth. No change of the Hadamard singularity occurs. \end{proof} \begin{lemma}[Bounded Weyl expectation continuity] \label{lem:weylcontinuity} For \(\omega\in\mathfrak S_C\), \[ |\omega(W(u))-\omega(W(l))| \le\sqrt C\,\|u-l\|_{\mathrm U} +\frac{|\sigm(u,l)|}{2}. \] Consequently, for either the sine or cosine effect, \begin{equation} \left|\omega(E_{\bullet,u})-\omega(E_{\bullet,l})\right| \le H_C(l)\|u-l\|_{\mathrm U},\qquad H_C(l)=\frac{\sqrt C+\|l\|_{\mathrm U}}2. \label{eq:weylcontinuity} \end{equation} \end{lemma} \begin{proof} Use the Weyl relation to write \(W(u)\) as a phase times \(W(l)W(u-l)\). Cauchy--Schwarz for the state and spectral calculus give \[ |\omega(W(l)(W(u-l)-I))| \le\sqrt{\omega(X(u-l)^2)}. \] The phase costs at most \(|\sigm(u,l)|/2\). Finally, \[ |\sigm(u,l)| =|\sigm(u-l,l)| \le2\|u-l\|_{\mathrm U}\|l\|_{\mathrm U}. \] Taking either real or imaginary parts and dividing by two proves the effect bound. \end{proof} Let an alternative admissible compact profile and coupling have coefficient \(\widehat a=\widehat g\,k_{\widehat F}\). Put \(r=\|\widehat a-a\|_{\mathrm U}\). \begin{theorem}[Separate profile channel and observable estimates] \label{thm:profile} On every fixed finite outgoing packet family, \[ \ddiam(\NN_{\boldsymbol h}^{\widehat a}, \NN_{\boldsymbol h}^{a}) \le\min\{1,\sqrt{1-e^{-r^2}}\}\le\min\{1,r\}. \] For the fixed target sine effect, \[ |\omega_s^{\widehat a}(E_{\sin,l}) -\omega_s^a(E_{\sin,l})| \le e^{-\|l\|_{\mathrm U}^2/2} \|\widehat a-a\|_0\|l\|_0. \] \end{theorem} \begin{proof} The conditional vector states in the quasifree realization have overlap modulus \(e^{-r^2/2}\); their pure-state trace distance is \(\sqrt{1-e^{-r^2}}\). Restriction to packets contracts it. A measure-and-prepare channel with the same source projectors has half-diamond difference at most the maximal conditional state distance: pinching a referenced input into source blocks and applying the triangle inequality proves this bound. Thus it remains valid with references. For the observable estimate, use the exact sine formula, \(|\sin x-\sin y|\le|x-y|\), and \(|\sigm(\widehat a-a,l)|\le 2\|\widehat a-a\|_0\|l\|_0\). \end{proof} These are genuine, but different, comparisons. The channel estimate uses the fixed initial field state and exact controlled-Weyl structure. It is not a theorem for arbitrary unbounded interaction perturbations. A scattering approximation used in the witness must separately certify its coefficient error in the common weighted norm. A fitted transmission probability alone does not control the complex scattering overlap \(c_h\). %===================================================================== \section{Compact bulk approximation of the outgoing witness} \label{sec:compactpacket} %===================================================================== The witness coefficient \(l\) is a phase-adjusted compact-band outgoing quadrature coefficient. Its Fourier packet is not compact in time. Choose a measurement center \(t_{\mathrm m}\) and a fixed \(T>0\) with \[ t_{\mathrm m}-T-1>\sup\{t:(t,x)\in\supp f\}. \] All probe coupling below is thus later than the source switching. Let \[ \mathcal C=\mathcal U(\text{complex conjugation})\mathcal U^{-1}. \] It is an antiunitary involution commuting with real functions of \(\omega\). It is not componentwise conjugation of incoming mode coefficients. For real Cauchy data at \(t_{\mathrm m}\), define \[ k_{t_{\mathrm m}}(q,p) =e^{\ii\omega t_{\mathrm m}} \frac{\mathcal Up-\ii\omega\mathcal Uq}{\sqrt{2\omega}}. \] Put \[ b=e^{-\ii\omega t_{\mathrm m}}\sqrt{2\omega}\,\omega^{-2}l, \] \[ \mathcal Up=\frac{b+\mathcal Cb}{2},\qquad \mathcal Uq=\frac{\mathcal Cb-b}{2\ii\omega}. \] These are real smooth Sobolev data with compact spectral support away from zero, and \(k_{t_{\mathrm m}}(q,p)=\omega^{-2}l\). Choose \(\chi\in C_c^\infty((-2,2))\), real and equal to one on \([-1,1]\). For \(L\ge1\), set \[ q_L=\chi(x/L)q,\qquad p_L=\chi(x/L)p, \] and let \(s_L\) be their free Cauchy solution at \(t_{\mathrm m}\). Choose a smooth switch \(\eta_T(t-t_{\mathrm m})\), zero before \(-T\) and one after \(T\), and define \begin{equation} b_L=-\partial_t^2\Dop(\eta_Ts_L),\qquad B_L=\Bulk[b_L]. \label{eq:compactwitness} \end{equation} It is compact, with \[ |t-t_{\mathrm m}|\le T,\qquad |x|\le2L+T. \] Write \(\delta q=q_L-q,\ \delta p=p_L-p\), and define \begin{equation} \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} \label{eq:packeterror} \end{equation} \begin{theorem}[Filtered compact packet bridge] \label{thm:packetbridge} The compact test coefficient \(l_L=k(b_L)\) satisfies \[ \|l_L-l\|_{\mathrm U}\le\delta_L,\qquad \delta_L\longrightarrow0. \] For \(\omega\in\mathfrak S_C\), \[ |\omega(E_{\sin,l_L})-\omega(E_{\sin,l})| \le H_C(l)\delta_L. \] \end{theorem} \begin{proof} For any free field solution, integration by parts gives \[ \phi(\Bulk[\Dop(\eta_Ts_L)]) =\int(p_L\Psi(t_{\mathrm m})-q_L\partial_t\Psi(t_{\mathrm m})) \dd x. \] The conserved Cauchy symplectic pairing and \(\int\eta_T'=1\) make this identity exact. Applying \(-\partial_t^2\) to the test multiplies its coefficient by \(\omega^2\). The coefficient difference is therefore \[ e^{\ii\omega t_{\mathrm m}}\omega^2 \frac{\mathcal U\delta p-\ii\omega\mathcal U\delta q} {\sqrt{2\omega}}. \] 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 \eqref{eq:packeterror}. The cutoffs converge in \(H^2\) for \(p\) and \(H^3\) for \(q\). This follows from derivative tails and \(\partial_x^j\chi(x/L)=O(L^{-j})\). The bounded smooth potential gives convergence of all displayed graph norms. For an ordinary-derivative bound one can use \[ \begin{split} D_p&=\|\delta p\|_2^2+\|\Lop\delta p\|_2^2,\\ D_q&=\|\delta q\|_2^2+\|\Lop\delta q\|_2^2 +\|\partial_x(\Lop\delta q)\|_2^2 +\int V|\Lop\delta q|^2\dd x, \end{split} \] and \(\delta_L^2\le(1+2/\beta)(D_p+D_q)\). The fractional-power inequalities follow from \(\lambda^{3/2},\lambda\le1+\lambda^2\) and \(\lambda^{5/2}\le1+\lambda^3\). Finally apply Lemma~\ref{lem:weylcontinuity}. \end{proof} The derivative filter controls the thermal infrared weight after spatial truncation. Mere energy completeness would not establish this weighted convergence. For each finite \(L\), the compact support is strictly exterior. As \(L\) grows, the shell reaches toward both radial ends. This is not fixed-zone arbitrary-accuracy access. %===================================================================== \section{A local scalar probe for the sine witness} \label{sec:localprobe} %===================================================================== \subsection{Exact local construction} The independent probe is another real massless scalar \(\psi\). Both fields initially have the same free operator \(P\). The local linear system--probe framework supplies the coupled Green operators and Weyl scattering interpretation \cite{FewsterVerch,FewsterJubbRuep}. We now give the particular construction and estimates. Fix \(L,T\) and abbreviate \(B=B_L,\ b=b_L\). Choose a spacelike-compact free probe solution \(v_{\mathrm{bulk}}\) equal to one near \(\supp B\). To construct it, prescribe compact Cauchy data at \(t_{\mathrm m}-T-1\) equal to the constant solution and its zero velocity on the causal predecessor of \(\supp B\). Finite propagation and uniqueness give the desired equality. Write \(v_{\mathrm{bulk}}=Y_0v/r\). Choose a future switch \(\eta_+\), zero before \(t_{\mathrm m}+T+1\) and one after \(t_{\mathrm m}+T+2\), and set \[ z_{\mathrm{red}}=-\Dop(\eta_+v),\qquad z=\Bulk[z_{\mathrm{red}}]. \] Then \[ G_{\mathrm A}z=(1-\eta_+)v_{\mathrm{bulk}}=1 \quad\text{near }\supp B. \] Couple the fields by \[ S_{\mathrm{probe,int}} =-\epsilon\int\rho\phi\psi\,\dd\mathrm{vol}_g, \qquad \rho=-B,\qquad \epsilon>0, \] with coupled operator \[ T_\epsilon= \begin{pmatrix}P&\epsilon\rho\\ \epsilon\rho&P\end{pmatrix}. \] The final bounded probe effect is \begin{equation} O_{\sin,\epsilon} =\frac12\left(I+\Impart W_\psi(z/\epsilon)\right). \label{eq:probeeffect} \end{equation} Let \[ w_\epsilon=G_{\mathrm A,T_\epsilon}(0,z)^{\mathsf T}, \] \[ B_\epsilon=-\rho(w_\epsilon)_\psi,\qquad z_\epsilon=z/\epsilon-\rho(w_\epsilon)_\phi. \] \begin{theorem}[Exact induced sine effect] \label{thm:induced} For a centered quasifree independent probe state \(\sigma_\psi\), \[ \Theta_\epsilon(I\otimes W_\psi(z/\epsilon)) =W_\phi(B_\epsilon)\otimes W_\psi(z_\epsilon). \] Its induced sine effect is \begin{equation} \boxed{ \varepsilon_{\sigma_\psi,\epsilon}(O_{\sin,\epsilon}) =\frac12\left[ I+c_\epsilon\Impart W_\phi(B_\epsilon)\right], \qquad c_\epsilon=e^{-V_{\sigma_\psi}(z_\epsilon,z_\epsilon)/2}. } \label{eq:inducedsine} \end{equation} \end{theorem} \begin{proof} For a future-supported test vector \(H\), advanced comparison gives the scattering-test transformation \[ \theta_\epsilon H =H-(T_\epsilon-P\oplus P)G_{\mathrm A,T_\epsilon}H. \] Green's identity, evaluated in a past Cauchy region where the theories agree, verifies the transformation. The conserved coupled symplectic current shows that it is symplectic and therefore induces the Weyl scattering morphism. For \(H=(0,z/\epsilon)\), its components are exactly \(B_\epsilon,z_\epsilon\). The two fields are separate tensor factors, so no mixed Weyl phase occurs. The centered probe characteristic function is real and nonnegative. Taking the imaginary part after applying \(\id\otimes\sigma_\psi\) gives \eqref{eq:inducedsine}. Thus the sine identity is not inferred by using a cosine probability for the opposite source signs. \end{proof} \subsection{Finite advanced estimates} In reduced coordinates put \[ a_{\mathrm p}=\fmet(r)\rho=-Y_0b/r,\qquad A_{\mathrm p}u=G_{\mathrm A,\Dop}(a_{\mathrm p}u), \qquad v_{\mathrm A}=G_{\mathrm A,\Dop}z_{\mathrm{red}}. \] On the coupling support, \(-a_{\mathrm p}v_{\mathrm A}=b\). Use the time interval \[ I=[t_{\mathrm m}-T-1,t_{\mathrm m}+T+3],\qquad \Delta=|I|, \] and the norm \[ \|u\|_{\mathcal X}^2 =\sup_{t\in I}\sum_{j=0}^2 \left(\|\partial_t\partial_x^ju\|_2^2 +\|\partial_x^{j+1}u\|_2^2+\|\partial_x^ju\|_2^2\right). \] Conservative explicit constants are \[ c_2=64\left(1+\sum_{j=0}^2\|\partial_x^jV\|_\infty\right), \] \[ M_2=8\sup_{t\in I}\sum_{j=0}^2 \|\partial_x^ja_{\mathrm p}(t,\cdot)\|_\infty, \qquad \Lambda=\Delta e^{c_2\Delta}M_2. \] Let \(X=2L+T\), \(V_K=2X\Delta\), and define \[ a_0=\|a_{\mathrm p}\|_\infty,\qquad a_t=\|\partial_ta_{\mathrm p}\|_\infty, \] \[ L_{\mathrm p} =\frac{2V_KC_X}{\sqrt{2\pi}} \sqrt{A_0a_0^2+A_1(a_t+a_0)^2}. \] \begin{lemma}[Advanced and covariance bounds] \label{lem:advanced} The preceding constants satisfy \[ \|A_{\mathrm p}u\|_{\mathcal X}\le\Lambda\|u\|_{\mathcal X}, \qquad \|k(a_{\mathrm p}u)\|_{\mathrm U} \le L_{\mathrm p}\|u\|_{\mathcal X}. \] \end{lemma} \begin{proof} Differentiate the spatial-derivative wave energies through order two. Integration by parts cancels the principal terms; the potential commutators involve only \(V,V',V''\). Cauchy--Schwarz gives the backward-time inequality \[ \frac{\dd}{\dd t}\sqrt{\mathcal E_2} \le c_2\sqrt{\mathcal E_2}+\|J\|_{H^2}. \] The product rule gives \(\|a_{\mathrm p}u\|_{H^2}\le M_2\sqrt{\mathcal E_2(u)}\). Backward Gronwall inequality with zero final data proves the advanced bound. The one-dimensional Sobolev inequality bounds both \(\sup|u|\) and \(\sup|\partial_tu|\) by \(2\|u\|_{\mathcal X}\). Hence \[ \|a_{\mathrm p}u\|_1\le2V_Ka_0\|u\|_{\mathcal X}, \] \[ \|\partial_t(a_{\mathrm p}u)\|_1 \le2V_K(a_t+a_0)\|u\|_{\mathcal X}. \] Lemma~\ref{lem:compact} proves the covariance bound. \end{proof} For \(\epsilon^2\Lambda^2<1\), the exact reduced coupled solution is \[ (w_\epsilon)_\psi =(I-\epsilon^2A_{\mathrm p}^2)^{-1}v_{\mathrm A}, \qquad (w_\epsilon)_\phi=-\epsilon A_{\mathrm p}(w_\epsilon)_\psi. \] This follows from the two advanced equations; the convergent Neumann series solves them with zero final data, and uniqueness identifies it with the exact solution. Consequently, \begin{equation} \|k(B_\epsilon)-l_L\|_{\mathrm U} \le d_{\mathrm p}(L,\epsilon) :=L_{\mathrm p} \frac{\epsilon^2\Lambda^2}{1-\epsilon^2\Lambda^2} \|v_{\mathrm A}\|_{\mathcal X}, \label{eq:localtransfer} \end{equation} where \(k(B_\epsilon)\) denotes its reduced-test coefficient. The residual probe test has reduced form \[ (z_\epsilon)_{\mathrm{red}} =z_{\mathrm{red}}/\epsilon+\epsilon j_\epsilon, \] \[ j_\epsilon =a_{\mathrm p}A_{\mathrm p} (I-\epsilon^2A_{\mathrm p}^2)^{-1}v_{\mathrm A}, \] \[ \|k(j_\epsilon)\|_{\mathrm U} \le L_{\mathrm p}\frac{\Lambda}{1-\epsilon^2\Lambda^2} \|v_{\mathrm A}\|_{\mathcal X}. \] \subsection{Independent probe squeezing and its cost} Take an independent probe reference state with the same Unruh covariance. A fixed reference would give \(V(z_\epsilon,z_\epsilon)\sim\epsilon^{-2}V(z,z)\); its attenuation would tend to zero. Weak coupling alone does not remove probe noise. Choose a compact real test \(p_0\) with \(c_0=\sigm_\psi(z,p_0)\ne0\). Such a test exists by nondegeneracy of the free scalar test symplectic quotient. Set \[ p_\epsilon= p_0/\sigm_\psi(z_\epsilon,p_0). \] For sufficiently small \(\epsilon\) this is defined and \(\sigm_\psi(z_\epsilon,p_\epsilon)=1\). On their Darboux plane scale \(z_\epsilon\) by \(\epsilon^2\) and \(p_\epsilon\) by \(\epsilon^{-2}\), fixing its symplectic orthogonal complement. Explicitly, \[ \begin{split} S_\epsilon u={}&u- \left[\sigm_\psi(u,p_\epsilon)z_\epsilon -\sigm_\psi(u,z_\epsilon)p_\epsilon\right]\\ &+\epsilon^2\sigm_\psi(u,p_\epsilon)z_\epsilon -\epsilon^{-2}\sigm_\psi(u,z_\epsilon)p_\epsilon. \end{split} \] Prepare \[ \sigma_{L,\epsilon}(W_\psi(u)) =\sigma_{\mathrm{ref}}(W_\psi(S_\epsilon u)). \] This is a centered regular quasifree state. Its covariance differs from the reference by finite sums of products of smooth solutions: smearing one variable of a Hadamard bisolution with a compact smooth test gives a smooth solution in the other variable. Thus its Hadamard singularity is unchanged. Choose \(X_z\) containing the support of \(z_{\mathrm{red}}\), and put \[ b_z=\mathcal B_{X_z}(z_{\mathrm{red}}),\qquad b_J=2L_{\mathrm p}\Lambda\|v_{\mathrm A}\|_{\mathcal X}, \qquad N_L=(b_z+b_J)^2. \] For \(\epsilon\le1,\ \epsilon^2\Lambda^2\le1/2\), \[ V_{\sigma_{L,\epsilon}}(z_\epsilon,z_\epsilon) =\epsilon^4V_{\mathrm{ref}}(z_\epsilon,z_\epsilon) \le N_L\epsilon^2, \] \begin{equation} \frac{1-c_\epsilon}{2}\le N_L\epsilon^2/4. \label{eq:localnoise} \end{equation} A sufficient nonzero-denominator condition is \[ \epsilon^2\,2b_J \sqrt{V_{\mathrm{ref}}(p_0,p_0)}<|c_0|/2. \] It follows by expanding \(\sigm_\psi(z_\epsilon,p_0) =c_0/\epsilon+\epsilon\sigm_\psi(j_\epsilon,p_0)\) and applying the covariance bound to the symplectic form. The conjugate variance satisfies \[ V_{\sigma_{L,\epsilon}}(p_\epsilon,p_\epsilon) \ge\frac1{4N_L\epsilon^2}. \] This is the uncertainty inequality for a pair of commutator \(\ii I\). The squeeze magnitude is \(2\log(1/\epsilon)\); the processing test is \(z/\epsilon\). No bounded-effort preparation limit is asserted. \subsection{The local probability theorem} \begin{theorem}[Local source-sign probability certificate] \label{thm:local} Use the same independently prepared probe for both source signs. For either sign, \begin{equation} \boxed{ \left| (\omega_s\otimes\sigma_{L,\epsilon}) \Theta_\epsilon(I\otimes O_{\sin,\epsilon}) -\omega_s(E_{\sin,l}) \right| \le\varepsilon_{\mathrm{loc}}, } \label{eq:localcertificate} \end{equation} where \[ \varepsilon_{\mathrm{loc}} =\min\left\{1,\, \frac{1-c_\epsilon}{2} +H_{C_{\mathrm{src}}}(l) \bigl(\delta_L+d_{\mathrm p}(L,\epsilon)\bigr) \right\}. \] In the small-coupling regime, \[ \varepsilon_{\mathrm{loc}} \le\min\left\{1,\, H_{C_{\mathrm{src}}}(l)\delta_L+ \left(H_{C_{\mathrm{src}}}(l)T_L+N_L/4\right)\epsilon^2 \right\}, \] \[ T_L=2L_{\mathrm p}\Lambda^2\|v_{\mathrm A}\|_{\mathcal X}. \] The local probability gap is at least \[ \max\{0,\Delta_{\sin}-2\varepsilon_{\mathrm{loc}}\}. \] \end{theorem} \begin{proof} The exact induced effect is \eqref{eq:inducedsine}. Removing its attenuation changes an expectation by at most \((1-c_\epsilon)/2\). The packet and transfer estimates give \[ \|k(B_\epsilon)-l\|_{\mathrm U} \le\delta_L+d_{\mathrm p}. \] Proposition~\ref{prop:sourceclass} verifies the same moment class for both source states, so Lemma~\ref{lem:weylcontinuity} gives \eqref{eq:localcertificate}. Equations~\eqref{eq:localtransfer}--\eqref{eq:localnoise} give the simplified form. The gap follows by the reverse triangle inequality. \end{proof} For every fixed source, packet, and positive local tolerance, finite parameters exist: first choose \(L\) to make \(\delta_L\) small, then choose \(\epsilon\) to meet the transfer, noise, and denominator inequalities. All constants are finite at that fixed \(L\). This is a mathematical local measurement scheme with a declared final probe effect and preparation. It does not construct laboratory squeezing equipment or a final probe-POVM device. %===================================================================== \section{Observable, readout, and statistical error ledgers} \label{sec:records} %===================================================================== Let the implemented source coefficient be \(\widehat a\). Use \[ C=1+4\|\widehat a\|_0^2 \] for its two displaced states. If a scattering or packet calculation supplies a separate weighted coefficient error \(r_{\mathrm{sc}}\), include it in the target comparison using Lemma~\ref{lem:weylcontinuity}. A sufficient per-preparation observable allowance is \begin{equation} \begin{split} e_s\le\min\{1,\, & e^{-\|l\|_{\mathrm U}^2/2} \|\widehat a-a\|_0\|l\|_0\\ &+H_C(l)(r_{\mathrm{sc}}+\delta_L+d_{\mathrm p}) +(1-c_\epsilon)/2+e_{\mathrm{read},s}\}. \end{split} \label{eq:obsledger} \end{equation} The terms respectively concern source-profile error, certified coefficient error, compact packet approximation, local transfer, probe noise, and final readout. A final readout allowance must be justified for a complete common record alphabet. Ordinary binary outcomes and all declared reset, assignment, and missing-record flags are retained. Assign every failure a fixed score in \([0,1]\); no trial is removed. A complete-law total-variation error changes such a score expectation by at most that error. \begin{theorem}[Complete observed-gap certificate] \label{thm:observed} If the two attempted-run laws have the allowances \(e_+,e_-\) in \eqref{eq:obsledger}, their score-expectation gap is at least \[ \Delta_{\sin}-e_+-e_-, \] replaced by its positive part. For \(n_+,n_-\) independent attempted trials with fixed parameters, let \[ u_s=\sqrt{\frac{\log(4/\beta)}{2n_s}},\qquad 0<\beta<1. \] With probability at least \(1-\beta\), \begin{equation} \boxed{ |\widehat p_+-\widehat p_-| \ge\Delta_{\sin}-e_+-e_--u_+-u_-. } \label{eq:sample} \end{equation} The same statement holds for predetermined scores in \([0,1]\). \end{theorem} \begin{proof} An effect or bounded-score expectation changes by at most its complete-law allowance. Apply this to both preparations. For independent variables in \([0,1]\), the centered log moment-generating function is bounded by \(t^2/8\): its tilted second derivative is a variance at most \(1/4\). Exponential Markov inequality gives \(\Pr(|\widehat p-p|>u)\le2e^{-2nu^2}\). Apply this to both ensembles and take a union bound. Then combine with the expectation-gap estimate. \end{proof} Independence and fresh apparatus preparations are additional repeated-experiment hypotheses. One-shot correctness does not establish them. For the benchmark of Section~\ref{sec:benchmark}, a nonvacuous schedule is \[ e_+,e_-\le\Delta_{\sin}/8,\qquad n_s\ge\frac{32}{\Delta_{\sin}^2}\log(4/\beta). \] Then \eqref{eq:sample} is at least \(\Delta_{\sin}/2\). This schedule is attainable only if the separately supplied readout and preparation error floors permit its allowances. \paragraph{No mixed-metric channel ledger.} The profile theorem provides a referenced packet-channel estimate. The local theorem provides an effect expectation estimate. The latter is not inserted into a coherent-transfer or recovery diamond-distance budget. No coherent local scalar access channel has been constructed here. %===================================================================== \section{Causal support and the exterior placement of the source} \label{sec:causality} %===================================================================== \begin{theorem}[Spacelike observation invariance] \label{thm:causality} If a compact local field test \(G\) is spacelike separated from \(K=\supp F\), then \[ [\phi(F),\phi(G)]=0, \] and the source interaction leaves every observable generated by such local Weyl operators unchanged: \[ U_F^\dagger(I\otimes W(k_G))U_F =I\otimes W(k_G). \] The statement is independent of the initial source density. The chronological propagator is the identity before switching. For future local tests after switching, the conditional mean is the retarded solution of the driven equation. \end{theorem} \begin{proof} Causal support of \(E\) gives \[ \sigm(k_F,k_G)=\int F\,EG\,\dd\mathrm{vol}_g=0 \] at spacelike separation. The Weyl commutation relation proves the invariance, and products generate the corresponding local algebra. Before switching, \(k_t=0\), so \eqref{eq:propagator} is identity. In a source branch, the field equation is \[ P\phi=-sgF. \] Its mean with zero driven data before switching is \(-sgG_{\mathrm R}F\). For a test to the future of the source, \(EF=-G_{\mathrm R}F\). The displacement shift is \[ -sg\int F\,EG =sg\int (EF)G, \] where antisymmetry of \(E\) was used. It equals the retarded mean. \end{proof} The completed scattering displacement is a statement about outgoing identification. It is not used to modify an observable in the physical past; earlier observations use the truncated propagator. The outgoing packet has a noncompact temporal profile. Its compact approximants in Section~\ref{sec:compactpacket} have their own support and growing-shell ledger. A packet label is not a point in spacetime. \paragraph{Source placement.} The source is exterior. This construction demonstrates information transfer through exterior scalar radiation. It does not demonstrate escape from a black-hole interior. Moving the source behind the horizon would change the geometry, support, propagation, and observable problem and would require a new model. The single qubit is spatially extended over a shell. The theorem establishes the printed field-observation support statements, not a generally covariant microscopic theory of that extended source. %===================================================================== \section{Excess radiative energy and switching work} \label{sec:energy} %===================================================================== The stationary baseline need not have finite total radiation energy. Accordingly, we define finite excess energy from the change of the two-point kernel, not by subtracting two divergent total expectations. The branch incoming amplitude is \(-\ii sgk_F\). Stationary scattering maps it to \[ -\ii sg(d_\infty,d_{\mathrm{hor}}). \] The added occupation kernel is the rank-one kernel of this amplitude. The sign does not affect its energy. \begin{theorem}[Finite excess scalar energy] \label{thm:energy} The excess outgoing-infinity and future-horizon energies are \begin{equation} \boxed{ \begin{aligned} \delta E_\infty&= g^2\int_0^\infty\omega|d_\infty(\omega)|^2\dd\omega,\\ \delta E_{\mathrm{hor}}&= g^2\int_0^\infty\omega|d_{\mathrm{hor}}(\omega)|^2\dd\omega,\\ \delta E_{\mathrm{tot}}&= g^2\int_0^\infty\omega (|k_{F,\mathrm H}|^2+|k_{F,\mathrm I}|^2)\dd\omega. \end{aligned} } \label{eq:energy} \end{equation} They are finite, nonnegative, and \(\delta E_{\mathrm{tot}} =\delta E_\infty+\delta E_{\mathrm{hor}}\). For a packet supported in \(B=[\omega_-,\omega_+]\), \[ \delta E_{\infty,B} =g^2\int_B\omega|d_\infty|^2\dd\omega \ge\omega_-|\alpha_{s,h}|^2. \] The exact energy of the projected classical signal \(\alpha_{s,h}h\) is \[ |\alpha_{s,h}|^2\int_B\omega|h(\omega)|^2\dd\omega. \] This is not the energy of the whole band and does not imply that a nonmonochromatic packet has Hamiltonian \(\omega_0N_h\). \end{theorem} \begin{proof} Weyl displacement adds the rank-one mean-amplitude kernel to the centered occupation kernel. Pairing this positive change with frequency yields \eqref{eq:energy}. Pointwise unitarity of the scattering matrix preserves the weighted squared norm. Lemma~\ref{lem:compact}, specifically \eqref{eq:energybound}, proves integrability. Cauchy--Schwarz gives \[ |c_h|^2\le\int_B|d_\infty|^2\dd\omega, \] so the band lower bound follows. The energy of the specified projected signal is its ordinary frequency-weighted norm. Since frequency multiplication need not commute with the packet projection, energy cross terms with the complementary signal are not silently discarded. \end{proof} \subsection{The switching apparatus supplies the work} The source has \(H_S=0\); its internal energy cannot pay the positive energy in \eqref{eq:energy}. Let \(v_{\mathrm R}=G_{\mathrm R,\Dop}f\). The driven conditional reduced mean is \[ m_s=-sgv_{\mathrm R}. \] Define its classical Killing energy by \[ E_{\mathrm{cl}}(t) =\frac12\int \left[(\partial_tm_s)^2+(\partial_xm_s)^2+Vm_s^2\right]\dd x. \] It is finite because finite propagation gives compact Cauchy support at each finite time. \begin{proposition}[Exact switching-work ledger] \label{prop:work} The control work in this prescribed driving model is \[ W_{\mathrm{ctrl}} =-g^2\int(\partial_tf)v_{\mathrm R}\dd t\,\dd x =g^2\int f\,\partial_tv_{\mathrm R}\dd t\,\dd x =\delta E_{\mathrm{tot}}. \] This is a finite excess-energy statement. It is not a black-hole mass-loss law or an autonomous source-energy conservation theorem. \end{proposition} \begin{proof} The reduced mean solves \(\Dop m_s=-sgf\). Differentiating its classical energy and integrating by parts gives \[ \dot E_{\mathrm{cl}} =\int(\partial_tm_s)\Dop m_s\dd x =g^2\int f\,\partial_tv_{\mathrm R}\dd x. \] It has zero driven energy before switching. Integrate in time. Since \(f\) vanishes at the temporal endpoints, integration by parts gives the first equality. In the Schr\"odinger description the explicitly time-dependent interaction is \(gZ_S\int f(t,x)\Psi(x)\dd x\). Its explicit derivative has expectation \(g\int(\partial_tf)\langle Z_S\Psi\rangle\dd x\). The conditional means give \(\langle Z_S\Psi\rangle=-gv_{\mathrm R}\), independently of the source populations. Thus its work is the displayed integral. After switching, mode expansion of the real driven solution, or the equivalent displaced mean-amplitude calculation, gives \(E_{\mathrm{cl}}=\int\omega g^2\sum_j|k_{F,j}|^2\dd\omega\). This is \(\delta E_{\mathrm{tot}}\). \end{proof} The time-ordering scalar phase does not alter this ledger. The background energy remains the fixed stationary baseline; no backreaction equation is introduced. %===================================================================== \section{Resources, physical interpretation, and successors} \label{sec:resources} %===================================================================== \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{longtable}{@{}p{0.24\textwidth}p{0.66\textwidth}@{}} \toprule Resource & Specification and boundary\\ \midrule \endfirsthead \toprule Resource & Specification and boundary\\ \midrule \endhead \bottomrule \endfoot Source & One degenerate qubit with a fixed source basis and independent initial field. Its full-shell extension is an effective model.\\ Source support & One fixed compact exterior shell suffices for the constructed nonzero signal. The profile and its time slicing are declared.\\ Field & The actual massless scalar of the Schwarzschild scattering model. No separate logical radiation payload is appended.\\ Signal calculation & Compact-profile coefficient \(k_F\), full complex scattering amplitudes, \(d_\infty,d_{\mathrm{hor}}\), and packet overlaps \(c_h\). Transmission probabilities alone are insufficient.\\ Witness & A bounded sine-Weyl effect with a specified phase reference. Its Gaussian noise factor and source-dependent phase are exact.\\ Local approximation & Compact Cauchy-data cutoffs and derivative filtering. Full angular-shell access; growing radial support at increasing packet accuracy.\\ Probe & One independent massless scalar, compact bilinear coupling, future processing test, and a specified quasifree preparation.\\ Probe effort & Processing test \(z/\epsilon\), squeeze magnitude \(2\log(1/\epsilon)\), and conjugate variance at least \(1/(4N_L\epsilon^2)\).\\ Final readout & The two effects \(O_{\sin,\epsilon},I-O_{\sin,\epsilon}\). Laboratory preparation and final readout require their own calibration and visible failure accounting.\\ Quantum recovery & The radiation channel is entanglement breaking. The exact packet recovery optimum uses an abstract Helstrom decoder; no local execution or coherent field capture is claimed.\\ Energy & Finite excess energy in both radiation ends, supplied by the prescribed switching apparatus. The degenerate source does not provide internal-energy fuel.\\ Repeated trials & Fresh independent preparations and no parameter drift are additional statistical hypotheses.\\ \end{longtable} \end{center} \subsection{A response-matching programme, not a derived source law} The benchmark supplies an explicit effective interaction. It does not explain why a black hole should possess this qubit or coupling. A future physically motivated source model could be constrained by absorption and emission response functions. Quantum-horizon effective theory provides one framework in a stated long-wavelength, short-relative-to-lifetime regime \cite{HorizonEFT}. It is not a derivation of the present unknown-input source channel. Required additional data include: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item a source spectrum and its Hamiltonian; \item transition matrix elements and state-dependent correlation functions; \item absorption and emission matching in one stated regime; \item higher source correlations relevant to coherent inputs; \item a controlled remainder and the relation to exterior field observables; \item energy exchange and a dynamical control model. \end{enumerate} \subsection{Noncommuting source pulses are a separate problem} A possible successor uses \[ e^{-\ii g_2X_S\phi(F_2)} e^{-\ii g_1Z_S\phi(F_1)}. \] The immediate single-source-observable channel proof no longer applies in the same form. That fact is not a proof of quantum transmission. A release for such a successor would require all source matrix-unit outputs, a referenced packet-output test, the retained source and environmental data, packet-selection errors, and a stated recovery or obstruction criterion. No such result is claimed here. %===================================================================== \section{Logical and sign repairs} \label{sec:debug} %===================================================================== \begin{enumerate}[label=\textbf{D\arabic*.},leftmargin=*] \item \textbf{The Hamiltonian and displacement signs agree.} The printed interaction \(H_I=gZ_S\Phi_F\) gives \(W(-sgk_F)\), not \(W(sgk_F)\). The conditional characteristic phase and packet amplitude use that sign consistently. \item \textbf{Time ordering has a proved domain and phase.} The central phase is derived from chronological Weyl products. No bounded-generator theorem is applied to an unbounded field. \item \textbf{The initial field is independent of the message.} Every source-dependent displacement follows from the specified interaction, not from an alternative initial radiation state. \item \textbf{The scalar channel is normal only after restriction.} The field state functional is defined first. The finite-mode trace-class densities are then constructed explicitly. No global physical radiation density is assumed. \item \textbf{Tracing the source removes its off-diagonal blocks.} This is why the accessible radiation channel is measure-and-prepare. A visible classical signal is not unknown-qubit recovery. \item \textbf{Cosine and sine witnesses serve different tasks.} The cosine expectation is even in the source sign. The sine witness supplies the nonzero sign-sensitive certificate. \item \textbf{Equal source-label counts are not equal radiation states.} Parity proves equality of the whole selected joint number law, while the sine effect proves a state difference. The common displaced law can still differ from the Hawking baseline. \item \textbf{The selected packet is not the entire signal.} Unselected outgoing modes, the future horizon, and the retained source are not replaced by vacuum or omitted from interpretation. \item \textbf{The retained source is not the full complement.} No environmental decoupling distance is inferred without the unselected field and a specified background purification. \item \textbf{Local probe access is a probability construction.} It is not a coherent field-to-memory channel or an instrument comparison retaining quantum outputs. \item \textbf{The local scheme is after source switching.} A future measurement time slice is explicitly chosen. The scattering endpoint is not used to signal into the physical past. \item \textbf{Growing access and squeezing are priced.} Arbitrary mathematical accuracy is not a fixed-shell, bounded-effort, or fixed-hardware theorem. \item \textbf{Excess energy is not a difference of infinities.} The positive rank-one two-point change defines the finite energy. A nonmonochromatic packet is not assigned an exact single frequency. \item \textbf{Switching work is not source mass loss.} The source is degenerate and the drive is prescribed. No black-hole evaporation dynamics follows from this ledger. \item \textbf{The source is exterior.} No interior communication, gravitational origin, or resolution of the black-hole information problem is inferred. \end{enumerate} %===================================================================== \section{Final analytical release} \label{sec:release} %===================================================================== \begin{theorem}[Em03-F exterior scalar-source release] \label{thm:release} Under the stated stationary field input, scattering normalization, and effective exterior-source specification: \begin{enumerate}[label=\textup{(\roman*)},leftmargin=*] \item the compact source interaction has the exact unitary propagator and common time-ordering phase \eqref{eq:propagator}--\eqref{eq:UF}; \item its source-to-field map and all source matrix units are calculated, with the off-diagonal identities \eqref{eq:offdiagonalzero}; \item identified outgoing scalar packet restrictions are normal displaced Gaussian states with amplitudes \eqref{eq:alpha}; \item a compact exterior separable source profile and outgoing packet have a proved nonzero coupling; \item the bounded sine-Weyl effect has the exact source-sign gap \eqref{eq:singap}, while selected joint number probabilities are identical for opposite labels; \item the packet radiation channel is entanglement breaking, with the exact optimal deterministic recovery values \eqref{eq:recovery}; \item the source-displaced states satisfy the common moment bound \eqref{eq:Csrc}, and the compact local scalar probe gives \eqref{eq:localcertificate}; \item source-profile, observable, complete-readout, and independent-trial errors have the separate proved allowances of Theorem~\ref{thm:profile}, \eqref{eq:obsledger}, and Theorem~\ref{thm:observed}; \item spacelike field-observation invariance, retarded future response, and retained-source coherence are calculated; \item excess outgoing scalar energy is finite and has the switching-work ledger of Theorem~\ref{thm:energy} and Proposition~\ref{prop:work}. \end{enumerate} The construction is restricted to a massless s-wave source and full-shell access. The local approximation may enlarge its coupling region and requires increasing probe preparation effort. Independent repeated preparations are retained as an additional statistical hypothesis. The source law is an effective exterior specification. No microscopic quantum-gravity derivation, black-hole interior information escape, coherent local field transfer, Page curve, backreaction, mass-loss equation, or complete evaporation theorem is established. \end{theorem} \begin{proof} Theorem~\ref{thm:unitary} proves (i). Theorem~\ref{thm:fieldchannel} proves (ii). Proposition~\ref{prop:displacement}, Lemma~\ref{lem:gaussian}, and Theorem~\ref{thm:packetnormal} prove (iii). Theorem~\ref{thm:nonzero} proves (iv). Theorems~\ref{thm:witness} and \ref{thm:counts} prove (v). Theorems~\ref{thm:EB} and \ref{thm:recovery} prove (vi). Proposition~\ref{prop:sourceclass}, Theorem~\ref{thm:packetbridge}, and Theorem~\ref{thm:local} prove (vii). The cited profile and record arguments prove (viii). Theorems~\ref{thm:causality} and \ref{thm:backaction} prove (ix). The energy and work arguments prove (x). Every comparison retains its own domain and metric. The model contains neither a gravitational source derivation nor the interior, coherent-access, or evaporation dynamics required for the excluded physical conclusions. \end{proof} \paragraph{Final scope statement.} Em03-F constructs an exactly specified exterior quantum-source coupling to the massless scalar field on Schwarzschild and identifies source-dependent information in outgoing packet observables. It derives a bounded phase-sensitive discrimination witness and quantifies its relation to occupation statistics and local probe access. For the single-observable source coupling, it also proves that the radiation channel is entanglement breaking and cannot recover an arbitrary unknown qubit. The source dynamics is an effective model, not a derivation from microscopic quantum gravity, and no black-hole interior information recovery or evaporation theorem is established. %===================================================================== \section*{Declarations} %===================================================================== \paragraph{Verification and data availability.} All analytical specifications used in the release are printed here. No executed radial-scattering dataset, numerical control schedule, compilation record, formal proof certificate, experimental calibration, or apparatus demonstration is supplied or asserted. \paragraph{Attribution.} Simple-generated qubit--field interactions, Weyl displacement, Gaussian restrictions, and entanglement-breaking channel principles are established constructions. The contribution claimed here is the stated exterior-source realization, its outgoing scalar identification, its exact benchmark calculations, and its quantitative observable-access composition. \paragraph{Predecessor status.} The supplied Em03-A--E manuscripts are used only through the interfaces identified in this paper. The needed compact coefficient, packet, and probe arguments are reproduced in their stated specialization. Documentary registry accounts and unrelated analytic programmes are not formal premises. %===================================================================== \begin{thebibliography}{99} \bibitem{Em03A} N.~Cheung, \emph{Em03-A: A Fixed-Background Hawking Emission Theorem}, version 1, 8 October 2026. Supplied manuscript and excerpts. The selected stationary scalar prescription and packet occupation law are inherited inputs, not microscopic source dynamics. \bibitem{Em03B} N.~Cheung, \emph{Em03-B: A Certified Apparatus-Model Readout of One Hawking Packet}, version 2, 8 October 2026. Supplied manuscript account. Effective capture and finite readout context; not a premise for local coherent scalar access here. \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}. Joint Gaussian states, bounded diagnostics, and complete-record comparison context. \bibitem{Em03D} N.~Cheung, \emph{Em03-D: A Local System--Probe Realization of a Hawking-Packet Observable}, version 1, 8 October 2026. Supplied source \code{Em03-D-v1.tex}. Filtered compact s-wave packet approximation and growing full-shell local probe architecture. The sine-effect adaptation and source-state verification are proved here. \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}. Finite payload model used only for the architectural comparison. \bibitem{Tjoa} E.~Tjoa, \emph{Non-perturbative simple-generated interactions with a quantum field for arbitrary Gaussian states}, Physical Review D \textbf{108} (2023), 045003. arXiv:2207.01141. doi:10.1103/PhysRevD.108.045003. Exact qubit--field and Gaussian-state methodological precedent; not the Schwarzschild packet or local-error proof of this paper. \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. Declared bulk state existence and Hadamard input. \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 scattering and horizon/infinity channels. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}, European Mathematical Society, 2007. arXiv:0806.1036. Normally hyperbolic Cauchy theory, Green operators, and causal support. \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. Local system--probe scattering and induced-observable framework. \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. Weyl-algebra measurement and asymptotic local-observable context; not an independently assumed outgoing-packet bridge. \bibitem{EntanglementBreaking} M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai, \emph{General Entanglement Breaking Channels}, Reviews in Mathematical Physics \textbf{15} (2003), 629--641. arXiv:quant-ph/0302031. doi:10.1142/S0129055X03001709. Measure-and-prepare and separability characterization. The qubit recovery constants for this channel are proved directly. \bibitem{HorizonEFT} W.~D.~Goldberger and I.~Z.~Rothstein, \emph{An Effective Field Theory of Quantum Mechanical Black Hole Horizons}, arXiv:1912.13435. Response-matching programme in a stated effective-theory regime; not a derivation of the exterior qubit or its message channel. \end{thebibliography} \end{document}