% Em01-A.tex % Intended compilation: pdflatex Em01-A.tex, run twice. % No external bibliography, figures, or manuscript files are required. % This manuscript proves conditional compatibility and non-implication % results. It does not claim to resolve the registered graph problem. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage[margin=1in]{geometry} \usepackage{enumitem} \usepackage{booktabs,tabularx} \usepackage{hyperref} \hypersetup{ hidelinks, pdftitle={Em01-A: Conditional Compatibility and Witness-to-Graph Limits}, pdfauthor={} } \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\Test}{\mathcal D} \newcommand{\Dist}{\mathcal D'} \newcommand{\SolE}{\mathscr S_{\mathrm E}} \newcommand{\SolW}{\mathscr S_{\Box}} \newcommand{\SolH}{\mathscr S_{\mathrm{dD}}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Pres}{\mathscr P} \newcommand{\Kint}{\mathscr K_{\mathrm{int}}} \newcommand{\Ucan}{U_{\mathrm{can}}} \newcommand{\pair}[2]{\langle #1,#2\rangle} \newcommand{\Eplus}{E_{+}} \newcommand{\Eminus}{E_{-}} \newcommand{\GR}{G_{\mathrm R}} \newcommand{\GA}{G_{\mathrm A}} \newcommand{\jetins}{\mathsf E} \newcommand{\jettest}{\mathsf J} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\PV}{PV} \title{\textbf{Em01-A}\\[0.4em] Conditional Compatibility of Linear Representatives\\ and the Limits of Witness-to-Graph Inference} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} Two proposed manuscript bridges are examined without identifying their distinct objects or domains. In a fully specified flat linearized-gravity setting, a compact tensor distribution admitting a smooth compact wave-shell representative has a smooth compact conserved representative on a vacuum solution space containing compact pure gauges if and only if the original distribution is conserved. The causal-propagator smoothness required by the conserved-current construction follows from the wave-shell relation once the bundles and operators have actually been identified. An explicit residual-gauge example shows that smooth compact wave-shell representability alone is insufficient. For the proposed prepared-response-to-joint-graph bridge, we prove precise information-loss, reconstruction-descent, extension, and normal-jet results. Partial witness contraction is noninjective even on smooth compact kernels. Canonical-domain agreement does not establish global descent, and a bound on normal order does not by itself make supported ambiguities finite-dimensional. These are completed mathematical results about bridge conditions. They neither identify the actual VII-B tensor with a conserved V-current nor construct the unresolved XV joint graph. The paper-specific conclusions are restricted to the supplied passages and registry descriptions. \end{abstract} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Flat linearized setting}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Conserved representative}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Joint graph}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The equivalence does not identify the two proposed bridges.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{A conserved representative is not the joint graph. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:scope-Em01-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Flat linearized setting}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Joint graph}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The equivalence does not identify the two proposed bridges.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{A conserved representative is not the joint graph. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:split-Em01-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Flat linearized setting}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Conserved representative}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Joint graph}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The equivalence does not identify the two proposed bridges.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (c) -- (s); \end{tikzpicture} \caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:chain-Em01-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Flat linearized setting}; \node[align=center,font=\small] (m) at (5.2,0) {not the same object}; \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {Joint graph}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Conserved representative. The equivalence does not identify the two proposed bridges.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Flat linearized setting is not Joint graph. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:pair-Em01-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Flat linearized setting}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Conserved representative}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Joint graph}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The equivalence does not identify the two proposed bridges.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:gate-Em01-A} \end{figure} Figure~\ref{fig:split-Em01-A} records one view of the same separation. Figure~\ref{fig:chain-Em01-A} records one view of the same separation. Figure~\ref{fig:pair-Em01-A} records one view of the same separation. Figure~\ref{fig:gate-Em01-A} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-Em01-A} records that separation. The drawing adds no hypothesis. \tableofcontents \section{Claim, evidence, and scope} \subsection{What is being proved} The original plan has two different targets: \begin{enumerate}[label=\textup{(\roman*)}] \item determine when a linear wave-shell representative can be compared with a compact conserved-current representative; \item determine whether a prepared witness response supplies the missing joint-graph construction. \end{enumerate} Neither target permits replacing an unidentified distribution by a similarly named object from another manuscript. The first target admits a sharp conditional theorem in a common flat setting. The second admits rigorous non-implication and interface theorems, but not a construction of the actual graph from the supplied information. The principal results are Theorem~\ref{thm:compatibility}, Proposition~\ref{prop:counterexample}, Theorem~\ref{thm:descent}, and Theorem~\ref{thm:jetextension}. Their proofs are given here. They do not depend on treating the registered manuscripts as a single theory. \subsection{Source statements used for comparison} The supplied passage from V states a smoothing theorem for a compact symmetric conserved tensor distribution, subject to smoothness of its specified causal propagation. Its conclusion is a smooth compact conserved representative on its declared vacuum solution space \cite[Theorem \texttt{thm:smoothing}, lines 1178--1245]{V}. The supplied VII-B passage states a different result: \begin{equation} q-q_{\mathrm{sm}}=\Box r, \qquad q_{\mathrm{sm}}\in C_c^\infty, \qquad r\in\Dist_c, \label{eq:source-wave-repair} \end{equation} and consequently equality of pairings on smooth solutions of \(\Box h=0\) \cite[Theorem \texttt{thm:onshell}, lines 1089--1144]{VIIb}. The same manuscript excludes the original nonzero-smearing bilinear insertion from its declared off-shell microcausal polynomial domain \cite[Theorem \texttt{thm:main}, lines 842--903]{VIIb}. For the graph track, XIV-I constructs the named internal amplitude map but explicitly leaves the characteristic composition with the quantum cross-lines unresolved \cite[Theorems \texttt{thm:I2}, \texttt{thm:I3}]{XIVi}. XIV-J constructs a supported-sector reference on its stated off-characteristic domain and leaves the full joint map unresolved \cite[Theorems \texttt{thm:reference}, \texttt{thm:J2}]{XIVj}. The supplied XV-F record and its patches do not pass F0 \cite{XVf,Registry}. XVI-D reports a prepared witness response and a conditional coordinate identification. XVI-H constructs a compact three-slot kernel under explicit inherited inputs; its specified partial contraction is not the full kernel or quantum admission \cite{XVId,XVIh}. \begin{remark}[Evidence limitation] The mathematical proofs below are self-contained relative to the explicitly stated Green-operator package. Comparisons with the registered manuscripts use only the supplied quotations and registry descriptions. No unquoted manuscript theorem, no unregistered manuscript, and no external-model output is used as a premise. \end{remark} \section{A common classical setting} \subsection{Bundles, pairing, and operators} Let \[ M=\R^{1+3}, \qquad \eta=\operatorname{diag}(1,-1,-1,-1), \qquad \Box=\partial_t^2-\sum_{i=1}^3\partial_i^2. \] All tensor operators below have constant coefficients. Indices are raised and lowered with \(\eta\). Let \(H=\operatorname{Sym}^2T^*M\). A symmetric contravariant tensor distribution \(C\) is identified with its lowered tensor \(c\). Its pairing with a smooth covariant tensor \(h\) is \[ \pair c h=\pair C h. \] In the smooth case this means \[ \pair c h =\int_M c_{\mu\nu}h^{\mu\nu}\dd^4x =\int_M C^{\mu\nu}h_{\mu\nu}\dd^4x. \] Compact support of \(c\) makes this pairing meaningful for arbitrary smooth \(h\), without a decay assumption on \(h\). Define trace reversal, the gauge operator, and the de Donder operator by \begin{align} (Th)_{\mu\nu} &=h_{\mu\nu}-\frac12\eta_{\mu\nu}\tr h, \label{eq:trace-reversal}\\ (K\xi)_{\mu\nu} &=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \label{eq:gauge-operator}\\ (Dh)_\nu &=\partial^\mu(Th)_{\mu\nu}. \label{eq:de-donder} \end{align} In four dimensions, \(T^2=1\). Trace reversal is self-adjoint for the stated pairing. The linearized Einstein operator used in this paper is \begin{equation} 2\mathcal Lh=-\Box(Th)+T K(Dh). \label{eq:einstein-operator} \end{equation} This equation fixes the normalization and sign convention. The formal gauge adjoint is \begin{equation} (K^*c)_\nu=-2\partial^\mu c_{\mu\nu}. \label{eq:gauge-adjoint} \end{equation} We call \(c\) conserved when \(K^*c=0\). Define three spaces of smooth tensors: \begin{align} \SolE&=\{h:\mathcal Lh=0\},\\ \SolW&=\{h:\Box h=0\},\\ \SolH&=\{h:\mathcal Lh=0,\ Dh=0\}. \end{align} They are not identified with any manuscript's named solution space without an additional, explicit domain identification. \begin{lemma}[Operator identities] \label{lem:identities} The operators above satisfy \[ DK\xi=\Box\xi,\qquad \mathcal LK\xi=0,\qquad \partial^\mu(\mathcal Lh)_{\mu\nu}=0. \] Moreover, \[ \SolH=\{h:\Box h=0,\ Dh=0\}. \] \end{lemma} \begin{proof} Expanding \(DK\xi\), the two gradient-divergence terms cancel, leaving \(\Box\xi\). Substitution into \eqref{eq:einstein-operator}, together with commutation of the constant-coefficient operators, gives \(\mathcal LK\xi=0\). For any one-form \(d\), \[ \partial^\mu(TKd)_{\mu\nu}=\Box d_\nu. \] Taking the divergence of \eqref{eq:einstein-operator} therefore gives \[ 2\partial^\mu(\mathcal Lh)_{\mu\nu} =-\Box(Dh)_\nu+\Box(Dh)_\nu=0. \] If \(Dh=0\), equation \eqref{eq:einstein-operator} reduces to \(2\mathcal Lh=-\Box Th\). Since \(T\) is invertible and commutes with \(\Box\), the last assertion follows. \end{proof} \begin{remark}[The solution-space mismatch is substantive] The wave equation \(\Box h=0\) is not the vacuum Einstein equation \(\mathcal Lh=0\) on unrestricted tensor fields. For example, let \[ h_{00}=t^2+(x^1)^2,\qquad h_{\mu\nu}=0 \quad\text{for all other components}. \] Then \(\Box h=0\), whereas direct substitution gives \[ (\mathcal Lh)_{22}=(\mathcal Lh)_{33}=1. \] Conversely, every compact pure gauge \(K\xi\) belongs to \(\SolE\), but a nonzero compact pure gauge cannot solve the homogeneous wave equation, by uniqueness for zero past data. Thus an on-shell equality on \(\SolW\) does not automatically give an equality on \(\SolE\). \end{remark} \subsection{The analytic Green package and its signs} \begin{assumption}[Flat Green package] \label{ass:green} The componentwise retarded and advanced Green operators \(\GR,\GA\) act on compact tensor distributions and satisfy: \begin{enumerate}[label=\textup{(\roman*)}] \item \[ \Box\GR a=a=\Box\GA a,\qquad \GR\Box a=a=\GA\Box a \quad(a\in\Dist_c); \] \item \[ \supp(\GR a)\subseteq J^+(\supp a),\qquad \supp(\GA a)\subseteq J^-(\supp a); \] \item they commute with constant-coefficient differential operators and with \(T\); \item they take compact smooth sources to smooth solutions; \item the corresponding retarded, advanced, and smooth Cauchy uniqueness statements hold. \end{enumerate} \end{assumption} These are the usual componentwise wave Green operators on Minkowski space. We retain them as an explicit analytic input rather than importing an unidentified manuscript propagator. We use both causal signs: \begin{equation} \Eplus=\GR-\GA,\qquad \Eminus=\GA-\GR=-\Eplus. \label{eq:causal-signs} \end{equation} The wave-shell construction below uses \(\Eplus\). The conserved-current formula uses \(\Eminus\). This prevents a hidden sign identification between two source conventions. \begin{lemma}[Compact wave primitive] \label{lem:compact-primitive} For \(a\in\Dist_c(M;H)\), \[ \Eplus a=0 \quad\Longleftrightarrow\quad a=\Box r\text{ for some }r\in\Dist_c(M;H). \] The same equivalence holds with \(\Eminus\). \end{lemma} \begin{proof} If \(\Eplus a=0\), then \[ r:=\GR a=\GA a. \] The support inclusions give \[ \supp r\subseteq J^+(\supp a)\cap J^-(\supp a). \] For compact \(\supp a\), this intersection is compact in Minkowski space. Hence \(r\) is compact and \(\Box r=a\). Conversely, if \(a=\Box r\) with \(r\) compact, both Green operators give \(r\), so their difference vanishes. The \(\Eminus\) statement follows from \eqref{eq:causal-signs}. \end{proof} \begin{lemma}[Wave-shell equality] \label{lem:wave-equality} If \(c,c_{\mathrm{sm}},r\) are compact tensor distributions and \[ c-c_{\mathrm{sm}}=\Box r, \] then \[ \pair c h=\pair{c_{\mathrm{sm}}}h \qquad(h\in\SolW). \] If \(c_{\mathrm{sm}}\) is smooth, then \[ \Eminus Tc=\Eminus T c_{\mathrm{sm}} \] is smooth and spacelike compact. \end{lemma} \begin{proof} Formal self-adjointness and compact support give \[ \pair{c-c_{\mathrm{sm}}}h =\pair r{\Box h}=0 \qquad(h\in\SolW). \] Commutation with \(T\) and the Green identities give \[ \Eminus Tc =\Eminus T c_{\mathrm{sm}} +\Eminus\Box Tr =\Eminus T c_{\mathrm{sm}}. \] The last expression is smooth by Assumption~\ref{ass:green}. Its support is contained in the causal hull of a compact set, hence is spacelike compact. \end{proof} \begin{proposition}[Linear time-slice construction] \label{prop:wave-timeslice} Let \(c\in\Dist_c(M;H)\), and suppose \(u=\Eplus c\) is smooth. Choose a smooth temporal cutoff \(\rho\) equal to zero sufficiently far in the past and one sufficiently far in the future. Then \begin{equation} c_{\mathrm{sm}} =[\Box,\rho]u =\rho''u+2\rho'\partial_tu \label{eq:our-wave-representative} \end{equation} is smooth and compact, and \[ c-c_{\mathrm{sm}}=\Box r \] for some compact tensor distribution \(r\). \end{proposition} \begin{proof} The field \(u\) solves \(\Box u=0\) and is spacelike compact. The derivatives of \(\rho\) are supported in a bounded time slab, whose intersection with the causal hull of a compact set is compact. Thus \eqref{eq:our-wave-representative} is smooth and compact. Since \(c_{\mathrm{sm}}=\Box(\rho u)\), uniqueness gives \[ \GR c_{\mathrm{sm}}=\rho u,\qquad \GA c_{\mathrm{sm}}=(\rho-1)u. \] Therefore \(\Eplus c_{\mathrm{sm}}=u=\Eplus c\). Lemma~\ref{lem:compact-primitive} supplies the compact \(r\). \end{proof} \section{The conserved-current bridge} \subsection{A conserved-current smoothing theorem in the common setting} \begin{theorem}[Conserved-current smoothing] \label{thm:current-smoothing} Let \(c\in\Dist_c(M;H)\) be symmetric and conserved, and suppose \[ v=\Eminus Tc \] is smooth. Let \(\chi\) be a smooth temporal cutoff, zero sufficiently far in the past and one sufficiently far in the future. Define the covariant representative \(j\) and its raised current \(J\) by \begin{equation} j=2\mathcal L(\chi v),\qquad J=\sharp j. \label{eq:conserved-representative} \end{equation} Then \(j\), and hence \(J\), is smooth, compact, and conserved. Moreover, \begin{equation} \pair jh=\pair ch \qquad(h\in\SolE). \label{eq:full-vacuum-representation} \end{equation} \end{theorem} \begin{proof} Since \(v=\Eminus Tc\), \[ \Box v=0,\qquad Dv=\partial^\mu(Tv)_{\mu\nu} =\Eminus\partial^\mu c_{\mu\nu}=0. \] Lemma~\ref{lem:identities} gives \(\mathcal Lv=0\). Consequently \[ \mathcal L(\chi v)=[\mathcal L,\chi]v. \] The right-hand side is supported in the transition slab of \(\chi\). The spacelike compactness of \(v\) therefore makes its support compact. Smoothness is immediate, and the Bianchi identity in Lemma~\ref{lem:identities} gives conservation. First suppose \(h\in\SolH\). Put \[ d=D(\chi v),\qquad F=\Box(\chi Tv). \] Both are smooth and compact. Equation \eqref{eq:einstein-operator} gives \[ j=-F+TKd. \] Integration by parts yields \[ \pair{TKd}h=\pair{Kd}{Th}=-2\pair d{Dh}=0. \] Hence \(\pair jh=-\pair Fh\). As \(Tv=\Eminus c\), the retarded and advanced identities are \[ \GR F=\chi Tv,\qquad \GA F=(\chi-1)Tv. \] It follows that \[ \Eminus F=-Tv=-\Eminus c, \qquad \Eminus(F+c)=0. \] By Lemma~\ref{lem:compact-primitive}, \(F+c=\Box A\) for a compact distribution \(A\). Since \(\Box h=0\), \[ \pair{F+c}h=\pair A{\Box h}=0. \] Thus \(\pair jh=\pair ch\) on \(\SolH\). Now let \(h\in\SolE\) be arbitrary. Solve \[ \Box w=Dh,\qquad w|_{t=0}=0,\qquad \partial_tw|_{t=0}=0. \] A global smooth solution exists without spatial decay: componentwise, the zero-data formula is \[ w(t,x)=\frac1{4\pi} \int_0^t(t-s) \int_{S^2}(Dh)(s,x+(t-s)\omega) \dd\omega\dd s. \] For negative \(t\), the integral has its usual oriented meaning. Each value depends on a finite spatial region, so arbitrary smooth sources cause no convergence issue. Set \(h_0=h-Kw\). The operator identities give \[ Dh_0=0,\qquad \mathcal Lh_0=0. \] Thus \(h_0\in\SolH\). Compact conservation of \(c\) and \(j\) implies \[ \pair c{Kw}=0=\pair j{Kw}. \] The equality on \(h_0\) therefore extends to \(h\), proving \eqref{eq:full-vacuum-representation}. \end{proof} \begin{remark}[Sign audit] If the convention \(\Eplus=\GR-\GA\) is used throughout, the same representative is \[ j=-2\mathcal L(\chi\Eplus Tc). \] Changing the causal-propagator sign without changing this prefactor would reverse the represented functional. Equation~\eqref{eq:conserved-representative} uses the \(\Eminus\) convention fixed here. \end{remark} \subsection{The sharp compatibility criterion} \begin{definition}[Vacuum comparison space] A vacuum comparison space is a linear subspace \(\mathscr X\subseteq\SolE\) containing every compact pure gauge: \[ K\xi\in\mathscr X \qquad \bigl(\xi\in C_c^\infty(M;T^*M)\bigr). \] \end{definition} \begin{theorem}[Conditional linear bridge] \label{thm:compatibility} Let \(\mathscr X\) be a vacuum comparison space. Suppose \(c\in\Dist_c(M;H)\) is symmetric and has a wave-shell repair \begin{equation} c-c_{\mathrm{sm}}=\Box r, \qquad c_{\mathrm{sm}}\in C_c^\infty(M;H), \qquad r\in\Dist_c(M;H). \label{eq:bridge-wave-relation} \end{equation} Then the following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item \(K^*c=0\); \item \(\pair c{K\xi}=0\) for every compact smooth one-form \(\xi\); \item there exists a smooth compact conserved tensor \(j\) satisfying \[ \pair jh=\pair ch \qquad(h\in\mathscr X). \] \end{enumerate} When these assertions hold, the representative in \eqref{eq:conserved-representative} is available. For this representative, \begin{equation} \pair{c_{\mathrm{sm}}}h-\pair jh =-\pair r{\Box h} \qquad(h\in\SolE). \label{eq:full-domain-defect} \end{equation} In particular, \(c_{\mathrm{sm}}\) and \(j\) agree on \(\SolE\cap\SolW\), but their equality on all of \(\mathscr X\) requires the additional condition \[ \pair r{\Box h}=0\qquad(h\in\mathscr X). \] \end{theorem} \begin{proof} The equivalence of (i) and (ii) is exactly the definition of the distributional formal adjoint. If (iii) holds, compact conservation of \(j\) gives \[ \pair c{K\xi}=\pair j{K\xi}=0, \] because \(K\xi\in\mathscr X\). Hence (iii) implies (ii). Assume (i). By Lemma~\ref{lem:wave-equality}, \[ \Eminus Tc=\Eminus T c_{\mathrm{sm}} \] is smooth. Thus the propagator-smoothness hypothesis of Theorem~\ref{thm:current-smoothing} follows from the wave-shell repair rather than being an additional unchecked assumption. That theorem supplies \(j\), proving (iii). Finally, on \(\SolE\) the representative satisfies \(\pair jh=\pair ch\). Pairing \eqref{eq:bridge-wave-relation} with \(h\) gives \[ \pair{c_{\mathrm{sm}}}h-\pair jh =-\pair{\Box r}h=-\pair r{\Box h}, \] which proves the remaining claims. \end{proof} \begin{remark}[Why this is conditional] To apply Theorem~\ref{thm:compatibility} to VII-B and V, one must first prove that the actual \(q=L^*a\), after the stated bundle and duality identification, is the tensor \(c\) in \eqref{eq:bridge-wave-relation}; that the relevant wave operator is this componentwise \(\Box\); and that V's selected solution space is identified with the comparison space being used. Only after these identifications does the theorem reduce the remaining full-space representability test to conservation. The supplied passages do not perform those identifications or prove \(K^*q=0\). The theorem therefore supplies a complete conditional bridge, not an unconditional bridge for the registered observable. \end{remark} \begin{remark}[Restricted spaces] If a proposed comparison space omits compact pure gauges, the necessity argument need not apply. In particular, this theorem does not identify a finite-plane representability construction with a full-space admissibility criterion. V distinguishes those constructions \cite[Object ledger, lines 247--281]{V}. \end{remark} \subsection{A counterexample to the bare smoothing inference} \begin{proposition}[Smooth compact does not imply conserved representability] \label{prop:counterexample} There exists a smooth compact symmetric tensor \(c\) such that: \begin{enumerate}[label=\textup{(\roman*)}] \item \(c-c_{\mathrm{sm}}=\Box r\) with \(c_{\mathrm{sm}}=c\) and \(r=0\); \item \(\Eminus Tc\) is smooth; \item no smooth compact conserved tensor represents \(h\mapsto\pair ch\) on \(\SolE\), or even on \(\SolH\). \end{enumerate} \end{proposition} \begin{proof} Choose \(g\in C_c^\infty(\R^3)\) with \(g(x_0)>0\). Let \(\zeta\) solve \[ \Box\zeta=0,\qquad \zeta(0,x)=0,\qquad \partial_t\zeta(0,x)=g(x). \] It is smooth and spacelike compact. For example, \[ \zeta(t,x)=\frac{t}{4\pi} \int_{S^2}g(x+t\omega)\dd\omega. \] By continuity, \(\partial_t\zeta>0\) on some nonempty spacetime neighborhood, which may be chosen with \(t>0\). Choose a nonzero nonnegative \(\psi\in C_c^\infty(M)\) supported in that neighborhood. Set \[ C^{00}=\psi,\qquad C^{\mu\nu}=0 \quad\text{for all other components}, \] and let \(c\) be its lowered tensor. Define \[ \xi_0=\zeta,\qquad \xi_i=0,\qquad h=K\xi. \] Then \[ Dh=\Box\xi=0,\qquad \mathcal Lh=0,\qquad \Box h=0. \] Thus \(h\in\SolH\). Nevertheless, \[ \pair ch =2\int_M\psi\,\partial_t\zeta\dd^4x>0. \] Every compact conserved tensor \(j\) annihilates this smooth pure gauge: \[ \pair j{K\xi}=\pair{K^*j}\xi=0. \] Consequently no such \(j\) represents \(c\) on \(\SolH\) or \(\SolE\). Assertions (i) and (ii) hold because \(c\) itself is smooth and compact. \end{proof} \begin{remark} This example disproves an implication between the stated properties. It is not claimed to belong to VII-B's actual range \(\{L^*a\}\), and it is not a counterexample to either manuscript's theorem. \end{remark} \subsection{The bilinear insertion remains a different question} \begin{proposition}[A linear bridge does not admit an excluded coefficient] \label{prop:bilinear} Let \(\mathcal P\) be an off-shell polynomial domain, and let \(\mathcal B\) be a functional with \(\mathcal B\notin\mathcal P\). Suppose an admissible quantum algebra requires the classical coefficient of every admitted element to belong to \(\mathcal P\), as an actual functional on the full off-shell configuration space. Then no admitted element can have classical coefficient equal to \(\mathcal B\) on that full space. A coefficientwise formal construction requiring \(\mathcal B\) as an input cannot be repaired merely by proving an on-shell equality for a linear functional. \end{proposition} \begin{proof} Such an admitted element would have a classical coefficient in \(\mathcal P\) equal to \(\mathcal B\), contradicting \(\mathcal B\notin\mathcal P\). An on-shell linear identity supplies neither membership of the bilinear functional in \(\mathcal P\) nor equality of off-shell classical coefficients. It therefore does not remove the contradiction. \end{proof} For the hypotheses of VII-B, the supplied exclusion theorem provides precisely the excluded coefficient for nonzero smearing. Its subsequent formal retarded identity is conditional on existence of its terms and stops at the insertion-domain prerequisite \cite[Theorem \texttt{thm:main}; consequences, lines 1216--1325]{VIIb}. Proposition~\ref{prop:bilinear} preserves that scope. It is not a universal obstruction to all larger domains or modified observables. \section{Witness data versus complete distributions} \subsection{The relevant distinction} A witness response is an observation of a distribution. A joint-graph construction is a distribution, or a map producing one, on a specified full test space. These coincide only if the observation is proved to determine the object. Let \(W\) be a smooth manifold and \(V\) a finite-rank residual bundle. A distributional section is a continuous functional on the corresponding compactly supported dual tests. For a linear test subspace \(\mathscr H\), define \[ \mathscr H^\perp =\{T\in\Dist(W;V):T(\varphi)=0 \text{ for all }\varphi\in\mathscr H\}. \] \begin{lemma}[Witness determination] \label{lem:witness} Agreement of two distributions on \(\mathscr H\) determines their equality exactly when \(\mathscr H^\perp=\{0\}\). In particular, density of \(\mathscr H\) in the full test space is sufficient. \end{lemma} \begin{proof} The difference of the distributions belongs to \(\mathscr H^\perp\). This proves the first assertion. If \(\mathscr H\) is dense, continuity makes that difference vanish on every test. \end{proof} \subsection{Partial contraction loses information} \begin{proposition}[Noninjective witness contraction] \label{prop:partial-contraction} Let \(M_i=\R^{d_i}\), \(d_i\ge1\), and let \(w_1,w_2\) be fixed smooth scalar functions on \(M_1,M_2\). For smooth compact kernels \(T\) on \(M_1\times M_2\times M_3\), define the witness response \[ \mathcal W(T)(\varphi) =\pair T{w_1\otimes w_2\otimes\varphi}, \qquad \varphi\in C_c^\infty(M_3). \] Then \(\mathcal W\) is not injective. Its kernel contains a nonzero smooth compact kernel. \end{proposition} \begin{proof} Choose a nonzero \(v_1\in C_c^\infty(M_1)\) with \[ \int_{M_1}v_1w_1=0. \] Such a function exists: the displayed integral is one linear functional on an infinite-dimensional test space. Explicitly, two linearly independent test functions yield a nonzero linear combination annihilating it. Choose nonzero \(v_2\in C_c^\infty(M_2)\) and \(v_3\in C_c^\infty(M_3)\), and set \[ B=v_1\otimes v_2\otimes v_3. \] Then \(B\ne0\), but \[ \mathcal W(B)(\varphi) =\left(\int v_1w_1\right) \left(\int v_2w_2\right) \left(\int v_3\varphi\right)=0 \] for every \(\varphi\). All these kernels are smooth and compact. \end{proof} \begin{corollary} A prescribed partial contraction does not, by itself, identify the full kernel, even when compact support, finite-order continuity, and absence of wavefront singularities are already available. \end{corollary} \begin{proof} If a kernel has the prescribed contraction, adding the nonzero \(B\) from Proposition~\ref{prop:partial-contraction} preserves that contraction and those regularity properties. \end{proof} \begin{remark}[Scope of the example] The proposition is an explicit information-loss model. It does not assert that such an addition respects the actual XV contact conditions, tensor incidences, or output wavefront restrictions. Those additional frozen conditions may remove ambiguities, but that must be proved. \end{remark} Accordingly, a relation obtained from XVI-D's witness response, or XVI-H's specified partial contraction, cannot be promoted to equality of the actual graph distributions without an identification theorem or a proved determining test class \cite{XVId,XVIh}. \section{Reconstruction descent and canonical-domain agreement} \subsection{Presentations are not reconstructed kernels} Let \(\Pres\) be a vector space of internal presentations and \[ \Rec:\Pres\longrightarrow\Dist(W;V_{\mathrm{int}}) \] a linear reconstruction map. Define \[ \Kint:=\im\Rec. \] A presentation-level candidate has the form \[ \widetilde{\mathsf M}:\Pres\longrightarrow\Dist(W;V). \] \begin{theorem}[Exact descent criterion] \label{thm:descent} There exists a unique linear map \[ \mathsf M:\Kint\longrightarrow\Dist(W;V) \] satisfying \[ \widetilde{\mathsf M}=\mathsf M\Rec \] if and only if \begin{equation} \ker\Rec\subseteq\ker\widetilde{\mathsf M}. \label{eq:descent} \end{equation} If \(\Kint\) carries the quotient topology induced by \(\Pres/\ker\Rec\), a continuous \(\widetilde{\mathsf M}\) satisfying \eqref{eq:descent} induces a continuous \(\mathsf M\). \end{theorem} \begin{proof} If the factorization exists, every null presentation is mapped to zero. Conversely, if \(\Rec A=\Rec A'\), then \(A-A'\in\ker\Rec\). Condition \eqref{eq:descent} gives \(\widetilde{\mathsf M}(A)=\widetilde{\mathsf M}(A')\). Thus \[ \mathsf M(\Rec A):=\widetilde{\mathsf M}(A) \] is well-defined and linear. Surjectivity onto \(\Kint\) gives uniqueness. Continuity is the defining universal property of the quotient topology. \end{proof} This is the same algebraic interface made explicit in the supplied XIV-J passage \cite[Proposition \texttt{prop:descent}, lines 813--846]{XIVj}. Here its topological qualification is stated separately: algebraic descent alone does not prove continuity for an independently prescribed topology on the reconstructed space. \subsection{Restriction does not establish global descent} Let \(\Ucan\subseteq W\) be open, and put \[ Z=W\setminus\Ucan. \] No identification of \(Z\) with a particular attachment set is assumed. \begin{proposition}[Canonical-domain agreement is insufficient] \label{prop:restriction-descent} Agreement with canonical contraction on \(\Ucan\) can show \[ \widetilde{\mathsf M}(A)|_{\Ucan}=0 \qquad(A\in\ker\Rec), \] without proving \(\widetilde{\mathsf M}(A)=0\) on \(W\). \end{proposition} \begin{proof} An explicit model suffices. Let \(W=\R\), \(\Ucan=\R\setminus\{0\}\), and \(\Pres=\R^2\). Fix a nonzero smooth distribution \(k\), and set \[ \Rec(a,b)=ak. \] Define \[ \widetilde{\mathsf M}(a,b)=b\delta_0. \] Its restriction to \(\Ucan\) is the zero canonical map. However, \((0,1)\in\ker\Rec\) and \[ \widetilde{\mathsf M}(0,1)=\delta_0\ne0. \] Thus global descent fails despite canonical-domain agreement. All maps in this example are continuous on the finite-dimensional presentation space. \end{proof} \begin{lemma}[An intrinsic null relation] \label{lem:null-relation} For a normal coordinate \(z\), \[ z\delta'(z)+\delta(z)=0. \] Consequently a presentation splitting that treats these two summands independently need not descend to distributions. \end{lemma} \begin{proof} For a test \(\varphi\), \[ \pair{z\delta'}\varphi =-\partial_z(z\varphi(z))|_{z=0} =-\varphi(0). \] This is \(-\pair\delta\varphi\). Each summand is nonzero, although their sum vanishes. \end{proof} Tensoring this identity with an existing internal distribution does not change it. The supplied XIV-J adapted-chart example uses exactly this mechanism to show why presentation degree is not automatically an intrinsic joint-symbol filtration \cite[Lemma \texttt{lem:null}, lines 853--883]{XIVj}. \subsection{Extension ambiguity and extension existence} \begin{proposition}[The affine extension space] \label{prop:affine-extension} Let \(t^\circ\in\Dist(\Ucan;V)\). If one extension \(T_0\in\Dist(W;V)\) exists, all extensions are exactly \[ T_0+\Dist_Z(W;V), \] where \(\Dist_Z\) denotes distributions supported in \(Z\). This statement does not assert existence of an extension. \end{proposition} \begin{proof} Two extensions have difference zero on \(\Ucan\), equivalently difference supported in \(Z\). Conversely, adding a distribution supported in \(Z\) does not change the restriction. \end{proof} \begin{proposition}[Smooth off-set data need not extend] \label{prop:nonextension} There is a smooth function on \((0,1)\) that has no distributional extension to \((-1,1)\). \end{proposition} \begin{proof} Take \(u(x)=e^{1/x^2}\). Suppose a distribution \(T\) on \((-1,1)\) restricts to \(u\) on \((0,1)\). On a fixed compact neighborhood of zero, continuity of \(T\) gives constants \(C\) and \(m\) such that \[ |T(\varphi)|\le C\max_{0\le j\le m}\|\varphi^{(j)}\|_\infty \] for tests supported in that neighborhood. Choose a nonnegative nonzero \(\varphi\in C_c^\infty((-1/4,1/4))\), and for small \(\varepsilon>0\) put \[ \varphi_\varepsilon(x) =\varphi\!\left(\frac{x-\varepsilon}{\varepsilon^2}\right). \] Its support lies in \((0,1)\), so \[ T(\varphi_\varepsilon) =\int e^{1/x^2}\varphi_\varepsilon(x)\dd x. \] For sufficiently small \(\varepsilon\), points in this support satisfy \(0