% XVI-B.tex % % Single-file pdfLaTeX document. % Compile twice: % pdflatex XVI-B.tex % pdflatex XVI-B.tex % % No external bibliography, figures, or included files. % % Mathematical scope: % Source-level consequences are proved from explicitly stated inputs. % The complete-insertion identification and class-admission programme % remain unresolved. % % A successful compiler execution is not claimed. \documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Landmark XVI-B: Prepared-source diagnostics}, pdfsubject={Source-level proofs and unresolved insertion compatibility} } \setlength{\parindent}{0pt} \setlength{\parskip}{0.4em} \setlength{\emergencystretch}{2em} \setlist[itemize]{ leftmargin=*, itemsep=0.2em, topsep=0.3em } \setlist[enumerate]{ leftmargin=*, itemsep=0.25em, topsep=0.3em } \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\EL}{EL} \newcommand{\Unresolved}{\textbf{Unresolved}} \newcommand{\mc}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Isol}{\mathcal I_{\mathrm{sol}}} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\scopebox}[2]{% \par\medskip \noindent \fbox{% \begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}% }% \par\medskip } \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}[lemma]{Proposition} \newtheorem{corollary}[lemma]{Corollary} \theoremstyle{definition} \newtheorem{problem}[lemma]{Open obligation} \numberwithin{equation}{section} \title{Landmark XVI-B\\[0.3em] \large Prepared-source diagnostics and the complete-insertion boundary\\ \normalsize Revised manuscript --- programme status: unresolved} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We prove the source-level consequences of the supplied density-weighted cubic preparation on the XI-G pure-trace plane-wave pair. The polarized correction is first order in the frequency, its spatial traceless projection vanishes, and the prepared bulk source retains the stated second-order coefficient. We also prove the distinction between comparison with the unramped source and comparison with a sharp-onset source, and derive the weighted-representative residual under compact interior variations. These results do not identify a source with its retarded response, a response with the completed insertion, or an insertion with a compact free-shell kernel. The XI-G obstruction transfers only after a same-class comparison of actual observable kernels. The supplied later sheets are recorded separately: XVI-D constructs the witness response, XVI-E supplies a polarized root pairing, and XVI-F-v2 supplies an independently defined auxiliary evaluation. Their remaining compatibility checks are not discharged by the source calculation. No new quantum retarded coefficient or admission theorem is claimed. \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) {Prepared source}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Comparison identities}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A new retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {No new gravitational retarded coefficient is constructed.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Diagnostics are not a new coefficient. 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-XVI-B} \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) {Prepared source}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A new retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {No new gravitational retarded coefficient is constructed.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Diagnostics are not a new coefficient. 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-XVI-B} \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) {Prepared source}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Comparison identities}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A new retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {No new gravitational retarded coefficient is constructed.}; \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-XVI-B} \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) {Prepared source}; \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) {A new retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Comparison identities. No new gravitational retarded coefficient is constructed.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Prepared source is not A new retarded coefficient. 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-XVI-B} \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: Prepared source}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Comparison identities}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A new retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {No new gravitational retarded coefficient is constructed.}; \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-XVI-B} \end{figure} Figure~\ref{fig:split-XVI-B} records one view of the same separation. Figure~\ref{fig:chain-XVI-B} records one view of the same separation. Figure~\ref{fig:pair-XVI-B} records one view of the same separation. Figure~\ref{fig:gate-XVI-B} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XVI-B} records that separation. The drawing adds no hypothesis. \scopebox{Scope and release status.}{% The regulator width $\varepsilon>0$ is fixed. Elementary source identities and conditional comparison statements are proved below. No new gravitational retarded coefficient is constructed. X-G and XI-G remain in force on their respective hypotheses. Applicability of XI-G to a prepared observable class is not decided by changing the source. The complete-insertion and class-comparison programme remains \Unresolved. The source, response, root pairing, auxiliary evaluation, and class-kernel pairing are distinct objects. An unverified identification is not an assertion that their values differ. } \section{Provenance and the documentary finding} \label{sec:finding} \subsection{Available material} This manuscript uses: \begin{itemize} \item the supplied XVI-B draft and revised XVI-A; \item the supplied XVI-D, XVI-E, XVI-F, and XVI-F-v2 texts; \item the supplied predecessor excerpts and source locators, especially XI-A, XI-B, XI-G, XII-B, XII-C, XII-D, and XII-E; \item the supplied retrieval coverage report. \end{itemize} The coverage report identifies corpus \texttt{b0c93438}, retrieval \texttt{retrieval-3}, and reports retrieved passages across the series through XV-F. That is a report of passage retrieval, not a reproducible, exhaustive search of every complete manuscript. The supplied XVI-C text is absent. Versions of the supplied XVI drafts are not inferred from unprovided metadata. The distinction between the two supplied XVI-F texts matters. For the support of the integrated kernel and for direct fixed-data asymptotics, this note uses \texttt{XVI-F-v2.tex}, not the earlier assertion that the integrated kernel is automatically compactly supported. \subsection{What counts as a replacement} For the documentary question, a replacement would have to be an explicitly identified alternative to: \begin{enumerate} \item the bilinear source symbol of $J_3$, or \item the selected inverse $-\mathcal P E_0^{\mathrm R}$, \end{enumerate} together with a calculation showing removal of the named XI-G witness mechanism while stating the retained normalization, domains, and observation data. The preparation $J_D=\sigma_\varepsilon J_3+\mathcal D_\sigma$ is not described here as a replacement of the frozen bulk symbol or inverse. It is the existing density-prepared source to be analyzed. The defensible documentary conclusion is: \begin{quote} No verified replacement satisfying this description is identified in the supplied record. \end{quote} This statement is not a theorem that no such replacement can exist, and it is not a reason to stop analysis of the existing prepared source. \subsection{Kinds of statement} We distinguish: \begin{itemize} \item \textbf{Imported input:} a formula or theorem reported in supplied predecessor material; \item \textbf{Proved consequence:} a deduction whose proof appears in this manuscript; \item \textbf{Reported later result:} a result stated in a supplied later XVI sheet, not silently imported into an earlier source calculation; \item \textbf{Open obligation:} a missing construction, estimate, or identification. \end{itemize} A citation to a predecessor label does not transfer every theorem of that manuscript. \section{Setting, notation, and explicit inputs} \label{sec:notation} \subsection{Geometry and the switch} Work on ambient Minkowski space \[ M=\mathbb R^4,\qquad y=(t,\mathbf x),\qquad \eta=\operatorname{diag}(1,-1,-1,-1), \] and, where a slab is needed, set \[ M_T=[0,T]\times\mathbb R^3,\qquad M_T^\circ=(0,T)\times\mathbb R^3,\qquad 0<\varepsilon0$. Put \begin{equation} k_1=(1,\mathbf n),\qquad k_2=(1,-\mathbf n),\qquad \gamma_a^\lambda =\eta\,e^{\ii\lambda k_a\cdot y}, \label{eq:witness} \end{equation} where $k\cdot y=\eta_{\mu\nu}k^\mu y^\nu$. The supplied XI-G symbol calculation is retained as input: \begin{align} \mathsf q^{00}&=-2, & \mathsf q^{0i}&=0, & \mathsf q^{ij}&=n^in^j-3\delta^{ij}, \label{eq:qcounter}\\ J_3(\gamma_1^\lambda,\gamma_2^\lambda) &=\lambda^2\mathsf q\,e^{2\ii\lambda t}. \label{eq:bulk-eval} \end{align} The tensor $\mathsf q$ in these displays is the unscaled coefficient. It does not itself contain $\lambda$ \cite{XIG}. \section{The correct XI-G transfer statement} \label{sec:class} A source tensor and an observable kernel are different kinds of object. In particular, changing $J_3$ to $J_D$ does not by itself determine a change of free-shell class. Let $\mathcal K$ be the space of compact distributional kernels on $M^3$ with two symmetric-tensor slots and one scalar slot, symmetric under interchange of the tensor slots. Set $P_m=\Box+m^2$, with $m\ge0$, and define \begin{equation} \begin{split} \Isol=\bigl\{z\in\mathcal K:\ &\langle z,\gamma_1\otimes\gamma_2\otimes\phi\rangle=0\\ &\text{for all smooth } \Box\gamma_1=\Box\gamma_2=0,\quad P_m\phi=0 \bigr\}. \end{split} \label{eq:solution-nullspace} \end{equation} Compact support makes these pairings with smooth solutions meaningful. For $w\in\mathcal K$, write \begin{equation} [w]_{\mathrm{sol}}=w+\Isol. \label{eq:class} \end{equation} Let $\mathcal K_{\mathrm{mc}}\subset\mathcal K$ denote the kernels representing the predecessor's microcausal polynomial target. The reported XI-G conclusion, for its frozen kernel $w_0$ and its stated hypotheses, is \begin{equation} [w_0]_{\mathrm{sol}}\cap\mathcal K_{\mathrm{mc}} =\varnothing. \label{eq:old-obstruction} \end{equation} This is the kernel formulation of its stated non-admission into $\mc$ \cite{XIG}. \begin{proposition}[Conditional transfer of the frozen obstruction] \label{prop:transfer} Suppose an actual prepared observable kernel $w_D\in\mathcal K$ has been constructed. If \begin{equation} w_D-w_0\in\Isol, \label{eq:same-class} \end{equation} then \[ [w_D]_{\mathrm{sol}}=[w_0]_{\mathrm{sol}}, \] and \eqref{eq:old-obstruction} transfers to $w_D$. \end{proposition} \begin{proof} Linearity of the pairings makes $\Isol$ a vector subspace. Consequently \[ w_D+\Isol =w_0+(w_D-w_0)+\Isol =w_0+\Isol. \] The asserted obstruction follows from \eqref{eq:old-obstruction}. \end{proof} Distinct distributions can satisfy \eqref{eq:same-class}. Conversely, proving $w_D-w_0\notin\Isol$ would remove this particular same-class transfer, but would not prove that the new class meets $\mathcal K_{\mathrm{mc}}$. One differing pairing on a triple of free solutions would prove different classes, once both kernels and the compared evaluations exist. Agreement on only one witness family would not prove \eqref{eq:same-class}, whose quantifier is all free triples. Neither the source identity \eqref{eq:actualsource} nor the calculation of its value on \eqref{eq:witness} constructs $w_D$. XI-G applicability is therefore left unresolved, not categorically asserted or categorically excluded. X-G's source-preserving variational obstruction is a separate statement. Changing the action or preparation is not, without a comparison theorem, an upgrade of the original model \cite{XG,XVIA}. \section{Source support and preparation memory} \label{sec:support} The comparisons in this section are made after evaluation on fixed smooth inputs. They are not prescriptions for multiplying an arbitrary distributional kernel by a discontinuous switch. Let \[ \Lambda_\varepsilon =\{(t,\mathbf x):0\le t\le\varepsilon\} \] be the closed temporal layer. It is not spatially compact. \begin{proposition}[Two different source comparisons] \label{prop:support} Assume the imported layer property $\supp\mathcal D_\sigma[\gamma]\subset\Lambda_\varepsilon$, with the correction vanishing at the plateaus as in the supplied XII-C/XII-D preparation. Then, for smooth $\gamma$, \begin{align} J_D[\gamma]&=0 &&(t\le0),\label{eq:pastzero}\\ J_D[\gamma]&=J_3[\gamma] &&(t\ge\varepsilon). \label{eq:plateau-source} \end{align} Comparison with the unramped source gives \begin{equation} J_D-J_3=(\sigma-1)J_3+\mathcal D_\sigma, \qquad J_D-J_3=-J_3\quad(t<0). \label{eq:unramped-comparison} \end{equation} It need not be supported in $\Lambda_\varepsilon$. Define the comparison function \[ H(t)= \begin{cases} 0,&t\le0,\\ 1,&t>0. \end{cases} \] Then the locally integrable tensor \begin{equation} J_D-HJ_3=(\sigma-H)J_3+\mathcal D_\sigma \label{eq:sharp-comparison} \end{equation} has distributional support in $\Lambda_\varepsilon$. \end{proposition} \begin{proof} For $t\le0$, $\sigma=0$ and the imported correction vanishes. For $t\ge\varepsilon$, $\sigma=1$ and the correction vanishes. This proves \eqref{eq:pastzero} and \eqref{eq:plateau-source}, and substitution proves \eqref{eq:unramped-comparison}. For the witness pair, the unramped $00$ source equals $-2\lambda^2e^{2\ii\lambda t}$ even at negative times. Thus \eqref{eq:unramped-comparison} is genuinely not a layer-supported comparison on this family. The coefficient $\sigma-H$ vanishes for $t<0$ and $t>\varepsilon$. Since the evaluated $J_3$ is smooth, $(\sigma-H)J_3$ is locally integrable and supported in the closed layer. Adding the layer-supported correction proves \eqref{eq:sharp-comparison}. \end{proof} The value assigned to $H$ at $0$ does not affect its associated distribution. The product $HJ_3$ above is justified by the smooth evaluated source. No product of $H$ with an arbitrary source kernel is declared. General distribution products require a justified construction; smooth multiplication and singular multiplication must not be conflated \cite{BDH}. The comparison with $HJ_3$ does not introduce a sharp variational model or assert a sharp-width limit. \begin{lemma}[Late source agreement does not force response agreement] \label{lem:memory} There are smooth scalar forcing terms that agree for $t\ge\varepsilon$ but whose zero-data responses do not agree there. \end{lemma} \begin{proof} Choose $F\in C_c^\infty((0,\varepsilon);\mathbb R)$, $F\ge0$, $F\not\equiv0$. The sources $F$ and $0$ agree for $t\ge\varepsilon$. Their zero-data response difference is \[ z(t)=\int_0^t(t-s)F(s)\dd s. \] For $t\ge\varepsilon$, \[ z(t)=tM_0-M_1,\qquad M_0=\int_0^\varepsilon F(s)\dd s>0,\qquad M_1=\int_0^\varepsilon sF(s)\dd s. \] In fact $z(t)>0$ for $t\ge\varepsilon$. Thus the responses differ after their sources have become equal. \end{proof} This scalar example proves the invalidity of the general inference. It is not an evaluation of the gravitational response. A retarded support bound places a response in a causal future, not necessarily in the source's temporal layer \cite{XIID}. \section{Weighted representatives and the boundary question} \label{sec:boundary} Here $\EL$ denotes the bulk Euler--Lagrange derivative under compact interior field variations. The weight $\sigma$ is fixed and is not varied. \begin{proposition}[Weighted divergence residual] \label{prop:boundary} For local densities related by \[ \ell_3'=\ell_3+\partial_\mu B^\mu, \] one has \[ \EL(\ell_3')=\EL(\ell_3), \] whereas \begin{equation} \EL(\sigma\ell_3')-\EL(\sigma\ell_3) =-\EL\bigl((\partial_\mu\sigma)B^\mu\bigr). \label{eq:boundaryrepresentative} \end{equation} \end{proposition} \begin{proof} A total divergence has zero bulk Euler derivative under compact interior variations: its variation is again a divergence whose integral has no boundary contribution. Use the product rule \[ \sigma\partial_\mu B^\mu =\partial_\mu(\sigma B^\mu)-(\partial_\mu\sigma)B^\mu. \] Apply the linear bulk Euler derivative. The first term on the right is a total divergence and contributes zero. This proves \eqref{eq:boundaryrepresentative}. \end{proof} This is the supplied XII-B weighted-representative relation, now derived from the stated variation convention \cite{XIIB}. The XI-A statement that the linear Einstein--Hilbert coefficient is a divergence, together with its exclusion of metric boundary variation, does not supply a boundary action in this setup \cite{XIA}. The narrow citation question is therefore settled: \begin{quote} The cited XI-A passage does not add a boundary action or authorize a metric boundary variation to the prepared cubic model. \end{quote} This does not settle every possible boundary problem. XII-C fixes its cubic representative and uses compact interior variations \cite{XIIC}. Fixing that representative excludes an unannounced change of model; it does not prove the weighted residual in \eqref{eq:boundaryrepresentative} vanishes for every other representative. No boundary charge, boundary action, or additional variational principle is constructed here. \section{The pure-trace source calculation} \label{sec:symbol} Define the constant-coefficient differential expression \begin{equation} \mathcal T^{\mu\nu} =\eta^{\mu\nu}\partial_t +\delta^\mu_0\partial^\nu +\delta^\nu_0\partial^\mu. \label{eq:trace-operator} \end{equation} \begin{proposition}[Arbitrary pure-trace polarization] \label{prop:trace} For arbitrary smooth scalar functions $\psi_1,\psi_2$, with $p=\psi_1\psi_2$, the imported correction satisfies \begin{equation} \mathcal D_\sigma^{\mu\nu}(\psi_1\eta,\psi_2\eta) =-\frac{\sigma'}2\,\mathcal T^{\mu\nu}p. \label{eq:trace-identity} \end{equation} The same identity holds under complex bilinear extension. The coefficient of $\sigma''$ is zero on this sector, including its polarization. \end{proposition} \begin{proof} Since $\sigma$ depends only on $t$, \[ \partial_\rho\sigma=\delta_\rho^0\sigma', \qquad \partial^\mu\sigma=\delta^\mu_0\sigma'. \] Equation \eqref{eq:trace-input} becomes \[ \mathcal D_\sigma^{\mu\nu}[\psi\eta] =-\sigma'\psi\,\mathcal T^{\mu\nu}\psi. \] Apply \eqref{eq:polarization}. The diagonal terms cancel, leaving \[ -\frac{\sigma'}2\bigl( \psi_1\mathcal T^{\mu\nu}\psi_2 +\psi_2\mathcal T^{\mu\nu}\psi_1 \bigr). \] Each summand of $\mathcal T^{\mu\nu}$ is a first derivative with constant coefficient, so the product rule identifies this with the right side of \eqref{eq:trace-identity}. The formula contains no conjugation and is bilinear over $\mathbb C$ after complexification. The reported $C_2$ coefficient vanishes for every pure-trace input. The sum of two pure-trace inputs is pure trace. Consequently its polarization also vanishes. \end{proof} \begin{proposition}[Evaluated correction and prepared source] \label{prop:pair} On \eqref{eq:witness}, define \begin{equation} B^{\mu\nu} =\eta^{\mu\nu}+2\delta^\mu_0\delta^\nu_0, \qquad \mathfrak d_1^{\mu\nu}(t) =-\ii\sigma'(t)B^{\mu\nu}. \label{eq:d1} \end{equation} Then \begin{equation} \begin{aligned} \mathcal D_\sigma^{\mu\nu} (\gamma_1^\lambda,\gamma_2^\lambda) &=\mathfrak d^{\mu\nu}(t,\lambda)e^{2\ii\lambda t},\\ \mathfrak d^{\mu\nu}(t,\lambda) &=\lambda\mathfrak d_1^{\mu\nu}(t) =-\ii\lambda\sigma'(t)B^{\mu\nu}. \end{aligned} \label{eq:Dsymbol} \end{equation} In components, \begin{equation} \mathfrak d^{00}=-3\ii\lambda\sigma', \qquad \mathfrak d^{0i}=0, \qquad \mathfrak d^{ij}=\ii\lambda\sigma'\delta^{ij}. \label{eq:d-components} \end{equation} The prepared evaluated source is \begin{equation} \begin{aligned} S_\lambda^{\mu\nu} &:=J_D^{\mu\nu} (\gamma_1^\lambda,\gamma_2^\lambda)\\ &=\bigl[ \lambda^2\sigma(t)\mathsf q^{\mu\nu} +\lambda\mathfrak d_1^{\mu\nu}(t) \bigr]e^{2\ii\lambda t}. \end{aligned} \label{eq:JDeval} \end{equation} The coefficient of $\sigma''$ in this evaluated correction is zero. \end{proposition} \begin{proof} Put $\psi_a=e^{\ii\lambda k_a\cdot y}$. Then \[ \psi_1\psi_2=e^{2\ii\lambda t}, \quad \partial_t(\psi_1\psi_2) =2\ii\lambda e^{2\ii\lambda t}, \quad \partial^\nu(\psi_1\psi_2) =2\ii\lambda\delta^\nu_0e^{2\ii\lambda t}. \] Substitution in \eqref{eq:trace-identity} gives \eqref{eq:Dsymbol}. The signature gives $B^{00}=3$, $B^{0i}=0$, and $B^{ij}=-\delta^{ij}$, proving \eqref{eq:d-components}. Add $\sigma$ times \eqref{eq:bulk-eval} to \eqref{eq:Dsymbol}, using \eqref{eq:actualsource}. This gives \eqref{eq:JDeval}. \end{proof} The coefficient $\mathfrak d$ includes $\lambda$. The coefficient $\mathfrak d_1$ does not. Likewise, $\mathsf q$ is not multiplied by an additional frequency factor inside its definition. At fixed $t$, after removing the common exponential, \eqref{eq:JDeval} is a polynomial in $\lambda$. Its degree-two coefficient is $\sigma(t)\mathsf q$. Where that coefficient is nonzero, a degree-one correction cannot cancel it as a polynomial family. This is a source statement, not a conclusion about a retarded response or complete insertion. At $\lambda=0$, the displayed source vanishes. For $\lambda>0$ and $t\ge\varepsilon$, \begin{equation} S_\lambda^{00}(t,\mathbf x) =-2\lambda^2e^{2\ii\lambda t}. \label{eq:source-noncompact} \end{equation} Thus the source is not in $C_c^\infty(M_T^\circ)$: it is nonzero at every spatial point at such a slab time. \subsection{Spatial traceless channel} Let \begin{equation} h_{ij}=n_in_j-\frac13\delta_{ij}, \qquad n_i=\delta_{ij}n^j. \label{eq:h} \end{equation} This is a Euclidean spatial tensor, not Lorentzian lowering of the spatial components of $k_a$. \begin{corollary}[Source-level traceless diagnostic] \label{cor:tf} The spatial traceless projections and contractions satisfy \begin{align} \mathsf q_{\mathrm{TF}}^{ij} &=n^in^j-\frac13\delta^{ij}, & (\mathfrak d_1)_{\mathrm{TF}}^{ij}&=0, \label{eq:tf}\\ h_{ij}\mathsf q^{ij}&=\frac23, & h_{ij}\mathfrak d_1^{ij}&=0. \label{eq:tf-contractions} \end{align} Consequently \begin{equation} h_{ij}S_\lambda^{ij} =\frac23\lambda^2\sigma(t)e^{2\ii\lambda t}. \label{eq:source-contraction} \end{equation} \end{corollary} \begin{proof} Equation \eqref{eq:qcounter} gives $\delta_{ij}\mathsf q^{ij}=1-9=-8$. Subtracting one third of that trace gives \[ \mathsf q_{\mathrm{TF}}^{ij} =n^in^j-3\delta^{ij}+\frac83\delta^{ij} =n^in^j-\frac13\delta^{ij}. \] Equation \eqref{eq:d-components} shows that the correction is spatially proportional to $\delta^{ij}$, so its traceless projection is zero. Also $h_{ij}\delta^{ij}=0$ and \[ h_{ij}n^in^j =|\mathbf n|^4-\frac13|\mathbf n|^2 =\frac23. \] The contractions and \eqref{eq:source-contraction} follow. \end{proof} This does not prove that the completed insertion detects this channel. The insertion can select tensor contractions and preparation-dependent terms not determined by this source projection alone. \subsection{Input covectors and output frequency} The phase covectors are the lowered vectors \[ (k_1)_\flat=(1,-\mathbf n),\qquad (k_2)_\flat=(1,\mathbf n). \] They are co-oriented future null covectors, one in each metric slot. Their vector sum is \begin{equation} K_{\mathrm{sum}}=k_1+k_2=(2,\mathbf0), \qquad K_{\mathrm{sum}}^2=4. \label{eq:sum} \end{equation} Therefore \begin{equation} \Box e^{2\ii\lambda t} =-4\lambda^2e^{2\ii\lambda t}. \label{eq:box-phase} \end{equation} The generated phase is noncharacteristic for the massless wave operator. This does not combine the two metric-slot covectors into a replacement for the three-slot XI-G criterion. The arithmetic multiplier $-1/(4\lambda^2)$ sends a coefficient $\lambda^2 a$ to $-a/4$ and a coefficient $\lambda b$ to $-b/(4\lambda)$. This is only frozen-frequency arithmetic. For time-dependent coefficients it omits coefficient derivatives, initial data, and preparation-generated homogeneous terms. It is not an asymptotic theorem for $E_0^{\mathrm R}$. For each fixed $\lambda$, $S_\lambda$ is smooth. That does not establish smoothness or microcausality of the distributional kernel of the underlying polynomial functional. Wavefront properties of source evaluations and of functional kernels are different questions \cite{BDH}. \section{Integral and scalar diagnostics} \label{sec:diagnostics} \subsection{Integration by parts can introduce a second switch derivative} \label{sec:ibp} \begin{lemma}[A smooth-amplitude integral identity] \label{lem:ibp} Let $a$ be smooth on a neighborhood of $[0,\varepsilon]$, independent of $\lambda$, and let $\lambda>0$. Define the illustrative integral \[ I_{\mathrm{toy}}^{\mu\nu}(\lambda) =-\ii\lambda B^{\mu\nu} \int_0^\varepsilon a(t)\sigma'(t)e^{2\ii\lambda t}\dd t. \] Then \begin{equation} I_{\mathrm{toy}}^{\mu\nu}(\lambda) =\frac{B^{\mu\nu}}2 \int_0^\varepsilon \bigl(a'\sigma'+a\sigma''\bigr) e^{2\ii\lambda t}\dd t. \label{eq:ibp} \end{equation} \end{lemma} \begin{proof} Set $b=a\sigma'$. By \eqref{eq:flat}, $b(0)=b(\varepsilon)=0$. One integration by parts gives \[ \int_0^\varepsilon b e^{2\ii\lambda t}\dd t =-\frac1{2\ii\lambda} \int_0^\varepsilon b'e^{2\ii\lambda t}\dd t. \] Since $b'=a'\sigma'+a\sigma''$, multiplication by $-\ii\lambda B^{\mu\nu}$ proves \eqref{eq:ibp}. \end{proof} The absence of $\sigma''$ in \eqref{eq:Dsymbol} is compatible with its appearance in \eqref{eq:ibp}. The latter derivative arose from transferring a derivative in an integral, not from changing the source symbol. This calculation assumes a smooth scalar amplitude. It does not prove that an amplitude obtained from an unidentified singular retarded kernel is smooth, and it does not evaluate the completed gravitational insertion. \subsection{A smooth-switch scalar residual} \label{sec:ode} \begin{lemma}[Residual of the local oscillatory candidate] \label{lem:ode-residual} For the real switch $\sigma$ and $\lambda>0$, set \[ v_\lambda(t)=-\frac{\sigma(t)}4e^{2\ii\lambda t}. \] Then \begin{equation} v_\lambda''-\lambda^2\sigma e^{2\ii\lambda t} =\left(-\ii\lambda\sigma'-\frac{\sigma''}4\right) e^{2\ii\lambda t}. \label{eq:ode-residual} \end{equation} On an interval, $v_\lambda$ solves $v_\lambda''=\lambda^2\sigma e^{2\ii\lambda t}$ if and only if $\sigma$ is constant there. \end{lemma} \begin{proof} Direct differentiation gives \[ v_\lambda' =-\left(\frac{\sigma'}4+\frac{\ii\lambda\sigma}2\right) e^{2\ii\lambda t} \] and \[ v_\lambda'' =\left(\lambda^2\sigma-\ii\lambda\sigma' -\frac{\sigma''}4\right)e^{2\ii\lambda t}. \] This proves \eqref{eq:ode-residual}. If the residual vanishes on an interval, its imaginary part is $-\lambda\sigma'=0$, because $\sigma$ is real and $\lambda>0$. Thus $\sigma$ is constant there. Conversely, a constant $\sigma$ makes both derivative terms zero. \end{proof} The restrictions ``real switch'' and ``$\lambda>0$'' are essential to the stated converse. They prevent an overbroad reading of the original scalar illustration. \subsection{A sharp-switch scalar solution} \begin{lemma}[Sharp-onset scalar solution] \label{lem:sharp-ode} The function \begin{equation} u_H(t) =H(t)\left( -\frac14e^{2\ii\lambda t} +\frac14+\frac{\ii\lambda t}{2} \right) \label{eq:sharp-ode} \end{equation} is a distributional solution of \[ u_H''=\lambda^2H(t)e^{2\ii\lambda t}, \] vanishes for $t<0$, and has zero initial position and velocity. \end{lemma} \begin{proof} Let $b(t)=-e^{2\ii\lambda t}/4+1/4+\ii\lambda t/2$. Then \[ b(0)=0,\qquad b'(0)=0,\qquad b''(t)=\lambda^2e^{2\ii\lambda t}. \] Distributional differentiation of $Hb$ therefore produces neither a delta term nor a derivative of a delta: \[ (Hb)''=Hb''+\delta\,b'(0)+\delta'\,b(0)=Hb''. \] This proves the equation. The traces at $0$ and the negative-time vanishing follow from the same formulas. \end{proof} For $t>0$, the terms $1/4+\ii\lambda t/2$ solve the homogeneous scalar equation. The linear term is $O(\lambda)$ on a fixed bounded time interval. This $H$ is not $\sigma_\varepsilon$. Neither Lemma~\ref{lem:ode-residual} nor Lemma~\ref{lem:sharp-ode} is a model or evaluation of the gravitational insertion, a value of the selected graviton inverse, or a cancellation of the XI-G witness. \subsection{Level of each calculation} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.29\linewidth}Y Y@{}} \toprule Calculation & Level & Conclusion not supplied\\ \midrule Pure-trace polarization & Source identity & Response or kernel admission\\ Prepared plane-wave value & Evaluated source & Complete insertion value\\ Spatial traceless contraction & Source projection & Detection by the insertion\\ Equation \eqref{eq:ibp} & Smooth-amplitude illustrative integral & Smoothness of a retarded-kernel amplitude\\ Scalar ODE formulas & Scalar illustrations & Gravitational response or cancellation\\ Timelike sum frequency & Phase arithmetic & Three-slot wavefront criterion\\ \bottomrule \end{tabularx} \end{center} \section{The actual completion and the root reading} \label{sec:completion} \subsection{The two completion legs} For an admissible response tensor $r$, the supplied completion uses \cite{XIID,XVID} \begin{align} a_1^{\mu\nu}[r] &=\frac12\eta^{\mu\nu}\operatorname{tr}_\eta r-r^{\mu\nu}, & b_1[r]&=\frac12\operatorname{tr}_\eta r, \label{eq:coefficients}\\ X_1^\mu[r] &=-E_0^{\mathrm R} \partial_\nu\bigl(\sigma a_1^{\nu\mu}[r]\bigr), \label{eq:coordinate-leg}\\ V_{1,g}[r]\phi &=\partial_\mu \bigl(g a_1^{\mu\nu}[r]\partial_\nu\phi\bigr) +m^2g b_1[r]\phi, \label{eq:scalar-forcing}\\ \phi_1[r;\phi] &=-E_m^{\mathrm R}\bigl(V_{1,g}[r]\phi\bigr). \label{eq:scalar-leg} \end{align} Here \[ f\in C_c^\infty(M_T^\circ),\qquad g\in C_c^\infty(M_T^\circ;[0,1]), \] and $\phi$ is real and smooth in the reported later witness calculations. The completion functional is \begin{equation} \mathcal A_1[f,g](r;\phi) =\int f(x)\bigl( \phi_1[r;\phi](x) -X_1^\mu[r](x)\partial_\mu\phi(x) \bigr)\dd^4x. \label{eq:Aone} \end{equation} The original diagonal insertion is defined by \begin{equation} \mathcal C_{3,\mathrm D}(f,g)[\gamma,\phi] =\mathcal A_1[f,g](u_{\mathrm D}[\gamma];\phi) \label{eq:CD} \end{equation} on its inherited domain. The divergence in \eqref{eq:scalar-forcing} retains derivatives of $g$. The mass term is retained. No factor $g$ is inserted into the coordinate forcing \eqref{eq:coordinate-leg}. \subsection{The root is a separate verification route} The supplied XII-D theorem states a continuous linear compact root map \[ a_{f,g}:C^\infty(M)\longrightarrow C_c^\infty(M;\operatorname{Sym}^2T^*M) \] for fixed observation data and collar, and the root identity \begin{equation} \mathcal C_{3,\mathrm D}(f,g)[\gamma,\phi] =\int a_{f,g}[\phi]_{\mu\nu}(y) J_D^{\mu\nu}[\gamma](y)\dd^4y \label{eq:root-identity} \end{equation} on its stated domain \cite{XIID}. The reported existence of this scalar identity is not replaced by the source contraction \eqref{eq:source-contraction}. Conversely, its polarization should not be rejected merely because the theorem is printed using a single diagonal metric. \begin{lemma}[Conditional polarization of the root identity] \label{lem:root-polarization} Suppose \eqref{eq:root-identity} holds for every metric in a real vector space of smooth inputs, with fixed $f,g,\phi$ and root. Suppose $J_D$ is homogeneous quadratic there. Polarization of the diagonal insertion then equals the root paired with the polarized source. Its complex bilinear extension satisfies, on the witness, \begin{equation} T^{\mathbb C}(\gamma_1^\lambda,\gamma_2^\lambda;\phi) =\int a_{f,g}[\phi]_{\mu\nu}S_\lambda^{\mu\nu}\dd^4y, \label{eq:polarized-root} \end{equation} provided the real cosine and sine inputs and their required sums belong to that real domain. \end{lemma} \begin{proof} Apply \eqref{eq:root-identity} to $\gamma_1+\gamma_2$, $\gamma_1$, and $\gamma_2$. Subtract and divide by two. Linearity of the integral gives the polarized source \eqref{eq:polarization}. For real inputs $a,b,c,d$, complex bilinear extension has the unique four-term form \[ T^{\mathbb C}(a+\ii b,c+\ii d) =T(a,c)-T(b,d)+\ii T(a,d)+\ii T(b,c). \] Apply the same expansion to the polarized source and use Proposition~\ref{prop:pair}. \end{proof} This lemma is a consequence of a valid root identity on the specified real domain. It does not independently verify the coordinate-field construction in \eqref{eq:coordinate-leg}, nor extend a compact-test Green theorem to a noncompact forcing. The supplied XVI-E makes precisely this scalar root reading on the real cosine and sine family and denotes the resulting witness value by $Q_\lambda$ \cite{XVIE}. \section{Smooth root tests and their limited asymptotics} \label{sec:root-tests} The following result proves a test-pairing statement, not an admission statement. \begin{proposition}[Fixed compact smooth amplitude] \label{prop:root-decay} Let $a_{\mu\nu}\in C_c^\infty(\mathbb R^4;\operatorname{Sym}^2T^*M)$ be independent of $\lambda$. Define \[ Q_a(\lambda) =\int_{\mathbb R^4} a_{\mu\nu}(y)S_\lambda^{\mu\nu}(y)\dd^4y. \] The integral exists, and \[ Q_a(\lambda)=O(\lambda^{-\infty}) \qquad(\lambda\to+\infty) \] at fixed $\varepsilon$, $\mathbf n$, and $a$. \end{proposition} \begin{proof} The integrand is smooth and compactly supported. Since $S_\lambda$ is spatially independent, put \[ A(t)=\int_{\mathbb R^3} a_{\mu\nu}(t,\mathbf x)\sigma(t)\mathsf q^{\mu\nu} \dd^3x, \] \[ B(t)=\int_{\mathbb R^3} a_{\mu\nu}(t,\mathbf x)\mathfrak d_1^{\mu\nu}(t) \dd^3x. \] Both belong to $C_c^\infty(\mathbb R)$ and are independent of $\lambda$. Equation \eqref{eq:JDeval} gives \[ Q_a(\lambda) =\lambda^2\int_{\mathbb R}A(t)e^{2\ii\lambda t}\dd t +\lambda\int_{\mathbb R}B(t)e^{2\ii\lambda t}\dd t. \] For each integer $M\ge0$, integration by parts yields \[ \left|\int A(t)e^{2\ii\lambda t}\dd t\right| \le(2\lambda)^{-M}\|A^{(M)}\|_{L^1}, \] and the analogous bound for $B$. Given $N\ge0$, choose $M=N+2$. For $\lambda\ge1$, \[ |Q_a(\lambda)| \le2^{-(N+2)}\lambda^{-N} \left(\|A^{(N+2)}\|_{L^1} +\|B^{(N+2)}\|_{L^1}\right). \] This proves the stated rapid decay. \end{proof} For the reported compact root $a=a_{f,g}[\phi]$, this reproduces the fixed-data estimate of XVI-E. It does not establish: \begin{itemize} \item the same estimate for a frequency-dependent $\phi$; \item uniformity as $\varepsilon\downarrow0$; \item pointwise decay of the response; \item a kernel wavefront bound or quantum admission; \item identification with an independently constructed auxiliary completion. \end{itemize} In particular, rapid decay of a smooth compact test pairing can coexist with a nonzero source or response contraction. \section{Update from the supplied later sheets} \label{sec:later} The original XVI-B statement that the entire successor chain was unanswered must be localized. The supplied later sheets report progress on named objects. Those results are not reclassified as absent merely because they were not proved in the original XVI-B draft. \subsection{Response: XVI-D} XVI-D constructs \begin{equation} V_\lambda^{\mu\nu}(t) =\int_0^t(t-s)S_\lambda^{\mu\nu}(s)\dd s, \qquad R_\lambda=-\mathcal P V_\lambda, \label{eq:later-response} \end{equation} and, using the inherited smooth zero-data uniqueness theorem, identifies \[ U_D[\gamma_1^\lambda,\gamma_2^\lambda]=R_\lambda. \] It also computes the spatial traceless response contraction \cite{XVID}. These are reported later response results. They do not identify $V_\lambda$ with a Green symbol applied outside its proved compact-interior domain. The scalar completion forcing is compact because every term contains $g$ or a derivative of $g$. The coordinate forcing \[ F_{\mathrm{co}}^\mu =\partial_t(\sigma V_\lambda^{0\mu}) \] has no such factor $g$ and is spatially noncompact. Thus the compact-interior Green lemma alone does not verify its use in the named coordinate leg. \subsection{Root and coordinate candidate: XVI-E} XVI-E constructs the root pairing $Q_\lambda$, retains the polarized scalar root identity $T^{\mathbb C}=Q_\lambda$, and constructs the explicit field \[ X_{\mathrm{cand}}^\mu(t) =-\int_0^t\sigma(s)V_\lambda^{0\mu}(s)\dd s. \] It proves the corresponding wave equation and zero-data uniqueness comparison, but does not identify that candidate with an independently verified inherited coordinate field on the witness family \cite{XVIE}. Uniqueness of solutions to a Cauchy problem is not, on its own, a definition of the named Green operator on a larger source space. \subsection{Auxiliary evaluation and corrected support: XVI-F-v2} XVI-F-v2 independently defines \[ I_{\mathrm{cand}} =\int f\bigl( \phi_{\mathrm{cand}} -X_{\mathrm{cand}}^\mu\partial_\mu\phi \bigr)\dd^4x. \] It proves existence and a source representation involving an integrated kernel. The earlier claim that this integrated kernel is automatically compact was corrected in that version \cite{XVIFvTwo}. The relevant elementary support fact is as follows. \begin{lemma}[An integrated kernel can have an affine past tail] \label{lem:affine-tail} Let $\widehat\tau\in C_c^\infty(\mathbb R)$ be supported in $[0,t_*]$ and define \[ K_I(u)=\int_u^\infty(t-u)\widehat\tau(t)\dd t. \] Then $K_I$ is smooth, $K_I''=\widehat\tau$, and $K_I=K_I'=0$ for $u>t_*$. For $u<0$, \begin{equation} K_I(u)=M_1-uM_0, \qquad M_j=\int_0^\infty t^j\widehat\tau(t)\dd t \quad(j=0,1). \label{eq:affine-tail} \end{equation} Thus compact support of $\widehat\tau$ does not imply compact support of $K_I$. Nevertheless $\sigma K_I$ and $\sigma'K_I$ belong to $C_c^\infty(\mathbb R)$. The same statements hold componentwise for tensor coefficients. \end{lemma} \begin{proof} Differentiate under the integral: \[ K_I'(u)=-\int_u^\infty\widehat\tau(t)\dd t, \qquad K_I''(u)=\widehat\tau(u). \] The future vanishing follows from the source support. For $u<0$, expand $(t-u)$ over the fixed source support to obtain \eqref{eq:affine-tail}. The product $\sigma K_I$ is smooth, vanishes for $u\le0$, and vanishes for $u>t_*$. The product $\sigma'K_I$ is smooth and supported in the fixed ramp. \end{proof} Consequently, a representation \[ -\int K_I^{ab}(u)S_{\lambda,ab}(u)\dd u \] with fixed smooth coefficients of this type has rapid decay by the same integration-by-parts proof as Proposition~\ref{prop:root-decay}. This is the direct fixed-data mechanism used for $I_{\mathrm{cand}}$ in XVI-F-v2. It does not require transferring decay from $Q_\lambda$. XVI-F-v2 leaves the compatibility residual \[ Q_\lambda-I_{\mathrm{cand}} =\int\rho^{ab}(u)S_{\lambda,ab}(u)\dd u \] unresolved, with \[ \rho^{ab}(u) =\int_{\mathbb R^3}a_{f,g}[\phi]^{ab}(u,\mathbf x)\dd^3x +K_I^{ab}(u). \] Its explicit advanced-field construction settles a spatial-integral identity for that constructed field. The remaining named check is its identification with the inherited advanced symbol on the boundary-flat source class. The pairings are not proved equal and are not asserted to differ. Two rapidly decreasing scalar families need not be equal. \subsection{Reported predecessor admission} The supplied XII-E handoff reports compact microcausal membership of the polarized kernel of the completed XII-D insertion on its stated sheet \cite{XIIE}. This note neither re-proves nor rescinds that reported theorem. It is not silently identified with XVI-C's class-kernel pairing, an independently defined auxiliary evaluation, or a quantum retarded coefficient. Any proposed use of it must state the matching object, domain, and normalization. \section{The complete-pairing specification and status ledger} \label{sec:chain} The original reference to ``the complete pairing of XVI-C'' is a programme pointer. Since XVI-C was not supplied, this manuscript does not invent its section labels or definitions. The following chain is a local specification of what must be checked: \[ \text{prepared source} \ \longrightarrow\ \text{response} \ \longrightarrow\ \text{completed insertion} \ \longrightarrow\ \text{compact kernel/class} \ \longrightarrow\ \text{class-kernel evaluation}. \] A smooth root pairing or auxiliary evaluation can be a separate verification route, but an identification with this chain must be proved. \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{0.20\linewidth}Y Y@{}} \toprule Stage/object & Established or reported & Remaining check\\ \midrule Prepared source $J_D$ & Imported from XII-C; pure-trace witness value proved here & No full observable kernel follows from the source value\\ \addlinespace Witness response $U_D$ & Constructed in supplied XVI-D & Not automatically the completed insertion\\ \addlinespace Scalar completion leg & Verified in supplied XVI-D & Does not verify the coordinate leg\\ \addlinespace Inherited coordinate leg & Formula retained; compact-source theorem distinguished & Identification on the witness family remains unverified\\ \addlinespace Root pairing $Q_\lambda$ & Constructed in XVI-E; fixed smooth-root decay proved above & Not automatically the independently defined completion\\ \addlinespace Auxiliary $I_{\mathrm{cand}}$ & Constructed in XVI-F-v2; direct fixed-data decay reported & Equality with $Q_\lambda$ and inherited completion not verified\\ \addlinespace Compact class kernel $w_D$ & No new kernel constructed here & Exact object and free-shell comparison required\\ \addlinespace Class pairing $A_D(\lambda)$ & XVI-C definition not supplied & Specification, existence, and identification required\\ \addlinespace XI-G transfer & Proposition~\ref{prop:transfer} & Same-class condition not evaluated for an actual matching kernel\\ \bottomrule \end{tabularx} \end{center} The complete-pairing test must retain separately: \begin{enumerate} \item the bulk source and the full correction $\mathcal D_\sigma$; \item forced response terms and preparation-generated homogeneous terms; \item both completion legs, including derivatives of $g$ and the mass term; \item the tensor contractions actually selected by the insertion; \item any gauge companion and Einstein-equation term produced by the relevant complete derivation; \item support, operator-domain, polarization, and complexification hypotheses at each stage; \item the final comparison with the frozen observable kernel. \end{enumerate} The gauge-fixed free-shell condition is not substituted for an Einstein-shell hypothesis. A cancellation is used only after proof. The decisive question is which preparation-dependent terms the complete observable detects, not whether the correction vanishes as an uncontracted source. \begin{problem}[Inherited completion compatibility] \label{prob:completion} Verify the inherited coordinate-field construction on the real cosine and sine response family and its required sums. Prove that its polarization and complexification give the same field used by the independently constructed candidate, if that identification is claimed. State the exact source space, zero-data prescription, and cutoff-independence or operator-domain theorem used. A compact final observation test does not by itself make the coordinate forcing a compact-interior source. Status: \Unresolved. \end{problem} \begin{problem}[Root--auxiliary identification] \label{prob:root-compatibility} Resolve the residual comparison reported in XVI-F-v2. Specify whether the inherited advanced symbol on the boundary-flat class denotes the independently constructed smooth zero-final-data field. Prove equality of the required pairings if it follows. Do not infer equality from their separate rapid decay, and do not infer inequality from their different definitions. Status: \Unresolved. \end{problem} \begin{problem}[Compact-kernel and class comparison] \label{prob:class} Specify the actual observable to be compared with XI-G, its compact kernel $w_D$, its slot structure and symmetry, and the free-shell pairing. Evaluate whether $w_D-w_0\in\Isol$. Apply Proposition~\ref{prop:transfer} only after that comparison. A different-class result would not, by itself, discharge admission. Status: \Unresolved. \end{problem} \begin{problem}[Full asymptotic and admission interface] \label{prob:admission} After the actual evaluation is defined and identified, establish its required high-frequency estimate with the complete tensor and branch summation retained. State whether the scalar test is fixed or frequency dependent, which constants are uniform, and which kernel wavefront or admission conclusion is sought. The fixed-data estimates for $Q_\lambda$ and $I_{\mathrm{cand}}$ are not substituted for that conclusion. Status: \Unresolved. \end{problem} These are compatibility and construction obligations, not proved obstructions. They do not discharge XVI-A's retarded-product obligations 4--6. \section{Debugging summary} \label{sec:debug} The revision makes the following corrections explicit: \begin{enumerate} \item \textbf{Class transfer is conditional.} A changed source is not proof of a changed free-shell class. The exact same-class transfer is Proposition~\ref{prop:transfer}. \item \textbf{The documentary finding is bounded.} The supplied retrieval report is not an exhaustive nonexistence theorem. \item \textbf{The sharp comparison has a domain.} $HJ_3$ is used only for smooth evaluated sources. No discontinuous multiplier is applied automatically to an unidentified distributional kernel. \item \textbf{Temporal support is not compact support.} A closed ramp layer is spatially unbounded. \item \textbf{Late source agreement is not response agreement.} Lemma~\ref{lem:memory} gives an explicit counterexample to that inference. \item \textbf{The boundary conclusion is narrow.} No boundary action follows from the cited XI-A passage. Weighted representative changes still have \eqref{eq:boundaryrepresentative}. \item \textbf{Frequency factors are separated.} $\mathsf q$ and $\mathfrak d_1$ are independent of $\lambda$; $\mathfrak d=\lambda\mathfrak d_1$. \item \textbf{The ODE converse has its hypotheses.} The switch is real and $\lambda>0$. \item \textbf{Polarization is not rejected on typographical grounds.} A valid diagonal root identity on a real vector space polarizes by Lemma~\ref{lem:root-polarization}. That scalar operation does not independently verify an unconstructed coordinate-field representation. \item \textbf{Later progress is not erased.} XVI-D's response and XVI-E's root pairing are recorded. XVI-F-v2 supplies direct auxiliary decay, not a transfer from the root. \item \textbf{The integrated support error is corrected.} A compact source can generate an affine past tail. Effective switched amplitudes can be compact even when the integrated kernel is not. \item \textbf{No stale hypothesis is cited.} No null-survival assumption is imposed. The supplied revised XVI-A contains no adopted \texttt{hyp:null-survival} gate. \item \textbf{No compilation result is fabricated.} This is a single-file pdfLaTeX source. Compiler execution and full-corpus verification are not claimed. \end{enumerate} \section{Non-claims and final release} \label{sec:nonclaims} This manuscript does not claim: \begin{itemize} \item a repair of X-G or XI-G; \item a new admission into $\mc$ or $\mathscr F_{\mathrm{pol}}$; \item a new quantum retarded coefficient or a completed second-order relational construction; \item an interacting BRST differential or an unrestricted quantum master equation; \item a general extension of the named Green operators from compact-interior tests to every forcing; \item equality or inequality of $I_{\mathrm{cand}}$ and $Q_\lambda$; \item a complete inherited insertion evaluation from the source-level contraction alone; \item a sharp-width limit or estimates uniform in $\varepsilon$; \item a loop, state, positivity theorem, or ultraviolet subtraction. \end{itemize} The selected inverse retains coefficient $-1$, not $-4$. Full Einstein--Hilbert density weighting is not substituted for the selected cubic preparation. The source $\sigma J_3$ alone is not the prepared model \cite{XIIC,XVIA}. \scopebox{Final release statement.}{% The source-level identities, support comparisons, weighted-representative residual, and elementary diagnostics are proved from the stated inputs. The supplied later response, root, and auxiliary results are recorded on their named objects and domains. Their unresolved identifications are not replaced by a source symbol argument. X-G and XI-G retain their original quantifiers. Applicability of XI-G to an actual matching prepared class requires a kernel comparison. The complete-insertion, class-comparison, and admission programme remains \Unresolved. This is a completed source-diagnostic proof manuscript, not a completed proof of the unresolved retarded coefficient. } \begin{thebibliography}{99} \bibitem{XVIA} \emph{Landmark XVI-A}. Supplied revised \texttt{XVI-A.tex}. Used for the declared normalization, forcing sign, conditional class-transfer correction, and the unresolved retarded-product obligations. No external publication status is asserted. \bibitem{XVID} \emph{Landmark XVI-D}. Supplied \texttt{XVI-D.tex}. Internal locators: \texttt{eq:Slambda}, \texttt{eq:Vlambda}, \texttt{lem:identify}, \texttt{lem:contraction}, \texttt{lem:coord}, and \texttt{lem:scalar}. Used as a supplied later response and completion-domain record. \bibitem{XVIE} \emph{Landmark XVI-E}. Supplied \texttt{XVI-E.tex}. Internal locators: \texttt{lem:Q}, \texttt{lem:TD}, \texttt{lem:TC}, \texttt{lem:Xcand}, \texttt{lem:compat}, and \texttt{cor:decay}. Used for the root pairing, conditional polarized root reading, and delimited coordinate compatibility. \bibitem{XVIFvTwo} \emph{Landmark XVI-F}, revised supplied text. Filename \texttt{XVI-F-v2.tex}. Internal locators: \texttt{prop:Icand}, \texttt{prop:kernel}, \texttt{eq:Kaffine}, \texttt{prop:residual}, \texttt{prop:sEA}, \texttt{prop:sint}, \texttt{prob:missing}, and \texttt{prop:decay}. The earlier supplied \texttt{XVI-F.tex} is not used for its superseded compact-support assertion. \bibitem{XIA} \emph{Manuscript XI-A}, version 1. Supplied identifier \texttt{0662c6c3}. Supplied locator: ``Quadratic gravitational coefficient,'' \texttt{prop:FP}, lines 404--445. The boundary-variation and divergence statements are used only within the compact-interior setup reported in the supplied XVI-B material. \bibitem{XIB} \emph{Manuscript XI-B}, version 1. Supplied identifier \texttt{862627cd}. Supplied excerpts: ``Free-multiplet release,'' lines 205--225; ``Normalized graviton inverse,'' lines 563--586. The compact-interior scope of \texttt{lem:Green} is also quoted in the supplied XVI-D/E/F-v2 texts. \bibitem{XIG} \emph{Manuscript XI-G}, version 1. Supplied identifier \texttt{3cae2be9}. Supplied excerpts: ``Free-shell class,'' lines 494--507; ``Separation and the admission criterion,'' lines 954--987; ``The free-shell class of the cubic kernel is obstructed,'' lines 1107--1134. Witness-symbol locators reported in the supplied XVI-B draft: \texttt{def:symbol}, \texttt{eq:qcounter}, \texttt{lem:qvalues}, and \texttt{prop:planewave}. \bibitem{XG} \emph{Manuscript X-G}, version 1. Supplied identifier \texttt{d362d49d}. Used only for the reported source-preserving variational obstruction on its original hypotheses, as described in the supplied coverage material and revised XVI-A. \bibitem{XIIB} \emph{Manuscript XII-B}, version 1. Supplied identifier \texttt{db136021}. ``Boundary representatives matter after weighting,'' \texttt{lem:boundary}, \texttt{eq:boundaryrepresentative}, lines 499--521. \bibitem{XIIC} \emph{Manuscript XII-C}, version 1. Supplied identifier \texttt{e0611fd7}. ``Actual density-weighted cubic source,'' \texttt{thm:actualsource}, \texttt{eq:actualsource}, and \texttt{eq:time-layer}, lines 636--678. The pure-trace formula is supplied through XVI-B's quotation of \texttt{eq:full-trace-sector}. The full unquoted coefficient formulas are not reconstructed. \bibitem{XIID} \emph{Manuscript XII-D}, version 1. Supplied identifier \texttt{e3e3f67d}. Supplied excerpts: ``Local construction and cutoff independence,'' lines 406--446; ``Completed new insertion,'' lines 556--569; ``Completed compact root,'' lines 628--674. Completion-leg formulas are also reproduced in the supplied XVI-D/E/F-v2 texts. \bibitem{XIIE} \emph{Manuscript XII-E}, version 1. Supplied identifier \texttt{26f86eb3}. ``XII-D: the six handed objects,'' \texttt{input:XIID}, lines 235--288. Its reported admission is not re-proved or revoked here. \bibitem{BDH} C.~Brouder, N.~V.~Dang, and F.~H\'elein, \emph{A smooth introduction to the wavefront set}, Journal of Physics A: Mathematical and Theoretical \textbf{47} (2014), 443001; arXiv:1404.1778 [math-ph]. Used for the distinction between justified distribution multiplication, smooth multiplication, and kernel wavefront questions; not as a source-specific estimate for $J_D$. \end{thebibliography} \end{document}