% XVI-A.tex % % Compile with: % pdflatex XVI-A.tex % pdflatex XVI-A.tex % % Single-file document: no external bibliography or included files. % % IMPORTANT: % This is a revised, logically audited proof-obligation manuscript. % It is not a proof constructing the unresolved retarded coefficient. % The defining kernels and their wavefront estimates were not supplied. % No execution of a LaTeX compiler is represented by this source. \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-A: Fixed-width retarded coefficient obligations}, pdfsubject={Audited proof-obligation manuscript; coefficient unresolved} } \setlength{\parindent}{0pt} \setlength{\parskip}{0.45em} \setlength{\emergencystretch}{2em} \setlist[itemize]{leftmargin=*,itemsep=0.25em,topsep=0.35em} \setlist[enumerate]{leftmargin=*,itemsep=0.3em,topsep=0.35em} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\supp}{supp} \newcommand{\Unresolved}{\textbf{Unresolved}} \newcommand{\NotSpecified}{\textit{Not specified in the supplied material}} \newcommand{\Dprime}{\mathcal D'} \newcommand{\Eprime}{\mathcal E'} \newcommand{\mc}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Isol}{\mathcal I_{\mathrm{sol}}} \theoremstyle{definition} \newtheorem*{problem}{Open obligation} \numberwithin{equation}{section} \title{Landmark XVI-A\\[0.3em] \large Fixed-width retarded coefficient obligations\\ \normalsize Revised proof-obligation manuscript --- coefficient status: unresolved} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} This note fixes the declared normalization and source notation for Landmark XVI-A, separates those declarations from predecessor results, and records the missing data required to formulate and discharge three retarded-slot wavefront obligations. It also corrects a potentially invalid inference: a changed source distribution need not determine a different free-shell class. The X-G variational obstruction and the XI-G frozen-class obstruction remain in force on their respective hypotheses. Standard distributional criteria are supplied as a conditional verification interface, not as estimates for the unidentified XVI-A kernels. No retarded coefficient is constructed, and no theorem resolving obligations 4--6 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) {Fixed positive width}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Three wavefront obligations}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The obligations are recorded. The retarded coefficient is not constructed.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The obligations do not construct the 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-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) {Fixed positive width}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The obligations are recorded. The retarded coefficient is not constructed.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The obligations do not construct the 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-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) {Fixed positive width}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Three wavefront obligations}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The obligations are recorded. The retarded coefficient is not 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-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) {Fixed positive width}; \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) {Retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Three wavefront obligations. The obligations are recorded. The retarded coefficient is not constructed.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Fixed positive width is not 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-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: Fixed positive width}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Three wavefront obligations}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Retarded coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The obligations are recorded. The retarded coefficient is not 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-A} \end{figure} Figure~\ref{fig:split-XVI-A} records one view of the same separation. Figure~\ref{fig:chain-XVI-A} records one view of the same separation. Figure~\ref{fig:pair-XVI-A} records one view of the same separation. Figure~\ref{fig:gate-XVI-A} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XVI-A} records that separation. The drawing adds no hypothesis. \begin{center} \fbox{% \begin{minipage}{0.92\linewidth} \textbf{Scope and release status.} Throughout, the regulator width satisfies $\varepsilon>0$ and is held fixed. Neither X-G nor XI-G is repaired here. The new retarded coefficient remains \Unresolved. The displayed coefficient slots are schematic and are not an identity. A missing estimate is not a proof of obstruction. A verified estimate for one operation is not, by itself, construction of a retarded quantum product. \end{minipage}} \end{center} \section{Document identity, provenance, and logical status} \label{sec:provenance} The document identifier is \textbf{XVI-A}, corresponding to the supplied filename \texttt{XVI-A.tex}. The alternative spelling ``X-VI-A'' in the request is treated as an editorial spelling of this identifier, not as a separately established landmark. The available material consists of the supplied XVI-A draft, selected predecessor excerpts, and a supplied coverage summary. It does not include the complete predecessor manuscripts or the later XVI sheets. Consequently, references to predecessor labels below identify the supplied passages; they do not certify a full-series consistency check. We distinguish four kinds of statement: \begin{enumerate} \item \textbf{Declared input:} a relation retained from the supplied XVI-A draft. Retention is not an independent derivation. \item \textbf{Reported predecessor result:} a result stated in a supplied predecessor excerpt or coverage summary, confined to its stated hypotheses. \item \textbf{Audit deduction:} an elementary consequence whose justification is given here. \item \textbf{Open obligation:} a proposed analytic task for which the defining data or proof has not been supplied. \end{enumerate} A named source dependency, conceptual connection, or inherited interface does not transfer every theorem of the named manuscript. In particular, a free Green operator or free nilpotence statement does not discharge an interacting quantum admission obligation. \section{Frozen declarations and notation} \label{sec:frozen} \subsection{Declared relations} The following relations are retained exactly at the level of the supplied draft. No new coefficient, source formula, or wavefront estimate is inferred from them: \begin{align} S_{\mathrm{EH}} &= -\frac{2}{\kappa^{2}}\int\sqrt{-q}\,R, \label{eq:EH}\\ S_{\mathrm{gf}} &= \int D_{\nu}(\gamma)\,D^{\nu}(\gamma), \label{eq:gf}\\ \Psi &= -i\int\bar c_{\mu} \bigl(2D^{\mu}\gamma-b^{\mu}\bigr), \label{eq:gaugefermion}\\ G_{\mathrm{gr}}^{R} &= -P E_{0}^{R}, \label{eq:inverse}\\ S_{\mathrm{grav},D} &= S_{\mathrm{EH}} +\kappa\int(\sigma_{\varepsilon}-1)\,\ell_{3}, \label{eq:preparedaction}\\ J_{D} &= \sigma_{\varepsilon}J_{3}+D_{\sigma}. \label{eq:preparedsource} \end{align} The action--equation sign convention is \begin{equation} r=-u_D. \label{eq:sign} \end{equation} These formulas remain in the inherited integral notation. No unprovided measure, boundary condition, tensor contraction, or integration-by-parts prescription is silently reconstructed. \subsection{Symbol register} The register specifies the roles of the symbols, not unprovided local formulas. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.20\linewidth}Y@{}} \toprule Symbol & Role and limitation\\ \midrule $\kappa$ & Gravitational coupling appearing in the declared normalization.\\ $q,\sqrt{-q},R$ & Metric, metric-volume shorthand, and scalar curvature in the Einstein--Hilbert declaration.\\ $\gamma$ & Metric perturbation in the inherited free-field convention.\\ $D_\nu(\gamma)$ & Inherited gauge-fixing expression. Its local differential formula is not supplied here.\\ $\bar c_\mu,b^\mu,\Psi$ & Antighost, auxiliary field, and gauge-fermion symbols.\\ $P,E_0^R,G_{\mathrm{gr}}^R$ & Inherited tensor factor, retarded Green symbol, and graviton inverse in the selected normalization. Their kernel domains are not inferred from the notation.\\ $\sigma_\varepsilon$ & Fixed-width switch. Any use of smooth multiplication below explicitly requires $\sigma_\varepsilon\in C^\infty$.\\ $\ell_3$ & Cubic gravitational density label. No local expression is supplied.\\ $J_3,J_D,D_\sigma$ & Cubic source, prepared source, and switch-dependent correction labels, related by \eqref{eq:preparedsource}. Their defining kernels are not supplied.\\ $r,u_D$ & Response labels with the sign convention \eqref{eq:sign}. No Green-operator formula for $u_D$ is added.\\ \bottomrule \end{tabularx} \end{center} The notation $J_D$ and $D_\sigma$ is used consistently. No identification with $\mathcal J_D$ or $\mathcal D_\sigma$ in unprovided manuscripts is assumed. The symbol $D_\sigma$ denotes the correction in \eqref{eq:preparedsource}; it is not shorthand for applying $D_\nu$ to $\sigma_\varepsilon$. \subsection{Provenance of the declarations} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.23\linewidth}Y Y@{}} \toprule Item & Available provenance & Status here\\ \midrule \eqref{eq:EH} & Supplied XVI-A frozen sheet & Declared normalization\\ \eqref{eq:gf}, \eqref{eq:gaugefermion} & Supplied draft; XI-B free-multiplet and gauge-fermion statements \cite{XIB} & Declared relations; no new auxiliary-field derivation\\ \eqref{eq:inverse} & Supplied draft; XI normalization described in the supplied XI-B/XI-G coverage & Selected input; no identification of differently named Green kernels\\ \eqref{eq:preparedaction}, \eqref{eq:preparedsource} & Supplied XVI-A draft & Proposed preparation; not an established predecessor construction\\ \eqref{eq:sign} & Supplied XVI-A draft & Sign convention\\ \bottomrule \end{tabularx} \end{center} In particular, the displayed action and source relations do not constitute a proof that varying the complete prepared action yields $J_D$. That assertion would require the complete action, the definition of $\ell_3$, the switch dependence, and the variational conventions. \subsection{Excluded alternatives} The following exclusions remain part of the declared interface: \begin{itemize} \item Full Einstein--Hilbert weighting is not substituted for \eqref{eq:preparedaction}; the draft excludes it in order to retain the selected XI-B free operator. \item The X-sheet inverse $-4PE_0^R$ is not used. The X-F normalization reported in the supplied material is separately written $-4\,\mathrm{Ptr}\otimes E_0^R$ \cite{XF}. No identification of $P$ with $\mathrm{Ptr}$ is needed here. \item The factors $-1$ and $-4$ are not interchangeable. A transfer between the corresponding conventions would require an explicit normalization map. \end{itemize} \section{The predecessor obstructions remain separate} \label{sec:nonrepair} \subsection{X-G is not bypassed by formal quantum corrections} The supplied X-G material reports an obstruction to an exact source-preserving variational upgrade and the strictly positive classical witness \begin{equation} W_{\mathrm{XG}} :=\frac{5M_\chi}{8} \int_0^\varepsilon \zeta\,(\sigma'')^2\,dt >0. \label{eq:XGwitness} \end{equation} The symbols in this witness belong to the X-G data and hypotheses \cite{XG}. They are not silently identified with the XVI-A preparation. The strict inequality is retained as a reported predecessor fact, not derived from unspecified assumptions on $M_\chi$, $\zeta$, or $\sigma$. \paragraph{Audit deduction.} Corrections beginning at strictly positive formal powers of $\hbar$ cannot cancel this nonzero classical coefficient. \paragraph{Justification.} For arbitrary formal coefficients $a_n$, \begin{equation} W_{\mathrm{XG}}+\sum_{n\geq1}\hbar^n a_n\neq0 \quad\text{in }\mathbb C[[\hbar]], \label{eq:formalnoncancel} \end{equation} because its constant coefficient is $W_{\mathrm{XG}}\neq0$. Equality of formal series is coefficientwise. This argument concerns formal corrections of order at least $\hbar$; it makes no assertion about numerical cancellation at a chosen nonzero numerical value of $\hbar$. Changing the action may change the model or the hypotheses of the obstruction. Such a change is not a source-preserving upgrade of the original model merely because it is proposed later. XVI-A supplies neither a counterexample to X-G nor a proof of the excluded upgrade. \subsection{Changing a distribution does not prove a change of class} \label{sec:class} The supplied XI-G excerpt defines a free-shell class of a compact cubic kernel $u$ on $M^3$, with two symmetric-tensor slots and one scalar slot \cite{XIG}. Let $\mathcal K$ denote the vector space of compact kernels with this slot structure and tensor-slot symmetry. Write \begin{equation} \begin{split} \Isol :=\bigl\{w\in\mathcal K:\ &\langle w,\gamma_1\otimes\gamma_2\otimes\phi\rangle=0\\ &\text{for every smooth } \Box\gamma_1=\Box\gamma_2=0,\quad P_m\phi=0 \bigr\}. \end{split} \label{eq:solutionideal} \end{equation} Here $P_m$ is the scalar free operator used in that excerpt. Compact support allows the displayed pairing with smooth solutions. The class is \begin{equation} [u]_{\mathrm{sol}}=u+\Isol. \label{eq:affineclass} \end{equation} The reported XI-G result is \begin{equation} [u]_{\mathrm{sol}}\cap\mc=\varnothing \label{eq:XIGobstruction} \end{equation} for the frozen data and hypotheses of its obstruction theorem \cite{XIG}. The symbol $\mc$ denotes the predecessor's microcausal polynomial target space; no new admission theorem for that space is assumed. \paragraph{Audit deduction: the exact transfer condition.} If a future prepared observable kernel $u_D\in\mathcal K$ satisfies \begin{equation} u_D-u\in\Isol, \label{eq:sameclasscondition} \end{equation} then it belongs to the same class and inherits the class obstruction: \begin{equation} [u_D]_{\mathrm{sol}}=[u]_{\mathrm{sol}}, \qquad [u_D]_{\mathrm{sol}}\cap\mc=\varnothing. \label{eq:sameclasstransfer} \end{equation} \paragraph{Justification.} The set $\Isol$ is a vector subspace by linearity of the pairings. If $u_D-u\in\Isol$, then \[ u_D+\Isol=u+(u_D-u)+\Isol=u+\Isol. \] Equation \eqref{eq:XIGobstruction} then gives \eqref{eq:sameclasstransfer}. A nonzero difference $u_D-u$ does not exclude \eqref{eq:sameclasscondition}. Thus the sentence ``the distributions are different, so XI-G does not govern the new distribution'' is not justified without a free-shell comparison. There is a further distinction: $J_D$ is a source label, whereas $u$ in \eqref{eq:XIGobstruction} is a compact observable kernel on $M^3$. The supplied material does not define the map from $J_D$ to a corresponding kernel $u_D$. It is therefore not legitimate to identify these objects merely because both are described as distributions. The corrected conclusion is: \begin{quote} XI-G remains valid for its frozen class. Its applicability to a kernel induced by $J_D$ has not been established or excluded. A same-class proof would transfer the obstruction. A different-class proof would remove that particular transfer, but would not prove admission of the new class. \end{quote} Even if a kernel induced by $\sigma_\varepsilon J_3$ were admitted to some declared class, that would not establish admission of the kernel induced by $\sigma_\varepsilon J_3+D_\sigma$. The correction and the total kernel must be analyzed. \subsection{Gauge-fixed and Einstein-shell conditions are not identified} The supplied XI-F excerpt contains a cubic transformation law with both a gauge-fixing companion and an Einstein-equation contribution. The supplied XI-G coverage also states that its witness lies on the gauge-fixed shell and off the Einstein shell \cite{XIF,XIG}. Accordingly: \begin{itemize} \item the gauge companion is retained; \item the Einstein-shell contribution is retained; \item a gauge-fixed free-shell condition is not used as a substitute for an Einstein-shell hypothesis; \item a transformation law for the unramped response is not asserted for the prepared response without proof. \end{itemize} The relevant prepared terms have not been supplied as explicit kernels. Retaining them is a requirement on a future derivation, not a claim that they have already been constructed. \section{Schematic slots and their editorial indexing} \label{sec:slots} The supplied target display is retained as a list of visible slots: \begin{equation} \begin{aligned} &\underbrace{O_2}_{\text{local slot}} +\underbrace{R_1(V_1;O_1)}_{\text{first retarded slot}}\\ &\qquad +\underbrace{R_1(V_2;F_f)}_{\text{second retarded slot}} +\underbrace{\tfrac12 R_2(V_1,V_1;F_f)} _{\text{quadratic retarded slot}}. \end{aligned} \label{eq:schematic} \end{equation} \textbf{Equation \eqref{eq:schematic} is schematic, not an identity.} It is not assigned a coefficient name, because doing so could misrepresent an undefined sum as a constructed distribution. The symbol roles are: \begin{itemize} \item $F_f$: the baseline tested-observable label; \item $O_1,O_2$: correction labels in the proposed observable bookkeeping; \item $V_1,V_2$: interaction-argument labels in that bookkeeping; \item $R_1,R_2$: first and second retarded-product labels, not defined multilinear maps in this note. \end{itemize} No identification of $V_1$ or $V_2$ with a particular density or action is made. No coupling expansion, $\hbar$ expansion, factorial convention, or renormalization prescription is inferred beyond the literal factor $\tfrac12$ in the supplied display. For tracking purposes only, the obligations are indexed by \begin{align} 4&\longleftrightarrow R_1(V_1;O_1),\label{eq:index4}\\ 5&\longleftrightarrow R_1(V_2;F_f),\label{eq:index5}\\ 6&\longleftrightarrow \tfrac12 R_2(V_1,V_1;F_f). \label{eq:index6} \end{align} This is an editorial correspondence, not a proof that the source $J_D$ induces these slots. The local slot $O_2$ is also unconstructed in the supplied material. Discharging obligations 4--6 would not silently settle its status or prove \eqref{eq:schematic} to be an expansion identity. \section{Standard microlocal verification interface} \label{sec:interface} This section states general distributional tests. It does not provide a wavefront estimate for an XVI-A kernel. Let $X,Y,Z$ be smooth manifolds. Fix compatible smooth density conventions. For $v\in\Dprime(X)$, $\WF(v)$ is a conic subset of $T^*X\setminus 0$. Tensor-valued kernels can be checked in local smooth frames. \subsection{Pullback} For a smooth map $a:X\to Y$, define \begin{equation} N_a= \bigl\{(a(x),\eta): \eta\neq0,\ (da_x)^T\eta=0\bigr\}. \label{eq:normalset} \end{equation} For $v\in\Dprime(Y)$, a standard sufficient condition for $a^*v$ is \begin{equation} \WF(v)\cap N_a=\varnothing. \label{eq:pullbacktest} \end{equation} Under this condition, \begin{equation} \WF(a^*v) \subset \bigl\{(x,(da_x)^T\eta): (a(x),\eta)\in\WF(v)\bigr\}. \label{eq:pullbackbound} \end{equation} See \cite[Proposition 5.1]{BDHoperations}. \subsection{Product and distributional pairing} For $v,w\in\Dprime(X)$, define $-\WF(w)=\{(x,-\xi):(x,\xi)\in\WF(w)\}$. The standard product criterion is \begin{equation} \WF(v)\cap\bigl(-\WF(w)\bigr)=\varnothing. \label{eq:producttest} \end{equation} It suffices for the product $vw$. If \[ \WF_0(v):=\WF(v)\cup \{(x,0):x\in\supp v\}, \] then \begin{equation} \WF(vw)\subset \bigl\{(x,\xi+\eta): (x,\xi)\in\WF_0(v),\ (x,\eta)\in\WF_0(w),\ \xi+\eta\neq0\bigr\}. \label{eq:productbound} \end{equation} See \cite[Theorem 13]{BDHintro}. Pairing two distributions requires such a product criterion, or another explicitly justified construction, together with the appropriate support conditions. \subsection{Projection and support} For the projection $\pi:Y\times Z\to Y$, fiber integration of $h\in\Dprime(Y\times Z)$ is defined when $\pi|_{\supp h}$ is proper. Its wavefront bound is \begin{equation} \WF(\pi_*h)\subset \bigl\{(y,\eta): \eta\neq0,\ \exists z\ (y,z;\eta,0)\in\WF(h)\bigr\}. \label{eq:pushforwardbound} \end{equation} See \cite[Theorem 6.3]{BDHoperations}. Properness is a support requirement, not a wavefront transversality requirement. \subsection{What these tests do not prove} These are sufficient tests for the specified operations. An intersection in an unverified upper bound does not establish an intersection in the actual wavefront set. Failure of a sufficient test does not, by itself, prove that no alternative construction exists \cite{BDHintro}. Nor do these tests alone select extensions across singular diagonals or normalize retarded quantum products. Such extension and normalization problems are separate in perturbative constructions \cite{BDF}. \subsection{Conditional use of the interface} Suppose a future slot is given by a finite ordered recipe whose inputs are specified distributions. If every operation in that recipe is justified, every integration has the required support control, and any extension step is separately constructed, the recipe defines an output distribution. The verification proceeds in the order of the recipe: an operation is applied only after its inputs have been defined and its hypotheses checked. Its output estimate is then available for the next operation. This finite induction does not supply the missing input kernels, prove a required final cone condition, or establish a quantum admission theorem. \section{Fixed-width switching is not a smoothing theorem} \label{sec:switching} If the chosen switch is smooth at fixed $\varepsilon$, then smooth multiplication is defined and does not enlarge wavefront sets: \begin{equation} \WF(\sigma_\varepsilon J_3)\subset\WF(J_3). \label{eq:smoothmult} \end{equation} The wavefront set of a sum is contained in the union of the wavefront sets of its summands: \begin{equation} \WF(J_D) \subset \WF(\sigma_\varepsilon J_3)\cup\WF(D_\sigma). \label{eq:sumbound} \end{equation} These general facts are discussed in \cite{BDHintro}. They do not identify $\WF(D_\sigma)$ or the wavefront sets of the observable kernels derived from $J_D$. In particular, \eqref{eq:smoothmult} is not a statement that $\sigma_\varepsilon J_3$ is smooth. \paragraph{Audit deduction.} On an open set where a smooth multiplier $a$ is nowhere zero, \[ \WF(av)=\WF(v) \] after restriction to that open set. \paragraph{Justification.} One inclusion follows from smooth multiplication. Locally, $1/a$ is smooth and $v=(1/a)(av)$, which gives the reverse inclusion. Thus switching does not remove a singularity at a point where the switch is nonzero merely by being smooth. Possible cancellation in the total source must be proved for the actual distributions; the upper bound \eqref{eq:sumbound} neither proves nor excludes such cancellation. All statements here are at fixed $\varepsilon$. No uniform estimate as $\varepsilon\downarrow0$, and no sharp-width limit, is asserted. \section{Required specification packet} \label{sec:packet} An obligation is not mathematically testable until the relevant objects and operations are identified. For each $j\in\{4,5,6\}$, a future proof must provide the following packet. \begin{enumerate}[label=\textbf{P\arabic*.}] \item \textbf{Exact objects.} Give the distributions or kernels to be estimated, including their dependence on $J_D$, their tensor or graded slots, symmetry, density conventions, and relevant formal orders. Names such as ``the required kernel'' are insufficient. \item \textbf{Spaces and maps.} Give every input and output manifold, bundle, test space, pullback map, diagonal embedding, and projection used by the slot. \item \textbf{Operation recipe.} Specify the ordered contractions, products, pullbacks, pairings, and integrations. State whether any kernel is initially defined only away from a diagonal. \item \textbf{Exact covector statements.} State the actual wavefront inclusions to be proved. For each operation, identify the corresponding normal-set, opposite-covector, or projection test. Specify the final allowed cone rather than referring to ``the usual estimate.'' \item \textbf{Support and extension.} Prove the support condition for each integration and the declared retarded support condition. Construct any needed extension and state its normalization. Compact tests do not automatically prove every internal projection to be proper. \item \textbf{Terms retained.} Show how the full correction $D_\sigma$ enters. Retain the gauge companion and the Einstein-shell term whenever the relevant derivation produces them. A cancellation may be used only after it has been proved. \item \textbf{Parameters and scope.} State that $\varepsilon$ is fixed. Specify any other parameter dependence and any topology needed for a claimed functional or parameter-family result. Pointwise estimates are not silently promoted to uniform estimates or algebraic closure. \item \textbf{Discharge statement.} State exactly which operation and output estimate the proof establishes. Distinguish that result from construction, normalization, covariance, variational compatibility, and quantum admission. \end{enumerate} The supplied material does not contain a completed packet for any of obligations 4--6. Accordingly, no slot-specific covector estimate is fabricated here. \section{Proof obligations 4--6} \label{sec:obligations} The obligations below preserve the original numbering. The indexing is that of \eqref{eq:index4}--\eqref{eq:index6}. \begin{problem}[Obligation 4: first retarded slot] Specify the kernel realization of $R_1(V_1;O_1)$ associated with the prepared source $J_D$, and complete the packet of Section~\ref{sec:packet} for that realization. Prove the identified wavefront estimates and support conditions needed for every operation in that slot, with $\varepsilon$ fixed. State the final output distribution and its required cone bound. \textbf{Present gap.} The kernel realization, its input and output manifolds, its maps, and its slot-specific discharge criterion have not been supplied. \textbf{Status:} \Unresolved. \textbf{Stop.} No first retarded slot is constructed by this note. \end{problem} \begin{problem}[Obligation 5: second retarded slot] Specify the kernel realization of $R_1(V_2;F_f)$ associated with $J_D$, and complete the packet of Section~\ref{sec:packet}. Retain the gauge companion in the complete kernel derivation. Prove the identified wavefront estimates and support conditions for the full expression, with $\varepsilon$ fixed. Do not replace the complete expression by one obtained by setting the gauge companion to zero. \textbf{Present gap.} Neither the complete prepared kernel nor the estimate controlling its gauge-companion contribution has been supplied. \textbf{Status:} \Unresolved. \textbf{Stop.} No second retarded slot is constructed by this note. \end{problem} \begin{problem}[Obligation 6: quadratic retarded slot] Specify the kernel realization of $\tfrac12 R_2(V_1,V_1;F_f)$ associated with $J_D$, and complete the packet of Section~\ref{sec:packet}. Retain the Einstein-shell term in the complete derivation. Prove the identified estimates for repeated contractions, pullbacks, pairings, integrations, and any required diagonal extensions, with $\varepsilon$ fixed. Do not substitute a gauge-fixed free-shell condition for an unproved Einstein-shell hypothesis. \textbf{Present gap.} The quadratic kernel, its repeated-operation recipe, its extension data, and its slot-specific discharge criterion have not been supplied. \textbf{Status:} \Unresolved. \textbf{Stop.} No quadratic retarded slot is constructed by this note. \end{problem} Absence of any listed estimate leaves the corresponding obligation unresolved. It is not a proof of nonexistence. Conversely, a later proof of one estimate discharges only the task identified by that proof. \clearpage \section{Obligation ledger} \label{sec:ledger} The ledger records the present state of specification and proof. ``Not specified'' is a defect to be supplied, not a completed discharge criterion. \begin{center} \small \setlength{\tabcolsep}{4pt} \renewcommand{\arraystretch}{1.15} \begin{tabularx}{\linewidth} {@{}p{0.05\linewidth}p{0.19\linewidth}Y Y p{0.14\linewidth}@{}} \toprule No. & Schematic slot & Required result & Conditions retained & Status\\ \midrule 4 & $R_1(V_1;O_1)$ & Exact kernel, operation recipe, wavefront bounds, and support proof & Fixed $\varepsilon$; full prepared source & Unresolved\\ \addlinespace 5 & $R_1(V_2;F_f)$ & Bounds and support proof for the complete second slot & Fixed $\varepsilon$; gauge companion retained & Unresolved\\ \addlinespace 6 & $\tfrac12 R_2(V_1,V_1;F_f)$ & Bounds for repeated operations and any diagonal extension & Fixed $\varepsilon$; Einstein-shell term retained & Unresolved\\ \bottomrule \end{tabularx} \end{center} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.08\linewidth}Y Y@{}} \toprule No. & Exact object and maps & Slot-specific discharge criterion\\ \midrule 4 & \NotSpecified & \NotSpecified\\ 5 & \NotSpecified & \NotSpecified\\ 6 & \NotSpecified & \NotSpecified\\ \bottomrule \end{tabularx} \end{center} The generic criteria in Section~\ref{sec:interface} do not fill these cells. An actual kernel and recipe are needed before one can determine which criteria apply. \subsection*{Additional stops} Even a complete microlocal packet for each retarded slot would not, without further proof: \begin{itemize} \item construct the local slot $O_2$; \item establish the schematic display as a coefficient identity; \item select or construct renormalized retarded products; \item establish an interacting BRST or BV statement; \item establish covariance or variational compatibility of $J_D$; \item provide a state, positivity result, loop construction, higher-order construction, or sharp-width limit; \item repair the X-G or XI-G obstruction. \end{itemize} \clearpage \section{Materials required for a construction manuscript} \label{sec:materials} A subsequent manuscript seeking an actual coefficient theorem must supply, at minimum: \begin{enumerate} \item the complete prepared action and its variational conventions, including the formula for $\ell_3$; \item explicit definitions of $J_3$ and $D_\sigma$, and a proof relating their sum to that action if such a relation is claimed; \item the normalized free operator, its field bundles, source domains, and Green kernels; \item definitions of $F_f$, $O_1$, $O_2$, $V_1$, $V_2$, and the proposed retarded maps; \item the full gauge-companion and Einstein-shell contributions appearing in the prepared derivation; \item the compact observable kernel induced by $J_D$, together with its free-shell comparison to the frozen XI-G kernel; \item completed specification packets and proofs for obligations 4--6; \item the extension, normalization, and admission results needed to turn the schematic bookkeeping into a coefficient theorem. \end{enumerate} Neither the provenance note nor the general analytic interface is a substitute for these materials. \section{Consistency audit and final status} \label{sec:conclusion} The following corrections are incorporated: \begin{enumerate} \item \textbf{Notation is fixed.} $J_D$ and $D_\sigma$ are used throughout; no unverified calligraphic aliases are introduced. \item \textbf{Normalizations remain separate.} The selected inverse is \eqref{eq:inverse}, and the X-sheet factor $-4$ is excluded. \item \textbf{The sign is unchanged.} The convention remains $r=-u_D$. \item \textbf{Class transfer is stated correctly.} Different distributions can represent the same free-shell class. The exact conditional transfer argument is given in Section~\ref{sec:class}. \item \textbf{Missing estimates are not obstructions.} Generic sufficient criteria are not asserted for unspecified kernels. \item \textbf{No term is silently deleted.} The gauge companion and Einstein-shell term remain explicit requirements. \item \textbf{The schematic is not promoted to an identity.} No coefficient name or equality is attached to \eqref{eq:schematic}. \item \textbf{Compilation claims are limited.} The source is arranged as one pdfLaTeX document with an internal bibliography. A successful compiler execution is not claimed. \end{enumerate} \begin{center} \fbox{% \begin{minipage}{0.92\linewidth} \textbf{Final release statement.} X-G and XI-G remain in force on their respective hypotheses. Their applicability to a changed model or changed observable class must be checked, not assumed. Obligations 4--6 remain open. The retarded coefficient is \Unresolved. No coefficient construction, interacting quantum admission, or sharp-width limit is claimed. \medskip The manuscript is complete as an audited statement of scope, conditional verification criteria, and missing proof data. It is not a completed proof of the proposed retarded coefficient. No theorem resolving the XVI-A target is claimed. \end{minipage}} \end{center} \begin{thebibliography}{99} \bibitem{Draft} \emph{Landmark XVI-A}. User-supplied draft, filename \texttt{XVI-A.tex}. Source of the frozen declarations, the sign convention, obligations 4--6, and the schematic slot display. No external publication status is asserted. \bibitem{XIB} \emph{Manuscript XI-B}, version 1, \texttt{XI-B.tex}, supplied identifier \texttt{862627cd}. Supplied excerpts: ``Free-multiplet release,'' lines 205--225; ``Off-shell nilpotence,'' lines 383--396; ``Free causal polynomial algebra,'' lines 743--762. The associated supplied coverage summary is used only within its stated scope. \bibitem{XIF} \emph{Manuscript XI-F}, version 2, \texttt{XI-F-v2.tex}, supplied identifier \texttt{ef5fb4a8}. Supplied excerpt: ``The cubic response obeys the complete second-order law,'' lines 954--980. This reference does not assert the corresponding law for the prepared XVI-A response. \bibitem{XIG} \emph{Manuscript XI-G}, version 1, \texttt{XI-G.tex}, 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. The obstruction is used only for the frozen class and hypotheses reported there. \bibitem{XF} \emph{Manuscript X-F}, version 1, \texttt{X-F.tex}, supplied identifier \texttt{4c1ccd1e}. Supplied coverage summary and source locators: field content and second-order metric, lines 450--486; fixed-regulator release, lines 1425--1469. The complete manuscript was not supplied. \bibitem{XG} \emph{Manuscript X-G}, version 1, \texttt{X-G.tex}, supplied identifier \texttt{d362d49d}. Supplied coverage summary of the source-preserving variational obstruction and its positive classical witness. Source locators include the exact-upgrade definition, lines 245--261, and the qualified release, lines 1841--1885. The full proof was not supplied. \bibitem{BDHintro} C.~Brouder, N.~V.~Dang, and F.~H\'elein, \emph{A smooth introduction to the wavefront set}, arXiv:1404.1778 [math-ph] (2014). In particular, Theorem 13 and the discussion of fundamental wavefront-set properties. \bibitem{BDHoperations} C.~Brouder, N.~V.~Dang, and F.~H\'elein, \emph{Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker)}, arXiv:1409.7662v2 [math.FA] (2016). In particular, Proposition 5.1 and Theorems 6.1 and 6.3. \bibitem{BDF} R.~Brunetti, M.~D\"utsch, and K.~Fredenhagen, \emph{Perturbative Algebraic Quantum Field Theory and the Renormalization Groups}, arXiv:0901.2038v2 [math-ph] (2009). Used for the distinction between distributional operations and the separate extension and normalization problems of perturbative quantum products. \end{thebibliography} \end{document}