% !TeX program = pdflatex %===================================================================== % XV-F-patch.tex % % Standalone source for pdfLaTeX. % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XV-F-patch.tex % pdflatex -interaction=nonstopmode -halt-on-error XV-F-patch.tex % % Scope-corrected release: % - the four supplied source sheets are audited and transcribed; % - source definitions, patch conventions, and unresolved parameters % are distinguished; % - the local XIV-I amplitude module is fully transcribed; % - its coefficientwise reconstruction is C^\infty-linear by its % displayed definition, without importing a separate theorem; % - the XIV-J target and all five joint-map clauses are transcribed; % - the polarized stress convention is fixed from the displayed % XIII-C vertex; % - residual-species notation and contact-role formulas are supplied % only with their actual parameter dependence exposed; % - no template is represented as a recovered frozen coefficient; % - no complete reconstruction kernel or presentation-level % correction space is asserted; % - the canonical-domain complement equality remains refused; % - no preliminary lift, null defect, characteristic prescription, % new estimate, or quantum admission is constructed. % % Important corrections to the proposed outline: % 1. A metric residual test on Z is not a section of an unspecified % output bundle over W. % 2. Writing a generic differential operator or naming a spectator % does not recover the frozen operator or spectator. % 3. The bound six concerns the contact normal multiindex, not the % order of D_Gamma. % 4. XIV-J's descent criterion is global; canonical agreement checks % it only after restriction to U_can. % 5. The supplied sheets reproduce K_8 and K_9. Their formulas are % not unavailable merely because XIV-H is a non-input. % 6. XIV-I explicitly identifies R as the kernel of E_0^R and displays % its retarded cone formula. Its kernel support is consequently % available within that copied namespace. % 7. The interior-support diagnostic in XV-F Remark 21 overlooks the % displayed b derivative in K_8. % 8. Completion of four documentary ledgers does not imply that all % four corresponding frozen-object gaps pass the original F0. % % No successful compiler execution, proof-assistant verification, % independent symbolic computation, independent review, or acceptance % is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,longtable,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\tabcolsep}{4pt} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{2} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{inputstatement}[theorem]{Source statement} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\sig}{\sigma_\eps} \newcommand{\ER}{E_0^{\mathrm R}} \newcommand{\EAm}{E_m^{\mathrm A}} \newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}} \newcommand{\PP}{\mathcal P} \newcommand{\Vcal}{\mathcal V_g} \newcommand{\Ocal}{\mathcal O} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\HT}{H_T} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\Ucan}{U_{\mathrm{can}}} \newcommand{\Nlift}{\widetilde{\mathcal N}_{\mathrm{att}}} \newcommand{\Nbar}{\overline{\mathcal N}_{\mathrm{att}}} \newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}} \newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}} \newcommand{\Cross}{\mathscr C_{\mathrm{cross}}} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Ksup}{K_{\mathrm{supp}}} \newcommand{\Kint}{\mathscr K_{\mathrm{int}}} \newcommand{\Ppres}{\mathscr P_{\mathrm{int}}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Ar}{\mathfrak A_{\mathrm R}} \newcommand{\Bres}{\mathbb B_{\mathrm{res}}} \newcommand{\Eres}{E_{\mathrm{res}}} \newcommand{\Amap}{\mathsf A} \newcommand{\bmap}{\mathsf b} \newcommand{\Tform}{\mathcal T_{\mathrm{form}}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\Ds}{\mathsf D_{\mathrm s}} \newcommand{\Dp}{\mathsf D_{\mathrm p}} \newcommand{\Tpl}{\mathsf T} \newcommand{\Open}{\mathsf O} \newcommand{\Refused}{\mathsf R} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\Contr}{Contr} \DeclareMathOperator{\Diff}{Diff} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\Sym}{Sym} \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 } \newcommand{\source}[4]{% \par\smallskip\noindent {\small\textbf{#1.}\enspace \texttt{#2}; \texttt{#3}; #4.} \par\smallskip } \title{\textbf{XV-F-patch}\\[0.5em] Definitional completions for the frozen joint graph\\[0.4em] \large Residual species, correction roles, the amplitude module,\\ and the joint-map clauses: a scope-corrected source patch} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We patch the source manifest supplied as \texttt{XV-F-vav3.md}, using only the full supplied texts of \texttt{XIV-G.tex}, \texttt{XIII-C.tex}, \texttt{XIV-I.tex}, and \texttt{XIV-J.tex}. The local amplitude generators, chart-function coefficient class, reconstruction rule, formal target, and five joint-map clauses are now reproduced rather than paraphrased. A polarized stress convention is fixed from the displayed metric--scalar vertex. Two proposed completions require a narrower release. The metric residual tests in XIV-G do not identify an output bundle over the common graph space. Likewise, generic formulas for a graph differential operator, external weight, and spectator coefficient do not recover their frozen graphwise values. We therefore distinguish source definitions, definitions introduced by the patch, and parameterized formulas. The polynomial weight convention is not attributed to XIII-C, and the normal contact bound six is not used as a bound on the graph differential operator. The displayed reconstruction is coefficientwise $C^\infty$-linear on the transcribed local module. Its chart-function relations are genuine local null presentations; they do not determine the complete six-entry reconstruction kernel. XIV-J's global descent criterion is transcribed, while canonical agreement verifies only its restricted version. The canonical-domain complement equality remains refused. The four documentary patch ledgers are completed within these boundaries. The stronger assertion that every corresponding frozen-object gap passes XV-F's original F0 is not established. No preliminary lift, null-defect calculation, characteristic prescription, new analytic estimate, or admission of $Q_H\Rquant$ is supplied. Gates F1--F4 remain unentered. \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) {Local amplitude module}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Five joint-map clauses}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Residual output fiber}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The residual fiber stays open. The patch does not pass F0.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{A transcribed package still stops at F0. 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-XV-F-patch} \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) {Local amplitude module}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Residual output fiber}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The residual fiber stays open. The patch does not pass F0.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{A transcribed package still stops at F0. 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-XV-F-patch} \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) {Local amplitude module}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Five joint-map clauses}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Residual output fiber}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The residual fiber stays open. The patch does not pass F0.}; \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-XV-F-patch} \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) {Local amplitude module}; \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) {Residual output fiber}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Five joint-map clauses. The residual fiber stays open. The patch does not pass F0.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Local amplitude module is not Residual output fiber. 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-XV-F-patch} \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: Local amplitude module}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Five joint-map clauses}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Residual output fiber}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The residual fiber stays open. The patch does not pass F0.}; \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-XV-F-patch} \end{figure} Figure~\ref{fig:split-XV-F-patch} records one view of the same separation. Figure~\ref{fig:chain-XV-F-patch} records one view of the same separation. Figure~\ref{fig:pair-XV-F-patch} records one view of the same separation. Figure~\ref{fig:gate-XV-F-patch} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XV-F-patch} records that separation. The drawing adds no hypothesis. \scopebox{Exact release}{% \textbf{Completed:} the four owned transcription and convention ledgers, including the local amplitude definition, the displayed target, the full joint-map clauses, and the chosen stress polarization. \smallskip \textbf{Explicitly parameterized, not recovered:} the residual output fiber and the graphwise correction coefficients. \smallskip \textbf{Still missing:} a fully identified frozen codomain, a complete presentation package, the complete reconstruction kernel, the actual presentation-level correction space, and the remaining inherited coefficient definitions. \smallskip \textbf{Ending:} the owned documentary patch is complete to its stated scope; the actual graph remains \emph{Unresolved at F0}. This is not ``lift defined and descended.''} \tableofcontents %===================================================================== \section{Scope, evidence, and the stop rule} \label{sec:scope} %===================================================================== \subsection{The question being answered} The proposed patch asks whether four source sheets can be completed, definitionally, so that XV-F's Proposition \texttt{prop:F0} no longer fails on the gaps those sheets own. The question has two distinct readings. The first is documentary. A formula which is present in a supplied full text, but was previously available only as a paraphrase, can now be reproduced. Its source notation, coefficient class, and scope can be inspected directly. This is the appropriate reading for XIV-I's amplitude definition and XIV-J's target and map-class definition. The second is mathematical. A role such as ``the specified graph differential operator'' might be replaced by a generic formula for a finite differential operator. That replacement gives a mathematical template, but it does not identify the specified operator. Similarly, a species name does not by itself construct a vector bundle, and a named spectator does not determine its distributional coefficient. The second reading therefore requires information beyond the existence of a displayed syntactic form. The patch must not conflate these readings. Its central distinction is between completing a documentary ledger and completing the frozen-object package needed for a lift. The former is accomplished below. The latter is not claimed merely because every row has acquired a formula or an explicit boundary statement. This distinction is necessary for a proof-ready release. If an unknown frozen coefficient is replaced by an arbitrary parameter and the resulting expression is called the actual coefficient, the model has been changed without a theorem. If instead the parameter remains exposed, the expression is a well-defined family, and its status can be stated honestly. A source patch is useful precisely when it makes that difference visible. \subsection{Inputs and non-inputs} The mathematical manuscript inputs are the full supplied texts of XIV-G, XIII-C, XIV-I, and XIV-J \cite{XIVG,XIIIC,XIVI,XIVJ}. The supplied XV-F manifest is the record being corrected \cite{XVF}. It is not treated as an independent verification of all claims in the manuscripts it describes. The supplied file is named \texttt{XV-F-vav3.md}. The accompanying retrieval account also reports a registration under \texttt{XV-F-v3.tex}. This patch does not assert that those two files are byte-identical or that their line numbers coincide. References to XV-F's labels refer to the supplied manifest and its reported registration, not to an independently inspected missing source file. XIV-D, XIV-E, XIV-H, XII-A, XII-B, and XI-B are not direct inputs. Their names may occur in copied handoffs, in source bibliographies, or in the list of future dependencies. That occurrence does not import their full contents. A premise explicitly copied into one of the four supplied sheets is usable with the scope of that copied premise; an uncopied predecessor formula is not recovered by a citation to its owner. The coverage account is documentary context only. In particular, a declared continuation path is not a theorem identifying every object appearing along that path. The patch does not use comparison tracks, classical master-equation statements, or earlier contact certificates to discharge a distributional composition obligation. \subsection{Locators and the meaning of ``defines''} Reported line numbers are retained as documentary locators. They have not been independently recounted against a version-controlled source archive. Source labels and the supplied passage contents are therefore reported together. Where an exact label has no separately supplied line range, that absence is stated rather than repaired by an invented number. The word \emph{defines} is qualified throughout: \[ \Ds=\text{defines in a supplied source},\qquad \Dp=\text{defines by an explicit patch convention}. \] A source definition and a patch convention are not interchangeable. The latter can remove an ambiguity in the notation of this paper, but it cannot retrospectively prove that the source used the same convention. We also use \[ \Tpl=\text{parameterized formula},\qquad \Open=\text{unidentified frozen datum},\qquad \Refused=\text{explicitly refused assertion}. \] A template can define a family perfectly well while remaining insufficient to identify the member frozen in the predecessor. Thus $\Tpl$ is not a weaker kind of $\Ds$ which can be promoted by formatting. It records a different mathematical object. A refusal is also not a definition of an alternative object. The refusal of a particular complement equality closes that proposed identification. It does not provide a set equation for the canonical domain, and it does not prohibit a later independent description of that domain. \subsection{Patch gates and their corrected interpretation} The proposed gates are retained, with the distinction between documentary completion and frozen-object identification made explicit. \begin{center} \small \begin{tabularx}{\linewidth}{@{}P{.07\linewidth} P{.25\linewidth} X@{}} \toprule Gate & Content & Pass condition and limitation\\ \midrule P0 & Four patch ledgers & Every owned request is disposed of as a source definition, patch convention, parameterized formula, refusal, or explicit open datum. An open datum is not relabelled ``defines.''\\[0.4em] P1 & Residual codomain & The residual species is named. A distribution-valued bundle codomain is an object only after its fiber and pairing have been specified. Naming the species alone is a limited pass.\\[0.4em] P2 & Correction roles & The contact syntax is displayed. The actual graph operator, weight, and spectator remain unidentified unless their parameters are supplied. A template is not an exhaustive parametrization of the frozen class.\\[0.4em] P3 & Amplitude module & Generators, chart-function coefficients, reconstruction, and inherited coefficient seminorms are transcribed. The local module is distinguished from the complete graph presentation space.\\[0.4em] P4 & Joint-map clauses & The target, all five clauses, the grouped blocks, the test-support clause, and the stress convention are displayed. No map satisfying those clauses is thereby constructed.\\ \bottomrule \end{tabularx} \end{center} Under the original stronger reading of P0, every owned core slot would have to be an actual definition or a genuine source refusal. That stronger pass is not established: residual output fibers and frozen graph coefficients are not supplied by the permitted texts. The present P0 is a completed audit, not an assertion that those missing objects have appeared. The stop rule is accordingly exact. When a completion would require a non-input or an unidentified parameter, the patch stops at that item. It may display the dependence on the missing parameter, but it may not choose the parameter and call the choice inherited. No analytic gate is opened by this documentary work. %===================================================================== \section{Non-transfer ledger} \label{sec:nontransfer} %===================================================================== The following ledger records ownership without transferring theorems between namespaces. The action column has only the three operations specified in the proposed plan: \emph{transcribe}, \emph{define from stated roles}, and \emph{leave open}. The status of a result of ``define from stated roles'' is determined separately; it can be a patch convention or a parameterized formula rather than a recovered source object. \begin{longtable}{@{}P{.22\linewidth} P{.15\linewidth} P{.25\linewidth} P{.29\linewidth}@{}} \caption{Ownership and permitted patch actions.} \label{tab:nontransfer}\\ \toprule Object & Owner & XV-F record & Patch action\\ \midrule \endfirsthead \toprule Object & Owner & XV-F record & Patch action\\ \midrule \endhead \bottomrule \endfoot $W$ and $N$ & XIV-G; copied local geometry & Dimensions named; characteristic incidence incompletely recorded in the manifest & \emph{Transcribe} the common space and the selected locus with their copied scope.\\[0.4em] Smooth metric residual test $G$ & XIV-G & Residual testing discussed; output fiber absent & \emph{Transcribe} its actual domain and tensor type.\\[0.4em] $\Eres$ & XV-F codomain slot & Unidentified & \emph{Define from stated roles} a parameterized species-bundle notation; \emph{leave open} its identification with the actual output fiber.\\[0.4em] Collision sets $\Delta_I$ & XIII-C & Transcribed in graph variables & \emph{Transcribe} their definition and choose local relative coordinates without asserting a lift to $W$.\\[0.4em] $D_\Gamma,w_\Gamma,S_{\Gamma/I}$ & XIII-C roles & Roles, not actual coefficient formulas & \emph{Define from stated roles} the parameterized contact syntax; \emph{leave open} the frozen graphwise parameters.\\[0.4em] $\Cpart$ & XIII-C & Class of contact values & \emph{Transcribe}; do not identify it with a space of maps on presentations.\\[0.4em] $T_0$ and the vertex & XIII-C & Stress displayed; polarization ambiguous & \emph{Transcribe}; \emph{define from stated roles} the unique symmetric polarization used in this patch.\\[0.4em] Adapted charts and $\Ar^m(V)$ & XIV-I & Chart and rule transcribed; generators and coefficient class paraphrased & \emph{Transcribe} all three parts of the local definition.\\[0.4em] Local reconstruction relations & XIV-I rule; XIV-J null identity & One relation known; chart-function action incompletely recorded & \emph{Transcribe} the null identity and unpack the coefficientwise module action.\\[0.4em] Complete $\Ppres$, $\Kint$, and $\ker\Rec$ & XIV-J common presentation interface & Incomplete enumeration and kernel & \emph{Leave open} the complete six-entry specification; do not substitute one local amplitude module.\\[0.4em] $R$, $R*R$, and kernel support & XIV-I copied normalization & Displays available; a cross-sheet support identification requested & \emph{Transcribe} the displays and their internal kernel identification; do not transfer to an external namespace.\\[0.4em] $K_8$ and $K_9$ & Copied in XIV-I and XIV-J & Displays available; coefficient ingredients incomplete & \emph{Transcribe} the displays; \emph{leave open} their unrecovered coefficient ingredients.\\[0.4em] $m^\circ$ and the target & XIV-J & Target paraphrased & \emph{Transcribe} the contraction recipe on its proved canonical domain.\\[0.4em] Grouped blocks & XIV-J & Replacement symbol disputed & \emph{Transcribe} $H_T$ and $H$; do not replace $H_T$ by a hatted block or an admissible Hadamard replacement.\\[0.4em] Joint-map class & XIV-J & Four clauses paraphrased & \emph{Transcribe} all five actual clauses, including the complete-graph wavefront requirement.\\[0.4em] Descent criterion & XIV-J & Necessary interface & \emph{Transcribe} the global criterion and the limitation of its verification by canonical agreement.\\[0.4em] $\Ucan$ & XIV-I canonical proposition & Named domain; complement equality refused & \emph{Transcribe} the named domain and retain the refusal.\\[0.4em] Presentation-level $\mathcal A$ & XV-F interface; not supplied by XIII-C & Unavailable & \emph{Leave open}. Contact values do not determine a correction map on every presentation.\\ \end{longtable} Three non-transfers deserve emphasis. First, XIV-G remains unresolved at G2. Its source differential symbol is not the joint principal symbol of a composed response. Second, XIV-J's descent criterion is a global algebraic criterion; only the information obtained from agreement on $\Ucan$ is restricted to that domain. Third, XIII-C's class remains a class of contact changes in graph coefficients. It is not, without another statement, a vector space $\mathcal A\subseteq\operatorname{Hom}(\Ppres,Y)$. These distinctions are not merely terminological. They concern different domains, different codomains, and different identifications. A kernel can be tested against a smooth residual direction without having a pointwise slice. A coefficient change can be supported on a collision without specifying its value on every null presentation. A formal target can be written without defining the characteristic multiplication occurring in that target. No row in Table~\ref{tab:nontransfer} asserts that an XIV theorem proves an unstated XV identity. Where the patch adopts a convention, the adoption is visible. Where the source has a formula, the formula is reproduced. Where the source has only a role, that role remains a parameter until its actual member is recovered. %===================================================================== \section{Patch G: the common space and residual species} \label{sec:G} %===================================================================== \subsection{The copied common-space release} \source{Defines/source statement}{XIV-G.tex}{thm:release(i),(iii),(iv)} {reported lines 1710--1751; detailed source typing also appears in \texttt{prop:sourceoperator}} \begin{inputstatement}[The parts of XIV-G used here] \label{input:G} For the frozen retained instance, the chosen common space and characteristic test locus are $W$ and $N=\Nlift$, of dimensions sixteen and eight, respectively. For a smooth residual test $G$, the full differentiated source $\JD'[G]$ is a differential operator of order at most two. Smooth residual testing can hide nonzero attachment-diagonal wavefront directions even for compact balanced kernels. XIV-G supplies no complete internal symbol relation, joint section, controlled remainder, or characteristic prescription for its grouped target. Its ending remains \emph{Unresolved at G2}. \end{inputstatement} The common space is \begin{equation} W=A\times B\times Z\times Y,\qquad (a,b,z,y),\qquad \dim W=16, \label{eq:W} \end{equation} where the four factors are open spacetime neighborhoods. The supplied XIV-G text copies the retained characteristic configuration in the form \begin{equation} \begin{split} N=\bigl\{(a,b,z,y)\in W:\;&a=z,\quad y=a+\alpha\ell_{\mathbf n},\\ &b=a+\beta\ell_{\mathbf n},\quad \ell_{\mathbf n}=(1,\mathbf n),\\ &\mathbf n\in S^2,\quad 0<\alpha<\beta\bigr\}. \end{split} \label{eq:N} \end{equation} This is the selected retained locus, with its copied neighborhood restrictions. It is not the assertion that every characteristic configuration of the global selected pair is represented by this local formula. The dimensions have the stated bookkeeping interpretation: $a$ contributes four parameters, $\mathbf n$ contributes two, and $\alpha,\beta$ contribute two. The attachment equality determines $z$. Thus the copied local locus has eight parameters in the sixteen-dimensional common space. No transverse degree follows from this count, and no amplitude order is assigned to a future extension by subtracting the dimensions. \subsection{The actual type of a residual metric test} XIV-G's tested source uses \begin{equation} G\in C_c^\infty\bigl(Z;\Sym^2T^*M\bigr), \qquad \mathsf Q_Gh=\JD'[G]h. \label{eq:Gtest} \end{equation} Its local operator representation is \begin{equation} \mathsf Q_Gh(s) =\sum_{|\nu|\le2}A_\nu[G](s)\partial_s^\nu h(s), \qquad q_{G,2}(s,\xi) =\sum_{|\nu|=2}A_\nu[G](s)(i\xi)^\nu. \label{eq:Goperator} \end{equation} The coefficients are smooth and supported in $\supp G$. These are the source's local differential-operator statements. They retain the actual source coefficients and ramp derivatives within that copied operator statement. The test in \eqref{eq:Gtest} is a metric direction on $Z$. It is not a compact test section on all of $W$. Pulling it back by the projection to $Z$ gives a smooth object depending only on $z$, but does not generally give compact support in $a,b,y$. Moreover, the mixed residual output also retains a scalar slot at $y$. The metric direction alone therefore cannot be declared to be the full test space for the output coefficient. This matters for duality. A vector-valued distribution with values in a bundle $E$ pairs with compact sections of the \emph{dual} bundle, with an appropriate density. It does not, without a chosen fiber pairing, pair canonically with sections of the same bundle. A metric on $E$ could identify those two spaces, but no such metric on an output bundle is supplied by XIV-G. The input metric bundle itself is not rank-mysterious. In four spacetime dimensions a symmetric covariant two-tensor has ten components. What is not identified in XIV-G is a separate bundle called $E_{\mathrm{res}}$ which packages the output residual species with the required value/test duality. The two questions must not be confused. \subsection{A species-bundle convention with its parameter exposed} \begin{definition}[Residual species and parameterized bundle notation] \label{def:Eres} The phrase \emph{residual species} in this patch records the field slots left after the specified contractions. In the retained mixed sector those slots are the metric slot at $z$ and the scalar slot at $y$, as recorded in the supplied graph and joint-map statements. Let $\Bres$ be a finite-dimensional complex value space supplied as an additional parameter. Define \begin{equation} \Eres[\Bres]=W\times\Bres\longrightarrow W, \label{eq:Eres} \end{equation} and use the flat coordinate density on $W$ to set \begin{equation} \mathcal D'(W;\Eres[\Bres]) := \mathcal L_{\mathrm{cont}} \bigl(C_c^\infty(W;\Bres^*),\C\bigr). \label{eq:vectorD} \end{equation} For every specified $\Bres$, this is a named vector-valued distribution space. The fiber rank is $\dim\Bres$. This convention is introduced here. XIV-G does not identify $\Bres$, its rank, a fiber metric, or an output symbol for a bundle named $\Eres$. Accordingly, the unparameterized notation $\Eres$ is only a species placeholder until such an identification is supplied. It is not an unidentified bundle magically made concrete by writing an arrow to $W$. The metric tests \eqref{eq:Gtest} are part of the inherited testing interface. They are not asserted to be exactly $C_c^\infty(W;\Bres^*)$. \end{definition} Definition~\ref{def:Eres} is a genuine definition of a family. For each parameter it identifies the total space, projection, dual test space, and scalar pairing. It does not choose the parameter which the frozen joint-map target uses. This is the strongest codomain completion justified without silently inserting an output fiber convention. One could choose an explicit tensor-product value bundle for the mixed residual slots and compute its rank. That would be a legitimate additional mathematical convention. It would also have to specify the covariant or contravariant value convention, its density, and its identification with the target's contraction indices. This patch does not make that extra choice under the label ``from XIV-G.'' The reason for leaving the parameter visible is not that a printed integer is intrinsically necessary for defining a distribution space. A vector bundle defined by explicit bundle operations is an object even if its rank is not separately printed. The missing point here is the identification of the actual output fiber and pairing, not the typography of a dimension statement. \subsection{Residual testing and the information it does not preserve} The testing limitation in Input statement~\ref{input:G} remains in force. Smooth testing can remove attachment-diagonal wavefront directions from a tested kernel even when those directions are present in the joint kernel. The source illustrates this with a compactly weighted diagonal delta whose residual test becomes a smooth function. The present patch neither enlarges that example into a model of the characteristic target nor derives a new estimate from it. Its role is negative and precise: regularity after every smooth metric test cannot be used as a substitute for an untested joint composition theorem. A source symbol of order two is a polynomial on the local operator relation. It is not the symbol of the gravitational response after propagation, and it is not the principal symbol of the full grouped cross-line composition. Likewise, a definition of $\Eres[\Bres]$ adds no output wavefront bound. Bundle typing specifies what a value would be if constructed. It does not construct that value, control its characteristic directions, or make it microcausal after pushforward. \begin{proposition}[Exact status of Patch G] \label{prop:Gstatus} Patch G reproduces the common-space and residual-testing statements. It defines a parameterized species-bundle codomain. It does not identify the frozen output fiber, and it does not enter G3--G5 or any of XV-F's analytic gates. \end{proposition} \begin{proof} Equations \eqref{eq:W}--\eqref{eq:Goperator} reproduce the supplied common-space and testing interface. Equations \eqref{eq:Eres}--\eqref{eq:vectorD} define a family for a specified parameter. No supplied statement identifies that parameter with an output fiber. The source's unresolved ending is retained in Input statement~\ref{input:G}; none of the definitions here adds a composition or wavefront theorem. \end{proof} Thus P1 passes as a named-species and conditional-codomain interface. It does not pass as an identification of the actual frozen bundle. This limitation is an owned boundary, not a nonexistence statement about possible residual bundles. %===================================================================== \section{Patch C: collision coordinates and correction-role formulas} \label{sec:C} %===================================================================== \subsection{Collision sets and the displayed vertex} \source{Defines in the source}{XIII-C.tex}{sec:class, Collision sets} {reported lines 338--349; the definition is unlabelled in the supplied full text} \begin{definition}[Collision sets; transcribed] \label{def:collisions} For each inventory graph, let $y$ be its interaction vertex and let $x,u,v$ denote its displayed observable vertices when present. For a connected vertex subset $I$ containing $y$, with $|I|\ge2$, \begin{equation} \Delta_I=\{x_a=x_b\text{ for every }a,b\in I\}. \label{eq:Delta} \end{equation} Intersections and loci obtained after contracting such subsets are included. Diagonals involving only a pre-existing observable subgraph retain their inherited normalization. \end{definition} The connected subsets are those associated with XIII-C's cycle inventory. They live in the graph variables of that inventory. Nothing in Definition~\ref{def:collisions} identifies every graph variable with one of $a,b,z,y$. In particular, a formula for a collision in a graph with internal vertices is not automatically a formula for its lift or pushforward locus in $W$. \source{Defines in the source}{XIII-C.tex}{eq:vertex} {reported lines 416--421} The supplied interaction and stress are \begin{equation} \Vcal[\gamma,\phi] =-\frac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}[\phi]\dd^4x, \qquad T_0^{\mu\nu}[\phi] =\partial^\mu\phi\partial^\nu\phi -\frac12\eta^{\mu\nu} \bigl(\partial_\rho\phi\partial^\rho\phi-m^2\phi^2\bigr). \label{eq:vertex} \end{equation} Here $\eta=\operatorname{diag}(1,-1,-1,-1)$, consistently with the four supplied sheets. The vertex is a functional, not an Euler operator. Its mixed derivative will be computed algebraically in Patch J; no additional vertex from a non-input is used. \subsection{The frozen contact class} \source{Defines in the source}{XIII-C.tex}{def:Cpart, eq:partial-counterterm} {reported lines 358--392; the shorter range 358--385 does not include the whole closing qualification} \begin{definition}[Partial-diagonal ambiguity class; transcribed] \label{def:Cpart} The class $\Cpart$ consists of finite sums of graphwise contact changes at order $\hbar^2$ in the time-ordered product, hence at order $\lambda\hbar$ in the selected retarded expansion. Before pushforward they have the form \begin{equation} D_\Gamma\left[ w_\Gamma\,\partial_r^\alpha\delta(r_I)\, S_{\Gamma/I}\right], \qquad |\alpha|\le6. \label{eq:Cpartsource} \end{equation} Here: \begin{enumerate}[label=\textup{(\roman*)}] \item $r_I$ is a system of relative coordinates transverse to $\Delta_I$; \item $S_{\Gamma/I}$ is an inherited spectator graph coefficient, with the two remaining field slots retained; \item $w_\Gamma$ is built from $g,\partial g,\sig,\sig',\sig''$ and the smooth external graph weights already present; \item $D_\Gamma$ is the specified finite graph differential operator; \item the change is even, of ghost number zero and field degree two, and is supported in the existing compact graph closure; \item it leaves the tree datum and the inherited observable-subgraph normalizations unchanged. \end{enumerate} Only well-defined products and pushforwards are members. The bound six is a conservative frozen bound; an established scaling degree can impose a smaller bound on a particular locus. The total-diagonal subclass is $\Ctot\subseteq\Cpart$. \end{definition} The source additionally warns that a spectator can remain singular in its tangential variables. Smoothness concerns the external weights, not the entire spectator distribution. It also warns that differentiating a diagonal distribution can differentiate its multiplying weights upon pushforward. This is not permission to add independent arbitrary higher-jet coefficient functions. These qualifications are retained. In particular, a polynomial weight convention introduced below is not inferred from the words ``built from.'' Those words do not specify a polynomial degree, a coefficient field beyond the ordinary constants, or a complete list of admissible graphwise combinations. \subsection{An explicit choice of local collision coordinates} Let a graph have vertex-coordinate space $X_\Gamma$ and fix $I=\{i_0,i_1,\ldots,i_k\}$. In a coordinate neighborhood choose \begin{equation} r_I= \bigl(x_{i_1}-x_{i_0},\ldots,x_{i_k}-x_{i_0}\bigr) \in\R^{4k}. \label{eq:rI} \end{equation} Let $s_I$ consist of $x_{i_0}$ and the coordinates of vertices outside $I$. Then \begin{equation} \chi_I:X_\Gamma\supset V_{\Gamma,I} \longrightarrow V_{s_I}\times V_{r_I}, \qquad x\longmapsto(s_I,r_I), \qquad \Delta_I\cap V_{\Gamma,I}=\{r_I=0\}. \label{eq:collisionchart} \end{equation} The inverse recovers $x_{i_j}=x_{i_0}+r_{I,j}$ and leaves the other vertices unchanged. This is an explicit coordinate completion of role (i), not a new collision prescription. The codimension is $4k=4(|I|-1)$ in the graph coordinate space. Coordinate-density conversions, if needed for a different local chart, are smooth coefficient factors. They do not alter which collision is being represented and do not identify this graph codimension with the codimension-eight locus $N$ in $W$. \subsection{Displayed templates for the three remaining roles} The following formulas expose every parameter that the source role leaves unspecified. \begin{definition}[Parameterized contact-role realization] \label{def:contacttemplate} Fix a graph and a collision chart \eqref{eq:collisionchart}. Choose the following data. First, choose a finite-order differential operator \begin{equation} D_\Gamma =\sum_{|\nu|\le d_\Gamma} d_{\Gamma,\nu}(s_I,r_I) \partial_{s_I,r_I}^{\nu}, \qquad d_\Gamma\in\mathbb N,\quad d_{\Gamma,\nu}\in C^\infty(V_{\Gamma,I}). \label{eq:Dtemplate} \end{equation} Its actual order and coefficient functions are parameters. The bound $|\alpha|\le6$ in \eqref{eq:Cpartsource} does not bound $d_\Gamma$. Second, list the fixed smooth weight generators already present for that graph, denoted \begin{equation} h_{\Gamma,1},\ldots,h_{\Gamma,\ell_\Gamma}. \label{eq:externalweights} \end{equation} They are placeholders for inherited external weights, not freely chosen additional functions. Include the graphwise evaluations of the source-named functions $g,\partial g,\sig,\sig',\sig''$ in this list. No formula for $\sig$ is introduced. For the \emph{polynomial realization convention} of this patch, choose a finite set $F_\Gamma\subset\mathbb N^{\ell_\Gamma}$ and constant coefficients $a_{\Gamma,\kappa}$, and define \begin{equation} w_\Gamma =\sum_{\kappa\in F_\Gamma} a_{\Gamma,\kappa} \prod_{j=1}^{\ell_\Gamma} h_{\Gamma,j}^{\kappa_j}. \label{eq:wtemplate} \end{equation} This is an explicit polynomial family. The source does not state that this family equals its entire admissible weight class. Third, let \begin{equation} S_{\Gamma/I} =s_{\Gamma/I}\in \mathcal D'(V_{s_I};\text{two residual field slots}) \label{eq:Stemplate} \end{equation} be the spectator parameter. Its name records the contracted graph and retained slots. Equation \eqref{eq:Stemplate} does not determine the inherited spectator distribution. For $|\alpha|\le6$, define the local template value \begin{equation} \begin{split} C_{\Gamma,I,\alpha} =D_\Gamma\Bigl[ w_\Gamma\, \chi_I^*\bigl( s_{\Gamma/I}(s_I)\otimes \partial_{r_I}^{\alpha}\delta(r_I) \bigr)\Bigr]. \end{split} \label{eq:contacttemplate} \end{equation} Only choices satisfying the source's parity, ghost-number, field-degree, support, normalization, product, and pushforward requirements are candidates for membership in $\Cpart$. \end{definition} The tensor product in \eqref{eq:contacttemplate} is meaningful for a singular tangential spectator: it combines independent $s_I$ and $r_I$ variables. Smooth multiplication and the displayed finite differential operator are then meaningful local distributional operations. This supplies a coherent local formula without pretending that a singular spectator is a smooth function. A spectator whose inherited realization requires further products or pushforwards is not recovered merely by this local notation. Those products remain subject to the source membership condition. Nor is a coordinate expression on one collision chart a proof of compatibility on intersections of several collision charts. \subsection{What the templates prove} \begin{lemma}[Support and tree-order properties] \label{lem:contactproperties} Suppose the parameters in Definition~\ref{def:contacttemplate} are chosen so that the seed in \eqref{eq:contacttemplate} is supported in the existing compact graph closure $\mathcal K_{\mathrm{gr}}$. Then \begin{equation} \supp C_{\Gamma,I,\alpha} \subseteq \Delta_I\cap\mathcal K_{\mathrm{gr}} \quad\text{locally}. \label{eq:contactsupport} \end{equation} A well-defined pushforward has support in the image of that set. If this value is used as a contact change at order $\hbar^2$ in the time-ordered product, it does not alter the selected tree coefficient. \end{lemma} \begin{proof} The tensor factor $\partial^\alpha\delta(r_I)$ is supported on $r_I=0$. Multiplication by a smooth function and application of a finite differential operator do not enlarge distributional support. The support hypothesis supplies the compact graph closure. A pushforward, when defined, has support contained in the image of the original support. For the last assertion, a change $\delta\mathsf T_2=\hbar^2 C$ changes the retarded expression $(i/\hbar)(\mathsf T_2-\Vcal\star_H\Ocal)$ by $i\hbar C$. It has no coefficient at $\hbar^0$. The inherited tree datum is consequently unchanged. \end{proof} The compact support in this lemma is a hypothesis inherited from the class, not a consequence of writing a polynomial in smooth weights. A polynomial can have noncompact support. The patch therefore does not insert a new cutoff to manufacture membership. Similarly, preservation of an observable-subgraph normalization is a retained admissibility requirement, not a consequence of differentiation alone. \begin{proposition}[The contact template is not the frozen parametrization] \label{prop:Cstatus} Equations \eqref{eq:Dtemplate}--\eqref{eq:contacttemplate} define a parameterized local contact realization. They do not identify the actual frozen $D_\Gamma$, $w_\Gamma$, or $S_{\Gamma/I}$. They prove neither exhaustiveness of the frozen class nor an identity between that class and a presentation-level correction space $\mathcal A$. \end{proposition} \begin{proof} The source roles specify that there is a graph operator, an admissible external weight, and an inherited spectator. They do not supply the parameters in \eqref{eq:Dtemplate}, \eqref{eq:wtemplate}, and \eqref{eq:Stemplate}. A family indexed by unspecified parameters does not identify one prescribed member. Exhaustiveness would require a theorem that every permitted frozen change is represented by the displayed parameter family, together with a statement of its admissible parameters. Neither is supplied. Finally, the values in \eqref{eq:contacttemplate} are distributions. A map in $\mathcal A$ would require a value for each presentation, and possibly additional linearity, continuity, covariance, and tensor restrictions. No such assignment follows from a class of values. \end{proof} P2 is therefore complete as an explicit role-formula ledger. It does not pass the stronger frozen-parameter recovery condition. Calling the displayed parameters ``specified'' would not change this conclusion. The actual values must be supplied. %===================================================================== \section{Patch I: the local amplitude module and reconstruction} \label{sec:I} %===================================================================== \subsection{Source notation and the adapted charts} A notation correction is needed before transcription. The supplied XIV-I full text defines $\Ar^m(V)$, not a module written $\mathcal A^m(V)$. The latter notation occurs in the XV-F retrieval account. This patch uses the actual source notation and does not postulate an additional equality between two differently named modules. \source{Defines in the source}{XIV-I.tex}{sec:amplitudespace, Actual coordinates for the two placements} {reported lines 828--855; the final chart is reported at lines 852--855} The two placements use \begin{equation} \begin{gathered} d_+=b-a,\qquad r_+=a-z,\qquad u_+=a,\\ d_-=b-z,\qquad r_-=z-a,\qquad u_-=z,\\ w_\pm=(u_\pm,\zeta_\pm,q_\pm,r_\pm,y), \qquad q_\pm=d_\pm^2. \end{gathered} \label{eq:adaptedchart} \end{equation} The maps to $(a,d_+,r_+,y)$ and $(z,d_-,r_-,y)$ are diffeomorphisms onto their images. On the retained nonzero future cone patches, $(q_\pm,\zeta_\pm)$ are ordinary regular coordinates for $d_\pm$. The residual separation $r_\pm$ is retained rather than tested away. The dimension is $4+3+1+4+4=16$. The charts are local adapted charts on their stated domains. They are not an identification of a local chart with every point of the global graph space, and they do not transport the collision inventory of XIII-C into $W$ without further maps. \subsection{The existing internal kernel} \source{Defines in the source}{XIV-I.tex}{input:convolution, eq:R} {reported display lines 329--335; the preceding subsection identifies $R$ as the kernel of $E_0^{\mathrm R}$} Within the copied XIV-I namespace, $R$ is the kernel of $\ER$, and the displayed normalization is \begin{equation} R(t,\mathbf x) =\frac{\Theta(t)}{2\pi}\delta(t^2-|\mathbf x|^2), \qquad B:=R*R =\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2). \label{eq:Rsource} \end{equation} The source records $\Box\ER=\id$. The convolution display is a premise concerning existing internal kernels, not a new product prescription for the quantum graph. Consequently, the support of this copied kernel is not a missing sentence: \begin{equation} \supp R=\partial J^+,\qquad J^+=\{(t,\mathbf x):t\ge|\mathbf x|\}. \label{eq:Rsupport} \end{equation} Indeed its cone formula has the measure pairing \begin{equation} \pair{R}{\varphi} =\frac1{4\pi}\int_0^\infty t\int_{S^2}\varphi(t,t\omega)\dd\omega\dd t. \label{eq:Rpair} \end{equation} It vanishes off the forward cone boundary and is nonzero on every open neighborhood meeting that boundary. This elementary support reading introduces no estimate and no uniqueness theorem. The identification is internal to the supplied XIV-I statement. It does not identify the same printed symbol in an excluded manuscript. Thus a request for the support of the copied kernel is resolved; a request for a separate cross-sheet identification can still be open. \subsection{Generators, coefficients, and the reconstruction rule} \source{Defines in the source}{XIV-I.tex}{eq:generators, def:amodule, eq:amodule, eq:reconstruction} {the registered definition is reported at lines 873--901; the generator display is adjacent to that definition} Use the source Fourier convention \[ \widehat u(\omega,\xi) =\int e^{-i(t\omega+\mathbf x\cdot\xi)} u(t,\mathbf x)\dd t\dd^3x, \qquad \kappa=(\omega,\xi). \] The inverse transform has factor $(2\pi)^{-4}$. Let $M_{\mathrm R}$ denote the Fourier transform of the existing kernel $R$. Its source pole representation is \begin{equation} M_{\mathrm R} =-\frac{s}{2\rho}(\omega-s\rho-i0)^{-1} +\frac{s}{2\rho(\omega+s\rho)}, \qquad \rho=|\xi|>0,\quad |\omega-s\rho|