% ix-e-3.tex % Compile with: pdflatex ix-e-3.tex % Run pdflatex a second time to resolve cross-references. % % IMPORTANT AUDIT: % The supplied statements do not give the defining equation or the % boundary-trace kernel of the inherited operator E_0. % This manuscript does not silently replace that operator. % % It proves: % (1) an unconditional support characterization for W_g; % (2) a no-cancellation lemma for the sharp surface distribution; % (3) a complete conditional bridge theorem under an explicit % boundary-trace certificate; % (4) that the proposed solid-ball lower-support assertion is false % for the standard flat massless zero-Cauchy realization; % (5) that null spheres nevertheless suffice for the bridge. % % Supplied-data release: % Unresolved at verification of the boundary-trace certificate for % the inherited E_0, not at another evaluation of Im K. % % No time-ordered product is constructed. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm} \usepackage{mathrsfs} \usepackage[margin=1in]{geometry} \usepackage{array,booktabs,longtable} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, citecolor=blue, urlcolor=blue, pdftitle={Landmark IX-E-3: Functional support and the two-way commutator bridge}, pdfauthor={} } \setlength{\emergencystretch}{3em} \newcommand{\dd}{\mathrm d} \newcommand{\R}{\mathbb R} \newcommand{\Cplx}{\mathbb C} \newcommand{\supp}{\operatorname{supp}} \newcommand{\fsupp}{\operatorname{supp}_{\mathrm{fun}}} \newcommand{\Span}{\operatorname{span}_{\Cplx}} \newcommand{\Imag}{\operatorname{Im}} \newcommand{\id}{\mathrm{id}} \newcommand{\Mplus}{M_+} \newcommand{\Mint}{M_+^\circ} \newcommand{\Padm}{\mathscr P_{\mathrm{adm}}} \newcommand{\Dfirst}{D_{\mathrm{first}}} \newcommand{\Dtwo}{\mathcal D_2} \newcommand{\Otwo}{\mathcal O_2} \newcommand{\Ttwo}{\mathsf T_2} \newcommand{\Vthree}{\mathscr V^{[3]}} \newcommand{\Cfg}{\mathcal C_{f,g}} \newcommand{\Bsurf}{\mathsf B} \newcommand{\Ezero}{\mathsf E_0} \newcommand{\Nset}{\mathcal N_F} \newcommand{\Sset}{\mathcal S_F} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{hypothesis}[theorem]{Hypothesis} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \title{\textbf{Landmark IX-E-3}\\[0.4em] Functional support of the sharp insertion\\ and the two-way commutator bridge} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We isolate the actual functional-support calculation needed to apply IX-E-2's evaluated commutator to the retained two-way sector. The local quadratic functional \(W_g\) has functional support exactly \(\supp g\). For the sharp insertion, a nonzero smooth surface coefficient cannot be cancelled by the locally bounded bulk term in its distributional metric derivative. A correction is necessary to the proposed solid-ball argument. For the explicitly specified flat massless zero-Cauchy realization, the initial-surface support is the union of the time-zero null spheres generated by \(\supp f\), not necessarily the union of their solid balls. We prove this support statement, exhibit an open missed region, and show that the retained null spheres nevertheless suffice to exclude the commutator slab. Thus a gap inside a solid ball does not by itself provide an obstruction witness. The bridge proof is complete under a displayed boundary-trace certificate. The supplied prerequisite statements, however, do not specify that certificate for the inherited operator \(\mathsf E_0\). We do not identify a standard Green operator with an unspecified inherited operator by notation alone. The supplied-data release is therefore \emph{Unresolved} at that specific identification. If the certificate is supplied, the bridge follows without another support calculation. No time-ordered product is constructed. \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) {Bridge proved, one class}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Bridge open}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Bridge obstructed}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The three legal outcomes are not interchangeable.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One support outcome does not settle the others. 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-IX-E-3} \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) {Bridge proved, one class}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Bridge obstructed}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The three legal outcomes are not interchangeable.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One support outcome does not settle the others. 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-IX-E-3} \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) {Bridge proved, one class}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Bridge open}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Bridge obstructed}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The three legal outcomes are not interchangeable.}; \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-IX-E-3} \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) {Bridge proved, one class}; \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) {Bridge obstructed}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Bridge open. The three legal outcomes are not interchangeable.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Bridge proved, one class is not Bridge obstructed. 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-IX-E-3} \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: Bridge proved, one class}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Bridge open}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Bridge obstructed}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The three legal outcomes are not interchangeable.}; \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-IX-E-3} \end{figure} Figure~\ref{fig:split-IX-E-3} records one view of the same separation. Figure~\ref{fig:chain-IX-E-3} records one view of the same separation. Figure~\ref{fig:pair-IX-E-3} records one view of the same separation. Figure~\ref{fig:gate-IX-E-3} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-IX-E-3} records that separation. The drawing adds no hypothesis. \section{Release contract and frozen domain} \subsection{Three bridge outcomes} The legal outcomes for this support problem are: \begin{enumerate} \item \textbf{Bridge proved.} The implication (SB) holds for every retained first-stage pair, including the declared conjugate sector. The double-ordering commutator gate clears, but the overall causal-product release remains Unresolved. \item \textbf{Proved obstructed.} A retained pair has both actual support orderings certified and a nonzero commutator. Every symmetric causally factorizing map on the fixed domain is then excluded. \item \textbf{Unresolved.} A named support calculation, cancellation check, or ordering certificate remains unfinished. Absence of a proof is not an obstruction. \end{enumerate} There is no Constructed outcome for \(\Ttwo\) within this paper. Let \(\sigma\) be complex-linear flip and \(\gamma\) componentwise antilinear conjugation. Retain exactly \begin{equation} \Dtwo = \Span\left\{ \sigma^a\gamma^b d: d\in\Dfirst,\quad a,b\in\{0,1\} \right\}, \qquad \Otwo=\{0\}. \label{eq:frozen-domain} \end{equation} Here \[ \Dfirst = (\mathscr B\otimes\mathscr W) +(\mathscr W\otimes\mathscr B) \] is the inherited space of actual first-stage tensors. No tensor is removed, including the strongly separated pairs whose commutator was evaluated as zero in IX-E-2. \begin{center} \fbox{\begin{minipage}{0.92\textwidth} \textbf{Release box.} On the supplied information, the bridge release is \textbf{Unresolved at the inherited boundary-trace certificate} \textup{(TC)} stated below. \smallskip A complete conditional support proof is given. For the explicitly specified flat massless zero-Cauchy realization, the bridge is proved, although the proposed solid-ball lower-support assertion is false. \smallskip That realization is not silently substituted for the inherited \(L=\mathsf E_0C\). Verification that the inherited operator has the displayed trace remains necessary. \smallskip No certified nonzero two-way pair is produced. No time-ordered replacement or retarded construction is begun. \end{minipage}} \end{center} \subsection{The necessary debugging distinction} The notation \(L=\Ezero C\), the sharp adjoint identity, and an upper causal support bound do not, by themselves, specify the action of \(\Ezero^*\) on arbitrary compact sources. In particular, the statement that advanced propagation is supported in a ball is an upper-support statement. It does not say whether propagation has a tail inside the ball, is supported on its boundary, or has a nonzero coefficient at a particular point. We therefore distinguish: \begin{itemize} \item results following from the explicitly copied inputs; \item a sufficient boundary-trace certificate for the inherited operator; \item a fully specified flat realization for which that certificate, the support alternative, and the bridge are proved. \end{itemize} This separation is essential to avoid changing the fixed functionals while ostensibly keeping their names. \section{Copied inputs and conventions} Work on \[ \Mplus=[0,\infty)\times\R^3, \qquad \Mint=(0,\infty)\times\R^3, \qquad \Sigma=\{0\}\times\R^3. \] Configurations are unrestricted smooth off-shell scalar and symmetric metric configurations. Complex test functions are allowed; detection by real configurations is addressed explicitly. The inherited sharp insertion and local quadratic functional are \begin{align} B_f[h,\phi] &=-\int_{\Mplus} f(x)(Lh)^\mu(x)\partial_\mu\phi(x)\,\dd^4x, \qquad L=\Ezero C, \label{eq:B}\\ W_g[h] &=\int_{\Mplus}g(y)h_{00}(y)^2\,\dd^4y. \label{eq:W} \end{align} \subsection{Evaluated kernel and commutator} Write \[ x=(t,\mathbf x),\qquad y=(s,\mathbf y),\qquad \tau=t-s,\qquad u=t+s,\qquad R=|\mathbf x-\mathbf y|. \] IX-E-2 supplies, on \(t,s>0\) and \(R>0\), \begin{equation} \begin{aligned} \Imag K(x,y) ={}& \frac{t}{16\pi R} \bigl[\delta(\tau-R)-\delta(\tau+R)\bigr]\\ &+\frac{1}{32\pi R} \bigl[\Theta(R-|\tau|)-\Theta(R-u)\bigr]. \end{aligned} \label{eq:imagK} \end{equation} Away from the cone boundaries it is zero for \(R<|\tau|\), equals \(1/(32\pi R)\) for \(|\tau|u\). The direct delta contribution is separate; the reflected cone is a jump, not another delta. The Fourier definition governs the spatial apex. The same input supplies \begin{align} B_f\star_cW_g-W_g\star_cB_f &=\hbar\Cfg,\\ \Cfg &=-4i\pair{f(x)g(y)\Imag K(x,y)} {\partial_t\phi(x)h_{00}(y)}. \label{eq:commutator} \end{align} These identities are not re-evaluated here~\cite{IXE2}. The compact-detection lemma is also imported: a nonzero compactly supported distribution \(fg\Imag K\) in the open half-space gives a nonzero actual commutator on the unrestricted configuration space. No transfer to a constrained quotient is asserted. The strongly separated witness class has exact zero commutator. The observation-spacelike control strictly inside \[ |t-s|0\), \begin{equation} \Cfg =-\frac{i\alpha}{8\pi} \int\frac{f(x)g(y)}{|\mathbf x-\mathbf y|} \,\dd^4x\,\dd^4y\ne0 \label{eq:control} \end{equation} for the nonnegative tests specified in IX-E-2. That control is not an obstruction witness without both actual ordering certificates. \subsection{Sharp adjoint and support bounds} For \(a\) supported in \(\Mint\), the supplied VII-B identity is \begin{equation} L^*a =\theta C^*b+\delta(t)\Bsurf(b|_\Sigma), \qquad b=\Ezero^*a, \label{eq:adjoint} \end{equation} where \begin{equation} \Bsurf^{\alpha\beta}(v) =-n^{(\alpha}v^{\beta)} +\frac12\eta^{\alpha\beta}n^\mu v_\mu. \label{eq:Bsurface} \end{equation} There is no \(\delta'(t)\) term. The off-shell surface contribution is retained; an on-shell smoothing does not replace it~\cite{VIIB}. For \(F=\supp f\) and \(G=\supp g\), the copied bounds are \begin{equation} \fsupp B_f\subset J^-(F)\cap\Mplus\ \cup F, \qquad \fsupp W_g\subset G. \label{eq:upper} \end{equation} \subsection{Actual ordering and derivative support} The series definition is \begin{equation} F_1\gtrsim F_2 \quad\Longleftrightarrow\quad \fsupp F_1\cap J^-(\fsupp F_2)=\varnothing. \label{eq:ordering} \end{equation} Null relation and coincidence are included. Two-way ordering means that both orderings hold. We use IX-E-2's derivative-support characterization: on the unrestricted affine configuration space, functional support is the closure of the union of the supports of all first-derivative distributions, over all configurations and field components. This characterization is an imported result, not a replacement by the upper set in~\eqref{eq:upper}~\cite{IXE2}. The obligation is \begin{equation} \boxed{ B_f\gtrsim W_g,\quad W_g\gtrsim B_f \quad\Longrightarrow\quad f(x)g(y)\Imag K(x,y)=0. } \tag{SB} \label{eq:SB} \end{equation} \section{Derivative support and surface non-cancellation} The two variations of the sharp insertion are \begin{align} \mathrm D_hB_f[h,\phi][\dot h] &=-\int f(x)(L\dot h)^\mu(x) \partial_\mu\phi(x)\,\dd^4x, \label{eq:metric-var}\\ \mathrm D_\phi B_f[h,\phi][\dot\phi] &=-\int f(x)(Lh)^\mu(x) \partial_\mu\dot\phi(x)\,\dd^4x. \label{eq:scalar-var} \end{align} With \(a_\mu=-f\partial_\mu\phi\), the metric derivative is represented by the full \(L^*a\) in~\eqref{eq:adjoint}. The scalar variation samples \(Lh\) on \(F\). \begin{proposition}[Exact support of the local quadratic slot] For every \(g\), \[ \fsupp W_g=G. \label{eq:Wexact} \] \end{proposition} \begin{proof} At the allowed configuration \(h_{00}=1\), the metric derivative has kernel \(2g\), whose support is \(G\); combine this with the upper bound in~\eqref{eq:upper}. \end{proof} \begin{lemma}[Injectivity of the surface coefficient] If \(n\) is non-null, the map \(v\mapsto\Bsurf(v)\) is injective. \end{lemma} \begin{proof} Contract~\eqref{eq:Bsurface} with \(n_\alpha\): \[ n_\alpha\Bsurf^{\alpha\beta}(v) =-\frac12(n_\alpha n^\alpha)v^\beta. \] Thus \(\Bsurf(v)=0\) implies \(v=0\). \end{proof} \begin{lemma}[A smooth surface coefficient cannot be cancelled] Suppose, near a point of \(\Sigma\), \[ q=\theta A+\delta(t)D, \] where \(A\) is locally bounded and \(D\) is smooth on \(\Sigma\). If \(D\) is nonzero at that point, the point belongs to \(\supp q\). \label{lem:no-cancel} \end{lemma} \begin{proof} In any sufficiently small spatial neighbourhood choose a compact smooth tensor test \(\zeta\) such that \(\pair{D}{\zeta}\ne0\). Let \(\rho\) be a compact smooth temporal function with \(\rho(0)=1\), and test against \[ \zeta_\varepsilon(t,\mathbf z) =\rho(t/\varepsilon)\zeta(\mathbf z). \] The surface pairing is the fixed nonzero number \(\pair{D}{\zeta}\). The bulk pairing is \(O(\varepsilon)\) by local boundedness. Their sum is nonzero for sufficiently small \(\varepsilon\). Its test support lies in the prescribed neighbourhood. \end{proof} The convention for \(\delta(t)\) here is the inherited surface distribution: it pairs with the trace on \(\Sigma\). No additional factor or boundary normalization is introduced. Lemma~\ref{lem:no-cancel} does not prove that \(b|_\Sigma\) is nonzero. It proves that, once nonzero trace is established with the stated bulk regularity, the bulk term cannot remove its support. These are distinct obligations. \section{The boundary-trace certificate} \begin{hypothesis}[Boundary-trace certificate (TC)] For every compact smooth covector source \(a\) in \(\Mint\), the inherited \(b=\Ezero^*a\) is smooth up to \(\Sigma\), \(C^*b\) is locally bounded there, and its componentwise trace is \begin{equation} b_\mu(0,\mathbf z) = \int_{\Mint} \frac{\delta(t-|\mathbf w-\mathbf z|)} {4\pi|\mathbf w-\mathbf z|} a_\mu(t,\mathbf w)\,\dd t\,\dd^3w. \tag{TC} \label{eq:TC} \end{equation} A fixed overall nonzero sign in this formula would give the same support conclusions. \end{hypothesis} This certificate is more precise than an upper ball bound. Its verification for the inherited operator is not included in the supplied statements. We first prove exactly what it implies. \begin{theorem}[Null-sphere lower support under (TC)] Assume \textup{(TC)} and the copied adjoint identity. For every \(x=(t,\mathbf x)\in F\) with \(t>0\), \begin{equation} \{(0,\mathbf z):|\mathbf z-\mathbf x|=t\} \subset\fsupp B_f. \label{eq:sphere-lower} \end{equation} Moreover, \[ F\cap\Sigma\subset\fsupp B_f. \label{eq:boundary-F-lower} \] \label{thm:sphere} \end{theorem} \begin{proof} First suppose \(f(x)\ne0\), and fix \(\mathbf z\) with \(|\mathbf x-\mathbf z|=t\). Choose a nonnegative compact smooth cutoff \(\chi\), positive at \(x\), supported in a small interior neighbourhood of \(x\) on which \[ r_{\mathbf z}(\mathbf w)=|\mathbf w-\mathbf z| \] is smooth and nonzero. Define \begin{equation} \phi(v,\mathbf w) = \bigl(v-r_{\mathbf z}(\mathbf w)\bigr) \overline{f(v,\mathbf w)}\chi(v,\mathbf w). \label{eq:detecting-phi} \end{equation} This is a smooth compact configuration, and its support is away from \(\Sigma\). Consequently \(a=-f\,\dd\phi\) is an allowed source for~\eqref{eq:TC}. On the cone \(v=r_{\mathbf z}(\mathbf w)\), \[ \partial_v\phi =\overline f\,\chi. \] Its temporal trace component is therefore \begin{equation} b_0(0,\mathbf z) = -\int_{\R^3} \frac{|f(r_{\mathbf z}(\mathbf w),\mathbf w)|^2 \chi(r_{\mathbf z}(\mathbf w),\mathbf w)} {4\pi r_{\mathbf z}(\mathbf w)} \,\dd^3w \ne0. \label{eq:trace-detection} \end{equation} The integral is finite, and its integrand is positive on a nonempty open spatial set. This construction respects the restriction \(a=-f\,\dd\phi\); it does not treat its components as independently selectable sources. If configurations are required to be real, write \(\phi=\phi_1+i\phi_2\). Since the trace map is complex-linear and the displayed result is nonzero, at least one real configuration \(\phi_j\) gives nonzero trace. Injectivity of \(\Bsurf\), followed by Lemma~\ref{lem:no-cancel}, shows that \((0,\mathbf z)\) lies in the support of the actual metric derivative. The derivative-support characterization places it in \(\fsupp B_f\). For an arbitrary \(x\in F\) with \(t>0\), take \(x_j=(t_j,\mathbf x_j)\to x\) with \(f(x_j)\ne0\). For a given sphere point write \(\mathbf z=\mathbf x+t\mathbf n\), \(|\mathbf n|=1\), and set \(\mathbf z_j=\mathbf x_j+t_j\mathbf n\). The preceding argument applies to each \(\mathbf z_j\). Closedness of functional support proves \eqref{eq:sphere-lower}. Finally, any point of \(F\cap\Sigma\) is a limit of interior points where \(f\ne0\), unless \(f\) is identically zero near it, in which case it is not in \(F\). The same construction with \(t_j\to0\) gives sphere points converging to that boundary point. This proves~\eqref{eq:boundary-F-lower}. \end{proof} The theorem supplies a lower support statement for the fixed test \(f\), not merely for a freely chosen source concentrated near one observation. \section{A fully specified flat realization and the solid-ball error} This section is a diagnostic realization, not a replacement of the inherited operator. Let \[ \eta=\operatorname{diag}(1,-1,-1,-1), \qquad P=\partial_t^2-\Delta, \] and define \begin{equation} (Ch)^\mu = \partial_\nu h^{\nu\mu} -\frac12\partial^\mu(\eta_{\alpha\beta}h^{\alpha\beta}). \label{eq:C-flat} \end{equation} Let \(\Ezero\) be the componentwise solution operator for \(P\) with zero displacement and velocity at \(t=0\). Its retarded representation is \begin{equation} (\Ezero j)(t,\mathbf x) = \int_0^t\!\!\int_{\R^3} \frac{\delta(t-v-|\mathbf x-\mathbf w|)} {4\pi|\mathbf x-\mathbf w|} j(v,\mathbf w)\,\dd^3w\,\dd v. \label{eq:E-flat} \end{equation} This is the standard flat three-dimensional wave realization. Its propagation is concentrated on null spheres, rather than throughout the interiors of causal balls~\cite{Ivrii}. \begin{proposition}[Verification of (TC) for the flat realization] The operator~\eqref{eq:E-flat} satisfies \textup{(TC)}. \end{proposition} \begin{proof} Pair~\eqref{eq:E-flat} with a compact source \(a\), interchange the compactly supported distributional integrations, and collect the coefficient of \(j(v,\mathbf z)\). This gives \[ (\Ezero^*a)(v,\mathbf z) = \int_{t\ge v}\!\!\int_{\R^3} \frac{\delta(t-v-|\mathbf w-\mathbf z|)} {4\pi|\mathbf w-\mathbf z|} a(t,\mathbf w)\,\dd^3w\,\dd t. \] At \(v=0\) this is~\eqref{eq:TC}. Smoothness follows directly by angular integration: \[ (\Ezero^*a)(v,\mathbf z) = \frac1{4\pi} \int_0^\infty r \int_{S^2}a(v+r,\mathbf z+r\mathbf n) \,\dd\Omega(\mathbf n)\,\dd r. \] On compact sets in \((v,\mathbf z)\), the effective \(r\)-range is bounded. All derivatives can be passed under the integral. Since \(C^*\) is first order with constant coefficients, \(C^*b\) is smooth and locally bounded. \end{proof} \subsection{Exact initial-surface support} For \(f\in C_c^\infty(\Mint)\), define compact sets \begin{align} \Sset &= \left\{ \mathbf z: \exists(t,\mathbf x)\in F,\quad |\mathbf z-\mathbf x|=t \right\}, \label{eq:SF}\\ \Nset &= \left\{ (v,\mathbf z): \exists(t,\mathbf x)\in F,\quad 0\le v\le t,\quad |\mathbf z-\mathbf x|=t-v \right\}. \label{eq:NF} \end{align} The first is a union of initial null spheres. The second is the union of their truncated backward null cones. \begin{theorem}[Exact surface support for the flat realization] For \(f\in C_c^\infty(\Mint)\), \begin{equation} \fsupp B_f\cap\Sigma = \{0\}\times\Sset. \label{eq:exact-surface} \end{equation} \label{thm:surface-exact} \end{theorem} \begin{proof} The lower inclusion follows from Theorem~\ref{thm:sphere}. For the upper inclusion, every source \(a=-f\,\dd\phi\) has support in \(F\). The advanced representation above gives \(\supp b\cap\Mplus\subset\Nset\). The local differential operator \(C^*\) does not enlarge this support. The surface term has support in \(\{0\}\times\Sset\). The scalar derivative has support in \(F\), which is separated from \(\Sigma\). Thus all first-derivative distributions are supported in the fixed closed set \(\Nset\cup F\), whose intersection with \(\Sigma\) is \(\{0\}\times\Sset\). Their union and its closure cannot acquire another surface point. \end{proof} For this realization, the surface-ball alternative is decided exactly by~\eqref{eq:exact-surface}. A ball belongs to functional support precisely when it is contained in \(\Sset\). If a point \(\mathbf z_0\) of a proposed ball is outside \(\Sset\), the continuous function \[ d_F(\mathbf z) = \min_{(t,\mathbf x)\in F} \bigl|t-|\mathbf x-\mathbf z|\bigr| \] satisfies \(d_F(\mathbf z_0)>0\). Choose \(0<\varepsilon\frac{13}{20}. \] No backward null relation is possible. Therefore \(b=0\) throughout \(U\) for every source \(-f\,\dd\phi\), and the bulk and surface derivative terms both vanish there. The scalar derivative is supported in \(F\), disjoint from \(U\). Consequently \begin{equation} \fsupp B_f\cap U=\varnothing. \label{eq:explicit-gap} \end{equation} The time-zero ball associated with \((1,\mathbf0)\) contains \(B(\mathbf0,1/4)\). Thus the proposed universal solid-ball lower-support assertion is false for the flat realization. A standard wave support theorem does not repair this assertion: the sphere-versus-ball distinction is exactly the relevant three-dimensional propagation distinction~\cite{Ivrii}. \subsection{The scalar-slot alternative} \begin{proposition}[Observation support is visible in the scalar slot] For the flat realization, \[ F\subset\fsupp B_f. \label{eq:F-lower} \] Every nonempty open set meeting \(F\) contains scalar dependence at a suitable off-shell metric configuration. \end{proposition} \begin{proof} Choose an interior point \(x\) with \(f(x)\ne0\) in the given open set. Take \(w\) compactly supported in a smaller interior neighbourhood with \[ \dd w(x)=0,\qquad Pw(x)\ne0. \] For example, a temporal quadratic multiplied by a cutoff equal to one near \(x\) has these properties. Set \[ h^{\alpha\beta}=\eta^{\alpha\beta}Pw. \] Equation~\eqref{eq:C-flat} gives \[ Ch=-P\,\dd w. \] Because \(w\) and all its derivatives have zero initial data, uniqueness of the zero-Cauchy problem gives \[ Lh=-\dd w. \] The scalar derivative is therefore \[ \mathrm D_\phi B_f[h,\phi][\dot\phi] = \int f\,\partial^\mu w\,\partial_\mu\dot\phi = -\pair{\partial_\mu(f\partial^\mu w)}{\dot\phi}. \] At \(x\), its smooth kernel equals \(-f(x)Pw(x)\ne0\). It is detected by a scalar variation in the prescribed open set. If \(f\) is complex, the nonzero complex kernel is still detected by a real test, or by its real and imaginary decomposition. Taking closures gives all of \(F\), including its boundary points. Configurations with \(Lh=0\) do not defeat this argument: functional support uses all configurations, and the displayed configuration is a detector. \end{proof} \section{The geometric bridge and the missed-ball branch} The preceding gap does not provide a two-way pair. Null spheres alone already obstruct the relevant common-past geometry. \begin{lemma}[Sphere--ball intersection] Let \(t>0\), \(s\ge0\), and \(R=|\mathbf x-\mathbf y|\). Then \[ \{|\mathbf z-\mathbf x|=t\} \cap\{|\mathbf z-\mathbf y|\le s\}\ne\varnothing \quad\Longleftrightarrow\quad t-s\le R\le t+s. \label{eq:sphere-ball} \] \end{lemma} \begin{proof} The smallest distance from \(\mathbf y\) to the sphere centered at \(\mathbf x\) is \(|R-t|\), including \(R=0\). The claimed intersection is therefore equivalent to \(|R-t|\le s\). \end{proof} \begin{theorem}[Full exclusion from sphere and observation support] Suppose \[ \fsupp W_g=G,\qquad F\subset\fsupp B_f, \] and every initial null sphere from each positive-time point of \(F\) lies in \(\fsupp B_f\). If both actual orderings hold, then \begin{equation} |\mathbf x-\mathbf y|>t+s \quad\text{for every }x\in F,\ y\in G. \label{eq:full-exclusion} \end{equation} \label{thm:geometric} \end{theorem} \begin{proof} Suppose \(R\le t+s\). If \(t=0\), then \(R\le s\) makes \(x\in J^-(y)\). Since \(x\in\fsupp B_f\) and \(y\in\fsupp W_g\), \(B_f\gtrsim W_g\) fails. Let \(t>0\). If \(R0. \] Thus \(R\) is bounded away from zero and both direct-cone deltas are disjoint from the observation support. IX-E-2's Fourier-defined kernel vanishes in the open region \(R>t+s\); its common-regulator proof also supplies smooth time dependence with values in spatial distributions. The margin gives the same zero restriction at any initial-time boundary met by these smooth tests. There is no additional apex or boundary contact to assign. Hence \(fg\Imag K=0\). If either test is zero, the conclusion is immediate. \end{proof} The missed region~\eqref{eq:explicit-gap} therefore cannot carry a certified two-way counterexample in the open commutator slab. For any observation pair in that slab, a retained null-sphere point, possibly outside the missed solid-ball interior, lies in the causal past of \(y\). This is the required ordering certificate against the proposed control. The control is nonzero, but at least one ordering fails. \subsection{A minimal conditional bridge} The scalar-slot lower theorem is useful for full exclusion~\eqref{eq:full-exclusion}, but it is stronger than needed to annihilate the commutator. \begin{theorem}[Bridge from the boundary-trace certificate alone] Assume \textup{(TC)}, the copied sharp adjoint, the copied Fourier-defined kernel properties, and unrestricted configurations. Then \textup{(SB)} holds. \label{thm:minimal} \end{theorem} \begin{proof} Theorem~\ref{thm:sphere} gives the initial null spheres and \(F\cap\Sigma\) as lower supports. Proposition~\eqref{eq:Wexact} gives \(\fsupp W_g=G\). For \(t>0\), every observation pair in \[ |t-s|\le R\le t+s \] satisfies \(t-s\le R\le t+s\). Its initial sphere therefore meets the initial ball of \(y\). The corresponding actual support point of \(B_f\) lies in \(J^-(y)\), contradicting \(B_f\gtrsim W_g\). For \(t=0\), the closed slab reduces to \(R=s\); the lower support \(x\in F\cap\Sigma\) gives the same contradiction. Thus \(F\times G\) misses the closed commutator slab, including its direct cone, reflected boundary, and spatial apex. The copied Fourier support statement and its boundary continuation imply \(fg\Imag K=0\). No scalar controllability assumption is needed for this narrower conclusion. \end{proof} This identifies the exact sufficient certificate whose verification would complete the inherited bridge. \section{Conjugate transfer and fixed-domain obstruction} Let \(S(d)=\star_c(d)\) and \[ J(d)=S(d)-S(\sigma d). \] The inherited reversal identity gives \begin{equation} J(\gamma d)=-J(d)^*. \label{eq:conjugate-J} \end{equation} Conjugation preserves actual functional support on the unrestricted real configuration space: a dependence vanishes exactly when its conjugate vanishes. Consequently any proved original-sector bridge clears the double-ordering commutator test on the declared conjugates and flips. For the real-coefficient flat realization, \(B_f^*=B_{\overline f}\) and \(W_g^*=W_{\overline g}\); the distributional statement itself also applies directly to those tests. This transfer does not prove the bridge on the original pairs. It also does not evaluate relations among sums of actual tensors. \begin{proposition}[Retained-pair obstruction criterion] If a retained pair has both actual orderings and \(\Cfg\ne0\), no symmetric causally factorizing map \(\Ttwo:\Dtwo\to\Vthree\) exists. \end{proposition} \begin{proof} Factorization and symmetry would require \[ B_f\star_cW_g =\Ttwo(B_f,W_g) =\Ttwo(W_g,B_f) =W_g\star_cB_f. \] This contradicts~\eqref{eq:commutator}. A higher coefficient or contact cannot equalize unequal prescribed free values. \end{proof} No pair satisfying this proposition's premises is established on the supplied inherited data. \section{Supplied-data audit and decision} The logical status is now precise. The copied adjoint specifies how an already determined \(b=\Ezero^*a\) enters the derivative. It does not give \(b|_\Sigma\). The copied upper bound likewise does not give that trace. Finally, a mixed covariance formula is not, without a stated covariance-to-Green-operator identification, a formula for \(\Ezero^*\) on every source \(-f\,\dd\phi\). The flat realization proves that the proposed universal solid-ball assertion is not a safe substitute for this missing information. It also proves that solid-ball support is unnecessary: the null-sphere certificate is sufficient. \begin{longtable}{ >{\raggedright\arraybackslash}p{0.45\textwidth} >{\raggedright\arraybackslash}p{0.45\textwidth}} \toprule Evidence & Release\\ \midrule \endhead Inherited operator verified to satisfy \textup{(TC)} & Bridge proved by Theorem~\ref{thm:minimal}, including conjugate transfer; overall causal-product release remains Unresolved.\\ \addlinespace Certified retained two-way pair with nonzero actual commutator & Proved obstructed on the frozen \(\Dtwo\); stop.\\ \addlinespace Only an upper propagation ball and the adjoint decomposition are supplied & Unresolved at the trace-detection certificate for \(\Ezero^*(-f\,\dd\phi)\).\\ \addlinespace An open solid-ball gap is found & Not by itself an obstruction. Its null-sphere ordering consequences must be checked.\\ \addlinespace Only IX-E-2's strongly separated zero class is recomputed & No new bridge result.\\ \bottomrule \end{longtable} \begin{theorem}[Audited release] For the explicitly specified flat massless zero-Cauchy realization, the bridge is proved. For the inherited operator on the supplied prerequisite information alone, the release is \textbf{Unresolved at verification of \textup{(TC)}}. Neither a proved fixed-domain obstruction nor a constructed time-ordered product is released. \end{theorem} \begin{proof} The first statement follows from the verified trace, Theorem~\ref{thm:minimal}, and conjugate transfer. For the inherited operator, the supplied statements do not include the defining trace required by that theorem. The conditional calculation cannot identify two operators merely because both are denoted by \(\Ezero\). No alternative complete lower-support proof or certified two-way counterexample is supplied. This is the named Unresolved outcome. \end{proof} The outstanding request is specific: provide the defining Cauchy equation and initial conditions, or the equivalent boundary-trace formula, for the inherited \(\Ezero\). If it is~\eqref{eq:E-flat}, the proof above completes the bridge without reopening the kernel evaluation. \section{Retained boundaries and conclusion} Even a proved bridge leaves the actual-tensor relation test, coefficient and descendant admission, named fixed-support continuity, exact old overlap, contact equations, analytic involution, and conditional subtraction obligations unchanged. No replacement distribution is declared to be the sole remaining task. V's external \(R_{1,1}\) and Newton--wave cubic, VII-D's numerical finite part, VI's compact-switch defect, VII-A's plateau intertwiner, and the VII-E/F and VIII certificates remain non-identifications. No conical current or token is identified with the present commutator. The free-sector descent and word coherence remain closed. F9a is not promoted to hypocontinuity. VII-C remains in force under its joint hypotheses (E1)--(E4). G2 remains unopened. No finite-part subtraction is renamed as a product, and no \(R_1\), \(R_2\), \(\mathsf T_3\), or Feynman replacement is begun. The central correction is mathematical rather than terminological: in the flat massless realization, actual initial support is generated by null spheres, and solid-ball interiors can be invisible to every configuration. Nevertheless, those spheres meet the initial ball of every observation pair in the closed commutator slab. They therefore prohibit the relevant ordering and prove the bridge. For the inherited operator, the remaining burden is not another witness or kernel calculation. It is verification of the displayed boundary-trace certificate. Keeping that identification explicit is necessary to preserve the fixed target and the honest release status. \begin{thebibliography}{9} \bibitem{IXE} \emph{Landmark IX-E: Binary time ordering and causal subtraction on the declared sharp comparison sector}. Supplied manuscript \texttt{IX-E.tex}. Used for the fixed-domain obstruction and the distinction between the commutator gate and the complete causal interface. \bibitem{IXE2} \emph{Landmark IX-E-2: Exact imaginary part of the locked Cauchy kernel and the first-stage commutator on the retained two-way sector}. Supplied manuscript. Used for the evaluated kernel, commutator, compact detection, witness zero, observation-spacelike control, derivative-support characterization, and conjugate reversal identity. \bibitem{VIIB} \emph{Landmark VII-B}. Sharp adjoint statement as quoted in the supplied IX-E-3 specification. Only the displayed adjoint and surface coefficient are imported. An unprovided defining kernel for the inherited \(\mathsf E_0\) is not reconstructed. \bibitem{Ivrii} V.~Ivrii, \emph{Partial Differential Equations}, Chapter 9, Section 9.1, ``Wave equation in dimensions \(3\) and \(2\).'' University of Toronto online lecture text. Used for the standard flat Kirchhoff realization and its sphere-supported propagation, not to identify an unspecified internal operator. \end{thebibliography} \end{document}