% ix-e.tex % Compile with: pdflatex ix-e.tex % Run pdflatex a second time to resolve cross-references. % % This manuscript proves the consistency and conditional-subtraction % results available from the supplied premises. Its release outcome is % explicitly Unresolved: no unspecified kernel, old product, or analytic % extension is fabricated. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1in]{geometry} \usepackage{array,booktabs,longtable} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, citecolor=blue, urlcolor=blue, pdftitle={Landmark IX-E: Binary time ordering and causal subtraction}, pdfauthor={} } \setlength{\emergencystretch}{3em} \newcommand{\Padm}{\mathscr P_{\mathrm{adm}}} \newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}} \newcommand{\Dfirst}{D_{\mathrm{first}}} \newcommand{\Dsecond}{D_{\mathrm{second}}} \newcommand{\Dnew}{D_{\mathrm{new}}} \newcommand{\Dold}{D_{\mathrm{old}}} \newcommand{\Ddecl}{D_{\mathrm{decl}}} \newcommand{\Dcmp}{D_{\mathrm{cmp}}} \newcommand{\Dtwo}{\mathcal D_2} \newcommand{\Otwo}{\mathcal O_2} \newcommand{\Oact}{\mathcal O_{\mathrm{act}}} \newcommand{\Ttwo}{\mathsf T_2} \newcommand{\Told}{\mathsf T_2^{\mathrm{old}}} \newcommand{\Rhat}{\widehat{\mathsf R}} \newcommand{\fsupp}{\operatorname{supp}_{\mathrm{fun}}} \newcommand{\Span}{\operatorname{span}_{\mathbb C}} \newcommand{\im}{\operatorname{im}} \newcommand{\Ker}{\operatorname{ker}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\mathrm{id}} \newcommand{\Athree}{\mathbb C[\hbar]/(\hbar^4)} \newcommand{\Atwo}{\mathbb C[\hbar]/(\hbar^3)} \newcommand{\Vthree}{\Padm\otimes_{\mathbb C}\Athree} \newcommand{\Vtwo}{\Padm\otimes_{\mathbb C}\Atwo} \newcommand{\Vplus}{\hbar\Vthree} \newcommand{\Qker}{\mathcal Q_{f,g}} \newcommand{\QRker}{\mathcal Q^R_{g,f}} \newcommand{\DK}{K-\overline K} \newcommand{\Rel}{\mathrm{Rel}} \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}\\ Binary time ordering and causal subtraction\\ on the declared sharp comparison sector} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We formulate the binary causal-extension problem on a fixed actual tensor domain inherited from IX-Free-2 and prove its algebraic consistency tests, its first-stage commutator identity, its double-ordering obstruction, an algebraic extension criterion, and the conditional unnormalized subtraction theorem. We correct the double-ordering geometry: two support orderings require causal disjointness, not overlapping functional supports. Neither full support of the locked contraction kernel nor injectivity of the first-stage maps determines the commutator. The release outcome is \emph{Unresolved}. The supplied information does not specify the restriction of \(K-\overline K\) needed to evaluate the decisive causal test, an old time-ordered map on a nonzero comparison subspace, or the sharp distributional extension and continuity data. Consequently this manuscript does not claim a constructed causal product or a universal obstruction. Its proved results identify exactly what a subsequent construction or no-go proof must discharge. Degree-four free descent is not reopened. \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) {Declared two-point domain}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Actual tensor map}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Full support bridge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Defining the map does not finish the bridge.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The tensor map is not the whole bridge. 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} \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) {Declared two-point domain}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Full support bridge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Defining the map does not finish the bridge.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The tensor map is not the whole bridge. 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} \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) {Declared two-point domain}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Actual tensor map}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Full support bridge}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Defining the map does not finish the bridge.}; \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} \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) {Declared two-point domain}; \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) {Full support bridge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Actual tensor map. Defining the map does not finish the bridge.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Declared two-point domain is not Full support bridge. 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} \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: Declared two-point domain}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Actual tensor map}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Full support bridge}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Defining the map does not finish the bridge.}; \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} \end{figure} Figure~\ref{fig:split-IX-E} records one view of the same separation. Figure~\ref{fig:chain-IX-E} records one view of the same separation. Figure~\ref{fig:pair-IX-E} records one view of the same separation. Figure~\ref{fig:gate-IX-E} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-IX-E} records that separation. The drawing adds no hypothesis. \section{Release contract and corrections} \subsection{Scope and legal outcomes} The target is a complex-linear actual tensor map \[ \Ttwo:\Dtwo\longrightarrow\Vthree, \qquad \Ttwo(d)=\sum_{j=0}^{3}\hbar^jT_j(d), \qquad T_j(d)\in\Padm, \] on one fixed flip-stable comparison domain. An ``actual'' tensor means an element of the algebraic tensor product of actual functionals, rather than an unquotiented presentation by test functions. There are three legal outcomes: \begin{enumerate} \item \textbf{Constructed:} all coefficients exist and every declared interface clause holds on the fixed domain. \item \textbf{Proved obstructed:} a contradiction excludes every map satisfying that interface on the fixed domain. \item \textbf{Unresolved:} at least one required consistency, extension, admission, contact, continuity, or overlap obligation remains undischarged. \end{enumerate} Failure of one proposed replacement kernel excludes that candidate only. Removing an offending tensor changes the target and does not discharge a claim covering that tensor. \begin{center} \fbox{\begin{minipage}{0.92\textwidth} \textbf{Release box.} The domain chosen below contains all of \(D_{\mathrm{first}}\) and its conjugates. No offending first-stage pair may subsequently be deleted. The outcome of the supplied-data audit is \textbf{Unresolved}. Free-word coherence does not imply causal factorization. No \(\mathsf T_3\), \(R_2\), general free-domain enlargement, G2 release, or repair of VII-C is asserted. \end{minipage}} \end{center} The inherited hard stop is retained: the causal-product release is Unresolved because the time-ordered extension and its remaining causal, admission, and overlap obligations are unfinished, not because the degree-four free word has failed to descend. \subsection{Corrections required before the gates} \paragraph{A. Injectivity is not an ordering comparison.} The inherited equations \[ \Ker q_L=\Ker q_R=\{0\}, \qquad \supp K=M_+^2, \] and the slab witness \(\QRker\neq0\) imply \[ W_g\star_cB_f\neq W_gB_f. \] They do not imply \(\Qker\neq\QRker\). For example, two injective linear maps can be the same map. The difference of the two contractions must be computed independently. \paragraph{B. Contacts cannot repair inconsistent prescribed values.} If both support orderings hold, symmetry and factorization require the new value to equal both fixed free values. If those values are unequal, no alteration of the new value can satisfy both equations. This argument applies even on the permitted contact set. \paragraph{C. Fixed old coefficients must agree exactly.} A contact is normalization freedom only where normalization is unfixed. A contact discrepancy from a supplied, fixed old coefficient remains a failed overlap until it is removed by a permitted normalization preserving all other conditions. \paragraph{D. Double ordering means causal disjointness.} The sentence that a two-way pair requires overlapping functional supports is incorrect. With the ordering used here, both orderings exclude every causal relation between the two supports, including coincidence and null relation. Proposition~\ref{prop:geometry} proves the precise statement. \paragraph{E. A time slab proves an ordering, not necessarily its uniqueness.} A later slab proves \(W_g\gtrsim B_f\). It does not, by itself, rule out \(B_f\gtrsim W_g\): supports in different time slabs can still be spacelike separated. Uniqueness of the ordering needs an actual causal relation between actual functional supports. \section{Prerequisites and entering state} \subsection{Inherited objects, rather than reconstructed sources} The following are imported premises. They are not proved anew in IX-E. The actual definitions of the sharp functional carrier, \(C\), \(\mathsf E_0\), the locked kernel \(K\), and the old products belong to the prerequisite sources. The labels C-I1--C-I9 are retained as inherited free-sector hypotheses. Their individual wording was not included in the supplied text; no purported verbatim reconstruction of those clauses is made here. The proofs below state the particular inherited properties they use explicitly. \begin{longtable}{ >{\raggedright\arraybackslash}p{0.25\textwidth} >{\raggedright\arraybackslash}p{0.31\textwidth} >{\raggedright\arraybackslash}p{0.34\textwidth}} \toprule Object & Entering status & IX-E use\\ \midrule \endhead \(\star_c\) on \(\Ddecl\) & Constructed in IX-Free-2, relative to its premises & Named free product, not time ordering\\ Both associations of \(B_f,W_g,B_{f'}\) & Constructed through \(\hbar^3\) & Free data only\\ \(\Ker p_L=\Ker p_R=\{0\}\) & Evaluated & Degree-four descent remains closed\\ \(\mathcal C_{\mathrm{desc}} =\mathcal C_{\mathrm{glue}}=\mathbb R\) & Evaluated & Not a causal normalization\\ F9a & Fixed-support families only & Not F9b or hypocontinuity\\ New \(\Ttwo,R_1,R_2\) & Absent & Binary construction obligation\\ VII-C under its hypotheses (E1)--(E4) & Retained & Not repaired by \(U_c\)\\ G2 & Untouched & Not opened by binary time ordering\\ \bottomrule \end{longtable} The word ``new'' distinguishes the absent sharp extension from any old time-ordered product that may separately be supplied. \subsection{Free domain and word} The inherited algebraic tensor domains are \begin{align} \Dfirst &= (\mathscr B\otimes\mathscr W) +(\mathscr W\otimes\mathscr B),\\ \Dsecond &= (\mathscr L\otimes\mathscr B) +(\mathscr B\otimes\mathscr R),\\ \Dnew&=\Dfirst+\Dsecond,\qquad \Dold=\Pmc\otimes\Pmc,\\ \Ddecl&=\Dnew+\Dold,\qquad \Dnew\cap\Dold=\{0\}. \end{align} The two associations of the inherited word agree with \begin{align} \mathcal A_c(f,g,f') ={}&B_fW_gB_{f'} +\hbar(\mathsf G_{12}+\mathsf G_{13}+\mathsf G_{23}) \notag\\ &+\hbar^2( \mathsf G_{12,23}+\mathsf G_{12,13}+\mathsf G_{13,23}) +\hbar^3\mathcal N_c(f,g,f')\,1. \label{eq:word} \end{align} Equation~\eqref{eq:word} is free-word coherence. It supplies neither a binary time-ordered extension nor causal factorization. No closure under intermediate multiplication is imported. In particular, an arbitrary expression \[ (F\star_cG)\otimes H \] need not belong coefficientwise to the declared free domain. \subsection{Conjugation conventions} Let \(\sigma(F\otimes G)=G\otimes F\) be complex-linear flip. Let \[ \gamma(F\otimes G)=F^*\otimes G^* \] be componentwise antilinear conjugation. Then \[ d^\dagger=\sigma\gamma d,\qquad \sigma^2=\gamma^2=\id,\qquad \sigma\gamma=\gamma\sigma. \] Conjugation fixes the formal parameter \(\hbar\). The inherited reversal property is \begin{equation} \Ddecl^\dagger=\Ddecl, \qquad \star_c(d)^*=\star_c(d^\dagger). \label{eq:free-star} \end{equation} The comparison domain is \[ \Dcmp=\Ddecl\cap\sigma(\Ddecl). \] Thus both free orderings exist on \(\Dcmp\). Flip stability of all of \(\Ddecl\) is not assumed. \subsection{Admission and support} Admission includes the IX-A requirement \begin{equation} \bigl(\mathrm D_{\boldsymbol\eta}F\bigr)'_h(\psi) \in\mathscr S_1, \qquad \mathscr S_1=\mathscr M_1+\mathscr Q_\Sigma, \label{eq:admission} \end{equation} at every smooth configuration and for every finite list of required directions. The admitted coefficient space is treated as a complex vector space stable under conjugation. Admission of required descendants is part of the coefficient requirement, not an automatic consequence of a pairing existing. The sharp adjoint identity is \begin{equation} C^*(\theta b) =\theta C^*b+\delta(t)\mathcal B(b|_\Sigma). \label{eq:sharp} \end{equation} Dropping its second term changes the functional; it is not a presentation rewrite. Functional support is the closed support of configuration dependence. The imported bounds are \begin{align} \fsupp B_f &\subset P_F\cup F, &P_F&=J^-(F)\cap\{t\ge0\}, &F&=\supp f,\\ \fsupp W_g&\subset G, &&& G&=\supp g. \label{eq:support-bounds} \end{align} These are upper bounds. Their nonempty intersections do not prove nonempty intersections of the actual functional supports. \section{E0: a fixed comparison domain} No nonzero old comparison subspace carrying an explicitly supplied old time-ordered map is present in the supplied data. The only old seed space on which a map is determined without additional information is \[ \Otwo=\{0\}, \] with its unique zero map. We therefore freeze the minimal full first-stage target \begin{equation} \Dtwo =\Span\{\sigma^a\gamma^b d: d\in\Dfirst+\Otwo,\ a,b\in\{0,1\}\}. \label{eq:fixed-domain} \end{equation} This choice does not claim a nonzero old time-ordered overlap. Any actual old intersection must still be audited: \begin{equation} \Oact=\Dtwo\cap\Dold. \label{eq:actual-old} \end{equation} One must not infer \(\Oact=0\) merely from \(\Dnew\cap\Dold=0\): conjugation saturation needs its own subspace information. \begin{proposition}[Domain stability]\label{prop:domain} Under~\eqref{eq:free-star}, the domain~\eqref{eq:fixed-domain} satisfies \[ \Dtwo\subset\Dcmp,\qquad \sigma(\Dtwo)=\Dtwo,\qquad \gamma(\Dtwo)=\Dtwo,\qquad \Dtwo^\dagger=\Dtwo. \] It contains every tensor in \(\Dfirst\). \end{proposition} \begin{proof} By definition, \(\sigma(\Dfirst)=\Dfirst\). Since \(\Dfirst\subset\Ddecl\), this implies \(\Dfirst\subset\Dcmp\). Also \[ \sigma(\Dcmp) =\sigma(\Ddecl)\cap\Ddecl=\Dcmp. \] Because dagger commutes with flip and preserves \(\Ddecl\), \[ \Dcmp^\dagger =\Ddecl\cap\sigma(\Ddecl)=\Dcmp. \] Consequently \(\gamma=\sigma\dagger\) preserves \(\Dcmp\). Every generator in~\eqref{eq:fixed-domain} therefore belongs to \(\Dcmp\). Applying either involution permutes these generators; antilinearity of \(\gamma\) preserves their complex span. \end{proof} \begin{center} \begin{tabular}{p{0.29\textwidth}p{0.60\textwidth}} \toprule Domain item & Decision\\ \midrule First-stage pairs & All retained\\ Flips and conjugates & All retained\\ Old seed \(\Otwo\) & Zero; no nonzero supplied old map\\ Actual old intersection & \(\Oact\); not evaluated by the excerpt\\ Intermediate products & No closure asserted\\ Second-stage binary pairs & Not added unless already in the span\\ \bottomrule \end{tabular} \end{center} The interface is classical multiplication, symmetry, coefficient and descendant admission, actual-presentation independence, exact specified old overlaps, causal factorization, anti-time-ordering involution, and one named fixed-support continuity statement. A nonzero old seed may be added only by a separately declared change of target, accompanied by its actual old map. \section{E1: ordered-data consistency} \subsection{Correct support geometry} \begin{definition} For actual functionals \(F,G\), set \[ F\gtrsim G \quad\Longleftrightarrow\quad \fsupp F\cap J^-(\fsupp G)=\varnothing. \] The causal relations include their null boundaries and coincidence. \end{definition} \begin{proposition}[Double-ordering geometry]\label{prop:geometry} Let \(A=\fsupp F\) and \(B=\fsupp G\). Both \(F\gtrsim G\) and \(G\gtrsim F\) hold exactly when no point of \(A\) is causally related in either direction to a point of \(B\). In particular, \(A\cap B=\varnothing\). \end{proposition} \begin{proof} The first condition excludes pairs \(a\in A,b\in B\) with \(a\in J^-(b)\). The second excludes pairs with \(b\in J^-(a)\). These are precisely the two directions of causal relation. If \(a=b\), reflexivity gives both relations, so coincidence is excluded as well. \end{proof} A null-related pair in the actual supports violates at least one ordering condition. An ordered prescription cannot be extended to such a pair merely by taking a limit. Conversely, sufficiently separated spacelike supports satisfy both orderings and are part of the consistency test, not an unordered exceptional region. \begin{lemma}[A spacelike sharp comparison region]\label{lem:spacelike} In the flat half-space \(M_+=[0,\infty)\times\mathbb R^n\), suppose \(f,g\) are supported at times in \([0,T]\), with spatial supports in balls \(\overline B(x_0,r)\), \(\overline B(y_0,s)\), respectively. If \[ |x_0-y_0|>2T+r+s, \] then the upper support sets in~\eqref{eq:support-bounds} are causally disjoint. Hence both \(B_f\gtrsim W_g\) and \(W_g\gtrsim B_f\) hold. \end{lemma} \begin{proof} A point of \(P_F\cup F\) has time in \([0,T]\) and spatial distance at most \(T+r\) from \(x_0\). A point of \(G\) has spatial distance at most \(s\) from \(y_0\). Their spatial separation is therefore greater than \(T\), whereas their time separation is at most \(T\). They are strictly spacelike. The actual functional supports are subsets of these sets. \end{proof} Thus two-way first-stage pairs can be specified without knowing exact functional supports. What remains unknown is the locked kernel difference on their observation supports. For the inherited slab witness, the bounds \[ P_F\cup F\subset\{0\le t\le a\}, \qquad G\subset\{t>b\}, \qquad b>a, \] give \(W_g\gtrsim B_f\): no later point can lie in the causal past of an earlier point. They do not alone decide the reverse ordering. \subsection{The complete first-stage commutator calculation} The locked first-stage coefficients are \begin{align} \Qker &=-2\left\langle f(x)g(y)K(x,y), \partial_t\phi(x)h_{00}(y) \right\rangle,\\ \QRker &=-2\left\langle g(y)f(x)\overline{K(x,y)}, h_{00}(y)\partial_t\phi(x) \right\rangle. \label{eq:locked-q} \end{align} The brackets denote the supplied distributional pairing, including its configuration arguments. Classical configuration factors commute; no interchange of singular distributions is being made. \begin{proposition}[First-stage commutator]\label{prop:commutator} On the declared first-stage sector, \begin{align} B_f\star_cW_g-W_g\star_cB_f &=\hbar\mathcal C_{f,g},\\ \mathcal C_{f,g} &=-2\left\langle f(x)g(y)\bigl(K-\overline K\bigr)(x,y), \partial_t\phi(x)h_{00}(y) \right\rangle \label{eq:commutator}\\ &=-4i\left\langle f(x)g(y)\operatorname{Im}K(x,y), \partial_t\phi(x)h_{00}(y) \right\rangle. \notag \end{align} \end{proposition} \begin{proof} Subtract the two inherited first-stage free products. Their classical terms agree. Reorder only the smooth configuration factors and the scalar test factors in the second expression of~\eqref{eq:locked-q}. Distributional linearity gives \eqref{eq:commutator}. Finally, \(K-\overline K=2i\operatorname{Im}K\). \end{proof} Equation~\eqref{eq:commutator} is the calculation determined by the supplied kernel data. A further evaluation requires a formula, conjugation identity, or support theorem for \(\operatorname{Im}K\). Full support of \(K\) says nothing of this kind: a real distribution can have full support while \(K-\overline K=0\). \begin{lemma}[Compact detection]\label{lem:detection} Let \[ D_{f,g}=f(x)g(y)(K-\overline K)(x,y) \] be a nonzero compactly supported distribution whose support is contained in the interior of \(M_+^2\). Suppose the configuration space permits arbitrary smooth scalar and metric configurations. Then \(\mathcal C_{f,g}\neq0\) as an actual functional. \end{lemma} \begin{proof} There are compactly supported smooth functions \(u(x),v(y)\) with \[ \langle D_{f,g},u\otimes v\rangle\neq0. \] Indeed, finite sums of tensor-product test functions are dense in the test-function space on a product. This can be seen by a partition of unity into product charts followed by smooth Fourier approximation on compact rectangles. Continuity of a distribution would force it to vanish if it vanished on every such tensor. Extend \(u\) smoothly by zero where necessary and choose \[ \phi(t,\boldsymbol x) =\int_{t_*}^{t}u(s,\boldsymbol x)\,\dd s. \] Then \(\partial_t\phi=u\). Choose the smooth metric configuration with \(h_{00}=v\) and all other components zero. Equation~\eqref{eq:commutator} gives a nonzero value. If only real configurations are used, decomposition into real and imaginary parts supplies real \(u,v\) detecting the complex distribution. \end{proof} No compactness of \(\phi\) is required: compactness belongs to the functional kernel, while configurations are smooth. If the configuration space is constrained, the realization assumption in this lemma must instead be proved within that space. \begin{theorem}[Fixed-domain obstruction]\label{thm:obstruction} If \(B_f\otimes W_g\in\Dtwo\), both support orderings hold, and \(\mathcal C_{f,g}\neq0\), no symmetric causally factorizing map \(\Ttwo\) exists on \(\Dtwo\). \end{theorem} \begin{proof} Factorization and symmetry would give \[ B_f\star_cW_g =\Ttwo(B_f,W_g) =\Ttwo(W_g,B_f) =W_g\star_cB_f. \] This contradicts Proposition~\ref{prop:commutator}. The nonzero coefficient at order \(\hbar\) remains nonzero in \(\Athree\). The argument is independent of every contact choice and every higher coefficient. \end{proof} Theorem~\ref{thm:obstruction} is conditional, not an evaluated obstruction. Lemma~\ref{lem:spacelike} supplies geometric test regions; the supplied excerpt does not evaluate \(K-\overline K\) there. \begin{proposition}[Classical multiplication is not a solution] The symmetric candidate \(\Ttwo^{\mathrm{cl}}(F,G)=FG\) fails factorization on the inherited ordered slab witness. \end{proposition} \begin{proof} On that witness \(W_g\gtrsim B_f\) and \[ W_g\star_cB_f=W_gB_f+\hbar\QRker, \qquad \QRker\neq0. \] The candidate has no such coefficient. \end{proof} This proves the copied separation statement: free coherence, including~\eqref{eq:word}, does not itself furnish causal factorization. \section{Actual tensor relations and an algebraic criterion} Double-ordering tests are necessary but not sufficient. Every actual linear relation among prescribed tensors must also be respected. This obligation is distinct from the already closed degree-four descent. Write \(S(d)=\star_c(d)\) on \(\Dtwo\) and define \begin{equation} A(d)=\frac12\bigl(S(d)+S(\sigma d)\bigr). \label{eq:A} \end{equation} The inherited classical coefficient of \(A\) is multiplication. Equation~\eqref{eq:free-star} implies \[ A(\sigma d)=A(d),\qquad A(\gamma d)=A(d)^*. \] Let \(\{(d_i,t_i)\}_{i\in I}\) be the collection of all prescribed values: \(t_i=S(d_i)\) for every ordered pair, together with \(t_i=\Told(d_i)\) for every tensor on a supplied old overlap. The index set need not be finite. Assume these values and their required descendants are admitted, and their classical coefficients equal multiplication. Put \[ x_i=t_i-A(d_i)\in\Vplus. \] Let \[ Q=\Dtwo/\im(\id-\sigma),\qquad [d]\in Q. \] Conjugation induces an antilinear involution on \(Q\). \begin{theorem}[Algebraic consistency criterion]\label{thm:algebraic} There exists a complex-linear map \(\Ttwo:\Dtwo\to\Vthree\) satisfying all prescribed values, classical multiplication, symmetry, and the binary involution identities if and only if the following finite-relation condition holds: \begin{equation} \sum_{i\in I_0}a_i[d_i] +\sum_{i\in I_0}b_i[\gamma d_i]=0 \quad\Longrightarrow\quad \sum_{i\in I_0}a_ix_i -\sum_{i\in I_0}b_ix_i^*=0 \label{eq:relation-test} \end{equation} for every finite \(I_0\subset I\) and every complex choice of coefficients. This is an algebraic theorem. It does not assert fixed-support continuity or identify a distributional time-ordering construction. \end{theorem} \begin{proof} Write \(\Ttwo=A+X\). Symmetry is equivalent to \(X(\sigma d)=X(d)\), so \(X\) factors through \(Q\). The binary involution is equivalent to \[ X(\gamma d)=-X(d)^*. \label{eq:X-involution} \tag{5.3} \] For prescribed tensors this requires the assignments \[ [d_i]\longmapsto x_i,\qquad [\gamma d_i]\longmapsto -x_i^*. \] Any linear map must respect their finite relations, proving necessity of~\eqref{eq:relation-test}. Conversely, that condition defines a complex-linear map \(x\) on \[ U=\Span\{[d_i],[\gamma d_i]:i\in I\}\subset Q. \] The space \(U\) is conjugation-stable, and the assignments give \(x(\gamma u)=-x(u)^*\). Set \(y=ix\). Then \[ y(\gamma u)=y(u)^*. \] On the real fixed-point space \(U_{\mathbb R}=\{u\in U:\gamma u=u\}\), the map \(y\) takes values in the real fixed-point space of \(\Vplus\). Choose a real vector-space complement of \(U_{\mathbb R}\) in \(Q_{\mathbb R}\), extend \(y\) by zero there, and complexify. Every \(q\in Q\) has the decomposition \[ q=\frac{q+\gamma q}{2} +i\,\frac{q-\gamma q}{2i} \] into real fixed-point vectors. The extended \(y\) therefore remains conjugation-equivariant. Define \(X=-iy\), pulled back to \(\Dtwo\), and \(\Ttwo=A+X\). The range of \(X\) lies in \(\Vplus\), so the classical coefficient is unchanged. The construction gives symmetry, prescribed values, and~\eqref{eq:X-involution}. All outputs are finite linear combinations of admitted coefficient functionals. Linearity of the carrier conditions preserves their required descendants. \end{proof} For a two-way pair the quotient identifies \([d]\) and \([\sigma d]\), while the forced corrections have opposite signs. Condition~\eqref{eq:relation-test} therefore requires their common correction to vanish, reproducing Theorem~\ref{thm:obstruction}. The arbitrary complement in the proof is not a continuity argument. It can produce a discontinuous map even when every prescribed coefficient is admitted. The theorem consequently cannot be used to label the complete IX-E interface Constructed without the remaining analytic proof. \section{E2--E5: analytical obligations and contacts} \subsection{E2: replacement contractions} Only after consistency of the forced data is established should an unordered extension be proposed. For every contraction, the construction must specify: \begin{enumerate} \item the exact inherited free distribution and slot convention; \item the proposed time-ordered distribution; \item its differential identity and source-contact normalization; \item its restrictions to every ordered region; \item the singular set requiring extension; \item the residual actual functional after contraction. \end{enumerate} The flat contractions mentioned in the prerequisite sheet include \(K\) and \[ S_{\mu\nu}=\partial_\mu\partial_\nu\Delta_m^+. \] Their names do not specify a time-ordered replacement. In particular, replacing a differentiated contraction requires deciding whether differentiation is applied to a fundamental solution or whether a distinct differentiated time-ordering convention is used. The sharp metric slot must be derived from the actual functional derivative through \(L=\mathsf E_0C\), retaining \eqref{eq:sharp}. Time ordering by observation times alone is not justified by~\eqref{eq:support-bounds}. Nor may differentiation, sharp Cauchy inversion, or the sharp adjoint be silently commuted with time ordering. The standard perturbative algebraic framework distinguishes time-ordering on regular functionals from the renormalized extension needed for local functionals~\cite{FR}. Its Feynman-kernel formulas must be used with their distributional hypotheses~\cite{FR}. Those results do not, without a separate verification of hypotheses, construct this nonlocal sharp sector. A wavefront analysis must determine the actual unresolved set. Observation diagonals, the Cauchy surface, and the reflected branch of \(K\) are candidates until excluded. The binary target does not require importing the degree-four triangle solely because that triangle exists in the inherited word. \subsection{E3: admission and named continuity} For every coefficient, the proof must establish existence of its distributional pairings, compact configuration kernels, the metric derivative condition~\eqref{eq:admission}, all required descendants, and annihilation of actual-presentation kernels. Existence on a test family does not establish admission. In particular, an expression omitting the boundary contribution in~\eqref{eq:sharp} cannot be justified by referring to a different presentation of the same functional. For definiteness, a possible weak fixed-support target is the following. Give \(\Padm\) the locally convex topology generated by \[ F\longmapsto \left| \mathrm D_{\eta_1}\cdots\mathrm D_{\eta_r}F(\psi) \right| \] for all configurations and finite direction lists under consideration. For every named test-function presentation with supports in fixed compact sets, require each \(T_j\) to be continuous from the product of the corresponding \(C^\infty_K\) spaces into this topology. This is one explicit continuity strength; it is not asserted to follow from the supplied F9a statement. It neither claims normal-topology continuity on the full carrier nor hypocontinuity of multiplication. A different topology may be chosen, but it must be named and proved, rather than inferred from the absence of the joint VII-C assumptions. \subsection{E4: contact equations and old overlap} If a finite-scaling-degree point-extension theorem applies in dimension \(n\), uniqueness holds below degree \(n\); at higher degree the degree-preserving freedom is a finite sum of derivatives of delta of order at most \(\lfloor\operatorname{sd}(u)-n\rfloor\)~\cite{BF}. Application to a submanifold requires the corresponding transverse and microlocal hypotheses~\cite{BF}. No such hypotheses or degrees have been evaluated here for the sharp Cauchy or reflected loci. The parent identity \[ u_\mu^{\rho_a}-u_\mu^{\rho_b}=c_{ab}r \] classifies its triangle finite part. It does not classify binary time-ordered ambiguities. On every supplied old overlap one must compute \[ \Ttwo^{\mathrm{cand}}-\Told \] and solve for permitted contacts so that this difference is \emph{exactly zero}. A smooth nonzero spacelike discrepancy away from the named contact set is not a contact. All coefficients of \(\star_c\), all supplied old free and time-ordered coefficients, and the parent parameter \(c\) remain fixed. No subtraction inside \(\star_H\) is authorized. If source slots are included, differentiating \[ P_xH_F=c_F\delta \] shows why the simultaneous prescription \[ j=Ph,\qquad H_{jh}^{\mathrm{TO}}=0 \] cannot retain that fundamental-solution identity when \(c_F\neq0\). This excludes that simultaneous prescription; it does not exclude every possible sharp causal extension. \subsection{E5: involution is a separate equation} Define anti-time-ordering by \begin{equation} \overline{\Ttwo}(d) =S(d)+S(\sigma d)-\Ttwo(d). \label{eq:antitime} \end{equation} The required identity is \begin{equation} \Ttwo(d)^*=\overline{\Ttwo}(\gamma d). \label{eq:time-star} \end{equation} Using~\eqref{eq:A}, equations~\eqref{eq:antitime}--% \eqref{eq:time-star} are exactly \[ X(\gamma d)=-X(d)^*. \] Symmetry instead requires \(X(\sigma d)=X(d)\). The two equations are different. Contacts must solve the overlap, ordered-restriction, admission, symmetry, and involution equations simultaneously. Averaging a candidate is not proof that its old overlap or causal restrictions survive. The algebraic theorem isolates these constraints, but does not solve the sharp distributional contact problem. \section{E6: conditional unnormalized subtraction} The next result audits directly the binary implication quoted from IX-D. It does not presume that its hypotheses have been discharged. \begin{theorem}[Conditional causal subtraction]\label{thm:subtraction} Suppose \(\Ttwo\) is constructed on \(\Dtwo\) with admitted coefficients and required descendants, classical multiplication, factorization, exact supplied old overlaps, and \eqref{eq:antitime}--\eqref{eq:time-star}. Suppose the inherited free map has the corresponding admission and old-overlap properties. Define \begin{equation} \Rhat(d)=i\bigl(\Ttwo(d)-S(d)\bigr). \label{eq:Rhat} \end{equation} Then: \begin{enumerate} \item \(\Rhat(d)\in\Vthree\), including required descendants; \item its classical coefficient vanishes; \item its supplied old unnormalized retarded overlaps agree exactly; \item for \(d=V\otimes O\), it vanishes if \[ \fsupp O\cap J^+(\fsupp V)=\varnothing; \] \item \(\Rhat(d)^*=\Rhat(\gamma d)\). \end{enumerate} If both input maps satisfy the same named fixed-support continuity statement into a locally convex coefficient space, their difference satisfies that statement. \end{theorem} \begin{proof} Admission and descendant admission follow from linearity of the admitted carrier. The classical coefficients of the two maps are both multiplication, so they cancel. Exact old overlap follows by subtracting the two exact old values. The stated support condition is equivalent to \[ \fsupp V\cap J^-(\fsupp O)=\varnothing. \] Thus \(V\gtrsim O\), and factorization makes the two terms in~\eqref{eq:Rhat} equal. For conjugation, use~\eqref{eq:free-star} and \eqref{eq:time-star}: \begin{align*} \Rhat(d)^* &=-i\bigl(\Ttwo(d)^*-S(d)^*\bigr)\\ &=-i\bigl( S(\gamma d)+S(\sigma\gamma d)-\Ttwo(\gamma d) -S(\sigma\gamma d)\bigr)\\ &=i\bigl(\Ttwo(\gamma d)-S(\gamma d)\bigr) =\Rhat(\gamma d). \end{align*} Finally, addition and scalar multiplication are continuous in a locally convex vector space. No multiplication-continuity upgrade is used. \end{proof} The support assertion concerns the stated functional-support ordering of a pair. It is not a distributional support theorem for independently unsmeared argument slots. \subsection{Normalization audit} \begin{proposition}[Order loss under coefficient shifting] The linear coefficient-shift map \[ L:\hbar\Vthree\longrightarrow\Vtwo, \qquad L(\hbar N_1+\hbar^2N_2+\hbar^3N_3) =N_1+\hbar N_2+\hbar^2N_3 \] is well defined. Order-three numerator data do not determine an order-three normalized coefficient in \(\Vthree\). \end{proposition} \begin{proof} The coefficient expansion in \(\Vthree\) is unique, so \(L\) is well defined. Within \(\Vthree\), multiplication by \(\hbar\) annihilates every \(\hbar^3C\). Hence, for every \(C\in\Padm\), \[ \hbar\bigl(N_1+\hbar N_2+\hbar^2N_3+\hbar^3C\bigr) =\hbar N_1+\hbar^2N_2+\hbar^3N_3. \] The coefficient \(C\) is undetermined. Equivalently, \(\hbar\) has no inverse in \(\Athree\). \end{proof} If the conditional construction is completed, the clean normalized output is the coefficient shift \[ R_1=L(\Rhat)\in\Vtwo, \] with any phase convention stated explicitly. A zero choice for an additional \(\hbar^3\) coefficient is a truncation convention, not a coefficient derived from order-three numerator data. No normalized product is released here because \(\Ttwo\) has not been constructed. Higher retarded identities require higher time ordering and the relevant intermediate products. Consequently neither \(R_2\) nor \(\mathsf T_3\) is defined. \section{E7: decision, missing data, and exclusions} \subsection{Evaluated status} \begin{longtable}{ >{\raggedright\arraybackslash}p{0.18\textwidth} >{\raggedright\arraybackslash}p{0.72\textwidth}} \toprule Gate & Status on the fixed target\\ \midrule \endhead E0 & Domain containment and flip, conjugation, and reversal stability are proved. The old seed is explicitly zero. The actual old intersection and any required nonzero old map are not supplied.\\ E1a & The exact commutator identity is proved. Its kernel restriction on two-way sharp pairs is not evaluated.\\ E1b & A necessary and algebraically sufficient finite-relation criterion is proved. It is not evaluated for the complete forced data.\\ E1c & The corrected support geometry, later-slab implication, and a spacelike sufficient condition are proved.\\ E2 & No locked sharp time-ordered replacement distribution or extension is supplied. Construction remains unfinished.\\ E3 & No candidate coefficient family exists to which the admission and fixed-support continuity proofs can be applied.\\ E4 & Binary contact freedom and exact nonzero old-overlap equations are unevaluated.\\ E5 & The involution constraint is proved algebraically. It is not discharged for an analytic candidate.\\ E6 & The conditional subtraction and normalization audit are proved. Their construction hypotheses remain undischarged.\\ \bottomrule \end{longtable} \begin{theorem}[Release decision] Relative to the supplied prerequisite information, the IX-E release outcome on~\eqref{eq:fixed-domain} is \textbf{Unresolved}. No Constructed or Proved obstructed outcome is established. \end{theorem} \begin{proof} A Constructed outcome requires a map satisfying every declared clause. No sharp replacement distribution, admitted coefficient construction, complete old-overlap solution, or continuity proof has been supplied. A Proved obstructed outcome requires an evaluated incompatibility. Theorem~\ref{thm:obstruction} supplies such a criterion, but its nonzero kernel premise has not been established on a two-way pair. Likewise, no violation of~\eqref{eq:relation-test} has been evaluated. The ordered slab excludes classical multiplication only. Therefore the remaining obligations have precisely the third legal status. \end{proof} The earliest unevaluated causal calculation is the restriction of \[ f(x)g(y)(K-\overline K)(x,y) \] to admitted pairs for which the support geometry supplies both orderings, followed by the full actual-relation test. The exact locked kernel formula or an appropriate conjugation-support theorem is needed to finish it. Independently, a nonzero old-overlap claim requires the actual old map, not merely its name. \subsection{Retained non-identifications} The inherited product \(\star_c\) is not \(\Ttwo\) and is not \(R_1(V_1;B_f)\). The triangle scalar \(U_c\) is not a causal subtraction. V's \(R_{1,1}\), the oscillator coefficient, and the Newton--wave cubic remain different objects. VI's relational coefficient and VII-A's plateau intertwiner are not the map sought here. VII-E/F and VIII certificates have no supplied map into \(\Padm\). A cutoff or numerical modification of \(U_c\) supplies none of the missing order-\(\hbar\) and order-\(\hbar^2\) binary coefficients. The parent finite-part conversion also supplies no binary overlap classification. \subsection{G2 and VII-C remain closed} The compact-switch witness copied from IX-A and IX-D is \[ B_f[h+\mathcal K\xi,\phi]-B_f[h,\phi] =-\int f\,X_\xi\phi, \qquad g\mathcal K\xi=0, \] with nonzero dressing variation under its stated hypotheses. In the coupling expansion its nonzero order-\(\kappa\) coefficient cannot be canceled by a correction beginning at order \(\kappa^2\): coefficient extraction at order \(\kappa\) annihilates every such correction. Binary time ordering therefore does not open G2. VII-C remains in force under its original joint hypotheses (E1)--(E4), which are distinct from the gate labels of this paper. Subtracting a scalar finite part does not prove normal-topology admission continuity. F9a and separate pointwise continuity are not promoted to F9b or hypocontinuity. \section{Conclusion} The first-stage commutator is controlled by the imaginary part of the locked kernel, not by full support or injectivity. Double ordering means causal disjointness of functional supports. A nonzero commutator on any retained two-way pair is a universal obstruction on the fixed domain, and no contact can remove that contradiction without changing prescribed free data. If the prescribed values instead satisfy all actual relations, Theorem~\ref{thm:algebraic} supplies an algebraic extension criterion. Its arbitrary linear extension is not a substitute for a sharp distributional construction or continuity proof. Admission, contacts, old overlap, and continuity must still be established at the declared analytical strength. Once those obligations are discharged, Theorem~\ref{thm:subtraction} supplies the unnormalized binary retarded difference through \(\hbar^3\). Normalization shifts the determined order to \(\hbar^2\). With the information presently supplied, the honest release remains Unresolved; the free-sector descent and every stated external boundary remain unchanged. \begin{thebibliography}{9} \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428 [math-ph], 2012. Used for the distinction between regular time-ordering formulas and renormalized extensions; not invoked as an existence theorem for the sharp sector. \bibitem{BF} R.~Brunetti and K.~Fredenhagen, \emph{Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds}, Communications in Mathematical Physics \textbf{208} (2000), 623--661; arXiv:math-ph/9903028. Used only for extension principles under their stated scaling-degree and microlocal hypotheses. \bibitem{Internal} \emph{Internal prerequisite statements: IX-A, IX-Free-2, and IX-D}, as quoted in the supplied IX-E specification. The full source manuscripts, the individual wording of C-I1--C-I9, the locked formula for \(K\), and a nonzero old time-ordered comparison map were not supplied. Statements imported from these prerequisites are identified as premises rather than reconstructed proofs. \end{thebibliography} \end{document}