% !TeX program = pdflatex %===================================================================== % XII-A.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XII-A.tex % pdflatex -interaction=nonstopmode -halt-on-error XII-A.tex % % Revision: % - equation operator distinguished from action Hessian; % - separation uses reduced, componentwise wave propagators; % - free-shell descent uses smooth multisolutions, not mollification; % - one contraction orientation throughout; % - parity and ghost number included in admissible product data; % - Grassmann-valued evaluation specified; % - tree normalization and support sets made explicit; % - formal completion of the composite sector specified; % - ordering corrections and named-sector stability proved; % - detected observation class proved robust; % - regular ambient--slab time-slice identification proved. % % External mathematical input: % XI-G's obstruction theorem is stated explicitly and cited. % It is not independently rederived from the unavailable predecessor. % % No enlargement, state, interacting differential, time ordering, % quantum retarded product, sharp-width limit, or Einstein-subshell % no-go is claimed. % % No successful compilation or independent verification 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,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{inputtheorem}[theorem]{Input theorem} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \newcommand{\sig}{\sigma_\eps} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \newcommand{\Keq}{\mathsf K_{\mathrm{eq}}} \newcommand{\Kact}{\mathsf K_{\mathrm{act}}} \newcommand{\Pred}{\mathbf P} \newcommand{\Ered}{\mathbf E} \newcommand{\Ezero}{E_0} \newcommand{\Emass}{E_m} \newcommand{\Gact}{G_{\mathrm{act}}} \newcommand{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\Pmass}{P_m} \newcommand{\Cconf}{\mathscr C} \newcommand{\Csol}{\mathscr C_{\mathrm{sol}}} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Fsol}{\mathscr F^{\mathrm{sol}}_{\mathrm{mc,pol}}} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\Nreg}{\mathscr N_{\mathrm{reg}}} \newcommand{\Acl}{\mathscr A^{\mathrm{cl}}_{\mathrm{off}}} \newcommand{\Areg}{\mathscr A^{\mathrm{reg}}_{\mathrm{off},H}} \newcommand{\Afree}{\mathscr A_0} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\EalgP}{\mathscr E_{H'}} \newcommand{\Sect}{\mathscr S_H} \newcommand{\SectP}{\mathscr S_{H'}} \newcommand{\sfree}{s_0} \newcommand{\Bcal}{\mathcal B} \newcommand{\Tcal}{\mathcal T} \newcommand{\Cthree}{\mathcal C_3} \newcommand{\Obs}{\mathfrak O_{\mathrm G}} \newcommand{\Vp}{\widetilde V_+} \newcommand{\Vm}{\widetilde V_-} \newcommand{\Edist}{\mathcal E'} \newcommand{\Dtest}{\mathcal D} \newcommand{\alphad}{\alpha_d} \newcommand{\balphad}{\bar\alpha_d} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\formalideal}{\mathfrak m} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Alg}{Alg} \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 } \title{\textbf{Landmark XII-A}\\[0.4em] One graded polynomial carrier\\ for the finite-ramp multiplet,\\[0.2em] \large with ordering-stable composites\\ and a robust observation exclusion} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We construct a polynomial quantum carrier for the finite-ramp Einstein--scalar multiplet $(\gamma,\phi,c,\bar c,b)$ at fixed $\eps>0$. The equation operator, the action Hessian, and the Hadamard product matrix have distinct roles. Reduced solution coordinates provide a componentwise wave-propagator separation lemma. An oriented Hadamard smoothing lemma then proves that polynomials vanishing on the complete gauge-fixed free shell form a two-sided quantum ideal, without requiring microcausal equation primitives or a mollification argument. The regular quotient recovers the causal free-field algebra. Smooth admissible changes of product data yield continuous ordering isomorphisms. Their corrections on the quadratic insertion and the scalar-cubic tree are computed explicitly and lie in the regular subalgebra; consequently the named, formally completed composite sector is ordering-stable. The frozen free differential is checked in both ghost orderings and transported by conjugation. Continuity is stated at the established level: fixed-support kernel continuity, stagewise product hypocontinuity, a Hausdorff shell quotient, and global separate continuity. Global hypocontinuity and completeness are not released. A regular time-slice theorem identifies ambient and interior-slab free algebras. Finally, taking XI-G's observation-qualified obstruction as an explicit input theorem, we exclude the detected cubic shell classes for every admissible ordering. Continuous detection functionals make the detected class open and dense on every fixed-support test stage on which a detector is nonzero. The undetected set, the Einstein subshell, and any enlargement remain undecided. \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) {Finite-ramp multiplet}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Polynomial carrier}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Width sent to zero}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The carrier is constructed at fixed positive width, not at zero width.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The carrier is at fixed positive width. 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-XII-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Finite-ramp multiplet}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Width sent to zero}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The carrier is constructed at fixed positive width, not at zero width.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The carrier is at fixed positive width. 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-XII-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Finite-ramp multiplet}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Polynomial carrier}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Width sent to zero}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The carrier is constructed at fixed positive width, not at zero width.}; \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-XII-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Finite-ramp multiplet}; \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) {Width sent to zero}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Polynomial carrier. The carrier is constructed at fixed positive width, not at zero width.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Finite-ramp multiplet is not Width sent to zero. 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-XII-A} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Finite-ramp multiplet}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Polynomial carrier}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Width sent to zero}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The carrier is constructed at fixed positive width, not at zero width.}; \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-XII-A} \end{figure} Figure~\ref{fig:split-XII-A} records one view of the same separation. Figure~\ref{fig:chain-XII-A} records one view of the same separation. Figure~\ref{fig:pair-XII-A} records one view of the same separation. Figure~\ref{fig:gate-XII-A} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XII-A} records that separation. The drawing adds no hypothesis. \scopebox{Release and proof boundary}{% \textbf{Constructed:} the graded polynomial product, regular presentation, free-shell descent, frozen free differential, ordering transport, ordering-stable composite sector, the delimited continuity statements, observation robustness, and the regular ambient--slab identification. \smallskip \textbf{Proved obstructed, conditional on the stated XI-G input:} admission of the detected cubic classes into this carrier. \smallskip \textbf{Unresolved or not opened:} undetected observations, restriction to the Einstein subshell, global product hypocontinuity, completeness, and enlargement. Absence of an enlargement construction is not a no-go theorem. \smallskip The predecessor obstruction is copied, not reopened. No state, interacting differential, quantum retarded product, time-ordered product, loop construction, or sharp-width limit is introduced.} \tableofcontents %===================================================================== \section{Frozen conventions and the operator split} \label{sec:operators} %===================================================================== Let $M=\R^4$ with $\eta=\diag(1,-1,-1,-1)$, $\Box=\partial_t^2-\Delta_{\mathbf x}$, and $\Pmass=\Box+m^2$, $m\ge0$. Fix $T>0$, $\MT=[0,T]\times\R^3$, and $\MTo=(0,T)\times\R^3$. The ramp $\sig$ is smooth, has fixed width $\eps>0$, and vanishes for $t\le0$. The nonzero switch $g$ belongs to the frozen admissible class of XI-G; in particular its compact support lies in $\MTo$. Additional plateau hypotheses used by XI-G are part of that input, not inferred from compact support alone. There is one metric field and no second classical metric fluctuation. The species, parities, and ghost numbers are \[ \begin{array}{c|ccccc} A&\gamma&\phi&c&\bar c&b\\ \hline |A|&0&0&1&1&0\\ \mathrm{gh}(A)&0&0&1&-1&0 . \end{array} \] The symbols $f,g,\sig,\eta$ are external data, not field generators. Both $\hbar$ and $\kappa$ are even, ghost-number-zero formal parameters. Trace reversal, divergence, and the symmetrized gradient are \[ \PP r=r-\tfrac12\eta\,\tr_\eta r,\qquad D^\mu r=\partial_\nu(\PP r)^{\nu\mu},\qquad (\RR v)_{\mu\nu}=\partial_{(\mu}v_{\nu)} . \] Parentheses include the factor $\tfrac12$. With $A=2G_{\mathrm{lin}}$ and formal adjoint $D^*$, the frozen identities are \begin{equation} \PP^2=\id,\qquad A+2D^*D=-\Box\PP,\qquad \PP D^*=-\RR,\qquad D\RR=\tfrac12\Box,\qquad A\RR=0 . \label{eq:identities} \end{equation} They fix the normalization $\Ggr=-\PP E_0^{\mathrm R}$. \subsection{Equation geometry versus action normalization} The off-shell configuration space $\Cconf$ consists of smooth multiplets $\Phi=(\gamma,\phi,c,\bar c,b)$. Define the equation operator by \begin{equation} \Keq\Phi =(\Box\gamma,\Pmass\phi,\Box c,\Box\bar c,b-D\gamma). \label{eq:Keq} \end{equation} It is not asserted to be formally self-adjoint. Its kernel is the complete gauge-fixed free shell \begin{equation} \Csol =\{\Box\gamma=0,\ \Pmass\phi=0,\ \Box c=\Box\bar c=0,\ b=D\gamma\}. \label{eq:shell} \end{equation} The action Hessian is a different operator. In the reordered blocks $(\gamma,b)\oplus\phi\oplus(c,\bar c)$, \begin{equation} \Kact= \begin{pmatrix}A&2D^*\\2D&-2\id\end{pmatrix} \oplus(-\Pmass) \oplus \begin{pmatrix}0&-i\Box\\i\Box&0\end{pmatrix}. \label{eq:Kact} \end{equation} The ghost block is interpreted with the graded variational convention. Ordinary transpose identities for bosonic matrices are not substituted for graded transpose identities. Writing $\rho=b-D\gamma$, the first two action equations become \[ -\Box\PP\gamma+2D^*\rho=0,\qquad -2\rho=0. \] The scalar sign and the invertible ghost mixing do not change their kernels. Thus \begin{equation} \ker\Keq=\ker\Kact=\Csol . \label{eq:samekernel} \end{equation} For solution geometry only, use \begin{equation} Y=(\gamma,\phi,c,\bar c),\qquad JY=(\gamma,\phi,c,\bar c,D\gamma),\qquad \Pred=\diag(\Box,\Pmass,\Box,\Box). \label{eq:reduction} \end{equation} Then $J:\ker\Pred\to\Csol$ is an isomorphism. The variable $b$ remains independent off shell; $J$ is not imposed on the off-shell carrier. \subsection{Named inverses and causal brackets} Let $E_\mu^{\mathrm R/A}$ be the retarded/advanced inverse of $\Box+\mu^2$, and put $E_\mu=E_\mu^{\mathrm R}-E_\mu^{\mathrm A}$. On compact sources, \[ (\Box+\mu^2)E_\mu^{\mathrm R/A} =E_\mu^{\mathrm R/A}(\Box+\mu^2)=\id . \] These inverses have causal support. We use the standard distributional extension and smooth Cauchy theory \cite{BGP}. The reduced causal operator is \[ \Ered=\diag(E_0,E_m,E_0,E_0). \] It belongs to $\Pred$, not to $\Kact$. The retarded/advanced inverses of $\Kact$ are \begin{equation} \begin{aligned} \Gact_{\gamma b}^{\mathrm R/A} &=\begin{pmatrix} -\PP E_0^{\mathrm R/A}&\RR E_0^{\mathrm R/A}\\ -D\PP E_0^{\mathrm R/A}&0 \end{pmatrix},\\ \Gact_\phi^{\mathrm R/A}&=-E_m^{\mathrm R/A},\\ \Gact_{c\bar c}^{\mathrm R/A} &=\begin{pmatrix} 0&-iE_0^{\mathrm R/A}\\ iE_0^{\mathrm R/A}&0 \end{pmatrix}. \end{aligned} \label{eq:Gact} \end{equation} For example, for source $(U,v)$ the auxiliary solution is \[ \gamma=-\PP E_0^{\mathrm R/A}U+\RR E_0^{\mathrm R/A}v, \qquad b=-D\PP E_0^{\mathrm R/A}U . \] Indeed $b=D\gamma-v/2$ because $D\RR E_0^{\mathrm R/A}v=v/2$. Substitution verifies both inverse identities. The scalar and ghost inverse identities follow by direct multiplication. The action causal matrix $\Delta=\Gact^{\mathrm R}-\Gact^{\mathrm A}$ therefore has blocks \begin{equation} \begin{aligned} \Delta_{\gamma\gamma}&=-\PP E_0,& \Delta_{\phi\phi}&=-E_m,\\ \Delta_{\gamma b}&=\RR E_0,& \Delta_{b\gamma}&=-D\PP E_0,& \Delta_{bb}&=0,\\ \Delta_{c\bar c}&=-iE_0,& \Delta_{\bar cc}&=iE_0 . \end{aligned} \label{eq:Delta} \end{equation} All other blocks vanish. The causal algebra $\Afree(M)$ is the completed graded algebra generated by compact linear smearings, modulo the equations and \begin{equation} [\Phi_A(u),\Phi_B(v)]_{\mathrm{gr}} =i\hbar\pair{u}{\Delta_{AB}v}\mathbf1 . \label{eq:CCR} \end{equation} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.23\linewidth}X@{}} \toprule Object & Exclusive role in the proofs\\ \midrule $\Keq$ & Defines the shell and linear equation functionals\\ $\Pred,\Ered$ & Reduced separation and regular time-slice construction\\ $\Kact,\Gact^{\mathrm R/A},\Delta$ & Action normalization and causal brackets\\ $H$ & Quantum contractions and ordering\\ \bottomrule \end{tabularx} \end{center} %===================================================================== \section{Polynomial domain and admissible product data} \label{sec:domain} %===================================================================== \subsection{Graded kernels and evaluation} Odd fields are evaluated in arbitrary finite Grassmann algebras. An identity means equality of every Grassmann coefficient for every such evaluation. This convention applies to restriction, vanishing, polarization, and all evaluation maps below. A polynomial has the form \[ F(\Phi)=\sum_{n=0}^N\pair{u_n}{\Phi^{\otimes n}}, \] where $u_n$ is a compactly supported, graded-symmetric distribution on $M^n$, with finitely many species components. Graded symmetrization is normalized: \[ \Sym_n=\frac1{n!}\sum_{\pi\in S_n} \epsilon(\pi)\,\pi_* . \] The sign $\epsilon(\pi)$ is the Koszul sign. Our kernels contain no additional $1/n!$. Consequently \[ F^{(k)}(\Phi) =\sum_{n\ge k}\frac{n!}{(n-k)!} \pair{u_n}{\Phi^{\otimes(n-k)}\otimes\,\cdot^{\otimes k}} . \label{eq:functionalderivatives} \] All derivative and polarization identities are understood coefficientwise in the Grassmann algebra. Let $\Vp,\Vm$ be the closed causal covector cones, including zero. For $n\ge1$, define \begin{equation} G_n=\dot T^*(M^n)\setminus(\Vp^{\,n}\cup\Vm^{\,n}). \label{eq:Gn} \end{equation} The polynomial microcausal carrier $\Fpol$ consists of the above polynomials with \begin{equation} \WF(u_n)\subset G_n . \label{eq:microcausal} \end{equation} Degree zero consists of scalars. The regular classical algebra $\Acl$ is the subalgebra generated by compact smooth linear smearings; its kernels are finite sums of tensor products of tests. ``Regular'' here means finite-rank regular, not every smooth compact kernel. Define the vector spaces \begin{equation} \Nsol=\{F\in\Fpol:F|_{\Csol}=0\},\qquad \Fsol=\Fpol/\Nsol,\qquad \Ealg=\Fsol[[\hbar,\kappa]] . \label{eq:shellquotient} \end{equation} A product on $\Ealg$ will be justified in Section~\ref{sec:quotients}; it is not assumed in this definition. \subsection{The frozen Hadamard matrix} Put $h_\mu=-W_\mu$, where \[ W_\mu(x,y) =\int_{\R^3} \frac{e^{-i\omega_\mu(\mathbf p)(x^0-y^0) +i\mathbf p\cdot(\mathbf x-\mathbf y)}} {(2\pi)^3\,2\omega_\mu(\mathbf p)} \dd^3\mathbf p, \qquad \omega_\mu(\mathbf p)=\sqrt{|\mathbf p|^2+\mu^2}. \] These distributions are bisolutions and satisfy $h_\mu-h_\mu^{\mathsf T}=iE_\mu$. Their wavefront relation has one fixed orientation: \[ (x,p;y,-p)\in\WF(h_\mu) \quad\Longrightarrow\quad p\in V_\sigma\setminus\{0\}, \qquad \sigma\in\{+,-\}. \] The relation includes the null-geodesic condition. The same orientation is used for every species. This is product data; no state or positivity condition is selected. The full matrix is \begin{equation} \begin{aligned} H_{\gamma\gamma}&=-\PP h_0,& H_{\phi\phi}&=-h_m,\\ H_{\gamma b}&=D_yH_{\gamma\gamma},& H_{b\gamma}&=D_xH_{\gamma\gamma},& H_{bb}&=D_xD_yH_{\gamma\gamma},\\ H_{c\bar c}&=-ih_0,& H_{\bar cc}&=ih_0 . \end{aligned} \label{eq:H} \end{equation} All other blocks vanish; the ghost blocks include the vector identity. Slot differentiation includes the corresponding tensor index contraction. For the translation-invariant frozen data, \begin{equation} H_{\gamma b}=\RR_xh_0,\qquad H_{b\gamma}=\RR_yh_0,\qquad H_{bb}=\tfrac12\Box_xh_0\,\id=0. \label{eq:Hfrozen} \end{equation} These identities follow from $\partial_y=-\partial_x$, $\PP^2=\id$, and $D\RR=\tfrac12\Box$. A replacement matrix need not have zero $bb$ block. \begin{lemma}[Equations and causal part] \label{lem:H} The matrix $H$ solves $\Keq$ in both slots and \begin{equation} H_{AB}(x,y)-(-1)^{|A||B|}H_{BA}(y,x) =i\Delta_{AB}(x,y). \label{eq:Hantisymmetric} \end{equation} It is even and has ghost number zero. \end{lemma} \begin{proof} The auxiliary identities in \eqref{eq:H} impose $b=D\gamma$ in each slot; the other equations are the scalar wave equations. The causal identity follows from $h_\mu-h_\mu^{\mathsf T}=iE_\mu$ and slot differentiation. For the odd pair, \[ H_{c\bar c}+H_{\bar cc}^{\mathsf T} =-i(h_0-h_0^{\mathsf T})=E_0=i\Delta_{c\bar c}. \] The reversed pair gives $-E_0$. The displayed nonzero blocks have the stated parity and ghost number. \end{proof} \begin{definition}[Admissible replacement] \label{def:admissible} An admissible $H'$ is a complex-valued block bidistribution with: \begin{enumerate}[label=\textup{(\roman*)}] \item the same shell equations in both slots; \item the same graded antisymmetric part $i\Delta$; \item smooth difference $d=H'-H$; \item $H'_{AB}=0$ unless $|A|+|B|=0\pmod2$ and $\mathrm{gh}(A)+\mathrm{gh}(B)=0$. \end{enumerate} \end{definition} Thus $d$ is a smooth graded-symmetric bisolution. Smoothness does not mean compact support. Compact kernels make all its contractions proper on their supports. The wavefront sets of $H'$ and $H$ agree. %===================================================================== \section{The product and one contraction convention} \label{sec:product} %===================================================================== Use graded left derivatives and the graded tensor-product rule. Normalize the even contraction operator $\Gamma_H$ by \[ \Phi_A(u)\star_H\Phi_B(v) =\Phi_A(u)\Phi_B(v)+\hbar\pair{u}{H_{AB}v}. \] Define \begin{equation} F\star_HG=m\exp(\hbar\Gamma_H)(F\otimes G). \label{eq:star} \end{equation} For homogeneous kernels of degrees $n,m$, its $k$-contraction is \[ \frac{\hbar^k}{k!}\frac{n!\,m!}{(n-k)!\,(m-k)!} \Sym_{n+m-2k}\pi_* \bigl[(u\otimes v)H^{\otimes k}\bigr], \qquad 0\le k\le\min(n,m). \] Here $\pi$ drops the paired variables; indices and Koszul signs are contracted as prescribed above. The orientation convention is fixed once: \begin{equation} (y,p;y',-p)\in\WF(H),\ p\in V_\sigma \quad\Longrightarrow\quad q=-p\in V_{-\sigma},\qquad q'=p=-q \label{eq:orientation} \end{equation} for covectors of $u,v$ canceled by that line. Zero entries occur when individual line factors are smooth. This convention is also used for the continuity stages. \begin{theorem}[Polynomial product] \label{thm:product} Every contraction is defined. Its support is compact, and its positive-degree kernel satisfies \eqref{eq:microcausal}. A complete contraction is a scalar. The product is associative, unital, even, and ghost-number preserving. It preserves the regular subalgebra and extends coefficientwise to formal series. \end{theorem} \begin{proof} A forbidden product cancellation would require zero covectors in all unpaired variables and $q\in V_{-\sigma}^{\,k}$, $q'=-q\in V_\sigma^{\,k}$ in the paired variables. The corresponding nonzero input wavefront tuple would lie wholly in a forbidden causal cone, contrary to \eqref{eq:microcausal}. Hence the distribution product exists. The projection is proper on the compact support of $u\otimes v$. Suppose a nonzero output tuple were wholly future causal. If $\sigma=+$, the second input tuple, consisting of its future external entries and $q'$, would be wholly future causal. It must therefore be zero. This forces $q'=q=0$ and the second external tuple to vanish. The first tuple is now wholly future causal and must also vanish, a contradiction. For $\sigma=-$, start with the first input. Past-causal output tuples are excluded by the reversed argument. Symmetrization preserves the exclusion. When no variables remain, proper pushforward gives a scalar; there is no degree-zero wavefront claim. On three factors the even contraction operators $\Gamma_{12},\Gamma_{13},\Gamma_{23}$ commute: their coefficients are independent of the configuration variables and the graded signs are included. Both bracketings equal \[ m^{(3)} \exp\!\bigl(\hbar(\Gamma_{12}+\Gamma_{13}+\Gamma_{23})\bigr). \] The required multi-contractions satisfy the same wavefront cancellation argument, so this identity involves defined distributions, not merely formal symbols. Constants give the unit. Finite-rank kernels contract to finite-rank kernels. The grading assertions follow from Definition~\ref{def:admissible}. For a fixed formal coefficient only finitely many orders of contraction and input coefficients contribute. \end{proof} The wavefront calculus underlying these constructions is the one used for extended Wick algebras \cite{HW,Hormander}. In particular, graded signs change coefficients, not the wavefront-existence conditions. %===================================================================== \section{Reduced separation and Hadamard smoothing} \label{sec:separation} %===================================================================== \begin{lemma}[Reduced kernel separation] \label{lem:reducedseparation} Let $w$ be a compact distribution on $M^n$ with reduced species indices. The following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item $w$ annihilates every tensor tuple of smooth solutions of $\Pred$; \item $\Ered_1\cdots\Ered_n w=0$; \item $w=\sum_{j=1}^n\Pred_j r_j$ with compact distributions $r_j$. \end{enumerate} If $w$ is smooth, the $r_j$ can be smooth. Every such $w$ annihilates every smooth multisolution, not only factorized solution tuples. \end{lemma} \begin{proof} All assertions reduce componentwise to $P=\Box+\mu^2$. The wave causal operator maps compact smooth tests to smooth spatially compact solutions. Pairing (i) against these solutions and using $E^{\mathsf T}=-E$ proves (ii); tensor products of tests separate distributions. Choose a smooth time function $\chi$ equal to zero sufficiently early and one sufficiently late. For compact $w$, define \[ r=(1-\chi)E^{\mathrm R}w+\chi E^{\mathrm A}w . \] Causal support and the bounded transition interval make $r$ compact. Direct differentiation, using the inverses of $P$, gives \begin{equation} w-Pr=P(\chi Ew)=[P,\chi]Ew=:Tw . \label{eq:cutoff} \end{equation} Every operator here belongs to $P$; the action causal matrix $\Delta$ is absent. Apply this construction in each slot. The $T_j$ commute, and \[ T_1\cdots T_nw =\Pred_1\cdots\Pred_n [\chi_1\cdots\chi_n\,\Ered_1\cdots\Ered_nw]. \] If (ii) holds, this expression vanishes. Telescoping $w-T_1\cdots T_nw$ with $\id-T_j=\Pred_jR_j$ gives (iii), with compact primitives. The construction preserves smoothness. Finally (iii) annihilates every smooth multisolution by transposition. It also proves (i). \end{proof} \begin{corollary}[Full-shell separation] \label{cor:fullseparation} A compact full-multiplet kernel $w$ annihilates all smooth shell tuples if and only if \begin{equation} w=\sum_j\Keq_j^{\mathsf T}r_j \label{eq:fullseparation} \end{equation} with compact distributional primitives. Smooth $w$ admits smooth primitives. Such a kernel annihilates every smooth full-shell multisolution. \end{corollary} \begin{proof} Use the invertible differential change of variables $\Phi\mapsto(Y,\rho)$, $\rho=b-D\gamma$. Expand each slot using $\Phi=JY+e_b\rho$. Terms containing a $\rho$ slot already factor through that slot's equation $b-D\gamma$. The remaining reduced kernel is $(J^{\mathsf T})^{\otimes n}w$ and satisfies Lemma~\ref{lem:reducedseparation}. Integration by parts expresses the resulting terms as \eqref{eq:fullseparation}. The transformations are differential and preserve compact support and smoothness. The converse and the multisolution statement follow by transposition. \end{proof} \begin{lemma}[Oriented Hadamard smoothing] \label{lem:smoothing} If $z\in\Edist(M^k)$ is microcausal, then \[ S(x_1,\ldots,x_k) =\int H(x_1,y_1)\cdots H(x_k,y_k) z(y_1,\ldots,y_k)\dd y \] is a smooth full-shell multisolution. The statement holds for $H'$ and for the transposed line orientation as well. \end{lemma} \begin{proof} The input projection is proper on $\supp z$ over every compact output set. The tensor wavefront bound for $H^{\otimes k}$ has entries $(p_j,-p_j)$ with all nonzero $p_j$ in $V_\sigma$; zero pairs cover smooth factors. An undefined product would require a nonzero input tuple of $z$ entirely in $V_\sigma^{\,k}$, which microcausality excludes. An output wavefront point would require the same type of input tuple when the input covectors are canceled for pushforward. It is therefore excluded too: $\WF(S)=\varnothing$. The bisolution equations imply the shell equations in each output slot. Adding smooth $d$ does not alter the argument. Transposition reverses the common orientation, and microcausality excludes both orientations. \end{proof} Smearing some slots of a microcausal kernel against a smooth configuration preserves microcausality: a forbidden remaining tuple would lift to a forbidden original tuple with zero covectors in the smeared slots. Thus the lemma applies to $G^{(k)}(\Phi)$. %===================================================================== \section{The regular quotient and free-shell descent} \label{sec:quotients} %===================================================================== \begin{lemma}[The undeformed source is not a quantum presentation] \label{lem:commutativesource} There is no parity-preserving unital algebra homomorphism from $\Acl[[\hbar]]$ to $\Afree(M)$ sending every linear smearing to its corresponding causal generator. \end{lemma} \begin{proof} The image of a graded-commutative source is graded commutative. But the scalar commutator is $-i\hbar\pair{u}{E_mv}\mathbf1$, nonzero for suitable tests. \end{proof} Define \[ \Areg=(\Acl[[\hbar,\kappa]],\star_H). \] Let $\Nreg$ be the ordinary graded ideal generated by \[ E_r(\Phi)=\pair{r}{\Keq\Phi} =\pair{\Keq^{\mathsf T}r}{\Phi}, \qquad r\in\Dtest(M). \] \begin{proposition}[Regular equation ideal] \label{prop:regularideal} One has $\Nreg=\Nsol\cap\Acl$. Its formal extension is also the two-sided $\star_H$-ideal generated by the equation functionals. \end{proposition} \begin{proof} A regular polynomial uses finitely many linear coordinates. By Corollary~\ref{cor:fullseparation}, the linear coordinates vanishing on the shell are precisely compact smooth equation functionals. Choose a homogeneous basis separating these coordinates from those whose restrictions are independent. On the independent even coordinates, the finite coordinate map on the solution space is surjective and ordinary polynomial evaluation is faithful. On the odd coordinates, Grassmann evaluation is faithful on the exterior algebra. Thus a polynomial vanishing on the shell belongs to the ideal generated by its vanishing linear coordinates. This proves the equality. A contraction hitting $E_r$ vanishes because $\Keq_xH=\Keq_yH=0$. Therefore an equation factor passes through a quantum product, with its Koszul sign: \[ (E_rF)\star_HG=E_r(F\star_HG),\qquad G\star_H(E_rF)=(-1)^{|G||E_r|}E_r(G\star_HF). \] Ordinary and quantum multiplication generate the same ideal. \end{proof} \begin{theorem}[Regular quantum presentation] \label{thm:regularquotient} There is a canonical surjective homomorphism \[ \pi_{\mathrm{free},H}:\Areg\longrightarrow\Afree(M)[[\kappa]] \] sending linear symbols to causal generators, with kernel $\Nreg[[\hbar,\kappa]]$. \end{theorem} \begin{proof} Let $V=\Dtest/\Keq^{\mathsf T}\Dtest$, with the species grading. The regular classical quotient is $\operatorname{Sym}(V_{\mathrm{even}}) \otimes\bigwedge(V_{\mathrm{odd}})$. The contractions descend to $V$ by the bisolution equations. Their graded antisymmetric part is $i\Delta$, so linear generators satisfy exactly \eqref{eq:CCR}. The graded symmetrization map from the causal presentation to this deformed symmetric algebra is bijective: on finite words its highest field-degree part is ordinary graded symmetrization, and all contraction corrections lower degree. The triangular map is invertible degree by degree. Every word uses a finite-dimensional coordinate span; no finite-dimensional assumption on $V$ is needed. Extending coefficientwise gives the formal isomorphism. Composing its inverse with the quotient map defines $\pi_{\mathrm{free},H}$ and proves the kernel claim. \end{proof} \begin{theorem}[Free-shell descent] \label{thm:descent} The space $\Nsol[[\hbar,\kappa]]$ is a two-sided quantum ideal. Hence \[ Q_H:(\Fpol[[\hbar,\kappa]],\star_H)\longrightarrow\Ealg \] is a surjective algebra homomorphism. There is an injective homomorphism $j_H:\Afree(M)[[\kappa]]\to\Ealg$ making \[ \begin{array}{ccc} \Areg&\xrightarrow{\ \iota\ }&(\Fpol[[\hbar,\kappa]],\star_H)\\ \big\downarrow{\scriptstyle\pi_{\mathrm{free},H}} &&\big\downarrow{\scriptstyle Q_H}\\ \Afree(M)[[\kappa]]&\xrightarrow{\ j_H\ }&\Ealg \end{array} \] commute. \end{theorem} \begin{proof} Fix $F\in\Nsol$, $G\in\Fpol$, and $\Phi\in\Csol$. Graded polarization of the identically vanishing restriction gives that $w=F^{(k)}(\Phi)$ annihilates every solution tuple. By Corollary~\ref{cor:fullseparation}, $w=\sum_i\Keq_i^{\mathsf T}r_i$. By Lemma~\ref{lem:smoothing}, $S=H^{\otimes k}G^{(k)}(\Phi)$ is a smooth multisolution. Consequently \[ \pair{w}{S} =\sum_i\pair{r_i}{\Keq_iS}=0. \] The primitives need not be microcausal: they are paired with a smooth multisolution, not inserted into a distributional product. The $k=0$ term is $F(\Phi)G(\Phi)=0$. Every higher contraction vanishes by the preceding argument. For $G\star_HF$, use the transposed orientation from Lemma~\ref{lem:smoothing}; no symmetry of the noncommutative product is assumed. Kernel evaluation and the displayed derivative contraction are the same proper composition, so no approximation step is needed. This proves the ideal property and formal descent. Finally the restriction of $Q_H$ to $\Areg$ has kernel $\Nreg[[\hbar,\kappa]]$ by Proposition~\ref{prop:regularideal}. The induced map $j_H$ is therefore injective, and the diagram commutes. \end{proof} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.28\linewidth}X@{}} \toprule Carrier & Role\\ \midrule $\Acl$ & Undeformed finite-rank polynomial symbols\\ $\Areg$ & Regular quantum source of the causal quotient\\ $\Afree(M)[[\kappa]]$ & Regular on-shell causal algebra\\ $(\Fpol[[\hbar,\kappa]],\star_H)$ & Extended off-shell polynomial carrier\\ $\Ealg$ & Extended on-shell quantum carrier\\ $\mathscr A_\ast$ & Unspecified enlargement; not opened\\ \bottomrule \end{tabularx} \end{center} %===================================================================== \section{The frozen free differential} \label{sec:differential} %===================================================================== Define the odd, ghost-number-one symbol derivation by \[ \sfree\gamma=\RR c,\qquad \sfree\bar c=ib,\qquad \sfree\phi=\sfree c=\sfree b=0. \] It acts on kernels by differential operators and species replacement, and satisfies $\sfree^2=0$. \begin{proposition}[Quantum compatibility] \label{prop:differential} The operator $\sfree$ preserves $\Fpol$ and $\Nsol$. It is a derivation of $\star_H$ precisely when \begin{equation} (\delta H)_{AB} :=H_{\sfree A,B}+(-1)^{|A|}H_{A,\sfree B}=0 \quad\text{for every pair }A,B. \label{eq:differentialcondition} \end{equation} The condition holds for the frozen matrix. Thus $\sfree$ descends to a nilpotent derivation of $\Ealg$. \end{proposition} \begin{proof} Differential operators do not enlarge wavefront sets. On the shell, $\RR c$ is a wave solution and $D\RR c=\tfrac12\Box c=0$; also $\Box b=0$. Hence the odd linear variation is tangent to the shell and preserves its vanishing ideal. Commuting the linear symbol variation through a contraction gives $\delta H$. The quantum Leibniz defect vanishes if $\delta H=0$. Conversely, applying the defect to two linear smearings detects every component of $\delta H$ as a scalar, proving necessity. For the frozen data, both nontrivial ghost orderings cancel: \[ \begin{aligned} (\gamma,\bar c):\quad& \RR_xH_{c\bar c}+iH_{\gamma b} =-i\RR_xh_0+i\RR_xh_0=0,\\ (\bar c,\gamma):\quad& iH_{b\gamma}-\RR_yH_{\bar cc} =i\RR_yh_0-i\RR_yh_0=0. \end{aligned} \] The pairs $(b,\bar c)$ and $(\bar c,b)$ vanish because $H_{bb}=0$. Every remaining pair has only zero contraction blocks. \end{proof} %===================================================================== \section{Named composites and their formal sector} \label{sec:composites} %===================================================================== Set \[ \mathfrak a^{\alpha\mu,ab} =\tfrac12\eta^{\alpha\mu}\eta^{ab} -\eta^{\alpha(a}\eta^{b)\mu}, \qquad \Lambda^{\mu,ab}(x,z) =\sig(z)\mathfrak a^{\alpha\mu,ab} \partial_\alpha^zE_0^{\mathrm R}(x-z). \] Define \[ X_1^\mu[\gamma](x)=\int\Lambda^{\mu,ab}(x,z)\gamma_{ab}(z)\dd z \] and \begin{equation} \Bcal_f=-\int fX_1^\mu[\gamma]\partial_\mu\phi,\qquad b_f^{ab}(x,z)=\partial_\mu^x[f(x)\Lambda^{\mu,ab}(x,z)]. \label{eq:B} \end{equation} Thus $\Bcal_f=\int b_f^{ab}(x,z)\gamma_{ab}(z)\phi(x)\dd x\dd z$. The kernel of $\sfree\Bcal_f$ is obtained by applying $\RR_z^{\mathsf T}$ to its metric slot. For the scalar stress, let $\tau^{ab}(y;y_1,y_2)$ be the symmetric bilinear local kernel of the frozen quadratic stress $T_0^{ab}$: \[ T_0^{ab}[\phi](y) =\pair{\tau^{ab}(y;\cdot,\cdot)}{\phi\otimes\phi}. \] Equivalently $\tau$ is a fixed finite sum of products of derivatives of $\delta(y-y_1)\delta(y-y_2)$ with constant coefficients. This convention includes no polarization factor beyond the one displayed. The coordinate response to the metric Green operator is \[ Q^\mu{}_{ab}(x,y) =\int\Lambda^{\mu,cd}(x,z) (\Ggr)_{cd,ab}(z,y)\dd z. \] For compact external sets the integration is over a compact causal region, because $z\in J^-(x)\cap J^+(y)$ and $t_z\ge0$. The retarded wavefront relations imply $\WF(Q)\subset\{(x,p;y,-p)\}$. The frozen tree and its symmetric kernel are \begin{equation} \begin{aligned} \Tcal(f,g)&=\tfrac12\int f(x)Q^\mu{}_{ab}(x,y)g(y)T_0^{ab}[\phi](y) \partial_\mu\phi(x)\dd x\dd y,\\ t_{f,g}&=\Sym_3\Theta_{f,g},\\ \Theta_{f,g}&=-\tfrac12\partial_\mu^x \pi_{y*}\bigl[(f\otimes g\otimes1\otimes1)\Xi^\mu\bigr],\\ \Xi^\mu(x,y,y_1,y_2)&=Q^\mu{}_{ab}(x,y) \tau^{ab}(y;y_1,y_2). \end{aligned} \label{eq:T} \end{equation} The factor $\tfrac12$ is retained after integration by parts. The product defining $\Xi$ exists: a canceled external covector would force $p=0$, hence the $Q$ covector to vanish; the other external covectors then force the $\tau$ covector to vanish. The input relations and pushforward give \begin{equation} \begin{aligned} \WF(b_f),\ \WF(\RR_z^{\mathsf T}b_f) &\subset Z_2:=\{p+q=0\}\setminus0,\\ \WF(t_{f,g}) &\subset Z_3:=\{p+r_1+r_2=0\}\setminus0 . \end{aligned} \label{eq:sumzero} \end{equation} A tuple in one closed causal cone cannot have zero sum unless it is zero. These kernels are therefore microcausal. For tests supported in $K_f,K_g$, valid compact support bounds are \begin{equation} K_B=K_f\times(J^-(K_f)\cap\{t\ge0\}),\qquad K_T=\bigcup_{\pi\in S_3}\pi(K_f\times K_g\times K_g). \label{eq:compositesupports} \end{equation} Support and cone bounds for a graded-symmetric quadratic kernel are understood after the corresponding finite permutations. \begin{definition}[Completed named sector] \label{def:sector} Let $\mathscr W_H$ be the algebra of finite $\star_H$-words in compact linear fields and \[ Q_H\Bcal_f,\qquad Q_H\sfree\Bcal_f,\qquad Q_H\Tcal(f,g), \qquad f\in C_c^\infty(\MTo;\R), \] with coefficients in $\C[\hbar,\kappa]$. Set $\formalideal=(\hbar,\kappa)$. The sector $\Sect$ consists of $X\in\Ealg$ such that, for every $p\ge1$, some $W_p\in\mathscr W_H$ satisfies $X-W_p\in\formalideal^p\Ealg$. This is the formal coefficientwise completion, not closure in the analytic topology introduced later. For admissible $H'$, define safely $\SectP=\balphad(\Sect)$ once the ordering map is established. \end{definition} The completion allows coefficients with no uniform word-length bound. It is not identified with finite words having arbitrary formal scalar coefficients. It contains $j_H(\Afree(M)[[\kappa]])$. \begin{proposition} \label{prop:sector} The sector is graded, unital, closed under $\star_H$, and $\sfree$-stable. Every coefficient admits a compact microcausal lift. \end{proposition} \begin{proof} These claims first hold for finite words by Theorem~\ref{thm:product}. The differential preserves the linear fields, maps $\Bcal_f$ to its included image, annihilates that image, and annihilates the scalar-only tree. It is a quantum derivation. Products and this linear differential preserve every formal order, so the claims pass to the stated completion. \end{proof} %===================================================================== \section{Explicit ordering transport and sector stability} \label{sec:transport} %===================================================================== For admissible $H'=H+d$, define \begin{equation} \alphad =\exp\!\left(\frac{\hbar}{2} \left\langle d,\frac{\delta^2}{\delta\Phi^2}\right\rangle\right). \label{eq:alpha} \end{equation} For a polynomial, its self-contraction expansion terminates. For formal series it is defined coefficientwise. \begin{theorem}[Ordering equivalence and descent] \label{thm:transport} The map $\alphad$ preserves $\Fpol[[\hbar,\kappa]]$ and $\Nsol[[\hbar,\kappa]]$, and \begin{equation} \alphad(F\star_HG)=\alphad F\star_{H'}\alphad G . \label{eq:intertwining} \end{equation} It fixes constants and linear fields, is the identity modulo $\hbar$, has inverse $\alpha_{-d}$, and satisfies $\alpha_{d_1}\alpha_{d_2}=\alpha_{d_1+d_2}$. It induces $\balphad:\Ealg\to\EalgP$ with \[ \balphad Q_H=Q_{H'}\alphad,\qquad \balphad j_H=j_{H'} . \] Under the causal presentation, the regular descended map is the identity of $\Afree(M)[[\kappa]]$. \end{theorem} \begin{proof} A smooth self-contraction is multiplication followed by a proper projection. Output wavefront tuples lift with zero entries in the contracted slots, so microcausality is preserved. The self-contraction operators commute and lower field degree by two, proving the inverse and cocycle identities. Let $D_d=(\hbar/2)\langle d,\delta^2\rangle$. Graded symmetry of $d$ gives the Leibniz identity \[ D_d m=m(D_d^{(1)}+D_d^{(2)}+\hbar\Gamma_d). \] Exponentiating commuting operators proves \eqref{eq:intertwining}, the smooth-change ordering equivalence \cite{FR}. For $F\in\Nsol$ and $\Phi\in\Csol$, $F^{(2k)}(\Phi)$ annihilates solution tuples. The kernel made from paired copies of $d$ is a smooth multisolution in all $2k$ slots. Corollary~\ref{cor:fullseparation} therefore makes its pairing with $F^{(2k)}(\Phi)$ zero. Thus $\alphad\Nsol\subset\Nsol$; the inverse proves equality. Descent follows. Constants and linear fields have too few slots to self-contract, and every nonidentity term contains $\hbar$. The regular assertion follows because the descended homomorphisms agree on their topological generators. \end{proof} \begin{proposition}[Corrections on the seeds] \label{prop:seedcorrections} For the quadratic and cubic seeds, \begin{equation} \begin{aligned} \alphad\Bcal_f &=\Bcal_f+\hbar c_d(f),& c_d(f)&=\tfrac12\pair{d}{\Bcal_f^{(2)}},\\ \alphad\sfree\Bcal_f&=\sfree\Bcal_f,\\ \alphad\Tcal(f,g) &=\Tcal(f,g)+\hbar\,\phi(\ell_d(f,g)),& \ell_d(f,g)&=3\,\pi_{3*} [d_{\phi\phi}(x_1,x_2)t_{f,g}(x_1,x_2,x_3)] . \end{aligned} \label{eq:seedcorrections} \end{equation} Here $c_d(f)$ is a scalar and $\ell_d(f,g)\in C_c^\infty(M)$. \end{proposition} \begin{proof} The quadratic formula is the terminating exponential. The image $\sfree\Bcal_f$ has species $(c,\phi)$; its only possible self-contraction is forbidden by ghost number. For the scalar cubic, the second derivative contributes the factor $3\cdot2$, canceled in part by the $\tfrac12$ in \eqref{eq:alpha}. No fourth derivative remains. For the projected kernel $\ell_d$, a nonzero output wavefront point would come from $(0,0,p)\in\WF(t_{f,g})$. The sum-zero bound \eqref{eq:sumzero} forces $p=0$. Thus $\ell_d$ is smooth, and proper projection makes it compact. \end{proof} \begin{theorem}[Ordering stability of the named sector] \label{thm:sectorstability} The transported sector equals the formal coefficientwise completion generated by the unchanged seeds: \[ \SectP =\widehat{\Alg}_{\star_{H'}} \bigl(j_{H'}(\Afree(M)),\ Q_{H'}\Bcal_f,\ Q_{H'}\sfree\Bcal_f,\ Q_{H'}\Tcal(f,g)\bigr). \] \end{theorem} \begin{proof} Every ordering correction in Proposition~\ref{prop:seedcorrections} lies in the regular part: constants and compact smooth linear fields. The regular part is transported onto the regular part. Hence each transported seed belongs to the unchanged-seed algebra. Applying the inverse ordering map proves the opposite inclusion. Formal-order preservation gives equality of the completions. \end{proof} This is a theorem about this named sector, not general invariance of unchanged Wick formulas. \begin{proposition}[Transported differential] \label{prop:transporteddifferential} The operator $\sfree^{H'}=\balphad\sfree\balphad^{-1}$ is a nilpotent, ghost-number-one derivation and preserves $\SectP$. It equals the symbol rule $\sfree$ exactly when $\delta d=0$, with $\delta$ defined in \eqref{eq:differentialcondition}. \end{proposition} \begin{proof} Conjugation gives nilpotence, the derivation property, and sector stability. The symbol rule is a derivation for $H'$ exactly when $\delta H'=\delta d=0$. In that case its commutator with the self-contraction generator vanishes. Conversely equality on the quotient would make the symbol rule a quantum derivation there; its defect on two regular linear fields is a scalar testing $\delta d$. Scalars are not killed by the shell quotient, so all these tests must vanish. \end{proof} %===================================================================== \section{Continuity at the named levels} \label{sec:continuity} %===================================================================== For a closed cone $\Gamma\subset\dot T^*X$, use the normal topology on $\Dtest'_\Gamma(X)$: the strong distribution seminorms \[ p_B(u)=\sup_{\varphi\in B}|\pair{u}{\varphi}|, \qquad B\subset\Dtest(X)\ \text{bounded}, \] and the Fourier seminorms \[ \|u\|_{N,V,\chi} =\sup_{\xi\in V}(1+|\xi|)^N|\widehat{\chi u}(\xi)|, \qquad(\supp\chi\times V)\cap\Gamma=\varnothing. \] For compact $K\subset M^n$, set \[ \Edist_{K,\Gamma} =\{u:\supp u\subset K,\ \WF(u)\subset\Gamma\}. \] Finite species components and graded symmetry carry their induced topologies. The normal-topology facts used here are hypocontinuity of tensor products and compatible distribution products, continuity of transversal pullback and support-proper pushforward, and hypocontinuity of smooth multiplication \cite{BDH}. Fixing one factor gives a continuous linear map. Constant-coefficient differentiation is continuous, as follows from \[ \widehat{\chi\partial_j u} =i\xi_j\widehat{\chi u} -\widehat{(\partial_j\chi)u}. \] \subsection{Kernel maps and contraction stages} \begin{proposition}[Fixed-support kernel continuity] \label{prop:kernelcontinuity} On $C^\infty_{K_f}(M)$, the maps $f\mapsto b_f$ and $f\mapsto\RR_z^{\mathsf T}b_f$ are continuous into their stages with cone $Z_2$. The bilinear map $(f,g)\mapsto t_{f,g}$ is jointly continuous from $C^\infty_{K_f}\times C^\infty_{K_g}$ to the stage $\Edist_{K_T,Z_3}$. \end{proposition} \begin{proof} For $b_f$, multiply the fixed $\Lambda$ by $f\otimes1$ and differentiate. Each operation is continuous. For $t_{f,g}$, first use the smooth tensor map. Its seminorm estimate is directly \[ \sup_{K_f\times K_g}|\partial_x^\alpha\partial_y^\beta(fg)| \le \sup_{K_f}|\partial^\alpha f|\, \sup_{K_g}|\partial^\beta g|. \] Then multiply the fixed $\Xi$, push forward over $y$, differentiate, and symmetrize. The projection is proper on $M\times K_g\times M^2$. The support bounds are \eqref{eq:compositesupports}. \end{proof} For closed cones $\Gamma_1\subset G_n$, $\Gamma_2\subset G_m$ over compact input supports, put $N=n+m-2k$. For $N>0$, define the unsymmetrized contraction cone by \begin{equation} \begin{split} \Gamma_1\circ_k\Gamma_2 =\{(x,\zeta;x',\zeta')\ne0:\ &\exists\,y,y',q\in V_{-\sigma}^{\,k},\\ &(x,\zeta;y,q)\in\Gamma_1\cup0_{K_1},\\ &(x',\zeta';y',-q)\in\Gamma_2\cup0_{K_2}\}. \end{split} \label{eq:contractioncone} \end{equation} Here $0_K$ is the zero section over $K$. Let $K_{\mathrm{out}}$ and $\Gamma_{\mathrm{out}}$ be the finite unions of permutations of the projected support and this cone. \begin{lemma}[Closed target cone] \label{lem:targetcone} The cone in \eqref{eq:contractioncone} is closed and lies in $G_N$. It bounds the unsymmetrized output wavefront set. The permuted target cone is also closed and microcausal. \end{lemma} \begin{proof} The wavefront bound is the product--pushforward bound with convention \eqref{eq:orientation}; dropping the null-geodesic restriction only enlarges it. The causal exclusion is the argument of Theorem~\ref{thm:product}. For closedness, take convergent nonzero output covectors and witnesses $(y_j,y'_j,q_j)$. The base variables stay compact. If $q_j$ were unbounded, divide the first input tuple by $|q_j|$. A subsequence would limit to a nonzero tuple $(0,q_\infty)$ wholly in $V_{-\sigma}$ and in the closed cone $\Gamma_1$, a contradiction. Thus the witnesses have a convergent subsequence. The closed input cones then give a witness for the limit. Finite permutation unions preserve the assertions. \end{proof} \begin{theorem}[Stagewise product hypocontinuity] \label{thm:stagecontinuity} Each symmetrized contraction is hypocontinuous from the two named input stages to $\Edist_{K_{\mathrm{out}},\Gamma_{\mathrm{out}}}$. For $N=0$ its target is $\C$. The zero-contraction term is included using the usual tensor cone and its permutations. \end{theorem} \begin{proof} Tensoring the two varying inputs is hypocontinuous. Multiplication by the fixed line kernel is continuous linear on the compatible tensor stage. Proper pushforward and finite permutation pullbacks are continuous linear. Their composition is hypocontinuous. Compatibility was checked in Theorem~\ref{thm:product}, and the uniform target cone is Lemma~\ref{lem:targetcone}. Complete contraction is the same proper pushforward to a point. \end{proof} \subsection{Assembled topology and quotient} Let $\mathscr T_n$ be the locally convex inductive limit over all compact supports and closed cones inside $G_n$. Larger cones remove Fourier seminorms, so the stage inclusions are continuous. Give $\Fpol$ the locally convex direct-sum topology $\bigoplus_{n\ge0}\mathscr T_n$. Give formal series the product topology of their coefficients and $\Ealg$ the quotient topology. \begin{theorem}[Hausdorffness and separate continuity] \label{thm:globalcontinuity} The carrier is Hausdorff and $\Nsol$ is closed. The formal shell quotient is Hausdorff. Quantum multiplication is globally separately continuous. The maps $\sfree$, $\alphad$, $Q_H$, and their descended maps are continuous; $\balphad$ is a homeomorphism. \end{theorem} \begin{proof} On a fixed support stage the map to $\Edist(M^n)_\beta$ is continuous: for bounded $B\subset C^\infty(M^n)$ choose a compact cutoff $\chi=1$ near the support, and use the bounded test family $\chi B\subset\Dtest(M^n)$. The universal property gives a continuous injection of each inductive limit into the Hausdorff distribution space. The direct sum is consequently Hausdorff. Every Grassmann coefficient of evaluation at a fixed smooth shell configuration is continuous on every stage, by the same cutoff argument, hence on the direct sum. Their common kernel is $\Nsol$, which is closed. The coefficientwise quotient map is the product of open linear quotient maps; its kernel is $\Nsol[[\hbar,\kappa]]$. This identifies the formal quotient topology with the product topology on $\Fsol[[\hbar,\kappa]]$. Fix a polynomial $G$. On each input stage, multiplication by $G$ is a finite sum of continuous linear contractions into target stages, by Theorem~\ref{thm:stagecontinuity}. The inductive-limit and direct-sum universal properties give global continuity of that linear map. For a fixed formal $G$, any output coefficient depends on finitely many input coefficients and finite contraction orders. The same argument proves formal separate continuity. The differential acts stagewise by differential operators. Each coefficient of $\alphad$ acts by smooth multiplication and proper pushforward. These maps are globally continuous by the same universal properties. They preserve the null ideal, so continuity descends through the quotient. The inverse ordering map has the same construction. \end{proof} \begin{center} \small \begin{tabularx}{\linewidth}{@{}X p{.34\linewidth}@{}} \toprule Claim & Status\\ \midrule Fixed-support kernel continuity & Proved\\ Named-stage product hypocontinuity & Proved\\ Global separate continuity & Proved by a separate linear argument\\ Hausdorff free-shell quotient & Proved by coefficient evaluations\\ Global product hypocontinuity & Not released\\ Completeness of the assembled carrier & Not released\\ \bottomrule \end{tabularx} \end{center} The remaining hypocontinuity question concerns bounded-set control in the chosen inductive limit, not the cardinality of its indexing family. A possible countable exhaustion does not itself establish that control. No identification with a differently topologized open-cone distribution space is used. %===================================================================== \section{Robust observation exclusion} \label{sec:exclusion} %===================================================================== \subsection{Explicit predecessor input} The cubic response $\Cthree(f,g)$ is the frozen XI-F/XI-G functional, not an element already admitted to $\Fpol$. Write its polarized kernel as $u(f,g)$. For the detector, the frozen source normalization can be written \[ \alpha[f,g,\phi]_{ab}(z) =-\int f(x)\Lambda^{\mu}{}_{ab}(x,z) \partial_\mu\phi(x)\dd x, \qquad \beta[f,g,\phi]=E_0^{\mathrm A}(\PP\alpha[f,g,\phi]). \] The label $g$ records the frozen predecessor data. The source is smooth and compact; no cubic source is multiplied by a new ramp in this revision. For a free scalar $\phi_0$, unit $\mathbf n$, and $\mathbf p\in\R^3$, set \begin{equation} \begin{aligned} \mathsf b_{f,g,\phi_0}(\mathbf n,\mathbf p) &=-2\check\beta_{00}(\mathbf p) +(n^in^j-3\delta^{ij})\check\beta_{ij}(\mathbf p),\\ \check\beta_{\mu\nu}(\mathbf p) &=\int\beta_{\mu\nu}(0,\mathbf x) e^{-i\mathbf p\cdot\mathbf x}\dd^3\mathbf x . \end{aligned} \label{eq:detector} \end{equation} Define \[ \Obs(g) =\{f\in C_c^\infty(\MTo;\R): \exists\,\phi_0\ \text{free with } \mathsf b_{f,g,\phi_0}\not\equiv0\}. \] \begin{inputtheorem}[XI-G, with observation scope] \label{input:XIG} For the frozen admissible predecessor data: \begin{enumerate}[label=\textup{(\alph*)}] \item If $f\ge0$, $f\not\equiv0$, $\supp f\subset(0,t_g)\times\R^3$, where $t_g=\min\{t:(t,\mathbf x)\in\supp g\}>0$, and $\cos(mt)>0$ on $\supp f$, then $f\in\Obs(g)$. XI-G's scalar witness gives $\mathsf b(\mathbf n,0)=-2\Lambda_f$ with $\Lambda_f\ne0$. Such observations exist for every frozen admissible $g$ and $m\ge0$. \item If $f\in\Obs(g)$, no $F\in\Fpol$ satisfies \[ F|_{\Csol}=\Cthree(f,g)|_{\Csol}. \] Equivalently the shell class of the polarized cubic kernel has no microcausal polynomial representative. \end{enumerate} \end{inputtheorem} This is the observation-qualified obstruction and reduction theorem of XI-G \cite{XIG}; its proof is an external input. No sign of $\Lambda_f$ is needed here. The wave-propagator criterion is compatible with the cone convention: its primed bicharacteristic relation transports covectors with their causal character and time orientation preserved. In Minkowski coordinates they are the same transported covectors. Thus a forbidden causal tuple in the wavefront set of the trisolution must pull back to a forbidden source tuple. This establishes the necessary cone-preservation implication; it does not evaluate the criterion for undetected observations. \begin{theorem}[Ordering-independent exclusion] \label{thm:exclusion} Let $f\in\Obs(g)$ and $C=\Cthree(f,g)|_{\Csol}$. Then $C\notin\Fsol$. No classical-symbol coefficient of an element of $\Ealg$ equals $C$, and the same holds for every admissible $H'$. If $Y\in\Fsol$, then $Y+C\notin\Fsol$. Universal admission for all observations is excluded. \end{theorem} \begin{proof} The first assertion is Input theorem~\ref{input:XIG}(b). Every classical-symbol coefficient belongs to $\Fsol$, proving the second. The underlying quotient vector space is the same for $H'$; also $\balphad$ is the identity modulo $\hbar$. Either fact proves ordering independence. If $Y,Y+C$ belonged to the vector space $\Fsol$, their difference $C$ would belong to it. Input theorem~\ref{input:XIG}(a) supplies excluded observations for every frozen admissible data set, ruling out universal admission into this carrier. \end{proof} \subsection{Continuous detectors and robustness} \begin{lemma}[Detector continuity] \label{lem:detectorcontinuity} For fixed $g,\phi_0,\mathbf n,\mathbf p$, the map \[ L_{\phi_0,\mathbf n,\mathbf p}(f) =\mathsf b_{f,g,\phi_0}(\mathbf n,\mathbf p) \] is continuous real-linear on every $C_K^\infty(M;\R)$ with $K\subset\MTo$ compact. \end{lemma} \begin{proof} The map $f\mapsto f\partial_\mu\phi_0$ is continuous on the fixed test stage. Convolution with the advanced wave Green operator and differentiation produce a smooth function continuously. Multiplication by the fixed $\sig$ localizes the source to $J^-(K)\cap\{t\ge0\}$, a fixed compact set. A further advanced propagation, restriction to $t=0$, and contraction with $\PP$ are continuous smooth operations. Finite propagation bounds the spatial support of that restriction by a fixed compact set. Its Fourier evaluation at fixed $\mathbf p$ is therefore a continuous smooth-function pairing. These are exactly the operations in \eqref{eq:detector}. \end{proof} \begin{theorem}[Robust detected class] \label{thm:robustness} Set \[ \mathfrak Z_{g,K} =\bigcap_{\phi_0,\mathbf n,\mathbf p} \ker L_{\phi_0,\mathbf n,\mathbf p}. \] It is a closed real linear subspace, and \[ \Obs(g)\cap C_K^\infty(M;\R) =C_K^\infty(M;\R)\setminus\mathfrak Z_{g,K}. \] If one detector is nonzero on this stage, the detected class is open and dense there. Each detected observation remains detected, by the same witness, under sufficiently small stage perturbations. \end{theorem} \begin{proof} Continuity gives closed kernels. A symbol is nonzero exactly when it is nonzero at some $(\mathbf n,\mathbf p)$, proving the complement identity. If a detector is nonzero, the common kernel is proper. A proper linear subspace of a topological vector space has empty interior; otherwise scaling an interior neighborhood would fill the space. Its closed complement is consequently dense and open. Continuity preserves a fixed nonzero detector value in a neighborhood of its witness. \end{proof} \scopebox{Scope of exclusion}{% Detector vanishing is not admission. It does not establish the trisolution cone criterion. The Einstein subshell is a smaller restriction space and remains undecided. A full-shell obstruction does not by itself obstruct restriction to that smaller shell. \smallskip For unadmitted $Y$, the status of $Y+C$ is undecided: cancellation requires its own theorem. The ordering map is not applied to the excluded kernel $u$. No Hadamard line is contracted against $\Cthree$, and no enlargement is opened.} %===================================================================== \section{Regular ambient--slab identification} \label{sec:timeslice} %===================================================================== A time band containing a full Minkowski Cauchy surface cannot be relatively compact in spacetime. Accordingly choose \[ N=(a,b)\times\R^3,\qquad 0