% !TEX program = pdflatex %===================================================================== % IX-C-ST.tex % % One-file delivery, as explicitly requested. % Three ordered mathematical stages are retained: % Part I : IX-C-ST % Part II : IX-Free-ST % Part III : IX-Free-2-ST % % This packaging does not identify their intermediate release boundaries. % No external figures, bibliography database, or source files are needed. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error IX-C-ST.tex % pdflatex -interaction=nonstopmode -halt-on-error IX-C-ST.tex % % No successful compilation, machine check, numerical execution, % experimental implementation, or independent execution is asserted. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={IX-C-ST: A supported partial product for one sharp ordered word}, pdfsubject={Sequential local Grade S adaptations of IX-C, IX-Free, and IX-Free-2} } \setlength{\emergencystretch}{3em} \setlist{itemsep=3pt,topsep=5pt} \setcounter{tocdepth}{1} \numberwithin{equation}{section} \allowdisplaybreaks \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\HH}{\mathcal H} \newcommand{\DD}{\mathcal D} \newcommand{\EE}{\mathcal E} \newcommand{\AA}{\mathcal A} \newcommand{\Padm}{\mathscr P_\Sigma} \newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}} \newcommand{\taus}{\tau_\Sigma} \newcommand{\Bspace}{\mathscr B} \newcommand{\Wspace}{\mathscr W} \newcommand{\Tspace}{\mathscr T} \newcommand{\Lspace}{\mathscr L} \newcommand{\Rspace}{\mathscr R} \newcommand{\Dnew}{D_{\mathrm{new}}} \newcommand{\Dold}{D_{\mathrm{old}}} \newcommand{\Ddecl}{D_{\mathrm{decl}}} \newcommand{\Dhat}{\widehat D_{\mathrm{decl}}} \newcommand{\Cdesc}{\mathcal C_{\mathrm{desc}}} \newcommand{\Cglue}{\mathcal C_{\mathrm{glue}}} \newcommand{\Rel}{\mathcal R} \newcommand{\btrace}{\mathfrak b} \newcommand{\Ddir}{\mathrm D} \newcommand{\Ezero}{\mathsf E_0} \newcommand{\GA}{G_{\mathrm A}^{\mathrm{chart}}} \newcommand{\dd}{\mathrm d} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\proj}[1]{\kb{#1}{#1}} \newcommand{\loc}[1]{\texttt{\detokenize{#1}}} \newcommand{\ddia}{d_\diamond} \newcommand{\dtr}{d_{\mathrm{tr}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\clip}{clip} \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{IX-C-ST: A supported partial product\\ for one sharp ordered word}\\[5pt] \large Sequential local Grade S adaptations of\\ IX-C, IX-Free, and IX-Free-2\\[4pt] \normalsize Actual-functional descent, boundary-charge separation,\\ and finite certificate transport on a prescribed cone} \author{} \date{} \begin{document} \maketitle \begin{abstract} We adapt the sequential IX-C--IX-Free--IX-Free-2 programme to a causally saturated regular laboratory on the prescribed I-B conical spacetime. The three gates are retained in order: first-stage products and exact descent criteria; canonical two-edge outputs and degree-two second-stage maps; and degree-four descent with simultaneous overlap agreement. The carrier is the already adapted IX-A-ST space of actual off-shell polynomial functionals. The covariance and the VII-D triangle finite part are supplied chart data. For the selected word \(B_f,W_g,B_{f'}\), we prove injectivity of both degree-two and both degree-four presentation maps. A chart-local advanced-trace witness separates every nonzero sharp generator from the partial contractions and the old microcausal algebra. An anchored tensor-domain argument then eliminates the old overlap of the entire new-domain sum, not merely its pairwise intersections. Consequently every fixed real common parent parameter passes both scalar descent and gluing: \(\mathcal C_{\mathrm{desc}}=\mathcal C_{\mathrm{glue}}=\mathbb R\). The prescribed coefficients define one truncated partial product modulo \(\hbar^4\), with presentation independence, reversal, and coherence of both associations of this word through order \(\hbar^3\). The potential value of the result is a controlled example in which actual-functional factorization is achieved without assuming continuity of unchanged sharp approximants or altering the old product. The algebraic findings are adapted from the cited Cheung manuscripts; the added construction is their supported conical transfer and a complete finite certificate interface, including destructive-clock readout and reference-preserving causal transport. No global product, actual-stage continuity, causal lift, interacting Ward identity, or spacetime emergence follows. The inherited ultraviolet analysis remains an explicit premise and is not re-proved. \end{abstract} \scopebox{Packaging and exact release boundary}{% The explicit request for one combined \loc{.tex} file is followed. Parts I--III retain the ordered stages IX-C-ST, IX-Free-ST, and IX-Free-2-ST. A later part uses only results already established in the earlier part and the shared stated inputs. Part I ends with an unresolved factorization gate. Part II advances that gate but still releases neither complete association. Part III constructs the declared truncated partial product. These intermediate verdicts are not contradictory and are not retrospectively erased. This is not IX-D-ST, IX-E-ST, or a unified IX-ST. No registry modification is asserted.} \tableofcontents %===================================================================== \section{Introduction, attribution, and the Enter interface} \label{sec:intro} %===================================================================== The selected programme concerns the exact off-shell insertion \(B_f\), a quadratic metric observable \(W_g\), and one ordered three-factor word. Its source papers have distinct releases: IX-C gives criteria; IX-Free constructs the degree-two blocks; IX-Free-2 completes the declared descent and gluing problem \cite{IXC,IXFree,IXFreeTwo}. A numerical three-test pairing is not, by itself, a binary map on actual intermediate functionals. That distinction determines the order of the proofs below. The operational background is Enter's distinction between a consumed quantum input name, a classical readout, and an unmeasured residual \cite{EnterArticle,EnterI}. The philosophical essay emphasizes that an output record is not the return of an unknown quantum state and that a finite interface does not identify its mathematical labels with a gravitational object \cite{EnterPhilosophy}. We use finite Born instruments with explicitly proved complete channel identities; no new measurement-uniqueness claim is made. Spacetime Enter adds locations, calibration, complete channels, and declared causal routes to a prescribed spacetime. The clock is a finite encoding. Its destructive readout does not remove the geometric time coordinate or derive the metric \cite{IB,En09}. The finite operational layer in Section~\ref{sec:operational} is independent of the distributional factorization proofs. \begin{definition}[Local Grade S] \label{def:grade} A local Grade S package consists here of: \begin{enumerate}[label=\textup{(S\arabic*)}] \item a prescribed spacetime and a supported regular laboratory; \item mathematical objects with stated domains and explicit inputs; \item complete finite channels, including arbitrary untouched references; \item one destructive clock with a declared calibration; \item computational support bounds and a separate physical causal-containment certificate; \item one nonempty finite transport realization with resources and durations accounted for. \end{enumerate} Grade E, meaning a derivation of spacetime or its metric from records or quantum data, is not constructed. \end{definition} \paragraph{Dependency discipline.} The carrier, covariance, old product, and parent finite part are the explicit inputs in Section~\ref{sec:inputs}. Their cited proofs are not independently certified here. In contrast, the presentation injectivity, graph-specific leaf arguments, local charge witness, and algebraic gluing are proved on the supported chart below. A documentary separation is not a theorem premise. \paragraph{Novelty boundary.} The main algebraic mechanism is attributed to Cheung's IX-C, IX-Free, and IX-Free-2. This manuscript does not claim priority for those results. Its contribution is a sequential, supported adaptation with a common host contract and a complete certificate interface. The latter does not strengthen a partial product into an interacting field theory. %===================================================================== \section{The prescribed host and the supported laboratory} \label{sec:host} %===================================================================== Fix a deficit \(0<\delta_{\mathrm c}<2\pi\), set \[ \alpha=1-\frac{\delta_{\mathrm c}}{2\pi}, \] and fix a conversion speed \(c_{\mathrm{light}}>0\). The regular host is the prescribed product \begin{equation} M_{\delta_{\mathrm c}} =\mathbb R_T\times C_{\delta_{\mathrm c}}^\circ\times\mathbb R_z, \qquad g_{\delta_{\mathrm c}} =-c_{\mathrm{light}}^2\dd T^2 +\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2, \qquad r>0, \label{eq:host} \end{equation} with \(\varphi\) periodic of period \(2\pi\). The stipulated completion has defect \[ \mathcal W=\mathbb R_T\times\{o\}\times\mathbb R_z. \] It is a worldsheet, not a worldline. The lapse in \(T\) is one; static proper time is \(T\). These data are prescribed, as in I-B \cite{IB}. \begin{proposition}[Failure of global hyperbolicity] \label{prop:nonGH} The regular product \eqref{eq:host} is not globally hyperbolic. \end{proposition} \begin{proof} Fix a regular spatial point of radius \(r_*>0\). Take two events there separated in \(T\) by \(4r_*/c_{\mathrm{light}}\). At their middle time, points on the same radial ray with radii tending to zero lie in the causal diamond between the two events. They have no convergent subsequence in the regular product. The diamond is not compact. \end{proof} \begin{definition}[Supported developing presentation] \label{def:support} Choose a product-adapted isometric developing embedding \[ \iota:U=(-\epsilon_0,H)\times B_R \longrightarrow M_{\delta_{\mathrm c}}, \qquad t=c_{\mathrm{light}}(T-T_\Sigma), \] with developed metric \(\eta=\operatorname{diag}(-1,1,1,1)\). Choose \[ V=(0,T_0)\times B_{r_0},\qquad R>r_0+T_0,\qquad K_*=[0,T_0]\times\overline B_{r_0+T_0}\Subset U. \] Observation tests belong to \[ \Tspace=\mathcal D(V;\mathbb C). \] For each such test \(f\), the whole truncated developed past \[ J^-_\eta(\supp f)\cap\{t\ge0\} \] is included in \(K_*\). The compact laboratory contains a neighbourhood of \(\iota(K_*)\), and its closure is disjoint from \(\mathcal W\). The response has zero value and zero first \(t\)-derivative at \(t=0\). No additional incoming homogeneous or lateral boundary data are supplied. All compact test families have a positive smoothing margin inside the supported presentation. \end{definition} The spacetime host is never identified with \(\mathbb R^{1+3}\). An auxiliary flat extension can be used to write comparison formulae, but only the supported data and pairings are transported to the host. \begin{proposition}[Supported response and chart operations] \label{prop:saturation} The IX-A-ST zero-Cauchy response on \(V\) is independent of auxiliary field extensions that agree near \(K_*\). The same applies to its fixed smooth derivatives and compact observation pairings. For a smooth vector field \(v\) compactly supported in \(V\), the supported prescription satisfies \begin{equation} \Ezero\Box v=v\quad\text{on }V, \qquad \Box=-\partial_t^2+\Delta_{\mathbf x}. \label{eq:compactidentity} \end{equation} This identity constructs no global conical inverse. \end{proposition} \begin{proof} The difference of two extensions has zero source on the complete retained past and zero initial response data. The local uniqueness and finite-propagation argument of IX-A-ST therefore gives zero response on \(V\) \cite[{\loc{lem:saturation}}]{IXAST}. For \eqref{eq:compactidentity}, \(v\) itself is a zero-data solution with source \(\Box v\). The same local uniqueness identifies it with the prescribed response. Only a supported Cauchy statement is used; the global Green-operator hypotheses are different \cite{BGP}. \end{proof} \begin{proposition}[Catalogue replacement is not metric evolution] \label{prop:fixedhost} Changing the regulator label, subtraction scale, stage label, or parent parameter \(c\) changes a description register. It leaves the host, lapse, initial surface, worldsheet, and installed causal relation unchanged. \end{proposition} \begin{proof} Those geometric objects are fixed inputs of Definition~\ref{def:support}. No metric-update map is an output of a catalogue command. \end{proof} This is the fixed-host separation used in VIII-ST \cite{STInterfaces}. %===================================================================== \section{Shared analytical inputs and notation} \label{sec:inputs} %===================================================================== Configurations \(\Psi=(h,\phi)\) and configuration directions are smooth and off shell on the supported presentation. Metric configurations are symmetric covariant tensors. All compact coefficient kernels are understood with their actual tensor symmetries and retained boundary terms. Set \begin{align} (Ch)^\mu &=\partial_\nu h^{\mu\nu} -\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}), & L&=\Ezero C,& w&=Lh,\nonumber\\ B_f[h,\phi] &=-\int_V f(x)w^\mu(x)\partial_\mu\phi(x)\,\dd^4x, & W_g[h]&=\int_V g(y)h_{00}(y)^2\,\dd^4y. \label{eq:generators} \end{align} The notation \(\Ezero\) refers only to the supplied supported comparison response. Write \[ x=(t,\mathbf x),\qquad y=(s,\mathbf y),\qquad x'=(t',\mathbf x'),\qquad X=V^3. \] Distributional pairings are complex linear in their test. Complex tests give complex-valued functionals on real configurations; complexified directions are equivalent by polarization. \subsection{The already adapted carrier} \begin{assumption}[IX-A-ST carrier and topology] \label{ass:carrier} The carrier is the already adapted \((\Padm,\taus)\) of IX-A-ST \cite{IXAST}, not a reconstruction of the original global Minkowski carrier. It has the following properties. \begin{enumerate}[label=\textup{(A\arabic*)}] \item Its elements are actual finite polynomial functionals with compact symmetric distributional coefficient kernels. The admission condition is \[ (\Ddir_{\boldsymbol\eta}F)'_h(\Psi)\in \mathscr S_1=\mathscr M_1+\mathscr Q_\Sigma \] for every smooth configuration and every finite list of smooth directions. \item The sharp linear kernels and intrinsic trace are \[ q_a=L^*a=C^*(\theta\,\GA a),\qquad \mathscr Q_\Sigma=\{q_a:a\in\mathcal D(V;T^*U)_{\mathbb C}\}, \] \[ \btrace(m+q_a)=(\GA a)|_\Sigma. \] Here \(\GA\) is the supplied advanced chart comparison prescription restricted to the retained support. Elements of \(\mathscr M_1\) have no nonzero nonspacelike wavefront covectors; compact smooth kernels belong to \(\mathscr M_1\). The trace is well defined on actual linear kernels. \item The sharp adjoint retains \begin{equation} C^*(\theta b) =\theta C^*b+\delta(t)\mathcal B(b|_\Sigma). \label{eq:sharpadjoint} \end{equation} No surface term is dropped. \item Classical pointwise multiplication, conjugation, and fixed smooth directional differentiation preserve \(\Padm\). Classical multiplication is separately continuous. The Hausdorff topology \(\taus\) records strong compact coefficient kernels and all intrinsic trace coordinates. \item The generating maps \(f\mapsto B_f\), \(g\mapsto W_g\) are fixed-support continuous, and \(B_f\), \(W_g\), the selected mixed coefficient below, and the selected scalar-contracted self coefficient belong to \(\Padm\). Their specified fixed-support families and trace coordinates are continuous. \item For every nonzero real \(f\), the unchanged regular approximants satisfy \[ B_{f,\varepsilon}\not\longrightarrow B_f \quad\text{in }\taus. \] \end{enumerate} No quantum product on all of \(\Padm\), no joint classical continuity, and no quantum hypocontinuity is included. \end{assumption} The intrinsic trace is called a charge below. It is not an electric charge, an energy, or a stress tensor. \subsection{Covariance and old product} The comparison covariance is \[ H_{hh,\alpha\beta\rho\sigma} =P_{\alpha\beta\rho\sigma}\Delta_0^+,\qquad H_{\phi\phi}=\Delta_m^+,\qquad H_{h\phi}=H_{\phi h}=0, \] \[ P_{\alpha\beta\rho\sigma} =\eta_{\alpha(\rho}\eta_{\sigma)\beta} -\frac12\eta_{\alpha\beta}\eta_{\rho\sigma}, \qquad m\ge0. \] Use \[ \Delta_m^+(z) =\int_{\mathbb R^3} \frac{\dd^3p}{(2\pi)^3\,2\omega_m(p)} e^{-i\omega_m(p)z^0+ip\cdot\mathbf z}, \qquad \omega_m(p)=\sqrt{|p|^2+m^2}, \] and \[ S_{\mu\nu}(x,x') =\partial_{x^\mu}\partial_{x'^\nu} \Delta_m^+(x-x'). \] These are supplied chart formulae, not covariances derived from the conical metric. \begin{assumption}[Locked mixed kernel and old product] \label{ass:old} Retain the VII-C tensor identities \[ H_{ww}=0,\qquad C_x^\mu H_{hh,\rho\sigma} =\delta^\mu_{(\rho}\partial^x_{\sigma)}\Delta_0^+. \] The nonzero mixed component with \(h_{00}\) is \begin{equation} \begin{split} K(x,y)= \int_{\mathbb R^3}\frac{\dd^3p}{(2\pi)^3} e^{ip\cdot(\mathbf x-\mathbf y)} \left[ \left(-\frac{t}{4|p|}+\frac{i}{8|p|^2}\right) e^{-i|p|(t-s)} -\frac{i}{8|p|^2}e^{i|p|(t+s)} \right]. \end{split} \label{eq:K} \end{equation} The reverse ordered mixed component is \(\overline{K(x,y)}\); the contraction selects vector index \(0\). The displayed infrared powers are locally integrable in three momentum dimensions. High-frequency integrals are distributional. The full old compact polynomial microcausal algebra \(\Pmc\) is included faithfully in \(\Padm\). Its unchanged coefficients are \[ C_j^H:\Pmc\otimes\Pmc\longrightarrow\Pmc,\qquad 0\le j\le3, \] with \(C_0^H(F,G)=FG\), supplied by \[ F\star_HG=\sum_{j\ge0}\frac{\hbar^j}{j!} \pair{F^{(j)}}{H^{\otimes j}G^{(j)}}. \] The old convention is Hermitian: \[ C_j^H(G^*,F^*)=C_j^H(F,G)^*. \] The microcausal wavefront condition excludes joint covectors entirely in either closed causal cone, including zero individual covectors in those cones. \end{assumption} These inputs retain the VII-C and IX-A-ST domain distinctions \cite{VIIC,IXAST}; the functional microcausal convention is the one reviewed in \cite{FR}. In particular \(H_{ww}=0\) does not imply \(K=0\). The selected already admitted comparison outputs are \begin{align} Q_{f,g} &=-2\pair{f(x)g(y)K(x,y)} {\partial_t\phi(x)h_{00}(y)},\label{eq:Q}\\ T_{f,f'} &=\pair{f(x)f'(x')S_{\mu\nu}(x,x')} {w^\mu(x)w^\nu(x')}. \label{eq:Tself} \end{align} Their admission is the explicit selected-output input in Assumption~\ref{ass:carrier}, not closure under an unspecified new contraction. \subsection{The parent numerical calibration} \begin{assumption}[VII-D parent, in the IX-B placement] \label{ass:parent} Fix the radial regulator \(\rho\), its prescribed placement on the two original field slots of each outer insertion, and \(\mu>0\). The middle \(W_g\) is unsmoothed. For that placement, \[ T_\varepsilon^\rho =2K_\varepsilon(x,y) \overline{K_\varepsilon(x',y)}D_\varepsilon(x-x'), \] where \(D_\varepsilon\) is the corresponding smoothed \(S_{00}\) comparison block. There are \(u_\mu^\rho,r\in\mathcal D'(X)\) with \[ u_\mu^\rho=U_m+(\kappa_\rho+\log\mu)r \] and, for every compact \(A\Subset X\), \begin{equation} \left|\pair{ T_\varepsilon^\rho-\log((\mu\varepsilon)^{-1})r-u_\mu^\rho }{\Phi}\right| \le C_{A,\rho,m}\varepsilon(1+|\log\varepsilon|) p_{A,64}(\Phi). \label{eq:parentestimate} \end{equation} The estimate holds on arbitrary joint tests supported in \(A\), including those crossing \(\Rel\cap\{x=x'\}\). The allowed positive widths are also restricted by the supported smoothing margin. The residue is the positive parameter measure \[ r=\frac1{64(2\pi)^3}\Psi_*\nu, \] restricted to \(X\), where \[ \Psi(\mathbf z,\mathbf n,t,s,t') =\bigl((t,\mathbf z+t\mathbf n), (s,\mathbf z-s\mathbf n), (t',\mathbf z+t'\mathbf n)\bigr), \] \[ \dd\nu=\dd^3z\,\dd\Omega(\mathbf n)\,\dd t\,\dd s\,\dd t'. \] Its support is the reflected observation relation \(\Rel\), not an observation equality diagonal. For allowed regulator changes, \[ u_\mu^{\rho_a}-u_\mu^{\rho_b}=c_{ab}r,\qquad c_{ab}\in\mathbb R, \] with no extra normal derivative, nonconstant weight, or independent intersection term in that parent conversion. Also \[ u_{\mu_2}^{\rho}-u_{\mu_1}^{\rho} =\log(\mu_2/\mu_1)r. \] The parent distributions have dagger symmetry under conjugation and exchange of \(x,x'\). \end{assumption} This is the stated VII-D input, as used by IX-B \cite{VIID,IXB}. The supplied IX-B-ST draft is the preferred supported presentation of the same calibration, but no public registration of that draft is assumed. Neither its existence nor this paper re-proves the ultraviolet ledger. Fix one real number \(c\) for an entire mathematical construction and put \begin{equation} U_c=u_\mu^\rho+cr,\qquad \mathcal N_c(f,g,f') =\pair{U_c}{f\otimes g\otimes f'}. \label{eq:parentclass} \end{equation} All logarithms use fixed units, so their arguments and scale ratios are dimensionless. One uses the same \(c\) for all presentations, both associations, and declared numerical observation descendants. \begin{lemma}[Numerical derivative transfer] \label{lem:transfer} For a fixed smooth multiplier \(a\) and a differential operator \(D\) of order \(d\) on \(X\), the parent finite part transfers to \(aDu_\mu^\rho\), with residue \(aDr\) and test order \(64+d\). Conversion of \(U_c\) becomes \(c\,aDr\), with the same \(c\). \end{lemma} \begin{proof} Transpose \(aD\) onto the test in \eqref{eq:parentestimate}. For a common compact support, \[ p_{A,64}\bigl(D^{\mathrm t}(a\Phi)\bigr) \le C_{A,a,D}p_{A,64+d}(\Phi). \] The constant is bounded by a finite polynomial in the \(C^{64+d}\)-seminorms of \(a\) and the coefficients of \(D\). The resulting bound tends to zero uniformly on bounded joint-test sets. The conversion assertion is linearity. \end{proof} This is numerical observation differentiation, not configuration differentiation, exposed functional-slot differentiation, host evolution, or a binary-product theorem. %===================================================================== \section{Shared finite clock, certificates, and regular transport} \label{sec:operational} %===================================================================== The construction in this section may be instantiated with any one of the three stage tags \[ \sigma\in\{\mathrm C,\mathrm{Free},\mathrm{Free2}\}. \] One finite experiment has one clock factor. A stage tag is classical programme data, not an additional geometric time coordinate. \subsection{Complete destructive readout} Let \(C\cong\mathbb C^{2^m}\), \(m\ge1\), have calibration basis \(\{\ket{\tau_k}\}\). Let \(S\) contain the surviving finite registers, and define \[ D_k=\bra{\tau_k}_C\otimes I_S,\qquad P_k=\proj{\tau_k}. \] \begin{theorem}[Clock instrument] \label{thm:instrument} The channel \begin{equation} \mathcal M_C(X) =\sum_k\proj k_R\otimes D_kXD_k^\dagger =\sum_k\proj k_R\otimes \Tr_C[(P_k\otimes I_S)X(P_k\otimes I_S)] \label{eq:clock} \end{equation} is CPTP and satisfies \[ \Tr_R\mathcal M_C(X)=\Tr_CX. \] All identities hold with every untouched reference. The quantum clock names are consumed. \end{theorem} \begin{proof} The recorded Kraus operators are \(\ket k_R\otimes D_k\). Their squared moduli sum to \(I_{CS}\). Expansion in the calibration basis identifies the \(k\)-th output block with the \(kk\)-input block and proves both formulae and the record-discard identity. Tensoring those identities proves the reference extension. There is no quantum \(C\)-output. \end{proof} The construction agrees with the I-B and En09-A interface \cite{IB,En09}. Consumption concerns the clock encoding, not \(T\), \(t\), the lapse, or the prescribed cone. \begin{proposition}[Fixed event and recovery limits] \label{prop:clocklimits} If distinct calibrated values \(t_k\) must all equal the time of one fixed event whenever their outcomes occur, at most one outcome has positive probability. The clock channel has no left inverse on all clock states. \end{proposition} \begin{proof} Distinctness proves the first assertion. For the second, the clock superpositions \((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), with identical surviving input, give the same output in \eqref{eq:clock}. They cannot both be recovered by a left inverse. \end{proof} A nontrivial clock catalogue is therefore a programme/history calibration unless a separate time-observable contract is supplied. \subsection{Finite evaluation records and a normed comparison} A finite catalogue entry records: \[ (\sigma,\hbar\text{-order},c\text{-identifier}, f,g,f'\text{-identifiers},\rho,\mu, \text{coefficient type},\text{error enclosure}). \] Only finitely many entries are installed in one experiment. For an arbitrary real \(c\), a finite entry may store its symbolic specification or a declared finite approximation. There is no finite-bit exact encoding of all arbitrary real numbers or all smooth functions. Let \(i\) be a classical catalogue index and let \(P\) be an arbitrary finite payload. Pinch the catalogue in its designated basis and write \[ X_{ki}= (\bra{\tau_k}\bra i\otimes I_P) X(\ket{\tau_k}\ket i\otimes I_P). \] For installed values \(v_{ki}\in[-1,1]\), set \[ p_y(k,i)=\frac{1+yv_{ki}}2,\qquad y=\pm1, \] and let \(r(k,i)\) be the fixed report word. Define \begin{equation} \mathcal I_v(X) =\sum_{k,i,y}\proj{k,i,y,r(k,i)}_W \otimes p_y(k,i)X_{ki}. \label{eq:tablechannel} \end{equation} \begin{theorem}[Fixed-interface evaluation certificate] \label{thm:table} The map \eqref{eq:tablechannel} is CPTP, reference preserving on its surviving factors, and has conditional sign mean \(v_{ki}\). For the same labels, pinching, report words, and payload interface, \begin{equation} \ddia(\mathcal I_v,\mathcal I_{\widetilde v}) =\frac12\max_{k,i}|v_{ki}-\widetilde v_{ki}|, \qquad \ddia(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond. \label{eq:tabledistance} \end{equation} \end{theorem} \begin{proof} Kraus operators are \[ \sqrt{p_y(k,i)}\, \ket{k,i,y,r(k,i)} (\bra{\tau_k}\bra i)\otimes I_P. \] Completeness follows by summing first over \(y\). This also proves the complete surviving-block statement. For the distance, include a reference and use orthogonality of all recorded flags. The trace norm of the output difference is at most \[ \max_{ki}|v_{ki}-\widetilde v_{ki}| \sum_{ki}\|X_{ki}\|_1 \le \max_{ki}|v_{ki}-\widetilde v_{ki}|\|X\|_1. \] The last inequality is trace-norm contractivity of pinching, which is an average of unitary conjugations, on arbitrary operators. A basis input at a maximizing index attains the bound. \end{proof} For a supplied complex coefficient evaluation \(z_{ki}\), one may install \[ v_{ki}=\clip(\operatorname{Re}z_{ki}/M_{ki}), \qquad M_{ki}>0,\qquad \clip(u)=\max\{-1,\min\{1,u\}\}. \] If \(|z_{ki}-\widehat z_{ki}|\le E_{ki}\), then \[ \ddia(\mathcal I_v,\mathcal I_{\widehat v}) \le\min\left\{1,\max_{ki}\frac{E_{ki}}{2M_{ki}}\right\}. \] For the parent numerical coefficient, an enclosure can use \[ E_\varepsilon \le e_T+|\log((\mu\varepsilon)^{-1})|e_r +C\varepsilon(1+|\log\varepsilon|) p_{A,64+d}(\Phi), \] when \(e_T,e_r\) are supplied enclosures of the regular pairing and residue pairing. This is a triangle-inequality consequence of Lemma~\ref{lem:transfer}. No executed quadrature, field acquisition, charge measurement, or residue measurement is asserted. \begin{proposition}[Record agreement is weaker than residual agreement] \label{prop:residual} For finite instruments \(\{\mathcal A_w\}\), \(\{\mathcal B_w\}\), record equality on all inputs is equivalent to \(\mathcal A_w^*(I)=\mathcal B_w^*(I)\). Equality on all recorded residual observables requires the stronger identities \(\mathcal A_w^*(B)=\mathcal B_w^*(B)\) for every declared residual observable \(B\). Equal records can coexist with residual trace-distance difference \(1/2\). \end{proposition} \begin{proof} Pull each output effect back to the input and use separation by density matrices. For the example, append the same one-outcome record to identity and computational dephasing. On \(\proj+\), their residuals are \(\proj+\) and \(I/2\). Their difference has eigenvalues \(1/2,-1/2\). \end{proof} The complete-instrument distinction agrees with the VII-E, VII-F, and VIII comparison records \cite{STInterfaces}. A finite table does not determine a residual source functional. \subsection{Scheduled locality} \begin{theorem}[Computational support and separate causal containment] \label{thm:locality} For a finite layered network of complete block channels, the backwards ancestor set \(N_{s,t}(Y)\) of a joint output \(Y\) satisfies \[ (\Phi_{s+1}^*\cdots\Phi_t^*)(\AA(Y)) \subseteq\AA(N_{s,t}(Y)). \] An intervention outside that set cannot affect the joint output. Reference-assisted comparison uses the joint ancestor--reference reduction. If every dependency port is placed in the regular laboratory and every dependency edge has a supplied future-causal route, then scheduled ancestry is contained in the prescribed physical causal past. \end{theorem} \begin{proof} A block adjoint is unital. A block disjoint from the current observable support acts on identity and changes nothing; a meeting block enlarges support only within that block. Reverse induction proves the inclusion, and testing output observables proves the intervention statement. Carry the untouched reference algebra throughout. Concatenating the supplied future-causal routes proves the last assertion. \end{proof} Thus actual influence is bounded by the complete scheduled ancestry; equality is not assumed. A distributional contraction graph, or a parenthesization schedule, is not itself a physical port network \cite{En05,STInterfaces}. \subsection{One nonempty transport realization} \begin{assumption}[Finite operational permissions] \label{ass:permissions} The regular laboratory permits declared finite preparations, local unitaries, classical processing and storage, destructive readout, and exact finite-register carriage along specified future-causal paths. These are ideal finite-channel permissions, not derived field dynamics or a microscopic Hamiltonian implementation. \end{assumption} Choose a developed spatial segment of length \(L_*>0\), with a compact regular tube in the laboratory. Install \(n+1\) stations along it, each with a \(p\)-qubit register, \(p\ge1\). Initialize all but the sender in a fixed blank. Read the clock and produce the complete finite report at the sender. Retain every outcome and wait for a positive buffer interval. Forward the classical certificate on explicit causal wires. At each hop exchange neighbouring payload contents by opposite carriage paths taking developed time \(2L_*/n\), not by an instantaneous remote gate. \begin{theorem}[Reference-preserving certificate transport] \label{thm:transport} The route transmits the unknown finite payload exactly, including every untouched reference, and forwards its complete classical certificate history. Every non-output payload register ends in a payload-independent blank. All communication dependencies are causal in the supplied metric. Orthogonal sender payloads give a receiver trace-distance contrast one. \end{theorem} \begin{proof} After hop \(j\), the sender payload factor and all its reference correlations occupy station \(j\); every other payload register is blank. Tensor-factor exchange proves this by induction. Copying an already classical history preserves its joint correlations and is not cloning an unknown quantum state. Each carriage speed is \(1/2\) in developed units; local processing and buffering are timelike. The receiver controls are used only after their classical inputs arrive. For the contrast use \(\proj0,\proj1\) on one payload qubit. \end{proof} The transported payload is a finite register, not the continuum field. The unit signalling witness is an installed transport operation, not entanglement generated by the cone \cite{IB,En08}. If the report has \(b\) bits, the catalogue \(g\) bits, and lookup uses \(w\) work bits, a conservative reserve is \[ m+g+1+(n+1)p+w+(n+1)b \] basis-state qubit carriers, plus separately declared external reference storage. There are \(n\) exchanges, \(n\) history transmissions, at most \(nb\) elementary history-copy operations, and the declared clock/probe readouts. All local processing has positive charged duration. Carriage takes \(2L_*\); preprocessing, copying, and buffering are added to it before fixing the laboratory time window. No reset or thermodynamic cost theorem is claimed. For compatible implementations on the same complete interfaces, telescoping gives \[ \ddia(\widetilde\Gamma,\Gamma) \le\min\{1,\epsilon_{\mathrm{table}} +\epsilon_{\mathrm{enc}} +\epsilon_{\mathrm{dec}} +\textstyle\sum_j\epsilon_j\}. \] This follows by replacing one channel factor at a time and using diamond-norm contraction \cite{Watrous}. It is not a norm bound for the continuum partial product. %===================================================================== \part{IX-C-ST: first stages and exact criteria} \label{part:C} %===================================================================== \section{Both first stages and the placement ledger} \label{sec:first} %===================================================================== \begin{lemma}[Injective generating tests] \label{lem:injective} The maps \(f\mapsto B_f\), \(g\mapsto W_g\) are injective. \end{lemma} \begin{proof} Sharp-test injectivity is included in the adapted carrier input; it also follows from the stronger local separation proved in Lemma~\ref{lem:restricted} below. For \(W_g\), polarization of \(\int g\,v^2=0\) gives \(\int g\,uv=0\). Choose \(v=1\) near \(\supp g\); arbitrary compact \(u\) then gives \(g=0\). \end{proof} \begin{proposition}[Reverse coefficient and its sign] \label{prop:reverse} The reverse first-stage coefficient is \begin{equation} Q^R_{g,f'} =-2\pair{g(y)f'(x')\overline{K(x',y)}} {h_{00}(y)\partial_{t'}\phi(x')}. \label{eq:QR} \end{equation} It belongs to \(\Padm\), and \[ Q_{f,g}^*=Q^R_{\overline g,\overline f}. \] \end{proposition} \begin{proof} The locked reverse covariance is the conjugate mixed kernel. The middle observable has two metric slots, giving factor \(2\); the single sharp insertion supplies one minus sign. Conjugation of \eqref{eq:Q} gives the displayed reversal. Admission follows from the carrier's conjugation closure. \end{proof} This retains IX-C's reverse normalization \cite[{\loc{prop:reverse}}]{IXC}. Define actual-functional subspaces \[ \Bspace=\{B_f:f\in\Tspace\},\qquad \Wspace=\{W_g:g\in\Tspace\}, \] \[ \Lspace_4=\Span\{B_fW_g\},\quad \Lspace_2=\Span\{Q_{f,g}\},\qquad \Rspace_4=\Span\{W_gB_{f'}\},\quad \Rspace_2=\Span\{Q^R_{g,f'}\}. \] Let \[ \Lspace=\Lspace_4\oplus\Lspace_2,\qquad \Rspace=\Rspace_4\oplus\Rspace_2. \] The sums are direct by total field degree. The actual spaces \(\Lspace_4,\Rspace_4\) coincide, but their ordered roles differ. \begin{theorem}[Admitted first-stage products] \label{thm:first} The prescriptions \begin{equation} B_f\star_1W_g=B_fW_g+\hbar Q_{f,g},\qquad W_g\star_1B_{f'}=W_gB_{f'}+\hbar Q^R_{g,f'} \label{eq:first} \end{equation} define bilinear maps on \(\Bspace\times\Wspace\) and \(\Wspace\times\Bspace\), with values in \[ \Padm^{[3]}=\Padm\otimes_{\mathbb C}\mathbb C[\hbar]/(\hbar^4). \] They satisfy classical normalization, locked first-stage normalization, and reversal. \end{theorem} \begin{proof} Injective generating tests make the displayed bilinear assignments unambiguous on the actual argument spaces. Classical closure and the admitted mixed outputs place their coefficients in \(\Padm\). Proposition~\ref{prop:reverse} proves reversal. \end{proof} \begin{proposition}[Fixed-support first-stage continuity] \label{prop:firstcontinuity} On fixed compact observation supports, \((f,g)\mapsto Q_{f,g}\) and its reverse are jointly continuous into the strong coefficient topology. Their metric-derivative trace coordinates vanish, also after all required directional insertions. The first-stage generating families are continuous into \((\Padm^{[3]},\taus)\). \end{proposition} \begin{proof} The coefficient of \(Q\) is obtained by multiplying the fixed distribution \(K\) by \(f\otimes g\) and taking one fixed scalar-slot derivative. On common supports, finite-order distribution estimates bound this operation by finitely many smooth seminorms of \(f,g\), uniformly on bounded smooth test sets. Contracting its scalar slot with a smooth configuration leaves a compact smooth metric kernel, by the selected output property in the carrier input. Replacing the scalar configuration by a fixed scalar direction has the same property. A metric insertion removes the only metric slot. All higher derivatives vanish. Thus every relevant trace coordinate is zero. The classical first-stage term has the inherited generating test-family continuity. Combining its coordinates gives the last assertion. \end{proof} This is continuity along specified test families, not hypocontinuity on arbitrary carrier stages \cite[{\loc{prop:firstcontinuity}}]{IXC}. For the selected tests set \begin{equation} G_{12}=Q_{f,g}B_{f'},\qquad G_{23}=B_fQ^R_{g,f'},\qquad G_{13}=T_{f,f'}W_g. \label{eq:oneedges} \end{equation} \begin{proposition}[One-edge outputs] \label{prop:oneedges} All three outputs in \eqref{eq:oneedges} exist in \(\Padm\), with all required directional descendants. They have field degree four. \end{proposition} \begin{proof} Every factor is already admitted. Their products are classical pointwise products, so the carrier's classical closure applies. No new multiplication in singular variables is used. \end{proof} At this stage the desired two-edge placements are only labels: \begin{align} G_{12,23}^{\mathrm{lab}} &=2\pair{V_3K(x,y)\overline{K(x',y)}} {\partial_t\phi(x)\partial_{t'}\phi(x')}, \nonumber\\ G_{12,13}^{\mathrm{lab}} &=2\pair{V_3K(x,y)S_{0\nu}(x,x')} {h_{00}(y)w^\nu(x')}, \label{eq:twolab}\\ G_{13,23}^{\mathrm{lab}} &=2\pair{V_3S_{\mu0}(x,x')\overline{K(x',y)}} {w^\mu(x)h_{00}(y)}, \nonumber \end{align} where \(V_3=f\otimes g\otimes f'\). \begin{proposition}[Signs and multiplicities] \label{prop:placementcount} The three two-edge placements have insertion sign \(+1\), assignment multiplicity \(2\), and residual degree two. Their fully contracted completion has the parent coefficient \(T_\varepsilon^\rho\). \end{proposition} \begin{proof} The only available nonzero edges are metric edges \(12,23\) and scalar edge \(13\). Each two-edge choice removes four of the six slots. Both outer insertion signs occur, giving \(+1\). For \(12+23\), the two middle slots have two assignments. For \(12+13\), either middle slot is used, again giving two; the third case is its reverse. The tensor identity fixes index \(0\) on each mixed metric edge. Full contraction adds the remaining edge and gives the supplied triangle. \end{proof} This counts slots; it does not yet prove the products in \eqref{eq:twolab} exist \cite{IXC}. %===================================================================== \section{Descent, scalar tests, and coherence equations} \label{sec:criteria} %===================================================================== All spans and tensor products below are algebraic over \(\mathbb C\). Let \[ E_L=\Tspace\otimes\Tspace,\qquad E_R=\Tspace\otimes\Tspace \] and define \[ p_L(f\otimes g)=B_fW_g,\quad q_L(f\otimes g)=Q_{f,g}, \] \[ p_R(g\otimes f')=W_gB_{f'},\quad q_R(g\otimes f')=Q^R_{g,f'}. \] \begin{theorem}[Algebraic descent criterion] \label{thm:descent} Let \(v:E\to A\) be surjective and let \(\lambda:E\otimes Z\to Y\) be linear. There is a unique bilinear map \(\ell:A\times Z\to Y\) with \[ \ell(v(e),z)=\lambda(e\otimes z) \] if and only if \[ \lambda((\ker v)\otimes Z)=0. \] For two presented arguments, the condition is vanishing on \[ (\ker v)\otimes F+E\otimes(\ker w). \] \end{theorem} \begin{proof} Necessity follows by inserting a zero argument. For sufficiency, two presentations of the same argument differ by an element of the indicated kernel; the condition makes their assigned values equal. Linearity gives bilinearity and surjectivity gives uniqueness. Apply the argument to both slots for the last assertion. \end{proof} This is the IX-C actual-functional criterion \cite[{\loc{thm:descent}}]{IXC}. It is not a topological quotient theorem. Define the scalar relation space \[ Z_{\mathrm{rel}} =(\ker q_L)\otimes\Tspace \ \oplus\ \Tspace\otimes(\ker q_R). \] After the natural ordering of its three test factors, let \(\alpha,\beta\) be its pairings with \(u_\mu^\rho,r\). \begin{theorem}[Scalar relation tests] \label{thm:scalarfactor} The calibrated scalar assignments \[ \mathsf L_{2,c}(Q_{f,g},B_{f'}) =\mathcal N_c(f,g,f')\,1, \quad \mathsf R_{2,c}(B_f,Q^R_{g,f'}) =\mathcal N_c(f,g,f')\,1 \] descend exactly when \[ \alpha+c\beta=0\quad\text{on }Z_{\mathrm{rel}}. \] The permitted real set is empty, a singleton, or all of \(\mathbb R\). \end{theorem} \begin{proof} The descent criterion gives precisely those linear relation equations, using injectivity of the remaining sharp argument. Each equation is affine in \(c\). If two distinct real values solve every equation, their difference forces \(\beta=0\), and then \(\alpha=0\); all real values solve. If there are not two solutions, the solution set is empty or a singleton. \end{proof} Part I does not evaluate this test. Part II does. Let \(\mathsf L_j,\mathsf R_j\) denote proposed second-stage coefficient maps on the left and right intermediate spaces, with order zero classical. Write \(P=B_fW_g\), \(Q=Q_{f,g}\), \(P'=W_gB_{f'}\), \(Q'=Q^R_{g,f'}\). \begin{theorem}[Coefficientwise coherence criterion] \label{thm:criterioncoherence} Assuming that the displayed second-stage maps exist on actual arguments and that the scalar coefficient is supplied only by the degree-two block, the two associations agree modulo \(\hbar^4\) exactly when \begin{align} \mathsf L_1(P,B_{f'})+QB_{f'} &=\mathsf R_1(B_f,P')+B_fQ', \label{eq:assoc1}\\ \mathsf L_2(P,B_{f'})+\mathsf L_1(Q,B_{f'}) &=\mathsf R_2(B_f,P')+\mathsf R_1(B_f,Q'),\label{eq:assoc2}\\ \mathsf L_{2,c}(Q,B_{f'}) &=\mathsf R_{2,c}(B_f,Q').\label{eq:assoc3} \end{align} \end{theorem} \begin{proof} Expand the first stage \(P+\hbar Q\) and then its second multiplication, and do the same for \(P'+\hbar Q'\). The order-zero equality is classical associativity. Equating the next three coefficients gives exactly \eqref{eq:assoc1}--\eqref{eq:assoc3}. \end{proof} The intended prescriptions are \[ \begin{array}{ll} \mathsf L_1(P,B_{f'})=G_{13}+G_{23}, &\mathsf R_1(B_f,P')=G_{12}+G_{13},\\ \mathsf L_2(P,B_{f'})=G_{13,23}, &\mathsf R_2(B_f,P')=G_{12,13},\\ \mathsf L_1(Q,B_{f'})=G_{12,23}+G_{12,13}, &\mathsf R_1(B_f,Q')=G_{12,23}+G_{13,23}. \end{array} \] Until existence, admission, descent, and overlap agreement are proved, these remain prescription equations \cite[{\loc{thm:coherence}}]{IXC}. \begin{proposition}[Conditional presentation independence] \label{prop:Cpresentation} If a coefficient prescription descends through every actual presentation kernel and glues consistently on its actual tensor domain, its value is independent of presentation. No such conclusion follows from agreement of formal labels alone. \end{proposition} \begin{proof} The difference of two presentations is a relation annihilated by descent. Any further change of domain decomposition is annihilated by the gluing condition. \end{proof} \subsection*{Part I release} \begin{theorem}[IX-C-ST stage release] \label{thm:Crelease} Relative to the shared inputs, Part I constructs both admitted first stages, their fixed-support continuity, the one-edge outputs, and exact descent, scalar, and coherence criteria. With Section~\ref{sec:operational}, it forms a local Grade S criteria package. At this stage neither sharp second multiplication is constructed. Its factorization release is \emph{Unresolved}, not a universal obstruction. \end{theorem} \begin{proof} The claims are Theorem~\ref{thm:first}, Propositions~\ref{prop:firstcontinuity}, \ref{prop:oneedges}, \ref{prop:placementcount}, and the criteria of this section. The common operational layer does not use an unconstructed product. The graph products and relation-kernel discharges needed for a second multiplication have not been used in Part I. \end{proof} %===================================================================== \part{IX-Free-ST: presentation kernels and degree-two maps} \label{part:Free} %===================================================================== \section{Concrete relations and evaluated scalar descent} \label{sec:relations} %===================================================================== For \(v=\sum_i a_i f_i\otimes g_i\in E_L\), use the faithful algebraic embedding \[ v(x,y)=\sum_i a_if_i(x)g_i(y)\in\mathcal D(V^2). \] Faithfulness follows by choosing linearly independent first factors and applying dual linear functionals on their finite span. No completed tensor product is substituted. \begin{proposition}[Degree-two relation kernels] \label{prop:relations} The exact relation equations are \begin{align} q_L(v)=0 &\Longleftrightarrow \partial_t(vK)=0,\label{eq:qrelation}\\ q_R(v_R)=0 &\Longleftrightarrow \partial_{t'}(v_R(y,x')\overline{K(x',y)})=0. \nonumber \end{align} \end{proposition} \begin{proof} Observation-variable integration by parts gives \[ q_L(v)=2\pair{\partial_t(vK)}{\phi(x)h_{00}(y)}. \] There is no boundary term because \(v\) is compactly supported inside \(V^2\). The independent scalar and metric slots recover the coefficient on all product tests. Product-test density identifies the joint distribution. The reverse calculation is identical. \end{proof} Let \(J^\mu_{\alpha\beta}(x,z)\) be the supported kernel of \(L\), and let \(e_{00}^{\alpha\beta}h_{\alpha\beta}=h_{00}\). Define \begin{equation} \begin{split} \mathsf P[v](x,z_1,z_2,z_3) =\Sym_{(z_j,\alpha_j,\beta_j)} \partial_{x^\mu}\bigl[ v(x,z_2)J^\mu_{\alpha_1\beta_1}(x,z_1) \delta(z_2-z_3) e_{00}^{\alpha_2\beta_2}e_{00}^{\alpha_3\beta_3} \bigr]. \end{split} \label{eq:Prelationkernel} \end{equation} Symmetrization is averaging over the three metric slots. \begin{proposition}[Degree-four relation kernels] \label{prop:prelations} The kernel \(\mathsf P[v]\) is defined and compact in its field variables, and \[ p_L(v)=\pair{\mathsf P[v]} {\phi(x)h(z_1)h(z_2)h(z_3)}. \] Consequently \[ p_L(v)=0\Longleftrightarrow \mathsf P[v]=0. \] The right relation is the exchanged equation with \(v(x,y)=v_R(y,x)\). \end{proposition} \begin{proof} Before smooth multiplication, \(J(x,z_1)\) and \(\delta(z_2-z_3)\) are tensor factors in disjoint variables. Observation integration by parts gives the positive derivative in \eqref{eq:Prelationkernel}. Saturated finite propagation gives compact field support. One scalar derivative and three metric derivatives recover the symmetric kernel with factor \(3!\). Polarization and product-test density prove the equivalence. \end{proof} These are the concrete IX-Free relations \cite[{\loc{prop:relations}}, {\loc{prop:prelations}}]{IXFree}. Neither equation has yet been solved for \(p_L,p_R\). \begin{lemma}[Full distributional support of \(K\)] \label{lem:fullsupport} The restriction of \eqref{eq:K} to \(V^2\) has \[ \supp K=V^2. \] \end{lemma} \begin{proof} For \(a=t-s\), distributional differentiation of \eqref{eq:K} gives \[ \Box_xK=\partial_a\Delta_0^+(a,\mathbf x-\mathbf y). \] The terms without the explicit factor \(t\) solve the homogeneous wave equation. The \(t\)-term yields the displayed derivative. Away from \(a=0\) and \(|\mathbf x-\mathbf y|^2=a^2\), Abel damping of the radial Fourier integral gives \[ \Delta_0^+(a,\mathbf z) =\frac1{4\pi^2}\frac1{|\mathbf z|^2-(a-i0)^2}, \qquad \partial_a\Delta_0^+ =\frac{a}{2\pi^2(|\mathbf z|^2-a^2)^2}\ne0. \] That set is dense in every open product chart. Thus \(\Box_xK\) has full support, and a differential operator cannot enlarge support. Hence \(K\) has full support as well. \end{proof} This is chart support, not a theorem about the defect \cite[{\loc{lem:fullsupport}}]{IXFree}. \begin{lemma}[Compact primitive] \label{lem:primitive} If \(I\) is a connected open interval and \(T\in\mathcal E'(I\times N)\) satisfies \(\partial_tT=0\), then \(T=0\). \end{lemma} \begin{proof} A compact test of integral zero on \(I\) has a compact primitive. Thus \(T\) depends on a time test only through its integral. Choose an integral-one test disjoint from the compact time projection of \(\supp T\). Its pairing is zero, so every such constant-in-time pairing is zero. Product-test density gives the result. \end{proof} \begin{theorem}[Injective partial-contraction presentations] \label{thm:qinjective} Both \(q_L:E_L\to\Lspace_2\) and \(q_R:E_R\to\Rspace_2\) are algebraic isomorphisms onto their images. \end{theorem} \begin{proof} Equation~\eqref{eq:qrelation} and Lemma~\ref{lem:primitive} give \(vK=0\). If \(v\) were nonzero at a point, it would have a smooth reciprocal near that point, making \(K=0\) on a nonempty open set. Lemma~\ref{lem:fullsupport} excludes this. Hence \(v=0\), including as an algebraic tensor. Conjugation and variable exchange give the right case. \end{proof} \begin{corollary}[Scalar descent is unrestricted within the parent class] \label{cor:cdesc} For this sector, \[ Z_{\mathrm{rel}}=0,\qquad \Cdesc=\mathbb R. \] For every fixed real \(c\), both calibrated scalar maps in Theorem~\ref{thm:scalarfactor} exist uniquely as algebraic maps on the actual degree-two intermediate blocks. \end{corollary} \begin{proof} Both relation kernels vanish by Theorem~\ref{thm:qinjective}. The actual sharp argument has its unique observation test. Thus the parent pairing descends, and constants belong to \(\Padm\). \end{proof} This evaluates descent only. The overlap problem remains separate \cite[{\loc{cor:cdesc}}]{IXFree}. %===================================================================== \section{Canonical two-edge distributions and full admission} \label{sec:trees} %===================================================================== \begin{lemma}[Nonzero comparable endpoint covectors] \label{lem:endpoints} Every wavefront covector of \(K,\overline K\), or \(S_{\mu\nu}\) has nonzero covectors in both endpoint slots. Their coordinate Euclidean norms are equal. \end{lemma} \begin{proof} Split each Fourier integral into bounded and unbounded momenta. The bounded-momentum part is smooth, since every coordinate derivative remains integrable there. The two high-frequency mixed phases have endpoint covectors \[ (-|p|,p;\ |p|,-p),\qquad (|p|,p;\ |p|,-p). \] The massive scalar phase has conic limiting directions given by opposite nonzero null covectors. Oscillatory integration by parts excludes directions outside these phase-generated cones. Multiplication by smooth amplitudes, differentiation, and conjugation preserve the asserted endpoint property. \end{proof} \begin{lemma}[Two-edge leaf criterion and one-vertex smoothing] \label{lem:forest} Pull back two such endpoint kernels to a three-vertex two-edge tree. Their product exists. After multiplication by a compact smooth joint coefficient, integration over all vertices except any one gives a smooth function at the remaining vertex. \end{lemma} \begin{proof} Each edge has a leaf not belonging to the other edge. At that slot, only its own nonzero edge covector occurs. The two wavefront covectors therefore cannot sum to zero, which verifies the distributional product criterion. For pushforward to one vertex, every integrated slot must have zero covector. Remove an integrated leaf different from the retained vertex. Its incident edge must contribute zero; endpoint norm equality forces its other endpoint contribution to vanish as well. Repeat for the remaining edge. No nonzero covector survives at the retained vertex. The same norm comparison excludes one-slot directions in closed conic limits. Compact multiplication makes the pushforward proper, so its wavefront set is empty. \end{proof} The graph-specific checks are the IX-Free leaf argument \cite[{\loc{lem:endpoints}}, {\loc{lem:forest}}]{IXFree}. The distributional product, submersion pullback, and pushforward theorems used here are standard microlocal operations; their topologies must be specified separately \cite{Hormander,BDH}. Define \[ E_a=2V_3K(x,y)\overline{K(x',y)},\qquad E_{b,\nu}=2V_3K(x,y)S_{0\nu}(x,x'), \] \[ E_{d,\mu}=2V_3S_{\mu0}(x,x')\overline{K(x',y)}. \] \begin{proposition}[Existence of the three canonical outputs] \label{prop:existence} These are well-defined compact distributions on all of \(X\). The labels in \eqref{eq:twolab} now define actual quadratic functionals \(G_{12,23},G_{12,13},G_{13,23}\). Their compact field-variable coefficient kernels exist. \end{proposition} \begin{proof} Apply Lemma~\ref{lem:forest} to leaves \(x,x'\), then \(y,x'\), then \(x,y\). For the scalar-only output, its coefficient kernel is \[ \Sym_{x,x'}\partial_t\partial_{t'}(\pi_{13})_*E_a. \] For \(G_{12,13}\), first form \[ R_\nu(y,x')=(\pi_{23})_*E_{b,\nu}, \] then transpose \(L\) in the \(x'\)-variable and symmetrize the metric slots. This transposition is defined: applying \(L\) to a smooth joint test in one variable is continuous in the joint smooth topology, including parameter derivatives. Its transpose therefore acts on compact distributions. The third output uses the corresponding first-variable transpose. Finite propagation confines all field supports to the retained compact laboratory. \end{proof} \begin{theorem}[Admission of all three two-edge outputs] \label{thm:treeadmission} Every output in Proposition~\ref{prop:existence} belongs to \(\Padm\), including all metric admission tests after arbitrary finite smooth directional insertions. \end{theorem} \begin{proof} The scalar-only output has no metric derivatives. For \(G=G_{12,13}\), set \[ m_h(y) =\int E_{b,\nu}(x,y,x')w^\nu[h](x') \,\dd^4x\,\dd^4x', \] \[ a_{h,\nu}(x') =\int E_{b,\nu}(x,y,x')h_{00}(y) \,\dd^4x\,\dd^4y. \] These expressions denote proper distributional pushforwards. Lemma~\ref{lem:forest} makes them compact smooth functions. Direct differentiation gives the actual linear metric kernel \begin{equation} G'_h[h]=e_{00}m_h+L^*a_h=e_{00}m_h+q_{a_h}. \label{eq:treeadmission} \end{equation} It lies in \(\mathscr M_1+\mathscr Q_\Sigma\). A first metric directional insertion replaces \(h\) and \(w[h]\) in these formulae by the fixed direction and its smooth response. The same decomposition holds. Two directional insertions into the quadratic functional give a constant; a further metric derivative is zero. Scalar directions have no effect. This checks every finite direction list. For \(G_{13,23}\), integrate its tree to \(y\) and to \(x\) instead of \(y\) and \(x'\), and repeat the same argument. \end{proof} This is full admission, not merely existence of a compact kernel \cite[{\loc{thm:treeadmission}}]{IXFree}. No extension of a failed product is hidden in the proof: the two-edge wavefront products exist on all \(X\). %===================================================================== \section{Degree-two maps and the second staged release} \label{sec:degree2} %===================================================================== \begin{theorem}[Actual degree-two second-stage maps] \label{thm:degree2maps} For every fixed \(c\in\mathbb R\), there are unique algebraic bilinear maps on \(\Lspace_2\times\Bspace\) and \(\Bspace\times\Rspace_2\) with \begin{align} \mathsf L_0(F,B)&=FB,& \mathsf R_0(B,F)&=BF,\nonumber\\ \mathsf L_1(Q_{f,g},B_{f'}) &=G_{12,23}+G_{12,13},& \mathsf R_1(B_f,Q^R_{g,f'}) &=G_{12,23}+G_{13,23},\label{eq:degree2maps}\\ \mathsf L_{2,c}(Q_{f,g},B_{f'}) &=\mathcal N_c(f,g,f')1,& \mathsf R_{2,c}(B_f,Q^R_{g,f'}) &=\mathcal N_c(f,g,f')1.\nonumber \end{align} All values are in \(\Padm\). \end{theorem} \begin{proof} Theorem~\ref{thm:qinjective} provides the unique algebraic presentation tensor of the intermediate argument. The other argument has a unique sharp test. The displayed values are linear in those presentations. Their target membership follows from classical closure, Theorem~\ref{thm:treeadmission}, and Corollary~\ref{cor:cdesc}. \end{proof} Reversal on these blocks follows by conjugating the kernels and exchanging \(x,x'\): \(G_{12,13}\) is exchanged with \(G_{13,23}\), while \(G_{12,23}\) reverses to itself with conjugated tests. The real-parent dagger symmetry gives scalar reversal. \begin{proposition}[F9a on the completed degree-two blocks] \label{prop:F9a2} The coefficient families in \eqref{eq:degree2maps} are jointly continuous on fixed-support generating test spaces, with values in \((\Padm,\taus)\). This does not assert continuity of \(q_L^{-1}\) or \(q_R^{-1}\) in the inherited subspace topology. \end{proposition} \begin{proof} Multiplication of each fixed graph distribution by a varying smooth joint coefficient is continuous in its wavefront-controlled distribution space. The proper pushforwards and fixed transpositions in Proposition~\ref{prop:existence} then give strong coefficient continuity \cite{BDH}. For trace coordinates use \eqref{eq:treeadmission}. The one-vertex pushforwards are continuous into smooth functions by the verified empty-wavefront pushforward bound. Applying the fixed smooth advanced trace map to \(a_h\) is continuous. Replacing \(h\) by any fixed direction proves the same for all nonzero directional trace coordinates. For the scalar parent, a fixed distribution has a finite order on every compact set, which bounds product-test pairings by finitely many test seminorms. No inverse presentation map appears in these estimates. \end{proof} \begin{theorem}[IX-Free-ST stage release] \label{thm:Freerelease} Part II proves the concrete relation equations, full distributional chart support of \(K\), injectivity of \(q_L,q_R\), \(\Cdesc=\mathbb R\), all three canonical two-edge outputs with full admission, and the actual degree-two maps with reversal and F9a continuity. Together with the common operational layer, this is a local Grade S partial construction. At this stage degree-four descent and simultaneous new-domain/old-domain overlap agreement remain undischarged. Neither complete sharp association is released. F9b, continuity on actual carrier stages, is unclaimed. \end{theorem} \begin{proof} The assertions are the results of Sections~\ref{sec:relations}--\ref{sec:degree2}. They do not yet evaluate \(\ker p_L,\ker p_R\) or prove gluing of all tensor domains. Those are separate obligations, not consequences of the degree-two injectivity theorem. \end{proof} %===================================================================== \part{IX-Free-2-ST: degree-four descent and simultaneous gluing} \label{part:FreeTwo} %===================================================================== \section{Restricted sharp separation and degree-four descent} \label{sec:degree4} %===================================================================== \begin{lemma}[Restricted sharp separation] \label{lem:restricted} If \(B_f[k,\varphi]=0\) for every compact smooth metric direction \(k\) supported in \(V\) with \(k_{00}=0\), and every smooth scalar \(\varphi\), then \(f=0\). \end{lemma} \begin{proof} For \(\chi\in\mathcal D(V)\), let \(H_{11}=\chi\) be the only nonzero component of \(H\), and take \(k=\Box H\), where \(\Box\) is the chart operator. Then \(k_{00}=0\) and \[ Ck=\Box(CH),\qquad Lk=CH \] by Proposition~\ref{prop:saturation}. In particular \((CH)^0=\frac12\partial_t\chi\). Choose a smooth scalar equal to \(t\) near \(\supp f\). The assumed vanishing gives \[ 0=-\frac12\int f\,\partial_t\chi\,\dd^4x \] for every \(\chi\), hence \(\partial_tf=0\) in \(V\). A time-independent smooth function compactly supported inside \((0,T_0)\times B_{r_0}\) is zero. \end{proof} This proof uses neither a general range theorem for \(C\) nor a host Green operator \cite[{\loc{lem:restricted}}]{IXFreeTwo}. \begin{theorem}[Injective degree-four presentations] \label{thm:pinjective} The maps \(p_L:E_L\to\Lspace_4\) and \(p_R:E_R\to\Rspace_4\) are algebraic isomorphisms onto their images. Thus \[ \ker p_L=\ker p_R=0,\qquad \mathsf P[v]=0\Longleftrightarrow v=0. \] \end{theorem} \begin{proof} Suppose \[ p_L(v)[h,\phi] =-\int v(x,y)(Lh)^\mu(x) \partial_\mu\phi(x)h_{00}(y)^2\,\dd^4x\,\dd^4y=0. \] Differentiate once in a scalar direction \(\varphi\) and three times in metric directions \(k,a,b\), at zero, where \(k_{00}=0\), \(a_{00}=u\), \(b_{00}=z\). Only the assignments placing \(k\) in the response slot survive. The result is \[ -2\int v(x,y)(Lk)^\mu(x) \partial_\mu\varphi(x)u(y)z(y) \,\dd^4x\,\dd^4y=0. \] For \[ f_{u,z}(x)=\int v(x,y)u(y)z(y)\,\dd^4y \] Lemma~\ref{lem:restricted} gives \(f_{u,z}=0\). Any compact \(y\)-test \(d\) is such a product: take \(u=d\) and \(z=1\) near \(\supp d\). Every \(y\)-test contraction of \(v\) is therefore zero, so \(v=0\). Faithfulness of the algebraic test embedding finishes the left case. The right case is the same classical product with exchanged test factors. Proposition~\ref{prop:prelations} gives the last equivalence. \end{proof} The surviving factor \(2\) in slot isolation is compatible with recovery of the completely symmetric coefficient by three metric derivatives with factor \(3!\). This is not the false general principle that products of injective generating maps are automatically tensor-injective \cite[{\loc{thm:pinjective}}]{IXFreeTwo}. \begin{theorem}[All four degree-four prescriptions descend] \label{thm:degree4} The assignments \begin{align} \mathsf L_1(B_fW_g,B_{f'})&=G_{13}+G_{23},& \mathsf R_1(B_f,W_gB_{f'})&=G_{12}+G_{13},\nonumber\\ \mathsf L_2(B_fW_g,B_{f'})&=G_{13,23},& \mathsf R_2(B_f,W_gB_{f'})&=G_{12,13} \label{eq:degree4maps} \end{align} define unique algebraic bilinear maps on \(\Lspace_4\times\Bspace\) and \(\Bspace\times\Rspace_4\), with values in \(\Padm\). \end{theorem} \begin{proof} The two degree-four presentation kernels are zero. All four descent tests in Theorem~\ref{thm:descent} therefore hold. Every value is a previously proved admitted graph or sum of such graphs. No new admission claim is inferred from generator admission alone. \end{proof} %===================================================================== \section{Boundary-charge separation in the regular chart} \label{sec:charge} %===================================================================== The charge argument takes place near the regular initial surface \(\Sigma\), whose conormal is timelike in the developed metric. It concerns no singularity on \(\mathcal W\). \begin{lemma}[A nonzero surface coefficient is not microcausal] \label{lem:surfaceWF} If \(b\) is smooth near \(\Sigma\) and \(b|_\Sigma\ne0\), then \(C^*(\theta b)\) has a nonzero timelike wavefront covector. \end{lemma} \begin{proof} In the fixed component convention the surface map in \eqref{eq:sharpadjoint} has \[ \mathcal B(b)^{00}=-\frac12b_0,\qquad \mathcal B(b)^{0i}=\frac12b_i,\qquad \mathcal B(b)^{ij}=-\frac12\delta_{ij}b_0. \] It is injective. Some component has local form \[ \delta(t)\beta(\mathbf x)+\theta(t)c(t,\mathbf x), \qquad \beta\ne0. \] Choose compact localizers whose spatial pairing with \(\beta\) is nonzero. Its Fourier transform along \((\tau,0)\) is a nonzero constant plus \(O(|\tau|^{-1})\); the latter bound follows by one-sided integration by parts on the smooth Heaviside amplitude. It is not rapidly decreasing. If no timelike wavefront covector occurred on that compact patch, a finite microlocal partition would imply rapid decay in a cone about the normal rays, a contradiction. \end{proof} The proof also shows why a smooth one-sided bulk term does not cancel the delta coefficient \cite[{\loc{lem:surfaceWF}}]{IXFreeTwo}. \begin{lemma}[Microcausal linear kernels have zero charge] \label{lem:MCcharge} If \(\ell\in\mathscr S_1\) has no nonzero nonspacelike wavefront covector, then \(\btrace(\ell)=0\). \end{lemma} \begin{proof} Write \(\ell=m+q_a\). Both \(\ell\) and \(m\) avoid nonspacelike wavefront covectors, so their difference \(q_a\) does too. The supplied smooth response \(b=\GA a\) is smooth near \(\Sigma\). Lemma~\ref{lem:surfaceWF} forces its trace to vanish. The intrinsic trace formula gives the conclusion. \end{proof} \begin{proposition}[Zero-charge comparison classes] \label{prop:zerocharge} At every smooth configuration and after every finite smooth directional insertion, \[ \btrace((\Ddir_{\boldsymbol\eta}F)'_h(\Psi))=0 \quad\text{for }F\in\Lspace_2\cup\Rspace_2\cup\Pmc. \] \end{proposition} \begin{proof} For \(Q,Q^R\), every nonzero remaining metric kernel is compact smooth, as in Proposition~\ref{prop:firstcontinuity}. For \(F\in\Pmc\), contraction of higher functional derivatives with smooth directions is a proper pushforward after compact cutoffs on the coefficient supports. A wavefront covector of the remaining metric slot would come from a joint covector with all integrated slots zero. A nonzero causal covector there would make that joint covector entirely future or entirely past causal, excluded by the old microcausal condition. The remaining kernel therefore avoids causal covectors. It belongs to \(\mathscr S_1\) by adapted admission, so Lemma~\ref{lem:MCcharge} applies. \end{proof} \begin{lemma}[Local advanced trace formula] \label{lem:advancedtrace} For a compact smooth source \(a\) in \(V\), its supplied chart advanced trace is \begin{equation} (\GA a)_\mu(0,\mathbf z) =-\frac1{4\pi}\int_{\mathbb R^3} \frac{a_\mu(|\mathbf x-\mathbf z|,\mathbf x)} {|\mathbf x-\mathbf z|}\,\dd^3x. \label{eq:advancedtrace} \end{equation} For source support in \(V\), every relevant cone segment and initial endpoint lie in the retained laboratory. \end{lemma} \begin{proof} The spatial transform of \(\delta(s-|\mathbf x|)/(4\pi|\mathbf x|)\), \(s>0\), is \(\sin(s|p|)/|p|\). For \(\Box=-\partial_t^2+\Delta\), the zero-future-data mode equation has value \[ -\int_0^\infty\frac{\sin(s|p|)}{|p|} \widehat a(s,p)\,\dd s \] at \(t=0\). Inverse spatial transformation gives \eqref{eq:advancedtrace}. Saturation and \(R>r_0+T_0\) give the support assertion. This calculation represents the prescribed local comparison response, not a global cone inverse. \end{proof} \begin{lemma}[Nonzero sharp charge for every nonzero test] \label{lem:sharpcharge} For every \(0\ne f\in\Tspace\), there is a real smooth scalar configuration \(\phi\), compactly supported in \(V\), with \[ \btrace((B_f)'_h[h,\phi])\ne0. \] \end{lemma} \begin{proof} The metric derivative is \(q_{a_{f,\phi}}\), with \[ a_{f,\phi,\mu}=-f\partial_\mu\phi. \] Choose \(x_0=(t_0,\mathbf x_0)\) with \(f(x_0)\ne0\). Choose an initial point \(\mathbf z\) satisfying \(|\mathbf x_0-\mathbf z|=t_0\). It lies in the retained initial ball. On a small neighbourhood \(O\Subset V\) of \(x_0\), \(r(\mathbf x)=|\mathbf x-\mathbf z|\) is smooth and positive. Choose a nonnegative real \(\chi\in\mathcal D(O)\), positive at \(x_0\), and set \[ \phi_{\mathbb C}(t,\mathbf x) =(t-r(\mathbf x))\overline{f(t,\mathbf x)}\chi(t,\mathbf x). \] On the cone \(t=r(\mathbf x)\), \(\partial_t\phi_{\mathbb C}=\overline f\,\chi\). Equation~\eqref{eq:advancedtrace} gives \[ (\GA a_{f,\phi_{\mathbb C}})_0(0,\mathbf z) =\frac1{4\pi}\int \frac{|f(r(\mathbf x),\mathbf x)|^2 \chi(r(\mathbf x),\mathbf x)} {r(\mathbf x)}\,\dd^3x>0. \] Write \(\phi_{\mathbb C}=\phi_1+i\phi_2\) with real compact smooth parts. Complex linearity implies that at least one part has nonzero trace. Choose it. \end{proof} The local witness is the IX-Free-2 charge mechanism \cite[{\loc{lem:sharpcharge}}]{IXFreeTwo}. It does not use nonconvergence of regular approximants as a substitute for a charge calculation. \begin{theorem}[Actual-space separation] \label{thm:separation} One has \[ \Bspace\cap\Lspace_2 =\Bspace\cap\Rspace_2 =\Bspace\cap\Pmc=\{0\}. \] Consequently \[ (\Lspace_2\otimes\Bspace) \cap(\Bspace\otimes\Rspace_2)=\{0\}. \] \end{theorem} \begin{proof} A nonzero element of \(\Bspace\) is \(B_f\) for a nonzero test \(f\). Lemma~\ref{lem:sharpcharge} detects nonzero intrinsic metric-derivative charge at some configuration. Every comparison element in Proposition~\ref{prop:zerocharge} has zero charge there. Equality of actual functionals would identify their derivatives and intrinsic charges, a contradiction. For algebraic tensor subspaces, \[ (A\otimes B)\cap(A'\otimes B') =(A\cap A')\otimes(B\cap B'). \] This follows by choosing complements to the intersections in each ambient factor. Apply it to the displayed degree-two domains. \end{proof} The result is local sharp/zero-charge separation. It does not resolve the worldsheet or discharge a classical insertion obstruction in another programme. %===================================================================== \section{Whole-domain decomposition and the truncated product} \label{sec:gluing} %===================================================================== Inside \(\Padm\otimes\Padm\), define \begin{align} D_{\mathrm{first}} &=(\Bspace\otimes\Wspace)+(\Wspace\otimes\Bspace),\nonumber\\ D_{\mathrm{second}} &=(\Lspace\otimes\Bspace)+(\Bspace\otimes\Rspace),\nonumber\\ \Dnew&=D_{\mathrm{first}}+D_{\mathrm{second}},& \Dold&=\Pmc\otimes\Pmc,& \Ddecl&=\Dnew+\Dold. \label{eq:domains} \end{align} The full old domain is retained. \begin{proposition}[Direct decomposition of the new sum] \label{prop:newdirect} The six new domains form an algebraic direct sum: \begin{equation} \begin{split} \Dnew={}& (\Bspace\otimes\Wspace)\oplus(\Wspace\otimes\Bspace) \oplus(\Lspace_4\otimes\Bspace) \oplus(\Bspace\otimes\Rspace_4)\\ &\oplus(\Lspace_2\otimes\Bspace) \oplus(\Bspace\otimes\Rspace_2). \end{split} \label{eq:newdirect} \end{equation} \end{proposition} \begin{proof} Count metric and scalar slots in each argument. The argument-species bidegrees are respectively \[ ((1,1);(2,0)),\quad ((2,0);(1,1)),\quad ((3,1);(1,1)),\quad ((1,1);(3,1)), \] \[ ((1,1);(1,1)),\quad ((1,1);(1,1)). \] Polynomial homogeneity projects every relation onto these components. Only the last two share a component; their intersection is zero by Theorem~\ref{thm:separation}. Thus the entire sum, not merely its pairwise overlaps, is direct. \end{proof} \begin{lemma}[Anchored sum lemma] \label{lem:anchored} For subspaces \(B,P\subset A\) with \(B\cap P=0\), \[ (A\otimes B+B\otimes A)\cap(P\otimes P)=0. \] \end{lemma} \begin{proof} Define \(\pi(p+b)=p\) on \(P\oplus B\), and extend it algebraically to \(A\). Then \(\pi\otimes\pi\) is identity on \(P\otimes P\) and zero on the anchored sum. An element in the intersection is both preserved and annihilated, hence zero. No continuity of \(\pi\) is asserted. \end{proof} \begin{theorem}[Whole-sum old gluing] \label{thm:oldgluing} The entire new-domain sum has zero old overlap: \[ \Dnew\cap\Dold=0,\qquad \Ddecl=\Dnew\oplus\Dold. \] Every prescribed new coefficient therefore glues uniquely with the unchanged old coefficient on this declared domain. \end{theorem} \begin{proof} Every new tensor domain has at least one argument in \(\Bspace\). Thus \[ \Dnew\subseteq \Padm\otimes\Bspace+\Bspace\otimes\Padm. \] Use Lemma~\ref{lem:anchored} with \(B=\Bspace\), \(P=\Pmc\) and Theorem~\ref{thm:separation}. Proposition~\ref{prop:newdirect} already handles all relations inside the new sum. \end{proof} This is the simultaneous anchored-domain argument of IX-Free-2 \cite[{\loc{lem:anchored}}, {\loc{thm:oldgluing}}]{IXFreeTwo}. No old coefficient is subtracted or repaired. \begin{corollary}[Evaluation of both parameter sets] \label{cor:cglue} For the fixed declared sector, \[ \boxed{\Cdesc=\Cglue=\mathbb R.} \] \end{corollary} \begin{proof} Scalar descent holds for every real \(c\). Degree-four descent is independent of \(c\). The direct new sum and its zero old overlap impose no further equation on \(c\). \end{proof} The conclusion concerns the supplied parent family only. It is not an all-normalization classification, a running coupling, or host-curvature evolution. \subsection{Coefficient definition} Let \(C_0(F\otimes G)=FG\). For positive orders use the following table; all unlisted positive entries on the new rows are prescribed zero. \begin{center} \small \renewcommand{\arraystretch}{1.25} \begin{tabularx}{\linewidth}{@{}p{.25\linewidth}XXp{.08\linewidth}@{}} \toprule Domain & \(C_1\) & \(C_2\) & \(C_3\)\\ \midrule \(\Bspace\otimes\Wspace\) & \(Q_{f,g}\) & \(0\) & \(0\)\\ \(\Wspace\otimes\Bspace\) & \(Q^R_{g,f'}\) & \(0\) & \(0\)\\ \(\Lspace_4\otimes\Bspace\) & \(G_{13}+G_{23}\) & \(G_{13,23}\) & \(0\)\\ \(\Bspace\otimes\Rspace_4\) & \(G_{12}+G_{13}\) & \(G_{12,13}\) & \(0\)\\ \(\Lspace_2\otimes\Bspace\) & \(G_{12,23}+G_{12,13}\) & \(\mathcal N_c(f,g,f')1\) & \(0\)\\ \(\Bspace\otimes\Rspace_2\) & \(G_{12,23}+G_{13,23}\) & \(\mathcal N_c(f,g,f')1\) & \(0\)\\ \(\Dold\) & \(C_1^H\) & \(C_2^H\) & \(C_3^H\)\\ \bottomrule \end{tabularx} \end{center} Theorem~\ref{thm:degree4}, Theorem~\ref{thm:degree2maps}, and the gluing results make this a table of maps on actual tensors, not merely on labels. \begin{definition}[One truncated partial product] \label{def:product} Let \(R_3=\mathbb C[\hbar]/(\hbar^4)\). Define \[ \star_c:\Ddecl\longrightarrow\Padm\otimes R_3, \qquad \star_c(d)=\sum_{j=0}^3\hbar^jC_j(d), \] and extend \(R_3\)-linearly to \[ \Dhat=\Ddecl\otimes_{\mathbb C}R_3. \] A pair of formal functionals is admissible precisely when its tensor over \(R_3\) lies in this declared formal tensor domain. Its partial product is the value of this linear map. \end{definition} The domain is not all of \((\Padm\otimes R_3)\otimes_{R_3}(\Padm\otimes R_3)\). It is not claimed closed under arbitrary intermediate multiplication. The definition is a free Wick-style formal partial product whose coefficients are actual classical polynomial functionals. It is not an interacting renormalized product, nor is it itself a finite physical quantum channel. %===================================================================== \section{Wordwise coherence, reversal, and continuity} \label{sec:coherence} %===================================================================== \begin{theorem}[Both selected associations] \label{thm:coherence} For every fixed real common parent \(c\), both selected associations are admissible in Definition~\ref{def:product} and satisfy \begin{equation} \begin{split} (B_f\star_cW_g)\star_cB_{f'} &=B_f\star_c(W_g\star_cB_{f'})\\ &=B_fW_gB_{f'} +\hbar(G_{12}+G_{13}+G_{23})\\ &\quad+\hbar^2(G_{12,23}+G_{12,13}+G_{13,23}) +\hbar^3\mathcal N_c(f,g,f')1 \pmod{\hbar^4}. \end{split} \label{eq:word} \end{equation} \end{theorem} \begin{proof} The first-stage output has degree-four component \(B_fW_g\) and \(\hbar\) times degree-two component \(Q_{f,g}\). Their tensors with \(B_{f'}\) lie in the declared left second-stage domains. The reverse assertion is identical. Insert the coefficient table and collect powers of \(\hbar\). The order-one sum contains exactly the three one-edge outputs. The order-two sum contains exactly the three two-edge outputs. The scalar term comes from \(\hbar Q\) followed by its order-two scalar coefficient, hence has order \(\hbar^3\), not \(\hbar^2\). Both sides use the same \(U_c\). All maps involved have already descended and glued. \end{proof} This is wordwise coherence after actual binary factorization, not an assignment that declares two ternary sums equal. It is nevertheless not associativity for unspecified triples. \begin{theorem}[Partial-product reversal] \label{thm:reversal} The declared tensor domain is stable under \[ (F\otimes G)^\ddagger=G^*\otimes F^*, \] and \[ \star_c(d^\ddagger)=\star_c(d)^* \] with real formal \(\hbar\). \end{theorem} \begin{proof} Conjugation exchanges the locked first stages and sends \(\Lspace_2\) to \(\Rspace_2\), and \(\Lspace_4\) to \(\Rspace_4\). Thus it exchanges each left second-stage domain with its right counterpart. Kernel conjugation and \(x,x'\)-exchange send the one-edge and two-edge entries to the corresponding reversed entries. In particular the two mixed metric-quadratic trees are exchanged. The parent dagger symmetry and \(c\in\mathbb R\) give \[ \overline{\mathcal N_c(f,g,f')} =\mathcal N_c(\overline{f'},\overline g,\overline f). \] The old entries obey the inherited Hermitian convention. The unique glued maps therefore satisfy reversal. \end{proof} \begin{proposition}[Actual-functional presentation independence] \label{prop:presentation} The partial product respects all finite relations between actual presentations in its declared domain. Kernel permutation, tensor symmetry, and compact integration by parts preserving the actual off-shell functional do not change the result. Boundary terms must be retained. Equation-of-motion identification is not included. \end{proposition} \begin{proof} The four presentation maps are injective. The six new domains form a direct sum, and their whole old overlap is zero. Thus every coefficient is a map on actual tensors, independent of both test presentation and domain decomposition. The inherited old coefficients already have that property. Valid functional rewritings represent the same actual argument; dropping a term in \eqref{eq:sharpadjoint} is not such a rewriting. \end{proof} \begin{theorem}[F9a on the complete declared new sector] \label{thm:F9a} Every new coefficient family has joint fixed-support test-family continuity into \((\Padm,\taus)\). For degree-four intermediate arguments, this means continuity of \[ (f,g,f')\longmapsto C_j(B_fW_g\otimes B_{f'}), \qquad (f,g,f')\longmapsto C_j(B_f\otimes W_gB_{f'}). \] The corresponding degree-two and first-stage families have the same property. F9b, continuity or hypocontinuity on the actual intermediate subspace topologies, is unclaimed. \end{theorem} \begin{proof} The new positive-order degree-four values are finite sums of already established one-edge and two-edge families. Their coefficient kernels and all trace coordinates are continuous by Proposition~\ref{prop:firstcontinuity}, the carrier's selected self-output input, and Proposition~\ref{prop:F9a2}. Classical words have the same fixed-support continuity by their explicit compact coefficient constructions and directional Leibniz formulae. The scalar entries have finite-order distribution bounds on the fixed joint compact support. These proofs estimate the pulled-back generating families. They never estimate \(p_L^{-1},p_R^{-1},q_L^{-1}\), or \(q_R^{-1}\) in the carrier topology. Algebraic injectivity supplies no such inverse estimates. \end{proof} \begin{proposition}[The retained continuity obstruction] \label{prop:obstruction} The construction does not remove the VII-C incompatibility under its original simultaneous requirements: unchanged old-product agreement, sharp admission with specified normal admission continuity, bounded-set hypocontinuous quantum multiplication, and continuous scalar extraction. In particular, it does not imply \(B_{f,\varepsilon}\to B_f\) in \(\taus\). \end{proposition} \begin{proof} The unchanged regular triangle has a positive logarithmic coefficient on a separated detecting test by the inherited VII-C record \cite{VIIC}. Under those simultaneous normal-continuity requirements, the converging regular input kernels and hypocontinuous multiplication would force convergence of that extracted scalar sequence, contradicting its logarithm. The present product asserts no such continuity. Its old coefficients are unchanged, while its sharp scalar assignment is \(U_c\). The carrier separator already excludes the asserted unchanged convergence in \(\taus\). No proof above alters either fact. \end{proof} The obstruction retains its quantifiers; it is not a universal nonexistence theorem for partial products. The supplied \(U_c\) is not an observation-diagonal contact term, and it is not subtracted from positive-cutoff old products in this construction. %===================================================================== \section{Supported transfer and final release} \label{sec:release} %===================================================================== \begin{theorem}[Transfer to the prescribed regular host] \label{thm:hosttransfer} Transport all stated chart objects, coefficient densities, and test families through the developing embedding. The first stages, canonical tree outputs, four presentation injectivity results, charge separations, and truncated partial product hold on the corresponding supported host presentation. Every response and charge witness used in the proof remains inside the retained regular laboratory. No global conical covariance, Green operator, or product around the worldsheet is constructed. \end{theorem} \begin{proof} Distribution transport through a diffeomorphism is defined by transposition on tests and densities. The developing embedding preserves the local metric, tensor species, and compact supports. The response identities are supported Cauchy identities by Proposition~\ref{prop:saturation}. The compact directions in Lemma~\ref{lem:restricted} are supported in \(V\). The sharp trace witness uses an initial point within distance \(t_0\) of an observation point, hence in \(K_*\). The remaining operations are compact distributional pairings, proper pushforwards, fixed transpositions, and algebraic maps on the actual polynomial spaces. Their identities are unchanged by this presentation transport. Compact coordinate changes modify finite seminorm constants but not the proved finite orders. No step defines anything across the defect. \end{proof} \begin{proposition}[No Einstein implication] \label{prop:noeinstein} Symmetry, conservation, and zero trace of a tensor on the prescribed metric do not imply Einstein dynamics. \end{proposition} \begin{proof} In a regular flat chart take a nonzero constant null covector \(k\), and set \(t_{\mu\nu}=k_\mu k_\nu\). It is symmetric, conserved, nonzero, and traceless. The chart Einstein tensor is zero, so \(G_{\mu\nu}=8\pi G_{\mathrm N}t_{\mu\nu}\) fails for \(G_{\mathrm N}\ne0\). \end{proof} \begin{theorem}[Combined IX-C-ST release, with ordered stage verdicts] \label{thm:release} Relative to the shared explicitly stated analytical inputs and finite operational permissions: \begin{enumerate}[label=\textup{(\roman*)}] \item Part I provides the local Grade S first-stage and criteria package. At that stage complete factorization is unresolved. \item Part II adds concrete relation kernels, full chart support of \(K\), injective degree-two presentations, \(\Cdesc=\mathbb R\), canonical admitted two-edge outputs, and degree-two second-stage maps. At that stage complete associations remain unreleased. \item Part III proves both degree-four presentation kernels zero, all four degree-four descents, chart-local boundary-charge separation, a direct new-domain decomposition, and zero overlap of the whole new sum with the full old domain. \item For every fixed real common parent, \[ \Cdesc=\Cglue=\mathbb R, \] and Definition~\ref{def:product} constructs one partial product on \(\Dhat\), modulo \(\hbar^4\). Both associations of the selected word satisfy \eqref{eq:word}. Reversal, actual-functional presentation independence, and F9a hold on that sector. \item Each stage can use the same explicit destructive-clock, fixed-table, residual, locality, and regular reference-preserving transport interface. \end{enumerate} The final declared partial-product ending is \emph{Constructed relative to the stated inputs}. F9b, associativity for unspecified triples, general intermediate-domain closure, a causal lift, G2, and IX-D-ST are not constructed. \end{theorem} \begin{proof} The intermediate releases are Theorems~\ref{thm:Crelease} and \ref{thm:Freerelease}. The degree-four and charge results are proved in Sections~\ref{sec:degree4}--\ref{sec:charge}. Section~\ref{sec:gluing} proves simultaneous gluing and defines the actual partial product. Section~\ref{sec:coherence} proves its selected coherence, reversal, presentation independence, and continuity. Theorem~\ref{thm:hosttransfer} supplies the supported host placement. Section~\ref{sec:operational} supplies every finite operational clause independently of the product gates. No result there defines a causal field product. \end{proof} \subsection{Exact next gate} The constant projection of the constructed word is \[ \operatorname{ev}_{h=\phi=0} \bigl((B_f\star_cW_g)\star_cB_{f'}\bigr) =\hbar^3\mathcal N_c(f,g,f')\pmod{\hbar^4}. \] It is not a definition of a time-ordered map, a retarded map, or higher causal factorization. IX-D's \loc{thm:nonlift} is the appropriate documentary next comparison, not a premise of this manuscript \cite{IXD}. A causal successor must declare its own actual domains, support conditions, normalization, overlap, and admission proofs. The present domain is not asserted to be the complete free-product domain needed by that successor. \scopebox{Standing boundaries}{% No sharp \(\mathrm{TO}_2\), normalized \(R_1\) or \(R_2\), time-ordered extension, retarded product, interacting BRST/BV differential, quantum Ward identity, radiative coefficient, or ultraviolet completion is constructed. The formal free-word coefficients proved here are admitted classical carrier elements. The separate inherited interacting characteristic quantum coefficient is not admitted. The quantum certificate remains \loc{RG-open}. Grade E is not constructed. O.1 stays \loc{NOT_CLOSED} in the stated inherited prescription. XV-B is not opened as a downstream quantum-admission exit. XVI-N is not discharged. IX-C-ST discharges neither historical H2 nor the full I3/J2 characteristic-map obligations. It does not withdraw XIV-I's narrower constructed internal amplitude. This is not spacetime emergence, Einstein dynamics, a Page curve, holographic identification, or gravity-generated entanglement.} %===================================================================== \appendix \section{Debugging and logical audit} \label{app:debug} %===================================================================== \begin{longtable}{@{}p{.31\linewidth}p{.64\linewidth}@{}} \toprule Potential error & Correction in this manuscript\\ \midrule \endfirsthead \toprule Potential error & Correction in this manuscript\\ \midrule \endhead Three stages treated as independent constructions & The same host, carrier, covariance, parent, and first-stage values are used in order. Intermediate unresolved releases are retained.\\[4pt] The one-file instruction erases stage boundaries & The file contains three ordered parts with separate mathematical release statements. It does not claim three new registry entries have been installed.\\[4pt] The host is Minkowski space & Minkowski expressions are supported developing-chart comparison data on \eqref{eq:host}.\\[4pt] The defect is a worldline & After adjoining the axial coordinate it is the two-dimensional worldsheet \(\mathcal W\).\\[4pt] A compact observation patch determines the response & The entire truncated Cauchy past and a support margin are retained.\\[4pt] \(\Ezero\Box v=v\) constructs a conical inverse & It is a local zero-data identity for compact chart tests.\\[4pt] \(H_{ww}=0\) kills the mixed kernel & \(K\) is explicitly retained and has full distributional chart support.\\[4pt] Full support of \(K\) gives support of its imaginary part & No such inference is used. Those are separate questions.\\[4pt] The reverse first-stage sign changes to \(+2\) & It stays \(-2\). The full word has two insertion signs, giving \(+1\), and two middle-slot assignments.\\[4pt] A labelled tree expression is already a distribution & Existence is proved by nonzero comparable leaf covectors before polynomial admission is addressed.\\[4pt] Compact kernel existence proves metric admission & The explicit decomposition \(G'_h=e_{00}m_h+q_{a_h}\) and all finite directional descendants are checked.\\[4pt] Injectivity of \(q_L\) proves injectivity of \(p_L\) & A separate compact spatial-diagonal direction and three-metric-slot isolation prove the latter.\\[4pt] Two metric derivatives recover the degree-four kernel & Recovery needs one scalar derivative and three metric derivatives; complete metric polarization gives \(3!\).\\[4pt] Regular nonconvergence proves sharp charge & An independent advanced-trace witness is constructed.\\[4pt] The old algebra has zero charge because it is called bulk & The microcausal condition and the timelike surface-wavefront argument prove that statement.\\[4pt] Pairwise zero overlaps prove whole-sum gluing & Species homogeneity proves the whole new sum direct; the anchored lemma separately kills its whole old overlap.\\[4pt] \(\mathcal C_{\mathrm{desc}}=\mathbb R\) already proves gluing & Scalar descent is evaluated in Part II. Gluing is evaluated only after Part III's separate overlap proof.\\[4pt] A scalar normalization repairs a nonconstant discrepancy & The coherence equations occur in different coefficient degrees and must each be satisfied.\\[4pt] The scalar parent enters the word at order two & It follows an order-one first-stage partial and an order-two second-stage coefficient, so enters at order three.\\[4pt] Every pair of carrier elements can be multiplied & Admissibility is membership of the formal tensor in \(\Dhat\); no general intermediate closure is asserted.\\[4pt] F9a gives actual-stage continuity & No inverse presentation estimates or quotient-topology identification are proved. F9b is unclaimed.\\[4pt] A table reads the charge, residue, or continuum field & It encodes supplied evaluations and enclosures. No acquisition map is supplied.\\[4pt] Close numerical report values imply close flagged channels & The norm bound assumes identical output labels and fixed interfaces; orthogonal relabelling is different.\\[4pt] Equal calibration records fix residual sources & Only full residual-effect identities give residual agreement.\\[4pt] Association ancestry is a physical causal past & Computational ancestry and physical causal embedding are separate certificates.\\[4pt] A finite transport witness derives a cone or generates entanglement by gravity & The metric and carriage permissions are inputs. Transport preserves existing reference correlations.\\[4pt] No quantum coefficient is admitted means no positive-order formal coefficient was constructed & The refusal concerns the separate inherited interacting characteristic coefficient. The displayed free-word coefficients are actual classical carrier elements.\\ \bottomrule \end{longtable} %===================================================================== \section{Routing records and retained obligations} \label{app:routing} %===================================================================== The following are proposed routing records, not an assertion that a catalogue or registry has been changed. \begin{verbatim} delivery: filename: IX-C-ST.tex form: one manuscript with three ordered stages shared: track: product bearingProblem: products host: supported regular IX-A-ST / I-B presentation nearUv: false emergenceSource: false quantumCertificate: RG-open authorField: empty compilationAsserted: false stages: - identifier: IX-C-ST localGrade: S timeLandmarks: 1 mathematicalTargetAtStageExit: Unresolved completed: first stages, descent and coherence criteria - identifier: IX-Free-ST localGrade: S timeLandmarks: 1 mathematicalTargetAtStageExit: Unresolved completed: degree-two descent and admitted tree outputs - identifier: IX-Free-2-ST localGrade: S timeLandmarks: 1 mathematicalTargetAtStageExit: Constructed_relative completed: degree-four descent, gluing, declared partial product notStarted: - IX-D-ST - IX-E-ST - unified IX-ST \end{verbatim} The \loc{nearUv: false} tag classifies the supported adaptation. It does not deny the ultraviolet logarithm in the supplied parent or reclassify the VII-D analysis. \paragraph{IX-C-ST emergence walk.} Both admitted first stages and the exact descent and coherence criteria are certified on a prescribed chart. At this stage neither sharp second multiplication is constructed, and no metric is derived. \paragraph{IX-Free-ST emergence walk.} Presentation kernels, full mixed-kernel support, degree-two scalar descent, and three admitted tree outputs are certified. Complete associations are not released at this stage. The cone remains prescribed. \paragraph{IX-Free-2-ST emergence walk.} Degree-four descent, vanishing overlaps, and one declared truncated partial product are certified through order \(\hbar^3\), with a finite clock and certificate transport. Wordwise coherence is not unspecified-triple associativity, a causal lift, or spacetime emergence. \paragraph{Obligations.} \begin{description}[style=nextline,leftmargin=1.5em] \item[\loc{first-stage-not-sharp-product}] The first-stage and criteria release does not pass the second-stage construction gate. \item[\loc{degree-two-not-complete-association}] Degree-two maps do not pass degree-four descent and simultaneous gluing. \item[\loc{partial-product-not-causal-lift}] A causal successor must construct its actual time-ordered and retarded interfaces. The constant projection of a free word is insufficient. \item[\loc{first-stage-not-emergence}, \loc{degree-two-not-emergence}, \loc{partial-product-not-emergence}] A derivation of the metric, lapse, or cone from quantum data or records is a further theorem. None of the three stages provides it. \end{description} \subsection{Documentary separations} The source releases IX-C, IX-Free, and IX-Free-2 are attribution records; only the explicitly restated analytical inputs are premises of the present conditional proof. IX-A-ST, I-B, IX-B, VII-D, and VII-C have the distinct roles listed in Section~\ref{sec:inputs}. VIII-ST and VII-J-ST are interface comparisons, not product-construction inputs. V's external retarded datum, finite oscillator expression, and Newton--wave cubic are different objects from this word. No theorem from V enters a proof \cite{V}. IX-E and IX-E-2 give later first-stage commutator and support comparisons, not universal causal no-go theorems and not premises here \cite{LaterIX}. VII-H and VII-I concern separate free benchmarks. Em01-B's moderate smoothing growth and Em01-C's separate scalar expansion are not this parent residue or an evaluation of the full characteristic coefficient. XIV-I's internal amplitude does not close the full characteristic map; technical XV-B does not turn this free partial product into quantum admission. The XVI-N and XVI-O stops remain prescription-qualified and unchanged \cite{BoundaryRecords}. En03-A, En05-A, En08-A/B, and En09-A distinguish scheduled dependency, prescribed causal geometry, and finite readout. En10-A/B concern source-weight estimation under their own access assumptions; they do not acquire this residue or derive a metric \cite{OperationalRecords}. \begin{remark}[The OpenAI catalogue is not a premise] \label{rem:not-premise} No OpenAI theorem or catalogue family is a premise. No family is independently verified here. Oai282-A/B and Oai069-A do not make this construction a fieldwise Ward theorem, a general scale-to-conformal theorem, or an ultraviolet completion. The intrinsic boundary trace is not a Ward certificate. Oai283-A is at most a comparison for a fixed finite-channel error interface. Oai365-A does not derive a metric from a certificate tag. Oai212-A, Family 265, Oai272, Oai276, Oai279, Oai227, and Oai119 are finite-certificate comparisons, not Page curves or entanglement tests for this transport. In particular a static Family 265 record is not a radiation history \cite{OpenAIRecord}. \end{remark} \paragraph{Version-qualified historical stops.} Historical H2 in XIV-H and the later constructed internal I2 amplitude are not the same release. The present work discharges neither the historical H2 obligation in its own manuscript nor the full I3/J2 map obligations. It does not withdraw the narrower XIV-I result. ``XV-B is not opened'' refers to the downstream quantum-admission exit, not to the nonexistence of a technical manuscript bearing that identifier. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Authorship.} The author field is intentionally empty. Citation of N.~Cheung does not assign authorship of this adaptation. Authorized authorship and submission metadata must be supplied for a signed submission. \paragraph{Inherited inputs.} The adapted carrier, old covariance/product convention, and parent ultraviolet finite-part theorem are explicit inputs. Their complete proofs are not independently certified here. The manuscript supplies its own supported presentation arguments and the stated staged deductions. \paragraph{Scope.} The result is one source-locked, truncated free partial product on one named tensor domain. It is not an intrinsic global conical field theory, a general renormalized product, or an interacting quantum coefficient. \paragraph{Operational status.} Finite table matrices represent supplied evaluations and certified enclosures. They are not acquired field data. The declared carriage and local-operation permissions are finite-model assumptions, not a microscopic field implementation. \paragraph{Verification status.} The file is written for conventional pdfLaTeX and contains its bibliography internally. No successful compilation, machine check, numerical execution, independent execution, hardware synthesis, experimental implementation, independent referee verification, or journal acceptance is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{IXC} N.~Cheung, \emph{Landmark IX-C: Actual-functional factorization and coefficientwise coherence for one sharp ordered word}. \loc{IX-C.tex}, version 1. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/IX-C}. Specific source statements: \loc{thm:first}, \loc{prop:firstcontinuity}, \loc{thm:descent}, \loc{thm:scalarfactor}, \loc{thm:coherence}, and \loc{thm:package}. \bibitem{IXFree} N.~Cheung, \emph{Landmark IX-Free: Actual-functional binary factorization of one sharp ordered word}. \loc{IX-Free.tex}, version 1. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/IX-Free}. Specific source statements: \loc{prop:relations}, \loc{prop:prelations}, \loc{lem:fullsupport}, \loc{lem:primitive}, \loc{thm:qinjective}, \loc{lem:endpoints}, \loc{lem:forest}, \loc{thm:treeadmission}, and \loc{thm:degree2maps}. \bibitem{IXFreeTwo} N.~Cheung, \emph{Landmark IX-Free-2: Degree-four descent and overlap agreement for one sharp ordered word}. \loc{IX-Free-2.tex}, version 1. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/IX-Free-2}. Specific source statements: \loc{lem:restricted}, \loc{thm:pinjective}, \loc{lem:surfaceWF}, \loc{lem:sharpcharge}, \loc{thm:separation}, \loc{prop:newdirect}, \loc{lem:anchored}, \loc{thm:oldgluing}, \loc{def:product}, and \loc{thm:release}. \bibitem{IXAST} \emph{IX-A-ST: Boundary-sensitive admission of sharp Cauchy insertions on a prescribed conical Enter spacetime}. \loc{IX-A-ST.tex}, version 1. Supplied anonymous local adaptation. Enter Calculus Companion. \url{https://enter.grok.me/browse/IX-A-ST}. Carrier, saturation, separator, trace, and separate-classical- continuity inputs. \bibitem{IXB} N.~Cheung, \emph{Landmark IX-B: Derivative-stable triangle calibration and counterterm constraints for sharp Cauchy insertions}. \loc{IX-B-v2.tex}, version 2. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/IX-B}. The supplied IX-B-ST draft is a supported adaptation of this calibration; no public registration of that draft is asserted here. \bibitem{VIID} N.~Cheung, \emph{Finite part of the regulated Cauchy-reflection triangle: A strong numerical distributional limit for the VII-C family}. \loc{VII-D.tex}, version 1. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/VII-D}. Parent estimate and conversion input only; the ultraviolet ledger is not re-proved here. \bibitem{VIIC} N.~Cheung, \emph{Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions}. \loc{VII-C.tex}, version 1. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/VII-C}. Tensor data and the original \loc{ass:extension}/\loc{cor:noalgebra} boundary. \bibitem{IB} N.~Cheung, \emph{I-B-v1: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}. \loc{I-B-v1.tex}, version 1. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/I-B}. \bibitem{EnterArticle} N.~Cheung, ``The Enter calculus: A linear quantum lambda calculus with an operational characterization of measurement,'' \emph{Frontiers in Physics} \textbf{14} (2026), article 1954943. DOI: \loc{10.3389/fphy.2026.1954943}. Used for the finite operational background, not as an analytical product theorem. \bibitem{EnterI} N.~Cheung, \emph{Born--L\"uders as the unique measurement strategy of a strictly linear quantum lambda-calculus: A self-contained finite characterization in the Enter calculus}. \loc{I.tex}, version 2. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/I}. \bibitem{EnterPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Introductory essay; not a theorem premise. \url{https://enter.grok.me/enter}. \bibitem{En09} N.~Cheung, \emph{En09-A: Relational Enter}. \loc{En09-A.tex}, version 1. Enter Calculus Companion, supplied manuscript record. \url{https://enter.grok.me/browse/En09-A}. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}. European Mathematical Society, 2007. \url{https://arxiv.org/abs/0806.1036}. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, second edition. Springer, 1990. Distributional products, tensor products, wavefront sets, and proper pushforward calculus. \bibitem{BDH} C.~Brouder, N.~V.~Dang, and F.~H\'elein, ``Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker),'' \emph{Studia Mathematica} \textbf{232} (2016), 201--226. DOI: \loc{10.4064/sm8316-3-2016}. \url{https://arxiv.org/abs/1409.7662}. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}. Lecture notes, revised version 2, 2013. \url{https://arxiv.org/abs/1208.1428}. Functional microcausality and the distinction between free products and causal interacting constructions. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author manuscript and errata interface: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. \bibitem{STInterfaces} \emph{VIII-ST}, \emph{VII-E-ST}, \emph{VII-F-ST}, and \emph{VII-J-ST}, version 1 adaptation records. Enter Calculus Companion, supplied sources. Fixed-host, observation, clock, and transport interfaces; not premises supplying a distributional product. \url{https://enter.grok.me/browse/VIII-ST}; \url{https://enter.grok.me/browse/VII-E-ST}; \url{https://enter.grok.me/browse/VII-F-ST}; \url{https://enter.grok.me/browse/VII-J-ST}. \bibitem{En05} N.~Cheung, \emph{En05-A}, version 1, and \emph{VII-J}, version 1. Enter Calculus Companion, supplied manuscript records. Scheduled locality and separate physical containment. \url{https://enter.grok.me/browse/En05-A}; \url{https://enter.grok.me/browse/VII-J}. \bibitem{En08} N.~Cheung, \emph{En08-A}, version 1, and \emph{En08-B}, version 1. Enter Calculus Companion, supplied manuscript records. Scheduler/cone separation and transport relative to a prescribed null boundary. \url{https://enter.grok.me/browse/En08-A}; \url{https://enter.grok.me/browse/En08-B}. \bibitem{IXD} N.~Cheung, \emph{Landmark IX-D: Causal-support interfaces for one sharp ordered word after construction of the declared partial free product}. \loc{IX-D-v3.tex}, version 3. Enter Calculus Companion, manuscript record. \url{https://enter.grok.me/browse/IX-D}. \loc{thm:nonlift} and the causal-interface release are documentary successor boundaries, not premises. \bibitem{V} N.~Cheung, \emph{Admissible currents and first retarded corrections for a fixed-time scalar response}. \loc{V.tex}, version 1. Enter Calculus Companion, supplied manuscript record. \url{https://enter.grok.me/browse/V}. Separation source only. \bibitem{LaterIX} N.~Cheung, \emph{IX-E}, version 1, and \emph{IX-E-2}, version 1. Enter Calculus Companion, supplied manuscript records. First-stage commutator and support comparisons only. \url{https://enter.grok.me/browse/IX-E}; \url{https://enter.grok.me/browse/IX-E-2}. \bibitem{BoundaryRecords} N.~Cheung, \emph{VII-H}, version 1; \emph{VII-I}, version 1; \emph{Em01-B}, version 3; \emph{Em01-C}, version 3; \emph{XIV-H}, \emph{XIV-I}, \emph{XIV-J}, version 1; technical \emph{XV-B}, version 2; \emph{XVI-N}, revision r12; and \emph{XVI-O}, revision o70-sync. Enter Calculus Companion, supplied manuscript records. Version-qualified documentary boundaries only. \url{https://enter.grok.me/browse}. \bibitem{OperationalRecords} N.~Cheung, \emph{En03-A}, \emph{En10-A}, and \emph{En10-B}, version 1. Enter Calculus Companion, supplied manuscript records. Separate locality, source-estimation, and acquisition contracts; not this partial product. \url{https://enter.grok.me/browse/En03-A}; \url{https://enter.grok.me/browse/En10-A}; \url{https://enter.grok.me/browse/En10-B}. \bibitem{OpenAIRecord} OpenAI, \emph{OpenAI mathematics catalogue}, as documented in the separate Enter Calculus Companion wrapper \loc{OpenAI.tex}. The wrapper's \loc{rem:not-premise} and the supplied Oai comparison records are documentary separations. No family proof or verification status is imported. \url{https://enter.grok.me}. \end{thebibliography} \end{document}