% !TEX program = pdflatex %===================================================================== % IX-A-ST.tex % % Single-file pdfLaTeX manuscript with an internal bibliography. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error IX-A-ST.tex % pdflatex -interaction=nonstopmode -halt-on-error IX-A-ST.tex % % No successful compilation, numerical execution, hardware synthesis, % proof-assistant verification, or independent certification is asserted. % % Principal corrections and restrictions: % 1. The conical defect in the 3+1 product is a worldsheet. % 2. The analytic construction is supported in a regular developing chart. % 3. The chart contains the complete truncated Cauchy past of observations. % 4. The auxiliary Minkowski Green operators are not global cone operators. % 5. Boundary traces are intrinsic to the actual off-shell functional. % 6. The trace used here does not classify every boundary singularity. % 7. Classical multiplication is separately continuous; stronger claims % are not inferred. % 8. Finite tables contain supplied classical evaluations, not acquired % values of an unknown continuum field. % 9. Fixed-interface channel approximation is not convergence of sharp % kernels or of orthogonal encodings of different real labels. % 10. Scheduled ancestry is not a spacetime causal past. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={IX-A-ST: Boundary-sensitive admission of sharp Cauchy insertions on a prescribed conical Enter spacetime}, pdfsubject={Local classical admission, boundary traces, finite destructive clocks, and certified transport} } \setlength{\emergencystretch}{3em} \setlist{topsep=5pt,itemsep=3pt} \allowdisplaybreaks \numberwithin{equation}{section} \setcounter{tocdepth}{1} \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} \newtheorem{example}[theorem]{Example} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\NN}{\mathbb N} \newcommand{\HH}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\AA}{\mathcal A} \newcommand{\EE}{\mathcal E} \newcommand{\DD}{\mathcal D} \newcommand{\WW}{\mathcal W} \newcommand{\PP}{\mathscr P} \newcommand{\Pdist}{\mathscr P_{\mathrm{dist}}} \newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}} \newcommand{\Preg}{\mathscr P_{\mathrm{reg}}} \newcommand{\Padm}{\mathscr P_{\Sigma}} \newcommand{\Pgen}{\mathscr P_{\mathrm{gen}}} \newcommand{\taus}{\tau_{\Sigma}} \newcommand{\Tcoeff}{\mathcal T_{\mathrm{coeff}}} \newcommand{\Mone}{\mathscr M_1} \newcommand{\Qsig}{\mathscr Q_{\Sigma}} \newcommand{\Sone}{\mathscr S_1} \newcommand{\btrace}{\mathfrak b} \newcommand{\Ezero}{\mathsf E_0} \newcommand{\GR}{\mathsf G_{\mathrm R}} \newcommand{\GA}{\mathsf G_{\mathrm A}} \newcommand{\dd}{\mathrm d} \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{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\loc}[1]{\texttt{\detokenize{#1}}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{#1}}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_{\diamond}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\cl}{cl} \newcommand{\statementbox}[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-A-ST: Boundary-sensitive admission of sharp\\ Cauchy insertions on a prescribed conical Enter spacetime}\\[5pt] \large A local Grade S construction with finite evaluation records,\\ a destructive clock, and reference-preserving transport} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct a supported local adaptation of the IX-A admission domain for the exact off-shell harmonic insertion \[ B_f=-\int f\,(\mathsf E_0Ch)^\mu\partial_\mu\phi \] on a prescribed \(3+1\)-dimensional conical spacetime. A causally saturated regular developing presentation contains the entire truncated Cauchy past of each admitted observation. This makes the local zero-Cauchy response independent of auxiliary field extensions without asserting a global Green operator on the non-globally-hyperbolic regular cone. The resulting space consists of actual compact-distribution polynomial functionals. A canonical trace on their linear metric descendants, taken modulo microcausal linear kernels, defines a Hausdorff locally convex topology. The carrier contains the bulk polynomials, exact sharp insertions, coordinate descendants, and arbitrary finite classical pointwise words. Directional differentiation and conjugation are continuous, and pointwise multiplication is separately continuous. For every nonzero real observation test, an explicit positive advanced-Cauchy moment produces a normalized continuous separator: it equals one on the sharp insertion and zero on every unchanged regular approximant. Thus faithful admission does not identify the sharp element with that regularization limit. The additional contribution is a supported transfer theorem and a complete finite operational interface around these classical objects. One destructive clock selects installed evaluation tables; their outputs and an arbitrary reference-entangled payload admit exact causal transport through a compact regular laboratory. Finite records can certify the displayed separator but cannot determine the full infinite-dimensional admission topology or the quantum residual. This combination provides a controlled benchmark for later deformation and acquisition proposals while keeping classical admission, statistical readout, and quantum multiplication distinct. The admission construction itself is attributed to IX-A; no priority is claimed for its underlying distributional methods. No renormalized quantum product, quantum associativity, Einstein dynamics, or spacetime emergence is constructed. \end{abstract} \statementbox{Release boundary}{% The metric, developing presentation, initial surface, lapse, calibration, and operational permissions are inputs. The sharp insertion is retained exactly as an off-shell classical functional. Finite tables encode already-defined evaluations; they do not acquire an unknown continuum field. This is local Grade S, not Grade E. The quantum certificate remains \loc{RG-open}. Separate continuity of classical multiplication is not a construction of quantum multiplication, joint continuity, hypocontinuity, or quantum associativity.} \tableofcontents %===================================================================== \section{Introduction, provenance, and the Enter distinction} \label{sec:intro} %===================================================================== IX-A separates three mathematical tasks: admitting an exact distributional functional, choosing a topology on its carrier, and constructing a quantum product on that carrier \cite{CheungIXA}. The present paper addresses the first two on a supported regular presentation of the conical product used in I-B \cite{CheungIB}. It then supplies a finite clock--evaluation--transport interface. The latter does not solve the third task. The operational background is the Enter discipline. Quantum names specify access to declared tensor factors; destructive readout consumes the measured names and returns a classical record with specified surviving factors \cite{CheungEnter,CheungPhilosophy}. This interpretation distinguishes a readout from recovery of an unknown quantum state. It also distinguishes a channel identity from a thermodynamic reset law and a mathematical analogy from a physical identification. We specify finite Born instruments directly. We do not rederive the measurement-characterization theorem of the Enter calculus, enlarge its hypotheses to continuum fields, or infer a spacetime metric from its operational conditions. \begin{definition}[Local Grade S] A local Grade S package in this paper comprises: \begin{enumerate}[label=\textup{(S\arabic*)}] \item a prescribed background and a compact regular support presentation; \item explicit classical functionals, domains, topologies, and proofs; \item complete finite instruments with declared input and output interfaces; \item a finite destructive clock and an explicit calibration contract; \item scheduled-support and independently certified causal-containment statements; \item a finite reference-preserving transport realization with a resource ledger. \end{enumerate} Grade E, understood here as an emergence claim for spacetime or its metric, is not part of the definition. \end{definition} \paragraph{Nature of the adaptation.} The analytic core follows IX-A's actual-functional construction. The new local work is to state and verify sufficient support saturation, remove any implicit appeal to a global conical inverse, and connect a normalized boundary separator to a complete finite instrument without identifying that instrument with continuum acquisition. General wavefront and Cauchy theory are used in their usual stated domains \cite{BGP,BDH,BDHIntro}. \paragraph{Documentary sources.} The supplied internal sheets are version-qualified source records, not assumptions that substitute for proofs. The propositions needed for the present release are established below. Comparisons with later sheets and with the OpenAI catalogue do not transfer their models, gates, or verification status into this manuscript. %===================================================================== \section{Prescribed cone and a causally saturated developing laboratory} \label{sec:geometry} %===================================================================== Fix \(0<\mu<2\pi\), \(c>0\), and \[ \alpha=1-\frac{\mu}{2\pi}. \] The regular background and its metric are \begin{equation} M_\mu=\RR_T\times C_\mu^\circ\times\RR_z,\qquad g_\mu=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2, \qquad r>0, \label{eq:cone} \end{equation} with \(\varphi\) of period \(2\pi\). The stipulated completion has singular locus \[ \WW=\RR_T\times\{o\}\times\RR_z. \] It is a worldsheet. Only its fixed-\(z\) sections are worldlines. No smooth metric chart or pointwise Einstein tensor is asserted on \(\WW\). The signature is \((-,+,+,+)\). The lapse in \(T\) is one, and static observers have proper-time increment \(\dd T\). These are prescribed I-B data \cite{CheungIB}. \begin{proposition}[Non-global hyperbolicity] \label{prop:nonGH} The regular product \(M_\mu\) is not globally hyperbolic. \end{proposition} \begin{proof} Choose a regular spatial point of radius \(r_0\), one angle, and \(z=0\). Let \(p,q\) have that spatial position and times \(T_*\mp2r_0/c\). Points at time \(T_*\), the same angle and axial coordinate, and radii \(r_n\downarrow0\) lie in \(J^+(p)\cap J^-(q)\) by radial causal curves. They have no convergent subsequence in the regular product. The causal diamond is noncompact. \end{proof} \subsection{The local analytic contract} Let \(t=c(T-T_\Sigma)\), and use developed spatial coordinates \(\mathbf x\). The local metric is \[ \eta=\diag(-1,1,1,1),\qquad \Box_\eta=-\partial_t^2+\Delta_{\mathbf x}. \] The choice of units and the normalization of \(t\) are fixed inputs. \begin{definition}[Supported developing presentation] \label{def:support} Choose positive numbers \(\delta,T_0,r_0,R,H\), with \(R>r_0+T_0\) and \(H>T_0\), and an isometric developing identification of \[ U=(-\delta,H)\times B_R \] with a regular open region of \(M_\mu\), whose compact subregions used below have closures disjoint from \(\WW\). Set \[ V=(0,T_0)\times B_{r_0},\qquad \Sigma=\{0\}\times B_R,\qquad K_*=[0,T_0]\times\overline{B}_{r_0+T_0}\Subset U. \] Observation and coordinate-source tests belong to \(\DD(V)=C_c^\infty(V)\), with the appropriate tensor indices. The analytic prescription has zero homogeneous Cauchy data at \(t=0\). All field dependence needed by these tests lies in \(K_*\). No undeclared lateral boundary data or incoming homogeneous solution is added. \end{definition} The strict margin \(R>r_0+T_0\) is essential. A chart containing only the observation support is not sufficient. Such presentations exist: place a sufficiently small developed ball around a regular point and choose a sufficiently short observation interval. Larger finite operational windows can be accommodated by first fixing their size and choosing the regular point sufficiently far from the defect. For \(f\in\DD(V)\), define its auxiliary truncated past \begin{equation} D_f=J^-_\eta(\supp f)\cap\{t\ge0\}. \label{eq:Df} \end{equation} This is compact and contained in \(K_*\). The symbol \(J_\eta\) refers to the developed Minkowski presentation. No equality with a globally minimal causal past on the cone is claimed. \begin{lemma}[Support saturation and independence of extension] \label{lem:saturation} Let \(h\in C^\infty(U;S^2T^*U)\). Choose a compact cutoff in \(U\) equal to one near \(K_*\), extend the cutoff field by zero to auxiliary Minkowski space, and solve the zero-data wave problem there. Its value on \(V\), when used in the response below, is independent of the cutoff. The same statement holds for its smooth derivatives and its pairings with observation tests supported in \(V\). \end{lemma} \begin{proof} Two such extensions agree on a neighbourhood of \(K_*\), hence their first differential expressions agree on every \(D_f\). The difference of their zero-data solutions has zero source in the truncated past of \(\supp f\) and zero Cauchy data there. Finite propagation and local uniqueness give zero difference near \(\supp f\). The strict support margin gives the same conclusion for derivatives. These are local consequences of ordinary Minkowski Cauchy theory \cite[Chapter 3]{BGP}. \end{proof} This use of auxiliary Minkowski space constructs no intrinsic global conical Green operator. %===================================================================== \section{The exact sharp response and compact polynomial functionals} \label{sec:carrier} %===================================================================== Raise indices with \(\eta\), and write \[ h=\eta^{\alpha\beta}h_{\alpha\beta},\qquad (Ch)^\mu=\partial_\nu h^{\mu\nu}-\frac12\partial^\mu h. \] For smooth sources, define the zero-Cauchy response on the supported region by \begin{equation} (\Ezero s)(t) =-\int_0^t \frac{\sin((t-u)\sqrt{-\Delta})}{\sqrt{-\Delta}}s(u)\,\dd u, \qquad t\ge0. \label{eq:E0} \end{equation} At zero spatial frequency the quotient equals \(t-u\). Thus \[ \Box_\eta\Ezero s=s,\qquad \Ezero s|_\Sigma=0,\qquad \partial_t\Ezero s|_\Sigma=0. \] For fields without spatial decay, Lemma~\ref{lem:saturation} defines the local interpretation. Set \begin{equation} L=\Ezero C,\qquad w[h]=Lh,\qquad B_f[h,\phi]=-\int_V f(x)w^\mu[h](x)\partial_\mu\phi(x)\,\dd^4x, \label{eq:B} \end{equation} and, for a smooth compact covector source \(a\) in \(V\), \begin{equation} J(a)[h]=\int_V a_\mu(x)w^\mu[h](x)\,\dd^4x. \label{eq:J} \end{equation} These definitions retain the exact IX-A normalization \cite{CheungIXA}. No equations are imposed on \(h\) or \(\phi\). \subsection{Polynomial kernels} Let \[ E=S^2T^*U\oplus(U\times\CC),\qquad \mathcal E=C^\infty(U;E). \] A configuration is \(\Psi=(h,\phi)\). The compact-distribution polynomial carrier consists of finite sums \begin{equation} F(\Psi)=\sum_{n=0}^N\frac1{n!} \pair{u_n}{\Psi^{\otimes n}}, \qquad u_n\in\EE'(U^n;(E^{\boxtimes n})^*)_{\rm sym}. \label{eq:polynomial} \end{equation} The degree-zero coefficient is a complex number. A compact distribution acts on a smooth function by inserting any cutoff equal to one near its support. Conjugation means the coefficient involution \[ F^\dagger(\Psi)=\overline{F(\overline\Psi)}. \] On real configurations this is ordinary complex conjugation. This convention preserves complex polynomial dependence. \begin{lemma}[Uniqueness of actual kernels] \label{lem:unique} The symmetric kernels in \eqref{eq:polynomial} are uniquely determined by the functional on all smooth off-shell configurations. \end{lemma} \begin{proof} Scaling separates homogeneous degrees. Polarization recovers the symmetric multilinear form in each degree. If that form vanishes on all smooth directions, its kernel annihilates all finite sums of elementary tensor tests. Those sums are dense in the smooth test space on each compact product region. Distributional continuity implies that the kernel is zero. \end{proof} Consequently \(\Pdist\) is a space of actual functionals, not a free symbol algebra. Give its coefficient carrier the locally convex direct-sum topology \begin{equation} \Tcoeff=\bigoplus_{n\ge0} \EE'(U^n;(E^{\boxtimes n})^*)_{\rm sym,strong}. \label{eq:Tcoeff} \end{equation} The strong topology is uniform convergence on bounded sets of \(C^\infty(U^n)\), componentwise in the bundles. \subsection{Bulk polynomials} Let \(\overline V_\pm\) be the closed nonspacelike cotangent cones, including zero in individual slots. Define \(\Pmc\) by \begin{equation} \WF(u_n)\cap \left(\overline V_+^{\,n}\cup\overline V_-^{\,n}\right) =\varnothing, \label{eq:microcausal} \end{equation} where the total zero covector is omitted. Let \(\Preg\) consist of polynomials with smooth compact kernels. For a closed wavefront cone \(\Gamma\), the normal topology used here combines strong distribution seminorms with the localized Fourier seminorms \[ p_{N,A,\chi}(u)= \sup_{\xi\in A}(1+|\xi|)^N |\widehat{\chi u}(\xi)|, \qquad (\supp\chi\times A)\cap\Gamma=\varnothing. \] All assertions involving that topology specify degree, compact support, and wavefront stage. We do not confuse the normal topology with weaker distribution topologies \cite{BDH}. \begin{lemma}[Bulk descendants and classical products] \label{lem:bulk} Every fixed smooth directional derivative of a bulk polynomial is bulk. At each smooth configuration, its first metric derivative has no nonzero nonspacelike wavefront covector. The classical pointwise product of two bulk polynomials is bulk. \end{lemma} \begin{proof} Directional differentiation and evaluation contract slots with smooth fields. A forbidden nonspacelike tuple in the remaining slots would give a forbidden tuple of the original kernel by adjoining zero covectors in contracted slots. The convention in \eqref{eq:microcausal} therefore gives the descendant statement. Pointwise products use symmetrized exterior tensor products. An all-future or all-past nonzero wavefront tuple of such a tensor product would already violate the condition in one nonzero factor. \end{proof} %===================================================================== \section{The sharp adjoint and its intrinsic boundary trace} \label{sec:trace} %===================================================================== Use the auxiliary advanced and retarded inverses of \(\Box_\eta\), denoted by \(\GA\) and \(\GR\). They are employed only in the supported prescription. For a covector source \(a\in\DD(V;T^*U)\), set \[ b=\GA a. \] The formal adjoint of \(C\), under the stated tensor pairing, is \begin{equation} (C^*b)^{\alpha\beta} =-\partial^{(\alpha}b^{\beta)} +\frac12\eta^{\alpha\beta}\partial_\mu b^\mu. \label{eq:Cstar} \end{equation} Let \(n_\mu=\partial_\mu t\), and define \begin{equation} \mathcal B^{\alpha\beta}(v) =-n^{(\alpha}v^{\beta)} +\frac12\eta^{\alpha\beta}n_\mu v^\mu. \label{eq:boundarytensor} \end{equation} \begin{proposition}[Sharp adjoint, with its surface term] \label{prop:adjoint} The compact kernel of \(J(a)\) is \begin{equation} q_a=L^*a=C^*(\Theta(t)b) =\Theta(t)C^*b+\delta(t)\mathcal B(b|_\Sigma). \label{eq:qa} \end{equation} There is no \(\delta'(t)\) term, and \[ \supp q_a\subseteq J^-_\eta(\supp a)\cap\{t\ge0\}\Subset U,\qquad \WF(q_a)\subseteq N^*\Sigma\setminus0. \] \end{proposition} \begin{proof} The zero-data prescription gives \(Lh=\GR(\Theta Ch)\) on the observation region. Transpose the retarded kernel against the compact source: \[ J(a)[h] =\pair{\GA a}{\Theta Ch} =\pair{C^*(\Theta b)}{h}. \] All relevant intersections are contained in \(K_*\). The product rule gives \eqref{eq:qa}. The adjoint has differential order one, so it differentiates \(\Theta\) at most once. The advanced response to a smooth compact source is smooth; only the surface factor can contribute wavefront. The support statement follows from finite propagation. \end{proof} The conventions and the boundary mechanism agree with the IX-A sharp-adjoint construction \cite[\loc{prop:adjoint}]{CheungIXA}. \begin{lemma}[Boundary injectivity and noncancellation] \label{lem:noncancellation} The map \(v\mapsto\mathcal B(v)\) is injective. If \(b(0,\mathbf x_*)\ne0\), at least one component of \(q_a\) has both nonzero normal directions in its wavefront set at \((0,\mathbf x_*)\). \end{lemma} \begin{proof} With lower components \(v_\mu\), direct substitution gives \[ \mathcal B^{00}(v)=-\frac12v_0,\qquad \mathcal B^{0i}(v)=\frac12v_i,\qquad \mathcal B^{ij}(v)=-\frac12\delta^{ij}v_0. \] These identities prove injectivity. Choose a component with nonzero delta coefficient near the point. After localization it has the form \[ \delta(t)v(\mathbf x)+\Theta(t)r(t,\mathbf x), \] with smooth coefficients and \(v(\mathbf x_*)\ne0\). The Fourier transform of its delta term is independent of the normal frequency and nonzero at some fixed tangential frequency. Integration by parts in \(t\) makes the step term \(O(|\tau|^{-1})\). It cannot cancel the nonzero order-zero term as \(\tau\to+\infty\) or \(\tau\to-\infty\). These frequency rays lie in arbitrarily narrow cones around the two normals, proving the assertion. \end{proof} Define \[ \Mone= \{m\in\EE'(U;S^2TU): \WF(m)\cap(\overline V_+\cup\overline V_-)=\varnothing\}, \qquad \Qsig=\{q_a:a\in\DD(V;T^*U)\}, \qquad \Sone=\Mone+\Qsig. \] \begin{proposition}[A well-defined trace modulo bulk linear kernels] \label{prop:trace} There is a well-defined linear map \begin{equation} \btrace:\Sone\longrightarrow C_c^\infty(\Sigma;T^*U|_\Sigma), \qquad \btrace(m+q_a)=(\GA a)|_\Sigma. \label{eq:trace} \end{equation} It vanishes on \(\Mone\) and commutes with conjugation. \end{proposition} \begin{proof} If \(m+q_a=m'+q_{a'}\), then \(q_{a-a'}=m'-m\in\Mone\). If its advanced trace were nonzero, Lemma~\ref{lem:noncancellation} would give a timelike conormal wavefront covector, forbidden for \(\Mone\). Thus the trace is independent of the decomposition. Compact trace support follows from finite propagation. Reality of the differential and Green operators proves the conjugation statement. \end{proof} \begin{remark}[What this trace does not record] The trace is computed from an actual metric derivative. It is not an independently adjoined boundary variable. It need not detect a step singularity whose delta coefficient vanishes. No complete classification of boundary microlocal singularities, and no identity \(\ker\btrace=\Mone\), is asserted. \end{remark} %===================================================================== \section{The admission domain and its classical algebra} \label{sec:admission} %===================================================================== For a smooth direction \(\eta\in\mathcal E\), put \[ D_\eta F(\Psi)= \left.\frac{\dd}{\dd s}F(\Psi+s\eta)\right|_{s=0}. \] For a finite list \(\boldsymbol\eta=(\eta_1,\ldots,\eta_k)\), write \(D_{\boldsymbol\eta}=D_{\eta_1}\cdots D_{\eta_k}\); the empty list is the identity. \begin{definition}[Boundary-admissible actual polynomials] \label{def:Padm} The space \(\Padm\subseteq\Pdist\) consists of all \(F\) such that \begin{equation} (D_{\boldsymbol\eta}F)'_h(\Psi)\in\Sone \quad\text{for every smooth }\Psi \text{ and every finite smooth list }\boldsymbol\eta. \label{eq:admission} \end{equation} Its boundary coordinates are \begin{equation} \btrace_{\Psi;\boldsymbol\eta}(F) =\btrace\!\left((D_{\boldsymbol\eta}F)'_h(\Psi)\right). \label{eq:jets} \end{equation} \end{definition} This quantifies over actual distributions, not over chosen representatives of equation-of-motion classes. \begin{proposition}[Exact basic admissions] \label{prop:generators} The domain contains \(\Pmc\), all \(J(a)\), and all \(B_f\). For \(\eta=(k,\sigma)\), \begin{align} (B_f)'_h(h,\phi)&=-q_{f\,\dd\phi},\nonumber\\ (B_f)'_\phi(h,\phi)&=\partial_\mu(fw^\mu[h]),\nonumber\\ D_\eta B_f &=-\int f(Lk)^\mu\partial_\mu\phi -J(f\,\dd\sigma), \label{eq:descendants}\\ D_{\eta'}D_\eta B_f &=-\int f\bigl((Lk)^\mu\partial_\mu\sigma' +(Lk')^\mu\partial_\mu\sigma\bigr).\nonumber \end{align} Third derivatives vanish. The mixed and linear kernels are compact distributions. \end{proposition} \begin{proof} Bulk admission is Lemma~\ref{lem:bulk}. The first metric derivative of \(J(a)\) is \(q_a\); its higher derivatives vanish. Differentiate \eqref{eq:B} and integrate the scalar variation by parts to obtain \eqref{eq:descendants}. For a fixed smooth \(k\), \(Lk\) is smooth on \(V\), so the first term in \(D_\eta B_f\) is a regular scalar linear functional. Every source in the remaining metric derivatives is smooth and compactly supported in \(V\). For compact distribution kernels, the bilinear expression in \eqref{eq:B} is continuous in smooth fields restricted to a fixed compact neighbourhood of \(K_*\). The kernel theorem gives its compact mixed kernel. Alternatively, multiply the retarded kernel by \(\Theta(z^0)\) and transfer the finite \(C\) and scalar derivatives to it. The product exists because a nonzero wavefront pair of the wave kernel has nonzero observation-slot covector, whereas a wavefront covector of the step factor has zero observation-slot component. The operations are therefore legitimate \cite{BDHIntro}. \end{proof} Every compact-distribution polynomial depending only on \(\phi\) also belongs to \(\Padm\), because its metric derivatives vanish. That observation is not a Wick-product theorem for arbitrary scalar distributions. \begin{theorem}[Classical closure] \label{thm:closure} The domain \(\Padm\) is a unital commutative associative algebra under classical pointwise multiplication. It is stable under smooth directional differentiation and conjugation. The subalgebra \(\Pgen\) generated by \(\Pmc\), all \(B_f\), and all \(J(a)\) is contained in \(\Padm\) and has the same differential and conjugation stability. There is no bound on the finite number of factors in a pointwise word. \end{theorem} \begin{proof} The coefficients of a product are finite sums of symmetrized exterior tensor products, so they remain compact distributions. For a direction list, the Leibniz rule expresses \(D_{\boldsymbol\eta}(FG)\) as a finite sum \((D_I F)(D_JG)\). Its metric derivative at a fixed configuration is \[ (D_IF)(\Psi)(D_JG)'_h(\Psi) +(D_JG)(\Psi)(D_IF)'_h(\Psi). \] The coefficients here are complex numbers, and the distributional factors belong to \(\Sone\). Thus \eqref{eq:admission} holds for the product. An additional direction adds an entry to the defining list. Conjugation preserves \(\Mone\), \(\Qsig\), and their trace. Finally, \eqref{eq:descendants} expresses derivatives of the sharp generators through coordinate generators, regular scalar linear functionals, and constants. The finite Leibniz rule therefore also preserves \(\Pgen\). \end{proof} The classical multiplication in this theorem identifies no singular kernel variables. Its closure is not closure under contractions. %===================================================================== \section{The boundary-sensitive topology} \label{sec:topology} %===================================================================== Give \(C^\infty(\Sigma;T^*U|_\Sigma)\) its compact-open smooth Fréchet topology, even though individual traces have compact support. \begin{definition}[Admission topology] \label{def:topology} The topology \(\taus\) on \(\Padm\) is the initial locally convex topology for the coefficient inclusion into \(\Tcoeff\) and all maps \[ \btrace_{\Psi;\boldsymbol\eta}: \Padm\longrightarrow C^\infty(\Sigma;T^*U|_\Sigma). \] Equivalently it is the topology of the graph embedding \[ F\longmapsto \left((u_n(F))_{n\ge0}, (\btrace_{\Psi;\boldsymbol\eta}(F))_{\Psi,\boldsymbol\eta}\right). \label{eq:graph} \] \end{definition} The added seminorms include, for every compact \(A\subset\Sigma\) and integer \(j\ge0\), \[ \sup_{\mathbf x\in A,\ |\beta|\le j} \left|\partial_{\mathbf x}^{\beta} \btrace_{\Psi;\boldsymbol\eta}(F)(\mathbf x)\right|. \] No completion, infinite polynomial degree, or new product is introduced by this definition. \begin{proposition}[Hausdorffness and differential continuity] \label{prop:topbasic} The topology \(\taus\) is Hausdorff. Every fixed-direction operator \(D_\eta\) and conjugation are continuous. Evaluation of \(F\), or any fixed directional descendant, at any fixed smooth configuration is continuous. \end{proposition} \begin{proof} Lemma~\ref{lem:unique} makes the coefficient inclusion injective into a Hausdorff space, so \eqref{eq:graph} is Hausdorff. Contraction of a compact distribution with a fixed smooth field is strongly continuous: insertion of that field maps bounded sets of smooth target tests to bounded sets of source tests. The finite degree shifts are continuous on the locally convex direct sum. The same argument handles fixed evaluations. For the boundary coordinates, \[ \btrace_{\Psi;\boldsymbol\eta}(D_\eta F) =\btrace_{\Psi;(\boldsymbol\eta,\eta)}(F). \] Thus differentiation shifts coordinates already present in the topology. Conjugation replaces the configuration and direction list by their conjugates and conjugates the trace. \end{proof} \begin{theorem}[Separate continuity of classical multiplication] \label{thm:separate} For each fixed \(G\in\Padm\), multiplication by \(G\) is continuous on \((\Padm,\taus)\). No joint continuity or hypocontinuity is inferred. \end{theorem} \begin{proof} At coefficient level, multiplication by \(G\) is a finite sum of tensor products with fixed compact distributions, followed by symmetrization and degree shifts. For a fixed compact distribution \(v\), partial application of \(v\) maps a bounded set of smooth joint tests to a bounded set of tests in the remaining variables. This proves strong continuity of \(u\mapsto u\otimes v\). For a boundary coordinate, the differentiated Leibniz formula gives a finite sum of terms \[ (D_IF)(\Psi)\btrace_{\Psi;\boldsymbol\eta_J}(G), \qquad (D_IG)(\Psi)\btrace_{\Psi;\boldsymbol\eta_J}(F). \] The \(G\)-values and \(G\)-traces are fixed. The remaining maps are continuous evaluations or defining boundary coordinates. Every coordinate of multiplication by \(G\) is continuous. \end{proof} The proof supplies no uniform estimate for arbitrary bounded sets of varying multipliers. This is the precise distinction between the asserted statement and a hypocontinuity theorem \cite{BDH,CheungIXA}. \begin{theorem}[Continuous admission of test families] \label{thm:testcontinuity} The maps \[ \DD(V)\longrightarrow(\Padm,\taus),\quad f\longmapsto B_f, \qquad \DD(V;T^*U)\longrightarrow(\Padm,\taus),\quad a\longmapsto J(a) \] are continuous and linear. For finitely many fixed compact test supports, every fixed finite pointwise word in these generators depends jointly continuously on those tests in their smooth fixed-support topologies. \end{theorem} \begin{proof} For coordinate functionals, smooth finite-time Cauchy estimates make \(a\mapsto\GA a\), its derivatives, and its initial trace continuous on each fixed compact source stage. Formula \eqref{eq:qa} then gives strong continuity of \(q_a\). For \(B_f\), the compact mixed kernel is a finite derivative of the fixed retarded-step distribution multiplied by \(f(x)\). On a common compact support stage, strong seminorms of this kernel are bounded by finitely many smooth seminorms of \(f\). This follows by applying its finite distributional order to the bounded family of joint tests multiplied by \(f\). The only nonzero types of boundary coordinates are \begin{equation} \btrace_{(h,\phi);\varnothing}(B_f) =-(\GA(f\,\dd\phi))|_\Sigma, \qquad \btrace_{(h,\phi);(k,\sigma)}(B_f) =-(\GA(f\,\dd\sigma))|_\Sigma. \label{eq:Btrace} \end{equation} Higher direction lists give zero. Each map in \eqref{eq:Btrace} is continuous on fixed test supports. The test-function LF topology is final over those stages, proving the linear continuity assertions. For a word with fixed supports, its coefficient kernel is an exterior tensor product of fixed distributions multiplied by products of the varying smooth test functions. Finite-order estimates on the common compact region prove joint continuity of this restricted parameterization. Its boundary coordinates follow from the finite Leibniz formula. This restricted statement does not establish joint continuity of multiplication on all of \(\Padm\). \end{proof} \begin{proposition}[Faithful continuous bulk inclusion] \label{prop:bulkinclusion} The inclusion \(\Pmc\hookrightarrow\Padm\) is injective and continuous on each fixed-degree, fixed-support, fixed-wavefront normal-topology bulk stage. All its boundary coordinates vanish. The topology induced on \(\Pmc\) is its strong coefficient subspace topology, not an asserted identification with the finer normal topology. \end{proposition} \begin{proof} Injectivity follows from uniqueness of actual kernels. Lemma~\ref{lem:bulk} puts every relevant linear metric descendant in \(\Mone\), where \(\btrace\) vanishes. Only the coefficient topology remains on the bulk. Normal stages contain those strong seminorms, giving the stated continuous inclusion. \end{proof} %===================================================================== \section{A nonzero boundary witness and failure of regular convergence} \label{sec:separator} %===================================================================== The following signature-sensitive identity is the explicit separation mechanism of IX-A \cite[\loc{lem:witness}, \loc{thm:IX1}]{CheungIXA}. \begin{lemma}[Positive advanced-Cauchy moment] \label{lem:positive} Let \(f\in\DD(V;\RR)\) be nonzero. Define \[ u_f=\partial_t(tf),\qquad \phi_f=u_f,\qquad a=f\,\dd\phi_f,\qquad b=\GA a. \] Then \begin{equation} \int_\Sigma b_0(0,\mathbf x)\,\dd^3\mathbf x =\int_V [\partial_t(tf)]^2\,\dd^4x =:\mathcal E_f>0. \label{eq:positive} \end{equation} The initial trace is compactly supported inside \(\Sigma\). \end{lemma} \begin{proof} All spatial supports are uniformly bounded on the finite time interval in question. Put \[ A_0(t)=\int_{\RR^3}a_0(t,\mathbf x)\,\dd^3\mathbf x,\qquad B_0(t)=\int_{\RR^3}b_0(t,\mathbf x)\,\dd^3\mathbf x. \] Integration of \(\Box_\eta b_0=a_0\) gives \(-B_0''=A_0\). The advanced solution has zero data at sufficiently late times. Consequently, \[ B_0(0)=-\int_0^\infty tA_0(t)\,\dd t =-\int_V tf\,\partial_tu_f\,\dd^4x =\int_Vu_f^2\,\dd^4x. \] There are no integration boundary terms because \(f\) is compact inside \(V\). If \(u_f=0\), then \(tf\) is independent of \(t\); compact time support implies \(f=0\), a contradiction. Support saturation places the entire trace support inside the initial ball. \end{proof} \begin{corollary}[Strict admission and injectivity] \label{cor:strict} For every nonzero real \(f\), \[ B_f\in\Padm\setminus\Pmc. \] The linear map \(f\mapsto B_f\) is injective, also after complexification. \end{corollary} \begin{proof} At \(\Psi_f=(0,\phi_f)\), the metric derivative is \(-q_a\). Lemmas~\ref{lem:positive} and \ref{lem:noncancellation} give a nonzero timelike wavefront component, forbidden for a bulk linear metric descendant. The derivative is nonzero, so \(B_f\ne0\). For a complex source, separate real and imaginary parts on real configurations. \end{proof} Choose a real even compact mollifier \(\rho\) of integral one. Extend the compact mixed kernel \(K_f\) by zero in the developing coordinates and define \begin{equation} K_{f,\varepsilon} =(\rho_\varepsilon\otimes\rho_\varepsilon)*K_f,\qquad \rho_\varepsilon(x)=\varepsilon^{-4}\rho(x/\varepsilon). \label{eq:regularization} \end{equation} For sufficiently small \(\varepsilon>0\), all supports remain in a common compact subset of \(U^2\). Let \(B_{f,\varepsilon}\) be the resulting actual polynomial. Equivalently, using compatible local extensions, \(B_{f,\varepsilon}[h,\phi] =B_f[\rho_\varepsilon*h,\rho_\varepsilon*\phi]\). Each approximant has smooth compact kernel and belongs to \(\Preg\). \begin{theorem}[Normalized continuous separator] \label{thm:separator} For every nonzero real \(f\), there is a continuous linear functional \(\mathcal Q_f:\Padm\to\CC\) such that \begin{equation} \mathcal Q_f(B_f)=1,\qquad \mathcal Q_f(F)=0\quad(F\in\Pmc). \label{eq:normalized} \end{equation} In particular, \[ B_{f,\varepsilon}\not\longrightarrow B_f \quad\text{in }(\Padm,\taus), \] although their coefficient kernels converge strongly. If two supplied numerical evaluations of \(\mathcal Q_f\) have absolute errors at most \(\epsilon_0,\epsilon_1\) on a bulk element and on \(B_f\), their reported separation is at least \(1-\epsilon_0-\epsilon_1\). \end{theorem} \begin{proof} Choose \(\chi\in C_c^\infty(\Sigma)\) equal to one on the trace support in Lemma~\ref{lem:positive}, and set \begin{equation} \mathcal Q_f(F) =-\frac1{\mathcal E_f} \int_\Sigma\chi(\mathbf x) [\btrace_{\Psi_f;\varnothing}(F)]_0(\mathbf x) \,\dd^3\mathbf x. \label{eq:Qf} \end{equation} This is a continuous coordinate of the graph topology. Equations \eqref{eq:Btrace} and \eqref{eq:positive} give its value one on \(B_f\); Proposition~\ref{prop:bulkinclusion} gives zero on the bulk. No net of bulk elements can converge to \(B_f\) in this topology. Strong convergence of the mollified compact kernel follows by transposition: convolution of a bounded set of smooth tests converges uniformly with all finitely required derivatives on the common compact set. The error assertion is the scalar triangle inequality. \end{proof} Thus the closure of the bulk in \(\Padm\) is contained in the closed subspace \(\bigcap_{\Psi,\boldsymbol\eta}\ker \btrace_{\Psi;\boldsymbol\eta}\), and excludes every nonzero real sharp insertion. Equality with that closed subspace is not claimed. \subsection{The normal-topology comparison} In the mixed variables \((x,z)\), define the translation-invariant angular cone \begin{align} \Lambda_{\rm d} &=\{(\xi,\zeta)\ne0:\zeta=-\xi\},\nonumber\\ \Lambda_{\partial} &=\{(\xi,\zeta)\ne0: \eta^{-1}(\xi,\xi)=0,\ \xi+\zeta\in\RR\,\dd t\},\nonumber\\ \Gamma_B&=(U\times U)\times (\Lambda_{\rm d}\cup\Lambda_{\partial}). \label{eq:GammaB} \end{align} The second set includes \(\xi=0\). \begin{proposition}[The dropped admission-continuity hypothesis] \label{prop:normal} The mixed kernel satisfies \(\WF(K_f)\subseteq\Gamma_B\). On a sufficiently large common compact support stage, \[ K_{f,\varepsilon}\longrightarrow K_f \quad\text{in the normal topology with cone }\Gamma_B. \] The identity from these admitted mixed kernels to their actual functionals in \((\Padm,\taus)\) is not sequentially continuous at \(K_f\) for nonzero real \(f\). \end{proposition} \begin{proof} The retarded kernel has diagonal conormals and null-propagation pairs \((\xi,-\xi)\). Multiplication by the step factor in the \(z\)-slot adds \[ (\xi,-\xi+\tau\,\dd t),\qquad (0,\tau\,\dd t). \] At this boundary branch \(x^0>0=z^0\), so a diagonal contribution cannot meet the boundary there. Finite differentiation and smooth coefficients preserve the upper cone \eqref{eq:GammaB}. Strong convergence was proved above. For a localized Fourier seminorm on a closed cone separated from \(\Gamma_B\), use a slightly larger separated angular cone. The localized kernel has arbitrary polynomial decay in that larger cone and polynomial growth globally. Convolution regularization introduces bounded smooth multipliers converging to one on fixed frequency balls. Split the Fourier integral into a fixed ball, a near-frequency region in the larger separated cone, and its angularly separated complement. The first part tends uniformly to zero. Arbitrary rapid decay controls the second, and rapid decay of the localization transform controls the complement. Choosing the decay order larger than the requested seminorm order and the distribution growth order gives a tail bound uniform in \(\varepsilon\). This proves normal convergence, the standard compact regularization argument for the stated constant angular cone \cite{BDH}. Theorem~\ref{thm:separator} gives nonconvergence of the image functionals and hence the last assertion. \end{proof} \paragraph{Comparison, not evasion by subtraction.} VII-C records a particular logarithmic Wick obstruction under normal-admission continuity and old-product agreement \cite{CheungVIIC}. The present topology explicitly drops the regular-to-sharp admission-continuity hypothesis. It does not subtract the old regular products or show that a replacement quantum algebra exists. VII-D's numerical finite part and IX-B's subsequent bookkeeping are separate records, not premises of this argument \cite{CheungVIID,CheungIXB}. %===================================================================== \section{Presentation independence and selected existing coefficients} \label{sec:coefficients} %===================================================================== \begin{theorem}[Classical presentation independence] \label{thm:presentation} If two admissible presentations agree on every smooth off-shell configuration, they define the same element of \(\Padm\) and the same topology coordinates. Kernel permutations, tensor identities, compact integration by parts with all surface terms retained, and valid bulk--surface decompositions preserve the admitted element. No equation-of-motion quotient is used. \end{theorem} \begin{proof} Lemma~\ref{lem:unique} identifies their symmetric coefficient kernels. Equality of actual polynomials identifies every directional and metric derivative. Proposition~\ref{prop:trace} therefore identifies every boundary coordinate. The stated operations preserve actual distributional pairings. In particular \eqref{eq:qa} is an equality, but deleting its delta term is not an allowed rewriting. \end{proof} \subsection{A local comparison covariance, not a conical quantum state} For the selected coefficient comparisons only, fix in the auxiliary developed coordinates \[ H_{hh,\alpha\beta\rho\sigma}(x,y) =P_{\alpha\beta\rho\sigma}\Delta_0^+(x-y),\qquad P_{\alpha\beta\rho\sigma} =\eta_{\alpha(\rho}\eta_{\sigma)\beta} -\frac12\eta_{\alpha\beta}\eta_{\rho\sigma}, \] \[ H_{\phi\phi}(x,y)=\Delta_m^+(x-y),\quad m\ge0,\qquad H_{h\phi}=0, \] where \[ \Delta_m^+(t,\mathbf x) =\int_{\RR^3}\frac{\dd^3p}{(2\pi)^3\,2\omega_m(p)} e^{-i\omega_m(p)t+i p\cdot\mathbf x}. \] The graviton block is not asserted positive. This is neither a global cone Hadamard state nor a represented interacting gravitational field. On its existing bulk domain, retain the usual contraction formula \[ F\star_HG =\sum_{r\ge0}\frac{\hbar^r}{r!} \pair{F^{(r)}}{H^{\otimes r}G^{(r)}}. \] For polynomials the sum is finite. The distinction between this old domain and a quantum extension is standard in the functional formulation \cite{FR}. \begin{lemma}[Smeared sharp metric response] \label{lem:smoothing} For \(a\in\DD(V;T^*U)\), smearing a massless metric covariance against \(q_a\) in one slot gives a smooth function of the other slot. Reverse ordering has the same property. \end{lemma} \begin{proof} The kernel \(q_a\) has only timelike conormal wavefront covectors. A nonzero covariance wavefront covector is null in each slot and nonzero in both. The distributional composition criterion therefore has no opposing pair in the contracted variable. Its output wavefront can contain a covector only if such a matching pair survives, which is impossible. Compactness of \(q_a\) supplies properness for the integration. The output is smooth \cite{BDHIntro}. \end{proof} \begin{proposition}[A specified collection of admitted coefficient outputs] \label{prop:Wickoutputs} The following already-defined local contraction outputs are actual elements of \(\Padm\): \begin{enumerate}[label=\textup{(\roman*)}] \item coefficients of \(B_f\) with a smooth compact linear metric or scalar functional, in either order; \item the scalar-contracted metric quadratic functional \begin{equation} T_{f,f'}[h] =\int f(x)f'(x')w^\mu[h](x)w^\nu[h](x') \partial_\mu^x\partial_\nu^{x'} \Delta_m^+(x-x')\,\dd^4x\,\dd^4x'; \label{eq:selfcoefficient} \end{equation} \item the single-contraction coefficient with \(W_g=\int g\,h_{00}^2\), \(g\in\DD(V)\), in either order. \end{enumerate} These are selected coefficient-membership statements. They do not define a quantum multiplication on \(\Padm\). \end{proposition} \begin{proof} A smooth linear metric smearing makes the contracted metric covariance smooth. Its image under the local \(L\) is smooth on \(V\). The remaining output is therefore a regular scalar linear functional. A smooth scalar linear smearing gives a smooth scalar function \(v\), and the output is, up to the fixed contraction sign, \(J(f\,\dd v)\). For \eqref{eq:selfcoefficient}, pair the differentiated scalar covariance against the smooth compact two-variable coefficient at each fixed \(h\). This defines a continuous quadratic functional with compact metric kernel. Varying one \(w\)-factor leaves the other factor smoothly smeared against the scalar covariance. The resulting source for the varied \(w\) is smooth and supported in \(\supp f\) or \(\supp f'\). Its first metric derivative is therefore a finite sum of \(q_a\)'s. Every further directional descendant has the same property. For (iii), define the mixed kernel \(H_{wh}=(L\otimes I)H_{hh}\) on the supported observation variables. It is a defined distribution: the retarded-step composition has the same transversality used in Proposition~\ref{prop:generators}. The left-ordered coefficient is \[ -2\int f(x)g(y) H_{wh}^{\mu}{}_{00}(x,y) \partial_\mu\phi(x)h_{00}(y)\,\dd^4x\,\dd^4y. \] At fixed smooth \(\phi\), smearing the \(x\)-slot is precisely a metric covariance smeared with \(q_{f\,\dd\phi}\). Lemma~\ref{lem:smoothing} makes the result smooth in \(y\). Multiplication by \(g\) gives a regular first metric derivative. Scalar differentiation replaces \(\phi\) by a fixed smooth direction and preserves the argument. Reverse ordering changes the covariance orientation, not this smoothness conclusion. \end{proof} The cases above match the selected admission tests identified in IX-A \cite[\loc{prop:knownoutputs}]{CheungIXA}. No subsequent multiplication of their outputs by another sharp insertion is assigned here. \begin{remark}[Formal slots are not a quantum algebra] One may form \[ \mathcal V_\Sigma^{[3]} =\Padm\otimes_\CC\CC[\hbar]/(\hbar^4) \simeq\Padm^4 \] with its finite product topology. Projection to the classical slot and field-independent coefficient evaluations are continuous. The old bulk polynomial space embeds linearly and injectively. This does not supply a deformed product, quantum associativity, a state, or interacting admission. \end{remark} \subsection{Supported transfer} \begin{theorem}[Local transfer to the prescribed cone] \label{thm:transfer} Transport the tensor fields, densities, kernels, and initial surface through the developing identification of Definition~\ref{def:support}. The supported construction on the cone has exactly the admission, topology, separation, and presentation properties proved above. It is independent of auxiliary field extensions equal near \(K_*\). A product-adapted change of developed coordinates \(t'=t\), \(\mathbf x'=Q\mathbf x+\mathbf a\), with \(Q\) orthogonal, gives the corresponding isomorphic construction on transformed support data. No chart through the worldsheet, global conical inverse, or boost-invariant identification of different initial prescriptions is implied. \end{theorem} \begin{proof} The developing map is an isometry on the complete analytic support. Its tensor pullbacks and density transformations carry all finite-order expressions and compact distribution pairings to the same local formulas. Causal support saturation and Lemma~\ref{lem:saturation} prove extension independence. The displayed product-adapted coordinate changes preserve \(\Box_\eta\), the zero-Cauchy surface, and the finite-propagation prescription. Distribution pullback by these diffeomorphisms is continuous, with continuous inverse, in the coefficient and smooth trace spaces. Hence it preserves the graph topology. The final limitations follow from the domains actually used. \end{proof} %===================================================================== \section{Finite evaluation tables and one destructive clock} \label{sec:finite} %===================================================================== The continuum theorem precedes the finite interface. A finite record can report a supplied evaluation without becoming a measurement of the continuum object from which that evaluation was defined. \subsection{Declared finite data} Choose a finite catalogue \[ \mathcal X=\{0,\ldots,2^g-1\},\qquad x\longmapsto F_x\in\Padm, \] and finitely many real linear diagnostics \(\ell_j\) on an appropriate real subspace. Allowed examples include: \begin{itemize} \item the real or imaginary part of a coefficient pairing with a fixed smooth joint test; \item evaluation of a fixed directional descendant at a fixed smooth configuration; \item a boundary pairing \(\int_\Sigma\chi\cdot\btrace_{\Psi;\boldsymbol\eta}(F)\); \item the normalized separator \(\mathcal Q_f\). \end{itemize} These maps are continuous by the preceding theorems. Each finite table entry \(\ell_j(F_x)\), or a certified enclosure for it, is supplied classical data. No general computability theorem for arbitrary smooth inputs is assumed. In particular, continuity alone is not an algorithm for evaluating an arbitrary real number exactly. Choose a finite bound \(M>0\) such that all selected entries have absolute value at most \(M\), and put \[ v_{a,x}\in[-1,1] \] for the installed normalized score selected by clock code \(a\). A finite report word \(r(a,x)\) may additionally store prescribed rounded values and exact catalogue flags. It is a report of installed data, not a universal admission test. For the catalogue containing \(B_f\) and any finitely many regular approximants, \(\mathcal Q_f\) gives exactly the entries one and zero. Its existence and nonzero normalization were proved, not inferred from numerical sampling. \subsection{Complete clock instrument} Let \(C\) have dimension \(2^m\) and an installed orthonormal basis \(\{\ket{\tau_a}\}\). Let \(S\) contain every surviving factor and let \(E\) be any untouched reference. Put \[ D_a=\bra{\tau_a}_C\otimes I_S,\qquad P_a=\proj{\tau_a}. \] \begin{theorem}[Destructive clock] \label{thm:instrument} Calibrated basis change followed by destructive Enter has channel \begin{equation} \mathcal M_C(X) =\sum_a\proj a_R\otimes D_aXD_a^\dagger =\sum_a\proj a_R\otimes \Tr_C[(P_a\otimes I_S)X(P_a\otimes I_S)]. \label{eq:instrument} \end{equation} It is CPTP, including with \(\id_E\), and \[ \Tr_R\mathcal M_C(X)=\Tr_CX. \] Its quantum output contains \(S\), not \(C\). On a branch of positive probability one may normalize the returned block; zero branches require no conditional state. \end{theorem} \begin{proof} The Kraus operators are \(\ket a_R\otimes D_a\). Their squared moduli sum to \(\sum_aP_a\otimes I_S=I\). Expansion of \(X\) in clock matrix units shows that both expressions return its diagonal \(aa\)-block. Summing the blocks gives the partial-trace identity. The calculation is unchanged with an additional reference. \end{proof} This is the finite interface compared with I-B, VI-B, En09-A, and the observation sheets \cite{CheungIB,CheungVIB,CheungEn09,CheungObservation}. The proof is a complete finite-instrument calculation \cite{Watrous}. \begin{proposition}[Calibration and clock limits] \label{prop:clocklimits} At one fixed actual reading event \(e\), distinct calibrated labels \(t_a\) satisfying \[ p_a>0\Longrightarrow t_a=t(e) \] permit at most one positive-probability outcome. A nontrivial multi-outcome history or programme label therefore requires a separate interpretation, not that event-time equality. The clock channel has no universal left inverse on its full input space. No finite matrices obey \([T_C,H_C]=i\hbar I\) for \(\hbar\ne0\). Clock consumption does not consume \(T\), \(t\), or a spacetime dimension. \end{proposition} \begin{proof} Distinct labels prove the first assertion. The two clock states \((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), with the same payload, give identical destructive outputs, excluding a universal inverse. Tracing the commutator equation gives a contradiction. The geometric coordinates remain fixed classical inputs. \end{proof} No stationarity or covariance hypothesis for a conditional unitary-time interpretation is imposed. The finite clock selects an externally scheduled finite programme; it does not generate unscheduled quantum time \cite{CheungEn09}. \subsection{Complete clock--table channel} Let \(G\) be the catalogue register, and let \(P\) be an arbitrary finite quantum payload. The intended classical catalogue interface is jointly block diagonal in \(G\), including its correlations with other systems. Its canonical extension to all matrices first pinches \(G\). For \[ X_{a,x}= (\bra{\tau_a}_C\bra x_G\otimes I_P) X (\ket{\tau_a}_C\ket x_G\otimes I_P), \qquad p_y(a,x)=\frac{1+yv_{a,x}}2,\quad y=\pm1, \] define the recorded channel \begin{equation} \mathcal I(X)= \sum_{a,x,y} \proj{a,x,y,r(a,x)}_W\otimes p_y(a,x)X_{a,x}. \label{eq:tablechannel} \end{equation} \begin{proposition}[Complete evaluation instrument] \label{prop:table} The map \eqref{eq:tablechannel} is CPTP and acts as identity on the surviving payload in each subnormalized classical branch. It preserves arbitrary untouched reference correlations in that channel sense. Its conditional sign mean on catalogue entry \(x\) and clock code \(a\) is \(v_{a,x}\). It is implementable by a controlled finite probe isometry, probe readout, and classical report generation. \end{proposition} \begin{proof} Kraus operators are \[ K_{a,x,y}= \sqrt{p_y(a,x)}\, \ket{a,x,y,r(a,x)}_W (\bra{\tau_a}_C\bra x_G)\otimes I_P. \] Since \(\sum_y p_y(a,x)=1\), their squared moduli sum to \(I_{CGP}\). The formula on matrix units proves the reference claim. For the probe, use the normalized vector \[ \sqrt{p_+(a,x)}\ket+ +\sqrt{p_-(a,x)}\ket- \] as a controlled preparation, followed by destructive readout. The resulting finite isometry extends to a unitary on a finite larger register. The sign expectation is \(p_+-p_-=v_{a,x}\). \end{proof} Conditional reference states may change upon selecting a correlated branch. The unconditional reference marginal is unchanged. Reference preservation does not mean absence of conditional steering. \begin{proposition}[Fixed-table complete-channel approximation] \label{prop:approx} Suppose a second normalized probe vector is installed for each \((a,x)\), with Euclidean distance at most \(\delta\) from the first. Keep the same input, output, pinching, and report interfaces. Then \[ \ddia(\widetilde{\mathcal I},\mathcal I) \le\min\{1,\delta\}. \] Here \(\ddia(\Phi,\Psi)=\tfrac12\norm{\Phi-\Psi}_\diamond\). \end{proposition} \begin{proof} Retain \((a,x)\) as orthogonal flags before common classical postprocessing. The two controlled isometries differ in operator norm by at most \(\delta\). For isometries \(V,\widetilde V\), subtraction into two terms gives \[ \frac12\norm{\Ad_V-\Ad_{\widetilde V}}_\diamond \le\norm{V-\widetilde V}_{\rm op}. \] Common channel postprocessing contracts the diamond norm, including reference extensions \cite{Watrous}. \end{proof} Approximating a real value by a different orthogonal label does not itself give small diamond distance. The proposition approximates normalized amplitudes on one fixed interface. \begin{proposition}[Finite diagnostics are not the full topology] \label{prop:finiteness} No finite family of linear scalar diagnostics on \(\Padm\) determines every admitted functional. In particular, the finite evaluation table does not decide all boundary traces, all wavefront conditions, or convergence of arbitrary families in \(\taus\). \end{proposition} \begin{proof} For \(N\) diagnostics choose \(N+1\) linearly independent regular scalar linear functionals with disjoint nonzero smearings in \(U\). They belong to \(\Preg\subset\Padm\). The map from their span to \(\CC^N\) has a nonzero kernel. Thus a nonzero admitted functional has all listed readings zero. A finite list cannot replace the defining family of topology coordinates. No universal decision procedure is introduced by storing finitely many values. \end{proof} A finite table can nevertheless encode the exact separator for a declared finite catalogue. That is a restricted certificate, not a continuum reconstruction. %===================================================================== \section{Residual compatibility, blindness, and recovery safeguards} \label{sec:residual} %===================================================================== A record and a surviving quantum interface are different outputs. The following finite comparisons sharpen that distinction around the evaluation channel. Let \(P_z\) be input projections, let \(q\) aggregate their fine outcomes, and put \[ \Delta_w(X)=\sum_{z:q(z)=w}P_zXP_z,\qquad R_w=\sum_{z:q(z)=w}P_z. \] For a channel \(\mathcal T\) and an output instrument \(\{\mathcal A_w\}\), compare \[ \mathcal F(X)=\sum_w\proj w\otimes\mathcal T(\Delta_w(X)), \qquad \mathcal G(X)=\sum_w\proj w\otimes\mathcal A_w(\mathcal T(X)). \] \begin{proposition}[Record and residual criteria] \label{prop:compatibility} The record laws agree on every input exactly when \[ \mathcal T^*(\mathcal A_w^*(I))=R_w. \] Agreement on a declared recorded residual algebra \(\mathcal O\) holds exactly when \[ \sum_{z:q(z)=w}P_z\mathcal T^*(B)P_z =\mathcal T^*(\mathcal A_w^*(B)), \qquad B\in\mathcal O. \] These identities are stable under untouched references. \end{proposition} \begin{proof} Pair each recorded output block with \(I\), or with \(B\), and move the maps to the input by adjoints. States separate Hermitian effects. Linear extension gives the residual-algebra identities; reference extension tensors the corresponding maps with identity. \end{proof} \begin{proposition}[Sharp coarse Lüders compatibility] \label{prop:luders} If \(\mathcal A_w(Y)=Q_wYQ_w\) and \(\mathcal T^*(Q_w)=R_w\) for projections \(Q_w,R_w\), then \[ \mathcal T^*(Q_wBQ_w)=R_w\mathcal T^*(B)R_w. \] Agreement with the aggregated fine residual instrument further requires, and is equivalent to, \[ P_z\mathcal T^*(B)P_{z'}=0 \quad(z\ne z',\ q(z)=q(z')=w) \] for the declared residual observables. \end{proposition} \begin{proof} Write \(\mathcal T^*(B)=V^\dagger(B\otimes I)V\) with \(V\) an isometry obtained from a Kraus family. Expansion shows \[ \bigl((Q_w\otimes I)V-VR_w\bigr)^\dagger \bigl((Q_w\otimes I)V-VR_w\bigr)=0. \] Thus \((Q_w\otimes I)V=VR_w\), giving the first identity. Insert it into Proposition~\ref{prop:compatibility}; multiplication by distinct \(P_z,P_{z'}\) proves the remaining equivalence. \end{proof} \begin{proposition}[Equal records leave a residual gap] \label{prop:gap} Two instruments can have identical classical records on every input while their residual states have trace distance \(1/2\) on one qubit input. Their residual source expectations can therefore differ. \end{proposition} \begin{proof} Use one coarse outcome and compare identity with computational dephasing. Both record channels are constant. On \(\proj+\), their residuals are \(\proj+\) and \(I/2\). The difference is \(X/2\), with eigenvalues \(1/2,-1/2\). The trace distance is \(1/2\), and the observable \(X\) has expectations one and zero. \end{proof} The same comparison can be tensored with an unchanged evaluation-table register. Equal admission-test records then still fail to determine the extra quantum residual. This is the instrument-level separation documented in the VII and VIII observation records \cite{CheungObservation,CheungVIIIA}. \begin{proposition}[Informative-record recovery obstruction] \label{prop:recovery} If a complete recorded finite instrument has a CPTP left inverse on the full input matrix algebra, each record effect is a scalar multiple of the identity. An informative record is therefore incompatible with exact recovery of every unknown input from that complete recorded output. \end{proposition} \begin{proof} Let \(A_i\) be the recorded channel's Kraus operators and \(B_j\) those of a left inverse. The operators \(B_jA_i\) form a Kraus family for identity. Its Choi matrix has rank one, so \(B_jA_i=c_{ji}I\). Trace preservation of the inverse implies \[ A_i^\dagger A_k =\sum_j\overline{c_{ji}}c_{jk}I. \] Summing the diagonal terms associated with a record gives a scalar effect. \end{proof} This statement concerns the full unknown quantum input. It does not prevent exact transport of the unmeasured payload in Proposition~\ref{prop:table}, and it does not prevent reversible processing of supplied classical catalogue labels. \paragraph{Spectator-blind control.} An installed controller is spectator blind when its chosen table, programme, route, and resource guards depend only on declared classical inputs and named accessible reductions. Equal such complete inputs then select equal classical outputs. Adjoining an inaccessible spectator with unchanged named reductions does not alter the selected programme. This is equality under a specified factorization; it is not a new source-acquisition law or an assertion about unknown continuum fields. %===================================================================== \section{Scheduled locality and regular causal transport} \label{sec:transport} %===================================================================== \subsection{The complete scheduled ancestor set} Every programme input, table selector, record, reset, encoder, decoder, and controller wire read by an operation is included in its declared interface. All maps in this section are unconditional CPTP maps. For a finite site set \(X\), put \[ \HH_X=\bigotimes_{x\in X}\HH_x,\qquad \AA(Y)=\BB(\HH_Y)\otimes I_{X\setminus Y}. \] \begin{theorem}[Schedule-locality] \label{thm:locality} Let prescribed partitions \(\mathcal B_r\) define layers \[ \Phi_r=\bigotimes_{B\in\mathcal B_r}\Phi_{r,B}. \] Set \[ N_{t,t}(Y)=Y,\qquad N_{u,t}(Y)= \bigcup_{\substack{B\in\mathcal B_{u+1}\\ B\cap N_{u+1,t}(Y)\ne\varnothing}}B. \] Then \[ (\Phi_{u+1}^*\cdots\Phi_t^*)(\AA(Y)) \subseteq\AA(N_{u,t}(Y)). \] If all block diameters are at most \(\Delta\), then \[ N_{u,t}(Y)\subseteq \{x:d_X(x,Y)\le(t-u)\Delta\}. \] Equal input reductions on the ancestor region give equal output reductions on \(Y\). For joint output with a reference \(E\), equality must hold on the joint ancestor--reference input reduction. \end{theorem} \begin{proof} The adjoint of a trace-preserving map is unital. A block disjoint from current observable support therefore acts on identity and leaves it unchanged. Every other block can enlarge support only within its own interface. Reverse induction gives the inclusion. Each step adds at most one block diameter. Testing all output observables proves the reduced-state statements, with \(\BB(\HH_E)\) carried unchanged in the referenced proof. \end{proof} The same proof applies to a general finite acyclic network by reversing its complete port dependencies. Joint outputs require their joint ancestor set. An upper bound may include gates whose actual channel has no influence on the chosen output. \begin{proposition}[Exact deterministic no-signalling criterion] \label{prop:nosignal} For input factor \(A\) and output channel \(\Phi_B\), the following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item every deterministic local intervention on \(A\) leaves the \(BE\) output unchanged for every input and untouched reference; \item \(\Phi_B=\Xi\Tr_A\) for some CPTP map \(\Xi\); \item \(\Phi_B^*(\BB(\HH_B)) \subseteq I_A\otimes\BB(\HH_{A^c})\). \end{enumerate} \end{proposition} \begin{proof} Factorization proves intervention invariance. Conversely, compare identity on \(A\) with replacement by its maximally mixed state \(\tau_A\). Invariance gives \[ \Phi_B(X)=\Xi(\Tr_AX),\qquad \Xi(Z)=\Phi_B(\tau_A\otimes Z) \] first on states and then on all operators. Taking adjoints proves the equivalence with (iii). \end{proof} Postselection is not a deterministic intervention in this criterion. \begin{theorem}[Physical containment is an additional certificate] \label{thm:causal} Suppose each port of the complete operational network is assigned a regular host event and every dependency edge is implemented by an actual future-causal curve inside the supported presentation. Then actual intervention influence is contained in scheduled ancestry, which is contained in the prescribed metric causal past of the joint output. \end{theorem} \begin{proof} The first inclusion is Theorem~\ref{thm:locality}. Concatenating the installed causal curves along a network path gives the second. Proposition~\ref{prop:nosignal} gives exact no-signalling outside that region. \end{proof} These are the finite interfaces compared with En03-A, En05-A, and VII-J \cite{CheungLocality,CheungVIIJ}. Neither the functional support \(D_f\), the boundary-trace support, nor a computational ancestor set is thereby identified with a derived physical causal past. \subsection{A nonempty target-adapted regular protocol} \begin{assumption}[Finite operational permissions] \label{ass:operations} The host permits independent finite preparations, installed finite local unitaries and instruments, positive-duration local classical processing, storage on regular static station worldlines, and exact finite-register carriage along declared future-causal regular curves. These are ideal finite-model inputs, not continuum-field derivations of communication. \end{assumption} Choose two stations \(A,B\) on an axial segment of developed spatial length \(L>0\), with \(LT_0+3L+4\), enlarging the temporal support presentation before the experiment if needed. Install a \(p\)-qubit payload at \(A\), blank equal-sized payload registers at the other \(n\) stations, the catalogue register, the single clock, and all table/programme data. The payload may be entangled with an untouched reference. Read the clock after \(t=T_0+1\) and complete the local finite table instrument by \(t=T_0+2\). Every elementary operation is assigned a positive subinterval. Let \(k=m+g+1+s\) be the bit length of the complete record, where \(s\) is the report-word length. Starting at \(T_0+2\), use \(n\) stages of duration \(3L/n\): \begin{enumerate}[label=\textup{(\roman*)}] \item in the first \(L/n\), copy the already classical \(k\)-bit record into a fresh outgoing buffer, using \(k\) local positive subintervals; \item in the remaining \(2L/n\), carry that buffer forward and exchange the two adjacent payload contents along opposite regular paths. \end{enumerate} The carriage speed is \(1/2\) in developed units. All old classical copies are retained. A separated exchange is implemented by carriage, not by an instantaneous spatially nonlocal SWAP. \begin{lemma}[Exact chain transport] \label{lem:swap} Successive adjacent SWAPs along this chain move the unknown payload factor to the final station and return all preceding payload factors to blank. The identity holds on all payload--reference operators. \end{lemma} \begin{proof} A SWAP exchanges complete tensor factors and all their correlations. Induction over the chain moves the original factor one step at a time; each factor left behind is the original blank. The resulting factor permutation gives the identity on every matrix unit, including a reference. \end{proof} \begin{theorem}[Complete clock--table--transport realization] \label{thm:operational} Under Assumption~\ref{ass:operations}, the preceding finite regular protocol realizes \begin{equation} \mathcal J(X)= \sum_{a,x,y} \left(\proj{a,x,y,r(a,x)}\right)^{\otimes(n+1)} \otimes p_y(a,x)\, \mathcal U_{A\to B}(X_{a,x}) \otimes\omega_{\rm blank}. \label{eq:transportchannel} \end{equation} Here \(\mathcal U_{A\to B}\) is exact transport of the payload factor and \(\omega_{\rm blank}\) is the returned blank payload/work state. The formula is valid with arbitrary untouched references. Its complete dependencies are causally embedded in the fixed host, and the output arrives no later than \(T_0+2+3L\). \end{theorem} \begin{proof} Proposition~\ref{prop:table} gives the subnormalized clock--catalogue--sign branches. Copying a classical record gives the displayed correlated record copies, not copies of an unknown quantum state. Lemma~\ref{lem:swap} proves exact payload transport in every branch. All branches are retained, so their sum is the stated complete channel. Local gates lie on regular timelike station worldlines. The communication paths have speed \(1/2\), remain inside the regular tube, and deliver every control before its next use. Theorem~\ref{thm:causal} applies. The arrival bound is the sum of the \(n\) stage durations. \end{proof} This realizes the I-B transport pattern with explicit local carriage and controller support \cite{CheungIB,CheungTransport}. No apex register or singularity detector is required. No particle-incidence event is assumed in IX-A-ST. \subsection{Resource and error ledger} With \(w\) clean lookup-work bits, sufficient reserves are \[ \begin{array}{l|l} \toprule \text{resource}&\text{reserve}\\ \midrule \text{single destructive clock}&m\text{ qubits}\\ \text{catalogue input}&g\text{ classical bits}\\ \text{diagnostic probe}&1\text{ qubit}\\ \text{payload registers}&(n+1)p\text{ qubits}\\ \text{lookup work and report reserve}&w+s\text{ bits}\\ \text{complete record copies}&(n+1)k\text{ bits}\\ \text{record-copy operations}&nk\text{ CNOTs}\\ \text{payload exchanges}&n\text{ block SWAPs}\\ \text{record transmissions}&n\\ \text{destructive readouts}&m+1\\ \text{resets during the run}&0\\ \bottomrule \end{array} \] If every classical storage bit is conservatively represented by a basis-state qubit carrier, a sufficient peak bound is \begin{equation} Q_{\rm peak}\le m+g+1+(n+1)p+w+s+(n+1)k. \label{eq:peak} \end{equation} Classical table storage, installed matrix descriptions, and finite local programme counters are additional declared resources. A truth table for an \(s\)-bit report needs at most \(s2^{m+g}\) output bits before instruction overhead. No bound on description complexity of arbitrary exact real probe amplitudes is asserted. \begin{proposition}[Complete-channel error accounting] \label{prop:error} For compatible finite channel factors, \[ \ddia(\widetilde\Phi_N\cdots\widetilde\Phi_1, \Phi_N\cdots\Phi_1) \le\min\left\{1,\sum_{j=1}^N \ddia(\widetilde\Phi_j,\Phi_j)\right\}. \] A complete-channel error \(\varepsilon\) changes any fixed reference-assisted intervention signalling value by at most \(2\varepsilon\). Exact physical no-signalling of the implemented network instead follows from its actual causal embedding. \end{proposition} \begin{proof} Telescope the difference, replacing one factor at a time. Pre- and postcomposition by channels contract diamond norm. For a signalling comparison, the two intervened outputs each change by trace distance at most \(\varepsilon\). Apply the triangle inequality and take suprema. Exact causal no-signalling is Theorem~\ref{thm:causal}, independently of approximation. \end{proof} The ledger counts finite storage and communication. It is not an energy balance, a thermodynamic reset cost, or a Landauer theorem. %===================================================================== \section{Release theorem and the exact remaining gates} \label{sec:release} %===================================================================== \begin{proposition}[Conservation and trace are not Einstein dynamics] \label{prop:noeinstein} On a prescribed metric, symmetry, conservation, and zero trace of a tensor do not imply Einstein's equation. \end{proposition} \begin{proof} In a regular flat chart choose a nonzero constant null covector \(k\) and set \(t_{\mu\nu}=k_\mu k_\nu\). It is symmetric, constant and conserved, nonzero, and traceless. The chart's Einstein tensor is zero. Thus \(G_{\mu\nu}=8\pi G_{\rm N}t_{\mu\nu}\) fails when \(G_{\rm N}\ne0\). No statement on a distributional stress tensor at the defect is needed. \end{proof} \begin{theorem}[IX-A-ST local Grade S release] \label{thm:release} For the prescribed cone and the supported data of Definition~\ref{def:support}, the following are established: \begin{enumerate}[label=\textup{(\roman*)}] \item a local zero-Cauchy response independent of auxiliary extensions on the complete truncated causal support, without a global conical inverse; \item the exact off-shell insertion \(B_f\), coordinate descendants \(J(a)\), and an actual compact-distribution polynomial admission domain; \item a canonical boundary trace modulo microcausal linear kernels, imposed on all directional descendants; \item faithful bulk inclusion, classical presentation independence, and closure under arbitrary finite pointwise words, smooth directional derivatives, and conjugation; \item a Hausdorff locally convex graph topology with continuous differentiation, conjugation, test admission, and separate classical multiplication; \item for every nonzero real \(f\), a normalized continuous separator proving that unchanged regular approximants do not converge to \(B_f\), despite strong and specified normal kernel convergence; \item membership of the selected already-defined coefficient outputs in Proposition~\ref{prop:Wickoutputs}, without a quantum closure claim; \item a complete finite destructive clock, fixed evaluation-table channels and their fixed-interface approximation bound, residual compatibility criteria, and recovery safeguards; \item scheduled locality, exact no-signalling criteria, and an explicit reference-preserving regular transport route with independently certified physical containment and resource accounting. \end{enumerate} These statements constitute local Grade S only. The free quantum-product gate is unresolved in this release. The quantum certificate remains \loc{RG-open}. \end{theorem} \begin{proof} Lemma~\ref{lem:saturation} proves (i). Sections~\ref{sec:carrier}--\ref{sec:admission} prove (ii)--(iv). Section~\ref{sec:topology} proves (v). Theorem~\ref{thm:separator} and Proposition~\ref{prop:normal} prove (vi). Section~\ref{sec:coefficients} proves (vii) and the supported transfer. Sections~\ref{sec:finite}--\ref{sec:residual} prove (viii). Section~\ref{sec:transport} proves (ix). None of those results defines a renormalized multiplication on the enlarged domain, so the final gate statement is unchanged. \end{proof} \subsection{Potential use and scope of novelty} The construction gives a same-model distinction between: \[ \text{exact classical admission},\qquad \text{topological approximation},\qquad \text{finite recorded evaluation},\qquad \text{quantum multiplication}. \] Its finite normalized separator can serve as a benchmark for an implementation or a proposed deformation: changing the underlying admission topology, omitting the Cauchy term, or mistaking a precomputed label for acquired field data becomes an explicit, testable mathematical change. The admission domain and its boundary-sensitive mechanism are attributed to IX-A. The present contribution is the supported conical transfer and the complete finite operational assembly. It does not claim that general distributional boundary methods, finite instruments, or SWAP transport are new. \subsection{What a later quantum construction must still supply} A later quantum-product proposal on this carrier must specify: \begin{enumerate}[label=\textup{(Q\arabic*)}] \item an actual domain for every contracted kernel operation, not merely a carrier for its hoped-for output; \item agreement with the fixed old product on its common domain; \item extension and normalization data for the selected singular graphs and all required subsequent multiplications; \item closure and the required topology estimates on the same output space; \item associativity and descent through all actual-functional relations; \item separate time-ordered, retarded, interacting, and switched-gauge constructions where claimed. \end{enumerate} No assertion that every candidate must fail is proved. Conversely, a finite part for one numerical expression or membership of one output in \(\Padm\) passes none of these remaining gates automatically. The broader distinction between free products, causal products, and interaction maps is retained \cite{FR,CheungIXB,CheungIXD}. No conclusion here reverses a separately versioned later partial construction; it simply makes no transfer from it. \statementbox{Standing non-claims}{% No intrinsic global conical Green or Cauchy theory, continuum field instrument, renormalized relational product, quantum Ward/BRST/BV identity, radiative coefficient, or quantum renormalization separator is constructed. No joint continuity or hypocontinuity of classical multiplication is inferred, and no quantum associativity is constructed. Classical associativity is already the ordinary pointwise identity and is not the missing quantum result. The metric is not reconstructed from records. No Einstein dynamics, spacetime emergence, Grade E, Page curve, island, interior, holographic identification, gravity-generated entanglement, ultraviolet completion, or quantized relational time without external scheduling is proved. The free quantum-product gate remains unresolved in this release. O.1 stays \loc{NOT_CLOSED} in the stated inherited prescription. XV-B is not opened. XVI-N is not discharged. Historical H2 and the full I3/J2 characteristic-map boundaries are not discharged here. No quantum coefficient is admitted in the inherited interacting/radiative admission sense. No compilation is asserted.} The selected free contraction outputs located in Proposition~\ref{prop:Wickoutputs} are already-defined comparison data. Locating them in a classical carrier is not the admission of a new interacting or radiative quantum coefficient. %===================================================================== \appendix \section{Debugging and support checklist} \label{app:checklist} \begin{longtable}{p{.31\linewidth}p{.62\linewidth}} \toprule Potential mistaken inference & Correct statement in IX-A-ST\\ \midrule \endfirsthead \toprule Potential mistaken inference & Correct statement in IX-A-ST\\ \midrule \endhead A conical singular line in four dimensions & The completed product defect is a two-dimensional worldsheet. Fixed axial sections are worldlines.\\[4pt] A chart around the observation is enough & The chart contains the entire truncated past \(D_f\), with a strict spatial margin.\\[4pt] A global cone inverse is available & Only auxiliary Minkowski inverses on the saturated supported prescription are used.\\[4pt] Zero Cauchy data constrain \(h\) and \(\phi\) & They constrain the prescribed response \(w=\Ezero Ch\), not the off-shell input fields.\\[4pt] An initial trace is an independent boundary field & It is determined by the actual linear metric derivative modulo the specified bulk kernels.\\[4pt] Zero trace means no surface singularity & The trace need not detect every step singularity.\\[4pt] Equal on-shell formulas are identified & Only equality on every smooth off-shell configuration is used.\\[4pt] Classical closure implies Wick closure & Classical products use exterior tensor products; contractions have separate product hypotheses.\\[4pt] Bulk topology is unchanged in every respect & The inclusion from each normal bulk stage is continuous; the induced bulk topology is the strong coefficient topology.\\[4pt] Regularization converges in the new topology & The normalized boundary coordinate is one on the nonzero sharp insertion and zero on every regular approximant.\\[4pt] A finite diagnostic decides the full domain & Its finite-dimensional kernel leaves infinitely many admitted functionals indistinguishable.\\[4pt] A table reads an unknown field & It contains supplied evaluations and certified enclosures. No field-acquisition map is present.\\[4pt] A rounded label gives a small channel error & A complete-channel approximation is made on a fixed interface, using normalized amplitude or channel estimates.\\[4pt] A finite clock is the time axis & It is one finite encoding; the supplied coordinate and metric remain after consumption.\\[4pt] Equal records fix residual sources & The complete residual-effect criterion is stronger than equality of record effects.\\[4pt] Scheduled support is physical causality & Physical containment requires causal curves for every complete controller and payload dependency.\\[4pt] A local quantum coefficient has been admitted & Only selected existing free coefficients are located in the classical carrier; the inherited quantum-admission gate is not passed.\\ \bottomrule \end{longtable} %===================================================================== \section{Routing note, problem connections, and emergence card} \label{app:routing} The following is proposed documentary metadata, not an assertion that an external registry has been changed. \paragraph{Routing note.} IX-A-ST places the IX-A actual-functional admission domain on a causally saturated regular presentation of the prescribed I-B cone. The regular product is not globally hyperbolic. The defect is a worldsheet, and the compact laboratory does not touch it. The boundary-sensitive topology preserves the exact off-shell insertion and excludes convergence of its unchanged regular approximants. Classical pointwise multiplication is separately continuous. A finite destructive clock, supplied evaluation tables, residual criteria, and exact causal payload transport form the operational layer. No global conical inverse, continuum acquisition, renormalized quantum product, quantum associativity, Ward identity, or radiative coefficient is constructed. This is local Grade S, not Grade E or spacetime emergence. The inherited quantum certificate remains \loc{RG-open}. O.1 stays \loc{NOT_CLOSED}; XV-B is not opened; XVI-N and the historical H2/full I3/J2 boundaries are not discharged. No quantum coefficient or compilation is asserted. The author field is intentionally empty for anonymous review. Citation of N.~Cheung does not assign authorship of this adaptation. \begin{verbatim} identifier: IX-A-ST version: 1 grade: local Grade S primarySourceRecords: IX-A@1 I-B@1 supportingComparisonRecords: VII-E VII-G VII-E-ST VI-B VII-C En09-A En05-A En03-A VII-J VII-J-ST En08-A En08-B documentaryOnly: VII-D later IX sheets remaining internal series OpenAI catalogue relationshipRole: separation relationshipDistinction: A documentary comparison is not a proof transfer. primaryProblemConnections: products: Classical admission and separate continuity are not a renormalized quantum product. relational: Observation tests and boundary traces are not quantized gravitational observables. boundaryProblemConnections: time: One destructive finite clock; external scheduling remains. singularities: The host defect is prescribed; the laboratory is regular. vacuum: No vacuum-energy stability result. entanglement: No gravity-generated entanglement claim. beyond-EFT: No ultraviolet completion. Page-curve: Not constructed. timeLandmarks: 1 includeInEmergenceSources: false emergenceStatus: not constructed quantumCertificate: RG-open freeQuantumProductGate: unresolved in this release standingBoundaries: O.1 remains NOT_CLOSED in the stated inherited prescription. XV-B is not opened. XVI-N is not discharged. Historical H2 and full I3/J2 are not discharged here. No inherited quantum coefficient is admitted. No compilation is asserted. \end{verbatim} \paragraph{Emergence card and walk sentence.} IX-A-ST certifies a local classical admission domain and boundary-sensitive topology on a prescribed conical product, with finite destructive-clock evaluation records and reference-preserving causal transport; it constructs neither a renormalized quantum product nor spacetime emergence. \paragraph{Version-qualified analytical status.} Historical H2 in XIV-H, the named internal amplitude construction in XIV-I, and the missing full characteristic maps at I3/J2 are different records. This adaptation discharges none of those external obligations \cite{CheungAnalytic}. The phrase ``XV-B is not opened'' refers to the downstream quantum-admission exit, not to a claim that no technical manuscript with that identifier exists. The scope of the stated O.1 prescription is likewise not enlarged by unrelated subsector statements \cite{CheungStops}. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Anonymous review.} The author field is intentionally empty. Authorized author, affiliation, contribution, and submission metadata must be supplied for a signed submission. Citing N.~Cheung's articles and documentary manuscript records does not assign authorship of this adaptation. \paragraph{Source and attribution status.} The bibliography distinguishes the journal article, standard mathematical literature, and internal manuscript records. Internal sheets are cited by the supplied versions or identifiers without invented publication dates or DOIs. The OpenAI catalogue is a separate source collection attributed to OpenAI \cite{OpenAIRecord}; no theorem from it is a premise, and no family is independently verified here. \paragraph{Mathematical scope.} The constructions are local, classical, off shell, and polynomial. The finite operational package acts on supplied encodings and matrices. It is not an instrument on an unknown continuum configuration, a quantum-field representation theorem, or a gravitational evolution law. \paragraph{Verification.} This is one conventional pdfLaTeX source with an internal bibliography and no external figures or bibliography files. No successful pdfLaTeX execution, independently executed symbolic or numerical calculation, hardware compilation, experiment, proof-assistant verification, independent referee certification, or journal acceptance is asserted. The mathematical release consists of the explicit statements and proofs above. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{CheungIXA} N.~Cheung, \emph{Landmark IX-A: An off-shell admission domain and boundary-sensitive topology for sharp Cauchy relational insertions}. \loc{IX-A.tex}, version 1, Enter Calculus Companion, manuscript record. In particular: \loc{prop:adjoint}, \loc{lem:charge}, \loc{thm:closure}, \loc{thm:separate}, \loc{thm:admission}, \loc{lem:witness}, \loc{thm:IX1}, and \loc{prop:knownoutputs}. \url{https://enter.grok.me/browse/IX-A}. \bibitem{CheungIB} N.~Cheung, \emph{I-B: A finite Enter spacetime on a prescribed conical product}. \loc{I-B-v1.tex}, version 1, Enter Calculus Companion, supplied manuscript record. Prescribed background, destructive clock, fixed-event calibration, and finite transport interface. \url{https://enter.grok.me/browse/I-B}. \bibitem{CheungEnter} 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: \doi{10.3389/fphy.2026.1954943}. Publisher record: accepted article; final formatted version identified as forthcoming in the consulted record. \url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}. \bibitem{CheungPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2, Enter Calculus Companion, introductory essay. Documentary and philosophical comparison, not a theorem premise. \url{https://enter.grok.me/enter}. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}. ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007. DOI: \doi{10.4171/037}. \url{https://arxiv.org/abs/0806.1036}. \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: \doi{10.4064/sm8316-3-2016}. \url{https://arxiv.org/abs/1409.7662}. \bibitem{BDHIntro} C.~Brouder, N.~V.~Dang, and F.~H\'elein, \emph{A smooth introduction to the wavefront set}. Author manuscript, 2014. \url{https://arxiv.org/abs/1404.1778}. Wavefront product, tensor, and pullback criteria are used only where their hypotheses are checked. \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 products, free contractions, and the distinct time-ordered and interacting constructions. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's manuscript and errata interface: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. \bibitem{CheungVIIC} N.~Cheung, \emph{Manuscript VII-C}. \loc{VII-C.tex}, version 1, Enter Calculus Companion, supplied manuscript record. The specified normal-topology mixed-Wick obstruction, \loc{thm:main}; documentary comparison only. \url{https://enter.grok.me/browse/VII-C}. \bibitem{CheungVIID} N.~Cheung, \emph{Manuscript VII-D}. \loc{VII-D.tex}, version 1, Enter Calculus Companion, supplied manuscript record. A numerical finite-part prescription, not an algebra construction. \url{https://enter.grok.me/browse/VII-D}. \bibitem{CheungIXB} N.~Cheung, \emph{Manuscript IX-B}. \loc{IX-B-v2.tex}, version 2, Enter Calculus Companion, supplied manuscript record. Two-factor admission and the separate subsequent sharp multiplication boundary; not a premise here. \url{https://enter.grok.me/browse/IX-B}. \bibitem{CheungIXD} N.~Cheung, \emph{Manuscript IX-D}. \loc{IX-D-v3.tex}, version 3, Enter Calculus Companion, supplied manuscript record. Version-qualified distinction between a relative partial free product and missing time-ordered or retarded constructions. Documentary comparison only. \url{https://enter.grok.me/browse/IX-D}. \bibitem{CheungVIB} N.~Cheung, \emph{Manuscript VI-B}. \loc{VI-B-v1.tex}, version 1, Enter Calculus Companion, supplied manuscript record. Finite encoding of already-defined classical evaluations and complete destructive-clock instruments. \url{https://enter.grok.me/browse/VI-B}. \bibitem{CheungEn09} N.~Cheung, \emph{En09-A: Relational Enter}. \loc{En09-A.tex}, version 1, Enter Calculus Companion, supplied manuscript record. Complete conditioning and separately required covariance hypotheses. \url{https://enter.grok.me/browse/En09-A}. \bibitem{CheungObservation} N.~Cheung, \emph{VII-E}, \loc{VII-E.tex}, version 1; \emph{VII-G}, \loc{VII-G.tex}, version 1; and \emph{VII-E-ST}, \loc{VII-E-ST.tex}, version 1. Enter Calculus Companion, supplied manuscript records. Record/residual compatibility, spectator-blind control, and finite hybrid instruments. \url{https://enter.grok.me/browse/VII-E}; \url{https://enter.grok.me/browse/VII-G}; \url{https://enter.grok.me/browse/VII-E-ST}. \bibitem{CheungVIIIA} N.~Cheung, \emph{Manuscript VIII-A}. \loc{VIII-A-v3.tex}, version 3, Enter Calculus Companion, supplied manuscript record. Instrument-level residual separation, including \loc{A:thm:residual}. \url{https://enter.grok.me/browse/VIII-A}. \bibitem{CheungLocality} N.~Cheung, \emph{En03-A}, \loc{En03-A.tex}, version 1; and \emph{En05-A}, \loc{En05-A.tex}, version 1. Enter Calculus Companion, supplied manuscript records. Complete scheduled-support and reference-preserving comparison interfaces. \url{https://enter.grok.me/browse/En03-A}; \url{https://enter.grok.me/browse/En05-A}. \bibitem{CheungVIIJ} N.~Cheung, \emph{VII-J}, \loc{VII-J.tex}, version 1; and \emph{VII-J-ST}, \loc{VII-J-ST.tex}, version 1. Enter Calculus Companion, supplied manuscript records. Timed dependency graphs, separate causal embedding, and regular-part buffered transport. \url{https://enter.grok.me/browse/VII-J}; \url{https://enter.grok.me/browse/VII-J-ST}. \bibitem{CheungTransport} N.~Cheung, \emph{En08-A}, \loc{En08-A.tex}, version 1; and \emph{En08-B}, \loc{En08-B.tex}, version 1. Enter Calculus Companion, supplied manuscript records. Scheduler-versus-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{CheungAnalytic} N.~Cheung, \emph{XIV-H}, version 1; \emph{XIV-I}, version 1; \emph{XIV-J}, version 1; and technical \emph{XV-B}, version 2. Enter Calculus Companion, supplied manuscript records. Historical H2, internal amplitude progress, and the still unconstructed full characteristic map are distinct version-qualified objects. \url{https://enter.grok.me/browse/XIV-H}; \url{https://enter.grok.me/browse/XIV-I}; \url{https://enter.grok.me/browse/XIV-J}; \url{https://enter.grok.me/browse/XV-B}. \bibitem{CheungStops} N.~Cheung, \emph{XVI-N}, revision r12; \emph{XVI-O}; and \emph{XVI-Q}. Enter Calculus Companion, supplied manuscript records. Conditional insertion obstruction and prescription-qualified status comparisons only. \url{https://enter.grok.me/browse/XVI-N}; \url{https://enter.grok.me/browse/XVI-O}; \url{https://enter.grok.me/browse/XVI-Q}. \bibitem{OpenAIRecord} OpenAI, \emph{OpenAI mathematics catalogue}. Separate documentary source collection supplied with the request and recorded by the Enter Calculus Companion. No theorem, formalization scope, or verification claim is imported. \url{https://enter.grok.me}. \end{thebibliography} \end{document}