% !TEX program = pdflatex %===================================================================== % X-A-conv-ST.tex % Single-file manuscript for pdfLaTeX. % No external figures, source files, or bibliography database. % Run pdfLaTeX twice to resolve references. % % Source status: proposed. % No compilation, repository installation, numerical execution, % machine verification, or independent verification is asserted. %===================================================================== \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={X-A-conv-ST: A fixed-width convolution obstruction on a prescribed conical Enter spacetime}, pdfsubject={Supported analytical transfer, positive covariance defect, complete destructive instruments, and prescribed causal transport} } \setlength{\emergencystretch}{3em} \setlist{itemsep=3pt,topsep=5pt} \setcounter{tocdepth}{1} \numberwithin{equation}{section} \allowdisplaybreaks \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\HH}{\mathcal H} \newcommand{\Alg}{\mathcal A} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Ezero}{\mathsf E_0} \newcommand{\Hzero}{\mathsf H_0} \newcommand{\gC}{\mathfrak g_C} \newcommand{\gR}{\mathfrak g_R} \newcommand{\Tge}{T_{g,\varepsilon}} \newcommand{\Se}{S_\varepsilon} \newcommand{\Co}{\mathsf{Co}} \newcommand{\Enc}{\mathsf{Enc}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Law}{\mathsf{Law}} \newcommand{\Lie}{\mathcal L} \newcommand{\X}[1]{X_{#1}} \newcommand{\Jop}{\mathscr J} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\proj}[1]{\kb{#1}{#1}} \newcommand{\loc}[1]{\texttt{\detokenize{#1}}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\dTV}{d_{\mathrm{TV}}} \newcommand{\ddia}{d_\diamond} \newcommand{\ord}{\mathcal O} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\artanh}{artanh} \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{X-A-conv-ST: A fixed-width convolution obstruction\\ on a prescribed conical Enter spacetime}\\[5pt] \large Supported analytical transfer, complete destructive reports,\\ and the separation from the ramp prescription} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We transfer the analytical obstruction of Cheung's X-A--Convolution to the supported I-B/X-A-ST setting without changing its regulator, gauge algebra, Cauchy data, or observation test. All wave operators remain on the declared Minkowski completion; the regular conical host is prescribed rather than reconstructed. The complete convolution data are stated explicitly, so the obstruction is proved rather than hidden in a positivity assumption. The decisive identity is \(\Ezero\mathsf C T_{g,\varepsilon}(K\xi)=S_\varepsilon\xi\), not the identity required for coordinate covariance. For the parent compact witness we obtain a strictly positive translation-energy formula for \(c_\varepsilon\) at every allowed fixed width. The same coefficient survives evaluation on the scalar solution of the switched action, despite that action's plateau Noether identity. We retain the second-order defects and the genuine pure-gauge two-jet without promoting either calculation to an all-orders statement. The supported transfer identifies precisely which compact sets must lie in the inherited developing chart. A change of variables, including its density Jacobian and transported two-point kernel, preserves the positive defect; it does not make convolution diffeomorphism natural. Laboratory field interpretation is conditional on this explicit support certificate, which is not implied by the ramp's causal reserve. We then connect the analytical result to the proposed X-Enter-ST operational package: scalar-only invariant readout is necessarily uninformative, while a complete destructive-clock instrument can report supplied finite evaluations and preserve every unmeasured finite residual. An independently charged second coefficient would permit a conditional relational circuit, but is not constructed. Scheduled locality and reference-preserving SWAP transport concern these finite reports, not acquisition of the continuum coefficient. The contribution is a controlled analytical--operational non-repair theorem on one retained host. The parent obstruction is credited to X-A--Convolution; the new supported audit makes its domain and interpretation explicit. No convolution repair, quantum characteristic coefficient, interacting differential, ultraviolet completion, or spacetime emergence is established. \end{abstract} \statementbox{Release in one sentence}{% The frozen convolution prescription is proved obstructed at first classical order. Its positive gap survives supported transport and cannot be made into an informative invariant scalar-only output by changing readout. The finite reporting interface is constructed under declared permissions. A laboratory interpretation of the continuum calculation additionally requires the source-specific chart certificate stated below.} \tableofcontents %===================================================================== \section{Question, sources, and the Enter interface discipline} \label{sec:introduction} %===================================================================== \subsection{The analytical and operational halves} The principal analytical source is N.~Cheung's \emph{X-A-conv}, version~1, here called X-A--Convolution \cite{XConv}. It fixes a spacetime convolution of switched metric arguments and proves that its coordinate prescription fails covariance on the gauge algebra declared before the calculation. The principal operational comparison is Cheung's \emph{X-Enter}, version~1 \cite{XEnter}. Its proposed supported companion, \emph{X-Enter-ST}, is the source supplied in the preceding development \cite{XEnterST}. That source already separates the coefficient orbit, the complete readout, and the supplied-report instrument. The present manuscript supplies the analytical half of this supported comparison. In particular, it states the actual mollifier and witness, proves the Cauchy and preflight identities, derives strict positivity, and retains the second-order defects. It does not infer these facts from an unspecified smoothing operator. The ramp construction of X-A and X-A-ST is a separate prescription \cite{XA,XAST}. No ramp identity is an input to the convolution obstruction. \subsection{Background and philosophy} Enter is a destructive readout interface. Its measured quantum names are consumed from the continuation's quantum context; its declared output includes the classical outcome and any unmeasured surviving quantum system. Consumption is an access statement, not annihilation of every physical degree of freedom used by an apparatus. Cheung's operational characterization concerns a specified strategy class. The journal article and Manuscript~I distinguish that characterization from a field instrument or a derivation of qubit kinematics \cite{EnterArticle,EnterI}. The philosophical account emphasizes that readout, quantum recovery, thermodynamic implementation, and gravitational identification are different constructions \cite{EnterPhil}. We use explicit finite Born instruments and prove their channel identities directly. No abbreviated reconstruction of the Enter characterization is a premise of the classical argument. The spacetime interface follows I-B: a classical conical background, finite quantum names, accessible algebras, locations, schedules, controls, and resources are separately declared \cite{IB}. The readout does not select or evolve that metric. \subsection{Four distinctions governing the proof} The following distinctions remain binding throughout: \begin{enumerate} \item Existence and smoothness of a classical coefficient do not prove its gauge invariance. \item A change of coordinates transporting a frozen regulator is not active covariance of that regulator. \item A supplied finite report is not acquisition of an unknown continuum functional. \item Equality of classical outcome laws is weaker than equality of complete instruments with residual systems and references. \end{enumerate} Relational-observable and perturbative-gravity frameworks provide context, but do not prove covariance of this particular switched convolution prescription \cite{BFR}. %===================================================================== \section{Stage A: the unchanged host and the analytical completion} \label{sec:host} %===================================================================== \begin{assumption}[Retained host] \label{ass:host} Retain the I-B/X-A-ST instance, including its deficit, time orientation, regular laboratory, developing chart, time calibration, operational permissions, and reserved finite transport tube. Write its regular conical product as \[ \mathcal M^\circ =\RR_T\times(C_\alpha\setminus\{o\})\times\RR_z, \qquad G=-c_{\mathrm h}^2\dd T^2+\dd r^2 +\alpha^2r^2\dd\varphi^2+\dd z^2, \] where \(c_{\mathrm h}>0\) and \(0<\alpha<1\) retain their inherited values. The completion defect is \[ \mathcal W=\RR_T\times\{o\}\times\RR_z. \] It is a worldsheet. The already declared analytical completion and chart are \[ \widehat M=\RR^{1+3},\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \iota:\mathcal L\longrightarrow\mathcal M^\circ, \] with \begin{equation} \iota^*G=-\eta,\qquad t=c_{\mathrm h}(T-T_\Sigma). \label{eq:signature} \end{equation} The chart is injective on its declared domain and the laboratory does not meet \(\mathcal W\). No host-update map is added. \end{assumption} The constant \(c_{\mathrm h}\) is a host calibration speed. It is not the coefficient \(c_\varepsilon\). The parent calculation below uses its original completion slab \[ M_8=[0,8]\times\RR^3,\qquad \Sigma=\{t=0\}. \] This does not reset the inherited host time reserve to a new value. If the fixed parent data do not fit the retained chart, the corresponding laboratory interpretation requires the additional certificate in Section~\ref{sec:support}. No translation, rescaling, chart enlargement, or change of deficit is silently performed to force a fit. \begin{proposition}[Why no global cone inverse is used] \label{prop:notGH} The regular product \((\mathcal M^\circ,G)\) is not globally hyperbolic. Consequently, the global Cauchy and Green operators used below are operators on \(\widehat M\), not global inverses on the regular cone. \end{proposition} \begin{proof} Fix a radial ray, \(r_0>0\), and \(z=z_0\). Choose \(a>r_0/c_{\mathrm h}\) and set \[ p=(-a,r_0,\varphi_0,z_0),\qquad q=(a,r_0,\varphi_0,z_0). \] The points \[ x_n=(0,r_0/n,\varphi_0,z_0) \] lie in \(J^+(p)\cap J^-(q)\): the radial trip can be made within the available time. They have no convergent subsequence in the regular product, because their only completion limit has \(r=0\). Thus this causal diamond is not compact. Global hyperbolicity would require compact causal diamonds. \end{proof} The formal metric tensors used in the completion calculation are analytical inputs. Transporting them does not replace the fixed host metric \(G\). In particular, the transported analytical background is \(-G\), in agreement with \eqref{eq:signature}, not an unnoticed substitution of the opposite signature. %===================================================================== \section{The fully specified convolution contract} \label{sec:freeze} %===================================================================== This section retains the definitions of X-A--Convolution \cite[convention sheet]{XConv}. All subsequent conclusions refer to this frozen choice. \subsection{Fields, formal order, and gauge algebra} The smooth off-shell inputs are \[ h,k\in C^\infty(\widehat M;S^2T^*\widehat M), \qquad \phi\in C^\infty(\widehat M;\RR). \] No decay, equation-of-motion quotient, gravitational constraint, or gauge-fixed quotient is imposed. The scalar is massless. The formal metric jet is \begin{equation} q=\eta+\kappa h+\kappa^2k\pmod{\kappa^3}. \label{eq:metricjet} \end{equation} There is no factor \(1/2\) in the definition of \(k\). Statements modulo \(\kappa^3\) mean coefficient identities, not convergence of a numerical perturbation series. Set \(x_*=(4,0,0,0)\) and \(r(x)=|x-x_*|_E\), where the Euclidean norm is used only for supports and convolution. Define \[ u(s)= \begin{cases} e^{-1/s},&s>0,\\ 0,&s\le0, \end{cases} \qquad \chi(s)=\frac{u(s)}{u(s)+u(1-s)}. \] Freeze \begin{equation} g(x)=1-\chi\!\left(\frac{r(x)^2-4}{5}\right),\qquad U_C=\{r<1\},\qquad O_C=\{r<1/4\}. \label{eq:switch} \end{equation} Then \(g=1\) on \(r\le2\), and \(\supp g\subset\{r\le3\}\Subset(0,8)\times\RR^3\). For a vector field \(\xi\), write \[ (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \qquad X_\xi=\xi^\mu\partial_\mu. \] The algebra fixed before the calculation is \begin{equation} \gC=C_c^\infty(U_C;T\widehat M). \label{eq:gaugealgebra} \end{equation} Indices are raised and lowered with \(\eta\). The switch is fixed background data. For \(\xi,\zeta\in\gC\), \[ F_\kappa^*=\exp(\kappa\Lie_\xi+\kappa^2\Lie_\zeta) \pmod{\kappa^3} \] gives \begin{equation} h'=h+K\xi,\qquad k'=k+\Lie_\xi h+K\zeta+\frac12\Lie_\xi^2\eta. \label{eq:gaugetransformation} \end{equation} The scalar transforms by the same formal pullback. \subsection{The fixed positive-width regulator} Let \[ \varrho(v)= \begin{cases} Z^{-1}\exp\!\left(-\dfrac1{1-|v|_E^2}\right),&|v|_E<1,\\ 0,&|v|_E\ge1, \end{cases} \qquad Z=\int_{|v|_E<1}\exp\!\left(-\frac1{1-|v|_E^2}\right)\dd^4v. \] Freeze one width \begin{equation} 0<\varepsilon\le\frac18,\qquad \varrho_\varepsilon(v)=\varepsilon^{-4}\varrho(v/\varepsilon), \qquad \Se a=\varrho_\varepsilon*a,\qquad \Tge a=\Se(ga). \label{eq:regulator} \end{equation} Convolution is componentwise in these completion coordinates. The auxiliary metric is \begin{equation} \widetilde q=\eta+\kappa H+\kappa^2J,\qquad H=\Tge h,\quad J=\Tge k. \label{eq:auxmetric} \end{equation} The regulator acts on the metric arguments of the coordinate functional. It is not inserted into the action, the scalar argument, or the Cauchy inverse. It is neither Lorentz invariant nor diffeomorphism natural. \subsection{The fixed witness} Define \[ \psi(x)= \begin{cases} \exp\!\left(1-\dfrac1{1-64r(x)^2}\right),&r(x)<1/8,\\ 0,&r(x)\ge1/8, \end{cases} \qquad N=\int_{\widehat M}\psi(x)\dd^4x>0. \] Freeze \begin{equation} f=\frac{\psi}{N},\qquad \xi_*=\psi\,\partial_1\in\gC,\qquad \rho_0=x^1. \label{eq:witness} \end{equation} The free equation \(\Box_\eta\rho_0=0\) holds. Its unboundedness at spatial infinity is allowed by the declared smooth domain; no Hilbert-space normalizability is being asserted. The same \(g,f,\xi_*,\varrho\) are used for every allowed fixed width. No witness is removed after its defect is found. %===================================================================== \section{Stage B: Cauchy solution and regulator preflight} \label{sec:cauchy} %===================================================================== \subsection{The completion Cauchy operator} For a smooth source \(s\), define the zero-data inverse on \(M_8\): \begin{equation} (\Ezero s)(t) =\int_0^t \frac{\sin((t-u)\sqrt{-\Delta})}{\sqrt{-\Delta}} s(u)\,\dd u. \label{eq:Ezero} \end{equation} The multiplier has its continuous value \(t-u\) at zero spatial frequency. \begin{lemma}[Cauchy identities] \label{lem:Cauchy} The zero-data wave problem has a unique smooth solution for every smooth source on a neighborhood of \(M_8\), locally interpreted by finite propagation. It satisfies \[ \Box_\eta\Ezero s=s,\qquad \Ezero s|_\Sigma=\partial_t\Ezero s|_\Sigma=0. \] For smooth \(a\), \begin{equation} \Ezero\Box_\eta a=a-\Hzero[a], \label{eq:leftinverse} \end{equation} where \(\Hzero[a]\) is the homogeneous solution with the Cauchy data of \(a\). \end{lemma} \begin{proof} For spatially compact smooth sources, differentiating \eqref{eq:Ezero} proves the equation and initial data. The wave energy estimate, after spatial differentiation, bounds the solution on finite slabs by the source and initial-data Sobolev norms. Applied to a difference, it gives uniqueness. Sobolev embedding gives smoothness and continuous dependence. On a compact observation region, choose compact spatial cutoffs equal to one on its truncated causal past. Finite propagation makes the resulting solutions agree on overlaps. This defines the solution for general smooth sources and preserves uniqueness. These are the standard completion Cauchy arguments \cite{BGP}. Finally, \(a-\Ezero\Box_\eta a\) is homogeneous with the Cauchy data of \(a\), proving \eqref{eq:leftinverse}. \end{proof} \subsection{Auxiliary coordinates through second order} For a symmetric tensor \(A\), set \begin{align} a_1(A)^{\mu\nu} &=\frac12(\tr A)\eta^{\mu\nu}-A^{\mu\nu}, \label{eq:a1}\\ a_2(A)^{\mu\nu} &=A^\mu{}_\rho A^{\rho\nu} -\frac12(\tr A)A^{\mu\nu} +\left(\frac18(\tr A)^2 -\frac14A_{\rho\sigma}A^{\rho\sigma}\right) \eta^{\mu\nu}, \label{eq:a2}\\ \mathsf C^\mu A &=\partial_\nu A^{\nu\mu} -\frac12\partial^\mu\tr A =-\partial_\nu a_1(A)^{\nu\mu}. \label{eq:C} \end{align} Thus \[ \sqrt{-\det\widetilde q}\,\widetilde q^{\alpha\beta} =\eta^{\alpha\beta} +\kappa a_1(H)^{\alpha\beta} +\kappa^2\bigl(a_1(J)^{\alpha\beta} +a_2(H)^{\alpha\beta}\bigr) \pmod{\kappa^3}. \] The scalar coordinate functions, with external label \(\mu\), are defined by \begin{equation} \partial_\alpha\!\left( \sqrt{-\det\widetilde q}\,\widetilde q^{\alpha\beta} \partial_\beta\mathsf X^\mu\right)=0, \qquad \mathsf X^\mu=x^\mu+\kappa w^\mu+\kappa^2v^\mu, \label{eq:harmonic} \end{equation} with the coordinate Cauchy data of \(x^\mu\). \begin{proposition}[Actual coordinate coefficients] \label{prop:coordinates} The unique coefficients of \eqref{eq:harmonic} are \begin{align} w^\mu[h]&=\Ezero\mathsf C^\mu H, \label{eq:w}\\ v^\mu[h,k] &=\Ezero\mathsf C^\mu J -\Ezero\!\left[ \partial_\alpha a_2(H)^{\alpha\mu} +\partial_\alpha\!\left( a_1(H)^{\alpha\beta}\partial_\beta w^\mu\right) \right]. \label{eq:v} \end{align} Their values and first time derivatives vanish at \(\Sigma\). \end{proposition} \begin{proof} Insert the displayed density expansion into \eqref{eq:harmonic}. The coefficients of \(\kappa\) and \(\kappa^2\) give respectively \[ \Box_\eta w^\mu=\mathsf C^\mu H \] and the source displayed in \eqref{eq:v}. Lemma~\ref{lem:Cauchy} gives the formulas and successive uniqueness. \end{proof} These are the parent Cauchy formulas \cite[\loc{prop:coordcoeff}]{XConv}. In particular, \[ \mathsf C^\mu H =\Se\!\left( g\,\mathsf C^\mu h +(\partial_\nu g) \left[h^{\nu\mu}-\frac12\eta^{\nu\mu}\tr h\right]\right). \] Switch derivatives are retained. The nonlinear expression is \(a_2(\Se(gh))\), not \(g\,a_2(h)\) or \(\Se(g\,a_2(h))\). \subsection{The decisive preflight stop} \begin{lemma}[Pure-gauge response] \label{lem:preflight} For every \(\xi\in\gC\), \begin{equation} w[K\xi]=\Ezero\mathsf C\Tge(K\xi)=\Se\xi. \label{eq:preflight} \end{equation} \end{lemma} \begin{proof} Because \(g=1\) on a neighborhood of \(\supp\xi\), \(gK\xi=K\xi\). Constant-coefficient derivatives commute with convolution, and direct contraction gives \[ \mathsf C Kb=\Box_\eta b. \] Consequently, \[ \mathsf C\Tge(K\xi) =\mathsf C K(\Se\xi)=\Box_\eta(\Se\xi). \] The support of \(\Se\xi\) lies in the Euclidean \(\varepsilon\)-neighborhood of \(\supp\xi\), separated from \(t=0\). Both Cauchy data therefore vanish. Equation~\eqref{eq:leftinverse} proves the result. \end{proof} Coordinate covariance would require \(w[K\xi]=\xi\). The actual value is \(\Se\xi\) \cite[\loc{lem:preflight}, \loc{thm:preflightstop}]{XConv}. This calculation makes no global factorization of \(w[h]\) through \(\mathsf C h\); the switch terms prevent that simplification. %===================================================================== \section{Strict positivity and the evaluated classical obstruction} \label{sec:positive} %===================================================================== \subsection{Relational expansion and insertion sign} The formal inverse of the coordinate map is \[ Y^\mu(z)=z^\mu-\kappa w^\mu(z) +\kappa^2\bigl(w^\nu\partial_\nu w^\mu-v^\mu\bigr)(z) \pmod{\kappa^3}. \] Define \[ \mathcal O_{f,g,\varepsilon}[h,k,\phi] =\int f(z)\phi(Y(z))\,\dd^4z. \] Here \(f\) is fixed in relational-label space. No numerical global inverse is assumed. Set \[ \Jop(w) =(w^\nu\partial_\nu w^\mu)\partial_\mu +\frac12w^\mu w^\nu\partial_\mu\partial_\nu. \] Taylor expansion gives \begin{equation} \mathcal O^{(0)}=\int f\phi,\qquad \mathcal O^{(1)}=-\int fX_w\phi,\qquad \mathcal O^{(2)}=\int f\bigl(\Jop(w)\phi-X_v\phi\bigr). \label{eq:Ocoefficients} \end{equation} The negative first insertion sign is essential \cite[\loc{prop:Ocoeff}]{XConv}. \begin{proposition}[First-order defect] \label{prop:firstdefect} With \[ \delta_\xi h=K\xi+\kappa\Lie_\xi h,\qquad \delta_\xi\phi=\kappa X_\xi\phi, \] the coefficient of \(\kappa\) in the variation is \begin{equation} [\kappa]\,\delta_\xi\mathcal O =\int fX_{\xi-\Se\xi}\phi. \label{eq:firstdefect} \end{equation} \end{proposition} \begin{proof} The zeroth coefficient varies by \(\kappa\int fX_\xi\phi\). The leading variation of \(w\) is \(w[K\xi]=\Se\xi\), so the first insertion contributes \(-\kappa\int fX_{\Se\xi}\phi\). Their sum is \eqref{eq:firstdefect}. \end{proof} \subsection{The exact positive gap} \begin{theorem}[Strict convolution gap] \label{thm:positive} For the frozen data \eqref{eq:witness}, \begin{align} c_\varepsilon &:=\int f(x)(\psi(x)-\Se\psi(x))\,\dd^4x \label{eq:ceps}\\ &=\frac1{2N}\int_{\RR^4}\varrho_\varepsilon(a) \int_{\RR^4} \bigl(\psi(x)-\psi(x-a)\bigr)^2\,\dd^4x\,\dd^4a >0. \label{eq:energy} \end{align} Therefore the frozen coordinate and relational prescriptions fail first-order covariance on \(\gC\). \end{theorem} \begin{proof} Expanding the squared difference and using translation invariance of Lebesgue measure gives \[ \int(\psi(x)-\psi(x-a))^2\dd^4x =2\int\psi^2-2\int\psi(x)\psi(x-a)\dd^4x. \] Integration against the normalized \(\varrho_\varepsilon\), followed by division by \(2N\), gives \eqref{eq:ceps}. For every \(a\ne0\), the inner integral is positive. If it vanished, continuity would imply \(\psi(x)=\psi(x-a)\) everywhere. Iteration would make a nonzero compactly supported function invariant under arbitrarily large translations, which is impossible. The mollifier is strictly positive on \(0<|a|_E<\varepsilon\), a set of positive measure. This proves strict positivity. For \(\xi=\xi_*\) and \(\phi=x^1\), \eqref{eq:firstdefect} is exactly \(c_\varepsilon\). \end{proof} This is the positive-energy mechanism of the parent \cite[\loc{lem:positivegap}]{XConv}. It concerns every allowed fixed positive width. No width-removal argument is used. \begin{corollary}[Order-preserving non-repair] \label{cor:nohigher} A modification starting at order \(\kappa^2\), or at a positive quantum order while preserving the classical first coefficient, cannot restore the failed formal identity. \end{corollary} \begin{proof} Formal equality requires equality of each coefficient. Such a modification leaves the nonzero classical coefficient \(c_\varepsilon\) unchanged. \end{proof} This excludes only the stated order-preserving repair. It does not exclude a changed classical prescription. \subsection{The action and its scalar solution} For completeness, retain the parent's first-derivative gravitational density representative \[ \ell_{\mathrm{gr}}(\kappa,h) =-\frac2{\kappa^2}\mathsf A(q)^{\mu\nu} \left( \Gamma^\rho_{\mu\sigma}\Gamma^\sigma_{\nu\rho} -\Gamma^\rho_{\mu\nu}\Gamma^\sigma_{\rho\sigma} \right), \qquad \mathsf A(q)^{\mu\nu}=\sqrt{-\det q}\,q^{\mu\nu}, \] and \[ \ell_{\mathrm{sc}}(\kappa,h,\phi) =\frac12\mathsf A(q)^{\mu\nu} \partial_\mu\phi\,\partial_\nu\phi. \] The Christoffel symbols and curvature use the convention \[ R_{\mu\nu} =\partial_\rho\Gamma^\rho_{\mu\nu} -\partial_\nu\Gamma^\rho_{\mu\rho} +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu} -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}. \] Write \[ \ell_{\mathrm{gr}}+\ell_{\mathrm{sc}} =\ell_0+\kappa\ell_1+\kappa^2\ell_2+\cdots \] using \(q=\eta+\kappa h\), and freeze \begin{equation} S_g^{[2]}=\int \bigl(\ell_0+\kappa g\ell_1+\kappa^2g\ell_2\bigr) \dd^4x. \label{eq:action} \end{equation} The free integral is interpreted variationally with compact variations. The chosen density is not replaced by another representative after switching \cite{XConv}. For a metric jet, its scalar equation through the retained order is \begin{equation} D_g[q]\rho =\partial_\mu\!\left[ \left(\eta^{\mu\nu} +\kappa g a_1(h)^{\mu\nu} +\kappa^2g(a_1(k)^{\mu\nu}+a_2(h)^{\mu\nu})\right) \partial_\nu\rho\right]=0. \label{eq:scalarequation} \end{equation} Thus the second vertex has one factor \(g\), not \(g^2\). Define \[ V_{j,g}[h]u =\partial_\mu(g\,a_j(h)^{\mu\nu}\partial_\nu u), \qquad W_g[h,k]=V_{1,g}[k]+V_{2,g}[h]. \] With free input \(\rho_0\) and zero positive-order Cauchy data, the selected scalar is \[ \rho_g=\rho_0+\kappa\rho_{1,g}+\kappa^2\rho_{2,g}, \] where \begin{align} \rho_{1,g} &=-\Ezero V_{1,g}[h]\rho_0, \label{eq:rho1}\\ \rho_{2,g} &=\Ezero V_{1,g}[h]\Ezero V_{1,g}[h]\rho_0 -\Ezero W_g[h,k]\rho_0. \label{eq:rho2} \end{align} These formulas follow by expanding \eqref{eq:scalarequation} and applying Lemma~\ref{lem:Cauchy}. In particular, \[ V_{j,g}[h]u =g\,\partial_\mu(a_j(h)^{\mu\nu}\partial_\nu u) +(\partial_\mu g)a_j(h)^{\mu\nu}\partial_\nu u. \] No switch derivative has been discarded. \begin{lemma}[Scalar covariance on the frozen plateau] \label{lem:scalarcov} For the formal transformations generated by \(\gC\), \[ \rho_g[F_\kappa^*q;\rho_0] =F_\kappa^*\rho_g[q;\rho_0]\pmod{\kappa^3}. \] \end{lemma} \begin{proof} Before truncation the scalar density coefficient can be written \[ \mathsf A_g[q]=(1-g)\eta+g\,\mathsf A(q), \] with \(\eta\) interpreted here as a contravariant density. The curved density is natural under pullback. The only possible mismatch is \((1-g)(F_\kappa^*\eta-\eta)\). It vanishes because \(g=1\) wherever the transformation acts, and the transformation is the identity elsewhere. Also \(F_\kappa^*g=g\). Divergence of a vector density is natural. The pulled-back solution has the same Cauchy data because the transformation is the identity near \(\Sigma\). Successive Cauchy uniqueness proves the assertion. \end{proof} \begin{theorem}[The gap survives evaluation] \label{thm:evaluated} Let \[ q_\lambda=\exp(\lambda\kappa\Lie_{\xi_*})\eta \pmod{\kappa^3} \] and use the fixed free scalar \(\rho_0=x^1\). For \[ \mathcal A_\lambda =\int f(z)\rho_g[q_\lambda;\rho_0](Y_{q_\lambda}(z)) \dd^4z, \] the first coefficient difference is \begin{equation} a(\lambda):=[\kappa](\mathcal A_\lambda-\mathcal A_0) =\lambda c_\varepsilon, \qquad \lambda\in\RR. \label{eq:orbit} \end{equation} \end{theorem} \begin{proof} Lemma~\ref{lem:scalarcov} gives \(\rho_{1,g}[q_\lambda]=\lambda X_{\xi_*}\rho_0 =\lambda\psi\). Lemma~\ref{lem:preflight} gives \(w[q_\lambda]=\lambda\Se\xi_*\). The evaluated first relational coefficient is therefore \[ \rho_{1,g}-X_w\rho_0 =\lambda(\psi-\Se\psi). \] Smearing gives \eqref{eq:orbit}. \end{proof} This is the parent's evaluated obstruction \cite[\loc{thm:evaluatedgap}]{XConv}. The scalar correction has been derived from the switched action, not appended as an external force or assumed to be a symmetry of an unchanged free scalar shell. \subsection{The independent action identity} \begin{proposition}[Action compatibility does not repair the coordinates] \label{prop:action} For \(\xi\in\gC\), \[ \delta_\xi S_g^{[2]}=\ord(\kappa^3) \] as a formal variational identity. This is compatible with \(c_\varepsilon>0\). \end{proposition} \begin{proof} The unswitched first-derivative gravitational representative differs from the Hilbert density by a divergence. Together with the natural scalar density, this gives local current coefficients \(n_j^\mu\) satisfying \[ \delta^{(0)}_\xi\ell_0=\partial_\mu n_0^\mu,\qquad \delta^{(0)}_\xi\ell_1+\delta^{(1)}_\xi\ell_0 =\partial_\mu n_1^\mu, \] \[ \delta^{(0)}_\xi\ell_2+\delta^{(1)}_\xi\ell_1 =\partial_\mu n_2^\mu. \] The currents are local in a finite jet of \(\xi\). Their support is compact in the gauge-support region. These are the coefficient Noether identities for the fixed representative \cite[\loc{prop:actionidentity}]{XConv}. Varying \eqref{eq:action} and integrating by parts gives \[ \begin{split} \delta_\xi S_g^{[2]} ={}&\int\partial_\mu n_0^\mu\\ &+\kappa\left[ \int\partial_\mu(gn_1^\mu) -\int(\partial_\mu g)n_1^\mu +\int(1-g)\delta^{(1)}_\xi\ell_0\right]\\ &+\kappa^2\left[ \int\partial_\mu(gn_2^\mu) -\int(\partial_\mu g)n_2^\mu\right] +\ord(\kappa^3). \end{split} \] The boundary terms vanish by compact support. Every remaining factor involving \(1-g\) or \(\dd g\) vanishes on the gauge support. The convolution regulator appears nowhere in the action. It appears in \eqref{eq:harmonic}, so the independent action identity cannot change \eqref{eq:preflight}. \end{proof} For the scalar sign check alone, \[ \delta^{(0)}_\xi\!\int g\ell_{\mathrm{sc},1} +\delta^{(1)}_\xi S_{\mathrm{sc},0} = \int(g-1)(\Box_\eta\phi)X_\xi\phi +\int(\partial_\mu g)T^{\mu\nu}\xi_\nu=0. \] The calculation is off shell and retains its switch terms. %===================================================================== \section{The retained second-order defects} \label{sec:second} %===================================================================== \subsection{Infinitesimal ledger} Let \[ \mathsf W(A)=\Ezero\mathsf C A, \] and let \(a_2(A,B)\) denote the symmetric polarization \[ a_2(A,B)=\frac12\bigl(a_2(A+B)-a_2(A)-a_2(B)\bigr). \] Define \[ \begin{split} \mathsf V^\mu(A,B) =-\Ezero\Bigl[ &\partial_\alpha a_2(A,B)^{\alpha\mu}\\ &+\frac12\partial_\alpha\bigl( a_1(A)^{\alpha\beta}\partial_\beta\mathsf W(B)^\mu +a_1(B)^{\alpha\beta}\partial_\beta\mathsf W(A)^\mu \bigr)\Bigr]. \end{split} \] Then \[ w=\mathsf W(H),\qquad v=\mathsf W(J)+\mathsf V(H,H). \] For the bare functional \(k=0\), write \(\delta_\xi=\delta_\xi^{(0)}+\kappa\delta_\xi^{(1)}\), where \[ \delta_\xi^{(0)}h=K\xi,\quad \delta_\xi^{(0)}\phi=0,\quad \delta_\xi^{(1)}h=\Lie_\xi h,\quad \delta_\xi^{(1)}\phi=X_\xi\phi. \] Set \[ b=\Se\xi,\qquad d_2=\mathsf W\Tge(\Lie_\xi h)+2\mathsf V(H,Kb),\qquad R_{2,\xi}=d_2-X_\xi w. \] \begin{proposition}[Exact retained infinitesimal variation] \label{prop:seconddefect} The variation is \begin{equation} \begin{split} \delta_\xi\mathcal O ={}&\kappa\int fX_{\xi-b}\phi\\ &+\kappa^2\int f\left( X_{X_{b-\xi}w}\phi -X_{R_{2,\xi}}\phi +X_wX_{b-\xi}\phi\right) +\ord(\kappa^3). \end{split} \label{eq:seconddefect} \end{equation} \end{proposition} \begin{proof} The leading variation of \(w\) is \(b\), while \[ \delta_\xi^{(1)}w=\mathsf W\Tge(\Lie_\xi h), \qquad \delta_\xi^{(0)}v=2\mathsf V(H,Kb). \] Thus their sum is \(d_2\). Differentiating \(\Jop\) gives \[ D\Jop(w)[b]=X_wX_b+X_{X_bw}. \] The second-order integrand from \eqref{eq:Ocoefficients} is consequently \[ -X_{d_2}\phi+X_wX_b\phi+X_{X_bw}\phi-X_wX_\xi\phi. \] Using \(d_2=R_{2,\xi}+X_\xi w\) yields \eqref{eq:seconddefect}. \end{proof} Equation~\eqref{eq:seconddefect} retains the parent's defects \cite[\loc{prop:infdefects}]{XConv}. It is not replaced by zero and is not asserted beyond the displayed order. \subsection{A genuine pure-gauge two-jet} For \(\xi\in\gC\), define \[ q_\lambda=e^{\lambda\kappa\Lie_\xi}\eta,\qquad \phi_\lambda=e^{\lambda\kappa X_\xi}\phi_0 \pmod{\kappa^3}. \] Thus \[ h_\lambda=\lambda K\xi,\qquad k_\lambda=\frac{\lambda^2}{2}\Lie_\xi^2\eta. \] The straight line \(\eta+\lambda\kappa K\xi\) is not substituted for this jet. Set \[ b=\Se\xi,\qquad d_\xi=\frac12\Tge(\Lie_\xi^2\eta) -\frac12\Lie_b^2\eta. \] \begin{proposition}[Complete pure-gauge diagnostic] \label{prop:pure} The coordinate coefficients are \[ w_\lambda=\lambda b,\qquad v_\lambda^\mu =\lambda^2\left( \frac12b^\nu\partial_\nu b^\mu +\Ezero\mathsf C^\mu d_\xi\right). \] The relational scalar coefficients are \begin{align} \Phi_\lambda^{[1]} &=\lambda X_{\xi-b}\phi_0, \label{eq:pureone}\\ \Phi_\lambda^{[2]} &=\lambda^2\left( \frac12X_{\xi-b}^2 +\frac12X_{[\xi,b]} -X_{\Ezero\mathsf C d_\xi}\right)\phi_0. \label{eq:puretwo} \end{align} \end{proposition} \begin{proof} For the auxiliary comparison metric \(e^{\lambda\kappa\Lie_b}\eta\), the scalar functions \(e^{\lambda\kappa X_b}x^\mu\) solve the harmonic equation. They have the frozen data because \(b\) vanishes near \(\Sigma\). Their coefficients are \(\lambda b^\mu\) and \(\lambda^2b^\nu\partial_\nu b^\mu/2\). The actual auxiliary metric has the same first coefficient and an additional second coefficient \(\lambda^2d_\xi\). Linearity of \eqref{eq:v} in \(J\) adds \(\lambda^2\Ezero\mathsf C d_\xi\). This proves the coordinate formulas. Substitute the scalar pullback expansion and these coordinates into the formal composition. The quadratic operator is \[ \frac12X_\xi^2-X_bX_\xi+\frac12X_b^2 -X_{\Ezero\mathsf C d_\xi}. \] Since \([X_\xi,X_b]=X_{[\xi,b]}\), this is \eqref{eq:puretwo}. \end{proof} This is the retained parent pure-gauge test \cite[\loc{lem:purecoordinates}, \loc{prop:pureobservable}]{XConv}. It records the defect, not its cancellation. For the witness \eqref{eq:witness}, the first smearing is already \(\lambda c_\varepsilon\). %===================================================================== \section{Support, kernels, and the source-specific chart certificate} \label{sec:support} %===================================================================== \subsection{Classical smoothing and the actual dependence region} Let \[ K_f=\supp f,\qquad D_f=J_\eta^-(K_f)\cap M_8,\qquad B_\varepsilon=\{v:|v|_E\le\varepsilon\}, \] and \begin{equation} Q_\varepsilon =\supp g\cap(D_f+B_\varepsilon). \label{eq:Qepsilon} \end{equation} These sets are compact. \begin{proposition}[Fixed-width regularity and dependence] \label{prop:kernels} At fixed \(\varepsilon>0\), the coefficients \eqref{eq:Ocoefficients} are continuous polynomial functionals on the stated smooth spaces. Their first- and second-order joint kernels are smooth and compactly supported. Through order \(\kappa^2\), dependence on \(h,k\) is contained in \(Q_\varepsilon\), and dependence on \(\phi\) is contained in \(K_f\). \end{proposition} \begin{proof} For a constant symmetric tensor basis \(e_A\), put \[ H_{A,y}(u)=g(y)\varrho_\varepsilon(u-y)e_A, \qquad \Lambda_A^\mu(z,y) =[\mathsf W(H_{A,y})]^\mu(z). \] This is a jointly smooth parameter family: parameter derivatives commute with the linear Cauchy problem, and differentiated local energy estimates give continuous dependence. The polarized second family \[ \Omega_{AB}^\mu(z;y,y') =\mathsf V^\mu(H_{A,y},H_{B,y'})(z) \] is jointly smooth by the same argument and the finite Leibniz rule. For example, integrating the scalar derivative by parts gives the first kernel \[ u_{1,A}(y,z) =\partial_{z^\mu}\bigl(f(z)\Lambda_A^\mu(z,y)\bigr). \] The second coefficient similarly involves finite derivatives of smooth products of \(\Lambda\) and \(\Omega\), with factors \(f(z)\), \(g(y)\), and \(g(y')\). This proves smooth compact support. Pairings with these kernels are bounded by finite products of compact-open smooth seminorms, proving continuity. A wave source affecting \(z\in K_f\) lies in \(D_f\). A convolution input affecting such a source lies within Euclidean distance \(\varepsilon\) and in \(\supp g\). In a nested second-order term the earlier wave source remains in \(D_f\) by causal transitivity. Each metric input is thickened only by its own single convolution. The scalar derivatives are evaluated on \(K_f\). \end{proof} This is the parent support mechanism \cite[\loc{lem:smoothkernel}, \loc{prop:support}]{XConv}. It is an \(\varepsilon\)-thickened dependence bound, not exact metric causality in the original field arguments. The convolution can sample a metric value slightly to the future of an intermediate wave-source point. These classical statements do not construct the separate interacting characteristic quantum product. \subsection{Two explicit certificate levels} \begin{definition}[Source-specific support certificates] \label{def:certificate} For the retained chart \(\mathcal L\), distinguish: \begin{enumerate}[label=\textup{(SC-\arabic*)}] \item \emph{Witness certificate:} \[ P_\varepsilon:=K_f+B_\varepsilon\Subset\mathcal L. \] Because \(K_f\subset\{r\le1/8\}\), \[ P_\varepsilon\subset\{r\le1/4\}\Subset U_C. \] This suffices to transport the numerical witness pairing, its compact gauge parameter, and its convolution dependence. \item \emph{Full analytical certificate:} \begin{equation} \mathcal R_\varepsilon :=D_f\cup Q_\varepsilon \cup(\overline U_C+B_\varepsilon) \Subset\mathcal L. \label{eq:fullcertificate} \end{equation} The required neighborhoods and the original time reserve are also retained. This sufficient certificate covers the causal computation, the convolution inputs, and the whole frozen supported gauge algebra used in its chart representation. \end{enumerate} A certificate is a geometric inclusion for the actual retained chart. Compactness of the displayed sets alone does not prove their inclusion in that chart. \end{definition} The original ramp reserve may cover \(D_f\) for its own test without covering \(Q_\varepsilon\), the parent convolution supports, or the parent gauge region. Therefore it does not automatically establish \eqref{eq:fullcertificate}. No claim is made that the supplied host description by itself proves either certificate for the numerical placement \(x_*=(4,0,0,0)\). The supported statements below have these explicit hypotheses. This is a restriction on the interpretation claimed, not a reduction of \(\gC\) after finding the defect. \subsection{Density transport and the nonlocal kernel} Let \(V=\iota(\mathcal L)\), let \(\chi=\iota^{-1}:V\to\mathcal L\), and let \[ J_\chi(y)=|\det D\chi(y)|. \] Here \(\chi\) denotes the inverse chart only in this subsection; it is unrelated to the switch profile. For a completion vector field \(v\), denote its transported field by \(v^\iota=\iota_*v\). For a scalar \(s\), write \(s^\iota=s\circ\chi\). Then \begin{equation} \begin{split} c_\varepsilon =\int_V &(f\circ\chi)(y) \bigl(\xi_*^\iota-(\Se\xi_*)^\iota\bigr)^a(y)\\ &\hspace{12mm}\cdot\partial_{y^a}\chi^1(y) J_\chi(y)\,\dd^4y, \end{split} \label{eq:transportpairing} \end{equation} whenever the witness certificate holds. The transported convolution is the two-point operator \begin{equation} \begin{split} (\Se^\iota v^\iota)^a(y) ={}&\left.\frac{\partial\iota^a}{\partial x^\mu} \right|_{x=\chi(y)} \int_V\varrho_\varepsilon(\chi(y)-\chi(y'))\\ &\quad\cdot \frac{\partial\chi^\mu}{\partial y'^b}(y') (v^\iota)^b(y')J_\chi(y')\,\dd^4y'. \end{split} \label{eq:transportkernel} \end{equation} For compact chart-supported inputs, their pullbacks are extended by zero outside the chart. For the full calculation, certificate (SC-2) guarantees that the relevant convolution inputs are contained there. \begin{theorem}[Supported transfer preserves the obstruction] \label{thm:hosttransfer} Under (SC-1), the pairing \eqref{eq:transportpairing} is well defined and equals the same strictly positive \(c_\varepsilon\). Under (SC-2), the relevant completion coefficient calculation and its gauge transformations admit the corresponding chart representation. In particular, with transported analytical operators, \[ w^\iota[K^\iota\xi^\iota] =\Se^\iota\xi^\iota, \] and the witness first coefficient is still \(\lambda c_\varepsilon\). Neither conclusion constructs a global conical Green operator or covariance of an intrinsically redefined cone convolution. \end{theorem} \begin{proof} The vector--covector contraction is preserved by pushforward and pullback. The change-of-variables formula supplies \(J_\chi\), proving \eqref{eq:transportpairing} and positivity. Substituting the component transformation of \(v\) into the original convolution and changing its input variable gives \eqref{eq:transportkernel}. Thus it is the same operator in the supported presentation. For (SC-2), finite propagation and Proposition~\ref{prop:kernels} place every required argument in the named reserve. Representing the already defined completion operators in that chart preserves the coefficient identities, including \eqref{eq:preflight}. No inverse on all of \(\mathcal M^\circ\) is needed. \end{proof} In a developed Cartesian chart the volume factor may be one. It cannot be deleted from an arbitrary host coordinate presentation. The formula for the two-point kernel also explains why passive coordinate transport is not a new active diffeomorphism identity for the regulator. %===================================================================== \section{Stage C: readout closure of the retained coefficient} \label{sec:readout} %===================================================================== This section records the operational half of X-Enter and the supplied proposed X-Enter-ST \cite{XEnter,XEnterST}. The proofs are included so the present manuscript has no external source-code dependency. \subsection{The encoding-independent statement} \begin{definition}[Complete scalar-input output] For an encoding \(\Enc:\RR\to\mathscr S\) and a fixed output map \(\Phi:\mathscr S\to\mathscr Y\), the scalar-input output is \(\Phi\circ\Enc\). Its declared boundary includes all reported classical words and every returned quantum factor. It is informative about \(a\) if this composite is nonconstant. Orbit invariance means constancy on \(\Enc(\lambda c_\varepsilon)\), \(\lambda\in\RR\). \end{definition} \begin{theorem}[Scalar-only negative closure] \label{thm:orbitclosure} If \[ \Phi(\Enc(\lambda c_\varepsilon)) =\Phi(\Enc(0)) \quad\text{for all }\lambda\in\RR, \] then \[ \Phi(\Enc(a))=\Phi(\Enc(0)) \quad\text{for all }a\in\RR. \] \end{theorem} \begin{proof} For any \(a\), choose \(\lambda=a/c_\varepsilon\). Theorem~\ref{thm:positive} guarantees that this is defined. \end{proof} No continuity, Born rule, finite dimension, or linear encoding is used. The conclusion concerns the composite on the encoded family, not constancy of \(\Phi\) on unrelated states. The charge \(c_\varepsilon\) is fixed during this comparison; \(a\) is the varying coefficient. Reporting fixed calibration metadata is not forbidden. \begin{proposition}[Approximate orbit bound] \label{prop:approx} Let \(\Omega(a)\) be normalized finite classical--quantum output states. If \[ \sup_\lambda \dtr(\Omega(\lambda c_\varepsilon),\Omega(0))\le\eta, \] then, for all \(a,b\), \[ \dtr(\Omega(a),\Omega(0))\le\eta,\qquad \dtr(\Omega(a),\Omega(b))\le2\eta. \] For any finite hypothesis family \(a_x\), prior \(\pi_x\), and estimator using only this output, \[ \Pr(\widehat x=x)\le\min\{1,\max_x\pi_x+\eta\}. \] \end{proposition} \begin{proof} The first assertion uses the same substitution as Theorem~\ref{thm:orbitclosure}; the second uses the triangle inequality. If \(M_x\) are the estimator effects, trace distance bounds each effect expectation. Hence the success probability is at most \[ \sum_x\pi_x\Tr(M_x\Omega(0))+\eta \le\max_x\pi_x+\eta. \] \end{proof} \subsection{The retained amplitude family} Freeze \(\sigma>0\) and define \[ p(a)=\frac{1+\tanh(a/\sigma)}2,\qquad \ket{e(a)} =\sqrt{1-p(a)}\ket0+\sqrt{p(a)}\ket1,\qquad \tau(a)=\proj{e(a)}. \] Set \[ D=\proj1-\proj0,\qquad X=\kb01+\kb10,\qquad t(a)=\tanh(a/\sigma),\quad s(a)=\sech(a/\sigma). \] Then \[ \tau(a)=\frac12(I+t(a)D+s(a)X). \] For any fixed finite instrument \(\{\Phi_y\}_y\), with every quantum residual included, put \[ A_y=\Phi_y(I/2),\qquad B_y=\Phi_y(D/2),\qquad C_y=\Phi_y(X/2). \] Its selective output is \begin{equation} \Phi_y(\tau(a))=A_y+B_yt(a)+C_ys(a). \label{eq:selective} \end{equation} \begin{proposition}[Selective visibility or erasure] \label{prop:amplitude} Complete orbit invariance for this encoding is equivalent to \(B_y=C_y=0\) for every \(y\). For the classical law alone the equivalent condition is \(\tr B_y=\tr C_y=0\). Direct Enter has the visible discrepancy \begin{equation} \Delta_\varepsilon =\dTV(\Law_{\mathrm{read}}(c_\varepsilon), \Law_{\mathrm{read}}(0)) =\frac12\tanh(c_\varepsilon/\sigma)>0. \label{eq:TV} \end{equation} \end{proposition} \begin{proof} Differentiating \eqref{eq:selective} along the orbit gives \[ \frac{\dd}{\dd\lambda}\Phi_y(\tau(\lambda c_\varepsilon)) =\frac{c_\varepsilon}{\sigma} (B_ys^2-C_yst). \] At zero, invariance forces \(B_y=0\); at a nonzero parameter, \(st\ne0\), forcing \(C_y=0\). The converse is immediate. Taking traces proves the law statement. Direct Enter has success probability \(p(a)\), so the binary-law distance is \eqref{eq:TV}. \end{proof} A constant output law with a varying returned qubit is not complete invariance. Nor does erasure of this real-amplitude family imply erasure of every Bloch-sphere direction. \begin{proposition}[Second-order encoding diagnostic] \label{prop:encodingsecond} For a fixed instrument and encoding, let \[ a(\kappa,\lambda)=\kappa\lambda c_\varepsilon +\kappa^2b(\lambda)+\ord(\kappa^3), \qquad b(0)=0. \] If its complete output agrees with that of \(\tau(0)\) through order two for every fixed \(\lambda\), then \(B_y=C_y=0\). \end{proposition} \begin{proof} Every selective difference has expansion \[ \frac{\kappa\lambda c_\varepsilon}{\sigma}B_y +\kappa^2\left( \frac{b(\lambda)}{\sigma}B_y -\frac{\lambda^2c_\varepsilon^2}{2\sigma^2}C_y \right)+\ord(\kappa^3). \] Successive coefficient comparison gives the result. \end{proof} For example, Hadamard followed by Enter has \[ \Pr(1\mid a)=\frac{1-\sech(a/\sigma)}2 =\frac{a^2}{4\sigma^2}+\ord(a^4). \] Its first derivative vanishes at zero, but it is not invariant \cite{XEnter}. This is an encoding diagnostic, not a new expansion of the convolution prescription. Finite-dimensional encoding of real amplitudes also does not mean orthogonal finite-basis storage or exact field-to-qubit preparation for arbitrary real data. %===================================================================== \section{Complete instruments for supplied evaluations} \label{sec:instrument} %===================================================================== \subsection{A nonempty finite dictionary} A supplied dictionary contains quantity tags, frozen-data identifiers, support-certificate status, optional certified numerical enclosures, and logical status words. One explicit four-entry dictionary is: \[ \begin{array}{c|p{.68\linewidth}} 00&\loc{GAP_POSITIVE}: equation~\eqref{eq:energy} proves the fixed coefficient is positive.\\[3pt] 01&\loc{ORBIT_CONSTANT}: a specified erasure output is constant on the full scalar orbit.\\[3pt] 10&\loc{INFORMATIVE_FORBIDDEN}: full-orbit invariance together with informativeness is excluded.\\[3pt] 11&\loc{NOT_CONSTRUCTED}: an independently charged second classical coefficient has not been supplied. \end{array} \] A finite list of sampled coincidences is not a certificate of universal orbit constancy. Optional rational enclosures of \(c_\varepsilon\) are supplied data; no numerical quadrature has been executed here. \subsection{The destructive task clock} Let \(C\cong\CC^{2^q}\) have orthonormal calibration basis \(\{\ket{\tau_j}\}\), with \(q\ge1\), and let \(A\) contain every surviving finite register. For each installed task, let \(q_j\) be its finite report word. Define \[ D_j=\bra{\tau_j}_C\otimes I_A \] and \begin{equation} \mathcal M_C(Z) =\sum_j\proj{j,q_j}_R\otimes D_jZD_j^\dagger. \label{eq:clock} \end{equation} \begin{theorem}[Complete supplied-report instrument] \label{thm:instrument} Equation~\eqref{eq:clock} is CPTP and satisfies \begin{equation} \Tr_R\mathcal M_C(Z)=\Tr_CZ \label{eq:discard} \end{equation} on all operators, including every untouched-reference extension. On a positive-probability branch, the normalized quantum output is its actual conditional \(A\)-state. No normalized state is assigned to a zero-probability branch. The returned quantum interface contains \(A\), not \(C\). \end{theorem} \begin{proof} The recorded Kraus maps are \[ L_j=\ket{j,q_j}_R\otimes D_j,\qquad \sum_jL_j^\dagger L_j=I_{CA}. \] This proves complete positivity and trace preservation. Expand \[ Z=\sum_{j,l}\kb{\tau_j}{\tau_l}_C\otimes Z_{jl}. \] Its \(j\)-th output block is \(Z_{jj}\), and summing gives \(\Tr_CZ\). The same calculation with \(A\otimes E\) proves every reference extension and the conditional block statement. The codomain specifies consumption of \(C\). \end{proof} This is the I-B/En09-A interface \cite{IB,En09}, used in the supplied X-Enter-ST. An untouched reference may be conditionally steered; \eqref{eq:discard} concerns its unconditional marginal. A task-conditioned amplitude program can be composed with \eqref{eq:clock}. Complete histories give path maps \(K_h\) with \[ \Phi_y(\rho)=\sum_{h:\,\mathrm{out}(h)=y}K_h\rho K_h^\dagger, \qquad \sum_hK_h^\dagger K_h=I. \] The completeness identity follows by backwards induction over the finite instruction tree, including every failure branch and every returned quantum register. \subsection{What the instrument does not acquire} \begin{proposition}[No field-acquisition inference] \label{prop:noacquisition} The instrument reports its installed finite input. It does not acquire \(c_\varepsilon\), \(a(\lambda)\), or a continuum gravitational observable. \end{proposition} \begin{proof} For displayed numerical values \(\widehat a_j\), the report observable \[ B_R=\sum_j\widehat a_j\proj{j,q_j}_R \] has pullback \[ \mathcal M_C^*(B_R\otimes I_A) =\sum_j\widehat a_j\proj{\tau_j}_C\otimes I_A. \] This is an observable of the supplied finite encoding. No map from unknown continuum field data to that encoding occurs in the input contract or Kraus operators. \end{proof} A gauge-dependent task selector or residual cannot be exported for free during a closure comparison. If it depends on \(a\), it belongs to the complete encoding and output to which Theorem~\ref{thm:orbitclosure} applies. \begin{proposition}[No universal clock return] \label{prop:noreturn} For \(\dim C\ge2\), \(\mathcal M_C\) has no channel left inverse on its full clock--residual input space. \end{proposition} \begin{proof} With a fixed residual \(\omega_A\), the distinct clock inputs \[ \frac{\ket{\tau_0}+\ket{\tau_1}}{\sqrt2}, \qquad \frac{\ket{\tau_0}-\ket{\tau_1}}{\sqrt2} \] give the same recorded output. A left inverse could not return both different inputs from that one output. \end{proof} This does not exclude recovery from a separately declared code in another surviving factor \cite{En09}. The geometric time axis, conical defect, and future relation are not consumed. %===================================================================== \section{Stage D: the conditional relational alternative} \label{sec:reference} %===================================================================== \begin{assumption}[Additional classical reference] \label{ass:reference} An independently defined coefficient \(r\), with proved nonzero charge \(r_\varepsilon\), is supplied such that \[ a(\lambda)=a_0+\lambda c_\varepsilon,\qquad r(\lambda)=r_0+\lambda r_\varepsilon. \] For whole-algebra comparisons, its first-order charges \(q_a(\xi)\), \(q_r(\xi)\) are defined on \(\gC\). \end{assumption} This is a hypothesis, not a construction supplied by the ramp, a later tree, a free BRST multiplet, or a quantum spectator. \begin{proposition}[Exact matching requirement] \label{prop:matching} The combination \[ I_{\mathrm{rel}}=r_\varepsilon a-c_\varepsilon r \] is invariant on the stated witness orbit. Its charge vanishes on the whole declared coefficient algebra exactly when \begin{equation} r_\varepsilon q_a(\xi)-c_\varepsilon q_r(\xi)=0 \qquad(\xi\in\gC). \label{eq:matching} \end{equation} \end{proposition} \begin{proof} The witness \(\lambda\)-coefficient cancels. For a general direction, the charge of the combination is exactly the left side of \eqref{eq:matching}. \end{proof} This is a first-order coefficient statement, not an all-orders nonlinear gauge theorem. The chosen baseline values \(a_0,r_0\) do not separately prove physical gauge invariance. Freeze \(\sigma>0\) and \(\sigma'=(r_\varepsilon/c_\varepsilon)\sigma\ne0\). Consider the abstract preparation \[ \ket{\Psi(a,r)} =\frac12(\ket0+e^{i\alpha}\ket1) \otimes(\ket0+e^{i\beta}\ket1), \qquad \alpha=\frac a\sigma,\quad \beta=\frac r{\sigma'},\quad \theta=\alpha-\beta. \] Apply CNOT, first qubit controlling the second, then Hadamard on the first, then destructively read both. Report odd-parity outcomes as \(+\) and \(-\), and merge both even-parity outcomes into a retained failure \(F\). Let \[ \ket{\chi_\pm}=\frac{\ket{01}\pm\ket{10}}{\sqrt2}. \] \begin{theorem}[Conditional complete CNOT--Hadamard instrument] \label{thm:relational} The selective maps are \[ \Phi_F(\rho)=\bra{00}\rho\ket{00}+\bra{11}\rho\ket{11}, \qquad \Phi_\pm(\rho)=\bra{\chi_\pm}\rho\ket{\chi_\pm}. \] They form a complete instrument, with unit signal quantum output, and \[ \Pr(F)=\frac12,\qquad \Pr(\pm)=\frac{1\pm\cos\theta}{4}. \] Each selective map is invariant under \[ G_\gamma=\operatorname{diag}(1,e^{i\gamma})^{\otimes2}. \] The map identities hold with arbitrary untouched references and arbitrary passive surviving finite factors. Therefore the declared coarse output is invariant on the witness orbit under Assumption~\ref{ass:reference}; condition \eqref{eq:matching} supplies the corresponding whole-algebra coefficient statement. \end{theorem} \begin{proof} The four destructive path bras after the gates are the four Bell bras. The two even-parity projectors sum to \(\proj{00}+\proj{11}\); the odd ones are \(\proj{\chi_+}\) and \(\proj{\chi_-}\). Their sum is the identity. The odd amplitudes are proportional to \(e^{i\beta}\pm e^{i\alpha}\), giving the probabilities. For the failure map use Kraus bras \(\bra{00},\bra{11}\); each acquires only a scalar phase under \(G_\gamma\). Both \(\chi_\pm\) acquire the same scalar phase. These phases cancel in the selective maps, including their reference extensions. The two witness phases increase by the same amount. Equation~\eqref{eq:matching} gives the same conclusion for each stated algebra direction. \end{proof} This is X-Enter's conditional circuit \cite{XEnter}. It detects \(\cos\theta\), not a globally faithful encoding of the real \(I_{\mathrm{rel}}\). The raw even-parity probabilities are \[ \frac{1+\cos(\alpha+\beta)}4,\qquad \frac{1-\cos(\alpha+\beta)}4, \] which need not be invariant. The proved boundary is the coarse output \(F,+,-\). If a further accessible log retains the raw distinction, that larger boundary must be tested separately. No failure is discarded or renormalized away. The finite circuit is constructed at its abstract input. Its physical application still lacks the independently charged classical reference and its preparation/acquisition interface. %===================================================================== \section{Bounded comparisons and the separate ramp prescription} \label{sec:ramp} %===================================================================== \subsection{A bounded segment is new comparison data} If a separate restriction \(|\lambda|\le\Lambda\) is imposed, let \(d=\Lambda c_\varepsilon>0\). Direct amplitude readout obeys \[ \sup_{|\lambda|\le\Lambda} \dTV(\Law_{\mathrm{read}}(\lambda c_\varepsilon), \Law_{\mathrm{read}}(0)) =\frac12\tanh(d/\sigma). \] For \(0<\delta,v<1/2\) and a separately specified benchmark magnitude \(A_{\mathrm{sig}}>0\), variation at most \(\delta\) and visibility at least \(v\) hold exactly when \[ \frac{d}{\artanh(2\delta)} \le\sigma\le \frac{A_{\mathrm{sig}}}{\artanh(2v)}. \] These statements follow by monotonicity of \(\tanh\) \cite{XEnter}. Compact support gives no such amplitude restriction: every real multiple of a supported vector has the same support. A bounded comparison set also need not be an algebra closed under composition. \subsection{The ramp is not a convolution repair} The ramp's retained first-order identity is \[ \Co_1[K\xi]=\xi,\qquad \xi\in\gR, \] on its own plateau algebra \cite{XA,XAST}. The ramp multiplies a metric-density deviation inside a divergence. It does not replace \(\Tge=\Se\circ(g\,\cdot)\). The ramp edge comparison has a different integral, in its source convention, \[ d_{\mathrm{edge}} =-\int_0^\varepsilon (\sigma_\varepsilon^{\mathrm R})'(s) \alpha_{\mathrm{edge}}(s)\,\dd s. \] That integral is not \(c_\varepsilon\), and its edge parameter is not an admitted member of the frozen ramp plateau algebra. \begin{proposition}[No cross-prescription repair] \label{prop:separation} The ramp plateau identity, the convolution obstruction, and scalar-only negative readout closure are compatible. Neither the ramp nor its later scalar, tree, or free BRST/BV sheets changes the frozen convolution prescription by any operation constructed here. \end{proposition} \begin{proof} The convolution obstruction follows from \eqref{eq:regulator}--\eqref{eq:preflight} and \eqref{eq:energy}, without a ramp premise. The ramp identity concerns a different response map and supported algebra. The orbit theorem concerns the scalar family \eqref{eq:orbit}, not every already invariant scalar. The finite instruments act on declared finite inputs. None changes \(w^\mu=\Ezero\mathsf C^\mu\Tge h\), \(\Co_1\), or either classical action prescription. \end{proof} The later ST records are documentary separation controls \cite{LaterST}. In particular: \begin{itemize} \item X-B-ST through X-D-ST do not turn their scalar maps or sharp-width test convergence into a convolution repair or sharp metric admission. \item X-E-ST and X-F-ST concern their own tree sums and equation-level identities; they do not authorize a variational replacement of the convolution. \item X-G-ST and X-H-ST distinguish free nilpotence and relative classical BV structure from an interacting differential on an unchanged carrier. \end{itemize} No \(\varepsilon\to0\) is taken in this manuscript. Convolution is not an ultraviolet regulator. A coordinate counterterm or changed response requires a newly stated classical problem and proof. %===================================================================== \section{Stage E: scheduled locality and finite report transport} \label{sec:locality} %===================================================================== \subsection{Complete network locality} Let \(X_{\mathrm{site}}\) be a finite site set, with finite-dimensional site spaces. At tick \(s\), let \(\mathcal B_s\) be a partition and \[ \Gamma_s=\bigotimes_{B\in\mathcal B_s}\Gamma_{s,B}, \] where each block map is a channel on its complete declared interface. Every control, task selector, record, workspace, and reset used by a block belongs to that interface. For terminal sites \(Y\), define \[ N_{t,t}(Y)=Y,\qquad N_{s,t}(Y)= \bigcup_{\substack{B\in\mathcal B_{s+1}\\ B\cap N_{s+1,t}(Y)\ne\varnothing}}B. \] \begin{theorem}[Complete scheduled support] \label{thm:locality} The network satisfies \[ (\Gamma_{s+1}^*\circ\cdots\circ\Gamma_t^*)(\Alg(Y)) \subseteq\Alg(N_{s,t}(Y)). \] Equal input reductions on \(N_{s,t}(Y)\) give equal output reductions on \(Y\). A joint referenced conclusion requires equal joint input reductions on \(N_{s,t}(Y)\) and the untouched reference. A deterministic intervention outside the complete backwards region cannot change that output. If every dependency edge has a certified regular future-causal host route, scheduled ancestors lie in the prescribed metric causal past of the output. \end{theorem} \begin{proof} A block disjoint from the pulled-back support acts on an identity factor. Its adjoint is unital and preserves that factor. Other blocks enlarge support only inside their own interfaces. Backwards induction gives the inclusion. Pairing with all output observables proves the reduced-state statement. Tensoring with the reference algebra proves the referenced version. The same unitality argument handles an excluded deterministic intervention. Concatenating the independently certified causal routes proves the geometric containment. \end{proof} This is the prescribed-network mechanism used in En03-A and VII-E \cite{En03,VIIE}. Ancestry is an upper bound, not necessarily minimal influence or the entire physical causal past. Scheduler consistency alone does not determine a Lorentz cone \cite{En08A}. The metric inclusion in the theorem is an additional embedding certificate. It is not inferred from a drawing of a network or from one scalar speed bound. \subsection{A nonempty regular SWAP route} Within the retained regular developed tube, choose a nonzero straight spatial segment of length \(\ell\). Choose a departure time after readout, lookup, and the declared processing buffer, with the whole route in the existing time reserve. For \(n\ge1\), place equal payload registers \(A_0,\ldots,A_n\) at equally spaced stations. The payload includes the finite report and every quantum residual to be returned. Prepare \(A_1,\ldots,A_n\) in independent blanks \(\omega\), and apply successive SWAPs across developed durations \(\ell/n\). \begin{theorem}[Reference-preserving regular route] \label{thm:route} For any initial payload--reference state \(\rho_{A_0E}\), after hop \(j\) the complete route state is \[ \rho_{A_jE}\otimes \bigotimes_{i\ne j}\omega_{A_i}. \] The logical payload channel is the identity, including every reference. Arrival occurs after developed time \(\ell\), or host time \(\ell/c_{\mathrm h}\). All route registers remain in the complete resource ledger; the non-output ones are fixed independent blanks. \end{theorem} \begin{proof} At each hop, SWAP exchanges the entire current payload, including its reference correlations, with the next independent blank. Induction proves the tensor identity. Each adjacent separation and developed duration equals \(\ell/n\), giving a straight null certificate in the developed chart. The retained signature and calibration \eqref{eq:signature} give the host time. The chosen tube remains disjoint from the worldsheet. \end{proof} This is the target-adapted I-B/En08-B transport mechanism \cite{IB,En08B}. It does not assert exact null attainment in every direction of a previously fixed lattice. The SWAP permission is a declared finite-operation permission, not a derived microscopic Hamiltonian. The route preserves existing correlations. No entanglement is attributed to a gravitational mediator, and no continuum coefficient is transported. \subsection{Resource and error ledger} If the payload uses \(p\) qubits, the route reserves \((n+1)p\) qubits and uses \(n\) register SWAPs. Clock, signal, workspace, supplier, and reference holdings are separately charged. A conservative preallocated bound is \[ q_{\mathrm{clock}}+q_{\mathrm{signal}} +q_{\mathrm{workspace}}+(n+1)p. \] No optimality is claimed for this bound. For compatible ideal and implemented complete channels, \begin{equation} \ddia(\widetilde\Gamma,\Gamma) \le\min\left\{1,\, e_{\mathrm{prep}}+e_{\mathrm{clock}}+e_{\mathrm{lookup}} +e_{\mathrm{read}}+\sum_{j=1}^ne_j+e_{\mathrm{decode}} \right\}, \quad \ddia(\Lambda,\Gamma)=\frac12\norm{\Lambda-\Gamma}_\diamond. \label{eq:error} \end{equation} To prove this, replace one factor at a time and apply diamond-norm contraction under channel composition and the triangle inequality \cite{Watrous,En08B}. The comparison interfaces include all classical histories and quantum residuals. A small channel error is not exact causal containment. Nor does it validate a false supplied numerical enclosure or produce a missing field-acquisition map. %===================================================================== \section{Stage F: release and exact successor package} \label{sec:release} %===================================================================== \begin{definition}[Local Grade S in this manuscript] The supported release consists of the unchanged host, the explicit convolution contract, its written analytical obstruction, the scalar-input closure results, a nonempty finite supplied-report instrument, and a scheduled regular route. A laboratory interpretation of the full continuum calculation requires (SC-2). A laboratory interpretation of its numerical witness pairing requires only (SC-1). Installed finite reporting does not by itself require either continuum acquisition claim. \end{definition} \begin{theorem}[X-A-conv-ST release] \label{thm:release} For the fully specified completion data \eqref{eq:switch}--\eqref{eq:witness}, at fixed \(0<\varepsilon\le1/8\): \begin{enumerate}[label=\textup{(\roman*)}] \item the coordinate and scalar Cauchy coefficients exist uniquely through order two; \item the convolution response is \(\Se\xi\), and the fixed witness has the strictly positive coefficient \eqref{eq:energy}; \item the coefficient survives scalar-solution evaluation and has the real-surjective first-order orbit \eqref{eq:orbit}; \item the retained second-order defects are the displayed ones, and a higher-order addition preserving the first coefficient cannot repair its nonzero defect; \item under the appropriate source-specific chart certificate, density transport preserves that obstruction on the unchanged prescribed host; \item any scalar-only complete readout invariant on the full witness orbit is uninformative about that scalar; \item a complete destructive instrument reports finite supplied evaluations and statuses, with all unmeasured finite residuals and references retained; \item the two-coefficient circuit is constructed only conditionally on the independently supplied charge and matching hypotheses; \item scheduled locality and reference-preserving regular transport hold under the declared finite permissions. \end{enumerate} With (SC-2) and the retained operational reserves, this is local Grade S. No convolution repair, field-to-qubit acquisition, interacting differential, inherited characteristic quantum coefficient, or spacetime emergence is constructed. \end{theorem} \begin{proof} Lemma~\ref{lem:Cauchy}, Proposition~\ref{prop:coordinates}, and \eqref{eq:rho1}--\eqref{eq:rho2} prove (i). Lemma~\ref{lem:preflight} and Theorem~\ref{thm:positive} prove (ii). Theorem~\ref{thm:evaluated} proves (iii). Propositions~\ref{prop:seconddefect} and \ref{prop:pure}, with Corollary~\ref{cor:nohigher}, prove (iv). Theorem~\ref{thm:hosttransfer} proves (v). Theorem~\ref{thm:orbitclosure} proves (vi). Theorem~\ref{thm:instrument} and Proposition~\ref{prop:noacquisition} give the construction and its precise limitation in (vii). Proposition~\ref{prop:matching} and Theorem~\ref{thm:relational} prove (viii). Theorems~\ref{thm:locality} and \ref{thm:route} prove (ix). \end{proof} \begin{center} \small \renewcommand{\arraystretch}{1.13} \begin{tabularx}{\linewidth}{@{}p{.30\linewidth}p{.23\linewidth}X@{}} \toprule Object & Verdict & Exact qualification\\ \midrule Completion convolution coefficients & Constructed & Fixed smooth inputs and positive width\\ Covariance on \(\gC\) & Proved obstructed & Positive first classical coefficient\\ Witness pairing on the host & Supported transfer & Requires (SC-1)\\ Full chart representation & Supported transfer & Requires (SC-2); not implied by ramp support\\ Invariant informative scalar-only readout & Excluded & Complete output on the full real orbit\\ Finite supplied-report instrument & Constructed & Not continuum acquisition\\ Second charged coefficient & Not constructed & Conditional circuit does not supply it\\ Finite regular report route & Constructed under permissions & Identity transport of supplied finite data\\ Characteristic quantum coefficient & Not admitted & Separate analytical chain remains open\\ Spacetime emergence & Not constructed & Geometry and speed are inputs\\ \bottomrule \end{tabularx} \end{center} \subsection{Novelty and prospective usefulness} The positive-gap, preflight, evaluated-gap, and retained second-order mechanisms are credited to X-A--Convolution. The orbit, amplitude, and conditional relational mechanisms are credited to X-Enter \cite{XConv,XEnter}. The contribution of this supported conversion is the common, explicit interface between those mechanisms and one retained conical host. In particular: \begin{itemize} \item the regulator is transported as an actual two-point operator, not renamed an intrinsic cone convolution; \item the support certificate is separated from the ramp's different causal reserve; \item the positive analytical defect is kept distinct from a supplied finite report about it; \item complete residuals, classical controls, references, and resource holdings are included in the operational comparison. \end{itemize} The prospective value is a controlled diagnostic baseline: it identifies an obstruction before quantum admission or readout terminology can conceal it. It is not a new quantum-gravity prediction or a general impossibility theorem for regulated relational observables. \subsection{What a successor receives} A successor receives the frozen classical data, the Cauchy formulas, the positive energy identity, the evaluated coefficient orbit, the retained second-order defects, the explicit support-certificate template, the transported density and regulator, the scalar-only closure theorem, and the complete finite instrument and route contracts. It does not receive a repaired convolution coefficient, a constructed second reference, a continuum detector, or permission to substitute the ramp response. If the actual inherited chart has not been checked against (SC-1) or (SC-2), that inclusion remains a visible geometric input obligation. It cannot be declared proved by the existence of an abstract chart or by installing a finite task table. %===================================================================== \appendix \section{Debugging and non-identification audit} \label{app:debug} %===================================================================== \begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}} \toprule Potential error & Correction\\ \midrule \endfirsthead \toprule Potential error & Correction\\ \midrule \endhead Assuming the gap instead of proving it & The mollifier, switch, test, and gauge witness are explicit, and \eqref{eq:energy} proves positivity.\\[4pt] Replacing convolution by a ramp & They act on different objects and have different source equations and admissible algebras.\\[4pt] Using \(g^2\) in the second scalar vertex & The frozen action uses one switch factor at each retained interaction order.\\[4pt] Dropping switch derivatives & They remain inside the displayed divergences and in the explicit first source formula.\\[4pt] Treating the regular cone as globally hyperbolic & Proposition~\ref{prop:notGH} excludes this. All global analytical inverses stay on \(\widehat M\).\\[4pt] Calling the defect a worldline & The product defect is \(\RR_T\times\{o\}\times\RR_z\), a worldsheet.\\[4pt] Substituting host signature for analytical signature & The ledger is \(\iota^*G=-\eta\). Analytical contractions retain \(+---\).\\[4pt] Dropping the change-of-variables density & Equation~\eqref{eq:transportpairing} retains \(J_\chi\).\\[4pt] Calling transported convolution natural & Equation~\eqref{eq:transportkernel} retains the source coordinates and two-point kernel.\\[4pt] Inferring a convolution support reserve from the ramp & The convolution dependence is \(Q_\varepsilon=\supp g\cap(D_f+B_\varepsilon)\). Its chart inclusion is checked separately.\\[4pt] Shrinking the gauge algebra after the witness & The original \(\gC\) remains frozen. Support certificates qualify interpretation, not the analytical gauge algebra.\\[4pt] Using a straight metric line as a pure-gauge two-jet & The term \(\lambda^2\Lie_\xi^2\eta/2\) is retained.\\[4pt] Promoting a second-order check to an all-orders identity & Only the stated coefficient formulas are proved.\\[4pt] Treating \(\lambda c_\varepsilon\) as a uniform finite-coupling expansion & It is an orbit of a first-order coefficient; perturbative comparisons are pointwise in \(\lambda\).\\[4pt] Claiming that the whole instrument is constant & The orbit theorem constrains its composite with the scalar encoding on \(\RR\).\\[4pt] Confusing a fixed charge with the variable input & \(c_\varepsilon\) is fixed; \(a\) varies on its orbit.\\[4pt] Checking only outcome probabilities & Every returned quantum factor is part of the complete output.\\[4pt] Inferring exact real decoding from one qubit & The amplitude states are nonorthogonal; no field-to-qubit preparation or exact real decoder is constructed.\\[4pt] Exporting raw failure logs from the relational circuit & The proved invariant boundary is \(F,+,-\); the raw even-parity split is generally not invariant.\\[4pt] Identifying a spectator with the second charged coefficient & The second classical definition and its charges remain additional hypotheses.\\[4pt] Consuming geometric time & Only measured finite clock names are consumed.\\[4pt] Calling SWAP transport gravity-generated entanglement & The route preserves supplied correlations and uses a prescribed geometry.\\[4pt] Treating classical smoothing as a quantum-loop theorem & No interacting characteristic product, loop normalization, or quantum Ward coefficient is constructed.\\ \bottomrule \end{longtable} %===================================================================== \section{Standing analytical and documentary boundaries} \label{app:boundaries} %===================================================================== \paragraph{Smooth multiplication is not a general smoothing theorem.} For a smooth multiplier \(b\) and a distribution \(u\), \[ \WF(bu)\subseteq\WF(u). \] On an open set where \(b\ne0\), the reverse inclusion follows by multiplication by \(1/b\). This does not prove cancellation in a sum or define an otherwise missing characteristic product \cite{Hormander,XVIA}. The present convolution smooths its declared classical input slots at fixed width; it is not a substitute for the separate graph-product analysis. \paragraph{Distinct domain and product questions.} VII-C, VII-D, IX-A-ST, and IX-D-ST concern their own Wick-extension, finite-part, sharp-domain, topology, or support-bridge questions. XI-B and XI-G use different multiplet and admission contracts. None is identified with the convolution algebra by this conversion \cite{DomainRecords}. \paragraph{Characteristic chain.} The historical H2 statement in XIV-H, the narrower internal-amplitude construction at I2 in XIV-I, and the missing full joint maps at I3/J2 are different version-qualified records. This manuscript does not perform the missing characteristic composition, joint section, complete remainder, transverse estimates, contact compatibility, or output-wavefront theorem \cite{BoundaryRecords}. The downstream XV-B quantum-admission exit is not opened. This does not deny the existence of technical manuscripts bearing that identifier. Their interfaces are not an admitted full coefficient. \paragraph{Prepared-source boundaries.} XVI-N is not discharged. The scoped XVI-O kernel-subsector comparison is not bulk restoration or a repair of the present coefficient. O.1 remains \loc{NOT_CLOSED} within its inherited prescription \cite{BoundaryRecords}. \paragraph{Problem ledger.} In the supplied programme's labels, Problems 1, 2, and 8 remain unfinished; Problems 3--7 and 9--11 are not reached; Problem 12 remains unresolved. These inherited statuses are not independently rescanned or replaced by a universal impossibility theorem. \paragraph{Documentary finite-interface comparisons.} The supplied Families 282, 362, and 376 and the records Oai282-A, Oai283-A, Oai272-A, Oai276-A, Oai279-A, Oai119-A, and Oai365-A remain documentary comparisons \cite{FiniteRecords,OpenAICatalogue}. A finite Ward, constant-error, Choi, Holevo, lookup, Hellinger, or signature certificate is neither the orbit proof nor a convolution repair. Family 265's static entropy comparison is not a Page curve. No catalogue proof is used or independently verified here. \paragraph{Physical non-claims.} No quantum coefficient of the inherited characteristic graph is admitted. This does not deny the free carrier or the tree coefficients admitted in their own earlier contracts. There is no derived cone, Einstein dynamics from records, external-time-free quantum evolution, gravity-generated entanglement, Page curve, holographic identification, ultraviolet completion, or spacetime emergence. %===================================================================== \section{Proposed integration record} \label{app:metadata} %===================================================================== This is a proposed source file, not an executed repository transaction. A maintainer may install it as \loc{content/manuscripts/X-A-conv-ST.tex}, add the actual source and registry records, and run the repository's real tests. No schema, generated hash, passing test, or completed registration is invented. \begin{verbatim} identifier: X-A-conv-ST sourceStatus: proposed grade: local Grade S subject to stated support certificate primaryAnalyticSource: X-A-conv@1 operationalComparison: - X-Enter@1 - supplied proposed X-Enter-ST hostRetainedFrom: - I-B@1 - supplied X-A-ST widthStatus: fixed-positive-width analyticalCompletion: Minkowski globalConeGreenOperator: not constructed constructed: - explicit frozen convolution data - Cauchy formulas and pure-gauge response - strictly positive fixed-width coefficient - evaluated scalar-solution obstruction - retained second-order defect formulas - source-specific support-certificate template - conditional supported density and kernel transport - scalar-only negative readout closure - complete supplied-report clock instrument - scheduled locality and regular finite report route conditional: - laboratory witness interpretation requires SC-1 - full chart interpretation requires SC-2 - relational application requires independent charged reference - whole-algebra relational closure requires charge matching notConstructed: - convolution repair - field-to-qubit acquisition - second charged classical coefficient - interacting differential - inherited characteristic quantum coefficient - spacetime emergence registryEditsExecuted: false repositoryTestsExecuted: false compilationAsserted: false \end{verbatim} %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Authorship.} The author field is intentionally empty for anonymous review. Citation of N.~Cheung does not assign authorship of this adaptation. Authorized authorship and submission metadata must be supplied for a signed submission. \paragraph{Attribution.} The analytical obstruction and its defining calculations are attributed to X-A--Convolution, version~1. The readout obstruction and conditional alternatives are attributed to X-Enter, version~1. I-B, En09-A, En03-A, and En08-B supply the distinct host and operational patterns. Their relevant arguments are explicitly stated or proved here within the displayed hypotheses. \paragraph{Analytical status.} The positivity claim is proved from the explicit parent data. The chart interpretation is conditional on its stated support certificate. That condition is not hidden inside a claim that the unchanged inherited laboratory has already been checked. \paragraph{Operational status.} The instruments encode supplied finite data. They do not acquire unknown continuum fields, select a physical gravitational state, derive microscopic gate dynamics, or construct a thermal implementation. \paragraph{Bibliographic status.} Internal records are cited by identifier and supplied version without invented publication dates or DOIs. The journal Enter article is separately identified as an accepted-manuscript record. A documentary comparison is not a theorem premise or an independent proof certificate. \paragraph{Verification.} No successful \loc{pdflatex} execution, proof-assistant check, symbolic-program execution, numerical evaluation, repository installation, hardware synthesis, experiment, independent execution, independent referee verification, or journal acceptance is asserted. The mathematical release consists of the explicit definitions and written proofs in this source. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XConv} N.~Cheung, \emph{Landmark X-A: Classical covariance of a fixed-regulator switched relational scalar through order \(\kappa^2\): an explicit regulator obstruction}. \loc{X-A-conv.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-A-conv}. Especially \loc{lem:Cauchy}, \loc{prop:coordcoeff}, \loc{lem:preflight}, \loc{lem:positivegap}, \loc{thm:preflightstop}, \loc{prop:infdefects}, \loc{prop:support}, \loc{thm:evaluatedgap}, \loc{prop:actionidentity}, \loc{lem:purecoordinates}, and \loc{prop:pureobservable}. \bibitem{XEnter} N.~Cheung, \emph{Landmark X-Enter: Readout closure tests for the obstructed convolution coefficient: a named comparison with the ramp prescription}. \loc{X-Enter.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-Enter}. Surjective-orbit obstruction, selective amplitude tests, conditional relational instrument, and bounded-segment trade-off. \bibitem{XEnterST} \emph{X-Enter-ST: Readout closure of an obstructed convolution coefficient on a prescribed conical host}. Proposed single-file source supplied in the preceding development. Complete output, approximate-orbit estimate, supplied-evaluation instruments, support qualifications, and regular report transport. The supplied status is \loc{sourceStatus: proposed}; no independent registration or verification is inferred. \bibitem{XA} N.~Cheung, \emph{Landmark X-A: Classical covariance of one fixed-regulator switched relational scalar through order \(\kappa^2\)}. \loc{X-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-A}. Separate ramp prescription, plateau identity, and excluded-edge comparison. \bibitem{XAST} \emph{X-A-ST: Supported classical covariance of a fixed-ramp relational scalar}. Version 1 predecessor adaptation record supplied with the request. Retained developing chart, completion, signature ledger, plateau support, and finite Enter interface. No authorship or registration is inferred merely from citation of the analytical parent. \bibitem{IB} N.~Cheung, \emph{I-B: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}. \loc{I-B-v1.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/I-B}. Prescribed conical product and worldsheet, complete clock instrument, calibration, scheduled locality, and target-adapted transport. \bibitem{EnterArticle} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,'' \emph{Frontiers in Physics} \textbf{14} (2026), article 1954943. DOI: \loc{10.3389/fphy.2026.1954943}. Publisher's accepted-manuscript record; accepted September 28, 2026. \url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}. Operational characterization within its stated strategy class; not a classical covariance premise. \bibitem{EnterI} N.~Cheung, \emph{Manuscript I: Finite operational characterization of destructive measurement in the Enter calculus}. \loc{I.tex}, version 2. Enter Calculus Companion. \url{https://enter.grok.me/browse/I}. Full structural hypotheses are retained; no abbreviated characterization is reconstructed. \bibitem{EnterPhil} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Introductory essay, Enter Calculus Companion. \url{https://enter.grok.me/enter}. Interface orientation, not a theorem premise. \bibitem{En09} N.~Cheung, \emph{Realization of Relational Time by Destructive Enter on a Clock Register}. \loc{En09-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En09-A}. Complete instrument, reference extension, consumption, no universal clock return, and separate dynamical hypotheses. \bibitem{En03} N.~Cheung, \emph{Observation Boundaries for Destructive Quantum Readout and External Geometry}. \loc{En03-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En03-A}. Complete prescribed-network interfaces and schedule locality. \bibitem{VIIE} N.~Cheung, \emph{VII-E}. Version 1, supplied manuscript record, Enter Calculus Companion. \url{https://enter.grok.me/browse/VII-E}. Prescribed backwards support and complete observation interfaces. \bibitem{En08A} N.~Cheung, \emph{En08-A}. Version 1, supplied manuscript record, Enter Calculus Companion. \url{https://enter.grok.me/browse/En08-A}. Scheduler consistency versus a separately supplied geometric cone representation. \bibitem{En08B} N.~Cheung, \emph{Resource-controlled operational approximation of null boundary transport}. \loc{En08-B.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En08-B}. Reference-preserving SWAP transport, prescribed geometry and speed, resource accounting, and complete-channel errors. \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. \url{https://arxiv.org/abs/0806.1036}. Smooth Cauchy theory, finite propagation, and Green operators under their geometric hypotheses. \bibitem{BFR} R.~Brunetti, K.~Fredenhagen, and K.~Rejzner, ``Quantum gravity from the point of view of locally covariant quantum field theory,'' \emph{Communications in Mathematical Physics} \textbf{345} (2016), 741--779. \url{https://arxiv.org/abs/1306.1058}. External relational-observable and perturbative-gravity context; not a construction of the frozen convolution. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's manuscript and errata: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. Finite instruments, trace distance, and diamond-norm comparison. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, second edition, Springer, 1990. Smooth multiplication and distributional wavefront calculus. \bibitem{LaterST} \emph{X-B-ST}, \emph{X-C-ST}, \emph{X-D-ST}, \emph{X-E-ST}, \emph{X-F-ST}, \emph{X-G-ST}, and \emph{X-H-ST}. Version-specific predecessor source records supplied with the request, with their analytical parent constructions attributed therein to N.~Cheung. Scalar maps, scalar sharp-width tests, tree identities, free BRST compatibility, relative classical BV, and supplied-report comparisons. Documentary separations only; no common convolution repair or independent registration is inferred. \bibitem{XVIA} N.~Cheung, \emph{XVI-A}. Version 1, supplied manuscript record, Enter Calculus Companion. \url{https://enter.grok.me/browse/XVI-A}. Especially \loc{eq:smoothmult}: fixed-width smooth multiplication does not enlarge wavefront sets, but does not prove a smoothing, cancellation, or quantum-admission theorem. \bibitem{DomainRecords} N.~Cheung, \emph{VII-C}, version 1; \emph{VII-D}, version 1; \emph{XI-B}, version 1; and \emph{XI-G}, version 1; with the version-specific \emph{IX-A-ST} and \emph{IX-D-ST} adaptation records supplied with the request. Enter Calculus Companion. Distinct Wick-extension, finite-part, sharp-domain, free-multiplet, and microcausal admission questions. Comparison only. \bibitem{BoundaryRecords} N.~Cheung, \emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J}, version 1; technical \emph{XV-B}, version 2; \emph{XVI-N}, revision r12; and \emph{XVI-O}, revision o70-sync. Version-qualified supplied manuscript records, Enter Calculus Companion. Historical H2, the narrower internal amplitude at I2, the missing full I3/J2 maps, prepared-insertion obstructions, and the downstream admission exit retain their separate scopes. \bibitem{FiniteRecords} N.~Cheung, \emph{Oai282-A}, \emph{Oai283-A}, \emph{Oai272-A}, \emph{Oai276-A}, \emph{Oai279-A}, \emph{Oai119-A}, and \emph{Oai365-A}. Version-specific descriptions and excerpts supplied with the request, Enter Calculus Companion. Finite operational and signature comparisons only. No analytical or field-acquisition premise is imported. \bibitem{OpenAICatalogue} OpenAI, \emph{Mathematics catalogue}. Supplied documentary descriptions, including Families 282, 362, 376, and 265, recorded separately in the Enter Calculus Companion. No catalogue theorem is used or independently verified in this manuscript. \end{thebibliography} \end{document}