% !TEX program = pdflatex %===================================================================== % X-J-ST.tex % Standalone pdfLaTeX source. % No external figures, source files, or bibliography database. % % No compilation, numerical execution, machine verification, % repository registration, or independent review is asserted. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{mathrsfs} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={X-J-ST: A second charged coefficient and a selective amplitude package on a prescribed conical Enter spacetime}, pdfauthor={Ngo Cheung}, pdfsubject={Classical coefficient extraction, selective distinguishability, complete destructive instruments, and charge-matching boundaries} } \setlength{\emergencystretch}{3em} \setlist{itemsep=3pt,topsep=5pt} \setcounter{tocdepth}{1} \numberwithin{equation}{section} \allowdisplaybreaks \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\HH}{\mathcal H} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\Alg}{\mathcal A} \newcommand{\Fml}{\mathscr F_{\mathrm{ml}}} \newcommand{\Lie}{\mathcal L} \newcommand{\Ptr}{\mathsf P_{\mathrm{tr}}} \newcommand{\Cop}{\mathsf C} \newcommand{\Pgr}{\mathsf P_{\mathrm{gr}}} \newcommand{\Pm}{P_m} \newcommand{\grel}{\mathfrak g_U} \newcommand{\SI}{\mathcal S_{\lambda}^{[3]}} \newcommand{\Stilde}{\widetilde{\mathcal S}_{\lambda}} \newcommand{\QI}{Q_{\lambda}^{[3]}} \newcommand{\Hkin}{\mathcal H_s} \newcommand{\Cfirst}{\mathcal C_1} \newcommand{\Csecond}{\mathcal C_2} \newcommand{\Jcharge}{\mathcal J} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_\diamond} \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}}} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\gh}{gh} \DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\diag}{diag} \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-J-ST: A second charged coefficient\\ and a selective amplitude package}\\[4pt] \large Finite classical response data, positive report separation,\\ and complete destructive instruments on the prescribed I-B host} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct an explicit second charged scalar coefficient from the compact third-order interaction of X-I-ST, without changing its action, the prescribed conical host, or the fixed positive regulator width. Two disjoint third polarizations of the quartic vertex give independently variable compact moments. Their infinitesimal charge matrix has rank two. On a specified local common-flow family their charges agree exactly, whereas a separate compact gauge direction disproves charge matching on the whole relative algebra. The construction therefore supplies a genuine second coefficient for the present selective package, but not the missing universal reference for the earlier convolution observable. For a supplied value \(d\), an explicit destructive amplitude instrument retains a failure outcome of probability \(r\) and has the exact separation \[ \Delta(d)=\frac{1-r}{2} \left|\tanh\!\left(\frac{\nu d}{\sigma}\right)\right|. \] It is positive precisely when the signal is nonzero and the nonfailure probability is positive. A finite calibration instance has the exact gap \(3/16\), with no numerical quadrature. We prove the corresponding finite-dictionary formula, a complete residual-and-reference instrument identity, an encoding-level higher-order visibility test, and explicit error margins. An additional two-coefficient phase instrument is invariant under the specified common charge direction, including all selective outputs and untouched references, but retains periodic aliases and does not become invariant under independent charge directions. The principal contribution is the explicit separation between independent classical coefficient construction, restricted charge matching, and complete finite report distinguishability. This provides a controlled calibration target for later interfaces without treating encoded visibility as continuum acquisition or gauge restoration. The measurement identities are finite-dimensional, and the coefficient calculations are classical. The X-I-ST BV structure is unchanged through its retained order; the three-tree plateau is counted once; the reciprocity and convolution obstructions remain separate. No interacting quantum BRST differential, characteristic quantum coefficient, loop-anomaly cancellation, ultraviolet completion, or spacetime emergence is constructed. \end{abstract} \statementbox{Exact scope of the second coefficient}{% The second coefficient constructed here is a compact scalar response coefficient extracted from the X-I-ST interaction. It is independent of the first moment in this paper and is charged under explicitly specified relative scalar variations. A common charge is realized on a declared local comparison family. It is \emph{not} identified with a second reference having the required charge matching to the original convolution coefficient on its whole frozen gauge algebra. That stronger application remains unconstructed.} \tableofcontents %===================================================================== \section{Purpose, terminology, and the Enter discipline} \label{sec:introduction} %===================================================================== \subsection{What the construction adds} The plan requires a second coefficient, a positive selective total-variation gap, and complete instruments. These are different requirements. A nonzero number chosen to cancel another number is not an independently defined coefficient. A nonzero classical coefficient need not be visible in every encoding. A positive outcome-law gap does not determine the surviving quantum output. Finally, matching charges along one direction does not establish invariance under an entire gauge algebra. The construction below addresses these points separately: \begin{enumerate} \item two compact moments are obtained by finite polarization of the already specified X-I-ST vertex; \item their values and their charge directions are explicitly constructed rather than assumed; \item a fixed amplitude readout gives a positive finite-task gap, with all failure probability included; \item the complete selective maps, references, finite composition, and prescribed transport are proved; \item the precise failure of whole-algebra charge matching is exhibited. \end{enumerate} The scalar moment construction is elementary. The general theory of classical BV and finite-dimensional instruments is not new \cite{FRBV,DaviesLewis,Watrous}. The contribution is the supported assembly and the exact distinction between its positive result and the stronger reference construction that it does not provide. \subsection{Meaning of ``charged'' and ``second''} \begin{definition}[Charge convention] \label{def:charge} For a ghost-number-zero scalar functional \(F[\phi]\), and a compact vector field \(\xi\), define its scalar variation charge by \[ q_F(\phi;\xi) =\left.\frac{\dd}{\dd t}\right|_{t=0} F[\phi\circ F_t^\xi], \] where \(F_t^\xi\) is the flow of \(\xi\). The term \emph{charged} means that this derivative is nonzero on some declared data. It does not mean an electric charge, a conserved Noether charge, or a nonzero ghost number. \end{definition} The two coefficients in this paper are both ghost-number zero. Their BRST variations have ghost number one. The second coefficient is independent of the first moment constructed here. Neither moment is identified with the convolution coefficient of X-A-conv. The order of a coefficient in the classical action and the order of a signal in a finite encoding will also be kept distinct. \subsection{Enter: background and four non-identifications} Cheung's Enter calculus distinguishes a destructive measurement interface from a quantum-return interface \cite{CheungEnter,EnterI}. A measured quantum name is consumed from the accessible quantum output. The output contains a classical record and the unmeasured residual systems. This is an access statement, not a claim that every physical implementing degree of freedom is annihilated. The philosophical account \cite{EnterPhilosophy} keeps four distinctions: \begin{enumerate} \item consuming a measured register does not recover an unknown quantum code; \item a stored classical outcome is not a coherent decoder; \item specifying an input--output channel does not specify a heat, work, or reset model; \item resemblance between constructions does not establish a physical dictionary. \end{enumerate} For the present manuscript these imply, in particular, \[ \text{coefficient definition}\ne \text{continuum acquisition}, \qquad \text{record law}\ne \text{complete instrument}, \] and \[ \text{finite encoding}\ne \text{quantum field product}, \qquad \text{prescribed host}\ne \text{emergent spacetime}. \] The instruments below use specified Born semantics. No characterization theorem is rederived from shortened names for its hypotheses. \subsection{Source and dependency discipline} The supplied X-I-ST source is the classical predecessor \cite{XIST}. X-Enter and X-Enter-ST supply the amplitude, selective-output, and conditional-reference comparisons \cite{XEnter,XEnterST}. X-H-ST and X-G-ST distinguish relative classical BV, free causal BRST, and failure of automatic free-shell descent \cite{XHST,XGST}. All formulas needed for the new coefficient and report theorems are given below. A repository connection is not used as a proof. Conversely, an explicitly stated mathematical hypothesis does not cease to be a hypothesis merely because a documentary connection is called a separation. %===================================================================== \section{The unchanged host and its finite operational reserve} \label{sec:host} %===================================================================== \begin{assumption}[Retained I-B/X-A-ST host] \label{ass:host} Retain the same conical product, supported chart, width, time orientation, and finite operational permissions as in X-I-ST. In regular conical coordinates, \[ \mathcal M^\circ =\RR_T\times(C_\alpha\setminus\{o\})\times\RR_z, \qquad G=-c_{\mathrm h}^2\dd T^2+\dd r^2+ \alpha^2r^2\dd\varphi^2+\dd z^2, \] with the inherited \(0<\alpha<1\) and \(c_{\mathrm h}>0\). The completion defect is the worldsheet \[ \mathcal W=\RR_T\times\{o\}\times\RR_z. \] The independent analytical completion and supported chart are \[ \widehat M=\RR^{1+3},\qquad \eta=\diag(1,-1,-1,-1),\qquad \iota:\mathcal L\longrightarrow\mathcal M^\circ, \] \[ \iota^*G=-\eta,\qquad t=c_{\mathrm h}(T-T_\Sigma). \] The chart is injective on the retained domain, and its laboratory reserve is disjoint from \(\mathcal W\). Retain a nonempty relatively compact plateau \[ \overline U\Subset\mathcal L,\qquad g=\sigma_\varepsilon=1 \quad\text{near }\overline U,\qquad \varepsilon>0 \] with the same positive width as before. Any finite experiment released below must fit the already supplied time, space, register, and control reserve. Register allocation is charged; no enlargement of the chart or change of \(\varepsilon\) is inferred. \end{assumption} This is the prescribed-background meaning of I-B \cite{IB,XAST}. The analytical metric \(\eta\), formal field metrics, and host metric \(G\) remain different objects. \begin{proposition}[Non-global-hyperbolicity boundary] The regular conical product is not globally hyperbolic. The finite constructions in this paper require no global conical Green operator. \end{proposition} \begin{proof} Fix a radial ray and \(r_0>0\). For \(A>r_0/c_{\mathrm h}\), the diamond between \((-A,r_0,\varphi_0,z_0)\) and \((A,r_0,\varphi_0,z_0)\) contains \[ (0,r_0/n,\varphi_0,z_0),\qquad n\ge1. \] The sequence approaches the excluded apex and has no convergent subsequence in the regular product. The diamond is not compact. All coefficient calculations below are compact smooth integrals on \(\widehat M\), represented inside the retained chart. No inverse on \(\mathcal M^\circ\) is used. \end{proof} A compact density is transported by the full change-of-variables formula. Thus, for \(y=\iota(x)\), \[ L(x)\dd^4x =L(\iota^{-1}(y)) \abs{\det D\iota^{-1}(y)}\,\dd^4y. \] Dual antifield densities are transported compatibly. No coordinate Jacobian is silently dropped. %===================================================================== \section{The retained classical interface and its recovery} \label{sec:classical} %===================================================================== \subsection{Fields and field differential} Use the independent fields \[ \Phi=(r_{\mu\nu},\phi,c^\mu,\bar c_\mu,b_\mu), \] of respective parities \(0,0,1,1,0\) and ghost numbers \(0,0,1,-1,0\). The auxiliary field \(b\) remains independent. With fixed smooth backgrounds \(h,k\), write \[ \bar q=\eta+\kappa h+\kappa^2k,\qquad q=\bar q+\kappa r. \] The field differential is \begin{align} sr&=Kc+\kappa\Lie_c(h+r)+\kappa^2\Lie_ck, \label{eq:smetric}\\ s\phi&=\kappa c^\mu\partial_\mu\phi, &sc^\mu&=\kappa c^\nu\partial_\nu c^\mu, \label{eq:sscalar}\\ s\bar c&=b, &sb&=0, &sh&=sk=0, \label{eq:saux} \end{align} where \[ (Kc)_{\mu\nu}=\partial_\mu c_\nu+\partial_\nu c_\mu. \] The rules are left graded derivations commuting with total derivatives. \begin{lemma}[Field nilpotence] \label{lem:snil} The field differential satisfies \(s^2=0\) off shell. Its zeroth-order part is \[ s_0r=Kc,\qquad s_0\phi=s_0c=s_0b=0,\qquad s_0\bar c=b. \] \end{lemma} \begin{proof} For the odd vector field \(c\), set \(w^\mu=c^\nu\partial_\nu c^\mu\). The graded Lie identity is \(\Lie_c^2=\Lie_w\). Since \(sq=\kappa\Lie_cq\) and \(sc=\kappa w\), \[ s^2q=\kappa\Lie_{sc}q-\kappa\Lie_c(sq)=0. \] The fixed background implies \(s^2r=0\). The same identity on scalars gives \(s^2\phi=0\); direct expansion gives \(s^2c=0\). The nonminimal pair is nilpotent by definition. \end{proof} For compact relative directions use \[ \grel=C_c^\infty(U;T\widehat M). \] Compact test ghosts are not identified with nonzero compactly supported homogeneous free ghost solutions. \subsection{The finite action being retained} The following gives a self-contained coefficient specification of the X-I-ST interface. For a formal metric \(p\), let \(J_p=\sqrt{-\det p}\). Use the curvature convention \[ R_{\mu\nu} =\partial_\rho\Gamma^\rho_{\mu\nu} -\partial_\nu\Gamma^\rho_{\mu\rho} +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu} -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}, \] with the metric Christoffel formula, and put \[ \mathscr E[p]=J_p p^{\mu\nu}R_{\mu\nu}[p], \qquad L_p[\phi]=\frac12J_p (p^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-m^2\phi^2). \] Let \[ C_{\bar q}^{\mu}[r] =\bar\nabla_\nu\left( \bar q^{\nu\alpha}\bar q^{\mu\beta}r_{\alpha\beta} -\frac12\bar q^{\nu\mu}\bar q^{\alpha\beta}r_{\alpha\beta} \right), \] \[ \Psi_{\bar q} =\int J_{\bar q}\bar c_\mu (C_{\bar q}^{\mu}[r]-\bar q^{\mu\nu}b_\nu)\dd^4x. \] Define the local variational action \begin{equation} I=-\frac1{2\kappa^2} \int(\mathscr E[q]-\mathscr E[\bar q])\dd^4x +\int L_q[\phi]\dd^4x+s\Psi_{\bar q}. \label{eq:referenceaction} \end{equation} The sole negative-order coefficient is a divergence, \[ -\frac1{2\kappa} (\partial_\mu\partial_\nu r^{\mu\nu} -\Box\tr_\eta r). \] It is removed as a variational class before taking nonnegative coefficients. Adjoin canonical antifields with \[ |\Phi_A^*|=|\Phi^A|+1,\qquad \gh(\Phi_A^*)=-1-\gh(\Phi^A), \] and fix \[ (\Phi^A(x),\Phi_B^*(y)) =\delta^A_B\delta(x-y). \] For a left Hamiltonian differential, use the signed lift \begin{equation} \Hkin=\sum_A(-1)^{|\Phi^A|} \int\Phi_A^*s\Phi^A\,\dd^4x. \label{eq:Hkin} \end{equation} Then \((\Hkin,\Phi^A)=s\Phi^A\). In particular the antighost term is \(-\int\bar c^{*\mu}b_\mu\dd^4x\). Retain exactly the nonnegative coefficients through order three: \[ \mathcal S_{\mathrm{ref}}^{[3]} =\sum_{j=0}^3\kappa^j[\kappa^j](I+\Hkin). \] The order-three gravitational coefficient uses the order-five coefficient of \(\mathscr E[q]\). No inversion of a nilpotent \(\kappa\) is performed. Fix the inherited nonnegative, nonzero profile \[ a\in C_c^\infty(U;\RR),\qquad V_a=\frac1{4!}\int a\phi^4\dd^4x, \] and retain \begin{equation} \SI=\mathcal S_{\mathrm{ref}}^{[3]} +\lambda\kappa^3V_a \quad\text{over}\quad R_3=\RR[\lambda][\kappa]/(\kappa^4). \label{eq:SI} \end{equation} An ordinary polynomial representative of this expression is denoted \(\Stilde\). Local actions are understood modulo horizontal divergences; ordinary products of compact local functionals belong to the classical multilocal algebra \(\Fml(U)\), not necessarily to the single-integral local subspace \cite{FRBV,XHST,XIST}. \begin{proposition}[Retained third-order master identity] \label{prop:retainedCME} Equation~\eqref{eq:SI} satisfies \[ \frac12(\SI,\SI)=0\pmod{\kappa^4}, \qquad \QI F=(\SI,F),\qquad \QI^2=0\pmod{\kappa^4}. \] No term in this paper is added to that action. \end{proposition} \begin{proof} The signed lift obeys \((\Hkin,\Phi^A)=s\Phi^A\) on every generator. Its self-bracket is the cotangent lift of \(s^2\), so it vanishes by Lemma~\ref{lem:snil}. Naturality gives \[ s\mathscr E[q] =\kappa\partial_\mu(c^\mu\mathscr E[q]), \qquad sL_q=\kappa\partial_\mu(c^\mu L_q), \qquad s^2\Psi_{\bar q}=0. \] Thus the full formal reference Hamiltonian has zero master square modulo divergences. Its nonnegative truncation has zero coefficients through order three. The only new cross term from \(V_a\) starts at order four because \(s_0\phi=0\), and \((V_a,V_a)=0\). The odd Jacobi identity proves the differential claim. \end{proof} \subsection{Free-limit statements with the tadpole retained} For clarity, define \[ (\Ptr r)_{\mu\nu} =r_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta r, \qquad (\Cop r)^\mu=\partial_\nu(\Ptr r)^{\nu\mu}. \] The free normalization is \[ A=\frac12G_{\mathrm{lin}},\qquad AK=0,\qquad \Cop^*=-\frac12\Ptr K,\qquad \Cop K=\Box, \] \[ A+\frac12\Cop^*\Cop =\Pgr=-\frac14\Ptr\Box. \] The free action is \[ I_{\mathrm{free}} =\frac12\pair r{Ar} -\frac12\pair\phi{(\Box+m^2)\phi} +\int(b\cdot\Cop r-b^2-\bar c\cdot\Box c)\dd^4x. \] \begin{proposition}[Exact recovery and its necessary qualification] \label{prop:recovery} The classical structure in X-J-ST is exactly \eqref{eq:SI}; hence it agrees with X-I-ST at every retained order and with X-H-ST through order two. At \(\kappa=0\) its field differential is \(s_0\). Its complete BV Hamiltonian has \[ [\kappa^0]I=I_{\mathrm{free}}+\pair r{Ah}. \] It is the free BV Hamiltonian in the centered sector \(h=k=0\), and more generally whenever \(Ah=0\). An arbitrary off-shell background tadpole is not removed by taking \(\kappa=0\). Setting \(\lambda=0\) removes only the additional quartic interaction; it does not remove all reference Einstein--scalar interactions. \end{proposition} \begin{proof} There is no action modification after \eqref{eq:SI}. At order zero, the gravitational quadratic coefficient minus its background-only contribution is \[ \frac12\pair{h+r}{A(h+r)} -\frac12\pair h{Ah} =\frac12\pair r{Ar}+\pair r{Ah}. \] The other zero-order terms are those displayed in \(I_{\mathrm{free}}\). The field-limit and coupling statements follow directly from \eqref{eq:smetric}--\eqref{eq:SI}. \end{proof} The normalized free causal BRST theorem remains the X-G-ST theorem on the declared Minkowski completion \cite{XGST}. No interacting causal carrier is inferred from Proposition~\ref{prop:retainedCME}. A direct non-descent check is given in Section~\ref{sec:diagnostics}. %===================================================================== \section{Section A: two action-derived compact charged coefficients} \label{sec:coefficients} %===================================================================== \subsection{Finite polarization of the retained vertex} The open set \(\{a>0\}\) is nonempty. Choose two disjoint relatively compact balls in this set and nonzero nonnegative functions \[ u_i\in C_c^\infty(U),\qquad i=1,2, \] supported in those balls. Define \begin{equation} M_i=\int a u_i^3\dd^4x>0,\qquad w_i=\frac{a u_i^3}{M_i},\qquad \int w_i\dd^4x=1. \label{eq:weights} \end{equation} Their supports \(K_i=\supp w_i\) are disjoint. All choices occur inside the already fixed plateau. Freeze the calibration scalar \(\phi_{\mathrm{cal}}(x)=x^1\). It is an off-shell smooth function, not an assumed solution. \begin{definition}[The two coefficients] \label{def:coefficients} For \(i=1,2\), set \begin{align} P_i[\phi] &=3!\,[z^3][\kappa^3]\, \partial_\lambda \Stilde[r,\phi+zu_i,c,\bar c,b,\Phi^*], \label{eq:polarization}\\ \mathcal C_i[\phi] &=\frac{P_i[\phi]-P_i[\phi_{\mathrm{cal}}]}{M_i}. \label{eq:Ci} \end{align} Their order-labelled diagnostic insertions are \[ \mathcal O_i(\kappa)=\kappa^3\mathcal C_i. \] These insertions are not added to the action. \end{definition} \begin{theorem}[Explicit second coefficient] \label{thm:coefficient} The definition gives \begin{equation} P_i[\phi]=\int a u_i^3\phi\,\dd^4x,\qquad \mathcal C_i[\phi] =\int w_i(\phi-x^1)\dd^4x. \label{eq:moment} \end{equation} In particular, \(\Csecond\) is a compact, continuous, ghost-number-zero scalar functional defined by finite Taylor extraction from the X-I-ST interaction. The two functionals are linearly independent modulo constants. No distributional product, Green operator, or \(\hbar\)-dependent coefficient is used. \end{theorem} \begin{proof} The coupling derivative and order extraction in \eqref{eq:polarization} leave \(V_a\). The coefficient of \(z^3\) in \[ \frac1{24}\int a(\phi+zu_i)^4\dd^4x \] is \(\int a\phi u_i^3\dd^4x/6\). Multiplication by \(3!\) proves \eqref{eq:moment}. Smooth compact weights give continuity in the compact-open smooth topology. Choose a test variation equal to one near \(K_1\) and zero near \(K_2\); the derivative vector of \((\Cfirst,\Csecond)\) is then \((1,0)\). Interchanging the supports gives \((0,1)\), proving independence. \end{proof} The profile \(u_i\) is a fixed polarization direction. It is not a new dynamical field, compensator, or spurion. The calibration subtraction is fixed classical data and does not create invariance. \subsection{The charges and the formal-order correction} \begin{proposition}[Charge identity] \label{prop:charge} For every \(\xi\in\grel\), \begin{equation} q_i(\phi;\xi) :=q_{\mathcal C_i}(\phi;\xi) =\int w_i\xi^\mu\partial_\mu\phi\,\dd^4x. \label{eq:qi} \end{equation} On scalar functionals the retained BV differential gives \begin{equation} \QI\mathcal C_i =\kappa\Jcharge_i[\phi,c],\qquad \Jcharge_i[\phi,c] =\int w_i c^\mu\partial_\mu\phi\,\dd^4x. \label{eq:BRSTcharge} \end{equation} The polynomial field rule also gives exactly \begin{equation} s\mathcal O_i=\kappa^4\Jcharge_i. \label{eq:weightedcharge} \end{equation} \end{proposition} \begin{proof} Differentiate the scalar pullback and use \eqref{eq:moment}. For the BV action, only the scalar antifield coupling acts on \(\mathcal C_i\); its rule is \eqref{eq:sscalar}. The calibration scalar is not varied. Multiplication by \(\kappa^3\) gives \eqref{eq:weightedcharge}. \end{proof} \statementbox{Charge versus retained formal order}{% The coefficient \(\mathcal C_i\) is charged: its first scalar BRST variation is \(\kappa\Jcharge_i\). The order-labelled insertion \(\kappa^3\mathcal C_i\) has its first variation at order four. It is therefore closed \emph{modulo} \(\kappa^4\), but that truncated zero is not an exact invariance theorem. No real nonzero number is substituted for a nilpotent element of \(R_3\).} \subsection{Exact local charges and independent values} Choose \(t_*>0\) small enough that the compact swept sets \[ K_i+[-2t_*,2t_*]e_1 \] lie in disjoint open subsets of \(U\). Here \(e_1\) is the coordinate vector in the \(x^1\) direction. Choose smooth compact cutoffs \(\chi_i,\eta_i\) with disjoint supports for different \(i\), such that both equal one near these swept sets and \(\supp\eta_i\subset\{\chi_i=1\}\). Set \begin{equation} \xi_i=\eta_i\partial_1,\qquad \xi=\xi_1+\xi_2,\qquad \phi_{\mathbf b} =x^1+b_1\chi_1+b_2\chi_2, \quad \mathbf b=(b_1,b_2)\in\RR^2. \label{eq:witnessfamily} \end{equation} \begin{theorem}[Independent second charge and local matching] \label{thm:localmatching} The family \eqref{eq:witnessfamily} satisfies \begin{equation} \mathcal C_i[\phi_{\mathbf b}]=b_i,\qquad q_i(\phi_{\mathbf b};\xi_j)=\delta_{ij}. \label{eq:chargematrix} \end{equation} For \(|t|0\), set \begin{equation} p(z)=\frac{1+\tanh(z/\sigma)}2,\qquad \ket{e_\sigma(z)} =\sqrt{1-p(z)}\ket0+\sqrt{p(z)}\ket1, \qquad \tau_\sigma(z)=\proj{e_\sigma(z)}. \label{eq:amplitude} \end{equation} The scale \(\sigma\) has the units of \(z\). Let \[ D=\proj1-\proj0,\qquad X=\kb01+\kb10,\qquad t(z)=\tanh(z/\sigma),\qquad s(z)=\sech(z/\sigma). \] Then \begin{equation} \tau_\sigma(z) =\frac12(I+t(z)D+s(z)X). \label{eq:bloch} \end{equation} This is the amplitude family of X-Enter/X-Enter-ST, used here with the separately constructed coefficient \cite{XEnter,XEnterST}. Its input is a supplied finite encoding, not an unknown continuum field. \begin{definition}[Selective instrument convention] A finite instrument \(\{\Phi_y\}_{y\in Y}\) consists of completely positive maps with trace-preserving sum. Its complete recorded output is \[ \mathfrak R_\Phi(Z) =\sum_y\proj y_R\otimes\Phi_y(Z). \] Every \(\Phi_y(Z)\) is retained subnormalized. \end{definition} The output space of each block includes every declared surviving quantum register. A fixed auxiliary preparation may be incorporated into \(\Phi_y\). Reference extension means tensoring the actual maps with an identity, not assuming equality of unrelated reference preparations. Put \[ A_y=\Phi_y(I/2),\qquad B_y=\Phi_y(D/2),\qquad C_y=\Phi_y(X/2). \] Linearity gives the exact formula \begin{equation} \Phi_y(\tau_\sigma(z))=A_y+B_yt(z)+C_ys(z). \label{eq:selective} \end{equation} \begin{theorem}[Selective visibility or erasure] \label{thm:amplitude} For a fixed instrument and fixed \(\sigma\), its complete output on \(\tau_\sigma(z)\) is constant on a nonempty open interval containing zero if and only if \[ B_y=C_y=0\qquad(y\in Y). \] In that case it is constant on the entire declared real-amplitude family. For the classical law alone, the corresponding condition is \[ \tr B_y=\tr C_y=0\qquad(y\in Y). \] \end{theorem} \begin{proof} Differentiate \eqref{eq:selective}: \[ \frac{\dd}{\dd z}\Phi_y(\tau_\sigma(z)) =\frac1{\sigma}(B_ys(z)^2-C_ys(z)t(z)). \] At zero, constancy forces \(B_y=0\). At any nonzero point in the interval, \(s(z)t(z)\ne0\), so \(C_y=0\). The converse follows from the exact formula. Distinct classical blocks cannot cancel. Taking traces gives the outcome-law statement. \end{proof} Erasure on this family is not necessarily a replacement channel on all qubit states. It says nothing about a Bloch direction that never occurs in \eqref{eq:bloch}. \subsection{A higher-order encoding test} The formal interaction label \(\kappa\) is not a real preparation parameter. Introduce a separate real small-signal parameter \(g\) for the following encoding test. \begin{theorem}[Retained-order visibility test] \label{thm:secondorder} Fix the instrument and scale. Let \(N\ge1\) and \(q\ne0\), and suppose, for each fixed real \(\ell\), \[ z(g,\ell)=g^N\ell q+O(g^{N+1}). \] If \[ \mathfrak R_\Phi(\tau_\sigma(z(g,\ell))) -\mathfrak R_\Phi(\tau_\sigma(0)) =O(g^{2N+1}) \] in trace norm for every fixed \(\ell\), then \(B_y=C_y=0\) for all \(y\). In particular, for the third-order-labelled coefficient, \(N=3\), absence of response through the sixth encoding order forces erasure of this whole amplitude family. \end{theorem} \begin{proof} The leading order in each block is \[ g^N\frac{\ell q}{\sigma}B_y+O(g^{N+1}), \] so a nonzero fixed \(\ell\) forces \(B_y=0\). After that exact matrix vanishes, the leading term is \[ -g^{2N}\frac{\ell^2q^2}{2\sigma^2}C_y +O(g^{2N+1}), \] using \(\sech u=1-u^2/2+O(u^4)\). This forces \(C_y=0\). \end{proof} This strengthens the use of the predecessor's second-order diagnostic by explicitly distinguishing the order of the interaction from the order of its encoding response \cite{XEnterST}. It is not a sixth-order field-theoretic calculation. For example, Hadamard followed by destructive readout has \[ \Pr(1\mid z)=\frac{1-\sech(z/\sigma)}2 =\frac{z^2}{4\sigma^2}+O(z^4). \] A vanishing first derivative does not imply erasure. If \(z=g^3d\), this effect first appears at order \(g^6\). \subsection{General outcome-law discrepancy} Write \(\beta_y=\tr B_y\) and \(\gamma_y=\tr C_y\). For every fixed instrument, \begin{equation} \TV(P_z,P_0) =\frac12\sum_y \left|\beta_y\tanh(z/\sigma) +\gamma_y(\sech(z/\sigma)-1)\right|. \label{eq:generalTV} \end{equation} An endpoint equality can occur by cancellation, even when the instrument is not constant on the whole family. Thus positivity will be proved for a named instrument, not inferred for every readout. %===================================================================== \section{Section C: complete destructive instruments and the positive gap} \label{sec:instrument} %===================================================================== \subsection{A failure-retaining amplitude instrument} Let \(Q\cong\CC^2\) be the signal register, and let \(A_{\mathrm{fin}}\) contain every unmeasured finite system. Fix \(0\le r\le1\). On operators on \(Q\otimes A_{\mathrm{fin}}\), define \begin{align} \mathcal M_+(Z) &=(1-r)(\bra1\otimes I)Z(\ket1\otimes I), \label{eq:Mplus}\\ \mathcal M_-(Z) &=(1-r)(\bra0\otimes I)Z(\ket0\otimes I), \label{eq:Mminus}\\ \mathcal M_F(Z)&=r\,\Tr_QZ. \label{eq:MF} \end{align} The complete output is \[ \mathcal M(Z)=\sum_{y\in\{F,+,-\}} \proj y_R\otimes\mathcal M_y(Z). \] The failure branch is retained. Its two internal destructive outcomes are summed, not postselected. No raw failure bit is exported at this declared boundary. \begin{theorem}[Complete signal instrument] \label{thm:signal} The instrument \eqref{eq:Mplus}--\eqref{eq:MF} is complete. It satisfies, for every operator and every untouched reference \(E\), \begin{equation} \Tr_R[(\mathcal M\otimes\id_E)(Z)]=\Tr_QZ. \label{eq:signaldrop} \end{equation} The measured signal name \(Q\) is consumed. The quantum output consists of \(A_{\mathrm{fin}}\), with its conditional reference correlations. \end{theorem} \begin{proof} The Kraus maps are \[ \sqrt{1-r}(\bra1\otimes I),\quad \sqrt{1-r}(\bra0\otimes I),\quad \sqrt r(\bra0\otimes I),\quad \sqrt r(\bra1\otimes I), \] with the last two assigned the common outcome \(F\). Their adjoint products sum to the identity. Their sum after discarding the record is the partial trace over \(Q\). Both statements remain true after tensoring with \(I_E\). The output types give consumption. \end{proof} A finite realization prepares an independent binary failure flag, destructively reads it, and consumes the signal in either branch. Coarse-graining the two failure outcomes is a channel specification. If a larger accessible apparatus exports those outcomes, that larger boundary must be compared separately. \subsection{Exact positive gap} Use the same spectator preparation \(\rho_{A_{\mathrm{fin}}E}\) in both hypotheses, and supply \[ \tau_\sigma(\nu d)_Q\otimes\rho_{A_{\mathrm{fin}}E}, \] where \(\nu\in\RR\) is a fixed classical report gain. \begin{theorem}[Positive selective TV gap] \label{thm:TV} The complete classical law is \[ P_d(F)=r,\qquad P_d(\pm)=\frac{1-r}{2} \left(1\pm\tanh(\nu d/\sigma)\right). \] Its distance from the zero-coefficient law is exactly \begin{equation} \Delta(d):=\TV(P_d,P_0) =\frac{1-r}{2} |\tanh(\nu d/\sigma)|. \label{eq:TV} \end{equation} The complete output states, retaining the matched spectator and reference, also have trace distance \(\Delta(d)\). The gap is positive if and only if \(r<1\) and \(\nu d\ne0\). \end{theorem} \begin{proof} Apply the branch maps to the product input. The spectator factor is the same in every branch. The failure probability is unchanged. The two successful probabilities differ from their zero values by opposite quantities of magnitude \((1-r)|\tanh(\nu d/\sigma)|/2\). Summing their absolute differences and dividing by two gives \eqref{eq:TV}. For the complete outputs, orthogonality of the classical blocks makes the trace norm the sum of the block norms. The common normalized spectator factor has trace norm one, so the same expression results. \end{proof} The theorem compares specified supplied encodings. It does not claim a gap for arbitrary unrelated coefficient-dependent reference preparations, arbitrary faults, or an instrument that erases the signal. One may choose \(\nu=g^3\) to display the interaction-order label in the report calibration. Then, for fixed \(d\), \[ \Delta_g(d) =\frac{1-r}{2}|\tanh(g^3d/\sigma)| =\frac{1-r}{2\sigma}|g|^3|d|+O(|g|^9). \] This is an exact finite-state encoding of a chosen monomial, not an evaluation of \(R_3\) at a nonzero real \(\kappa\), nor a controlled remainder for full field dynamics. \subsection{An exact finite calibration instance} \begin{corollary}[A gap of \(3/16\)] \label{cor:benchmark} Take \[ \nu=1,\qquad r=\frac14,\qquad d=\frac{\sigma}{2}\log3, \] and evaluate \(\Csecond\) on \(\phi_{(0,d)}\). The signal state and laws are \[ \ket{e_\sigma(d)} =\frac12\ket0+\frac{\sqrt3}{2}\ket1, \] \[ (P_d(F),P_d(+),P_d(-)) =\left(\frac14,\frac9{16},\frac3{16}\right), \qquad (P_0(F),P_0(+),P_0(-)) =\left(\frac14,\frac38,\frac38\right). \] Consequently \(\Delta(d)=3/16\). \end{corollary} \begin{proof} The coefficient is \(d\) by Theorem~\ref{thm:localmatching}. Since \(\tanh((\log3)/2)=1/2\), substitution proves every displayed equality. \end{proof} This is an analytical calibration certificate. It does not assert a laboratory realization or a numerical run. \subsection{Finite dictionary and complete references} Let a task register \(J\) have a fixed orthonormal basis \(\ket j\), and let \(\pi_j\) be fixed task probabilities. For each valid task choose supplied values \(d_j\), scales \(\sigma_j>0\), failure probabilities \(r_j\), and gains \(\nu_j\). Prepare \begin{equation} \Omega_{\mathbf d} =\sum_j\pi_j\proj j_J\otimes \tau_{\sigma_j}(\nu_jd_j)_Q \otimes\rho_{A_{\mathrm{fin}}E,j}. \label{eq:dictionaryinput} \end{equation} All \(\pi_j\) and spectator preparations are identical in the zero comparison. The task labels are retained. \begin{theorem}[Positive TV gap on a finite dictionary] \label{thm:dictionaryTV} A controlled use of the signal instrument gives a complete instrument on every \(JQ A_{\mathrm{fin}}\) operator. For the preparations \eqref{eq:dictionaryinput}, \begin{equation} \TV(P_{\mathbf d},P_{\mathbf0}) =\sum_j\pi_j\frac{1-r_j}{2} |\tanh(\nu_jd_j/\sigma_j)|. \label{eq:dictionaryTV} \end{equation} The corresponding complete output trace distance, including \(A_{\mathrm{fin}}E\), is the same. The gap is positive exactly when at least one retained task has \(\pi_j>0\), \(r_j<1\), and \(\nu_jd_j\ne0\). \end{theorem} \begin{proof} Destructively read \(J\), then apply the instrument selected by its classical code. The Kraus maps are tensor products of \(\bra j_J\) with the branch maps in Theorem~\ref{thm:signal}. Completeness follows by summing first the branch maps and then the orthogonal task projectors. For the declared inputs, different \(j\)-blocks are orthogonal. Their trace norms add, and Theorem~\ref{thm:TV} applies in each block. \end{proof} For a non-power-of-two dictionary, pad the task register. An unused code returns an explicit invalid-task flag and still consumes its signal input. No probability is dropped. Retaining the task label matters. If it is forgotten, opposite signed successful-probability changes from two tasks can cancel. \subsection{The complete finite clock} Let \(C\cong\CC^{2^q}\) have a calibrated orthonormal basis \(\{\ket{\tau_k}\}\), and let \(B_C\ket{\tau_k}=\ket k\). For an arbitrary surviving interface \(A\), define \[ D_k=\bra{\tau_k}_C\otimes I_A, \qquad \mathcal T_C(Z) =\sum_k\proj k_{R_C}\otimes D_kZD_k^\dagger. \] \begin{theorem}[Destructive clock and combined instrument] \label{thm:clock} The clock channel is CPTP and, with every reference, \[ \Tr_{R_C}[(\mathcal T_C\otimes\id_E)(Z)] =\Tr_CZ. \] Composing it with the task-and-signal instrument gives a complete finite instrument. If all its records are discarded and the intervening operations act only on the consumed clock, task, and signal factors, the surviving output is their joint partial trace. Clock, task, and measured signal names are not quantum outputs of that instrument. \end{theorem} \begin{proof} The clock Kraus maps are \(\ket k_{R_C}\otimes D_k\). Their adjoint products sum to the identity. Clock-basis expansion proves the partial-trace identity. Composition with Theorem~\ref{thm:dictionaryTV} proves completeness; composing the displayed partial-trace identities proves the final statement. \end{proof} This is the complete destructive clock mechanism of I-B and En09-A \cite{IB,En09}. Conditional reference states may be steered by selecting an outcome; the unconditional reference marginal is unchanged. Calibrated readings are finite history labels unless a separate event-time correspondence is supplied. No claim is made that several stochastic readings all equal the coordinate time of one fixed event. %===================================================================== \section{An optional local relational phase package} \label{sec:relational} %===================================================================== The independent coefficients also instantiate a restricted two-coefficient comparison. This section does not replace the direct amplitude theorem or strengthen local charge matching to whole-algebra matching. For \(\sigma_{\mathrm p}>0\), prepare \begin{equation} \ket{\psi(b_1,b_2)} =\frac{\ket0+e^{i\alpha}\ket1}{\sqrt2} \otimes \frac{\ket0+e^{i\beta}\ket1}{\sqrt2}, \qquad \alpha=\frac{b_1}{\sigma_{\mathrm p}}, \quad \beta=\frac{b_2}{\sigma_{\mathrm p}}. \label{eq:phaseinput} \end{equation} Define \[ \ket{\chi_\pm}=\frac{\ket{01}\pm\ket{10}}{\sqrt2}. \] For input on the two signals and \(A_{\mathrm{fin}}\), put \begin{align} \mathcal N_F(Z) &=(\bra{00}\otimes I)Z(\ket{00}\otimes I) +(\bra{11}\otimes I)Z(\ket{11}\otimes I), \notag\\ \mathcal N_\pm(Z) &=(\bra{\chi_\pm}\otimes I) Z(\ket{\chi_\pm}\otimes I). \label{eq:phaseinstrument} \end{align} \begin{theorem}[Complete common-charge phase instrument] \label{thm:relational} The maps \eqref{eq:phaseinstrument} form a complete destructive instrument. For every \(\gamma\in\RR\), let \[ G_\gamma=\diag(1,e^{i\gamma}) \otimes\diag(1,e^{i\gamma}). \] Then \begin{equation} \mathcal N_y\circ\Ad_{G_\gamma\otimes I} =\mathcal N_y,\qquad y=F,+,-, \label{eq:phasecovariance} \end{equation} on all operators and with arbitrary untouched references. On \eqref{eq:phaseinput}, \[ P(F)=\frac12,\qquad P(\pm)=\frac{1\pm\cos\theta}{4}, \qquad \theta=\frac{b_2-b_1}{\sigma_{\mathrm p}}. \] For fixed \(b_1\), comparison of \(b_2=b_1+d\) with \(b_2=b_1\) has \begin{equation} \TV(P_d,P_0)=\frac{1-\cos(d/\sigma_{\mathrm p})}{4}. \label{eq:phaseTV} \end{equation} It is positive unless \(d/\sigma_{\mathrm p}\in2\pi\mathbb Z\). \end{theorem} \begin{proof} The two computational even-parity projectors and the two odd Bell projectors sum to the identity. Each Kraus bra in the failure sum acquires only a scalar phase under \(G_\gamma\); each \(\chi_\pm\) acquires the common phase \(e^{i\gamma}\). Their selective maps are therefore unchanged. Tensoring the operator identities with a reference identity proves the complete version. The odd-parity amplitudes are \((e^{i\beta}\pm e^{i\alpha})/(2\sqrt2)\). The even-parity total probability is \(1/2\). These give the laws and their exact TV distance. \end{proof} A CNOT followed by Hadamard and destructive readout implements the instrument, with the two even Bell outcomes merged into \(F\). No failure branch is removed \cite{XEnter,XEnterST}. \begin{corollary}[Exact scope of the local relational interpretation] On the common-flow family of Theorem~\ref{thm:localmatching}, the encoded phases both acquire \(t/\sigma_{\mathrm p}\). The complete declared phase instrument is therefore invariant under that comparison. It is not whole-\(\grel\) invariant on the two-coefficient family. \end{corollary} \begin{proof} The common-flow assertion follows from \eqref{eq:commonflow} and \eqref{eq:phasecovariance}. Under \(\xi_2\), only \(b_2\) changes, so the relative phase and, generically, its cosine change. \end{proof} The phase readout sees a cosine, not an exact real number. Evenness and periodicity produce aliases. Furthermore, exporting the raw even Bell outcome would enlarge the output boundary and can reveal \(\alpha+\beta\), destroying collective-phase invariance. The declared coarse failure map must not be confused with a larger logged instrument. %===================================================================== \section{Selectivity, orbit bounds, and non-acquisition} \label{sec:nonacquisition} %===================================================================== \subsection{The encoding-independent orbit statement} \begin{proposition}[Transitive-orbit erasure bound] \label{prop:orbit} Let \(T:\RR\to\mathcal D\) take values in a metric space \((\mathcal D,d)\). If \[ T(z+t)=T(z)\qquad(z,t\in\RR), \] then \(T\) is constant. If instead \[ \sup_{z,t\in\RR}d(T(z+t),T(z))\le\eta, \] then the diameter of its image is at most \(\eta\). \end{proposition} \begin{proof} For any \(z,z'\), take \(t=z'-z\). \end{proof} This argument requires neither a qubit encoding nor informational completeness. It is the elementary transitive-orbit mechanism used in the predecessor \cite{XEnterST}. The earlier convolution coefficient has its own whole-orbit contract. The local scalar flows in Theorem~\ref{thm:localmatching} are not substituted for that contract. A bounded comparison interval does not become a full real gauge orbit merely by being called an orbit. The positive amplitude gap detects dependence on a charged coefficient. It does not make that dependence gauge invariant. \subsection{Positive visibility does not mean informational completeness} \begin{proposition}[Explicit incompleteness] \label{prop:incomplete} The direct amplitude report is not informationally complete for arbitrary qubit inputs. Even the direct and Hadamard report settings together are not informationally complete for arbitrary qubits. The entire present coefficient package is also not informationally complete for the continuum scalar field. \end{proposition} \begin{proof} The direct effects lie in \(\operatorname{span}\{I,D\}\), so, for example, \(\proj+\) and \(\proj-\) have the same direct law. Adding the Hadamard setting gives effects in \(\operatorname{span}\{I,D,X\}\). The distinct states \((I+Y)/2\) and \((I-Y)/2\) still have the same laws. For the field claim choose a nonzero \(v\in C_c^\infty(U\setminus(K_1\cup K_2))\). For every \(\phi\) and every real \(t\), \[ \mathcal C_i[\phi+tv]=\mathcal C_i[\phi]. \] All encodings depending only on these two values therefore agree on infinitely many distinct fields. \end{proof} On the particular one-parameter family \(\tau_\sigma(z)\), the direct probability is injective in \(z\). This narrower statistical identifiability does not contradict Proposition~\ref{prop:incomplete}; it does not reconstruct arbitrary states or fields. \subsection{What is actually measured} \begin{proposition}[The instrument is not a field-acquisition map] \label{prop:noacquisition} The signal instrument measures effects of the supplied finite signal register. It does not acquire \(\mathcal C_2[\phi]\) from an unknown continuum field. \end{proposition} \begin{proof} Its signal effects are \[ E_F=rI,\qquad E_+=(1-r)\proj1,\qquad E_-=(1-r)\proj0. \] Their pullbacks act on \(Q\), not on a field configuration space or a field representation. The coefficient enters only through the separately supplied preparation \(\tau_\sigma(\nu d)\). No field--apparatus coupling, physical representation of the classical functional, or continuum-to-register channel has been defined. \end{proof} The same limitation applies to the phase preparation. Its finite two-qubit realization supplies no coherent acquisition of the two continuum moments. %===================================================================== \section{Errors, finite composition, locality, and transport} \label{sec:composition} %===================================================================== \subsection{Input precision and robust positive margins} Use the conventions \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1, \qquad \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \] \begin{lemma}[Lipschitz amplitude preparation] \label{lem:lipschitz} For all real \(z,z'\), \[ \dtr(\tau_\sigma(z),\tau_\sigma(z')) \le\frac{|z-z'|}{2\sigma}. \] \end{lemma} \begin{proof} The Bloch vector is \((s(z),0,t(z))\), whose derivative has Euclidean norm \[ \frac1{\sigma} \sqrt{s(z)^2t(z)^2+s(z)^4} =\frac{s(z)}{\sigma}\le\frac1{\sigma}. \] Trace distance of qubit states is half the Euclidean distance of their Bloch vectors. Integrate the derivative bound. \end{proof} \begin{theorem}[Robust selective certificate] \label{thm:robust} Suppose the two complete implemented comparisons have trace-distance errors at most \(e_d,e_0\) relative to the complete ideal outputs of Theorem~\ref{thm:TV}. Their actual record gap satisfies \begin{equation} \TV(\widetilde P_d,\widetilde P_0) \ge\max\{0,\Delta(d)-e_d-e_0\}. \label{eq:robustTV} \end{equation} If a positive task value is certified by \(d\in[L,U]\), \(L>0\), and the declared parameter bounds give \[ r\le r_{\max}<1,\qquad |\nu|\ge\nu_{\min}>0,\qquad \sigma\le\sigma_{\max}, \] then \begin{equation} \Delta(d)\ge \frac{1-r_{\max}}2 \tanh\!\left(\frac{\nu_{\min}L}{\sigma_{\max}}\right)>0. \label{eq:enclosuregap} \end{equation} \end{theorem} \begin{proof} Taking a record marginal is trace-distance contractive. Apply the reverse triangle inequality through the two ideal laws. The second result follows from monotonicity of \(\tanh\) on nonnegative arguments. \end{proof} A supplied-value approximation \(|d-\tilde d|\le\epsilon_d\) contributes at most \(|\nu|\epsilon_d/(2\sigma)\) to the preparation distance by Lemma~\ref{lem:lipschitz}. A channel error does not validate a false enclosure. These are complete-interface comparison statements \cite{Watrous,XIST}. They do not assert a finite-sample confidence level without an additional sampling model. \subsection{Finite adaptive composition} \begin{theorem}[Complete finite composition] \label{thm:composition} Any finite adaptive tree of the specified instruments, allowed finite gates, allocations, and compatible transports has complete path maps \(K_h\) satisfying \[ \Phi_y(Z)=\sum_{h:\,\mathrm{out}(h)=y}K_hZK_h^\dagger, \qquad \sum_hK_h^\dagger K_h=I. \] The identities hold with every untouched reference, and every failure history is included. Each destructive use consumes its input names. A later quantum use requires a separately supplied factor, including a new clock when needed. \end{theorem} \begin{proof} Compose the allocation isometries, unitaries, and destructive Kraus maps along each history. Backward induction on the finite tree proves completeness: each terminal identity is pulled back through a complete instrument or an isometry. Tensoring the path identities with the reference identity proves the complete statement. Input and output names enforce consumption. \end{proof} This is the finite complete-history mechanism of the predecessors and the resource-certified programme records \cite{XIST,XHST,En02B,En02C}. Completeness is not itself a distinguishability lower bound. An arbitrary later channel can erase a positive gap. \begin{corollary}[When a gap survives a continuation] \label{cor:gapcontinuation} If a continuation leaves the previously reported task, success/failure, and outcome record unchanged, the final joint output distance is at least the corresponding gap in \eqref{eq:dictionaryTV}. The identity report transport below preserves it. If the record is discarded or overwritten, no such general lower bound follows. \end{corollary} \begin{proof} The retained record is a marginal of the final output, so contraction bounds its distance by the final joint distance. For the negative statement, a replacement channel erases every input dependence. \end{proof} For compatible complete implementations, telescoping gives \begin{equation} \ddia(\widetilde\Gamma,\Gamma) \le\min\left\{1,\, e_{\mathrm{prep}}+e_{\mathrm{table}} +\sum_j e_j+e_{\mathrm{decode}}\right\}. \label{eq:telescoping} \end{equation} Each \(e_j\) concerns the complete fixed interface of one component. The estimate follows from channel contraction and the triangle inequality \cite{Watrous}. It does not prove exact causality of a faulty implementation. \subsection{Scheduled locality} Let \(X_{\mathrm{site}}\) be a finite site set, with finite site algebras. At tick \(s\), let the complete channel factor over a declared partition \(\mathcal B_s\): \[ \Gamma_s=\bigotimes_{B\in\mathcal B_s}\Gamma_{s,B}. \] All records, selectors, control memories, reset wires, and allocated workspace are included. For output 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 locality] \label{thm:locality} The network satisfies \[ (\Gamma_{s+1}^*\circ\cdots\circ\Gamma_t^*)(\Alg(Y)) \subseteq\Alg(N_{s,t}(Y)). \] Equal joint reductions on the ancestor input and an untouched reference give equal joint output reductions. A deterministic intervention outside the complete backwards region cannot change that output. If every dependency edge has a separately certified future-causal regular host route, scheduled ancestry is contained in the prescribed host causal relation. \end{theorem} \begin{proof} An excluded block acts on an identity factor in the pulled-back observable. Its adjoint is unital. A meeting block enlarges support only within itself. Backward induction proves the inclusion. Testing all output observables, including reference observables, proves the reduction and intervention claims. Concatenation of the supplied causal routes proves the geometric statement. \end{proof} This is the I-B/En03-A/VII-E support mechanism \cite{IB,En03,VIIE}. Scheduled ancestry is an upper bound, not necessarily minimal influence or an entire physical causal past. Scheduler consistency alone does not reconstruct a Lorentz cone \cite{En08A}. \subsection{Prescribed regular transport} Choose a straight developed spatial segment of length \(\ell>0\) inside the existing regular tube, and a departure time \(t_{\mathrm d}\) after readout and its declared buffers. For \(n\ge1\), place equal payload registers \(A_0,\ldots,A_n\) at equally spaced stations. The payload includes the finite report and every surviving quantum output required by the comparison. Initialize \(A_1,\ldots,A_n\) in independent blanks \(\omega\), and apply successive register SWAPs over developed durations \(\ell/n\). \begin{theorem}[Regular reference-preserving route] \label{thm:route} For every \(\rho_{A_0E}\), after hop \(j\) the state is \[ \rho_{A_jE}\otimes \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}. \] The logical payload channel is the identity including all references. Arrival is at \[ t_{\mathrm d}+\ell,\qquad T_{\mathrm d}+\ell/c_{\mathrm h}. \] The entire route lies inside the retained regular tube. \end{theorem} \begin{proof} A SWAP transfers the complete payload, including its reference correlations, into the next independent blank and leaves the previous factor blank. Induction proves the state formula. Equal station spacing and developed duration give the declared null cross-port displacement. The time calibration gives the host arrival time. \end{proof} This is the target-adapted I-B/En08-B route \cite{IB,En08B}. The SWAP permission is stroboscopic finite control, not a derived microscopic relativistic Hamiltonian. No exact-attainment theorem on an arbitrary previously fixed lattice is asserted. The route preserves existing correlations; it does not establish gravity-generated entanglement. \subsection{A finite resource ledger} A \(p\)-qubit payload uses \(n\) register SWAPs and \((n+1)p\) reserved route qubits. The experiment additionally charges: \begin{itemize} \item every fresh clock and task register; \item one supplied amplitude signal per direct trial, or two supplied phase signals per relational trial; \item the failure flag or equivalent finite ancilla; \item every classical report reserve and lookup workspace; \item source descriptions, numerical enclosures, supplier storage, resets, durations, and buffers. \end{itemize} Untouched external references are listed separately; their omission from laboratory operations does not remove them from the complete comparison. No repeated destructive trial is obtained by reusing a consumed quantum name. Arbitrarily many trials require an independently declared sequence of supplies and a resource ledger for that sequence. %===================================================================== \section{Preserved diagnostics and explicit separations} \label{sec:diagnostics} %===================================================================== \subsection{The earlier trees are not counted again} The X-F-ST packet remains \[ \kappa^2 \bigl(\mathcal T_{\mathrm{mat}} +\mathcal T_{\mathrm{mix}} +\mathcal T_{\mathrm{self}}\bigr). \] Its source normalization and its admission hypotheses remain those of its source record \cite{XFST}. \begin{proposition}[No action or lower-order interference] \label{prop:noninterference} X-J-ST modifies neither the classical action \eqref{eq:SI} nor its equations. The diagnostic insertions \(\kappa^3\mathcal C_i\) are not extra action vertices or extra trees. Consequently the retained tree packet, free Hessian, free propagators, and predecessor free fish and null-wavefront conditionals are unchanged. \end{proposition} \begin{proof} The only added constructions are coefficient extraction, specified comparison data, and finite report channels. None changes an action coefficient or a free numerical distribution. \end{proof} A mathematical instrument on the supplied encoding does not construct a graph containing quantum fields. \subsection{The X-I-ST next residual is not canceled} Let \[ \mathcal B_a =\frac16\int a\phi^3c^\mu\partial_\mu\phi\,\dd^4x. \] For the specified polynomial representative, \begin{equation} \frac12(\Stilde,\Stilde) -\frac12(\widetilde{\mathcal S}_0, \widetilde{\mathcal S}_0) =\lambda\kappa^4\mathcal B_a. \label{eq:oldresidual} \end{equation} Indeed, the additional vertex has no antifields, its self-bracket is zero, and its bracket with the kinematic lift is \(sV_a=\kappa\mathcal B_a\). No report channel modifies \eqref{eq:oldresidual}. A supplied real coefficient of this odd residual is a possible finite task, not a physical ghost observable. \subsection{Free-shell non-descent remains} For \(P_m=\Box+m^2\), \[ [P_m,c^\mu\partial_\mu]\phi =(\Box c^\mu)\partial_\mu\phi +2(\partial^\nu c^\mu) \partial_\nu\partial_\mu\phi. \] With \(p>0\), \(\omega=\sqrt{p^2+m^2}\), choose \[ \phi_0=\cos(\omega t-px^1),\qquad c=\vartheta t\,\partial_1. \] Then \(P_m\phi_0=\Box c=0\), but \[ P_m(c^\mu\partial_\mu\phi_0) =2\vartheta p\omega\cos(\omega t-px^1)\ne0. \] Thus the natural positive-order scalar rule does not preserve the unchanged free equation ideal. A compact relative ghost equal to this coefficient near a compact test supplies a local comparison, not a global compact free ghost solution \cite{XHST,XIST}. X-J-ST supplies no retarded interacting BV transport that would remove this distinction. \subsection{Reciprocity remains a classical diagnostic} The frozen X-G/X-G-ST source-switched metric operator retains its separate witness \[ D_{\mathrm{recip}} =\frac{5M_\chi}{8} \int_0^\varepsilon \zeta(t)(\sigma_\varepsilon''(t))^2\dd t>0 \] under its original support and free-input hypotheses \cite{XG,XGST,XIST}. Neither coefficient extraction nor a finite report changes that operator. The number is not \(\mathcal J_2\), not a master residual, and not a loop anomaly. Its exact source-preserving variational exclusion is not a universal no-go for every changed classical prescription. \subsection{The convolution obstruction is unrepaired} Retain the frozen convolution witness of X-A-conv/X-A-conv-ST: \[ c_\varepsilon =\frac1{2N}\int\varrho_\varepsilon(z) \int(\psi(x)-\psi(x-z))^2\dd^4x\,\dd^4z>0. \] Its kernel, gauge algebra, Cauchy prescription, and support certificates are unchanged \cite{XConv,XConvST}. There is no identification \[ \mathcal C_1=A_{\mathrm{conv}} \quad\text{or}\quad q_2=q_{\mathrm{conv}} \] in this paper. In particular, the present construction does not discharge X-Enter-ST's missing convolution reference and whole-algebra matching hypothesis. This refusal is not a nonexistence theorem for every future reference. It states the absence of the needed same-model identification here. \subsection{Characteristic, quantum, and emergence boundaries} The supplied XIV-H/I/J and XV-B records concern a different characteristic composition and reconstruction problem \cite{CharacteristicRecords}. A finite classical moment and its TV gap do not provide the missing full map, section, remainder, transverse estimates, contact compatibility, output wavefront theorem, or admitted characteristic coefficient. Accordingly, no downstream quantum-admission or interacting quantum Ward exit is opened. The existence of technical manuscripts with those identifiers is not denied. The prepared-insertion restrictions of XVI-N and the prescription-qualified XVI-O record are not changed \cite{PreparedRecords}. No blanket statement across other prescriptions is inferred. The following remain explicit non-claims: \begin{enumerate} \item no full interacting quantum BRST differential; \item no characteristic quantum coefficient or quantum field product; \item no loop-anomaly cancellation or ultraviolet completion; \item no repair of the classical convolution obstruction; \item no continuum acquisition of either coefficient; \item no informational completeness of an orbit, a qubit, or a continuum field; \item no host derivation, Einstein dynamics from records, or spacetime emergence. \end{enumerate} %===================================================================== \section{Release and prospective significance} \label{sec:release} %===================================================================== \begin{definition}[Local Grade S for X-J-ST] \label{def:grade} The release consists of: \begin{enumerate} \item the retained prescribed host, completion, positive width, and finite operational reserve; \item the two explicit compact coefficient definitions, their charge witnesses, and their matching boundary; \item a nonempty finite dictionary of supplied real tasks; \item complete destructive instruments including failures, surviving systems, and untouched references; \item a positive scoped TV theorem and its error margins; \item finite adaptive composition, scheduled locality, a regular reference-preserving route, and resource accounting. \end{enumerate} Continuum field acquisition is not included in this grade. \end{definition} \begin{theorem}[X-J-ST mathematical release] \label{thm:release} Under Assumption~\ref{ass:host}: \begin{enumerate}[label=\textup{(\roman*)}] \item the second charged coefficient is the compact ghost-number-zero scalar \[ \Csecond[\phi]=\int w_2(\phi-x^1)\dd^4x, \] obtained by the finite action polarization \eqref{eq:polarization}; \item its value is independent of the first moment, and the charge matrix is the identity on the declared two-parameter comparison family; \item a local common charge is realized exactly, while whole-relative-algebra matching fails on an explicit independent direction; \item the selective amplitude instrument has the exact positive gap \eqref{eq:TV}, with finite-dictionary formula \eqref{eq:dictionaryTV} and the exact \(3/16\) calibration instance; \item the clock, signal, task, and optional phase instruments are complete on arbitrary finite inputs and all untouched references; \item the higher-order encoding test and the robust gap margins hold with their stated hypotheses; \item finite adaptive composition, record-retaining continuations, scheduled locality, and regular report transport have the proved scopes; \item the X-I-ST action is unchanged, its free-limit qualification remains, and the earlier tree and obstruction diagnostics are neither recounted nor repaired. \end{enumerate} This is a local Grade S selective coefficient package, not a whole-algebra invariant convolution reference, not quantum BRST, and not a characteristic coefficient. \end{theorem} \begin{proof} Theorem~\ref{thm:coefficient} proves (i). Theorem~\ref{thm:localmatching} proves (ii) and the positive part of (iii); Theorem~\ref{thm:wholealgebra} proves its boundary. Theorems~\ref{thm:TV} and \ref{thm:dictionaryTV}, with Corollary~\ref{cor:benchmark}, prove (iv). Theorems~\ref{thm:signal}, \ref{thm:clock}, and \ref{thm:relational} prove (v). Theorems~\ref{thm:secondorder} and \ref{thm:robust} prove (vi). Theorem~\ref{thm:composition}, Corollary~\ref{cor:gapcontinuation}, and Theorems~\ref{thm:locality}--\ref{thm:route} prove (vii). Propositions~\ref{prop:recovery} and \ref{prop:noninterference}, together with Section~\ref{sec:diagnostics}, prove (viii). \end{proof} \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.33\linewidth}p{.22\linewidth}X@{}} \toprule Object & Verdict & Boundary\\ \midrule Second compact scalar coefficient & Constructed & Finite polarization of the retained vertex\\ Independent values and charges & Constructed & Specified off-shell data\\ Local common-charge comparison & Constructed & Not whole-algebra matching\\ Positive amplitude TV gap & Constructed & Supplied encoding and fixed interface\\ Complete residual/reference instrument & Constructed & All declared branches retained\\ Continuum field acquisition & Not constructed & No field-to-register channel\\ Whole convolution reference & Not constructed & No same-model charge identification\\ Free/classical recovery & Unchanged & Centered free BV comparison qualified\\ Three-tree plateau & Retained once & No extra tree or action term\\ Convolution gap & Unrepaired & Frozen positive witness retained\\ Quantum characteristic admission & Not established & Separate analytic gates untouched\\ Spacetime emergence & Not constructed & Metric and causal dictionary prescribed\\ \bottomrule \end{tabularx} \end{center} \subsection{What is new within this continuation} The positive content is not merely a new label for an assumed reference. The second coefficient has an explicit action-derived formula, independent scalar data, a nonzero compact charge, and a finite report experiment with an exact distinguishability margin. Equally important, the rank-two charge calculation shows exactly why the local positive construction does not become whole-algebra closure. The distinction survives the complete-instrument audit; it cannot be hidden by selecting a favorable outcome. The prospective usefulness is as a calibration and comparison module: later work can test a concrete second coefficient, a finite preparation, and a complete record interface without claiming that those operations have acquired continuum data or defined a quantum product. No general priority claim is made for polarization, hyperbolic-tangent encodings, instruments, or the classical BV method. \subsection{What a stronger successor must supply} A successor claiming the missing convolution application must identify: \begin{enumerate} \item the first and second coefficients in the same classical model and on the same domain; \item the complete allowed gauge algebra and both charge maps; \item the required matching on that whole algebra; \item any claimed equation-space descent; \item the continuum-to-apparatus preparation or acquisition map, if physical field measurement is claimed; \item the complete instrument comparison, allowed changes, errors, and resource budgets. \end{enumerate} None is supplied by a scalar TV gap alone. %===================================================================== \appendix \section{Debugging and quantifier audit} \label{app:debug} %===================================================================== \begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}} \toprule Potential overstatement & Correction used here\\ \midrule \endfirsthead \toprule Potential overstatement & Correction used here\\ \midrule \endhead ``Second'' means the missing convolution reference & The new second coefficient belongs to a named pair of action-derived moments. No convolution identification is asserted.\\[4pt] A compensating number is an independent coefficient & Independent variations and the rank-two charge matrix are explicitly exhibited.\\[4pt] Ghost number zero means uncharged & Charge is defined as a specified scalar variation, not ghost number or a conserved electrical charge.\\[4pt] An order-three insertion has a visible charge in \(\RR[\kappa]/(\kappa^4)\) & Its first variation is order four. The coefficient is charged, while the insertion is closed only at the retained truncated order.\\[4pt] Formal \(\kappa\) can be replaced by a real signal gain & The encoding uses a separate real gain. No nonzero real specialization of a nilpotent element is used.\\[4pt] One matching direction proves full gauge invariance & The independent direction \(\xi_2\) changes the purported relational difference.\\[4pt] Scalar comparison flows prove interacting solution transport & They are off-shell scalar comparisons. No Euler solution or free-shell identification is claimed.\\[4pt] A positive classical value gives a positive gap for every instrument & Visibility is proved for explicit named instruments. Erasing instruments remain possible.\\[4pt] A first-derivative test detects all dependence & The Hadamard example has a nonzero quadratic response. The encoding-level higher-order test is proved.\\[4pt] A successful branch is the complete experiment & Failure probability and every declared residual are retained.\\[4pt] A task label may reveal the tested hypothesis & The compared labels and all fixed control data are identical; the coefficient enters the amplitude.\\[4pt] A phase difference is an exact real-number decoder & The phase instrument sees a cosine and has aliases.\\[4pt] A coarse invariant failure outcome guarantees invariance of a raw failure log & Exporting the raw Bell failure distinction enlarges the boundary and can break the invariance.\\[4pt] An injective one-parameter response is tomography & The effects omit Bloch directions, and the field moments have an infinite-dimensional kernel.\\[4pt] Reference preservation means fixed conditional reference states & A conditional reference may be steered. The unconditional channel acts as the reference identity.\\[4pt] Finite adaptive completeness preserves every gap & A later replacement channel erases it. A retained prefix record gives the stated lower bound.\\[4pt] A channel error validates a numerical coefficient & An enclosure needs its own analytical or numerical justification.\\[4pt] The clock may be reused after destruction & Every new quantum input needs a separately supplied factor and resource charge.\\[4pt] A clock reading creates or consumes geometric time & Only a finite encoding is consumed. The prescribed time axis remains.\\[4pt] Scheduled ancestry reconstructs the metric cone & Causal embedding is a separate certificate on the unchanged host.\\[4pt] A finite route derives a microscopic interaction & The route uses declared stroboscopic finite controls.\\[4pt] X-J-ST repairs the fourth-order master residual & Its action is unchanged, so the residual is unchanged.\\[4pt] Setting \(\lambda=0\) gives the full free theory & It removes the added quartic vertex, not all reference interactions.\\[4pt] The centered sector is the only possible free-BV recovery condition & It is sufficient. The displayed tadpole also vanishes whenever \(Ah=0\).\\[4pt] A new source file is an executed registration & The companion metadata below is proposed only.\\ \bottomrule \end{longtable} %===================================================================== \section{Documentary comparisons not used as proof transfers} \label{app:documentary} %===================================================================== The characteristic records XIV-H, XIV-I, and XIV-J distinguish a historical internal-amplitude stop, the narrower I2 construction, and missing full characteristic maps at I3/J2. Technical XV-B interfaces do not themselves admit the full coefficient. Their version-specific statements are not advanced by the finite moment construction here \cite{CharacteristicRecords}. Likewise, the prepared-source and kernel-counterterm records have their own domains and prescription qualifications \cite{PreparedRecords}. This manuscript neither discharges XVI-N nor updates XVI-O's stated \loc{NOT_CLOSED} O.1 record. It makes no extrapolation to other versions or prescriptions. The supplied descriptions of OpenAI Families 282, 362, and 376 are documentary comparisons of finite-interface and finite-certificate styles \cite{OpenAIRecords}. No such paper is a premise of the coefficient construction, the TV identities, the instruments, or recovery. No catalogue theorem, reported formalization, or family-wide verification is asserted here. A finite dictionary is a finite collection of tasks, not a replacement for the infinite-dimensional field domain or for a missing distributional composition theorem. %===================================================================== \section{Proposed companion integration record} \label{app:integration} %===================================================================== This appendix is a proposal, not an executed repository transaction. It does not register, hash, compile, or test the source. \paragraph{Proposed routing note.} X-J-ST constructs a second charged coefficient and a selective amplitude package on the unchanged prescribed I-B/X-A-ST host. It proves a positive TV gap on finite dictionary inputs and complete destructive instruments. The result is not quantum BRST, not a characteristic coefficient, not continuum acquisition, and not a premise of an earlier sheet. The convolution obstruction is unrepaired. The trees are retained once. Common-direction charge matching is local and does not identify a reference for the whole convolution gauge algebra. \begin{verbatim} identifier: X-J-ST sourceStatus: proposed placement: after X-I-ST, before XI-A track: ramp problems: - relational - products nearUv: true classicalObject: type: ghost-number-zero compact scalar moment source: finite third polarization of X-I-ST V_a formalOrderLabel: kappa^3 firstWeightedChargeOrder: kappa^4 actionModified: false sameModelConvolutionReference: not constructed selectivePackage: finiteSuppliedDictionary: true positiveTVGap: proved for specified encodings failureOutcomeRetained: true untouchedReferencesIncluded: true informationalCompleteness: false continuumAcquisition: not constructed wholeGaugeMatching: not established localCommonDirectionMatching: constructed obligations: second-coefficient-not-quantum-product: Classical action polarization is not a quantum product. selective-TV-not-complete-orbit: A positive gap is not whole-orbit reconstruction. residual-instrument-not-acquisition: Supplied finite encodings are not acquired continuum fields. convolution-gap-unrepaired: No same-model whole-algebra reference is identified. local-match-not-whole-algebra: The independent charge direction remains visible. formal-order-not-real-gain: Encoding parameters do not specialize a nilpotent coupling. documentaryConnections: X-Enter-ST: selective and matching comparison; not a premise X-I-ST: explicit retained action interface X-H-ST: relative BV and complete-report comparison X-G-ST: unchanged free causal normalization X-F-ST: tree packet retained once X-A-conv-ST: independent unrepaired gap I-B: prescribed host and finite permissions En-series: complete clock, schedule, and route comparisons OpenAI: documentary separation only; not a premise timeLandmark: proposed: true reason: a complete finite destructive clock is explicitly defined emergenceStatus: not constructed includeInEmergenceSources: false registryEditsExecuted: false repositoryTestsExecuted: false compilationAsserted: false numericalExecutionAsserted: false machineVerificationAsserted: false independentExecutionAsserted: false \end{verbatim} Actual integration must use the repository's real schema, current source identifiers, and actual connection and index tests. No expected connection count, array-length increment, passing test result, or completed freeze is invented before that transaction occurs. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Analytical scope.} The new coefficients are compact classical response moments on smooth off-shell scalar data. Their independence and local charges are proved explicitly. No new interacting action, fourth-order completion, Koszul--Tate resolution, physical cohomology, or interacting free-carrier transport is claimed. \paragraph{Operational scope.} The finite instruments are specified on supplied register inputs, all declared selective outputs, and untouched references. They do not acquire the input coefficient from a continuum field or derive a microscopic detector. \paragraph{Attribution.} The classical vertex and BV interface are those of the supplied X-I-ST source. The amplitude and conditional phase-instrument mechanisms are credited to X-Enter/X-Enter-ST. The present continuation constructs the explicit moment pair, its independent charges and local matching boundary, and the failure-retaining finite-dictionary assembly. \paragraph{Verification.} No successful \loc{pdflatex} execution, numerical quadrature, symbolic-program execution, proof-assistant verification, hardware implementation, repository registration, independent execution, independent referee approval, or journal acceptance is asserted. The source is standalone and intended for pdfLaTeX; the mathematical claims are the written proofs above. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XIST} N.~Cheung, \emph{X-I-ST: A supported finite-order classical BV deformation on a prescribed conical Enter spacetime}. Supplied source \loc{X-I-ST.tex}, with author field Ngo Cheung. Third-order compact scalar vertex, polynomial-lift residual, centered free-BV comparison, and complete finite supplied-report instruments. The supplied source is the version used here; no independent compilation or registration is inferred. \bibitem{XEnterST} N.~Cheung, \emph{X-Enter-ST: Readout closure of an obstructed convolution coefficient on a prescribed conical host}. \loc{X-Enter-ST.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-Enter-ST}. Especially \loc{thm:amplitude}, \loc{thm:secondorder}, \loc{eq:TV}, \loc{prop:matching}, \loc{thm:relational}, \loc{thm:clock}, \loc{thm:locality}, and \loc{thm:route}. The second convolution reference is an additional hypothesis there, not an input silently supplied here. \bibitem{XEnter} N.~Cheung, \emph{X-Enter}. \loc{X-Enter.tex}, version 1. Version-qualified passages supplied with the request. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-Enter}. Selective amplitude visibility, retained-order tests, and conditional two-coefficient readout. \bibitem{XHST} N.~Cheung, \emph{X-H-ST: Relative classical BV structure and complete instruments for supplied residual reports}. \loc{X-H-ST.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-H-ST}. Signed classical lift, non-descent, background tadpole, and complete finite report interfaces. \bibitem{XGST} N.~Cheung, \emph{X-G-ST: Free causal BRST compatibility and a variational upgrade obstruction}. \loc{X-G-ST.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-G-ST}. Free auxiliary--ghost normalization, causal BRST, positive reciprocity diagnostic, and scoped fish and wavefront statements. \bibitem{XG} N.~Cheung, \emph{Landmark X-G: The normalized BRST multiplet and the one-loop background gate: proved compatibility conditions, an exact-upgrade obstruction, and the remaining nonlocal extension problem}. \loc{X-G.tex}, version 1. Version-qualified source record supplied with the request. \url{https://enter.grok.me/browse/X-G}. \bibitem{XFST} N.~Cheung, \emph{X-F-ST: Three fixed-width retarded trees and the background identity}. \loc{X-F-ST.tex}, version 1. Version-qualified supplied manuscript record. \url{https://enter.grok.me/browse/X-F-ST}. The retained three-tree packet is a comparison object, not reconstructed by the report instrument. \bibitem{XConv} N.~Cheung, \emph{X-A-conv}. \loc{X-A-conv.tex}, version 1. Version-qualified supplied manuscript record. \url{https://enter.grok.me/browse/X-A-conv}. Frozen classical convolution prescription and its positive covariance defect. \bibitem{XConvST} N.~Cheung, \emph{X-A-conv-ST}. Version 1, supplied manuscript record. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-A-conv-ST}. Independent convolution obstruction, support certificates, and complete finite report locality. \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 worldsheet, finite complete clock, calibration, scheduled locality, and regular transport. \bibitem{XAST} N.~Cheung, \emph{X-A-ST}. Version 1, predecessor source record supplied with the request. \url{https://enter.grok.me/browse/X-A-ST}. Supported completion chart, fixed positive width, signature ledger, and plateau contract. \bibitem{CheungEnter} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,'' \emph{Frontiers in Physics}, volume 14 (2026), doi:10.3389/fphy.2026.1954943. \url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}. Publisher advance article record; final pagination is not assumed. Specified Born semantics and the separate class-relative operational characterization are not identified. \bibitem{EnterI} N.~Cheung, \emph{Manuscript I}. \loc{I.tex}, version 2. Version-qualified supplied manuscript record. \url{https://enter.grok.me/browse/I}. Finite operational characterization of destructive measurement inside its stated structural class; not continuum field acquisition. \bibitem{EnterPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Introductory essay, Enter Calculus Companion. \url{https://enter.grok.me/enter}. Interpretive separation of readout, recovery, thermodynamic specification, and physical identification. \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 destructive-clock channel and arbitrary reference extension; dynamical interpretation has additional hypotheses. \bibitem{En03} N.~Cheung, \emph{En03-A}. \loc{En03-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En03-A}. Complete scheduled block interfaces and exact backwards-support theorem. \bibitem{VIIE} N.~Cheung, \emph{VII-E}. \loc{VII-E.tex}, version 1. Version-qualified supplied manuscript record. \url{https://enter.grok.me/browse/VII-E}. Observation boundaries and prescribed schedule locality. \bibitem{En08A} N.~Cheung, \emph{En08-A}. \loc{En08-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En08-A}. Scheduler consistency versus additional Lorentz-cone representation hypotheses. \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}. Finite reference-preserving transport relative to prescribed geometric and operational data. \bibitem{En02B} N.~Cheung, \emph{En02-B}. Version 1, supplied manuscript excerpts. Enter Calculus Companion. Finite complete-history channel soundness and live-name resource certificates. \bibitem{En02C} N.~Cheung, \emph{En02-C}. Version 1, supplied manuscript excerpts. Enter Calculus Companion. Reference-preserving sequential soundness and explicit storage ledgers. No radiation recovery task is imported here. \bibitem{FRBV} K.~Fredenhagen and K.~Rejzner, ``Batalin--Vilkovisky formalism in the functional approach to classical field theory,'' \emph{Communications in Mathematical Physics} \textbf{314} (2012), 93--127. \url{https://arxiv.org/abs/1101.5112}. Classical functional BV framework. Its global hypotheses are not transferred to the regular conical product. \bibitem{DaviesLewis} E.~B.~Davies and J.~T.~Lewis, ``An operational approach to quantum probability,'' \emph{Communications in Mathematical Physics} \textbf{17} (1970), 239--260. doi:10.1007/BF01647093. \url{https://link.springer.com/article/10.1007/BF01647093}. External instrument framework; the finite maps used in this paper are proved directly. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author manuscript and errata: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. Finite channels, complete references, trace distance, and diamond-norm comparison. \bibitem{CharacteristicRecords} N.~Cheung, \emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J}, version 1, and technical \emph{XV-B}, version 2. Version-qualified manuscript passages supplied with the request. Internal-amplitude constructions, reconstruction descent, partial characteristic references, and the unconstructed full-map/admission boundaries. \bibitem{PreparedRecords} N.~Cheung, \emph{XVI-N}, revision r12, and \emph{XVI-O}, revision o70-sync. Version-qualified passages supplied with the request. Prepared-insertion and prescription-qualified kernel boundaries only; no calculation from those programmes is performed here. \bibitem{OpenAIRecords} OpenAI, \emph{Mathematics catalogue and finite-interface manuscript descriptions}. Supplied documentary descriptions, including Families 282, 362, and 376. No family theorem or reported formalization is used as a premise or independently verified here. \end{thebibliography} \end{document}