\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-L-ST: Restricted classical BRST domains, exact free recovery, and complete residual-report instruments}, pdfsubject={Classical graded restriction, free causal BRST recovery, descent obstructions, and finite destructive instruments} } \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} \newtheorem{example}[theorem]{Example} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\NN}{\mathbb N} \newcommand{\HH}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\gh}{\operatorname{gh}} \newcommand{\im}{\operatorname{im}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\End}{\operatorname{End}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_\diamond} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\Ptr}{\mathsf P_{\mathrm{tr}}} \newcommand{\Cdiv}{\mathsf C} \newcommand{\Pgr}{\mathsf P_{\mathrm{gr}}} \newcommand{\Pm}{P_m} \newcommand{\ER}{\mathsf E_0^{\mathrm R}} \newcommand{\EA}{\mathsf E_0^{\mathrm A}} \newcommand{\Afree}{\mathfrak A_0} \newcommand{\Pfield}{\mathscr P} \newcommand{\Fml}{\mathscr F_{\mathrm{ml}}} \newcommand{\BValg}{\mathscr B} \newcommand{\Rmaster}{\mathcal R_{\mathrm{cl}}} \newcommand{\Rrec}{\mathscr R_{\mathrm{rec}}} \newcommand{\QBV}{\mathcal Q} \newcommand{\Mclock}{\mathcal M} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\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{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{X-L-ST: Restricted classical BRST domains,\\ exact free recovery, and complete residual reports}\\[5pt] \large A conditional graded package on the unchanged\\ I-B conical Enter spacetime} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct a restricted classical graded continuation on the fixed I-B/X-A-ST host, separating nonlinear field nilpotence, action compatibility, free-shell descent, and finite reporting. The normalized zero-coupling field rules, gauge fermion, auxiliary--ghost Green matrix, and graded causal relations are re-established explicitly. Free causal recovery is an identity of the resulting free presentation, not a claim that a classical product has become an interacting quantum product. The principal restricted construction is the centralizer of the classical master residual in the local variational BV algebra. Its Hamiltonian operator is nilpotent on that domain even when the master residual is nonzero. This gives a precise, nonempty restricted classical complex without asserting unrestricted symmetry restoration. A separate compactly tested free-solution witness proves that the prescribed nonlinear scalar rule does not descend to the unchanged unrestricted free-shell quotient. The witness works for every fixed scalar mass in this particular descent problem; it neither extends nor discharges the distinct prepared-insertion problem of XVI-N. Conditional correction and quantum-presentation theorems state the additional identities required for descent, nilpotence, and exact free-fiber recovery. Finite rank and left-kernel tests are valid only after complete coverage of the declared correction space has been proved. The odd, ghost-number-one master residual and the even reciprocity pairing remain different targets. Complete destructive residual-report instruments are recovered on identical installed finite interfaces, including arbitrary untouched references. Their adaptive composition, regular transport, resource fit, and error budget are proved locally. They encode supplied numerical evaluations, not Grassmann elements or acquired continuum fields. The contribution is an explicit compatibility and obstruction package, not a new general BV formalism. Its potential use is to prevent a restricted classical differential, a successful report, or one cancelled diagnostic from being mistaken for interacting quantum BRST. The ramp width remains fixed and positive; it is not a thickness assigned to the conical worldsheet. No aligned-null contraction, renormalized interacting product, inherited characteristic quantum coefficient, metric reconstruction, or clock-free relational time is constructed. \end{abstract} \scopebox{Exact status of the attempted continuation}{% Constructed: the normalized free package, the nonlinear off-shell field differential, its restricted classical BV centralizer, the explicit free-shell non-descent witness, and complete finite report channels. Conditional: correction-based quotient descent and a free-fiber-preserving quantum presentation. Not constructed: a nontrivial interacting quantum BRST algebra realizing the prescribed nonlinear rules on the unrestricted free causal carrier, a renormalized interacting product, or the inherited characteristic quantum coefficient.} \tableofcontents %===================================================================== \section{Purpose, source discipline, and the Enter interpretation} \label{sec:intro} %===================================================================== \subsection{What this manuscript attempts} The supplied development places X-L-ST after the free stage X-G-ST, the relative classical stage X-H-ST, and the later classical and finite-operational sheets X-I-ST, X-J-ST, and X-K-ST \cite{XGST,XHST,XIST,XJST,XKST}. That ordering is documentary. It does not transfer every predecessor theorem or identify their carriers. The present paper selects the restricted graded direction. Its nonlinear construction is classical. It also gives a conditional interface for a later quantum presentation, but does not assert that the missing quantum hypotheses have been satisfied. All mathematical inputs used below are displayed as definitions, explicit operator normalizations, or conditional hypotheses. The cited sheets identify the intended comparisons. Their titles and catalogue placement are not substituted for proofs. \subsection{The philosophy of Enter} The supplied philosophy essay treats Enter as destructive computational-basis readout and insists that output type, accessible systems, and the information task remain explicit \cite{EntPhil,EnterI}. In particular: \[ \begin{gathered} \text{readout}\ne\text{quantum recovery},\\ \text{classical residual}\ne\text{quantum product},\\ \text{report instrument}\ne\text{continuum acquisition},\\ \text{prescribed cone}\ne\text{spacetime emergence}. \end{gathered} \] Consumption removes a measured quantum name from the declared continuation. It does not annihilate the host time coordinate or every physical degree of freedom in a possible apparatus. A formal ghost is not a physical finite-register state. An odd classical residual cannot be inserted into a density matrix as if it were an ordinary real amplitude. Section~\ref{sec:reports} specifies the separate extraction of ordinary numerical coefficients before a report is installed. The classical and quantum BV frameworks already distinguish equation ideals, products, and master equations \cite{FRclassical,FRquantum}. The present paper makes no priority claim for those general structures. \subsection{Corrections required by the plan} Four points require explicit debugging. \begin{enumerate} \item The I-B conical defect is a codimension-two worldsheet. The positive width in the X track is a fixed analytical ramp or smoothing width, not a derived geometric core thickness. \item Off-shell field nilpotence does not imply invariance of a prescribed action or descent to its free equations. \item Ghost number is a grading. A filtration can be induced from it, but no convergence or spectral-sequence conclusion follows merely from saying ``filtered by ghost number.'' \item Cancelling a single shell witness or one reciprocity number is not sufficient to define a quantum product. Complete relation, product, domain, and nilpotence identities must still be supplied. \end{enumerate} %===================================================================== \section{Unchanged host and analytical completion} \label{sec:host} %===================================================================== \begin{assumption}[Retained I-B/X-A-ST host] \label{ass:host} Retain the prescribed time-oriented regular conical product \[ \mathcal M^\circ =\RR_T\times(C_\alpha\setminus\{o\})\times\RR_z, \] \[ G=-c_{\mathrm h}^{\,2}\dd T^2+\dd r^2 +\alpha^2r^2\dd\varphi^2+\dd z^2, \qquad 0<\alpha<1,\quad c_{\mathrm h}>0, \] with completion worldsheet \[ \mathcal W=\RR_T\times\{o\}\times\RR_z. \] The supported laboratory is disjoint from \(\mathcal W\). The independently declared analytical completion is \[ \widehat M=\RR^{1+3},\qquad \eta=\diag(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta_{\mathbf x}. \] On the supported developing chart \(\iota:\mathcal L\to\mathcal M^\circ\), retain \begin{equation} \iota^*G=-\eta,\qquad t=c_{\mathrm h}(T-T_\Sigma). \label{eq:signature} \end{equation} Retain the same switches, plateau \(U\Subset\mathcal L\), regulator prescription, and fixed positive width \[ \sigma_\varepsilon(t)=\chi(t/\varepsilon), \qquad \varepsilon>0. \] Neither \(\varepsilon\), \(\chi\), nor the host is varied. All analytical equations, free fields, and Green operators below are defined on \(\widehat M\), not on the regular cone. \end{assumption} These are the prescribed-host conventions of I-B and X-A-ST \cite{IB,XAST}. A formal tensor \(q\) introduced later is not the conical host metric \(G\). \begin{proposition}[The regular product supplies no global Cauchy inverse] \label{prop:host} The regular conical product is not globally hyperbolic. No global Green operator on that product is used here. \end{proposition} \begin{proof} Fix \(r_0>0\), a radial ray, and \(z_0\). Choose \(A>r_0/c_{\mathrm h}\). The causal diamond between the points with coordinates \((-A,r_0,\varphi_0,z_0)\) and \((A,r_0,\varphi_0,z_0)\) contains \[ (0,r_0/n,\varphi_0,z_0),\qquad n\ge1. \] This sequence approaches the excluded worldsheet and has no convergent subsequence in \(\mathcal M^\circ\). The diamond is not compact. The Green operators constructed below are instead the specified Minkowski operators on \(\widehat M\). \end{proof} \begin{remark}[Width and defect are different objects] Nothing in Assumption~\ref{ass:host} gives the worldsheet a positive physical thickness. Such a thick-core model would require additional geometric data and would not follow from retaining \(\varepsilon>0\). \end{remark} %===================================================================== \section{The normalized free auxiliary--ghost package} \label{sec:free} %===================================================================== \subsection{Fields, pairings, and operators} Use the field order \[ \Phi=(r_{\mu\nu},b_\mu,\phi,c^\mu,\bar c_\mu). \] The fields \(r,b,\phi\) are even and have ghost number zero. The ghosts \(c,\bar c\) are odd, with \[ \gh(c)=1,\qquad \gh(\bar c)=-1. \] All pairings use \(\eta\), the coordinate density, and the bilinear formal transpose, without complex conjugation. For a symmetric tensor \(r\), define \[ (\Ptr r)_{\mu\nu} =r_{\mu\nu}-\frac12\eta_{\mu\nu}r^\rho{}_\rho, \] \[ (Kv)_{\mu\nu}=\partial_\mu v_\nu+\partial_\nu v_\mu, \qquad (\Cdiv r)_\nu =\partial^\mu r_{\mu\nu} -\frac12\partial_\nu r^\mu{}_\mu. \] In four dimensions, \begin{equation} \Ptr^2=I,\qquad \Cdiv K=\Box,\qquad \Cdiv^*=-\frac12\Ptr K. \label{eq:operatoridentities} \end{equation} Set \begin{equation} \Pgr=-\frac14\Ptr\Box,\qquad A=\Pgr-\frac12\Cdiv^*\Cdiv,\qquad \Pm=\Box+m^2,\quad m\ge0. \label{eq:freeoperators} \end{equation} Then \[ AK=0,\qquad \Pgr=A+\frac12\Cdiv^*\Cdiv. \] \begin{proof}[Verification of the displayed identities] Trace reversal is involutive in dimension four. Substitution into \(\Cdiv Kv\) cancels the two divergence-gradient terms and leaves \(\Box v\). Compact integration by parts gives the transpose formula. Consequently, \[ \Pgr K=-\frac14\Ptr K\Box =\frac12\Cdiv^*\Box, \] and hence \[ AK=\frac12\Cdiv^*\Box -\frac12\Cdiv^*\Cdiv K=0. \] \end{proof} These formulas fix the X-E/X-F normalization retained in the supplied X-G-ST Green-matrix passage \cite{XGST}. No normalization from the distinct XII--XVI inverse convention is substituted. \subsection{Gauge fermion and free nilpotence} Define the odd derivation \(s_0\), commuting with spacetime derivatives, by \begin{equation} s_0r=Kc,\qquad s_0\phi=0,\qquad s_0c=0,\qquad s_0\bar c=b,\qquad s_0b=0. \label{eq:s0} \end{equation} It acts by the left graded Leibniz rule. \begin{lemma}[Off-shell free nilpotence] \label{lem:s0nil} The derivation \(s_0\) satisfies \(s_0^2=0\) on the polynomial jet algebra and on finite products of compactly smeared local polynomial functionals. \end{lemma} \begin{proof} Its square vanishes on every field generator. The square of an odd derivation is an even derivation, so vanishing on the generators proves the assertion. \end{proof} The normalized free gauge fermion is \[ \Psi_0=\int\bar c^\mu\bigl((\Cdiv r)_\mu-b_\mu\bigr) \dd^4x. \] As usual, an action expression is understood variationally, or with compact variations and the corresponding relative support convention. \begin{proposition}[Gauge-fixing normalization] \label{prop:gfnorm} The gauge-fixing expression is \begin{equation} s_0\Psi_0 =\int\left( b^\mu(\Cdiv r)_\mu-b^\mu b_\mu -\bar c^\mu\Box c_\mu\right)\dd^4x. \label{eq:gaugefix} \end{equation} Eliminating \(b\) gives \[ b=\frac12\Cdiv r,\qquad S_{\mathrm{gf},0}=\frac14\int(\Cdiv r)^2\dd^4x. \] The reduced metric Hessian is exactly \(\Pgr\). \end{proposition} \begin{proof} The odd Leibniz rule produces the minus sign in the ghost term. Equation~\eqref{eq:operatoridentities} gives \(s_0(\Cdiv r)=\Box c\). The algebraic \(b\)-equation gives the stated value. The Hessian of \(\frac14\int(\Cdiv r)^2\) is \(\frac12\Cdiv^*\Cdiv\), which added to \(A\) gives \(\Pgr\). \end{proof} The free gauge-fixed action is therefore represented by \begin{equation} S_{0,\mathrm{gf}} =\frac12\pair r{Ar} -\frac12\pair\phi{\Pm\phi}+s_0\Psi_0. \label{eq:freeaction} \end{equation} Its relative free BRST variation vanishes: the variation of \(b\Cdiv r\) cancels that of \(-\bar c\Box c\), and \(AK=0\). \subsection{The complete free Green matrix} Let \(\mathsf E_m^{\mathrm R/\mathrm A}\) be the Minkowski Green operators for \(\Pm\), and let \(\mathsf E_0^{\mathrm R/\mathrm A}\) be those for \(\Box\). Their compact-source inverse identities and causal support are the ordinary Minkowski wave-theory inputs \cite{BGP}. The bosonic \(r,b\) Hessian is \[ \mathcal L_{rb} =\begin{pmatrix}A&\Cdiv^*\\\Cdiv&-2I\end{pmatrix}. \] \begin{proposition}[Auxiliary Green matrix with the retained signs] \label{prop:Ghb} For either support choice, \begin{align} \mathcal G_{rb}^{\mathrm R/\mathrm A} &= \begin{pmatrix} \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} &\frac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\Cdiv^*\\ \frac12\Cdiv\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} &\frac14\Cdiv\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} \Cdiv^*-\frac12I \end{pmatrix} \nonumber\\ &= \begin{pmatrix} -4\Ptr\mathsf E_0^{\mathrm R/\mathrm A} &K\mathsf E_0^{\mathrm R/\mathrm A}\\ -2\Cdiv\Ptr\mathsf E_0^{\mathrm R/\mathrm A}&0 \end{pmatrix}, \label{eq:Ghb} \end{align} where \[ \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} =-4\Ptr\mathsf E_0^{\mathrm R/\mathrm A}. \] Both inverse identities hold on compact tests. \end{proposition} \begin{proof} The sourced equations are \[ Ar+\Cdiv^*b=J,\qquad \Cdiv r-2b=j. \] Eliminate \(b=(\Cdiv r-j)/2\). Then \[ \Pgr r=J+\frac12\Cdiv^*j. \] Apply the metric Green inverse and substitute back. The simplification follows from \[ \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\Cdiv^* =2K\mathsf E_0^{\mathrm R/\mathrm A}, \qquad \Cdiv\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\Cdiv^*=2I. \] These identities use \(\Ptr^2=I\), \(\Cdiv K=\Box\), and the compact-source Green identities. The same elimination on compact field pairs proves the left inverse. Differentiation preserves causal support. \end{proof} For the ghost ordering \((c,\bar c)\), use \[ \mathcal L_{\mathrm{gh}} =\begin{pmatrix}0&\Box\\-\Box&0\end{pmatrix}, \qquad \mathcal G_{\mathrm{gh}}^{\mathrm R/\mathrm A} =\begin{pmatrix} 0&-\mathsf E_0^{\mathrm R/\mathrm A}\\ \mathsf E_0^{\mathrm R/\mathrm A}&0 \end{pmatrix}. \] The quadratic ghost expression \(\frac12(c,\bar c)\mathcal L_{\mathrm{gh}}(c,\bar c)^T\) is \(-\int\bar c\Box c\), with the displayed ordering. Thus the full free matrix is \begin{equation} \mathcal G_0^{\mathrm R/\mathrm A} = \begin{pmatrix} -4\Ptr E_0&KE_0&0&0&0\\ -2\Cdiv\Ptr E_0&0&0&0&0\\ 0&0&-E_m&0&0\\ 0&0&0&0&-E_0\\ 0&0&0&E_0&0 \end{pmatrix}^{\mathrm R/\mathrm A}. \label{eq:fullGreen} \end{equation} \subsection{The free causal carrier} Put \[ \delta_m=\mathsf E_m^{\mathrm R}-\mathsf E_m^{\mathrm A}, \qquad \Delta_0=\mathcal G_0^{\mathrm R} -\mathcal G_0^{\mathrm A}. \] The free equations are \begin{equation} Ar+\Cdiv^*b=0,\qquad \Cdiv r-2b=0,\qquad \Pm\phi=0,\qquad \Box c=\Box\bar c=0. \label{eq:freeeq} \end{equation} Define \(\Afree\) over \(\CC[[\hbar]]\) by compactly smeared linear fields, equations~\eqref{eq:freeeq}, and the graded causal relations \begin{equation} [\Phi_A(f),\Phi_B(g)]_{\mathrm{gr}} =i\hbar\pair f{\Delta_{0,AB}g}\,\mathbf1. \label{eq:CCR} \end{equation} Only finite polynomial products are used. No coincident nonlinear Wick field is included by this definition. \begin{theorem}[Exact free causal BRST recovery] \label{thm:freeBRST} The presentation \eqref{eq:freeeq}--\eqref{eq:CCR} is a nonzero associative graded polynomial carrier. The derivation \(s_0\) preserves its defining relations and induces a nilpotent graded derivation of \(\Afree\). With the displayed normalization, the generator map to the supplied X-G-ST free presentation is the identity, and its inverse is the same generator map. \end{theorem} \begin{proof} The causal Green difference annihilates the free equations. It defines a graded antisymmetric central form on the compact-test quotient. The enveloping algebra of the corresponding central Lie superalgebra, with its central element identified with the unit and the form scaled by \(i\hbar\), gives the stated polynomial carrier. For equation stability, \(s_0(Ar+\Cdiv^*b)=AKc=0\), and \[ s_0(\Cdiv r-2b)=\Box c. \] The antighost equation maps to \(\Box b\). The metric Bianchi identity and \[ \partial^\mu(\Cdiv^*b)_{\mu\nu} =-\frac12\Box b_\nu \] show that \(\Box b=0\) follows from the bosonic equations. The nontrivial initially zero causal relation is the \(r,\bar c\) relation: \[ s_0[r,\bar c]_{\mathrm{gr}} =\{Kc,\bar c\}+[r,b] =-i\hbar K\delta_0+i\hbar K\delta_0=0. \] The remaining checks use the zero cross blocks and the zero \(cc\) and \(bb\) blocks. Nilpotence descends from Lemma~\ref{lem:s0nil}. The X-G-ST normalization in the supplied passages is exactly \eqref{eq:s0}, \eqref{eq:Ghb}, and the same free causal presentation \cite{XGST}. Identical generators and relations induce mutually inverse maps. This comparison does not verify other results of that manuscript. \end{proof} \begin{remark}[Free ghosts are not finite laboratory qubits] No positive physical Hilbert-space interpretation of the entire ghost carrier is asserted. The finite report systems of Section~\ref{sec:reports} are separate positive matrix algebras, not a CPTP realization of the ghost algebra. \end{remark} %===================================================================== \section{The nonlinear classical field differential} \label{sec:nonlinear} %===================================================================== \subsection{Domain and formal background split} Let \(\Pfield(U)\) be the graded algebra of smooth-coefficient polynomial jet expressions on unrestricted smooth germs in \(U\). Let \(\Fml(U)\) be the algebra generated by compactly smeared local polynomial functionals under finite pointwise products. The latter is multilocal: a product of two single-integral functionals is generally not a single-integral local functional. Both algebras are completed coefficientwise in the even formal parameter \(\kappa\). Fix smooth classical backgrounds \(h,k\) and write \begin{equation} \bar q=\eta+\kappa h+\kappa^2k,\qquad q=\bar q+\kappa r. \label{eq:split} \end{equation} The formal inverse-metric coefficients are permitted coefficientwise. Neither \(q\) nor \(\bar q\) is the host metric \(G\). Define \begin{align} sr&=Kc+\kappa\mathcal L_c(h+r)+\kappa^2\mathcal L_ck, \label{eq:sr}\\ s\phi&=\kappa c^\mu\partial_\mu\phi,\qquad sc^\mu=\kappa c^\nu\partial_\nu c^\mu, \label{eq:snonlinear}\\ s\bar c&=b,\qquad sb=0,\qquad sh=sk=0. \notag \end{align} The differential commutes with spacetime derivatives and uses the left graded Leibniz rule. In particular, \[ sq=\kappa\mathcal L_cq,\qquad s=s_0+\kappa s_1+\kappa^2s_2. \] \begin{theorem}[Exact off-shell field nilpotence] \label{thm:kinematic} The operator \(s\) is a ghost-number-one odd derivation, preserves the stated local and multilocal classical domains, and satisfies \(s^2=0\). Its zero-coupling field operator is exactly \(s_0\). \end{theorem} \begin{proof} For the odd vector field \(c\), set \[ w^\mu=c^\nu\partial_\nu c^\mu. \] The graded coordinate calculation gives \(\mathcal L_c^2=\mathcal L_w\). Terms with symmetric second derivatives cancel against antisymmetric ghost coefficients. Therefore \[ s^2q =\kappa\mathcal L_{sc}q-\kappa\mathcal L_c(sq) =\kappa^2(\mathcal L_w-\mathcal L_c^2)q=0. \] Since \(\bar q\) is fixed and formal multiplication by \(\kappa\) is injective before truncation, \(s^2r=0\). Similarly, \[ s^2\phi =\kappa(sc^\mu)\partial_\mu\phi -\kappa c^\mu\partial_\mu(s\phi)=0. \] Expanding \(s(c^\nu\partial_\nu c^\mu)\) gives zero after substituting \(sc=\kappa c\partial c\). The nonminimal pair is nilpotent by definition. Every fixed coefficient uses finitely many field factors and derivatives. Compactly smeared local terms remain compactly smeared local terms under \(s\), and the Leibniz rule preserves finite multilocal products. The leading coefficient is \eqref{eq:s0}. \end{proof} These are the relative field rules displayed in X-H-ST and X-I-ST \cite{XHST,XIST}. The proof does not infer nilpotence from either source's title. \begin{remark}[Support restrictions must be tangent] The unrestricted smooth-germ domain avoids a support mismatch in \(s\bar c=b\). If compact support is imposed on \(\bar c\), the same support requirement must be imposed on \(b\). A compact-antighost condition with unrestricted \(b\) does not define an \(s\)-stable configuration domain. \end{remark} \begin{remark}[Grading and filtration] Every fixed coefficient has a finite ghost-number decomposition. One may set \[ F^p=\bigoplus_{g\ge p}\Pfield^{\,g} \] on such finite expressions; \(sF^p\subseteq F^{p+1}\). No completeness, convergence, or spectral-sequence degeneration claim is attached to this filtration. \end{remark} %===================================================================== \section{A restricted classical BV construction} \label{sec:restrictedBV} %===================================================================== \subsection{Action input and signed Hamiltonian lift} The field rules do not determine an action. Let \(S_{\mathrm{ret}}(\kappa)\) be the separately specified, even, ghost-number-zero classical action of the retained relative problem, including its actual switches, backgrounds, and declared classical deformation terms. For the free-recovery statement its quadratic, zero-coupling part is required to be \eqref{eq:freeaction}. An equality modulo a relative support convention is used only on the support class where it has been established. Switch derivatives are not deleted by calling a region a plateau. Variational action notation is used in its usual local sense: compact variations determine the Euler derivatives. It is not an assertion that an integral of a noncompact configuration is a finite number \cite{FRclassical}. Introduce antifields \(\Phi_A^\ddagger\) of parity \(|\Phi_A|+1\) and ghost number \(-1-\gh(\Phi_A)\). Use the odd canonical bracket convention \[ (\Phi_A^\ddagger(x),\Phi^B(y)) =\delta_A^B\delta(x-y), \qquad (\Phi^A,\Phi^B)= (\Phi_A^\ddagger,\Phi_B^\ddagger)=0. \] The bracket extends by the odd biderivation rules. Antifields are written on the right in \begin{equation} S_{\mathrm{kin}} =\sum_A\int(s\Phi^A)\Phi_A^\ddagger\,\dd^4x. \label{eq:kinham} \end{equation} Consequently \[ (S_{\mathrm{kin}},\Phi^A)=s\Phi^A. \] Reordering \eqref{eq:kinham} with antifields on the left introduces the factor \((-1)^{|\Phi_A|+1}\). A different canonical bracket orientation requires its corresponding overall convention change. Let \(\BValg\) be the coefficientwise classical local variational BV algebra, together with finite products of its compactly supported functionals, on which these brackets are defined. No distributional quantum contraction is used. \begin{lemma}[Kinematic master identity] \label{lem:kinmaster} The Hamiltonian \eqref{eq:kinham} satisfies \[ \frac12(S_{\mathrm{kin}},S_{\mathrm{kin}})=0. \] \end{lemma} \begin{proof} Its Hamiltonian vector field restricts to \(s\) on the fields. The bracket of two antifield-linear Hamiltonians is antifield-linear, and its coefficient at \(\Phi_A^\ddagger\) is the coefficient of \(s^2\Phi^A\), with the fixed canonical ordering. Theorem~\ref{thm:kinematic} makes every coefficient zero. \end{proof} Define \begin{equation} S_{\mathrm{ext}}=S_{\mathrm{ret}}+S_{\mathrm{kin}}, \qquad \QBV=(S_{\mathrm{ext}},\,\cdot\,), \qquad \Rmaster=\frac12(S_{\mathrm{ext}},S_{\mathrm{ext}}). \label{eq:master} \end{equation} \begin{proposition}[The classical master residual] \label{prop:masterresidual} On the declared variational domain, \begin{equation} \Rmaster=sS_{\mathrm{ret}},\qquad \QBV^2F=(\Rmaster,F),\qquad \QBV\Rmaster=0. \label{eq:curvature} \end{equation} The residual has parity one and ghost number one. It is field-dependent and antifield-independent. It need not vanish merely because \(s^2=0\). \end{proposition} \begin{proof} Field-only functionals have zero mutual antibracket. Lemma~\ref{lem:kinmaster} removes the kinematic self-bracket. The two mixed brackets coincide because both Hamiltonians are even, and give \((S_{\mathrm{kin}},S_{\mathrm{ret}})=sS_{\mathrm{ret}}\). The other two identities are the odd Jacobi identities. The parity and ghost-number statements follow from the degree of the odd bracket. \end{proof} \subsection{The centralizer domain} \begin{definition}[Restricted classical domain] \label{def:centralizer} Define \begin{equation} \BValg_{\Rmaster} =\{F\in\BValg:(\Rmaster,F)=0\}. \label{eq:centralizer} \end{equation} All identities are coefficientwise in \(\kappa\), on the fixed relative variational domain. \end{definition} \begin{theorem}[Restricted classical nilpotent continuation] \label{thm:restricted} The domain \(\BValg_{\Rmaster}\): \begin{enumerate}[label=\textup{(\roman*)}] \item is a graded subalgebra for the classical product and the odd bracket; \item is preserved by \(\QBV\); \item carries the nilpotent differential \(\QBV|_{\BValg_{\Rmaster}}\); \item contains the complete field-only classical algebra, on which \(\QBV=s\); \item is the largest subdomain of \(\BValg\) on which the same Hamiltonian operator has square zero. \end{enumerate} This construction does not set \(\Rmaster=0\) in the ambient algebra. \end{theorem} \begin{proof} The map \((\Rmaster,\cdot)\) is an even derivation of both the product and bracket. Its kernel is therefore a graded subalgebra for both operations. By \eqref{eq:curvature} and Jacobi, \[ \QBV(\Rmaster,F) =(\QBV\Rmaster,F)+(\Rmaster,\QBV F) =(\Rmaster,\QBV F). \] Thus \((\Rmaster,F)=0\) implies \((\Rmaster,\QBV F)=0\). Equation~\eqref{eq:curvature} then proves nilpotence. Both \(\Rmaster\) and a field-only functional depend on no antifields, so their antibracket is zero. On such functionals \(\QBV=s\). Finally, if the same \(\QBV\) has square zero on an element \(F\), then \eqref{eq:curvature} forces \((\Rmaster,F)=0\). Every square-zero domain is contained in \eqref{eq:centralizer}. \end{proof} \begin{remark}[What is not asserted] The theorem supplies a restricted classical differential algebra. It does not prove that its cohomology is the physical observable algebra, that a Koszul--Tate resolution is acyclic, or that unrestricted bulk symmetry has been restored. The centralizer construction is an application of the classical Jacobi identity, not a new general BV theorem. \end{remark} \subsection{The exact meaning of free recovery} At \(\kappa=0\), the field rules are precisely \eqref{eq:s0}, and the normalized quadratic action and Green matrix are \eqref{eq:freeaction} and \eqref{eq:fullGreen}. The free causal presentation therefore recovers \(\Afree\) by Theorem~\ref{thm:freeBRST}. This recovery has three explicit steps: \[ \text{reduce the field rules and quadratic action} \ \longrightarrow\ \text{impose the displayed free equations} \ \longrightarrow\ \text{form the displayed graded causal algebra}. \] It is not an algebra homomorphism from a commutative classical field algebra to the noncommutative causal algebra. \begin{proposition}[No hidden quantum-family claim] \label{prop:freeboundary} The constructed field-only special fiber is exactly the free off-shell field algebra with differential \(s_0\). The free causal quantization of its displayed linear equations is exactly \(\Afree\). No flat family of interacting quantum algebras follows. In particular, the special fiber of the full antifield centralizer need not equal the centralizer of the specialized residual. \end{proposition} \begin{proof} The field-only algebra lies entirely in \(\BValg_{\Rmaster}\), so specializing its formal coefficients gives every free field expression, with the kernel consisting of multiples of \(\kappa\). Theorem~\ref{thm:freeBRST} gives its separate free causal presentation. Kernels need not commute with specialization. For example, the kernel of multiplication by \(\kappa\) on \(\CC[[\kappa]]\) is zero, whereas the specialized zero map has kernel \(\CC\). Hence no specialization theorem for the entire centralizer can be inferred without additional hypotheses. \end{proof} %===================================================================== \section{An explicit obstruction to unchanged free-shell descent} \label{sec:descentwitness} %===================================================================== Let \(\mathcal I_{\mathrm{eq}}\subset\Fml\) be the graded ideal generated by compact smearings of \eqref{eq:freeeq}. Let \(\mathcal I_{\mathrm{sol}}\) be the ideal of functionals vanishing on all smooth free solutions, interpreted with finite Grassmann coefficient algebras. Then \[ \mathcal I_{\mathrm{eq}}\subseteq\mathcal I_{\mathrm{sol}}. \] No equality between these ideals is needed. \begin{theorem}[Compactly tested nonlinear non-descent] \label{thm:nondescent} For every fixed \(m\ge0\), the nonlinear rule \[ s\phi=\kappa c^\mu\partial_\mu\phi \] does not preserve \(\mathcal I_{\mathrm{sol}}[[\kappa]]\). It therefore does not descend to the unchanged unrestricted classical free-shell quotient. A compactly smeared scalar equation detects the failure exactly at order \(\kappa\). \end{theorem} \begin{proof} Choose \((t_*,\mathbf x_*)\in U\) and a sufficiently small product neighborhood inside \(U\). Define \[ u_m(t)= \begin{cases} \sin(m(t-t_*))/m,&m>0,\\ t-t_*,&m=0. \end{cases} \] Thus \(u_m''+m^2u_m=0\), and \(u_m'>0\) on a sufficiently small time interval about \(t_*\). Take the Grassmann algebra with one odd generator \(\vartheta\), and the smooth free configuration \[ r=b=\bar c=0,\qquad c^1=\vartheta(t-t_*),\qquad c^\mu=0\quad(\mu\ne1), \] \[ \phi=(x^1-x_*^1)u_m(t). \] All equations~\eqref{eq:freeeq} hold. Choose a nonnegative \(f\in C_c^\infty(U)\), supported in that neighborhood, normalized by \[ \int f(x)u_m'(t)\,\dd^4x=1. \] The functional \[ F_f=\int f\,\Pm\phi\,\dd^4x \] belongs to \(\mathcal I_{\mathrm{eq}}\). On the displayed free configuration, \[ c^\mu\partial_\mu\phi =\vartheta(t-t_*)u_m(t), \] and therefore \[ \Pm(c^\mu\partial_\mu\phi)=2\vartheta u_m'(t). \] It follows that \begin{equation} (sF_f)\big|_{\mathrm{free}} =2\kappa\vartheta\ne0. \label{eq:witness} \end{equation} Thus \(sF_f\notin\mathcal I_{\mathrm{sol}}[[\kappa]]\). \end{proof} \begin{remark}[Compact test, not a compact spacetime wave] The detecting smearing is compact. The free solutions used in the proof are unrestricted smooth solutions and need not have compact spacetime support. No claim about XVI-N's separately specified Route B is made. \end{remark} \begin{corollary}[Equation-vanishing changes cannot repair this witness] \label{cor:restrictedcorrections} Suppose a corrected field derivation has \[ d\phi=\kappa(c^\mu\partial_\mu\phi+C_1\phi) +O(\kappa^2), \] commutes with spacetime derivatives, and satisfies \((C_1\phi)|_{\mathrm{free}}=0\) as a smooth function on every complete free configuration. Then it still fails unchanged free-shell descent on the witness of Theorem~\ref{thm:nondescent}. \end{corollary} \begin{proof} On a free configuration, the identically zero smooth function \(C_1\phi\) and all of its spacetime derivatives vanish. Hence \(\Pm(C_1\phi)=0\) there. The order-\(\kappa\) value remains \(2\kappa\vartheta\). \end{proof} This is an obstruction to a named correction class, not to every possible changed equation system or interacting carrier. \begin{corollary}[Semiclassically normalized quantum descent is not automatic] \label{cor:quantumobstruction} Suppose a proposed differential on an unchanged free quantum carrier is required to have the displayed nonlinear scalar rule as its classical normalization, including the compact equation functional \(F_f\). If its classical reduction preserves the unchanged free equations, such a proposal is incompatible with \eqref{eq:witness}. \end{corollary} \begin{proof} A well-defined differential sends the zero equation class to zero. Its assumed classical reduction would therefore make \(sF_f\) vanish on every free solution. Equation~\eqref{eq:witness} contradicts that requirement. \end{proof} No nonlinear quantum composite has been formed in this argument. The corollary is conditional on the proposed quantum map having the stated classical normalization. \begin{proposition}[Compact relative ghosts are not unrestricted free ghosts] \label{prop:ghostsupport} A smooth homogeneous Minkowski wave with compact spacetime support is zero. Consequently, restricting ghosts to compact spacetime support does not preserve the full unrestricted free ghost solution space. \end{proposition} \begin{proof} A compactly supported wave has zero Cauchy data on a sufficiently early time slice. Cauchy uniqueness makes it identically zero. \end{proof} The witness does not contradict relative off-shell nilpotence in X-H-ST. It tests a different operation: descent to the unrestricted free shell. %===================================================================== \section{Conditional corrections and quantum-presentation tests} \label{sec:conditional} %===================================================================== \subsection{An exact classical quotient criterion} \begin{theorem}[Conditional restricted quotient] \label{thm:conditionalquotient} Let \(\mathscr D\) be a declared graded subalgebra of \(\Fml[[\kappa]]\), and let \(J\subset\mathscr D\) be a graded ideal. Let \(C\) be an odd ghost-number-one derivation of \(\mathscr D\), beginning at positive order in \(\kappa\), and define \(d=s+C\). Then \(d\) induces a square-zero derivation of \(\mathscr D/J\) if and only if \begin{align} &(s+C)J\subseteq J, \label{eq:idealtest}\\ &(sC+Cs+C^2)\mathscr D\subseteq J. \label{eq:niltest} \end{align} Both conditions refer to the entire declared ideal and domain, not to a selected witness. \end{theorem} \begin{proof} A derivation descends through a quotient exactly when it preserves the defining ideal, proving \eqref{eq:idealtest}. Since \(s^2=0\), \[ d^2=sC+Cs+C^2. \] Its induced square vanishes exactly under \eqref{eq:niltest}. \end{proof} The same statement applies to a subalgebra of \(\BValg_{\Rmaster}\), with \(\QBV\) in place of \(s\), provided all operations are defined there. If a ghost-number-zero correction \(a\) is added to a classical action, it changes the master residual by \(sa\). The exact action-compatibility equation is \[ sS_{\mathrm{ret}}+sa=0 \] on the specified relative domain. Such a correction is not permitted merely because the equation can be written: its admissible support, normalization, and action-preservation requirements must be fixed first. \subsection{Order-by-order compatibility} Write \[ d=d_0+\kappa d_1+\cdots+\kappa^Nd_N \pmod{\kappa^{N+1}},\qquad d_0=s_0. \] Square-zero requires \begin{equation} \sum_{i+j=n}d_id_j=0 \pmod J,\qquad 0\le n\le N. \label{eq:MCorders} \end{equation} For example, \[ s_0d_1+d_1s_0=0,\qquad s_0d_2+d_2s_0+d_1^2=0. \] These are additional equations. A first-order linear solve does not discharge the second equation. \subsection{The conditional quantum presentation} Let \(V\) be the free graded compact-test generator space. Let \(\mathfrak T\) be its formal tensor algebra over \(\CC[[\hbar,\kappa]]\), completed coefficientwise. Let \(J_0\) denote the free equation and causal-relation ideal giving \(\Afree\). \begin{definition}[Additional quantum input] \label{def:quantuminput} A proposed interacting presentation supplies: \begin{enumerate}[label=\textup{(\roman*)}] \item an explicitly defined graded two-sided ideal \(\widehat J\subset\mathfrak T\); \item an odd ghost-number-one derivation \(\widehat d\) on \(\mathfrak T\), with actual values in that tensor algebra; \item the exact free-fiber identities \[ (\widehat J+\kappa\mathfrak T)/\kappa\mathfrak T=J_0, \qquad \widehat d\bmod\kappa=s_0; \] \item all analytical definitions needed to represent the proposed nonlinear field expressions on its intended domain. \end{enumerate} The symbol for a local nonlinear quantum field is not accepted as an element merely by analogy with a classical jet. \end{definition} \begin{theorem}[Conditional free-fiber-preserving quantum quotient] \label{thm:conditionalquantum} For Definition~\ref{def:quantuminput}, the quotient \[ \widehat{\mathfrak A}=\mathfrak T/\widehat J \] has a well-defined nilpotent induced derivation precisely when \[ \widehat d\,\widehat J\subseteq\widehat J, \qquad \widehat d^{\,2}\mathfrak T\subseteq\widehat J. \] Under the stated free-fiber identities, \[ \widehat{\mathfrak A}/\kappa\widehat{\mathfrak A} \cong\Afree, \] with the identity on free generators. If additionally \[ \kappa F\in\widehat J\Longrightarrow F\in\widehat J, \] the quotient has no \(\kappa\)-torsion. These conditions do not prove causal factorization, renormalized time ordering, positivity, or existence of the intended analytical representation. \end{theorem} \begin{proof} Ideal preservation is exactly representative independence. The second condition is exactly square-zero after quotienting. The free-fiber statement is the third isomorphism theorem applied to \(\mathfrak T/(\widehat J+\kappa\mathfrak T)\). The saturation condition is precisely the assertion that \(\kappa[F]=0\) implies \([F]=0\). The additional analytical and physical properties do not occur among these algebraic hypotheses. \end{proof} \scopebox{The interacting quantum gate is not passed}{% No nontrivial \(\widehat J,\widehat d\) satisfying the prescribed nonlinear classical normalization and all required analytical conditions is supplied here. The unchanged free-shell target is obstructed by Theorem~\ref{thm:nondescent}. A changed carrier would require its own construction and comparison. Transporting a known free algebra by an arbitrary invertible linear change would only give a transported free presentation until its relation to the retained interaction had been proved.} %===================================================================== \section{Master residuals, reciprocity, and finite certificates} \label{sec:separation} %===================================================================== \subsection{Two residuals with different types} The classical master residual satisfies \[ |\Rmaster|=1,\qquad \gh(\Rmaster)=1. \] By contrast, a reciprocity defect for an even classical source or response map \(J\), on even variations, has form \begin{equation} \Rrec(r;u,v) =\pair u{DJ(r)v}-\pair v{DJ(r)u}, \label{eq:reciprocity} \end{equation} and is even with ghost number zero. \begin{proposition}[Variational reciprocity is a separate test] \label{prop:reciprocity} If \(J\) is the variational derivative of a twice differentiable classical functional on the declared even variation domain, then \[ \Rrec(r;u,v)=0 \] for every admitted pair \(u,v\). Neither this identity nor its failure is a ghost-number-one master equation. \end{proposition} \begin{proof} The two terms are the mixed second variations of the same functional. Their equality is symmetry of the second derivative on even variations. The ghost-number and parity assignments follow from the arguments in \eqref{eq:reciprocity}; they differ from those of \(\Rmaster\). \end{proof} The supplied X-G-ST record contains a positive free-solution reciprocity witness and X-H-ST contains finite reports of its installed evaluation \cite{XGST,XHST}. For use here its exact source normalization is kept as a declared data package: \begin{equation} \mathfrak w_{\mathrm G} =(J_{\mathrm G},r_*,u_*,v_*,\ell_{\mathrm G}), \qquad \beta_{\mathrm G} :=\ell_{\mathrm G}(\Rrec_{\mathrm G})>0. \label{eq:recinput} \end{equation} No numerical value or unsupplied component formula is invented. Results using this positivity are conditional on that explicitly identified witness package. The newly proved scalar witness \eqref{eq:witness} is independent of it. \begin{proposition}[Preserved positive witness obstructs a restricted repair] \label{prop:positive} Suppose permitted source corrections have reciprocity changes in a declared vector space \(\mathcal B\). If \[ \ell_{\mathrm G}(B)=0\quad(B\in\mathcal B), \qquad \beta_{\mathrm G}>0, \] then no permitted correction restores reciprocity on a domain containing the witness. \end{proposition} \begin{proof} Every corrected pairing has evaluation \(\beta_{\mathrm G}+0>0\). It cannot be the zero bilinear defect. \end{proof} Cancelling \(\beta_{\mathrm G}\) at one pair would remove that detector only. It would not prove vanishing of the entire pairing. \subsection{No cancellation across ghost number} \begin{proposition}[Typed residual separation] \label{prop:degrees} A homogeneous ghost-number-one residual cannot cancel a homogeneous ghost-number-zero reciprocity defect. Their sum vanishes in a graded direct sum only if both components vanish. Neither residual is a loop anomaly merely because it is an obstruction. \end{proposition} \begin{proof} Distinct homogeneous summands of a graded direct sum are linearly independent. Both displayed residuals are classical constructions with no loop expansion. A quantum anomaly statement would require its own defined quantum product and master-Ward framework \cite{FRquantum}. \end{proof} \subsection{When a finite matrix test is legitimate} \begin{assumption}[Finite coverage before solving] \label{ass:coverage} For one specified linear correction problem: \begin{enumerate} \item all candidate corrections are parameterized by \(z\in\CC^n\), without omitted admissible directions; \item the complete residual belongs to a proved finite-dimensional vector space \(W\); \item a specified linear evaluation map \(E:W\to\CC^m\) is injective; \item the residual depends affinely on \(z\). \end{enumerate} Odd and even residual sectors are kept as separate graded summands. \end{assumption} \begin{theorem}[Exact finite feasibility certificate] \label{thm:finitecertificate} Under Assumption~\ref{ass:coverage}, the complete linear cancellation problem is equivalent to \[ Mz=b. \] It is solvable if and only if \[ \rank M=\rank[M\mid b], \] equivalently if and only if \[ \lambda^\mathsf T b=0 \quad\text{for every }\lambda \text{ with }\lambda^\mathsf T M=0. \] A nonzero \(\rank M\)-minor and a spanning left kernel give an exact algebraic certificate. If \(z_*\) is a solution, all solutions are \(z_*+\ker M\). \end{theorem} \begin{proof} Injective coverage makes zero evaluated residual equivalent to zero residual in \(W\). The matrix statement follows by choosing bases. The left-kernel condition is the finite-dimensional criterion \(b\in\im M\). The affine solution description follows by subtraction. \end{proof} This is the rank--minor--left-kernel shape compared with II-B \cite{IIB}. No source-specific rank, minor, coefficient, or numerical count is imported. Without proved coverage, a successful finite fit is only a fit. A normal-jet order bound does not by itself make a tangential distribution space finite dimensional. Even a complete linear fit does not supply the nonlinear equations \eqref{eq:MCorders}, the quantum relations, or an undefined graph product. %===================================================================== \section{Recovery of complete residual-report instruments} \label{sec:reports} %===================================================================== \subsection{From an installed evaluation to an ordinary number} A report package supplies: \begin{enumerate} \item a named compact evaluation configuration and all classical source data; \item a finite Grassmann coefficient convention; \item an ordinary real extraction from that evaluation; \item a finite encoding, calibration basis, and program; \item every returned quantum residual, failure, and reference. \end{enumerate} For example, if \[ \Rmaster(z) =\sum_I \rho_I(z)\,\vartheta^I \] in a specified finite Grassmann algebra, a report may use \[ a_{\mathrm{cl}} =\operatorname{Re}\rho_{I_*}(z) \] with \(I_*\) fixed in advance. A reciprocity report may instead use the ordinary real number \(\beta_{\mathrm G}\) in \eqref{eq:recinput}. These numbers are different reports. No map from an odd Grassmann element to a positive density matrix is asserted. \subsection{The destructive instrument} Let \[ \HH_C\cong\CC^{2^q},\qquad q\ge1, \] have a calibrated orthonormal basis \(\{\ket{\tau_k}\}\). Let \(S\) contain every unmeasured finite output, and set \[ P_k=\proj{\tau_k},\qquad D_k=\bra{\tau_k}_C\otimes I_S. \] \begin{theorem}[Complete residual-report readout] \label{thm:instrument} The channel \begin{equation} \Mclock_C(X) =\sum_k\proj k_R\otimes D_kXD_k^\dagger \label{eq:instrument} \end{equation} is CPTP and equals \[ \sum_k\proj k_R\otimes \Tr_C[(P_k\otimes I_S)X(P_k\otimes I_S)]. \] It satisfies \[ \Tr_R[(\Mclock_C\otimes\id_E)(X)]=\Tr_CX \] for every untouched reference \(E\). The quantum clock names are consumed. A positive-probability block is normalized only by dividing by its own trace; no normalized state is assigned to a zero-probability branch. \end{theorem} \begin{proof} The recorded Kraus maps are \(\ket k_R\otimes D_k\), and their squared moduli sum to \(\sum_kP_k\otimes I_S=I_{CS}\). Expansion of \(X\) in the calibration basis shows that both formulas return its \(kk\)-block. Summing gives the partial trace. Tensoring the Kraus maps with \(I_E\) proves the reference statement. The codomain has no quantum \(C\)-factor. \end{proof} This is the explicit I-B/En09-A interface \cite{IB,En09}, not a field-measurement theorem. If fixed complete continuations \(\Lambda_k\) are specified, then \begin{equation} \mathcal J(X)= \sum_k\proj k_R\otimes\Lambda_k(D_kXD_k^\dagger) \label{eq:continuation} \end{equation} is CPTP. Actual loss and rejection are retained as outcomes rather than postselected away. \begin{theorem}[Exact recovery of the installed predecessor reports] \label{thm:reportrecovery} Identify an X-H-ST residual-report or reciprocity-report interface with the same: \[ \{\ket{\tau_k}\},\quad \text{encoding},\quad S,\quad \{\Lambda_k\},\quad \text{record labels and failure convention}. \] Then the recovered channel is exactly \eqref{eq:continuation} on all input operators and every reference extension. If an X-K-ST finite continuation is appended on matching interfaces, its complete history channel is the same channel obtained by composing these maps. \end{theorem} \begin{proof} Theorem~\ref{thm:instrument} identifies each subnormalized block on all operators. Applying the identical \(\Lambda_k\) gives equality of every recorded output block. Finite induction gives equality after identical complete continuations. \end{proof} The theorem recovers the supplied X-H-ST report channels under identical installed data \cite{XHST}. It does not reconstruct an unavailable source evaluation or infer that its numerical calibration has been physically acquired. \subsection{A concrete amplitude report} For an ordinary supplied real number \(a\), fix an encoding scale \(\sigma_{\mathrm e}>0\), distinct from \(\varepsilon\), and define \[ p(a)=\frac{1+\tanh(a/\sigma_{\mathrm e})}{2}, \] \[ \ket{e(a)} =\sqrt{1-p(a)}\ket0+\sqrt{p(a)}\ket1, \qquad \tau(a)=\proj{e(a)}. \] This is the finite amplitude shape used in X-Enter-ST \cite{XEnterST}. It is a preparation from supplied data, not a continuum acquisition map. \begin{proposition}[Visibility and preparation error] \label{prop:encoding} Computational destructive readout of \(\tau(a)\) has probability \(p(a)\) for code \(1\). For every \(a,b\in\RR\), \begin{equation} \dtr(\tau(a),\tau(b)) \le\min\left\{1,\frac{|a-b|}{2\sigma_{\mathrm e}}\right\}. \label{eq:encodingerror} \end{equation} A fixed subsequent complete channel cannot enlarge this bound, including with an arbitrary independent input--reference state. \end{proposition} \begin{proof} The probability is the squared second amplitude. The Bloch vector is \[ \bigl(\operatorname{sech}(a/\sigma_{\mathrm e}), 0,\tanh(a/\sigma_{\mathrm e})\bigr). \] Its derivative has Euclidean norm \(\operatorname{sech}(a/\sigma_{\mathrm e})/ \sigma_{\mathrm e}\le1/\sigma_{\mathrm e}\). Qubit trace distance is half the Euclidean distance between Bloch vectors, proving \eqref{eq:encodingerror}. Tensoring with a state preserves trace norm, and channels contract trace distance. \end{proof} One attempted readout gives one random code, not the exact value \(a\) or its exact probability. No finite-sample inference theorem is included. \subsection{Report compatibility is weaker than residual compatibility} \begin{proposition}[Equal records can hide a surviving-state difference] \label{prop:recordresidual} There are complete one-outcome instruments with identical record laws and residual trace distance \(1/2\) on one qubit input. \end{proposition} \begin{proof} Compare the identity residual channel with computational dephasing, each carrying the same single classical symbol. On \(\proj+\), their outputs have residuals \[ \proj+,\qquad I/2. \] Their difference has eigenvalues \(1/2,-1/2\), so its trace distance is \(1/2\). \end{proof} This is the complete-residual distinction also documented in En03-A and related source records \cite{En03}. \begin{proposition}[No universal clock return] \label{prop:noreturn} The destructive channel has no left inverse on the full clock input space. \end{proposition} \begin{proof} With the same surviving input, the distinct clock states \[ \frac{\ket{\tau_0}+\ket{\tau_1}}{\sqrt2}, \qquad \frac{\ket{\tau_0}-\ket{\tau_1}}{\sqrt2} \] have identical outputs. No fixed left inverse can recover both. \end{proof} \begin{proposition}[No acquisition conclusion from the report definition] \label{prop:noacquisition} If two continuum configurations induce the same installed finite input state, every fixed report program has identical complete output on them. It cannot distinguish differing target values on that pair. This does not exclude every acquisition model. It states the missing input requirement of the present report package. \end{proposition} \begin{proof} A fixed linear channel has the same output on the same input. A claim to distinguish the target values would contradict that equality. An additional continuum-to-register acquisition map could change the premise, but no such map is constructed here. \end{proof} %===================================================================== \section{Finite composition, causal transport, and error accounting} \label{sec:composition} %===================================================================== \subsection{Complete adaptive histories} At each node \(v\) of a finite public-outcome tree, supply Kraus maps \[ L_{v,a,\mu}:\HH_v\to\HH_{va}, \qquad \sum_{a,\mu}L_{v,a,\mu}^\dagger L_{v,a,\mu}=I. \] The index \(a\) is public; \(\mu\) is not accessible unless separately reported. All actual memory, residuals, and failure flags belong to the node interfaces. Each clock input is a live, unconsumed factor or a new factor returned by an explicitly specified preparation or restoration map. Reusing the spelling of a consumed name does not supply a quantum input. \begin{theorem}[Complete finite composition] \label{thm:compose} The complete leaf-history channel is \begin{equation} \mathfrak C(X) =\bigoplus_{a\ \mathrm{leaf}} \sum_\mu K_{a,\mu}XK_{a,\mu}^\dagger, \qquad \sum_{a,\mu}K_{a,\mu}^\dagger K_{a,\mu}=I, \label{eq:history} \end{equation} where \(K_{a,\mu}\) is the compatible ordered product along that refined history. All failures and reference extensions are included. \end{theorem} \begin{proof} At each node, pull the completeness identity of its child continuations back through its Kraus maps. Backward induction over the finite tree gives the identity in \eqref{eq:history}. Tensor every map with \(I_E\) for the reference statement. \end{proof} This re-establishes the needed X-K-ST finite composition law \cite{XKST}. A single Kraus map per public history is not assumed. \subsection{Regular route} Choose a straight developed segment of length \(\ell>0\) inside the retained regular tube. Install \(n+1\) equal \(p\)-qubit payload registers \(A_0,\ldots,A_n\), with independent blanks at \(A_1,\ldots,A_n\). Use successive permitted SWAPs, each over developed duration \(\ell/(n\upsilon)\), where \(0<\upsilon\le1\). \begin{proposition}[Reference-preserving route] \label{prop:route} After hop \(j\), every payload--reference input has complete route state \[ \rho_{A_jE}\otimes\bigotimes_{i\ne j}\omega_{A_i}. \] Arrival occurs after developed duration \(\ell/\upsilon\), or host duration \(\ell/(c_{\mathrm h}\upsilon)\). The route stays in the supplied regular tube. \end{proposition} \begin{proof} Each SWAP moves the entire payload and all its correlations into the next independent blank, leaving the previous slot blank. Induction proves the formula. The specified spacings and durations certify future-causal cross-port dependencies. Equation~\eqref{eq:signature} gives the host duration. \end{proof} A record used at a distant controller must itself be delivered on a declared causal classical wire, or included in the transported payload. Treating a record as a mathematical reference preserves its correlations but does not deliver it. The SWAP permission is an operational input. No microscopic gravitational Hamiltonian is derived. \subsection{Locality and actual fit} For a complete block schedule, define the backwards region of an output by repeatedly taking the union of blocks meeting its current backwards region. Controllers and classical wires are included. \begin{proposition}[Complete scheduled support] \label{prop:locality} An output observable pulls back only to its scheduled ancestors. Equality of joint ancestor--reference reductions gives equality of the joint output reductions. Metric containment follows if every actual dependency edge has a separately certified future-causal path in the retained host. \end{proposition} \begin{proof} On a disjoint block the pulled-back observable is the identity, and a channel adjoint is unital. Only a meeting block can enlarge support. Backward induction proves the support claim. Concatenating the certified edge paths proves metric containment. \end{proof} At every branch cut \(c\), charge all live laboratory qubits, including reserved blanks and memory: \[ Q_{\mathrm{lab}}(c) =q_{\mathrm{clock}}(c)+q_{\mathrm{memory}}(c) +q_{\mathrm{work}}(c) +\sum_{\rho\in\mathcal R(c)}(n_\rho+1)p_\rho, \] with shared allocations counted only once. Returned storage, supplier storage, classical tables, and untouched references have separate ledgers. A host realization requires \[ \max_cQ_{\mathrm{lab}}(c)\le Q_{\mathrm{res}}, \] and the corresponding classical-storage, location, permission, and duration inequalities. Finiteness alone does not prove these inequalities. \begin{proposition}[No host or regulator enlargement] \label{prop:noenlarge} A certificate that passes these fit checks uses only its declared finite registers inside the retained reserve. Its operations change neither \[ (\mathcal M^\circ,G,\mathcal W,\iota,c_{\mathrm h}) \] nor the fixed switches, width, and regulator. \end{proposition} \begin{proof} None of the command types has a host-update or regulator-update output. Induction preserves that metadata. The fit checks give the location and capacity bounds. \end{proof} \subsection{Complete-channel error budget} Use \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1, \qquad \ddia(\Lambda,\Gamma) =\frac12\norm{\Lambda-\Gamma}_\diamond. \] These comparisons include arbitrary untouched references \cite{Watrous}. \begin{theorem}[Matched complete-channel stability] \label{thm:error} Suppose ideal and implemented stages are CPTP maps on matching complete interfaces, with \[ \ddia(\widetilde\Gamma_j,\Gamma_j)\le e_j. \] Then \[ \ddia(\widetilde\Gamma_m\cdots\widetilde\Gamma_1, \Gamma_m\cdots\Gamma_1) \le\min\{1,\textstyle\sum_j e_j\}. \] For a matched finite adaptive tree with node allowances \(e_v\), the bound is \[ \min\left\{1,\max_{a\ \mathrm{leaf}} \sum_{v\prec a}e_v\right\}. \] Preparation and final-output allowances are added once on their actual comparison interfaces. \end{theorem} \begin{proof} For a sequence, replace one factor at a time. Pre- and postcomposition by channels contract diamond norm; the triangle inequality gives the sum. For a tree, a controlled continuation on classical direct-sum blocks has error at most the largest child error. Thus the recursive bound is \[ b_v=e_v+\max_a b_{va},\qquad b_{\mathrm{leaf}}=0. \] Backward induction gives the maximum root-to-leaf sum. \end{proof} Hidden correlated memory cannot be priced by a carrier-only one-step marginal. It must be included in the actual step interface. A trace-distance bound for one tested input is not a diamond-norm bound. The same budget bounds retained event probabilities and record TV distance. It does not validate a false supplied enclosure, certify an unretained arrival flag, or provide a finite-sample confidence theorem. %===================================================================== \section{Analytical and interpretive boundaries} \label{sec:boundaries} %===================================================================== \subsection{What remains unchanged} The X-F three-tree plateau identity remains counted once \cite{XF}. Neither an additional report nor a second clock adds a duplicate tree coefficient. The X-A-conv/X-Enter-ST convolution obstruction is unrepaired \cite{XConvST,XEnterST}. The X-J-ST compact scalar coefficients remain classical supplied evaluations, not a whole-algebra convolution repair \cite{XJST}. The X-D-ST fixed-input sharp-limit result is not metric admission \cite{XDST}. No Einstein equation follows from trace preservation, a classical record, or reference-preserving transport. The host and action remain inputs. \subsection{The characteristic quantum chain is separate} The internal amplitude constructed at XIV-I's I2 does not define its joint cross-line composition. The supported-sector reference in XIV-J treats \(K_6+K_8+K_9\) on its stated off-locus domain, not the complete six-branch target \cite{XIVI,XIVJ}. For that target, the still-required chain is \[ \begin{gathered} \text{full joint characteristic map}\\ \longrightarrow\text{joint section and controlled remainder}\\ \longrightarrow\text{uniform transverse estimates}\\ \longrightarrow\text{contact compatibility}\\ \longrightarrow\text{output wavefront theorem}\\ \longrightarrow\text{normalized quantum admission}. \end{gathered} \] The present classical differential is not any arrow of this chain. No differential is applied to the absent inherited coefficient. XIV-H's H2 ending is historical and version-specific; XIV-I's narrower I2 advance is not withdrawn \cite{XIVH,XIVI}. The supplied XV-B version 2 has its own B1 full-map boundary, while some older series summaries use F0 for another technical stopping convention. These labels are not silently identified \cite{XVB}. Technical lifting theorems in XV-B do not constitute opening its downstream quantum-admission exit. A reconstruction-null criterion does not construct the missing graph. \subsection{O.1 and the prepared insertion} The standing value \[ \loc{O.1 = NOT_CLOSED} \] is retained outside each independently earned scope. The supplied XVI-Q Q11 description reports a limited massless one-loop \(\kappa^2\) two-point closure under its enlarged counterterm domain and XVI-P prescription; that record does not update XVI-O's different prescription or establish bulk symmetry restoration \cite{XVIrecords}. Accordingly, \[ \loc{bulk_symmetry_restored = false}. \] XVI-N and its Route B are not discharged. The mass parameter in Theorem~\ref{thm:nondescent} belongs to a different scalar equation-descent witness, not to the prepared insertion of XVI-N \cite{XVIN}. \subsection{External time and reference correlations} The external time axis, event locations, preparations, and durations remain after every destructive readout. A finite clock-conditioned report is not quantized relational time without an external clock. Reference-preserving transport preserves supplied correlations. It is not a product-to-entangled-state map generated by the metric. No Page curve, de Sitter state, cosmological bounce, vacuum-energy computation, ultraviolet completion, or holographic dictionary is constructed. \subsection{OpenAI comparisons} Oai282-A and Oai283-A are documentary comparisons for coverage, interface, and resource-accounting practices \cite{OaiRecords}. OpenAI Families 282, 362, and 376 are not premises. No family proof, catalogue scope note, or reported formalization is independently verified here. %===================================================================== \section{The twelve problem boundaries} \label{sec:problems} %===================================================================== The labels below retain the numbering supplied with the plan. A status is a statement about the scope of X-L-ST, not a universal impossibility theorem. \begin{longtable}{@{}p{.08\linewidth}p{.20\linewidth}p{.65\linewidth}@{}} \toprule Problem & Status here & Exact boundary\\ \midrule \endfirsthead \toprule Problem & Status here & Exact boundary\\ \midrule \endhead 1 & Unfinished & The restricted classical structure and exact free presentation do not construct a quantized gravitational observable.\\[4pt] 2 & Unfinished & Free causal recovery and report-channel recovery do not construct a renormalized interacting product.\\[4pt] 3 & Not decided & No conclusion about the problem is inferred from the present graded restriction or report channels.\\[4pt] 4 & Not reached & No additional physical or analytical construction for this problem is attempted.\\[4pt] 5 & Not reached & A destructive clock and its complete history are not a gravitational Page curve.\\[4pt] 6 & Excluded from scope & The host geometry and classical action are inputs. Their derivation from records is not attempted.\\[4pt] 7 & Not reached & The analytical Minkowski completion is not a de Sitter state or a cosmological observer algebra.\\[4pt] 8 & Unfinished & External preparation, calibration, events, and scheduling time remain. No clock-free quantized relational time is constructed.\\[4pt] 9 & Not reached & The conical worldsheet is not identified with a cosmological curvature singularity, and no continuation through such a singularity is proved.\\[4pt] 10 & Not reached & No vacuum-energy calculation or cancellation mechanism is supplied.\\[4pt] 11 & Not reached & Reference-preserving transport of supplied reports is not gravity-generated entanglement.\\[4pt] 12 & Unresolved & No complete aligned-null contraction or admitted inherited characteristic coefficient is produced.\\ \bottomrule \end{longtable} %===================================================================== \section{Local Grade S release} \label{sec:release} %===================================================================== \begin{definition}[Local Grade S in X-L-ST] \label{def:grade} A local Grade S instance consists of: \begin{enumerate} \item the fixed supported host and analytical completion; \item the normalized free presentation; \item the stated classical variational and restricted domains; \item separately typed residual evaluations and ordinary finite report encodings; \item complete report instruments and finite continuations; \item actual causal, resource, and duration fit certificates; \item explicitly conditional status for every additional descent, correction, or quantum-product hypothesis. \end{enumerate} A conditional algebraic theorem is not relabelled as an instantiated physical acquisition or interacting quantum construction. \end{definition} \begin{theorem}[X-L-ST mathematical release] \label{thm:release} For the data and domains explicitly fixed in this paper: \begin{enumerate}[label=\textup{(\roman*)}] \item the free field differential, gauge fermion, auxiliary--ghost Green matrix, and graded causal carrier have the retained normalization; \item the nonlinear field rules define an off-shell nilpotent classical derivation; \item the classical master residual gives the exact restricted nilpotent domain \(\BValg_{\Rmaster}\); \item unchanged unrestricted free-shell descent is obstructed by the explicit compactly tested witness \eqref{eq:witness}; \item quotient correction requires both ideal preservation and square-zero, and quantum free-fiber recovery requires the additional presentation data of Definition~\ref{def:quantuminput}; \item the odd master residual and even reciprocity pairing remain distinct, with restricted obstruction tests and qualified finite feasibility certificates; \item identical installed X-H-ST report interfaces are recovered exactly, with full residual and reference outputs; \item finite adaptive composition, regular transport, fit checks, and complete-channel error budgets retain their precise stated scope. \end{enumerate} The constructed release is local and classical/operational. The nontrivial interacting quantum presentation remains uninstantiated. XV-B is not opened as a downstream quantum-admission exit. XVI-N and Route B are not discharged. O.1 stays \loc{NOT_CLOSED} outside its independently earned scopes. No inherited characteristic quantum coefficient is admitted. \end{theorem} \begin{proof} Propositions~\ref{prop:gfnorm} and \ref{prop:Ghb}, with Theorem~\ref{thm:freeBRST}, prove (i). Theorem~\ref{thm:kinematic} proves (ii). Proposition~\ref{prop:masterresidual} and Theorem~\ref{thm:restricted} prove (iii). Theorem~\ref{thm:nondescent} proves (iv). Theorems~\ref{thm:conditionalquotient} and \ref{thm:conditionalquantum} prove (v). Section~\ref{sec:separation} proves (vi). Theorems~\ref{thm:instrument} and \ref{thm:reportrecovery} prove (vii). Section~\ref{sec:composition} proves (viii). The standing analytical exits require constructions not present among these results, as recorded in Section~\ref{sec:boundaries}. \end{proof} \subsection{Contribution and prospective impact} The restricted centralizer, the explicit non-descent witness, and the conditional compatibility tests make different claims testable without conflating them. In particular, the free-shell witness prevents an off-shell nilpotence calculation from being promoted to an unchanged interacting quantum carrier. The report recovery shows that this limitation does not prevent complete finite reporting of separately supplied classical data. The potential impact is methodological: a later continuation receives an explicit list of identities, domains, and accessible outputs to construct, rather than a claim of quantum restoration borrowed from shared terminology. No priority claim is made for the general BV, quotient, or channel-composition principles used in the assembly. %===================================================================== \appendix \section{Debugging and acceptance audit} \label{app:debug} %===================================================================== \begin{longtable}{@{}p{.37\linewidth}p{.58\linewidth}@{}} \toprule Potential inference & Correction in X-L-ST\\ \midrule \endfirsthead \toprule Potential inference & Correction in X-L-ST\\ \midrule \endhead Positive ramp width is a worldsheet thickness & The width is analytical metadata. No thick-core conical geometry is introduced.\\[4pt] A nilpotent field rule restores the action & The actual residual is \(\Rmaster=sS_{\mathrm{ret}}\). It is retained unless independently shown zero.\\[4pt] A square-zero restricted domain proves the ambient master equation & The centralizer restricts the domain; it does not set the ambient residual to zero.\\[4pt] A classical free limit is already the free quantum algebra & The field reduction, equation quotient, and graded causal presentation are separate operations.\\[4pt] A kernel specializes without qualification & The full antifield centralizer may have a jumping special fiber; no flatness is asserted.\\[4pt] A local field algebra is closed under products of local functionals & Finite products are placed in the multilocal algebra.\\[4pt] A compact antighost condition is automatically stable & The \(b\)-support must satisfy the same restriction under \(s\bar c=b\).\\[4pt] Relative compact ghosts are unrestricted free ghost solutions & A spacetime-compact homogeneous wave is zero.\\[4pt] The mass-uniform scalar witness solves XVI-N & It tests a different ideal and nonlinear scalar rule, not the prepared insertion or Route B.\\[4pt] Cancellation at one witness proves descent & The complete ideal must be preserved.\\[4pt] Cancellation of reciprocity proves the QME & The reciprocity pairing has different arguments, parity, and ghost number.\\[4pt] An obstruction is automatically a loop anomaly & Both residuals here are classical. A quantum master-Ward construction is additional.\\[4pt] A finite contact fit proves every identity & Finite-dimensional residual coverage and an exhaustive correction class are prerequisites.\\[4pt] A finite normal order is finite dimension & Tangential distribution spaces can remain infinite dimensional.\\[4pt] A Grassmann residual is a qubit amplitude & Only an explicitly extracted ordinary real coefficient is encoded.\\[4pt] One record returns an exact real parameter & One attempt returns a random finite code.\\[4pt] Equal records identify the surviving mechanism & Complete residual channels can differ while their record laws agree.\\[4pt] A record at the sender is available remotely & A causal classical delivery or enlarged payload is charged before remote use.\\[4pt] An untouched reference marginal proves identity transport & The SWAP proof establishes the full payload--reference identity instead.\\[4pt] A consumed clock is a second quantum input & A new factor and its actual preparation or restoration map are required.\\[4pt] Finite execution automatically fits the reserve & Occupancy, ownership, time, locations, and permissions require separate fit checks.\\[4pt] Stagewise marginals price hidden memory & The actual complete memory belongs to matching step interfaces.\\[4pt] A channel budget proves an enclosure & The numerical or analytical enclosure needs its own justification.\\[4pt] Historical H2 erases the constructed I2 advance & The source endings remain version-qualified.\\[4pt] A technical XV-B theorem opens quantum admission & Its interface results do not supply the complete inherited characteristic coefficient.\\[4pt] A scoped O.1 closure updates every prescription & Each independently earned scope retains its own action and counterterm-domain assumptions.\\[4pt] A proposed appendix is a completed repository edit & All installation, testing, and verification fields remain proposed or false.\\ \bottomrule \end{longtable} %===================================================================== \section{Proposed integration record} \label{app:integration} %===================================================================== This appendix is proposed metadata, not a registry transaction or a report of executed checks. \begin{verbatim} identifier: X-L-ST sourceStatus: proposed placement: documentaryAfter: - X-G-ST - X-H-ST - X-I-ST - X-J-ST - X-K-ST grade: local Grade S hostReconstructed: false hostUpdated: false secondConeDerived: false worldsheetThickened: false fixedPositiveRampWidthRetained: true regulatorReplaced: false analyticalEquationsOnMinkowskiCompletion: true constructed: normalizedFreeFieldRules: true normalizedFreeGaugeFermion: true auxiliaryGhostGreenMatrix: true freeCausalBRSTDifferential: true nonlinearOffShellClassicalFieldDifferential: true classicalMasterResidualCentralizer: true compactlyTestedFreeShellNondescentWitness: true completeFiniteResidualReportChannels: true finiteCompositionAndTransportProofs: true conditional: correctedQuotientDescent: additional input required interactingQuantumPresentation: not instantiated finiteCorrectionSolve: complete coverage required predecessorReciprocityEvaluation: exact named witness package required separations: freeNotInteracting: retained classicalResidualNotQuantumProduct: retained instrumentNotAcquisition: retained reciprocityNotLoopAnomaly: retained rampNotConvolutionRepair: retained prescribedConeNotEmergence: retained standingBoundaries: O.1: NOT_CLOSED outside independently earned scopes bulk_symmetry_restored: false XV-B-downstream-admission-exit: not opened XVI-N: not discharged Route-B: not discharged threeTreeIdentity: retained once convolutionObstruction: not repaired sharpWidthLimitIsMetricAdmission: false inheritedCharacteristicQuantumCoefficient: not admitted alignedNullContraction: not constructed quantizedRelationalTimeWithoutExternalClock: not constructed PageCurve: not constructed ultravioletCompletion: not constructed holographicIdentification: not constructed spacetimeEmergence: not constructed appendixStatus: proposed registryEditsExecuted: false repositoryTestsExecuted: false compilationAsserted: false numericalExecutionAsserted: false machineVerificationAsserted: false independentExecutionAsserted: false \end{verbatim} %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Authorship.} The author field is left blank for submission preparation. Citations attribute source records to N.~Cheung; they do not by themselves assign authorship of this draft. \paragraph{Provenance.} The bibliography distinguishes internal manuscript identifiers from published external literature. The supplied passages determine the internal normalization and version comparisons used here. Unavailable complete source texts are not represented as independently inspected proofs. In particular, the reciprocity witness remains an explicitly identified conditional input rather than an invented numerical result. \paragraph{Conditional status.} The conditional quantum-presentation theorem is not a construction of its additional inputs. No interacting quantum differential is claimed on the unrestricted X-G-ST causal algebra. \paragraph{Analytical scope.} All new differential calculations are classical local or multilocal calculations and compactly tested free equation comparisons. No singular interacting quantum graph is defined or extended. \paragraph{Verification.} No successful \loc{pdflatex} run, numerical computation, symbolic execution, proof-assistant verification, repository registration, physical calibration, independent execution, independent review, or journal acceptance is asserted. The mathematical claims are the explicit definitions and written proofs in this file. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{EntPhil} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Enter Calculus Companion. \url{https://enter.grok.me/enter}. Introductory interpretive map, not a theorem premise. \bibitem{EnterI} N.~Cheung, \emph{Manuscript I}. \loc{I.tex}, version 2. Enter Calculus Companion. Class-relative characterization of destructive readout. No broader uniqueness theorem is imported. \bibitem{IB} N.~Cheung, \emph{I-B}. \loc{I-B-v1.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/I-B}. Especially the prescribed conical host, \loc{thm:instrument}, \loc{prop:record-residual}, \loc{prop:fixed-event}, and the regular transport interface. \bibitem{XAST} N.~Cheung, \emph{X-A-ST}. Version 1, supplied manuscript passages. Supported completion chart, fixed-width host contract, and complete destructive instruments. \url{https://enter.grok.me/browse/X-A-ST}. \bibitem{XGST} N.~Cheung, \emph{X-G-ST}. Version 1, supplied manuscript passages. Especially \loc{lem:s0nil}, \loc{prop:gfnorm}, \loc{prop:Ghb}, \loc{thm:freeBRST}, \loc{thm:obstruction}, and \loc{cor:survival}. The displayed auxiliary Green matrix fixes the normalization used in the present paper. The positive reciprocity witness is retained only with its exact named source data. \url{https://enter.grok.me/browse/X-G-ST}. \bibitem{XHST} N.~Cheung, \emph{X-H-ST}. Version 1, supplied manuscript passages. Relative off-shell field rules, classical BV structure, free-shell non-descent comparisons, complete residual instruments, and acquisition boundaries. \url{https://enter.grok.me/browse/X-H-ST}. \bibitem{XIST} N.~Cheung, \emph{X-I-ST}. Version 1, supplied manuscript passages. Classical field deformation, signed BV lift, residual reporting, and finite adaptive composition. \url{https://enter.grok.me/browse/X-I-ST}. \bibitem{XJST} N.~Cheung, \emph{X-J-ST}. Version 1, supplied manuscript passages. Action-derived compact scalar coefficients, selective amplitude reports, and their non-identification with an interacting quantum product. \url{https://enter.grok.me/browse/X-J-ST}. \bibitem{XKST} N.~Cheung, \emph{X-K-ST: Certified composition of destructive clocks and finite instruments}. Version 1 in the supplied source record. Complete finite histories, fresh-clock provenance, causal record readiness, resource fit, and complete-channel error comparisons. \url{https://enter.grok.me/browse/X-K-ST}. \bibitem{XEnterST} N.~Cheung, \emph{X-Enter-ST}. Version 1, supplied manuscript passages. Complete destructive readout of supplied evaluations, amplitude visibility, regular transport, and the acquisition boundary. \url{https://enter.grok.me/browse/X-Enter-ST}. \bibitem{En09} N.~Cheung, \emph{En09-A}. Version 1, supplied manuscript passages. Complete relational clock extraction, destructive name semantics, and the retention of external preparation and scheduling time. \url{https://enter.grok.me/browse/En09-A}. \bibitem{En03} N.~Cheung, \emph{En03-A}. Version 1, supplied manuscript passages. Complete residual instruments and the distinction between record agreement and residual agreement. \url{https://enter.grok.me/browse/En03-A}. \bibitem{IIB} N.~Cheung, \emph{II-B}. \loc{II-B-v1.tex}, version 1. Supplied manuscript passages. Finite rank--minor--left-kernel certificate shape and destructive clock interface. No source-specific matrix entries are imported. \bibitem{XF} N.~Cheung, \emph{X-F}, version 1, and the separately identified \emph{X-F-ST} source record. Three-tree fixed-width background identity. The tree packet is retained once, not rederived or duplicated by the present construction. \bibitem{XConvST} N.~Cheung, \emph{X-A-conv-ST}, version 1, with the separately identified \emph{X-A-conv} source. Supplied documentary records. The fixed classical convolution obstruction is not repaired by reporting or graded restriction. \bibitem{XDST} N.~Cheung, \emph{X-D-ST}. Version 1, supplied source record. Finite reports of supplied limiting evaluations. Its sharp-width comparison is not metric admission. \bibitem{XIVH} N.~Cheung, \emph{XIV-H}. Version 1, supplied passages. Exact internal kernel formulas and the historical H2 internal-amplitude boundary. \bibitem{XIVI} N.~Cheung, \emph{XIV-I}. Version 1, supplied passages. Constructed I2 internal retarded-pole amplitude module and the unresolved I3 full joint characteristic operator map. \bibitem{XIVJ} N.~Cheung, \emph{XIV-J}. Version 1, supplied passages. Reconstruction descent, the supported-sector off-locus reference, and the unresolved J2 full map. No admitted inherited quantum coefficient is released. \bibitem{XVB} N.~Cheung, \emph{XV-B}. Version 2, supplied passages, especially \loc{thm:lifting}, \loc{thm:jetnull}, \loc{thm:pointcorrection}, and \loc{thm:release}. Technical lifting and extension interfaces are distinguished from the unconstructed full characteristic map and downstream quantum admission. \bibitem{XVIN} N.~Cheung, \emph{XVI-N}. Revision r12, supplied passages, especially \loc{thm:massless-obstruction}. Conditional prepared-insertion obstruction; Route B is not claimed discharged. The present scalar free-shell witness is a different problem. \bibitem{XVIrecords} N.~Cheung, \emph{XVI-O}, revision o70-sync; \emph{XVI-P}, revision p23; and \emph{XVI-Q}, revision Q11. Version-qualified supplied records. The reported one-loop two-point closure has its own action, external-leg, and counterterm-domain scope and does not update every O.1 prescription or establish bulk symmetry restoration. \bibitem{OaiRecords} N.~Cheung, \emph{Oai282-A} and \emph{Oai283-A}, version 1, supplied passages. Documentary comparisons for residual coverage, finite interfaces, and resource-accounted errors. The separate OpenAI catalogue and Families 282, 362, and 376 are not theorem premises or independently verified proofs here. \bibitem{FRclassical} 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. doi:10.1007/s00220-012-1487-y. \url{https://arxiv.org/abs/1101.5112}. External background for local/multilocal functionals, variational actions, and classical BV domains. \bibitem{FRquantum} K.~Fredenhagen and K.~Rejzner, ``Batalin--Vilkovisky formalism in perturbative algebraic quantum field theory,'' \emph{Communications in Mathematical Physics} \textbf{317} (2013), 697--725. doi:10.1007/s00220-012-1617-1. \url{https://arxiv.org/abs/1110.5232}. External distinction between renormalized products, quantum BV structures, and master-Ward anomalies. No interacting product for the present target is imported. \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}. External wave-equation and causal Green-operator background. All new free Green operations here are on the declared Minkowski completion. \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/}. External finite-channel, reference-extension, trace-distance, and diamond-norm framework. \end{thebibliography} \end{document}