% XX-F.tex % Self-contained pdfLaTeX source; no external inputs or BibTeX. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XX-F.tex % pdflatex -interaction=nonstopmode -halt-on-error XX-F.tex % % No successful compilation, proof-assistant verification, or % independent numerical or symbolic execution is asserted. % % RELEASE QUALIFICATION: % The native prerequisite package required by the proposed XX-F plan % has not been supplied in XX-A--XX-E. This source therefore does not % falsely announce a native quantum breaking, admissible cancellation, % interacting Ward identity, or closure of Gates Q4--Q5. % % New results: % 1. An explicit same-operator witness for the central-second-jet % obstruction on the unfiltered compact-test algebra. % 2. The complete first-order Leibniz defect of the coupled jet operation. % 3. A correction to the order interpretation: that centrality defect % starts at total s^3 in the weighted regular jet algebra, so it is % invisible after a common s^2 truncation. The free-equation quotient % obstruction nevertheless remains at order s. % 4. A weaker, explicit weighted Newton domain, with sharp threshold % for absolute convergence, local cutoff-error estimates, and % output-decay bounds. % 5. A complete normal-form/null test for the displayed contact module, % followed by its exact Newton image. % 6. A conditional breaking formula that includes changes of the Ward % operation as well as insertion counterterms. % 7. Constrained simultaneous repair, distinguishing an affine space % of formal corrections from nonlinear admissibility conditions. % % Debugging corrections: % - The prescribed compact geometric group is not the trivial compact % gauge-fixed residual subgroup. % - A formal classical symbol is not a pseudodifferential principal % symbol or an expectation-value limit. % - A positive-state constraint need not define a vector space. % - Zeroth-order changes of covariance require recomputation of the % construction; they are not automatically hbar-order counterterms. % - A distributional Newton-contact image is calculated before any % additional diagonal restriction. % - A finite normal derivative bound is not finite-dimensionality. % - The XIV characteristic graph is a separate target, not a Q3 input. % - All macros used below, including \abs, are explicitly defined. \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=25mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage{microtype} \usepackage{xurl} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={XX-F: Native Ward Admission for the Original-Rod Lapse}, pdfauthor={}, pdfsubject={Filtered quantum jet tests, weighted Newton domains, and conditional Q4--Q5 repair} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{3pt} \numberwithin{equation}{section} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\Ker}{\operatorname{ker}} \newcommand{\im}{\operatorname{im}} \newcommand{\Lie}{\mathcal L} \newcommand{\Dc}{C_c^\infty} \newcommand{\Sch}{\mathcal S} \newcommand{\barg}{\bar g} \newcommand{\barphi}{\bar\phi} \newcommand{\calA}{\mathcal A} \newcommand{\calB}{\mathcal B} \newcommand{\calC}{\mathcal C} \newcommand{\calF}{\mathcal F} \newcommand{\calH}{\mathcal H} \newcommand{\calL}{\mathcal L} \newcommand{\calR}{\mathcal R} \newcommand{\Gsp}{\mathsf G_{\mathrm{sp}}} \newcommand{\ret}{\mathrm{ret}} \newcommand{\adv}{\mathrm{adv}} \newcommand{\Ward}{\mathcal V} \newcommand{\pair}[2]{\langle #1,#2\rangle} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\sid}[1]{\texttt{\detokenize{#1}}} \newcommand{\wgt}[1]{\langle #1\rangle} \newcommand{\diag}[1]{\bigl[#1\bigr]_{\mathrm{diag}}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \title{\textbf{XX-F}\\[4pt] Native Ward Admission for the Original-Rod Lapse\\[4pt] \large Filtered Quantum-Jet Tests, Weighted Newton Domains,\\ and the Exact Conditional Q4--Q5 Repair Problem} \author{} \date{} \begin{document} \maketitle \begin{abstract} The proposed XX-F calculation assumes an admitted native Q3 insertion and a quantum Ward operation on its domain. Those objects are not supplied by XX-A--XX-E. An unconditional native breaking term, counterterm cancellation, or interacting identity therefore cannot be released from the attached package. We nevertheless sharpen the missing interfaces by proving new results on the identical background and free operator. The coupled classical jet transformation has a calculable first-order Leibniz defect on the centrally extended free algebra. A same-operator Cauchy witness proves that its antisymmetric part can be nonzero, excluding a first-order quantum derivation with that leading action on the unfiltered algebra. We then correct the order interpretation: the central-second-jet defect starts at total perturbative order three on a specified weighted regular jet algebra. It is invisible through total order two, while the free-equation quotient obstruction persists at order one. A weighted Newton theorem replaces unnecessarily strong Schwartz assumptions by explicit decreasing-source bounds, with cutoff-error and output-decay estimates. A finite spatial contact module is reduced to a unique normal form before its Newton image is taken. This prevents presentation-dependent contact counting and illegitimate diagonal readout. Finally, we derive the exact conditional breaking and repair equations when both the insertion and Ward operation may receive quantum corrections. Linear image membership is separated from nonlinear admissibility restrictions such as positivity. None of these interface theorems verifies the missing native covariance, infrared estimates, quantum operation, or interacting contribution inventory. Native Q3 remains open; Q4 and Q5 remain unentered and unclosed. No universal obstruction to a future native completion is asserted. \end{abstract} \tableofcontents %===================================================================== \section{The requested calculation and the application boundary} \label{sec:scope} %===================================================================== The requested deliverable is a native quantum breaking term, a calculated admissible counterterm image, cancellation, and a retained-order quantum identity. The plan explicitly requires a completed native prerequisite package before those calculations begin. XX-A supplies the classical freeze. XX-B supplies its stated compact-test free input. XX-C retains state and infrared choices. XX-D's native one-contraction construction is conditional. XX-E supplies domain tests and comparison images, not the missing quantum operation. The full texts of these five drafts are available in the present exchange \cite{XXA,XXB,XXC,XXD,XXE}. Accordingly, this manuscript is a complete proof of its \emph{stated interface results}, not a purported completion of the absent native quantum calculation. \paragraph{Gate numbering.} The original inventory assigns complete quantum remainder control to Q4 and the retained quantum identity to Q5. The proposed XX-F plan organizes the breaking calculation and its repair jointly under Q4--Q5. We distinguish these usages rather than declare the original remainder obligation closed by a renamed algebraic test. \paragraph{Source discipline.} Cheung's records provide comparison shapes: same-target admission, actual-image repair, and nontransfer \cite{CheungXVIIIA,CheungXIXD,CheungXIXE}. Their documentary status does not supply an unprinted covariance, anomaly, or counterterm coefficient. Theorems proved here use the displayed equations. \begin{definition}[Native entry package] \label{def:entry} Entry into the requested native Q4--Q5 calculation requires: \begin{enumerate}[label=\textup{(\roman*)}] \item a specified normalized covariance representative; \item verified products and infrared estimates for the insertion and every operation used by its Ward map; \item an admitted native insertion, its output space, normalization, and allowed Newton-contact images; \item a consistent treatment of second jets; \item a quantum Ward operation on the same domain, including its action on field relations, products, and normalization data; \item the complete contribution inventory at the orders claimed. \end{enumerate} Any operation with two noncompact rod legs also requires its actual two-endpoint covariance domain. \end{definition} No item is fulfilled merely by introducing its name. In particular, an undefined native breaking is not zero. \paragraph{Analytic frameworks.} Regular functional products provide the compact comparison algebra used below \cite{FR}. Normally hyperbolic Cauchy theory supplies the same-operator witness \cite{BGP}. Distribution-extension and quantum BV frameworks have separate hypotheses \cite{BF,FRBV,BDFrR}. They are not invoked as automatic constructions of the nonlocal lapse. %===================================================================== \section{Frozen geometric and free-field data} \label{sec:freeze} %===================================================================== \subsection{Background, clock, and output density} Use signature \((-,+,+,+)\) and \begin{equation} S[g,\phi]=\int\sqrt{-g} \left(\frac{M^2}{2}R[g] -\frac12g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi -V(\phi)\right)\dd^4x. \label{eq:action} \end{equation} The notation means the local action density with compact variations; no finite total background action is asserted. Fix \(M>0\), \(p>1\), \(t_*>0\), and \begin{align} A&=\sqrt{2p}\,M,& a(t)&=(t/t_*)^p,& H(t)&=p/t, \notag\\ \barphi(t)&=\phi_*+A\log(t/t_*),& \barg&=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j, \notag\\ V(\phi)&=\frac{M^2p(3p-1)}{t_*^2} e^{-2(\phi-\phi_*)/A}. \label{eq:background} \end{align} Work on \(I\times\R^3\), where \(I=(t_-,t_+)\) and \(01\), define decreasing-source seminorms \begin{equation} \norm{b}_{q,m;J} =\max_{j+\abs{\alpha}\le m} \sup_{t\in J,\mathbf x} \wgt{\mathbf x}^{q} \abs{\partial_t^j\partial_{\mathbf x}^{\alpha} b(t,\mathbf x)}, \qquad J\Subset I. \label{eq:weighted} \end{equation} Sources are smooth and possess the finite seminorms needed for each conclusion. These conditions are weaker than requiring every Schwartz seminorm to be finite. Let \[ \mathsf N b=\int P_i(\mathbf x-\mathbf y) b^i(t,\mathbf y)\dd^3y. \] For the rods, \(b^{ia}=(\calCz)^{ia}\). \begin{theorem}[Weighted native Newton realization] \label{thm:weighted} Suppose \(q>1\) and all required seminorms \eqref{eq:weighted} are finite. Then \(\mathsf N b\) is absolutely defined, smooth, and tends to zero at spatial infinity. It solves \[ \Delta\mathsf N b=-\partial_i b^i \] and is the unique smooth decaying solution. Uniformly for \(t\in J\), its undifferentiated output, and each fixed time derivative with the same source bound, obey \begin{equation} \abs{\mathsf N b(t,\mathbf x)} \le C\norm{b}_{q,0;J} \begin{cases} \wgt{\mathbf x}^{1-q},&13. \end{cases} \label{eq:weighteddecay} \end{equation} For a bounded cutoff profile \(\chi_R=\chi(\cdot/R)\) equal to one on the radius-\(R\) ball, put \(\mathsf N_Rb=\mathsf N(\chi_Rb)\). For every compact output region \(K\) and fixed derivative order \(m\), \begin{equation} \norm{\mathsf N b-\mathsf N_Rb}_{C^m(K)} \le C_{K,m}R^{1-q}\norm{b}_{q,m;J} \label{eq:cutoffestimate} \end{equation} for all sufficiently large \(R\). \end{theorem} \begin{proof} Near \(\mathbf y=\mathbf x\), the dipole kernel has locally integrable size \(C\abs{\mathbf x-\mathbf y}^{-2}\). At infinity, for compact \(\mathbf x\), its product with the source has radial majorant \(Cr^{-q}\dd r\), integrable for \(q>1\). Local smoothness follows by writing \(\mathbf r=\mathbf x-\mathbf y\), splitting with a smooth local cutoff in \(\mathbf r\), and transferring output derivatives to the smooth source in the singular part. The nonsingular part may be differentiated directly. Weighted source derivatives justify all these operations. For the decay estimate let \(r=\abs{\mathbf x}\) be large. In \(\abs{\mathbf y}\le r/2\), the kernel is bounded by \(Cr^{-2}\). The source integral is bounded by \(Cr^{3-q}\) for \(q<3\), \(C\log r\) for \(q=3\), and \(C\) for \(q>3\). In \(r/2<\abs{\mathbf y}<2r\), bound the source by \(Cr^{-q}\) and the dipole integral over a ball of radius \(3r\) by \(Cr\). In \(\abs{\mathbf y}\ge2r\), use the radial integral \(C\int_{2r}^{\infty}\rho^{-q}\dd\rho\). These give \eqref{eq:weighteddecay}. Time derivatives are treated by differentiating only the source. For compact output and sufficiently large \(R\), the cutoff difference is supported away from the kernel singularity. Each output spatial derivative of \(P_i\) improves its tail; time derivatives act on \(b\). Bounding all such terms by the undifferentiated radial estimate gives \eqref{eq:cutoffestimate}. Compact cutoff sources satisfy the distributional Poisson equation. The local smooth convergence, also for two output derivatives, passes it to the limit. A difference of smooth decaying solutions is entire harmonic and vanishes by the maximum principle. \end{proof} The \(q>1\) bound guarantees existence and decay, not necessarily an inverse-square rate. The stronger \(q>3\) condition gives that rate for the undifferentiated rod. \begin{proposition}[The absolute-convergence threshold cannot be lowered universally] \label{prop:threshold} At \(q=1\), the pointwise bound \(\abs{b(\mathbf y)}\le C\wgt{\mathbf y}^{-1}\) does not universally ensure absolute Newton-divergence convergence. \end{proposition} \begin{proof} Take a smooth source which outside a ball is \[ b^i(\mathbf y)=\frac{y_i}{\abs{\mathbf y}^{2}}. \] At \(\mathbf x=0\), \[ P_i(-\mathbf y)b^i(\mathbf y) =\frac1{4\pi\abs{\mathbf y}^{3}}. \] Its spatial integral diverges logarithmically. The source has the displayed rate and can be smoothly filled in inside the ball. \end{proof} This is a sharp threshold for the stated \emph{uniform sufficient absolute-convergence class}. Conditional cancellations or different domains can give other constructions. \subsection{Internal and external quantum uses remain hypotheses} Let \(D\) be the fixed symmetric subtraction remainder in the XX-D construction. Its internal equal-time rod integrand is \[ d^{ia}(t,\mathbf y,\mathbf x) =\left[ \calC_y^{ia}\partial_{x^a}\ell_xD(y,x) \right]_{t_y=t_x=t}. \] \begin{corollary}[Weighted internal normalization criterion] \label{cor:internal} Suppose \(d\) is smooth near equal-time coincidence and, for compact output \(K\), its required output derivatives satisfy \[ \abs{\partial_{t,\mathbf x}^{\beta}d^{ia} (t,\mathbf y,\mathbf x)} \le C_{K,\beta}\wgt{\mathbf y}^{-q}, \qquad q>1 \] outside a fixed ball. Then the internal rod correction \[ -2\int P_i(\mathbf x-\mathbf y) d^{ia}(t,\mathbf y,\mathbf x)\dd^3y \] exists, is cutoff independent, and has cutoff error \(O(R^{1-q})\) in the stated output differentiability norms. \end{corollary} \begin{proof} Near coincidence, change to relative spatial coordinates and use smoothness of \(d\) and local integrability of \(P_i\). At infinity, output differentiation is legitimate and the same radial estimate as \eqref{eq:cutoffestimate} applies. \end{proof} For an external compact probe \(g\), a sufficient alternative to XX-D's Schwartz condition is that \(\calC W(\cdot,g)\) have the decreasing-source bounds of Theorem~\ref{thm:weighted}, uniformly on bounded probe families. That theorem then defines its original rod. Any Ward-transformed kernel needs the same estimates for its \emph{actual transformed entries}. No selected native representative satisfying these conditions is constructed here. The weaker criteria are new usable sufficient conditions, not verification of Q3. %===================================================================== \section{Contact normal forms before Newton composition} \label{sec:contacts} %===================================================================== \subsection{The displayed finite contact module} For finite \(m\), consider \begin{equation} b^{ia}(t,\mathbf y;t',\mathbf z) =\delta(t-t') \sum_{\abs{\alpha}\le m} c_\alpha^{ia}(t,\mathbf y,\mathbf z) \partial_{\mathbf y}^{\alpha} \delta^3(\mathbf y-\mathbf z), \label{eq:contact} \end{equation} with smooth coefficients and compact tangential support. Let the source reconstruction map be \(c\mapsto b\). It is not assumed injective. Define \begin{equation} j_\gamma^{ia}(t,\mathbf z) =\sum_{\substack{\alpha\ge\gamma\\\abs{\alpha}\le m}} (-1)^{\abs{\alpha-\gamma}} \binom{\alpha}{\gamma} \left(\partial_{\mathbf y}^{\alpha-\gamma} c_\alpha^{ia}\right) (t,\mathbf z,\mathbf z). \label{eq:j} \end{equation} \begin{theorem}[Complete normal form and null test for this module] \label{thm:contactnormal} The reconstructed source has the exact unique form \begin{equation} b^{ia}=\delta(t-t') \sum_{\abs{\gamma}\le m} j_\gamma^{ia}(t,\mathbf z) \partial_{\mathbf y}^{\gamma} \delta^3(\mathbf y-\mathbf z). \label{eq:contactnormal} \end{equation} It is zero if and only if every \(j_\gamma^{ia}\) is zero. The original Newton-divergence image is \begin{equation} (\mathsf N b)^a(t,\mathbf x;t',\mathbf z) =\delta(t-t') \sum_{\abs{\gamma}\le m} j_\gamma^{ia}(t,\mathbf z) \partial_{\mathbf x}^{\gamma} P_i(\mathbf x-\mathbf z). \label{eq:Newtoncontact} \end{equation} Thus it depends on the reconstructed source, not on an arbitrary choice of coefficient presentation. \end{theorem} \begin{proof} Smooth multiplication of delta derivatives gives \[ c_\alpha(\mathbf y,\mathbf z) \partial_{\mathbf y}^{\alpha}\delta(\mathbf y-\mathbf z) =\sum_{\gamma\le\alpha} (-1)^{\abs{\alpha-\gamma}}\binom{\alpha}{\gamma} (\partial_{\mathbf y}^{\alpha-\gamma}c_\alpha) (\mathbf z,\mathbf z) \partial_{\mathbf y}^{\gamma} \delta(\mathbf y-\mathbf z). \] Collect coefficients to obtain \eqref{eq:contactnormal}. To extract a given coefficient, test in the relative variable \(\mathbf y-\mathbf z\) with \[ \theta(\mathbf y-\mathbf z) \frac{(\mathbf y-\mathbf z)^\gamma}{\gamma!}, \] where \(\theta=1\) near zero, and multiply by an arbitrary compact tangential test in \((t,\mathbf z)\). Every relative derivative at zero except \(\gamma\) vanishes. This extracts \(j_\gamma\), up to its fixed sign, proving uniqueness and the null test. For \eqref{eq:Newtoncontact}, the relative variables \(\mathbf x-\mathbf y\) and \(\mathbf y-\mathbf z\) are independent coordinates. Their tensor-product distribution is defined before pushforward. The contact restricts the integration to \(\mathbf y=\mathbf z\), making the operation proper on compact output tests. Its derivative signs give \[ \int P_i(\mathbf x-\mathbf y) \partial_{\mathbf y}^{\gamma} \delta(\mathbf y-\mathbf z)\dd^3y =\partial_{\mathbf x}^{\gamma} P_i(\mathbf x-\mathbf z). \] This proves the image formula. \end{proof} The normal-form shape agrees with the reconstruction discipline in Cheung's XV-B and XIV-J, but the present contact variables and Newton map are independently displayed and proved \cite{CheungXVB,CheungXIVJ}. \begin{remark}[What the image theorem does not authorize] The kernel in \eqref{eq:Newtoncontact} is a distribution across \(\mathbf x=\mathbf z\). An additional coincident-point restriction is not supplied. The zeroth derivative already has a noncompact dipole tail. Higher derivatives can include singular contact pieces as well as their off-diagonal tails. The finite bound \(\abs{\gamma}\le m\) does not make the coefficient space finite dimensional: the smooth tangential functions \(j_\gamma\) remain freely variable. Nor is the Newton map necessarily injective on sources; divergence-free source directions can be invisible. \end{remark} \subsection{A presentation-dependent highest term is not an invariant} In one relative variable \(q\), \[ q\delta'(q)+\delta(q)=0. \] Indeed, its value on a test \(\psi\) is \(-\partial_q(q\psi)(0)+\psi(0)=0\). Thus a degree-one presentation and a degree-zero presentation can reconstruct opposite nonzero terms. No leading counterterm, anomaly, or normalization space may be counted from presentation degree alone. The full reconstructed null relations must be respected. This local control does not construct the separate XIV characteristic graph \cite{CheungXIVI,CheungXIVJ,CheungXVB}. %===================================================================== \section{The conditional native breaking formula} \label{sec:breaking} %===================================================================== \subsection{What must be constructed before the formula is an evaluation} Suppose a future native package gives an admitted insertion and an operation on one specified domain: \begin{equation} I_f=O_f+\hbar I_{1,f}\pmod{s^3,\hbar^2},\qquad \mathcal D_\xi=\Ward_\xi+\hbar\mathcal A_\xi \pmod{\hbar^2}. \label{eq:expansion} \end{equation} The normalization data held fixed by \(\Ward_\xi\) must be declared. Any additional variations of those data belong to the specified \(\mathcal A_\xi\) or to another explicitly accounted operation. \begin{theorem}[Complete coefficient formula on an admitted domain] \label{thm:breaking} If \eqref{eq:expansion} is constructed on the same native domain and its classical symbol is \eqref{eq:O}, then the retained breaking is \begin{equation} \mathcal D_\xi I_f =\hbar B_{\xi,f}\pmod{s^3,\hbar^2},\qquad B_{\xi,f} =\Ward_\xi I_{1,f}+\mathcal A_\xi O_f. \label{eq:breaking} \end{equation} If, in fixed normalization coordinates, \[ I_{1,f}=\frac{s^2}{2}\Gamma_f+R_{1,f}, \] where \(\Gamma_f\) is a defined field-independent scalar Wick correction, then \[ B_{\xi,f} =\Ward_\xi R_{1,f}+\mathcal A_\xi O_f. \] Neither \(R_{1,f}\) nor \(\mathcal A_\xi O_f\) may be set to zero merely because the free scalar correction has zero ordinary field variation. \end{theorem} \begin{proof} Multiply the formal expansions. The order-zero term is \(\Ward_\xi O_f=0\) by Lemma~\ref{lem:classical}. The order-one term is exactly \eqref{eq:breaking}. Field independence gives \(\Ward_\xi\Gamma_f=0\) in the stated fixed coordinates. It says nothing about the other two terms. \end{proof} This theorem calculates a formal coefficient \emph{from constructed application data}. It does not calculate the native \(B_{\xi,f}\) from XX-A--XX-E, because neither the required operation nor its complete quantum insertion coefficient has been supplied. Quantum BV and anomalous Master Ward constructions distinguish the quantum operation from ordinary classical differentiation \cite{FRBV,BDFrR}. An anomaly-consistency framework is not a value for the present nonlocal \(B_{\xi,f}\). \subsection{Consistency is conditional on the operation} For a separately constructed Lie-algebra action \(d_x=d_{0,x}+\hbar d_{1,x}\) satisfying \([d_x,d_y]=d_{[x,y]}\), and an insertion with \(d_{0,x}O=0\), let \(d_xI=\hbar B_x\). Then \begin{equation} d_{0,x}B_y-d_{0,y}B_x=B_{[x,y]}. \label{eq:consistency} \end{equation} \begin{proof} Apply the representation identity to \(I\) and take the first quantum coefficient. \end{proof} For the pullback anti-action, the corresponding sign changes must be used. Equation \eqref{eq:consistency} is not imposed on an undefined defect. Nor does satisfying it prove that a defect is an admissible counterterm image. These distinctions are central to anomalous Ward analyses \cite{BDFrR}. %===================================================================== \section{Admissible images: insertion and operation changes} \label{sec:repair} %===================================================================== \subsection{The variation of the operation must also be counted} Let \(\mathscr C\) be a specified vector space of formal correction data after the native package has been constructed. Suppose its calculated insertion and operation changes are \[ I_f\longmapsto I_f+\hbar J_f(c),\qquad \mathcal D_\xi\longmapsto \mathcal D_\xi+\hbar M_\xi(c). \] Here \(J_f\) and \(M_\xi\) are actual linear maps on this correction space. \begin{theorem}[Calculated image including Ward-operation changes] \label{thm:image} The retained breaking changes by \begin{equation} B_{\xi,f}\longmapsto B_{\xi,f}+L_{\xi,f}c, \qquad L_{\xi,f}c =\Ward_\xi J_f(c)+M_\xi(c)O_f. \label{eq:image} \end{equation} If the Ward operation is fixed, \(M=0\). If the insertion correction is only a field-independent scalar in fixed normalization coordinates, its \(\Ward_\xi J_f\) image is zero. \end{theorem} \begin{proof} Expand the transformed operation on the transformed insertion through first order in \(\hbar\). The cross term \(M(c)J(c)\) has order \(\hbar^2\) and is discarded. The remaining new terms are exactly \eqref{eq:image}. \end{proof} Thus scalar reordering changes alone cannot cancel a nonzero breaking in the class with a fixed ordinary leading operation and no other image. This is a statement about that declared class, not a verdict about an unevaluated native anomaly. A zeroth-order covariance replacement \(W\mapsto W+S\) is a new choice of free construction. It must be recomputed or transported on a verified domain. It is not automatically a correction of the form \(\hbar J(c)\) to a fixed package. \subsection{Subsidiary defect and simultaneous repair} Retain the free mixed defect \[ D_{\mathrm{sub}}=-KW_Q-rW\calF^\dagger. \] For real symmetric smooth bisolution changes, \[ D_{\mathrm{sub}}\longmapsto D_{\mathrm{sub}} -\left(KS_Q+rS_{\calB}\calF^\dagger\right). \] Its smoothness does not imply either vanishing or compatibility with the native infrared domain. Suppose the calculated combined correction map is \[ \mathcal M:\mathscr C\longrightarrow V_{\mathrm{break}}\oplus V_{\mathrm{sub}}, \qquad \mathcal M c=(Lc,Tc). \] Its targets are the actual complete defects, with all tests and retained coefficients included. \begin{definition}[Constrained admissibility] \label{def:admissibility} Let \(\mathscr C_{\mathrm{allow}}\subseteq\mathscr C\) be the set of corrections satisfying all declared constraints: domains, normalization, reality, subsidiary conditions, infrared bounds, and any required physical positivity or quotient conditions. It need not be a vector space or an affine space. \end{definition} \begin{theorem}[Constrained simultaneous repair] \label{thm:repair} For constructed \(B,D_{\mathrm{sub}},L,T\), a permitted simultaneous correction exists exactly when \begin{equation} (-B,D_{\mathrm{sub})}\in \mathcal M(\mathscr C_{\mathrm{allow}}), \label{eq:constrained} \end{equation} where the notation on the left is understood as the ordered pair \((-B,D_{\mathrm{sub}})\). More explicitly, if \(c_*\in\mathscr C\) solves the unconstrained equation \(\mathcal M c_*=(-B,D_{\mathrm{sub}})\), then the complete permitted solution set is \begin{equation} \left(c_*+\Ker L\cap\Ker T\right) \cap\mathscr C_{\mathrm{allow}}. \label{eq:solutions} \end{equation} If the allowed set is the whole vector space, this reduces to the affine actual-image criterion. \end{theorem} \begin{proof} A simultaneous permitted repair means \(B+Lc=0\), \(D_{\mathrm{sub}}-Tc=0\), and \(c\in\mathscr C_{\mathrm{allow}}\). This is exactly membership in the displayed image. The difference of two unconstrained solutions belongs to \(\Ker L\cap\Ker T\), and every element of that common-kernel translate is an unconstrained solution. Intersect with the allowed set to obtain \eqref{eq:solutions}. \end{proof} \begin{remark}[Typographical clarification] Equation \eqref{eq:constrained} denotes \[ \boxed{\,(-B,D_{\mathrm{sub}}) \in\mathcal M(\mathscr C_{\mathrm{allow}})\,}. \] The ordered pair, not a quotient or image closure, is the exact target. \end{remark} \begin{proposition}[Linear solvability does not imply positivity-compatible solvability] \label{prop:positivity} Constraints on a covariance can make the allowed correction set nonaffine. Therefore solving a linear Ward equation in an enlarged vector space need not solve the constrained problem. \end{proposition} \begin{proof} Already for real symmetric matrices, start from \(\mu_0=\operatorname{diag}(0,1)\). The allowed changes preserving \(\mu_0+S\ge0\) do not form a vector space: \(S=\operatorname{diag}(1,0)\) is allowed, but \(-S\) is not. A linear equation demanding the latter direction is solvable in the unrestricted symmetric-matrix space and not in the positivity-compatible set. The example proves the logical distinction without identifying this matrix problem with the native physical quotient. \end{proof} The actual-image discipline is consistent with Cheung's XVIII-A and the supplied XV-B interfaces \cite{CheungXVIIIA,CheungXVB}. It is an application criterion, not a native cancellation proof here. \subsection{Finite probes and normalization freedom} If a finite probe map \(P\) satisfies \(P(B+Lc)=0\), the full identity follows only with independently proved coverage or injectivity on the allowed residual space. Otherwise a nonzero element of \(\Ker P\) is invisible. Pure-trace probes in particular miss the smooth compact metric-slot kernel \[ A^{\mu\nu}(x)A^{\rho\sigma}(y), \qquad A^{\mu\nu}\barg_{\mu\nu}=0,\quad A\ne0. \] This blind sector does not make a completely specified causal Green operator nonunique. It prevents selected contractions from certifying an unrestricted kernel \cite{CheungXVIIIA}. %===================================================================== \section{Formal orders, two-rod domains, and optional removal} \label{sec:orders} %===================================================================== \subsection{Formal order is not a numerical remainder estimate} A nonzero \(\hbar^0\) residual cannot be canceled by a correction divisible by \(\hbar\). This follows immediately from coefficientwise formal equality and applies to the classical cutoff residual of XX-E. Reduction modulo \(\hbar\), whenever the quantum insertion is admitted with the chosen symbol, recovers \eqref{eq:O}. The original one-rod term survives, and \(Y_2\) is absent for \(\partial_iN_0=0\). This is a formal classical-symbol statement, not a pseudodifferential principal-symbol theorem, an expectation-value limit, or a numerical \(O(\hbar^2)\) estimate. Finite self-contraction of a quadratic regular polynomial terminates algebraically. That fact does not bound the missing interacting coefficient, its complete quantum remainder, or a regulator-removal family. \subsection{Product-order bookkeeping} With one common amplitude parameter, \[ O_f^{(1)}=sL_f+s^2B_f(z,\cdot)\pmod{s^3}. \] Consequently the first quantum contraction of two insertions through total \(s^2\) is only \[ \hbar s^2W(L_f,L_g). \] A single noncompact quadratic derivative can enter at total order \(s^3\); two can enter at total order \(s^4\). Coefficientwise products of the quadratic coefficients require their own domains. The corrected total-order statement is XX-E's, not a result supplied by the earlier XX-D two-endpoint discussion. Whenever two rods actually enter, the canonical constraint from XX-C is retained: \begin{align} E_{RR}^{ab}(t_0+\delta,t_0;\mathbf r) &=-\frac{\delta}{16\pi M^2a(t_0)^3r} \left((p+9)\delta_{ab}-(p+1)n_an_b\right) \notag\\ &\quad+O(\delta^2r^{-1})+O_\delta(r^{-2}), \qquad n=\mathbf r/r. \label{eq:RR} \end{align} A domain preserving this nonzero canonical inverse-distance term cannot require both ordered two-rod covariances to be \(O(r^{-2})\). No two-endpoint covariance extension is proved by Theorem~\ref{thm:weighted}; its theorem concerns one Newton output with a controlled source. No two-rod Ward contribution is evaluated here. \subsection{The escaping-source control survives} For a nonzero nonnegative \(\rho\in\Dc((1,2))\), put \(\eta(\mathbf Y)=Y_1\rho(\abs{\mathbf Y})\). Let \(\tau\in\Dc(I)\), and take \[ h_{ij,R}=2a^2\tau R^{-1}\eta(\mathbf x/R)\delta_{ij}, \qquad h_{00,R}=h_{0i,R}=\varphi_R=0. \] These inputs vanish eventually on every fixed compact set, but \[ \calRz_R(t,\mathbf x) =\tau(t)\int P_a(-\mathbf Y+\mathbf x/R) \eta(\mathbf Y)\dd^3Y \] converges locally to a nonzero constant spatial vector times \(\tau(t)\). Its first component is \[ \frac{\tau(t)}{4\pi}\int \frac{Y_1^2}{\abs{\mathbf Y}^3} \rho(\abs{\mathbf Y})\dd^3Y. \] Each finite input has the decaying original rod. For the new weighted norm with \(q>1\), the source norm grows like \(R^{q-1}\). Thus the example is correctly excluded by the uniform weighted control needed for \eqref{eq:cutoffestimate}. Local convergence alone remains insufficient. \subsection{A future comparison theorem must be uniform} After an actual retained quantum identity has been proved, regulator removal may be considered. Keep independent parameters, for example \[ \lambda=(\epsilon,R,\Lambda_{\mathrm{IR}}). \] A removal theorem must state the output topology, uniform bounds for the actual products and Ward operations, and a Cauchy or direct convergence estimate. The rate \eqref{eq:cutoffestimate} is useful only when its source norms are uniform in the other parameters. If those norms grow with \(\epsilon^{-1}\) or an infrared cutoff, a joint limit does not follow. Two refinement families with separate limits need a cross-family estimate or a common characterization before regulator independence can be claimed. XIX-C provides the relevant non-determination comparison shape \cite{CheungXIXC}. No native quantum removal is attempted here. %===================================================================== \section{Native prerequisite audit and mathematical release} \label{sec:release} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.25\linewidth}p{.17\linewidth}Y@{}} \toprule Required native object & Present ending & Exact qualification\\ \midrule Selected covariance & Not supplied & A Hadamard class and smooth freedoms remain.\\[3pt] Native internal and external infrared bounds & Not verified & Section~\ref{sec:weights} gives weaker sufficient criteria, not a selected representative.\\[3pt] Admitted native insertion & Conditional only & XX-D's native theorem is not upgraded.\\[3pt] Consistent second-jet quantum treatment & Not supplied & Unfiltered obstruction and filtered qualification are now distinguished.\\[3pt] Native quantum Ward operation & Not supplied & The regular off-shell comparison does not descend to the gauge-fixed equation quotient.\\[3pt] Complete interacting inventory & Not evaluated & One contraction is not a complete interacting loop.\\[3pt] Native breaking and admissible repair & Undefined here & Conditional formulas do not constitute evaluation.\\[3pt] Q4 complete quantum remainder & Not closed & Formal truncation and classical cutoff rates are not its numerical or distributional remainder theorem.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[XX-F release] \label{thm:release} On the identical printed target, XX-F proves: \begin{enumerate}[label=\textup{(\roman*)}] \item the coupled classical identity including the original rod contribution; \item the explicit first-order Leibniz defect and a same-operator witness excluding the specified unfiltered central-second-jet derivation; \item the correction that this centrality defect begins at total \(s^3\) on the weighted regular jet algebra, while the free-equation quotient obstruction remains at order \(s\); \item weighted decreasing-source Newton existence, uniqueness, decay, and cutoff-error estimates; \item the complete null test and exact Newton image for the displayed spatial contact module; \item conditional native breaking and correction-image formulas, including changes of the Ward operation; \item constrained simultaneous repair and the distinction between unrestricted linear solvability and actual admissibility. \end{enumerate} Definition~\ref{def:entry} has not been discharged for the native quantum lapse. Its actual complete breaking, admissible quantum counterterm image, cancellation, interacting identity, and complete quantum remainder are not released. Native Q3 remains open. Under the proposed entry rule, native Q4 and Q5 remain unentered and unclosed. The specified comparison obstruction is not a universal nonexistence theorem for native completion. \end{theorem} \begin{proof} Item (i) is Lemma~\ref{lem:classical}. Item (ii) is Theorem~\ref{thm:Leibniz}, Lemma~\ref{lem:witness}, and Theorem~\ref{thm:unfiltered}. Item (iii) is Theorem~\ref{thm:filtered} and Proposition~\ref{prop:quotient}. Item (iv) is Theorem~\ref{thm:weighted}. Item (v) is Theorem~\ref{thm:contactnormal}. Item (vi) is Theorems~\ref{thm:breaking} and~\ref{thm:image}. Item (vii) is Theorem~\ref{thm:repair} and Proposition~\ref{prop:positivity}. The regular comparison does not admit the native noncompact Hessian. The weighted criteria do not select or authenticate a covariance. The breaking formula assumes the missing quantum operation and inventory. The repair theorem assumes their actual maps and defects. None of these assumptions is among the conclusions, proving the stated release boundary. \end{proof} %===================================================================== \section{Separation and negative-control ledger} \label{sec:ledger} %===================================================================== \small \begin{longtable}{@{}p{.24\linewidth}p{.67\linewidth}@{}} \toprule Record or correction & Scope retained; transfer refused\\ \midrule \endfirsthead \toprule Record or correction & Scope retained; transfer refused\\ \midrule \endhead XX-A & Independent original-rod variant; matter-scalar calibration; fixed label density; all two-jet terms; coupled classical identity. No full native Em02-E identification.\\[4pt] XX-B & Action normalization, full compact-test free input, causal inverses, and algebraic contractions. Rods are not propagating fields; retarded reachability does not enlarge the compact gauge group.\\[4pt] XX-C & Local subtraction does not select a state or global decay. Compact-source dipole tails are not covariance tails. The nonpositive representative, affine ambiguity, two-rod inverse-distance term, and subsidiary defect retain their stated scopes.\\[4pt] XX-D & Compact microcausal symbols and conditional native one-contraction admission remain distinct. Compact output smearing does not compactify the Hessian. Fixed classical pullback is not quantum substitution.\\[4pt] XX-E & Density conversion, coupled jets, cutoff residual, free quotient, contact images, simultaneous repair, and escaping sources are retained. Its unfiltered centrality test is now explicitly qualified by total-order bookkeeping.\\[4pt] XIX-A \cite{CheungXIXA} & VI harmonic scalar, original-rod lapse, VIII incidence parameter, and X-A density ramp remain different. Original and lapse-weighted rods are not silently interchanged; selected numerical agreements do not identify prescriptions.\\[4pt] XIX-B \cite{CheungXIXB} & The direct coordinate mismatch excludes that identification, not every VI construction. A classical residual is not a quantum anomaly.\\[4pt] XIX-C \cite{CheungXIXC} & Record, residual, finite support, and continuum refinement remain separate. Finite or local observations do not authenticate noncompact product limits.\\[4pt] XIX-D \cite{CheungXIXD} & Q1--Q5 inventory and same-target discipline are checklists. Q4 remainder and Q5 identity are not supplied by naming their objects.\\[4pt] XIX-E \cite{CheungXIXE} & Standing identifications, continuum, external-time, admission, route, and implementation boundaries remain.\\[4pt] XVIII-A \cite{CheungXVIIIA} & Pure-trace tests miss complete kernels; positive \(\hbar\)-orders do not repair fixed classical terms; cancellation needs the actual permitted image.\\[4pt] XVIII-B \cite{CheungXVIIIB} & The massive Minkowski scalar stress benchmark is not this coupled cosmological lapse or its loop inventory.\\[4pt] Em02-E \cite{CheungEm02E} & Its original-rod classical prescription and incomplete native quantum audit remain separate. Chart smoothing is not presumed gauge preserving.\\[4pt] XVI-Q \cite{CheungXVIQ} & Its narrow one-loop order-\(\kappa^2\) bulk two-point report is not lapse admission, a physical de Sitter state, bulk restoration, or expanded O.1 closure.\\[4pt] XVI-N; XV-B \cite{CheungXVIN,CheungXVB} & The stated special-family obstruction and technical lifting interfaces do not discharge spatially compact Route B or open downstream quantum admission.\\[4pt] XIV-H--XIV-J \cite{CheungXIVH,CheungXIVI,CheungXIVJ} & Internal amplitude construction, off-characteristic supported references, and reconstruction descent are not the cosmological Q3 package. The full characteristic graph remains a separate target.\\[4pt] Oai069-A; Oai282-B; catalogue \cite{CheungOai069,CheungOai282,Catalogue} & Finite/free Ward patterns and catalogue records supply no gravitational dictionary, counterterm values, native state, or independently authenticated proof.\\ \bottomrule \end{longtable} \normalsize %===================================================================== \clearpage \section*{One-page Track A release note} \addcontentsline{toc}{section}{One-page Track A release note} %===================================================================== \begingroup \fontsize{8}{9.4}\selectfont \paragraph{Exact prescription.} The target is the independently frozen original-rod lapse on \(\barg=-\dd t^2+(t/t_*)^{2p}\dd\mathbf x^2\), \(\barphi=\phi_*+\sqrt{2p}M\log(t/t_*)\), with \(p>1\) and \(0