% Oai282-A.tex % A single-file manuscript for pdfLaTeX. % No external bibliography, figures, datasets, or shell escape are required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error Oai282-A.tex % pdflatex -interaction=nonstopmode -halt-on-error Oai282-A.tex % % These commands have not been executed here. % The Family 282 source is recorded, not independently verified. % No proof-assistant execution or gravitational identification is asserted. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Oai282-A: Finite Local-Ward Certificates and Scale--Conformal Separation}, pdfsubject={Stress-tensor improvement, local Ward identities, finite certification, and operational separation} } \setlength{\emergencystretch}{3em} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \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} \newtheorem{record}[theorem]{Source record} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Rec}{Rec} \DeclareMathOperator{\States}{States} \DeclareMathOperator{\WF}{WF} \newcommand{\bbC}{\mathbb C} \newcommand{\bbR}{\mathbb R} \newcommand{\bbN}{\mathbb N} \newcommand{\B}{\mathcal B} \newcommand{\A}{\mathcal A} \newcommand{\D}{\mathcal D} \newcommand{\Sch}{\mathcal S} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert} \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{\ddia}{d_{\diamond}} \newcommand{\erec}{\varepsilon_{\mathrm{rec}}} \newcommand{\sourceid}[1]{\texttt{\detokenize{#1}}} \newcommand{\Iop}{\mathsf I} \title{\textbf{Oai282-A}\\[4pt] Finite Local-Ward Certificates and Scale--Conformal Separation:\\ Improvement, Global Actions, Modular Data, and Recovery} \author{Ngo Cheung} \date{Version 1 --- 12 October 2026} \begin{document} \maketitle \begin{abstract} We develop a self-contained companion to the supplied Family 282-01 record, \emph{Scale and conformal symmetry in four-dimensional operational quantum field theory}, dated 26 September 2026. Its conditional scale-to-local-conformal theorem is recorded, not independently verified and not used as a premise. Our mathematical input is instead a declared stress tensor, virial field, improvement scalar, common field domain, and specified Ward-insertion distributions. We prove the flat-space improvement identities, conservation of conformal-Killing currents, and conditional preservation of local affine Ward actions. We then give an exact finite local-Ward certificate with an explicit contact-defect ledger. A finite admissible correction space has a complete linear-algebraic feasibility test; extension from finite tests requires an independently proved coverage condition. An optional bounded-score validation theorem certifies only the declared finite residuals and tolerances. Explicit examples distinguish current conservation from Ward contact identities, infinitesimal actions from global implementation, modular standardness from invariant-vector uniqueness, and fixed scaling or modular records from full-code recovery and entropy histories. The free massless scalar improvement is calculated with its sign and normalization fixed. No new improvement scalar is reconstructed from the Family 282 hypotheses. All stages of the proposed comparison are addressed. No global conformal action on the entire bounded net, gravitational dictionary, Einstein dynamics, ultraviolet completion of the supplied series, Page curve, or inherited characteristic quantum coefficient is constructed. \end{abstract} \paragraph{Keywords.} Scale symmetry; conformal symmetry; stress tensor; improvement; Ward identity; contact term; finite certificate; modular inclusion; complete complement; operational comparison. \tableofcontents \section{Scope, evidence, and necessary corrections} \label{sec:scope} Stress-tensor improvement and the distinction between scale and conformal symmetry are established mathematical questions in quantum field theory. The improved tensor of Callan--Coleman--Jackiw and Polchinski's analysis are central background references \cite{CCJ,Polchinski}. Four-dimensional arguments based on dilatons, renormalization-group flows, and additional assumptions form a separate literature \cite{KomargodskiSchwimmer,LPR,DKST,DFKLP}. The supplied Family 282-01 record claims a local enhancement theorem in a specified bounded-local-net and physical-field framework \cite{OpenAI282}. This companion does not verify that claim. Its main results concern independently supplied distributions, finite tests, and explicit comparison interfaces. Cheung's supplied manuscripts distinguish an algebraic identity, a representation on a domain, a measurement construction, and a complete reference-preserving channel \cite{CheungII,CheungIII,CheungVI,CheungEn05C, CheungEn07A,CheungOai290A}. We retain those distinctions. Their model-specific conclusions are not imported as unchecked premises. \subsection{Debugging the proposed formulation} The following qualifications are indispensable. \begin{enumerate} \item \textbf{Conservation is not a complete Ward identity.} A symmetric conserved traceless tensor gives conserved conformal-Killing currents. It does not, without a specified contact or local-charge argument, prove that these currents implement the intended transformations on every field. \item \textbf{Finite validation is not universal symmetry.} A certificate on finitely many tests and matrix elements proves those tests and matrix elements. Extension to all tests requires coverage, continuity on a dense test family, or another independently proved completeness statement. \item \textbf{Local-to-global is an additional construction, not a universal impossibility.} The correct conclusion is that this finite certificate does not supply a global action. Some theories, including the free massless scalar, have independently constructed conformal implementations \cite{HislopLongo,BGL}. We do not assert that integration is impossible. \item \textbf{Global conformal geometry must be specified.} Special conformal flows are not everywhere-defined flows on one Minkowski chart. A global formulation may require the conformal compactification or its covering space. \item \textbf{Charge preservation needs a locality or boundary argument.} A total divergence does not automatically justify an unrestricted spatial integral of an operator-valued distribution. We prove local affine Ward-action preservation under an explicit spacelike-collar contract. Global charge equality remains conditional on the required limits. \item \textbf{Flat improvement is not a prohibition on gravitational coupling.} An improved matter tensor can enter a separately specified curved background or gravitational theory. What is absent here is the identification and dynamical construction needed for that use \cite{CCJ,HollandsWald}. \item \textbf{Deleting a scalar label is not necessarily deleting its descendants.} Removing the name \(Q=:\!\phi^2\!:\) while retaining \(\partial_\mu Q\) leaves the virial current available. A reduced field list must be examined explicitly. \item \textbf{Scaling dimensions are field labels, not automatically Hilbert-space energies.} Finite scaling multiplicity is not a finite code dimension, a phase-space estimate, or an entropy history. \item \textbf{Cyclic and separating is not unique invariant vacuum.} Modular standardness and uniqueness of an invariant vector are different assertions. \item \textbf{The En06-A theorem has an observable-limit contract.} Its \sourceid{thm:algmicro} uses product-preserving observable comparisons, not merely commuting channels \cite{CheungEn06A}. \item \textbf{No ultraviolet identification is inferred.} A conformal QFT can be relevant to an ultraviolet problem in an additional model. The Family 282 record does not itself construct an ultraviolet completion or matching theorem for Cheung's specified low-energy programme. \item \textbf{No reconstruction is hidden in our assumptions.} The scalar \(L\), its mixed products, and any required affiliated realizations are inputs to our local results. Their existence is not deduced from the Family 282 record. \end{enumerate} \subsection{Dependency convention} We distinguish: \[ \text{documentary attribution},\qquad \text{explicit hypotheses},\qquad \text{results proved below}. \] A source label does not convert a documentary record into a hypothesis satisfied by an application. The web catalogue confirms the Family 282 title, source date, local conclusion, and absence of a Lean scope note \cite{OpenAI282}. The accessible page also reports that its full manuscript source is loading. We therefore do not claim that a complete independent source audit has been performed. The detailed theorem and framework attribution below use the user-supplied statement and excerpts \cite{Dossier}. \section{Stage 0: the Family 282 statement as a source record} \label{sec:stage0} \subsection{The reported framework} The supplied \sourceid{ass:framework} concerns an already specified theory on \[ M=\bbR^{1,3},\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Box=\eta^{\mu\nu}\partial_\mu\partial_\nu. \] Its reported inputs include: \begin{enumerate} \item a local unitary positive-energy vacuum representation, with a Poincar\'e- and scale-invariant vacuum; \item an original physical field-label space \[ \mathcal F=\bigoplus_{\Delta\in\Sigma}\mathcal F_\Delta, \] with discrete, bounded-below, locally finite scaling grades and finite Lorentz multiplicities; \item finite scaling support for every original field; \item compatible realizations of the original fields and their adjoints affiliated with a fixed covariant causally commuting bounded local net; \item a physical local dilatation current implementing scale transformations through its local Ward identities; \item the source's original-product, domain, and physical-field admission conventions. \end{enumerate} The reported assumptions do not separately impose uniqueness of the invariant vector, Haag duality, the split property, a phase-space estimate, or field-operator energy bounds \cite{OpenAI282,Dossier}. This list is a scoped account of the supplied framework, not a replacement for every clause of the original assumption. \begin{record}[Family 282-01, \sourceid{thm:main}] \label{record:main} Under its full \sourceid{ass:framework}, the supplied preprint claims that the physical stress tensor has a symmetric conserved traceless local improvement with unchanged translation and Lorentz Ward charges. After selecting Hermitian dimension-four and dimension-three components \(T^{(4)}_{\mu\nu}\) and \(V^{(3)}_\mu\), it claims a Hermitian physical scalar \(L\) of dimension two satisfying \[ \partial^\mu V^{(3)}_\mu=\Box L, \] such that \begin{equation} t_{\mu\nu} =T^{(4)}_{\mu\nu} +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L \label{eq:record-improvement} \end{equation} is symmetric, conserved, and traceless. It further claims unbroken local Ward identities for \(X^\nu t_{\mu\nu}\), for conformal Killing fields \(X\). The conclusion is local. Integration to a global conformal action on the whole bounded net is left as a separate question \cite{OpenAI282,Dossier}. \end{record} The source claims to derive the scalar condition from a more general improvement problem, rather than assuming it. Our conditional calculations below do not reproduce that derivation. \subsection{Physical membership is a substantial conclusion} In the supplied statement, admitting \(L\) to the original physical field algebra requires sharply localized reconstructions, all finite mixed products, a common invariant domain containing \(\mathcal D_{\mathrm{old}}\), and compatible affiliation to the original net. A scalar distribution satisfying \(\Box L=T^\mu{}_\mu\) in some larger distribution space would not establish those requirements. A formal solution obtained by a Green operator need not be a local physical field. The reported proof route uses wedge modular data, half-sided modular inclusions, control of a critical scalar tail, cone modular identification, scalar-wave extraction, and mixed-product reconstruction \cite{Dossier}. Those operations are recorded here, not reconstructed or independently validated. \subsection{No formalization claim} The catalogue reports no Lean scope note for Family 282 \cite{OpenAI282}. This is a documentary statement about the reported repository scope, not a proof of the theorem or a proof that no other formalization could exist. Family 280's reported Lean file and Family 290-02's reported scope cannot be transferred to this theorem \cite{Dossier}. No checker, source build, or proof-assistant execution is asserted. \section{Stage 1: local currents, global actions, and bounded locality} \label{sec:stage1} \subsection{Three statements with different mathematical types} The following objects must not be identified: \begin{enumerate} \item local identities for field insertions and current smears; \item a strongly continuous unitary group representation, together with covariance of the bounded net; \item a construction of the spacetime metric from other data. \end{enumerate} A global implementation, if proposed, must specify its group or covering group, region action, Hilbert-space representation, domains of generators, and covariance: \[ U(g)\A(O)U(g)^*=\A(gO) \] whenever the geometric and algebraic expression is defined. If \(gO\) leaves the Minkowski chart, an appropriate extension of the net must be supplied. A local Ward certificate below does not supply these objects. \begin{proposition}[Special conformal flows can leave the Minkowski chart] \label{prop:flow} For \[ X_b^\mu(x)=2(b\cdot x)x^\mu-b^\mu x^2, \] the corresponding flow need not be complete on \(M\). \end{proposition} \begin{proof} Take \(b=(b^0,0,0,0)\), \(b^0>0\), and start on the time axis at \(x^0(0)=t_0>0\). The flow remains on that axis and solves \[ \frac{dt}{ds}=b^0t^2. \] Therefore \[ t(s)=\frac{t_0}{1-sb^0t_0}, \] which leaves the chart at \(s=(b^0t_0)^{-1}\). \end{proof} This is a geometric chart limitation, not an obstruction to a conformal action on a compactification. \begin{proposition}[Infinitesimal algebra alone need not exponentiate] \label{prop:nonintegrable} There is a densely defined symmetric infinitesimal generator satisfying the one-dimensional Lie-algebra relations that has no self-adjoint extension and hence no unitary one-parameter implementation extending it. \end{proposition} \begin{proof} On \(L^2(0,\infty)\), let \[ P=-i\frac{d}{dx},\qquad \operatorname{Dom}P=C_c^\infty(0,\infty). \] Integration by parts proves symmetry. Its adjoint deficiency equations are \[ -if'=if,\qquad -if'=-if. \] The first has the square-integrable solutions \(ce^{-x}\), while the second has no nonzero square-integrable solution. The deficiency indices are \((1,0)\), so no self-adjoint extension exists. Stone's theorem rules out the claimed unitary group. The Lie algebra has one generator and its only bracket is zero, so the formal bracket condition is satisfied. \end{proof} This elementary example is not a counterexample to the full Family 282 framework. It shows why a formal infinitesimal representation cannot substitute for the missing domain and integration argument. Analytic-vector criteria provide established sufficient integration tools under their own hypotheses \cite{Nelson}; those hypotheses are not silently added here. \subsection{Finite transport is not special conformal covariance} A finite schedule with site algebras and complete block channels has a backward support ledger. If \(N_T(Y)=Y\) and \[ N_s(Y)=\bigcup\{B:B\text{ is a block at }s+1,\ B\cap N_{s+1}(Y)\ne\varnothing\}, \] then \[ \Phi_{s+1}^*\cdots\Phi_T^*(\A(Y))\subseteq\A(N_s(Y)). \] Indeed, an adjoint block map on an identity factor preserves that identity; backward induction gives the inclusion. This is the finite comparison in the supplied En05-A and En03-A records \cite{CheungEn05A,CheungEn03A}. A diameter bound \(\Delta\) and tick duration \(\tau\) give the declared envelope speed \(\Delta/\tau\). They do not generate special conformal transformations or derive the Minkowski metric. En06-B's complete payload-transport identity likewise retains its prescribed geometry, routes, ports, and speed \cite{CheungEn06B}. \subsection{Channel commutation versus observable locality} \begin{proposition}[Commuting channels are not an observable-locality theorem] \label{prop:channels} Commuting CPTP maps do not imply that their associated observables or arbitrarily assigned region algebras commute. \end{proposition} \begin{proof} On one qubit, the conjugation channels \(\Ad_X\) and \(\Ad_Z\) commute because \(XZ=-ZX\). The observables \(X,Z\) do not commute. Assigning both to separately named regions does not alter that fact. A compatible observable localization is additional data. \end{proof} The actual En06-A observable-limit theorem requires eventually commuting bounded generators, norm-convergent comparisons, and asymptotic preservation of both cross-products \cite{CheungEn06A}. Under those requirements, \[ [a,b] =\lim_n\bigl(j_n(a_n)j_n(b_n)-j_n(b_n)j_n(a_n)\bigr)=0. \] Channel commutation alone supplies none of that comparison. Similarly, common-domain commutation of unbounded fields is not bounded-net causal commutation. The latter is an input in the Family 282 record, not an output of the finite transport ledger. \subsection{A tensor-free comparison, proved locally} Let \(A_{\mathrm{in}}\subseteq M_{\mathrm{in}}\) and \(A_{\mathrm{out}}\subseteq M_{\mathrm{out}}\) be unital \(C^*\)-subalgebras, and let \(T:M_{\mathrm{out}}\to M_{\mathrm{in}}\) be UCP. \begin{lemma}[Equal-snapshot separation] \label{lem:snapshot} For \(h=h^*\in M\) and a unital \(C^*\)-subalgebra \(A\subseteq M\), \[ h\in A \] if and only if \(\omega(h)=\nu(h)\) for every pair of states with \(\omega|_A=\nu|_A\). \end{lemma} \begin{proof} Membership gives equality by restriction. If \(h\notin A\), real Hahn--Banach separation gives a bounded Hermitian functional \(f\) vanishing on \(A\), with \(f(h)\ne0\). Write \(f=f_+-f_-\) using its Jordan decomposition. Because \(1\in A\), \[ f_+(1)=f_-(1)=m>0,\qquad f_+|_A=f_-|_A. \] The states \(f_+/m\) and \(f_-/m\) have equal snapshots and different values on \(h\), a contradiction. The Jordan decomposition used here is standard \(C^*\)-algebraic background \cite{Pedersen}. \end{proof} \begin{theorem}[Tensor-free snapshot comparison] \label{thm:snapshot} The following are equivalent: \begin{enumerate} \item \(T|_{A_{\mathrm{out}}}\) factors through a UCP map \(K:A_{\mathrm{out}}\to A_{\mathrm{in}}\); \item equal input restrictions to \(A_{\mathrm{in}}\) imply equal output restrictions to \(A_{\mathrm{out}}\); \item \(T(A_{\mathrm{out}})\subseteq A_{\mathrm{in}}\). \end{enumerate} The factor \(K\) is unique. No spatial complementary factor or state-extension channel is required. \end{theorem} \begin{proof} Factorization implies the other two assertions. State comparison applied to every self-adjoint output element, followed by Lemma~\ref{lem:snapshot}, gives containment. Under containment, restriction defines \(K\). Positivity at every matrix level is reflected by the subalgebra inclusion, so \(K\) is completely positive; it is unital and unique. \end{proof} This is the finite-interface comparison stated in En05-C \cite{CheungEn05C}. It compares already declared algebras and maps. It does not construct a local net, a stress tensor, a Ward current, or a conformal action. A restricted preparation class need not separate the algebra and cannot replace the universal state quantifier without another theorem. \section{Stage 2: conditional improvement and its affine charges} \label{sec:stage2} \subsection{The independent field contract} Let \(\mathcal E\) be a dense common invariant field domain. An operator-valued distribution below means a linear test-function map whose matrix elements on \(\mathcal E\) are distributions. All derivatives are distributional. We make no evaluation at a singular spacetime point. \begin{assumption}[Declared flat-space improvement data] \label{ass:improvement} Independently supply: \begin{enumerate} \item Hermitian field distributions \(T_{\mu\nu}=T_{\nu\mu}\), \(V_\mu\), and a Hermitian scalar \(L\), with their specified actions on \(\mathcal E\); \item the distributional identities \[ \partial^\mu T_{\mu\nu}=0,\qquad T^\mu{}_\mu=\partial^\mu V_\mu=\Box L; \] \item whenever physical-net conclusions are sought, the necessary mixed products, locality identities on \(\mathcal E\), and compatible closed affiliated realizations of the smears involved. \end{enumerate} When scaling grades are used, \(T,V,L\) are declared to have grades \(4,3,2\), respectively. \end{assumption} The existence and physical membership of \(L\) are assumed here. They are not deduced from Record~\ref{record:main}. Nor do distributional sums automatically prove the affiliated realization required in item three. Set \begin{equation} \Iop_{\mu\nu}[L] =\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L, \qquad t_{\mu\nu}=T_{\mu\nu}+\Iop_{\mu\nu}[L]. \label{eq:improvement} \end{equation} \begin{theorem}[Flat-space improvement identities] \label{thm:improvement} Under Assumption~\ref{ass:improvement}, \(t\) is Hermitian, symmetric, conserved, and traceless as a field distribution on \(\mathcal E\). More generally, without assuming \(T^\mu{}_\mu=\Box L\), \begin{equation} \partial^\mu t_{\mu\nu}=\partial^\mu T_{\mu\nu}, \qquad t^\mu{}_\mu=T^\mu{}_\mu-\Box L. \label{eq:improvement-identities} \end{equation} \end{theorem} \begin{proof} Distributional derivatives commute, so \[ \partial^\mu\Iop_{\mu\nu}[L] =\frac13(\Box\partial_\nu-\partial_\nu\Box)L=0. \] In four dimensions, \[ \eta^{\mu\nu}\Iop_{\mu\nu}[L] =\frac13(1-4)\Box L=-\Box L. \] The remaining assertions follow directly from the declared Hermitian and symmetry properties. \end{proof} The coefficient is the familiar four-dimensional scalar improvement normalization \cite{CCJ,Polchinski}. In dimension \(d>1\), the corresponding coefficient is \(1/(d-1)\). This theorem solves the improvement calculation \emph{once the scalar is supplied}. It does not classify all tensor-potential improvements and does not prove the source's scalar-extraction step. \subsection{Conformal-Killing currents} A conformal Killing field satisfies \[ \partial_\mu X_\nu+\partial_\nu X_\mu =2\sigma_X\eta_{\mu\nu}, \qquad \sigma_X=\frac14\partial_\rho X^\rho. \] Translations and Lorentz transformations have \(\sigma_X=0\); dilations have \(\sigma_X=1\); special conformal fields have \(\sigma_{X_b}=2b\cdot x\). \begin{proposition}[Current divergence] \label{prop:current} For any symmetric tensor distribution \(s_{\mu\nu}\), \begin{equation} \partial^\mu(X^\nu s_{\mu\nu}) =X^\nu\partial^\mu s_{\mu\nu} +\sigma_Xs^\mu{}_\mu. \label{eq:current-divergence} \end{equation} Consequently the currents \(j_{X,\mu}=X^\nu t_{\mu\nu}\) are conserved under Assumption~\ref{ass:improvement}. \end{proposition} \begin{proof} Apply the product rule. Symmetry of \(s\) annihilates the antisymmetric part of \(\partial^\mu X^\nu\), and the conformal Killing equation contracts the remaining part to \(\sigma_Xs^\mu{}_\mu\). \end{proof} Conservation is the conclusion of this proposition. Its action on other fields is a different question, addressed through the declared Ward-insertion contract in Section~\ref{sec:certificate}. \subsection{An explicit affine superpotential} For an affine Killing field \[ X^\mu=a^\mu+\omega^\mu{}_\rho x^\rho, \qquad \omega_{\mu\rho}=-\omega_{\rho\mu}, \] define \[ A_{\rho\mu\nu}[L] =\frac13\bigl(\eta_{\rho\nu}\partial_\mu L -\eta_{\mu\nu}\partial_\rho L\bigr), \] \begin{equation} B_{\rho\mu}[X,L] =X^\nu A_{\rho\mu\nu}[L] +\frac13(\partial_\rho X_\mu)L. \label{eq:affine-superpotential} \end{equation} \begin{lemma}[Affine current difference] \label{lem:affine} The distribution \(B_{\rho\mu}\) is antisymmetric, and \[ X^\nu\Iop_{\mu\nu}[L]=\partial^\rho B_{\rho\mu}[X,L]. \] \end{lemma} \begin{proof} The first term is \((X_\rho\partial_\mu L-X_\mu\partial_\rho L)/3\); the second is antisymmetric because \(X\) is Killing. Thus \(B_{\rho\mu}=-B_{\mu\rho}\). Moreover, \[ \partial^\rho A_{\rho\mu\nu}[L]=\Iop_{\mu\nu}[L]. \] Since \(\partial_\rho X_\mu\) is constant and \(\partial_\rho X^\rho=0\), differentiating \eqref{eq:affine-superpotential} gives \[ \partial^\rho(X^\nu A_{\rho\mu\nu}) =X^\nu\Iop_{\mu\nu} -\frac13(\partial^\rho X_\mu)\partial_\rho L. \] The derivative of the second term cancels the last expression. \end{proof} \begin{proposition}[Conditional preservation of local affine Ward actions] \label{prop:affine-ward} Let \(F\) be a declared local field polynomial on \(\mathcal E\). Choose \(h\in C_c^\infty(\bbR)\) and a spatial cutoff \(\chi\in C_c^\infty(\bbR^3)\) equal to one on the relevant spatial neighborhood. Suppose that: \begin{enumerate} \item the current smears and their commutators with \(F\) are defined on \(\mathcal E\); \item the support of \(h(t)\nabla\chi(\mathbf x)\) is spacelike separated from the localization of \(F\); \item the superpotential smears on that collar commute with \(F\) on \(\mathcal E\). \end{enumerate} Then the affine-current smears formed from \(T\) and \(t\) have the same commutator with \(F\). \end{proposition} \begin{proof} By Lemma~\ref{lem:affine}, their difference is \[ (\partial^\rho B_{\rho0})(h\chi) =-B_{\rho0}\bigl(\partial^\rho(h\chi)\bigr). \] Because \(B_{00}=0\), the time derivative of \(h\) contributes nothing. Every remaining term is supported on the spacelike collar. Item three makes its commutator with \(F\) vanish. \end{proof} Compatible affiliation in causally commuting regional algebras, together with the required invariant-domain products, is one way to justify item three. The proposition does not infer it from weak field commutation alone. If global affine charges are separately defined by suitable spatial limits and their collar terms vanish, the same argument gives equality of those charges. No unrestricted charge integral or boundary limit is constructed here. \subsection{Why this is not an Einstein equation} \begin{proposition}[A conserved traceless tensor need not source its declared flat metric] \label{prop:einstein} The conditions of symmetry, conservation, and zero trace do not imply \[ G_{\mu\nu}(g)=8\pi G_{\mathrm N}t_{\mu\nu} \] for the metric on which the tensor was calculated. \end{proposition} \begin{proof} Consider the classical massless scalar \(\phi=a x^0\), \(a\ne0\), on \(g=\eta\). Its canonical stress tensor, improved by \[ \frac16(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)\phi^2, \] has constant components \[ t_{00}=\frac{a^2}{2},\qquad t_{ii}=\frac{a^2}{6},\qquad t_{0i}=t_{ij}=0\quad(i\ne j). \] It is symmetric, conserved, and traceless. However \(G_{\mu\nu}(\eta)=0\), whereas \(t\ne0\). For \(G_{\mathrm N}>0\), the displayed Einstein equation fails. \end{proof} This is a counterexample to the algebraic implication, not a counterexample to the quantum framework of the source. The covariant replacement of the flat formula also requires care. With the curvature convention \[ [\Box_g,\nabla_\nu]L=R_{\nu\rho}\nabla^\rho L, \] one has \[ \nabla^\mu \bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box_g\bigr)L =R_{\nu\rho}\nabla^\rho L. \] Thus flat conservation cannot simply be promoted to curved conservation without additional terms and hypotheses. A curved-space matter action, its metric variation, a gravitational action, a state prescription, and renormalization conditions are additional inputs. The stress-tensor and time-ordered-product constraints in curved perturbative QFT illustrate this distinction \cite{HollandsWald}. Flat local conformal Ward identities do not assert the absence of Weyl anomalies on arbitrary curved backgrounds. \section{The finite local-Ward certificate} \label{sec:certificate} \subsection{A fixed insertion and contact prescription} The certificate must keep operator identities and insertion identities distinct. For each member \(r\) of a finite list, let \[ z=(x,\mathbf y_r) \] be the current coordinate and the declared spectator coordinates. Supply matrix-valued distributions \[ \mathscr T^r_{\mu\nu},\qquad \mathscr L^r,\qquad \mathscr P^r_\nu,\qquad \mathscr S^r,\qquad \mathscr C^r_X \] on an open joint-test domain \(U_r\). Their values lie in \(M_q(\bbC)\), for a specified finite probe space of dimension \(q\). They may be obtained from declared matrix elements or correlation insertions, but their product prescription is part of the input. The contact distributions \(\mathscr P,\mathscr S,\mathscr C_X\) include all contact terms that the claimed identity requires. If time-ordered products are used, differentiating a field inside time ordering is not automatically the same operation as differentiating the time-ordered distribution. The chosen derivative and contact prescription must justify the improved insertion below. Any discrepancy belongs in the contact ledger \cite{HollandsWald}. \begin{assumption}[Finite Ward-insertion contract] \label{ass:ward} For each \(r\): \begin{enumerate} \item \(\mathscr T^r_{\mu\nu}=\mathscr T^r_{\nu\mu}\); \item all displayed distributions, derivatives, and products with smooth functions are defined on \(U_r\); \item the improved insertion is declared by \[ \mathscr t^r_{\mu\nu} =\mathscr T^r_{\mu\nu} +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box_x) \mathscr L^r; \] \item a finite list of conformal Killing fields \(X_a\) and compact joint tests \(f_{raj}\in C_c^\infty(U_r)\) is fixed; \item the intended Ward-contact assignment \(X\mapsto\mathscr C^r_X\) is specified, and is linear on the tested real span of the \(X_a\). \end{enumerate} \end{assumption} This contract is not the full \sourceid{ass:framework} of Family 282. It neither reconstructs \(L\) nor asserts all physical mixed products. Define the divergence, trace, and contact-matching defects: \begin{align} \mathscr E^r_\nu &=\partial^\mu\mathscr T^r_{\mu\nu}-\mathscr P^r_\nu, \label{eq:divdefect}\\ \mathscr E^r_{\mathrm{tr}} &=\eta^{\mu\nu}\mathscr T^r_{\mu\nu} -\Box_x\mathscr L^r-\mathscr S^r, \label{eq:tracedefect}\\ \mathscr K^r_X &=X^\nu\mathscr P^r_\nu +\sigma_X\mathscr S^r-\mathscr C^r_X. \label{eq:contactdefect} \end{align} The finite Ward residual is \begin{equation} W_{raj} =\left[ \partial^\mu(X_a^\nu\mathscr t^r_{\mu\nu}) -\mathscr C^r_{X_a} \right](f_{raj}). \label{eq:wardresidual} \end{equation} \begin{theorem}[Exact finite local-Ward transfer certificate] \label{thm:ward} Under Assumption~\ref{ass:ward}, \begin{equation} W_{raj} =\sum_{\nu=0}^3 \mathscr E^r_\nu(X_a^\nu f_{raj}) +\mathscr E^r_{\mathrm{tr}}(\sigma_{X_a}f_{raj}) +\mathscr K^r_{X_a}(f_{raj}). \label{eq:wardtransfer} \end{equation} Consequently: \begin{enumerate} \item vanishing of the right side certifies exactly the named finite Ward residual; \item if the individual tested defects vanish, every named residual vanishes; \item if their operator norms are bounded respectively by \(a_{raj,\nu}\), \(b_{raj}\), and \(c_{raj}\), then \[ \|W_{raj}\|_{\mathrm{op}} \le\sum_\nu a_{raj,\nu}+b_{raj}+c_{raj}; \] \item the result extends to linear combinations of the declared tests and conformal Killing fields, with the corresponding linearly specified contacts. \end{enumerate} The theorem does not assert untested Ward identities, a global net action, Einstein dynamics, or recovery of a code. \end{theorem} \begin{proof} The distributional improvement identities give \[ \partial^\mu\mathscr t^r_{\mu\nu} =\partial^\mu\mathscr T^r_{\mu\nu},\qquad \eta^{\mu\nu}\mathscr t^r_{\mu\nu} =\eta^{\mu\nu}\mathscr T^r_{\mu\nu} -\Box_x\mathscr L^r. \] Apply Proposition~\ref{prop:current}, substitute \eqref{eq:divdefect}--\eqref{eq:contactdefect}, and evaluate on \(f_{raj}\). This proves \eqref{eq:wardtransfer}. The remaining claims follow by the triangle inequality and linearity. \end{proof} Multiplication by \(X_a^\nu\) and \(\sigma_{X_a}\) preserves compact support. Thus the complete finite input-test family must include \[ f_{raj},\qquad X_a^\nu f_{raj},\qquad \sigma_{X_a}f_{raj}, \] rather than only the original tests. The theorem retains contacts instead of inferring them from off-diagonal conservation. It permits cancellation among the tested defects; requiring each defect to vanish is sufficient, not necessary. \subsection{The exact finite matrix scope} Suppose the \(q\times q\) distributions are defined from an orthonormal family \(u_1,\ldots,u_q\subset\mathcal E\). Vanishing of \(W_{raj}\) proves the named matrix elements and, by sesquilinearity, every such matrix element between vectors in their span. It does not prove that the uncompressed residual vanishes on all of \(\mathcal E\), or that any closed generator has the expected graph domain. An entrywise estimate \[ |(W_{raj})_{kl}|\le\epsilon_{kl} \] gives \[ \|W_{raj}\|_{\mathrm{op}} \le\|W_{raj}\|_{\mathrm F} \le\left(\sum_{k,l}\epsilon_{kl}^2\right)^{1/2}. \] This is a finite matrix norm estimate, not a field-operator energy bound. \subsection{A complete finite correction test} Suppose all correction freedom allowed for this finite task has been independently parameterized by a real affine family \[ \mathscr W(c)=\mathscr W_0+\sum_{\alpha=1}^p c_\alpha\mathscr H_\alpha, \qquad c_\alpha\in\bbR. \] Reality preserves the specified Hermitian convention. Additional exact linear admissibility conditions may first be absorbed into a parameterization of the permitted affine space. Let \(E\) evaluate all named matrix residuals, splitting complex entries into real and imaginary parts. Write \[ w_0=E(\mathscr W_0)\in\bbR^m,\qquad N_{:\!,\alpha}=E(\mathscr H_\alpha). \] \begin{proposition}[Exact finite correction feasibility] \label{prop:finitefit} A permitted correction makes every named residual zero if and only if \[ -w_0\in\im N. \] Equivalently, \[ y^{\mathsf T}w_0=0 \quad\text{for every }y\in\ker N^{\mathsf T}. \] When \(c_*\) is one solution, all solutions are \(c_*+\ker N\). \end{proposition} \begin{proof} The evaluated residual is \(w_0+Nc\), so exact feasibility is the equation \(Nc=-w_0\). The orthogonal complement of \(\im N\) is \(\ker N^{\mathsf T}\). The final assertion is the usual affine solution-space description. \end{proof} This is a finite linear-algebra theorem. It does not select a physical counterterm class that has not been supplied. It does not classify every characteristic prescription or define an absent product. Cheung's Landmark II makes the analogous distinction between a finite evaluated fit and a global functional identity \cite{CheungII}. \subsection{What would make finite tests complete?} \begin{proposition}[Coverage in a declared defect space] \label{prop:coverage} Let \(\mathscr V\) be an independently specified finite-dimensional space containing every admissible residual distribution for the claim being tested. If \[ E|_{\mathscr V}:\mathscr V\to\bbR^m \] is injective, then \(E(W)=0\) implies \(W=0\) in \(\mathscr V\). Without such a coverage condition, vanishing of the finite evaluations does not generally imply a distributional identity. \end{proposition} \begin{proof} The first assertion is injectivity. For the second, any nonzero element of \(\mathscr V\cap\ker E\), when it exists, is invisible to every declared evaluation. \end{proof} Injectivity is a sufficient coverage condition. A particular target decision can sometimes be complete under weaker conditions. No statement about an unavailable defect family is made by writing down the evaluation matrix. \begin{proposition}[Finite tests can miss a conserved symmetric perturbation] \label{prop:invisible} For any finite family of compact linear test evaluations, there is a nonzero smooth compactly supported symmetric conserved tensor perturbation invisible to that family but with nonzero trace. This assertion concerns the unrestricted smooth tensor-distribution class, not the full Family 282 framework. \end{proposition} \begin{proof} Choose a nonempty relatively compact open set \(U\subset M\). For \(h\in C_c^\infty(U)\), put \[ S_{\mu\nu}[h] =(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)h. \] It is symmetric and conserved. Every declared finite evaluation of \(S[h]\), including a tested current divergence, is a linear functional of \(h\). The common kernel of finitely many such functionals in the infinite-dimensional space \(C_c^\infty(U)\) contains a nonzero \(h\). Its trace is \(-3\Box h\). A compactly supported smooth solution of \(\Box h=0\) is zero: it has zero Cauchy data before its compact time support, and uniqueness of the wave equation forces zero. Thus the chosen perturbation has nonzero trace. \end{proof} The perturbation is not claimed to be a translation-covariant physical stress tensor with the source's scaling spectrum. It establishes the finite-test limitation without manufacturing a counterexample to the conditional enhancement theorem. \subsection{Optional bounded-score validation} A Ward residual is not automatically a bounded observable. To obtain a statistical certificate, an independently justified measurement bridge is required. Index the real and imaginary parts of all evaluated matrix entries by \(k=1,\ldots,m\). Let the target real residual coordinate be \(w_k\). Supply a scale \(a_k>0\), complete scores \(Z_{ki}\in[-1,1]\), and allowances \(b_k\ge0\). \begin{assumption}[Bounded-score bridge] \label{ass:statistics} Conditional on a declared calibration record \(\mathcal Z\): \begin{enumerate} \item each \(n_k\ge1\) is fixed before validation data are read; \item scores within coordinate \(k\) are independent and have a common mean \(m_k\); \item on an event \(G\), with failure probability at most \(\beta\), \[ |w_k-a_km_k|\le b_k \quad\text{for every }k. \] \end{enumerate} All outcomes, losses, rejections, and failure flags have a predeclared score. No successful branch is substituted for the complete attempted-run law. \end{assumption} \begin{theorem}[Finite-confidence residual certificate] \label{thm:statistics} For \(0<\alpha_{\mathrm s}<1\), set \[ \widehat w_k=\frac{a_k}{n_k}\sum_i Z_{ki},\qquad \rho_k=a_k\sqrt{\frac2{n_k} \log\frac{2m}{\alpha_{\mathrm s}}}+b_k. \] Under Assumption~\ref{ass:statistics}, \[ \Pr\{|w_k-\widehat w_k|\le\rho_k \text{ for every }k\} \ge1-\alpha_{\mathrm s}-\beta. \] Therefore, if a tolerance \(\epsilon_k\) is certified only when \[ |\widehat w_k|+\rho_k\le\epsilon_k \quad\text{for every }k, \] the probability of certifying while some \(|w_k|>\epsilon_k\) is at most \(\alpha_{\mathrm s}+\beta\). \end{theorem} \begin{proof} Conditional Hoeffding inequalities give \[ \Pr\left\{ \left|\frac1{n_k}\sum_iZ_{ki}-m_k\right| >\sqrt{\frac2{n_k}\log\frac{2m}{\alpha_{\mathrm s}}} \,\middle|\,\mathcal Z\right\} \le\frac{\alpha_{\mathrm s}}m \quad\cite{Hoeffding}. \] A union bound, integration over \(\mathcal Z\), and intersection with \(G\) prove simultaneous coverage. Every invalid acceptance lies outside that event. \end{proof} Independence between coordinates is unnecessary for the union bound. Unaccounted adaptive stopping is not covered. The deterministic bridge allowance \(b_k\) is added to the statistical radius, not inferred from the data. The theorem is frequentist coverage, not a posterior probability conditional on acceptance. Positive finite radii certify tolerance, not exact equality. The bounded-statistic architecture is consistent with the scoped validation contracts in En01-B and Oai265-B \cite{CheungEn01B,CheungOai265B}. No field measurement bridge, calibration, score dataset, or laboratory execution is supplied here. An unbounded field smear cannot be treated as a bounded score merely by rescaling it. \section{Stage 3: modular null cuts and recovery are different interfaces} \label{sec:stage3} \subsection{The modular statement being compared} For a von Neumann algebra \(\mathcal M\) and a cyclic separating vector \(\Omega\), Tomita--Takesaki theory supplies modular unitaries \(\Delta_{\mathcal M}^{it}\). A half-sided modular inclusion is an inclusion \(\mathcal N\subseteq\mathcal M\) with a specified standard vector and a one-sided modular invariance condition. Under its precise hypotheses, this structure yields an associated one-parameter group and positivity properties \cite{Wiesbrock,ArakiZsido}. Wedge modular geometry has established QFT examples and hypotheses \cite{BisognanoWichmann,BGL}. The Family 282 record reports a further use of these structures in its scalar reconstruction. That use is not verified here. A modular group or positive null-cut generator is not yet a channel from a declared logical code to a declared accessible output. It does not select an environment, receiver, or deterministic quantum-return map. \subsection{Standardness does not imply uniqueness} \begin{proposition}[A standard modular pair with many invariant vectors] \label{prop:standard} Cyclicity and separation of \(\Omega\) for a von Neumann algebra do not imply that the modular unitary group has a unique invariant vector. \end{proposition} \begin{proof} Let \(\mathcal M=M_2(\bbC)\) act by left multiplication on the Hilbert--Schmidt space \(H=M_2(\bbC)\), and choose \[ \Omega=\rho^{1/2}, \] where \(\rho\) is a strictly positive density matrix. Invertibility makes \(\Omega\) cyclic and separating. The modular operator acts by \[ \Delta(X)=\rho X\rho^{-1}. \] Every matrix commuting with \(\rho\) is fixed by \(\Delta^{it}\). For diagonal \(\rho\), the diagonal subspace already has dimension two. \end{proof} This example separates two modular assertions. It is not a counterexample to a QFT vacuum-uniqueness theorem, and it does not identify modular invariance with Poincar\'e invariance. Those are additional groups and hypotheses. \subsection{The independent complete-channel criterion} Let \(C\cong\bbC^d\), \(d\ge2\), and supply a finite isometry \[ W:C\to R\otimes B. \] Define its accessible channel and complete complement: \[ \mathcal N(X)=\Tr_B(WXW^\dagger),\qquad \mathcal C(X)=\Tr_R(WXW^\dagger). \] The complement includes every inaccessible output and any preparation purification. \begin{theorem}[Exact full-code recovery] \label{thm:recovery} There exists a deterministic CPTP map \(D:\B(R)\to\B(C)\) with \(D\mathcal N=\id_C\) if and only if \[ \mathcal C(X)=\Tr(X)\tau_B \] for a fixed state \(\tau_B\). If \(r=\rank\tau_B\), then \[ \dim R\ge dr. \] Recovery preserves every independent external reference. \end{theorem} \begin{proof} Dilate an exact decoder by an isometry \(U:R\to C\otimes G\). Then \(Z=(U\otimes I_B)W\) dilates identity. Purity of the decoded basis outputs gives \[ Z\ket i=\ket i\ket{\chi_i}. \] Exact recovery of off-diagonal matrix units gives \(\langle\chi_j,\chi_i\rangle=1\); all normalized \(\chi_i\) coincide. Tracing \(CG\) proves constancy of \(\mathcal C\). Conversely, diagonalize \[ \tau_B=\sum_{a=1}^r\lambda_a\kb aa,\qquad \lambda_a>0. \] Write \[ W\ket i=\sum_a\sqrt{\lambda_a}\ket{u_{ia}}_R\ket a_B. \] The complementary matrix-unit equations imply \[ \langle u_{jb},u_{ia}\rangle=\delta_{ij}\delta_{ab}. \] Thus \(R\) contains \(dr\) orthonormal vectors. Define the isometry \(A:C\otimes\bbC^r\to R\) by \(A(\ket i\ket a)=u_{ia}\). For any fixed code state \(\omega_C\), \[ D(Y)=\Tr_{\bbC^r}(A^\dagger YA) +\Tr((I-AA^\dagger)Y)\omega_C \] is CPTP and recovers every matrix unit. Equality of linear maps extends after tensoring with every reference. \end{proof} This is the standard exact-correction structure \cite{KnillLaflamme,Watrous}, restated in the supplied En07-A/B, Hawk04-A/B, and Oai290-A interfaces \cite{CheungEn07A,CheungEn07B,CheungHawk04,CheungOai290A}. It is not a modular-Ward theorem. \subsection{An explicit record-only separation} \begin{definition}[Independent operational completion] \label{def:completion} Fix any declared Ward, scaling, or modular record. An independent finite completion appends an isometry \(W:C\to R\otimes B\) without a physical identification theorem connecting it to that record. An operational quantity is determined by the record alone only if it has the same value in every such completion with the logical and output dimensions fixed. \end{definition} \begin{proposition}[Fixed modular and Ward data do not determine recovery] \label{prop:recovery-separation} For any fixed record and \(d\ge2\), there are independent completions with identical \(d\)-dimensional accessible and complementary spaces, one exactly recoverable and one not recoverable. Their selected accessible entropy is zero for every pure logical input. \end{proposition} \begin{proof} Set \[ W_{\mathrm{good}}\psi=\psi_R\otimes\ket0_B,\qquad W_{\mathrm{bad}}\psi=\ket0_R\otimes\psi_B. \] The first accessible channel is identity and its complement is constant. The second accessible channel is constant and its complement is identity. Theorem~\ref{thm:recovery} separates their recovery verdicts. For each pure input, both accessible outputs are pure. All original Ward and modular record entries remain fixed. \end{proof} These completions are not claimed to be realizations of the source QFT or its null cuts. They specify the quantifiers of the record-only nonidentification claim. With \[ \ddia(\Phi,\Psi)=\tfrac12\|\Phi-\Psi\|_\diamond, \] every decoder of the bad completion has error at least \(1-d^{-2}\): its Bell-input output is a replacement product, with Bell overlap \(d^{-2}\). Thus the failure is reference-sensitive, not merely a distinguishability convention. \subsection{Family 290-02 is not instantiated} The supplied Oai290-A record distinguishes an exact finite-channel criterion from a modular-band witness under its own centralizer and spectral hypotheses \cite{CheungOai290A,Dossier}. The latter needs its specified algebra, state, centralizer, operator, modular action, averaging, and band conditions. A null-cut modular inclusion does not automatically verify those conditions. In Proposition~\ref{prop:standard}, for example, the faithful-state centralizer contains the diagonal algebra and is not scalar. The reported Lean note names Family 290-02 only \cite{Dossier}. It is not evidence that the lemma has been instantiated on the modular objects of Family 282. No such instantiation or checker execution is asserted. The four same-model endpoint obligations reported in En07-B remain additional: preparation and code, complete evolution, justified outgoing access and endpoint identification, and a constant or quantitatively decoupled complete complement \cite{CheungEn07B}. They are not constructed by the Ward certificate. \section{Stage 4: scaling multiplicities are not an information account} \label{sec:stage4} \subsection{Field grades and Hilbert spectra} The label equation \[ \mathcal F=\bigoplus_{\Delta\in\Sigma}\mathcal F_\Delta \] describes transformation properties of fields. It is not an assertion that the Hilbert-space dilation generator has that point spectrum. \begin{example}[A continuous dilation generator] On \(L^2(\bbR_+,dp/p)\), let \[ (U(s)f)(p)=f(e^sp). \] The coordinate \(y=\log p\) identifies this representation with translations on \(L^2(\bbR,dy)\). Its self-adjoint generator is unitarily equivalent, up to the chosen sign convention, to \(-i\,d/dy\), with spectrum \(\bbR\). \end{example} This is an elementary representation-theoretic illustration. It does not assign a new scaling spectrum to the source theory. It shows why field-label multiplicity cannot be read as a Hilbert-space energy multiplicity. \subsection{No entropy or tensor factor is specified} A scaling record contains no selected density matrix, radiation access factor, emission sequence, or decoder. The Family 282 assumptions also do not separately impose a split property. One cannot simply append a spatial partial trace to a bounded local algebra without declaring the relevant realization. For a finite code, the capacity inequality in Theorem~\ref{thm:recovery} counts actual orthonormal syndrome vectors. It is not an area law. A count of field dimensions is not that syndrome count either. \begin{proposition}[Fixed scaling data do not select an entropy history] \label{prop:history} A fixed scaling or Ward record does not determine the entropy history of an independently appended finite release process. \end{proposition} \begin{proof} Append qubits \(B,R_1,R_2,R_3\), all initially zero, and release \(R_j\) at tick \(j\). One completion uses identity evolution, giving accessible-prefix entropy \((0,0,0)\). Another first prepares a Bell pair on \(BR_1\), then swaps \(B\) with \(R_2\), and finally performs identity. Its accessible-prefix entropy is \((1,0,0)\). The initial spaces, preparation, release dimensions, and original record agree. Only the independently supplied evolution differs. \end{proof} This designed finite turnover is not a gravitational Page curve. The construction proves nonselection of a history, not an evaporation theorem. Cheung's Oai265-A records the analogous separation for a static lattice area-law record \cite{CheungOai265A}. Family 265's reported gapped ground-state setting and Family 260's reported initial-data enclosing area are different mathematical objects \cite{Dossier}. Neither is the trace \(T^\mu{}_\mu\), a scaling multiplicity, or a decoder criterion. \section{Stage 5: the free massless scalar and the field-list issue} \label{sec:stage5} \subsection{The normalization calculation} Independently supply the four-dimensional free massless scalar on Minkowski space, its vacuum Wick fields, and their invariant polynomial domain. Let \[ Q=:\!\phi^2\!:,\qquad K=:\!\partial_\rho\phi\,\partial^\rho\phi\!:, \] \[ T_{\mu\nu} =:\!\partial_\mu\phi\,\partial_\nu\phi\!: -\frac12\eta_{\mu\nu}K. \] The free equation is \(\Box\phi=0\). \begin{proposition}[Free-scalar improvement] \label{prop:free} On the supplied Wick domain, \[ T^\mu{}_\mu=-K,\qquad \Box Q=2K. \] Thus \[ V_\mu=-\frac12\partial_\mu Q,\qquad L=-\frac12Q \] satisfy \(T^\mu{}_\mu=\partial^\mu V_\mu=\Box L\), and \begin{equation} t_{\mu\nu} =T_{\mu\nu} +\frac16(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q \label{eq:free-improvement} \end{equation} is symmetric, conserved, and traceless. \end{proposition} \begin{proof} Contraction in four dimensions gives \(T^\mu{}_\mu=(1-2)K=-K\). The Wick derivative rule and the free equation give \[ \Box:\!\phi^2\!: =2:\!\partial_\rho\phi\,\partial^\rho\phi\!: +2:\!\phi\Box\phi\!: =2K. \] The definitions of \(V,L\) therefore give the trace relation. Substitution into \eqref{eq:improvement} gives \eqref{eq:free-improvement}. Conservation of \(T\) follows from the free equation, and Theorem~\ref{thm:improvement} completes the calculation. \end{proof} The grades are \[ [\phi]=1,\qquad[Q]=[L]=2,\qquad[V]=3,\qquad[T]=[t]=4. \] This is the standard scalar improvement example \cite{CCJ,Polchinski}. Its Wick and net realizations are supplied example data, not reconstructed by this proposition. No all-product Ward normalization is derived by the two displayed Wick identities alone. Such an assertion requires the contact or local-charge analysis appropriate to the chosen products. \subsection{Deleting a label versus changing the physical field algebra} \begin{proposition}[The descendant qualification] \label{prop:deletion} Deleting the label \(Q\) does not remove the virial current if the declared field list still contains \(\partial_\mu Q\). A list omitting both \(Q\) and the needed bilinear descendants can fail to contain the displayed physical virial current. \end{proposition} \begin{proof} The identity \[ V_\mu=-\tfrac12\partial_\mu Q \] proves the first assertion. For the second, a linear field-label list consisting only of the identity, the linear field \(\phi\) and its descendants, and \(T_{\mu\nu}\) and its descendants need not contain the bilinear dimension-three vector \(:\!\phi\partial_\mu\phi\!:\). It is neither a linear-field descendant nor a dimension-four stress-tensor descendant. This is a restricted label space, not a claim that it is closed under Wick products or coincident products. \end{proof} If a reduced list lacks the physical scale current, it no longer meets that reported hypothesis of Family 282. It is therefore not a counterexample to the conditional source claim. Conversely, a derivative-closed physical algebra requires a more careful deletion argument than removing one printed name. Physical-field membership, a distributional identity, and documentary source status are not identical notions. Their shared discipline is only that the necessary membership or hypothesis must be established before it is used. \subsection{The supplied gravitational scalar records are not this example} Em02-E's lapse is a relational metric quantity on a declared Einstein--scalar background. Its reported second-order coefficients include clock and spatial-rod corrections \cite{CheungEm02E}. They are not \(L=-Q/2\). Em03-A's scalar lives on a fixed normalized Schwarzschild exterior, with its specified state and radial scattering problem. Its reported surface gravity is \(\kappa=1/(4GM)\), and bulk Unruh identification is an external input in that manuscript \cite{CheungEm03A}. Those geometry and state data are not the Minkowski example. A local equivalence of field equations in one coordinate patch would not identify the states, local algebras, horizon geometry, or complete channels of the two constructions. No such identification is supplied here. The chiral circle-net correspondence reported by Family 280 is a different dimension and framework; the preceding Oai280-A draft retains it as a source record \cite{Oai280A,Dossier}. Family 211's planar-map and metric-measure limits and the Oai211-A comparison likewise do not supply this four-dimensional Ward identity \cite{Oai211A,Dossier}. \section{Stage 6: assembly and unchanged boundaries} \label{sec:stage6} \begin{theorem}[Oai282-A finite release] \label{thm:main} For the independently declared contracts in this paper: \begin{enumerate} \item a supplied scalar trace potential yields the exact flat-space improvement identities of Theorem~\ref{thm:improvement}; \item conformal-Killing current conservation follows from symmetry, conservation, and zero trace; \item affine local Ward actions are unchanged under the spacelike-collar contract of Proposition~\ref{prop:affine-ward}; \item the named finite Ward identities have the exact defect decomposition and quantitative certificate of Theorem~\ref{thm:ward}; \item finite admissible correction feasibility is decided by Proposition~\ref{prop:finitefit}, while extension beyond the tested family requires a coverage argument; \item the optional bounded-score contract gives the simultaneous finite tolerance certificate of Theorem~\ref{thm:statistics}; \item fixed Ward, modular, or scaling records do not determine the independently appended code recovery or entropy histories of Definition~\ref{def:completion}. \end{enumerate} These results do not prove the existence or physical admission of a new \(L\) under the Family 282 hypotheses. They do not construct a global conformal action on the entire bounded net, Einstein dynamics, an ultraviolet matching theorem, a gravitational endpoint, or the inherited characteristic quantum coefficient. \end{theorem} \begin{proof} Items one through six are the cited local calculations and finite-interface theorems. Item seven is Propositions~\ref{prop:recovery-separation} and \ref{prop:history}. The additional conclusions would require mathematical objects not constructed by those arguments: scalar reconstruction and all mixed-product admission, generator integration and net covariance, a gravitational action and identification, an ultraviolet-to-low-energy matching map, or the separately named characteristic graph construction. The Family 282 statement appears only as Record~\ref{record:main}. \end{proof} \subsection{The metric is an input, not an unknown solved by this certificate} Every derivative, trace, conformal Killing equation, support comparison, and improvement formula above uses the declared \(\eta\). The certificate does not determine a metric from arbitrary finite Ward values. It verifies identities relative to that supplied metric. This does not deny that geometric data can be encoded in sufficiently rich physical observations under additional reconstruction hypotheses. No such inverse problem or uniqueness theorem is part of this paper. \subsection{Classical relational descent remains a separate result} Cheung's III and VI use declared classical equation spaces, gauge groups, and relational constructions \cite{CheungIII,CheungVI}. VI's supplied second-order descent follows from the coordinate composition identity \[ \phi'\circ Y_{\gamma'} =\phi\circ F\circ F^{-1}\circ Y_\gamma =\phi\circ Y_\gamma \] on its stated formal domain. That mechanism is not a derivation from the current \(D_\mu\). The gravitational equations included in those models are not derived from the flat trace improvement. The supplied quantum finite-renormalization status of VI remains RG-open \cite{CheungVI}. No assertion here changes it. \subsection{No ultraviolet completion of the supplied series} An ultraviolet application would need a named high-energy theory, a low-energy target, matching observables and states, a regulator and renormalization prescription where relevant, approximation topology, error or remainder control, and symmetry compatibility. A scale-to-conformal statement about an already supplied QFT does not construct that package for another model. The prepared-source and reduced-sector records XVI-A, XVI-H, XVI-I, and XVI-P retain their stated domains and imported hypotheses \cite{CheungXVIPrepared,CheungXVIP}. A witness, compact kernel, extension family, two-point sector, or graph inventory is not the missing complete ultraviolet matching theorem. The source's improvement cannot alter the hypotheses or prepared insertion of the conditional XVI-N obstruction \cite{CheungXVIN}. The same word ``massless'' does not identify its observable with the free scalar of Section~\ref{sec:stage5}. \subsection{Standing refusals} The supplied status ledger remains: \[ \boxed{ \begin{gathered} \sourceid{O.1}:\ \sourceid{NOT_CLOSED},\\ \text{series Route B: not discharged},\\ \text{XVI-N remaining obligations: not discharged},\\ \text{XV-B downstream admission exit: not opened},\\ Q_HR_1^{(1)}\in\mathscr E_H:\ \text{not asserted}. \end{gathered}} \] XVI-P is a reported registered technical record, not a premise of this proof \cite{CheungXVIP}. The technical manuscript XV-B version 2 exists and proves specified lifting and extension interfaces. Its existence is distinct from opening a downstream quantum-admission exit \cite{CheungXVB}. The full inherited map remains unconstructed in the supplied I3/J2/B1 accounts \cite{CheungXIVI,CheungXIVJ,CheungXVB}. No tensor \(t_{\mu\nu}\) in this paper is identified with that characteristic graph. No differential is applied to its absent coefficient, and no quantum Ward defect for it is formed. Enter retains its declared destructive-readout meaning \cite{CheungI}. A returned classical record is not an arbitrary reference-entangled quantum return. The local Ward calculations construct no field instrument. \section{Mathematical successor questions within the source programme} \label{sec:successors} The comparison stages are complete as conditional local results and separations. The following mathematical extensions are not proved. \begin{enumerate} \item \textbf{Local-to-global implementation.} Construct the required generators or automorphisms, prove their integration, identify the appropriate conformal covering group, and establish covariance on the whole bounded net or its specified extension. The source explicitly leaves this separate. \item \textbf{Infinite scaling support or accumulation.} Dropping finite scaling support or local finiteness requires new convergence and domain control. Finite field-label expansions do not justify an unrestricted infinite expansion. \item \textbf{Vacuum uniqueness.} A conclusion about standard cone algebras does not establish a unique Poincar\'e- and scale-invariant vector. Any uniqueness theorem requires its own hypotheses. \item \textbf{Gauge-theory enlargement.} An application beyond the chosen gauge-invariant local algebra must identify physical fields, currents, constraints, domains, and observable-net realizations. Gauge-dependent formal potentials cannot replace physical membership. \item \textbf{Comparison with dilaton and \(a\)-theorem arguments.} Such a comparison should explicitly identify the state, scattering or analyticity assumptions, anomaly conventions, and improvement freedom. The papers \cite{KomargodskiSchwimmer,LPR,DKST,DFKLP} are external approaches, not automatic premises of the reported modular proof. \item \textbf{Comparison with Family 280.} Chiral circle nets and four-dimensional bounded nets have different geometric and representation-theoretic interfaces. A comparison may be useful, but neither source statement derives the other merely by sharing the word ``conformal''. \end{enumerate} No item is completed by naming it. None becomes a Page curve, metric derivation, or gravitational endpoint without additional identified constructions. \appendix \section{The separate characteristic-map chain} \label{app:chain} The supplied dossier also contains a six-step analytic programme. It is not a prerequisite of the finite Ward proof and is not discharged by it. \subsection{A necessary descent condition} Let \[ \Rec:\mathscr P\to\mathcal D'(W),\qquad \mathscr K=\im\Rec. \] \begin{proposition}[Presentation descent] \label{prop:descent} A linear map \(\widetilde M:\mathscr P\to\mathcal D'(W)\) defines a unique map \(M:\mathscr K\to\mathcal D'(W)\) with \(\widetilde M=M\Rec\) if and only if \[ \ker\Rec\subseteq\ker\widetilde M. \] \end{proposition} \begin{proof} Factorization annihilates the reconstruction kernel. Conversely, two equal reconstructions differ by an element of that kernel, so the condition makes their images equal and defines the unique induced map. \end{proof} For example, \[ q\delta'(q)+\delta(q)=0. \] With the supplied reconstruction convention, \[ \Rec((qi\theta+1)M_R)=0, \] although its degree-zero and degree-one terms reconstruct to nonzero opposite distributions. A representative-dependent highest degree is therefore not automatically an intrinsic reconstructed symbol \cite{CheungXIVJ,CheungXVB}. This proves a necessary algebraic interface. It constructs no cross-line characteristic product. \subsection{All six completion tests} {\small \renewcommand{\arraystretch}{1.15} \begin{longtable}{@{}p{0.08\linewidth}p{0.35\linewidth} p{0.51\linewidth}@{}} \toprule Step & Required construction & Completion test and present scope\\ \midrule \endfirsthead \toprule Step & Required construction & Completion test and present scope\\ \midrule \endhead \bottomrule \endfoot 1 & Full joint characteristic map. & A map on tests meeting the characteristic locus, respecting reconstruction descent, canonical-domain agreement, all retained branches and grouped cross-lines. The supplied I3/J2 full target remains unresolved. No such map is constructed here.\\[4pt] 2 & Joint section and complete remainder. & A result for the same complete map and frozen data. The internal I2 remainder is not this quantum remainder. Not constructed here.\\[4pt] 3 & Uniform transverse estimates. & Control in the actual transverse variables, including collision strata and their intersections. Internal frequency homogeneity does not supply it. Not constructed here.\\[4pt] 4 & Contact compatibility. & Compatibility with the permitted contacts and singular spectators, overlap agreement, and retained normalization. The finite Ward-fit matrix is not this contact class. Not constructed here.\\[4pt] 5 & Output wavefront theorem. & A proved bound after the actual full composition and proper pushforward into the named output space. A partial off-locus reference is insufficient. Not constructed here.\\[4pt] 6 & Normalized quantum admission. & Identification of the complete coefficient and proof of \(Q_HR_1^{(1)}\in\mathscr E_H\). No such admission is asserted. The downstream exit is not opened.\\ \end{longtable} } The supplied supported sector \(K_6+K_8+K_9\) is not the full sum \(K_{47}+K_5+K_6+K_8+K_9\). The two \(K_5\) placements and the remaining retained branches cannot be dropped \cite{CheungXIVI,CheungXIVJ}. The historical H2 stop, the reported internal advancement at I2, and the missing full map at I3/J2 are different statements \cite{CheungXIVH,CheungXIVI,CheungXIVJ}. Failure to construct the map is not a theorem that every permitted construction is impossible. Neither a finite-dimensional Ward correction fit nor a source improvement scalar supplies the absent distribution. The full analytic programme is preserved as an open contract. \section{Source attribution and verification declarations} \label{app:declarations} \subsection{What was consulted} The Family 282 catalogue page establishes its attribution, source date, stated scope, and reported absence of a Lean note \cite{OpenAI282}. The detailed source formulas and internal labels in this manuscript are attributed to the supplied dossier \cite{Dossier}. Cheung manuscripts are cited with identifiers and supplied versions. Where only excerpts or summaries are available, no independent verification of the entire manuscript is claimed. The Enter Calculus Companion describes itself as a searchable companion, not a proof checker \cite{CheungPortal}. Family 260, 265, 280, 290-02, 211, 362, and 376 are used only as the supplied documentary comparisons \cite{Dossier}. Their models, backgrounds, equations, regulators, or theorem dependencies are not identified with this one. \subsection{No unsupported release} This manuscript asserts no: \begin{itemize} \item independent verification of Family 282; \item reconstruction of a new physical \(L\) from its framework; \item global conformal implementation on every bounded local algebra; \item derivation of Minkowski space or another metric; \item Einstein equation, gravitational backreaction, or mass-loss law; \item ultraviolet completion or Wilson matching for the supplied series; \item holographic dictionary, island, or entanglement wedge; \item Page curve or gravitational endpoint; \item inherited characteristic quantum coefficient. \end{itemize} These are scope statements, not universal impossibility theorems. A later bridge theorem with additional identified hypotheses is not excluded. \subsection{Execution declaration} No successful \sourceid{pdflatex} execution, Lean checking, symbolic-code execution, statistical dataset validation, laboratory calibration, independent manuscript-proof verification, referee approval, or journal acceptance is asserted. The file is written for pdfLaTeX with a self-contained bibliography. Build commands are instructions, not execution evidence. The locally claimed mathematical results are the explicit proofs above. \section*{Conclusion} The completed finite implication is \[ \boxed{ \begin{gathered} \text{declared improvement and insertion data}\\ +\text{specified contacts} +\text{finite tests and probes}\\ \Longrightarrow \text{an exact finite local-Ward defect certificate}. \end{gathered}} \] With an independently supplied bounded-score bridge, the same finite task has a tolerance certificate with simultaneous coverage. The proof does not reconstruct the source's scalar, replace its bounded locality, or integrate its Ward currents. It does not determine a complete-complement channel or a gravitational model. All Stages 0--6 have been addressed with local calculations, finite certificates, explicit separations, and unchanged open boundaries. Family 282 remains an external source record. \sourceid{O.1} remains \sourceid{NOT_CLOSED}. \begin{thebibliography}{99} \bibitem{CCJ} C.~G.~Callan, Jr., S.~Coleman, and R.~Jackiw. A new improved energy-momentum tensor. \emph{Annals of Physics} \textbf{59} (1970), 42--73. doi:10.1016/0003-4916(70)90394-5. \bibitem{Polchinski} J.~Polchinski. Scale and conformal invariance in quantum field theory. \emph{Nuclear Physics B} \textbf{303} (1988), 226--236. doi:10.1016/0550-3213(88)90179-4. \bibitem{KomargodskiSchwimmer} Z.~Komargodski and A.~Schwimmer. \emph{On Renormalization Group Flows in Four Dimensions}. arXiv:1107.3987. Cited for the independently stated dilaton and anomaly framework, not as a premise of Family 282. \bibitem{LPR} M.~A.~Luty, J.~Polchinski, and R.~Rattazzi. \emph{The \(a\)-theorem and the Asymptotics of 4D Quantum Field Theory}. arXiv:1204.5221. The hypotheses and scope of that approach are retained. \bibitem{DKST} A.~Dymarsky, Z.~Komargodski, A.~Schwimmer, and S.~Theisen. \emph{On Scale and Conformal Invariance in Four Dimensions}. arXiv:1309.2921. \bibitem{DFKLP} A.~Dymarsky, K.~Farnsworth, Z.~Komargodski, M.~A.~Luty, and V.~Prilepina. \emph{Scale Invariance, Conformality, and Generalized Free Fields}. arXiv:1402.6322. Additional assumptions and improvement questions are not erased. \bibitem{HollandsWald} S.~Hollands and R.~M.~Wald. \emph{Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes}. arXiv:gr-qc/0404074. Wick products, time-ordered products, and their conservation and derivative-normalization conditions. \bibitem{Nelson} E.~Nelson. Analytic vectors. \emph{Annals of Mathematics} \textbf{70} (1959), 572--615. doi:10.2307/1970331. Cited for sufficient integration and operator-domain tools, not applied without their hypotheses. \bibitem{HislopLongo} P.~D.~Hislop and R.~Longo. Modular structure of the local algebras associated with the free massless scalar field theory. \emph{Communications in Mathematical Physics} \textbf{84} (1982), 71--85. doi:10.1007/BF01208372. \bibitem{BGL} R.~Brunetti, D.~Guido, and R.~Longo. Modular structure and duality in conformal quantum field theory. \emph{Communications in Mathematical Physics} \textbf{156} (1993), 201--219. arXiv:funct-an/9302008. \bibitem{BisognanoWichmann} J.~J.~Bisognano and E.~H.~Wichmann. On the duality condition for a Hermitian scalar field. \emph{Journal of Mathematical Physics} \textbf{16} (1975), 985--1007. doi:10.1063/1.522605. \bibitem{Wiesbrock} H.-W.~Wiesbrock. Half-sided modular inclusions of von-Neumann-algebras. \emph{Communications in Mathematical Physics} \textbf{157} (1993), 83--92. doi:10.1007/BF02098019. \bibitem{ArakiZsido} H.~Araki and L.~Zsid\'o. Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. \emph{Reviews in Mathematical Physics} \textbf{17} (2005), 491--543. arXiv:math/0412061. \bibitem{Pedersen} G.~K.~Pedersen. \emph{\(C^*\)-Algebras and Their Automorphism Groups}. Academic Press, 1979. \bibitem{KnillLaflamme} E.~Knill and R.~Laflamme. Theory of quantum error-correcting codes. \emph{Physical Review A} \textbf{55} (1997), 900--911. doi:10.1103/PhysRevA.55.900. arXiv:quant-ph/9604034. \bibitem{Watrous} J.~Watrous. \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Finite-dimensional channels, Choi matrices, references, and completely bounded trace norms. \bibitem{Hoeffding} W.~Hoeffding. Probability inequalities for sums of bounded random variables. \emph{Journal of the American Statistical Association} \textbf{58} (1963), 13--30. doi:10.1080/01621459.1963.10500830. \bibitem{OpenAI282} OpenAI, as attributed in the supplied source and catalogue. \emph{Scale and conformal symmetry in four-dimensional operational quantum field theory}. 26 September 2026. Family 282-01. Catalogue: \url{https://enter.grok.me/openai/282-01}. The catalogue and supplied \sourceid{thm:main} account are documentary records, not independently verified premises. The catalogue reports no Lean scope note for this family. \bibitem{Dossier} \emph{User-supplied Oai282-A development plan and comparative source dossier}. Includes Family 282 framework and theorem statements, source excerpts, standing status records, and comparative family descriptions. Retrieval label \sourceid{6f2e9611}. No comprehensive corpus scan or independent source-proof verification is asserted. \bibitem{CheungPortal} N.~Cheung. \emph{Enter Calculus Companion}. \url{https://enter.grok.me}. Searchable companion, not a proof checker. Individual manuscript identifiers and supplied versions govern the attributions below. \bibitem{CheungII} N.~Cheung. \emph{Landmark II}, version 2. Supplied manuscript \sourceid{II.tex}. Finite contact-membership certificate and its stated coverage boundaries. A finite fit is not a global functional identity. Enter Calculus Companion. \bibitem{CheungIII} N.~Cheung. \emph{Descent of first-order relational scalars and the free-product obstruction: Classical equation spaces, retarded sections, and compact Ward observables}. Landmark III, version 2. Supplied \sourceid{III.tex} excerpts. Classical quotient and coupled-jet results on their declared equation spaces; companion comparisons are not premises. Enter Calculus Companion. \bibitem{CheungVI} N.~Cheung. \emph{Relational Einstein--scalar observables at second order: Classical descent and a compact-switch obstruction}. Landmark VI, version 1. Supplied \sourceid{VI.tex} excerpts. Especially \sourceid{thm:descent}. Second-order classical relational descent; the supplied quantum certificate remains RG-open. Enter Calculus Companion. \bibitem{CheungEn05C} N.~Cheung. \emph{En05-C}, version 1. Supplied \sourceid{En05-C.tex} excerpts. Especially \sourceid{thm:comparison}. Tensor-free snapshot--channel--adjoint equivalence for declared unital local algebras. Enter Calculus Companion. \bibitem{CheungEn05A} N.~Cheung. \emph{En05-A}, version 1. Supplied \sourceid{En05-A.tex} excerpts. Especially \sourceid{thm:cone} and \sourceid{lem:comparison}. Schedule-past support and a prescribed metric envelope. Enter Calculus Companion. \bibitem{CheungEn03A} N.~Cheung. \emph{En03-A}, version 1. Supplied \sourceid{En03-A.tex} excerpts. Especially \sourceid{thm:locality}. Complete finite schedule support, not a derived field-theoretic causal structure. Enter Calculus Companion. \bibitem{CheungEn06A} N.~Cheung. \emph{En06-A}, version 1. Supplied \sourceid{En06-A.tex} excerpts. Especially \sourceid{thm:algmicro} and its localized observable-limit hypotheses. Cross-product preservation is not replaced by channel commutation. Enter Calculus Companion. \bibitem{CheungEn06B} N.~Cheung. \emph{En06-B}, version 1. Supplied \sourceid{En06-B.tex} excerpts. Especially \sourceid{thm:identity}. Complete finite instrument transport with prescribed routes, ports, and speed. Enter Calculus Companion. \bibitem{CheungEn01B} N.~Cheung. \emph{En01-B}, version 2. Supplied \sourceid{En01-B-v2.tex} excerpts. Especially \sourceid{thm:bounded}. Bounded complete-record validation under a declared statistical contract. Enter Calculus Companion. \bibitem{CheungOai265B} N.~Cheung. \emph{Oai265-B}, version 2. Supplied \sourceid{Oai265-B-v2.tex} record. Finite-confidence certification for one implemented qubit decoder; coverage is not a posterior or a field-instrument construction. Enter Calculus Companion. \bibitem{CheungEn07A} N.~Cheung. \emph{En07-A}, version 1. Supplied \sourceid{En07-A.tex} excerpts. Especially \sourceid{thm:exact}. Complete-complement recovery and the separation of operational discard, quantum return, and holographic terminology. Enter Calculus Companion. \bibitem{CheungEn07B} N.~Cheung. \emph{En07-B}, version 1. Supplied \sourceid{En07-B.tex} excerpts. Especially \sourceid{thm:exact} and \sourceid{thm:endpoint}. Finite endpoint statements retain their four same-model obligations. Enter Calculus Companion. \bibitem{CheungHawk04} N.~Cheung. \emph{Hawk04-A}, version 1, and \emph{Hawk04-B}, version 2. Supplied manuscript excerpts. Exact complete-complement recovery, complete preservation, and classical-record limitations. No gravitational evaporation interface is imported. Enter Calculus Companion. \bibitem{CheungOai290A} N.~Cheung. \emph{Oai290-A}, version 1. Supplied \sourceid{Oai290-A.tex} excerpts. Especially \sourceid{thm:exact}, \sourceid{thm:bounded}, and \sourceid{thm:assembly}. Finite channel decisions and a modular-band witness are separately typed. No application to the Family 282 null cuts is supplied. Enter Calculus Companion. \bibitem{CheungOai265A} N.~Cheung. \emph{Oai265-A}, version 1. Supplied \sourceid{Oai265-A.tex} excerpts. Especially \sourceid{def:determination} and \sourceid{thm:separation}. Static area-law records, selected entropy, recovery, and histories remain separate. Enter Calculus Companion. \bibitem{CheungEm02E} N.~Cheung. \emph{Em02-E}, version 1. Supplied \sourceid{Em02-E-v1.tex} excerpts. Especially \sourceid{thm:lapse} and \sourceid{thm:release}. Relational lapse on a declared Einstein--scalar background. Not the scalar improvement field \(L\). Enter Calculus Companion. \bibitem{CheungEm03A} N.~Cheung. \emph{Em03-A}, version 1. Supplied \sourceid{Em03-A-v1.tex} excerpts. Especially \sourceid{thm:main} and \sourceid{thm:release}. Fixed-background scalar emission and its stated exterior, state, scattering, and normalization inputs. Enter Calculus Companion. \bibitem{CheungXVIPrepared} N.~Cheung. \emph{XVI-A}, version 1; \emph{XVI-H}, version 1; and \emph{XVI-I}, version 1. Supplied manuscript excerpts. Prepared-source declarations, a conditional compact polarized kernel, and a reduced two-edge extension family. Their normalization and admission boundaries are retained. Enter Calculus Companion. \bibitem{CheungXVIN} N.~Cheung. \emph{XVI-N}, version r12. Supplied \sourceid{XVI-N.tex} excerpts. Especially \sourceid{thm:massless-obstruction}. Conditional prepared-insertion obstruction; the spatially compact Route B is not discharged. Enter Calculus Companion. \bibitem{CheungXVIP} N.~Cheung. \emph{XVI-P}, version p23. Supplied \sourceid{XVI-P.tex} excerpt and status record. Technical subsector and build-report provenance are documentary. \sourceid{O.1} remains \sourceid{NOT_CLOSED}. No execution is independently reproduced here. Enter Calculus Companion. \bibitem{CheungXIVH} N.~Cheung. \emph{Landmark XIV-H: Internal kernel relations on the operator route}, version 1. Supplied \sourceid{XIV-H.tex} excerpts. Historical H2 amplitude boundary and exact internal kernel maps. Enter Calculus Companion. \bibitem{CheungXIVI} N.~Cheung. \emph{Landmark XIV-I}, version 1. Supplied \sourceid{XIV-I.tex} excerpts. Especially \sourceid{thm:I2}, \sourceid{thm:I3}, and \sourceid{thm:release}. Internal reconstruction is not the full characteristic composition. Enter Calculus Companion. \bibitem{CheungXIVJ} N.~Cheung. \emph{Landmark XIV-J}, version 1. Supplied \sourceid{XIV-J.tex} excerpts. Especially \sourceid{prop:descent}, \sourceid{thm:reference}, \sourceid{thm:J2}, and \sourceid{cor:noadmission}. The full joint map remains unconstructed in the supplied release. Enter Calculus Companion. \bibitem{CheungXVB} N.~Cheung. \emph{XV-B}, version 2. Supplied \sourceid{XV-B.tex} excerpts. Relative lifting, finite normal-jet null tests, and conditional extension interfaces; the inherited full graph remains unresolved at B1 in the supplied account. The technical manuscript is distinct from the closed downstream quantum-admission exit. Enter Calculus Companion. \bibitem{CheungI} N.~Cheung. \emph{Landmark I}, version 2. Supplied \sourceid{I.tex} excerpts. Especially \sourceid{thm:main}. Born strategy and destructive projective residuals inside a stated finite strategy class. No field instrument or stress-tensor Ward theorem is imported. Enter Calculus Companion. \bibitem{Oai280A} \emph{Oai280-A: Sector-Label Certificates and VOA--Net Separation}. Earlier conversation draft. Cited only for its scoped source-record and operational comparison architecture. No independent registration, build, or proof verification is asserted. \bibitem{Oai211A} \emph{Oai211-A}. Earlier comparison draft as described in the supplied dossier. Conditional path-sum and metric-measure comparison only. No derivation of four-dimensional conformal symmetry is imported. \end{thebibliography} \end{document}