% Oai282-B.tex % Single-file manuscript for pdfLaTeX. % Suggested build: % pdflatex -interaction=nonstopmode Oai282-B.tex % pdflatex -interaction=nonstopmode Oai282-B.tex % % No successful compilation or proof-assistant verification is asserted. % The supplied Cheung manuscripts are documentary comparisons, not % premises of the free-field proofs below. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,longtable,array} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\D}{\mathcal D} \newcommand{\Sch}{\mathcal S} \newcommand{\Hil}{\mathcal H} \newcommand{\Dom}{\operatorname{Dom}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\im}{\operatorname{im}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\id}{\operatorname{id}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\Wick}[1]{\mathopen{:}#1\mathclose{:}} \newcommand{\pair}[2]{\langle #1,#2\rangle} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\Alg}{\mathfrak A} \newcommand{\Wal}{\mathcal W} \newcommand{\DF}{\D_{\mathrm{fin}}} \newcommand{\DE}{\D_E} \newcommand{\Dold}{\D_{\mathrm{old}}} \newcommand{\vx}{v_X} \newcommand{\sigX}{\sigma_X} \newcommand{\NOTCLOSED}{\texttt{NOT\_CLOSED}} \newcommand{\scopebox}[2]{% \begin{center} \fbox{\parbox{0.93\linewidth}{\textbf{#1}\par\smallskip #2}} \end{center}} \title{\textbf{Oai282-B}\\[3pt] From Finite Ward Certificates to Local Conformal Symmetry:\\ Physical Improvement, Explicit Contacts, and Complete Coverage\\ for the Four-Dimensional Free Massless Scalar} \author{} \date{October 12, 2026} \begin{document} \maketitle \begin{abstract} We give a self-contained, model-specific continuation of the improvement and finite-fit programme described in Oai282-A. The model is the vacuum Fock representation of the real massless scalar on four-dimensional Minkowski space, with its original Weyl local net. We construct the improvement scalar \(L=-\frac12\Wick{\phi^2}\), prove its mixed-product and common-domain properties, and establish affiliated realizations of its compact smearings in that same net. We calculate the elementary retarded Ward contact, its descendant contacts, and the local stress-current action on the complete Wick-polynomial field algebra. Translations, Lorentz transformations, dilations, and special conformal transformations are included. Successive actions obey the conformal Lie algebra with an explicitly fixed sign convention. A separate classification theorem identifies every conserved stress tensor in a declared homogeneous, Poincare-covariant local Wick-polynomial class. Its complete residual family is two-dimensional, and two explicitly normalized matrix-element probes separate it. Consequently finite vanishing gives the full local Ward conclusion within that class, rather than merely a fit on named tests. The conclusions concern local current implementation and ordered Wick-field products. They do not assert a general scale-to-conformal theorem, a complete renormalized time-ordered stress-tensor insertion scheme, global integration of the conformal generators, or an interacting improvement-scalar reconstruction. The distinct characteristic-product programme of XIV-I, XIV-J, and XV-B is not completed. No coefficient from that programme is admitted; O.1 remains \NOTCLOSED. \end{abstract} \tableofcontents %===================================================================== \section{The three targets and the exact scope} \label{sec:scope} %===================================================================== \subsection{The predecessor's split} The supplied description of Oai282-A separates three targets \cite{CheungA}. \paragraph{Target A: finite certification.} The predecessor's \texttt{thm:ward} and \texttt{prop:finitefit} concern named residuals: \[ w(\theta)=w_0+N\theta, \qquad w(\theta)=0 \quad\Longleftrightarrow\quad -w_0\in\im N. \] Its \texttt{thm:statistics} is an optional bounded-score tolerance statement, not an operator-valued distributional identity. \paragraph{Target B: local Ward implementation.} The predecessor's \texttt{prop:affine-ward} is conditional on a spacelike-collar contract and a declared polynomial. It is not, by itself, implementation by \(X^\nu t_{\mu\nu}\) on every field in a complete test class. The new free-field results below address precisely this additional issue. \paragraph{Target C: global implementation.} The predecessor's \texttt{prop:flow} and \texttt{prop:nonintegrable} separate local identities from global geometric flows and unitary integration. We reproduce the relevant elementary examples in Section~\ref{sec:global}. Global implementation is not attempted. This manuscript does not use a supplied series sheet as a premise. References to Cheung's manuscripts identify predecessor statements, comparison interfaces, and open boundaries. The mathematical inputs used in the proofs are the explicitly declared free Fock construction and standard functional-analytic results, with their hypotheses checked where applied. \subsection{What ``local Ward identity'' means here} \begin{definition}[The Ward conclusion used in this paper] \label{def:wardmeaning} A local conformal Ward conclusion consists of: \begin{enumerate} \item an invariant dense field domain and continuous ordered joint field distributions; \item a conserved current \(j_X^\mu=t^{\mu\nu}X_\nu\); \item a compact spacetime-smeared current whose commutator implements the declared infinitesimal transformation on every compactly localized Wick polynomial and every finite ordered product; \item elementary and descendant contact distributions with the normalization calculated below; \item compatibility of these actions with field relations and conformal Lie brackets. \end{enumerate} It does not mean that every renormalized time-ordered product containing arbitrarily many stress tensors has been constructed. \end{definition} The distinction is important. Distributional conservation of a current is weaker than its implementing a field transformation. Conversely, current implementation on an ordered field algebra does not construct a curved-background generating functional or remove the separate normalization issues of time-ordered products. General Ward normalization frameworks are discussed in \cite{DutschBoas,FR}; none is imported here as an unspecified completion of our insertion class. \subsection{Main statement} \begin{theorem}[Complete local free-scalar benchmark] \label{thm:main} Consider the real massless scalar in its vacuum Fock representation on \(M=\R^{1,3}\), with the Weyl net and domains specified in Section~\ref{sec:model}. Then: \begin{enumerate} \item \(Q=\Wick{\phi^2}\) and \(L=-Q/2\) belong to the same physical Wick-field extension of the original scalar theory. Their mixed ordered products exist on a common invariant dense domain containing the original vacuum-word domain. Every real compact smearing of \(L\) has a self-adjoint realization affiliated with the original local Weyl algebra containing its support. \item The improved tensor \[ t_{\mu\nu} = \Wick{\partial_\mu\phi\,\partial_\nu\phi} -\frac12\eta_{\mu\nu} \Wick{\partial_\rho\phi\,\partial^\rho\phi} +\frac16(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q \] is symmetric, conserved, and traceless on that domain. \item Its compact-slab currents implement all conformal Killing transformations on every compactly smeared Wick polynomial, all descendants, and all finite ordered mixed products. The elementary retarded contact is \[ \partial_\mu^x C_X^\mu(x,y) = \delta^{(4)}(x-y) \bigl(X^\nu\partial_\nu+\sigma_X\bigr)\phi(y), \qquad \sigma_X=\frac14\partial_\nu X^\nu. \] Descendant contacts and the corresponding complete field actions are given in Sections~\ref{sec:contacts} and \ref{sec:charges}. \item The sign-adjusted local derivations \(\delta_X\) defined in Section~\ref{sec:brackets} satisfy \[ [\delta_X,\delta_Y]=\delta_{[X,Y]} \] on the complete ordered Wick-field algebra. \item Within the stress-tensor class of Definition~\ref{def:stressclass}, the complete residual family has a proved finite-dimensional span. Two explicit probes are injective on that span. Their exact vanishing forces the displayed tensor \(t\), and therefore all the local Ward identities just stated. \end{enumerate} No assertion is made that the hypotheses of the supplied Family 282 claim have thereby all been verified. In particular, no global conformal action on the whole bounded net is constructed. \end{theorem} \begin{proof} Assertion (i) is proved in Propositions~\ref{prop:domain} and \ref{prop:physicalL}. Assertion (ii) is Proposition~\ref{prop:improvement}. The contact calculation is Proposition~\ref{prop:elementarycontact}; implementation and mixed products are Theorem~\ref{thm:localaction}. Lie compatibility is Proposition~\ref{prop:brackets}. Classification and complete separation are Theorems~\ref{thm:classification} and \ref{thm:coverage}. The exclusions are the scope of those statements, not additional enhancement claims. \end{proof} %===================================================================== \section{The free model, its test topology, and its domain} \label{sec:model} %===================================================================== \subsection{Conventions} Use \[ \eta=\operatorname{diag}(1,-1,-1,-1), \qquad \Box=\partial_0^2-\Delta_{\mathbf x}, \qquad \hbar>0. \] Let \[ C_+=\{p=(|\mathbf p|,\mathbf p):\mathbf p\in\R^3,\ \mathbf p\ne0\}, \qquad \dd\mu(p)=\frac{\dd^3\mathbf p}{(2\pi)^3\,2|\mathbf p|}. \] The one-particle space and Fock space are \[ \mathfrak h=L^2(C_+,\dd\mu), \qquad \Hil=\Gamma_s(\mathfrak h). \] Write \(\Omega\) for the Fock vacuum, \(N\) for particle number, and \(H=\mathrm d\Gamma(p^0)\). Creation and annihilation operators have the usual Fock normalization, and \[ \phi(x) = \sqrt{\hbar}\int_{C_+} \bigl(e^{-\ii p\cdot x}a(p)+e^{\ii p\cdot x}a^*(p)\bigr) \dd\mu(p). \] For test functions use \[ \widehat f(k)=\int_M e^{\ii k\cdot x}f(x)\dd^4x. \] The two-point distribution is \begin{equation} W(x-y) = \pair{\Omega}{\phi(x)\phi(y)\Omega} = \hbar\int_{C_+}e^{-\ii p\cdot(x-y)}\dd\mu(p). \label{eq:W} \end{equation} In particular, \[ W(z) = \frac{\hbar}{4\pi^2} \lim_{\epsilon\downarrow0} \frac{1}{|\mathbf z|^2-(z^0-\ii\epsilon)^2}. \] Let \(R\) be the retarded fundamental solution: \[ \Box R=\delta_0, \qquad \supp R\subseteq\overline V_+, \qquad R(z)=\frac1{2\pi}\Theta(z^0)\delta(z^2). \] With \(E=R-R_{\mathrm{adv}}\), our commutator convention is \begin{equation} [\phi(x),\phi(y)]=-\ii\hbar E(x-y), \qquad [\partial_0\phi(t,\mathbf x),\phi(t,\mathbf y)] =-\ii\hbar\delta^{(3)}(\mathbf x-\mathbf y). \label{eq:CCR} \end{equation} \subsection{The original bounded net} For every bounded open region \(O\), define \begin{equation} \Alg(O) = \{e^{\ii\overline{\phi(f)}}: f\in C_c^\infty(O;\R)\}''. \label{eq:net} \end{equation} The closure of \(\phi(f)\) is self-adjoint for real \(f\): finite-particle vectors are analytic for a Segal field. The same creation--annihilation calculation gives locality from \eqref{eq:CCR}. Thus \eqref{eq:net} is the original scalar Weyl net, not a net enlarged by adjoining \(Q\). \subsection{A common invariant domain} Set \[ \DF=\bigoplus_{n=0}^{\mathrm{alg}\,\infty} \mathfrak h^{\otimes_s n}, \qquad \DE = \bigoplus_{n=0}^{\mathrm{alg}\,\infty} \bigcap_{r\ge0}\Dom\bigl((1+H)^r\bigr) \cap\mathfrak h^{\otimes_s n}. \] The topology on each fixed-particle summand is given by the seminorms \(\norm{(1+H)^r\psi}\); the algebraic sum has the corresponding locally convex direct-sum topology. A local Wick field is a finite linear combination of \[ B_{\boldsymbol\alpha}(x) = \Wick{\prod_{a=1}^k\partial^{\alpha_a}\phi(x)}. \] Its smearings use \(\Sch(M)\), and its local smearings use \(C_c^\infty(M)\). These are vacuum normal products, with no self-contractions inside a Wick block. Define \(\Dold\) to be the span of the continuous joint-Schwartz vacuum-word vectors of the original field \(\phi\). This is the old domain of the benchmark just declared. It is not identified with an unspecified domain from another model. \begin{lemma}[Fixed-sector Wick estimates] \label{lem:wickest} For a fixed Wick field \(B\), input particle number \(n\), and integer \(s\ge0\), there exist an integer \(r\), a continuous Schwartz seminorm \(p\), and a constant \(C\) such that \begin{equation} \norm{(1+H)^s B(f)\psi_n} \le C p(f)\norm{(1+H)^r\psi_n}. \label{eq:wickest} \end{equation} Only finitely many output particle sectors occur. \end{lemma} \begin{proof} Expand the normal product into its finitely many creation--annihilation terms. On a participating set of momenta, each kernel has the form \[ P(p_1,\ldots,p_u,q_1,\ldots,q_v)\, \widehat f\left(\sum_{a=1}^u p_a-\sum_{b=1}^v q_b\right), \] where \(P\) is a fixed polynomial produced by the derivatives. The nonparticipating particles contribute identity kernels. For the participating variables put \[ P_0=\sum_a p_a^0, \qquad Q_0=\sum_b q_b^0. \] They are nonnegative, and \[ P_0\le Q_0+|P_0-Q_0|. \] An output energy weight can therefore be bounded by a polynomial in the input energy and the temporal Fourier variable. Schwartz decay absorbs the latter polynomial. More explicitly, after inserting a sufficiently large input weight \((1+Q_0)^{-r}\), the squared participating kernel is integrable: the radial mass-shell measure of each variable is a constant multiple of \(u\dd u\dd\omega\), the derivative factors have polynomial growth, and a sufficiently high power of \((1+|P_0-Q_0|)^{-1}\), together with \((1+Q_0)^{-r}\), makes the radial integrals finite. There is no infrared divergence, because \(u\dd u\) is locally integrable at zero. This gives a Hilbert--Schmidt bound for the weighted participating kernel. If one side has no participating variables, the same calculation uses the rapid decay in the sum of the positive energies. The finitely many Fock combinatorial factors depend on \(n\) and the degree of \(B\), and are absorbed in \(C\). The spectator energy weight is transferred to the input by the same inequality. Summing the terms proves \eqref{eq:wickest}. \end{proof} \begin{proposition}[Joint products, old-domain inclusion, and locality] \label{prop:domain} The following statements hold. \begin{enumerate} \item \(\DE\) is Hilbert-dense and invariant under every smeared Wick field and its formal adjoint. \item Every finite ordered list of Wick fields defines continuous joint-Schwartz matrix-element distributions on \(\DE\). The corresponding joint vacuum-vector maps are continuous. \item \(\Dold\subseteq\DE\), and \(\Dold\) is Hilbert-dense. \item All these Wick fields are closable on \(\DE\). Spacelike separated smearings commute on \(\DE\). \end{enumerate} \end{proposition} \begin{proof} Energy cutoffs in every finite-particle sector prove Hilbert density. Lemma~\ref{lem:wickest}, for every \(s\), proves invariance. Iterating the same lemma gives continuous multilinear maps in the individual Schwartz smearings, with bounds by products of finitely many Schwartz seminorms. Nuclearity identifies the completed projective tensor product of the Schwartz spaces with the joint Schwartz space. Hence the maps extend uniquely to arbitrary joint Schwartz kernels. The same estimates give continuous vacuum-vector maps. For old vacuum words, the Fourier variables creating particles have positive energies. A joint Schwartz kernel therefore gives rapid decay in their total energy, and hence finite moments of every order. This proves \(\Dold\subseteq\DE\). Old field vacuum words generate the scalar Fock representation: restrictions of spacetime Schwartz Fourier transforms to the positive mass shell are dense in \(\mathfrak h\), and their symmetric tensor products are dense in each particle sector. Thus \(\Dold\) is Hilbert-dense. The adjoint pairing of a normal product is the normal product of the adjoint factors, with the reversed operator order for an ordered word. It is defined on \(\DE\). A densely defined formal adjoint therefore gives closability. Finally, Wick's expansion of a commutator between spacelike separated blocks contains commutators of the elementary fields or their derivatives. Each vanishes by \eqref{eq:CCR}. The support-preserving joint-test extension preserves this identity. \end{proof} \begin{remark}[What was not inferred from Hilbert density] No arbitrary originally supplied closed realization has been identified merely because a domain is Hilbert-dense. A graph-core assertion requires an additional argument. For \(Q(f)\), that argument is supplied next. \end{remark} %===================================================================== \section{Physical membership of the improvement scalar} \label{sec:physical} %===================================================================== \subsection{The quadratic kernel} Let \(Q=\Wick{\phi^2}\). For real \(f\in\Sch(M)\), its quadratic Fock decomposition is \begin{equation} Q(f) = \hbar\bigl( a^*a^*(F_f)+aa(\overline{F_f})+2\,\mathrm d\Gamma(B_f) \bigr), \label{eq:Qfock} \end{equation} where, with the kernel convention of Section~\ref{sec:model}, \[ F_f(p,q)=\widehat f(p+q), \qquad B_f(p,q)=\widehat f(p-q). \] The pair kernel is symmetric, and \(B_f\) is self-adjoint for real \(f\). \begin{lemma}[Number bounds and essential self-adjointness] \label{lem:Qsa} For every real \(f\in\Sch(M)\), \(F_f\) is Hilbert--Schmidt, \(B_f\) is bounded, and \begin{equation} \norm{Q(f)\psi}\le C_f\norm{(N+1)\psi}, \qquad \psi\in\DF. \label{eq:Qnumber} \end{equation} The operator \(Q(f)\) is essentially self-adjoint on \(\DF\). Its restriction to \(\DE\) has the same self-adjoint closure. \end{lemma} \begin{proof} Schwartz decay in the temporal Fourier variable gives \[ |F_f(p,q)| \le C_M(1+p^0+q^0)^{-M}. \] The radial measure is proportional to \(p^0\dd p^0\) on each shell, so the square of this bound is integrable for sufficiently large \(M\). Thus \(F_f\) is Hilbert--Schmidt. For \(B_f\), Schwartz decay gives, for large \(M\), \[ |B_f(p,q)| \le C_M(1+|\mathbf p-\mathbf q|)^{-M}. \] The integral of this bound against \(\dd\mu(q)\) is uniformly bounded in \(p\): on \(|\mathbf q|<1\), the measure is integrable; on \(|\mathbf q|\ge1\), it is bounded by a constant times Lebesgue measure, and the translated integrable decay has constant integral. The corresponding column bound is identical. Schur's test proves boundedness. On the \(n\)-particle sector, the pair creation and annihilation terms are bounded by constants times \(\sqrt{(n+1)(n+2)}\norm{F_f}\) and \(\sqrt{n(n-1)}\norm{F_f}\). The number-preserving term is bounded by \(2n\norm{B_f}\). This proves \eqref{eq:Qnumber}. Repeated application changes particle number by at most two. For a vector with maximal particle number \(n\), \[ \norm{Q(f)^r\psi} \le C_f^r \prod_{j=0}^{r-1}(n+2j+1)\norm{\psi}. \] The analytic-vector series consequently converges for sufficiently small parameter. Nelson's analytic-vector theorem \cite{Nelson} gives essential self-adjointness on \(\DF\). On each fixed finite collection of particle sectors, \eqref{eq:Qnumber} bounds \(Q(f)\) by a constant times the Hilbert norm. Energy-cutoff vectors in \(\DE\) approximate any vector of \(\DF\), both in Hilbert norm and after applying \(Q(f)\). Thus \(\DE\) is a graph core for this particular closure. \end{proof} \subsection{Affiliation without enlarging the net} \begin{lemma}[Local quadratic approximation] \label{lem:localapprox} If \(f\in C_c^\infty(O;\R)\), then \(\overline{Q(f)}\) is affiliated with \(\Alg(O)\). \end{lemma} \begin{proof} Choose a real compact smooth mollifier \(\rho_\epsilon\), normalized to integral one, and supported in a ball of radius \(\epsilon\). For sufficiently small \(\epsilon\), every translated mollifier meeting \(\supp f\) is supported in \(O\). Put \[ Q_\epsilon(f) = \int f(x) \left( \phi(\rho_{\epsilon,x})^2 - \pair{\Omega}{\phi(\rho_{\epsilon,x})^2\Omega}I \right)\dd^4x. \] For fixed \(\epsilon\), the integral is first defined by its quadratic Fock kernels. Those kernels are Bochner integrals of rank-one and rank-two kernels over a compact parameter set. Riemann sums are finite real linear combinations of \[ \phi(g_j)^2-\pair{\Omega}{\phi(g_j)^2\Omega}I, \qquad g_j\in C_c^\infty(O;\R). \] Each summand is affiliated with \(\Alg(O)\), by the spectral calculus for the local Segal field. Each finite sum is essentially self-adjoint on \(\DF\), by the same bounded-number argument as in Lemma~\ref{lem:Qsa}. The unitary product formula for these self-adjoint summands, whose sum is essentially self-adjoint on the common finite-particle core, gives its unitary group as a strong limit of products of their local unitary groups \cite{Trotter}. Hence each finite sum is affiliated with \(\Alg(O)\). For fixed \(\epsilon\), the Riemann-sum pair kernels converge in Hilbert--Schmidt norm, and the number-preserving kernels converge in operator norm. The associated operators therefore converge on \(\DF\). This yields strong-resolvent convergence of their self-adjoint closures, so \(Q_\epsilon(f)\) is affiliated with \(\Alg(O)\). For completeness, the strong-resolvent implication used here has a short core argument. If self-adjoint \(A_j,A\) satisfy \(A_j\psi\to A\psi\) on a core of \(A\), then \[ (A_j-\ii)^{-1}(A-\ii)\psi-\psi = (A_j-\ii)^{-1}(A-A_j)\psi\longrightarrow0. \] The range of \(A-\ii\) on that core is dense and the resolvents are uniformly bounded, so convergence holds on the whole Hilbert space. Now remove the mollifier. On the mass shell the one-particle multiplier \(\widehat\rho_\epsilon(p)\) is uniformly bounded and converges pointwise to one. Thus the pair kernels converge to \(F_f\) in Hilbert--Schmidt norm by dominated convergence. The number-preserving kernels converge strongly to \(B_f\), with uniform operator-norm bounds, because they are obtained by placing these bounded multipliers on the two sides of \(B_f\). It follows that \(Q_\epsilon(f)\psi\to Q(f)\psi\) on \(\DF\). Lemma~\ref{lem:Qsa} and the same core argument give strong-resolvent convergence to \(\overline{Q(f)}\). The resolvents of all approximants belong to \(\Alg(O)\), and that von Neumann algebra is strongly closed. The limiting resolvents therefore belong to \(\Alg(O)\). This is affiliation of \(\overline{Q(f)}\) with the original net. \end{proof} \begin{proposition}[Physical \(L\) in the declared benchmark] \label{prop:physicalL} The field \[ L=-\frac12 Q \] has mixed ordered products with every original scalar field, every adjoint, and every Wick polynomial on \(\DE\supseteq\Dold\). It is local, and every real compact smearing has a self-adjoint affiliated realization in the original Weyl net. Complex compact smearings have compatible closed affiliated realizations whose action and adjoint action agree with the corresponding Wick insertions on \(\DE\). \end{proposition} \begin{proof} Mixed products, locality, and domain invariance follow from Proposition~\ref{prop:domain}. Real affiliated realizations follow from Lemma~\ref{lem:localapprox} and multiplication by \(-1/2\). For a complex smearing, write \(f=f_1+\ii f_2\) with real \(f_1,f_2\). Let \(A,B\) be the affiliated self-adjoint realizations of \(L(f_1),L(f_2)\). On \(\Dom A\cap\Dom B\), the sum \(A+\ii B\) is densely defined and closable: its adjoint contains \(A-\ii B\) on the same dense intersection. Every unitary in the commutant of the local algebra preserves both domains and commutes with both summands. It consequently preserves the closure of the sum. That closure is affiliated, agrees with \(L(f)\) on \(\DE\), and its Hilbert adjoint agrees there with \(L(\overline f)\). \end{proof} \begin{remark}[Why this is a genuine membership argument] No Green operator was used to define \(L\). The proof starts from the explicit Wick square, establishes its joint products, and constructs local affiliated realizations in the already declared net. It does not claim the same result for an interacting theory or a reduced derivative-only net. Essential self-adjointness of Wick squares is a substantive operator question in more general representations; see \cite{Sanders}. Our all-real-smearing conclusion uses the special bounded one-particle kernel of the flat free scalar and is not presented as that more general theorem. \end{remark} %===================================================================== \section{Improvement and conformal Killing fields} \label{sec:improvement} %===================================================================== Define \[ A_{\mu\nu}=\Wick{\partial_\mu\phi\,\partial_\nu\phi}, \qquad K=\Wick{\partial_\rho\phi\,\partial^\rho\phi}, \qquad T_{\mu\nu}=A_{\mu\nu}-\frac12\eta_{\mu\nu}K. \] The free equation \(\Box\phi=0\) holds as a field identity. \begin{proposition}[The signs and normalization of improvement] \label{prop:improvement} On \(\DE\), \begin{equation} T^\mu{}_\mu=-K, \qquad \Box Q=2K, \qquad V_\mu=-\frac12\partial_\mu Q, \qquad L=-\frac12Q. \label{eq:improvementdata} \end{equation} Consequently \begin{equation} t_{\mu\nu} = T_{\mu\nu} +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L = T_{\mu\nu} +\frac16(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q \label{eq:t} \end{equation} is symmetric, conserved, and traceless. \end{proposition} \begin{proof} Vacuum normal ordering commutes with constant-coefficient differentiation. Thus \[ \Box Q = 2\Wick{\partial_\rho\phi\,\partial^\rho\phi} +2\Wick{\phi\Box\phi} =2K. \] The trace of \(T\) in four dimensions is \(K-2K=-K\). The remaining identities in \eqref{eq:improvementdata} follow. The divergence of the improvement operator vanishes identically: \[ \partial^\mu (\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q=0. \] The canonical tensor is conserved by the free equation. The trace of the improvement is \(\frac12\Box Q=K\), which cancels the canonical trace. Symmetry is immediate. \end{proof} This is the free-field version of the improved tensor of Callan, Coleman, and Jackiw \cite{CCJ}. The calculation alone does not prove a fixed-point theorem. \subsection{The fifteen local generators} A conformal Killing field satisfies \begin{equation} \partial_\mu X_\nu+\partial_\nu X_\mu =2\sigma_X\eta_{\mu\nu}, \qquad \sigma_X=\frac14\partial_\rho X^\rho. \label{eq:CK} \end{equation} In four dimensions the relevant fields are \[ \begin{array}{lll} \text{translation:} &X^\mu=a^\mu, &\sigma_X=0,\\[2pt] \text{Lorentz:} &X^\mu=\omega^\mu{}_\nu x^\nu,\quad \omega_{\mu\nu}=-\omega_{\nu\mu}, &\sigma_X=0,\\[2pt] \text{dilation:} &X^\mu=\lambda x^\mu, &\sigma_X=\lambda,\\[2pt] \text{special conformal:} &X^\mu=2(b\cdot x)x^\mu-b^\mu x^2, &\sigma_X=2b\cdot x. \end{array} \] They obey \begin{equation} \Box X^\mu=-2\partial^\mu\sigma_X, \qquad \Box\sigma_X=0. \label{eq:CKaux} \end{equation} Put \[ d_X=X^\mu\partial_\mu+\sigma_X, \qquad v_X\phi=d_X\phi. \] A direct calculation gives \begin{equation} \Box d_X=(d_X+2\sigma_X)\Box. \label{eq:waveintertwine} \end{equation} Thus \(v_X\phi\) is again a free solution field. The improved current \[ j_X^\mu=t^{\mu\nu}X_\nu \] is conserved: \[ \partial_\mu j_X^\mu = (\partial_\mu t^{\mu\nu})X_\nu +t^{\mu\nu}\partial_\mu X_\nu =\sigma_X t^\mu{}_\mu=0. \] We now prove implementation, rather than stopping at this identity. %===================================================================== \section{The explicit current contact calculation} \label{sec:contacts} %===================================================================== \subsection{A useful Noether representative} Define \begin{equation} j_{N,X}^\mu = T^{\mu\nu}X_\nu +\sigma_X\Wick{\phi\partial^\mu\phi} -\frac12(\partial^\mu\sigma_X)Q. \label{eq:jN} \end{equation} Before imposing the free equation, the corresponding polynomial calculation gives \[ \partial_\mu j_{N,X}^\mu = \Wick{(\Box\phi)\,d_X\phi}. \] The identity uses \eqref{eq:CK} and \eqref{eq:CKaux}. Define the antisymmetric superpotential \begin{equation} B_X^{\rho\mu} = \frac16 \bigl(X^\mu\partial^\rho Q-X^\rho\partial^\mu Q\bigr) +\frac1{12}Q \bigl(\partial^\mu X^\rho-\partial^\rho X^\mu\bigr). \label{eq:Bsuper} \end{equation} \begin{lemma}[Exact current comparison] \label{lem:super} For every conformal Killing field, \begin{equation} j_X^\mu-j_{N,X}^\mu=\partial_\rho B_X^{\rho\mu}. \label{eq:superidentity} \end{equation} \end{lemma} \begin{proof} Differentiate \eqref{eq:Bsuper}. The conformal Killing equation combines the coefficients of \(\partial Q\) into \(-\frac12\sigma_X\partial^\mu Q\). The last coefficient is \[ \frac1{12} \bigl(\partial^\mu\partial_\rho X^\rho-\Box X^\mu\bigr)Q = \frac12(\partial^\mu\sigma_X)Q. \] The second-derivative terms are \[ \frac16 \bigl(X^\mu\Box Q-X_\nu\partial^\mu\partial^\nu Q\bigr). \] Since \(\Wick{\phi\partial^\mu\phi}=\frac12\partial^\mu Q\), their sum is exactly \(j_X^\mu-j_{N,X}^\mu\). \end{proof} \subsection{Retarded elementary insertion} The following insertion is defined directly, not by multiplying an unspecified time-ordered distribution by a step function. \begin{definition}[Elementary retarded current insertion] \label{def:C} Let \(C_X^\mu(x,y)\) be the polarization of \(j_X^\mu(x)\) obtained by replacing one elementary scalar factor by \(R(x-y)\): \begin{equation} \begin{split} C_X^\mu(x,y) = X_\nu(x)\Big[ &\partial^\mu\phi(x)\,\partial^\nu R(x-y) +\partial^\mu R(x-y)\,\partial^\nu\phi(x)\\ &-\eta^{\mu\nu} \partial_\rho\phi(x)\,\partial^\rho R(x-y)\\ &+\frac13 (\eta^{\mu\nu}\Box_x-\partial_x^\mu\partial_x^\nu) \bigl(\phi(x)R(x-y)\bigr) \Big]. \end{split} \label{eq:Cexplicit} \end{equation} It is understood as a weak field distribution on \(\DE\times\DE\). \end{definition} The existence here is elementary. Matrix elements of a single field between vectors in \(\DE\) are smooth functions of \(x\), with all derivatives locally bounded, by the energy-moment bounds. Equivalently, the variables \((x,x-y)\) are independent: \(\phi(x)R(x-y)\) is the corresponding tensor distribution. Differentiation is therefore legitimate. \begin{proposition}[Exact elementary conformal contact] \label{prop:elementarycontact} The insertion \eqref{eq:Cexplicit} has retarded support in \(x-y\) and satisfies \begin{equation} \partial_\mu^x C_X^\mu(x,y) = \delta^{(4)}(x-y)\,v_X\phi(y). \label{eq:contact} \end{equation} No conservation-to-implementation inference is used. \end{proposition} \begin{proof} Temporarily write \(u=\phi(x)\) and \(r=R(x-y)\). The polarized improved tensor in the square brackets is \[ s^{\mu\nu}(u,r) = \partial^\mu u\,\partial^\nu r +\partial^\mu r\,\partial^\nu u -\eta^{\mu\nu}\partial_\rho u\,\partial^\rho r +\frac13(\eta^{\mu\nu}\Box-\partial^\mu\partial^\nu)(ur). \] Commuting derivatives shows \[ \partial_\mu s^{\mu\nu} = (\Box u)\partial^\nu r+(\Box r)\partial^\nu u. \] Its trace is \[ s^\mu{}_\mu=u\Box r+r\Box u. \] Using \(\Box u=0\), \(\Box r=\delta^{(4)}(x-y)\), and \eqref{eq:CK}, we obtain \[ \begin{split} \partial_\mu(X_\nu s^{\mu\nu}) &= X_\nu\delta^{(4)}(x-y)\partial^\nu\phi(x) +\sigma_X\delta^{(4)}(x-y)\phi(x)\\ &= \delta^{(4)}(x-y) (X^\nu\partial_\nu+\sigma_X)\phi(y). \end{split} \] All products are the already defined tensor products or delta insertion maps. Retarded support follows from the support of \(R\) and its derivatives. \end{proof} In particular, the contact on the elementary field is explicitly \begin{equation} \begin{array}{ll} \text{translation:} &\delta(x-y)\,a^\mu\partial_\mu\phi(y),\\[2pt] \text{Lorentz:} &\delta(x-y)\, \omega^\mu{}_\nu y^\nu\partial_\mu\phi(y),\\[2pt] \text{dilation:} &\delta(x-y)\,\lambda(y\cdot\partial+1)\phi(y),\\[2pt] \text{special conformal:} &\delta(x-y)\, \left( [2(b\cdot y)y^\mu-b^\mu y^2]\partial_\mu +2b\cdot y \right)\phi(y). \end{array} \label{eq:fourcontacts} \end{equation} \subsection{Descendants and Wick-block contacts} For an elementary descendant, define \[ C_{X,\alpha}^\mu(x,y)=\partial_y^\alpha C_X^\mu(x,y). \] Then \begin{equation} \partial_\mu^x C_{X,\alpha}^\mu(x,y) = \partial_y^\alpha \bigl[\delta^{(4)}(x-y)v_X\phi(y)\bigr]. \label{eq:jetcontact} \end{equation} Thus derivative-of-delta contacts are retained: \[ \partial_y^\alpha[\delta(x-y)v_X\phi(y)] = \sum_{\beta\le\alpha} \binom{\alpha}{\beta} (\partial_y^{\alpha-\beta}\delta(x-y)) \partial_y^\beta v_X\phi(y). \] For a Wick block \(B_{\boldsymbol\alpha} =\Wick{\prod_a\partial^{\alpha_a}\phi}\), the single-block insertion is the sum over its participating factors: \begin{equation} \mathcal C_{X,B}^\mu(x,y) = \sum_{a=1}^k \Wick{ C_{X,\alpha_a}^\mu(x,y) \prod_{b\ne a}\partial^{\alpha_b}\phi(y) }. \label{eq:blockC} \end{equation} Here the colon orders all remaining quantum factors; it does not normal-order the numerical retarded kernel. \begin{lemma}[The single-block weak insertion is defined] \label{lem:blockweak} Equation~\eqref{eq:blockC} is a continuous sesquilinear distribution on \(\DE\times\DE\), and \begin{equation} \partial_\mu^x\mathcal C_{X,B}^\mu(x,y) = \sum_{a=1}^k \Wick{ \partial_y^{\alpha_a} [\delta(x-y)v_X\phi(y)] \prod_{b\ne a}\partial^{\alpha_b}\phi(y) }. \label{eq:blockcontact} \end{equation} This assertion is not an operator-domain claim for an arbitrary singularly smeared retarded insertion. \end{lemma} \begin{proof} Between two finite-particle energy-smooth vectors, every fully normal-ordered multipoint kernel is smooth. Indeed, its Fourier integrals contain only the external vector wave functions and finitely many polynomial momentum factors. Arbitrarily high energy moments make these integrals absolutely convergent after Cauchy--Schwarz; the locally integrable mass-shell measure controls zero momentum. The same argument applies to every derivative. Consequently each matrix element of \eqref{eq:blockC} is a finite sum of derivatives of \(R(x-y)\) multiplied by smooth functions. It is a distribution, with finite-order bounds on compact sets. The bounds depend continuously on energy seminorms of the two vectors. Differentiating in \(x\) and applying \eqref{eq:jetcontact} gives \eqref{eq:blockcontact}. \end{proof} \begin{remark}[The contact class is specified] The weak insertion above is a single-block retarded contraction. It is not declared to equal every possible renormalized time-ordered composite insertion. Products with additional singular spectators would require their own product or extension argument. The mixed-product Ward theorem below is instead proved directly by compact-current commutators on the common field domain. \end{remark} %===================================================================== \section{Compact-current implementation on the complete field algebra} \label{sec:charges} %===================================================================== \subsection{The spacelike-collar contract} Let \(K_0\subset M\) be a compact insertion support, and \(I\subset\R\) a compact time interval. Choose \[ a\in C_c^\infty(I;\R), \qquad \int a(t)\dd t=1. \] Choose \(\chi\in C_c^\infty(\R^3;\R)\), equal to one on a neighborhood of \[ \{\mathbf x: (t,\mathbf x)\in J(K_0) \text{ for some }t\in\supp a\}. \] This set is bounded. We also require the support of \(\nabla\chi\), throughout \(\supp a\), to be strictly spacelike separated from \(K_0\). Such a choice is obtained by making the one-region of \(\chi\) larger than the compact causal shadows. Define the actual spacetime-smeared operator \begin{equation} q_X[a,\chi] = \int a(t)\chi(\mathbf x)j_X^0(t,\mathbf x) \dd t\dd^3\mathbf x. \label{eq:q} \end{equation} It acts on \(\DE\) by Lemma~\ref{lem:wickest}. \begin{remark}[No unsupported sharp-time operator is used] A local quadratic Wick field need not admit a sharp-time operator smearing on the Fock domain. All charges in \eqref{eq:q} are smooth spacetime smearings. Equal-time calculations in the proof below calculate the weak commutator, whose contraction is linear; they do not assert existence of an unsmeared quadratic charge operator at a fixed time. \end{remark} \subsection{The elementary implementation calculation} \begin{lemma}[Action on scalar Cauchy data] \label{lem:cauchyaction} Under the collar contract, the weak current commutator has action \[ \frac{\ii}{\hbar}[q_X[a,\chi],\phi(f)] = (v_X\phi)(f), \qquad \supp f\subseteq K_0. \label{eq:elementaryaction} \] \end{lemma} \begin{proof} First use the Noether representative \eqref{eq:jN}. Its formal surface functional, solely for computing a weak commutator, is \[ \int_{\R^3} \left[ \frac{X^0}{2}\Wick{\pi^2+|\nabla\phi|^2} +X^i\Wick{\pi\partial_i\phi} +\sigma_X\Wick{\phi\pi} -\frac12(\partial_0\sigma_X)Q \right]\dd^3\mathbf x, \qquad \pi=\partial_0\phi. \] The canonical commutators give its action on \(\phi\) as \[ X^0\pi+X^i\partial_i\phi+\sigma_X\phi. \] Its action on \(\pi\) is \[ \nabla\cdot(X^0\nabla\phi) +\partial_i(X^i\pi)-\sigma_X\pi +(\partial_0\sigma_X)\phi. \] The conformal Killing identities imply \[ \partial_0X^0=\sigma_X,\qquad \partial_iX^i=3\sigma_X,\qquad \partial_0X^i=\partial_iX^0. \] Together with \(\partial_0\pi=\Delta\phi\), the latter expression is exactly \(\partial_0(v_X\phi)\). This weak surface commutator is legitimate independently of a sharp-time quadratic operator: one current-field contraction leaves a linear field, and the equal-time distributions are the canonical delta distributions in \eqref{eq:CCR}. They can equivalently be obtained by smooth time approximation before taking the weak commutator. The commutator with the full surface current, as a function of the insertion coordinate, satisfies the free wave equation. Equation~\eqref{eq:waveintertwine} shows that \(v_X\phi\) does too. The calculated Cauchy data therefore identify them by uniqueness of the scalar Cauchy problem. This argument is local in the causal shadow of the insertion. Replacing the full surface by \(\chi\) leaves the commutator unchanged because the omitted support is spacelike separated from the insertion. By Lemma~\ref{lem:super}, the improved and Noether currents differ on the surface by a spatial divergence. After multiplying by \(\chi\), the difference is supported where \(\nabla\chi\ne0\). Its commutator also vanishes by the collar contract. Finally average the weak surface commutator with \(a\). It is the same \(v_X\phi\) for every surface time in the slab, and \(\int a=1\). The resulting averaged operator is the well-defined operator \eqref{eq:q}. This proves \eqref{eq:elementaryaction} on \(\DE\). \end{proof} \subsection{Normal-ordering compatibility} \begin{lemma}[Infinitesimal covariance of the contraction] \label{lem:Wcov} For every conformal Killing field, \begin{equation} (d_X^x+d_X^y)W(x-y)=0 \label{eq:Wcov} \end{equation} as a tempered distribution. \end{lemma} \begin{proof} For the fifteen displayed fields, direct polynomial calculation gives \[ (X(x)\cdot\partial_x+X(y)\cdot\partial_y)(x-y)^2 = \bigl(\sigma_X(x)+\sigma_X(y)\bigr)(x-y)^2. \] Apply this identity to the rational inverse of the squared separation. The weight terms in \(d_X^x+d_X^y\) cancel its derivative. To pass the identity to \eqref{eq:W}, take independent complex point variables in a positive-energy tube. The same polynomial identity holds there, before taking a boundary value. The rational two-point function has the tempered boundary value \eqref{eq:W}; differentiation and polynomial multiplication commute with this boundary limit. This proves \eqref{eq:Wcov}, including the light-cone singularity. \end{proof} Define the prolonged evolutionary action by \begin{equation} v_X(\partial^\alpha\phi) = \partial^\alpha(d_X\phi), \qquad v_X B_{\boldsymbol\alpha} = \sum_{a=1}^k \Wick{ \partial^{\alpha_a}(d_X\phi) \prod_{b\ne a}\partial^{\alpha_b}\phi }. \label{eq:prolong} \end{equation} The explicit jet formula is \begin{equation} \begin{split} \partial^\alpha(d_X\phi) ={}& X^\mu\partial_\mu\partial^\alpha\phi\\ &+\sum_{0<\beta\le\alpha} \binom{\alpha}{\beta} (\partial^\beta X^\mu) \partial_\mu\partial^{\alpha-\beta}\phi\\ &+\sum_{\beta\le\alpha} \binom{\alpha}{\beta} (\partial^\beta\sigma_X) \partial^{\alpha-\beta}\phi. \end{split} \label{eq:jetaction} \end{equation} In particular, descendants are not silently treated as primaries. \begin{theorem}[Complete local Ward action and mixed products] \label{thm:localaction} Let \(B_1,\ldots,B_n\) be any Wick fields, including \(Q,L,t\) and their descendants. Let their compact insertion supports lie in \(K_0\), and choose \(q_X[a,\chi]\) with the collar contract. Then, on \(\DE\), \begin{equation} \frac{\ii}{\hbar} [q_X[a,\chi],B_i(f_i)] = (v_XB_i)(f_i), \label{eq:wickaction} \end{equation} and \begin{equation} \frac{\ii}{\hbar} \left[q_X[a,\chi],\prod_{i=1}^n B_i(f_i)\right] = \sum_{i=1}^n B_1(f_1)\cdots(v_XB_i)(f_i)\cdots B_n(f_n). \label{eq:mixedaction} \end{equation} The identity extends to every compact joint kernel with the same local support condition and every joint-Schwartz spectator vector. The ordered vacuum Ward identities are \begin{equation} \sum_{i=1}^n \pair{\Omega}{ B_1(x_1)\cdots(v_XB_i)(x_i)\cdots B_n(x_n)\Omega} =0 \label{eq:vacward} \end{equation} as joint tempered distributions. \end{theorem} \begin{proof} The elementary action is Lemma~\ref{lem:cauchyaction}. Differentiating it gives \eqref{eq:jetaction}. To calculate the action on a normal product, first use separated point variables and subtract the vacuum contractions defining normal ordering. The commutator is a derivation on the ordinary separated field product. Any possible extra numerical term from the subtracted contractions is a derivative of \((d_X^x+d_X^y)W(x-y)\), and therefore vanishes by Lemma~\ref{lem:Wcov}. Taking the normal-product limit gives \eqref{eq:prolong} and \eqref{eq:wickaction}. The limits can be taken on \(\DE\): their Fourier kernels converge under the weighted fixed-sector bounds used in Lemma~\ref{lem:wickest}. The common invariant domain makes the operator commutator Leibniz identity legitimate for every finite ordered product. This gives \eqref{eq:mixedaction}. For a compact joint kernel, cover its support by finitely many product patches and approximate in the compact-test topology by finite tensor sums with supports in those patches. Proposition~\ref{prop:domain} gives convergence both of the words and of their current insertions. The same argument applies to joint-Schwartz spectator vectors, using longer-word bounds. Thus the stated identities hold on the complete declared test class, not only on factorized tests. Finally expand an ordered vacuum Wick correlation into its contractions. On each contraction the summed infinitesimal action vanishes by \eqref{eq:Wcov}, including all differentiated contractions. Every contraction term, and hence the complete finite Wick sum, obeys \eqref{eq:vacward}. \end{proof} \begin{remark}[Mixed products are not ordinary products of distributions] The theorem concerns already constructed ordered field words and their joint-test extensions. It does not multiply arbitrary numerical distributions at a common singular point. Nor does it infer closure of an algebra of single-integral local functionals under ordinary multiplication: such a product is generally multilocal. \end{remark} %===================================================================== \section{Lie brackets and the sign convention} \label{sec:brackets} %===================================================================== The vector-field bracket is \[ [X,Y]^\mu=X^\nu\partial_\nu Y^\mu-Y^\nu\partial_\nu X^\mu. \] Its conformal factor obeys \[ \sigma_{[X,Y]} = X^\mu\partial_\mu\sigma_Y -Y^\mu\partial_\mu\sigma_X. \] The current commutator in \eqref{eq:wickaction} gives \(v_X\), with positive point-field action \(d_X\phi\). As an evolutionary action, it keeps the numerical coefficients fixed. Its bracket is therefore an anti-representation. To state a representation with the vector-field bracket, define \begin{equation} \delta_X B = -v_X B = -\frac{\ii}{\hbar}[q_X[a,\chi],B] \label{eq:deltasign} \end{equation} on a localized word. The result is independent of any choice satisfying the collar contract. \begin{proposition}[No Lie-bracket defect] \label{prop:brackets} The derivations \(\delta_X\) satisfy \[ [\delta_X,\delta_Y]=\delta_{[X,Y]} \] on every ordered Wick word, all joint compact kernels, and \(\DE\). \end{proposition} \begin{proof} For the scalar smearing, integration by parts gives \[ \delta_X\phi(f) = \phi(\mathcal B_Xf), \qquad \mathcal B_X=X^\mu\partial_\mu+3\sigma_X. \] Indeed, the transpose of \(d_X\) with respect to the flat measure is \(-X^\mu\partial_\mu-3\sigma_X\). The divergence cocycle gives \[ [\mathcal B_X,\mathcal B_Y]=\mathcal B_{[X,Y]}. \] Hence the claimed identity holds on every smeared scalar. It passes to derivatives because the action is prolonged, not assigned a new primary weight. It passes to normal products by Lemma~\ref{lem:Wcov}, which ensures compatibility with contraction relations. It passes to arbitrary ordered words because both sides are derivations. Continuity and support-preserving tensor density then give the complete joint-test identity. \end{proof} \scopebox{Sign debugging}{% With \(v_X\phi=(X\cdot\partial+\sigma_X)\phi\) and fixed numerical coefficients, one has \([v_X,v_Y]=-v_{[X,Y]}\). The representation statement is for \(\delta_X=-v_X\), as in \eqref{eq:deltasign}. Ignoring this distinction reverses a Lie-bracket sign.} The proposition proves Lie compatibility of the local field actions. It does not prove essential self-adjointness of the fifteen global charge operators, nor their simultaneous integration to a group. %===================================================================== \section{Coverage: classification before finite fitting} \label{sec:coverage} %===================================================================== \subsection{The admissible stress-tensor class} \begin{definition}[The homogeneous polynomial class] \label{def:stressclass} The admissible class consists of symmetric rank-two tensor fields \(S_{\mu\nu}\) satisfying all of the following: \begin{enumerate} \item \(S\) is a finite vacuum-normal-ordered local polynomial in \(\phi\) and its derivatives, on the declared free field domain. \item Its coefficients are constant, with no new dimensionful parameter or coordinate-dependent coefficient. \item It is Poincare covariant and parity even; tensor coefficients are built from \(\eta\). \item Every monomial has engineering dimension four, with \(\dim\phi=1\) and \(\dim\partial=1\). \item It is conserved on the free shell. \end{enumerate} Expressions are identified by the actual free field equation and normal-product identities, not by their presentation labels. \end{definition} This is a specified physical polynomial class. It does not contain every local distributional defect compatible with an unspecified renormalization problem. Put \[ I_{\mu\nu}=(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q. \] \begin{theorem}[Complete classification in the declared class] \label{thm:classification} Every \(S\) in Definition~\ref{def:stressclass} has a unique form \begin{equation} S_{\mu\nu}=\lambda T_{\mu\nu}+cI_{\mu\nu}, \qquad \lambda,c\in\R. \label{eq:classform} \end{equation} Its trace is \begin{equation} S^\mu{}_\mu=(6c-\lambda)K. \label{eq:classtrace} \end{equation} Its translation Ward action on the scalar is \(\lambda\) times the original translation action. Requiring the original translation normalization and zero trace forces \[ \lambda=1,\qquad c=\frac16. \] \end{theorem} \begin{proof} Dimension four and the available constant tensor contractions exhaust the monomials as follows. A degree-two term has two derivatives and is a linear combination of \[ A_{\mu\nu},\qquad \eta_{\mu\nu}K,\qquad C_{\mu\nu}=\Wick{\phi\partial_\mu\partial_\nu\phi}, \qquad \eta_{\mu\nu}\Wick{\phi\Box\phi}. \] The last is zero on shell. A degree-four term has no derivatives and is a multiple of \(\eta_{\mu\nu}\Wick{\phi^4}\). A degree-one term would need three derivatives and an invariant constant tensor of odd total rank; none is built from \(\eta\). A degree-three term has one derivative and the same odd-rank obstruction. A degree-zero term of dimension four would require a forbidden dimensionful constant; derivatives of the identity vanish. No higher field degree is compatible with dimension four. Thus, on shell, \[ S_{\mu\nu} = aA_{\mu\nu}+b\eta_{\mu\nu}K +dC_{\mu\nu} +e\eta_{\mu\nu}\Wick{\phi^4}. \] Conservation gives \[ \partial^\mu S_{\mu\nu} = \left(\frac{a+d}{2}+b\right)\partial_\nu K +e\,\partial_\nu\Wick{\phi^4}. \] The vacuum-to-four-particle matrix of the last term is nonzero, whereas every quadratic term has zero such matrix. Hence \(e=0\). The vacuum-to-two-particle matrix of \(\partial_\nu K\) is nonzero for noncollinear positive-energy momenta, so \[ b=-\frac{a+d}{2}. \] Since \[ I_{\mu\nu} = 2\eta_{\mu\nu}K-2A_{\mu\nu}-2C_{\mu\nu}, \] the unique parameters are \[ \lambda=a-d,\qquad c=-\frac d2. \] Uniqueness also follows from the independent quadratic tensor polynomials on noncollinear two-particle momenta. The trace is \(-\lambda K+3c\Box Q=(6c-\lambda)K\). For a translation, \(I_{\mu\nu}a^\nu\) is a divergence superpotential. Its compact-slab commutator vanishes under the collar contract. The canonical tensor therefore contributes exactly \(\lambda\) times the original translation action. The last assertion follows. \end{proof} \subsection{Two actual separating probes} Choose real compact \(f,h\) such that \[ u_P=(\partial_0\phi)(f)\Omega\ne0, \qquad u_K=K(h)\Omega\ne0. \] Such choices are explicit in the following sense. Take nonzero nonnegative compact bumps whose Fourier transforms are nonzero near zero. Then \[ \norm{u_P}^2 = \hbar\int_{C_+}(p^0)^2|\widehat f(p)|^2\dd\mu(p)>0. \] For \(K\), the pair-creation kernel is \(-p\cdot q\,\widehat h(p+q)\), and \begin{equation} \norm{u_K}^2 = 2\hbar^2 \int_{C_+\times C_+} (p\cdot q)^2|\widehat h(p+q)|^2\dd\mu(p)\dd\mu(q)>0. \label{eq:Kprobe} \end{equation} There are arbitrarily small noncollinear positive-energy momenta for which both factors are nonzero. Both vectors lie in \(\DE\). For \(S=\lambda T+cI\), define the translation residual \[ R_P(S) = \frac{\ii}{\hbar}[q_{P_0}^S[a,\chi],\phi(f)] -(\partial_0\phi)(f), \] where \(q^S\) uses \(S^{0\nu}X_\nu\), and its collar contains \(\supp f\). Define \begin{equation} \begin{split} e_P(S)&= \frac{\pair{u_P}{R_P(S)\Omega}}{\norm{u_P}^2},\\ e_{\mathrm{tr}}(S)&= \frac{\pair{u_K}{S^\mu{}_\mu(h)\Omega}}{\norm{u_K}^2}. \end{split} \label{eq:probes} \end{equation} These are genuine field matrix elements, not assumed abstract coordinate functionals. By Theorem~\ref{thm:classification}, \begin{equation} e_P(S)=\lambda-1, \qquad e_{\mathrm{tr}}(S)=6c-\lambda. \label{eq:probevalues} \end{equation} \subsection{The complete residual space} \begin{definition}[Complete residual family] \label{def:residual} For \(S\) in Definition~\ref{def:stressclass}, let \(\mathfrak W(S)\) be the family containing: \begin{enumerate} \item its trace field; \item every conformal-current conservation residual; \item every compact-slab implementation residual, for all conformal Killing fields, all Wick fields, all ordered products, all admissible collars, all compact joint kernels, and all matrix elements on \(\DE\); \item the elementary and descendant contact residuals, using the polarization prescription of Section~\ref{sec:contacts}. \end{enumerate} The comparison transformation in these residuals is the fixed free-scalar action already calculated. Thus \(\mathfrak W(t)=0\). \end{definition} Each listed residual is affine-linear in \(S\), with that fixed comparison action. The current and its polarization are linear in \(S\); the remaining comparison terms do not depend on \(S\). \begin{theorem}[Proved finite coverage, not a finite-test assumption] \label{thm:coverage} For the complete residual family of Definition~\ref{def:residual}, set \[ \mathscr V = \operatorname{span}_{\R} \{\mathfrak W(t+T),\,\mathfrak W(t+I)\}. \] Every admissible residual belongs to this known two-dimensional space. The evaluation \[ E=(e_P,e_{\mathrm{tr}}) \] is injective on \(\mathscr V\). Consequently, within Definition~\ref{def:stressclass}, \[ e_P(S)=e_{\mathrm{tr}}(S)=0 \quad\Longleftrightarrow\quad \mathfrak W(S)=0 \quad\Longleftrightarrow\quad S=t. \] The conclusion covers the whole declared residual family and test class, not only the two probe vectors. \end{theorem} \begin{proof} The classification theorem proves, rather than assumes, that \[ S=t+aT+bI, \qquad a=\lambda-1,\quad b=c-\frac16. \] Affine linearity and \(\mathfrak W(t)=0\) give \[ \mathfrak W(S) = a\,\mathfrak W(t+T)+b\,\mathfrak W(t+I). \] This proves the spanning assertion for every component of the complete family, including its infinitely many field and test labels. Equations~\eqref{eq:probevalues} give \[ E(\mathfrak W(S)) = \begin{pmatrix} 1&0\\ -1&6 \end{pmatrix} \begin{pmatrix}a\\ b\end{pmatrix}. \] The determinant is \(6\), so the evaluation is injective and the two displayed residual families are independent. If both probes vanish, then \(a=b=0\), hence \(S=t\). All its local residuals vanish by the contact and implementation calculations, not by an unsupported extrapolation from sampled tests. The converse is immediate because the two probes are themselves components of the complete residual family. \end{proof} \begin{remark}[Relation to the predecessor's coverage proposition] The abstract statement ``\(W\in\mathscr V\) and \(E|_{\mathscr V}\) injective imply \(E(W)=0\Rightarrow W=0\)'' is linear algebra. The substantive work here is the complete polynomial enumeration, free-shell conservation calculation, and construction of the two physical probes. Finite scaling multiplicity was not used as a substitute for those steps. This is the spanning-family discipline emphasized in the supplied description of Landmark II \cite{CheungII}. \end{remark} \begin{proposition}[The finite-fit interface] \label{prop:finitefit} Let an explicitly parameterized subfamily of the declared stress class produce \[ E(\mathfrak W(S_\theta))=w_0+N\theta. \] Then a candidate with complete vanishing exists exactly when \[ -w_0\in\im N. \] For a particular candidate, complete vanishing is equivalent to \(w_0+N\theta=0\). \end{proposition} \begin{proof} Membership in the actual image of \(N\) is equivalent to solving the displayed finite linear equation. Theorem~\ref{thm:coverage} then converts that solution into complete residual vanishing in the declared class. \end{proof} \begin{remark}[Tolerance is not exact coverage] Nonzero probe tolerances bound the two coefficient errors through the displayed inverse matrix. They do not give an exact field identity. A statistical confidence statement additionally needs a specified bounded-score experiment and sampling assumptions; none is constructed here. \end{remark} \subsection{The unrestricted failure mode} \begin{proposition}[Finite probes do not cover unrestricted tensors] \label{prop:invisible} Let \(O\) be a nonempty open spacetime region and let \(\ell_1,\ldots,\ell_m\) be any finite family of linear compact-test pairings on smooth symmetric tensors. There is a nonzero smooth compactly supported symmetric conserved tensor \(H_{\mu\nu}\), supported in \(O\), with nonzero trace and \[ \ell_j(H)=0,\qquad 1\le j\le m. \] \end{proposition} \begin{proof} Choose \(m+1\) disjoint small balls in \(O\), and nonzero real bumps \(u_1,\ldots,u_{m+1}\) supported in them. Define \[ H_{\mu\nu}^{(r)} = (\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)u_r. \] These tensors are symmetric and conserved, with traces \(-3\Box u_r\). No nonzero compact spacetime bump solves the homogeneous wave equation: its Cauchy data vanish sufficiently far in the past, so uniqueness would force it to vanish. Thus every displayed trace is nonzero. The map \[ (c_1,\ldots,c_{m+1}) \longmapsto \left(\ell_j\left(\sum_r c_rH^{(r)}\right)\right)_{j=1}^m \] has nontrivial kernel. Select a nonzero vector in that kernel. Disjoint supports make the traces linearly independent, so the resulting tensor has nonzero trace and is nonzero. \end{proof} This proposition does not contradict Theorem~\ref{thm:coverage}: the unrestricted smooth tensors do not belong to its homogeneous free-field polynomial class. It is also not a counterexample to the supplied Family 282 claim. %===================================================================== \section{What the local result does not integrate} \label{sec:global} %===================================================================== \begin{proposition}[A special-conformal flow leaves the chart] \label{prop:flow} The local flow of \[ X_b^\mu(x)=2(b\cdot x)x^\mu-b^\mu x^2 \] is \[ g_s(x)^\mu = \frac{x^\mu-sb^\mu x^2} {1-2s\,b\cdot x+s^2b^2x^2} \] where the denominator is nonzero. It need not define a flow \(\R\times M\to M\) on the entire Minkowski chart. \end{proposition} \begin{proof} Differentiating the rational formula verifies the flow equation and the initial condition. For \(b=(1,0,0,0)\) and \(x=(t,0,0,0)\), \(t\ne0\), it reduces to \[ g_s(x)^0=\frac{t}{1-st}. \] It has a pole at \(s=1/t\). A compactification or another chart is required to discuss continuation through that value. \end{proof} \begin{proposition}[Symmetry of an operator is not integration] \label{prop:nonintegrable} The symmetric operator \[ S=-\ii\frac{\mathrm d}{\mathrm dx}, \qquad \Dom S=C_c^\infty(0,\infty) \subset L^2(0,\infty), \] has deficiency indices \((1,0)\). It has no self-adjoint extension and cannot be the restriction of the generator of a unitary one-parameter group. \end{proposition} \begin{proof} The deficiency equation \((S^*-\ii)f=0\) has the square-integrable solution \(f(x)=Ce^{-x}\), giving index one. The equation \((S^*+\ii)f=0\) has only \(f(x)=Ce^x\), which is not square-integrable unless \(C=0\). The unequal deficiency indices exclude a self-adjoint extension. A unitary one-parameter group has a self-adjoint generator, so such a group is impossible. \end{proof} \begin{remark}[These are separation examples, not free-field obstructions] Proposition~\ref{prop:nonintegrable} is not a calculation of a deficiency index for one of our conformal charges. Proposition~\ref{prop:flow} does not exclude a conformal action on a compactification or its cover. The examples show why local Ward actions alone do not prove the additional global conclusion. They reproduce the logical refusals described for Oai282-A, not a claim that the standard free scalar lacks global conformal constructions. \end{remark} Nelson's criterion was applied to the particular Wick square in Lemma~\ref{lem:Qsa}. It was not applied simultaneously to the conformal generators. The free-scalar modular construction of Hislop and Longo \cite{HislopLongo} and the conformal-net framework of Brunetti, Guido, and Longo \cite{BGL} are relevant comparisons. They do not automatically integrate the compact currents constructed in this paper. \scopebox{The optional global gate}{% A separate global theorem must specify a group or covering group, its representation, operator domains and integration hypotheses, the region action, bounded-net covariance, and any extension beyond the Minkowski chart. None of these requirements is replaced by the local Lie-bracket calculation.} %===================================================================== \section{Exclusions and comparison interfaces} \label{sec:comparisons} %===================================================================== \subsection{No Einstein equation follows} \begin{proposition}[The gravitational dictionary is not implied] \label{prop:einstein} Symmetry, conservation, zero trace, and the local Ward theorem above do not imply \[ G_{\mu\nu}=8\pi G_{\mathrm N}t_{\mu\nu}. \] No metric equation or gravitational dictionary has been constructed. \end{proposition} \begin{proof} The declared background is the fixed Minkowski metric, whose Einstein tensor is zero. The stress tensor is nonzero: if it vanished, every compact translation charge would vanish, contrary to its nonzero action on the scalar established in Lemma~\ref{lem:cauchyaction}. Thus the displayed equation is not an identity in the declared model. It would require an additional metric, coupling prescription, and dynamical theorem. \end{proof} \subsection{No interacting scalar is reconstructed} In the free model we have constructed \(L\) by a normal product and local quadratic approximation. In an interacting theory, the formal expression \[ L=\Box^{-1}T^\mu{}_\mu \] would not establish locality, mixed products, domain compatibility, or affiliation. This paper supplies no such interacting construction. Likewise, deleting \(Q\) from a list of labels does not delete \(\partial_\mu Q\), or prove that \(Q\) is absent from a represented net. A reduced algebra has to be analyzed on its actual generators and closures. Renaming its list is not a membership argument. Four-dimensional scale-to-conformal arguments and their generalized-free-field loopholes are discussed in \cite{DKST,DFKLP}. Local renormalization-group identities with background sources provide a different framework \cite{Baume}. Those results are comparisons; our theorem is a free-model calculation with its own declared field algebra. \subsection{The susceptibility benchmark is separate} The supplied VII-H excerpt concerns a connected vacuum response of the Wick square, with a stated contact ambiguity \(\R\,\hbar\delta_0\), and evaluates an invariant nonzero pairing \cite{CheungVIIH}. Its response distribution is not \(T_{\mu\nu}\), not \(L\), and not the current insertion \eqref{eq:Cexplicit}. We do not transfer its response prescription into the Ward calculation. The relational lapse of Em02-E and the fixed-background emission law of Em03-A concern still different observables and geometries \cite{CheungEm02E,CheungEm03A}. Neither is an improvement-scalar reconstruction or a verification of fixed-point hypotheses. \subsection{Documentary comparison table} \begin{longtable}{@{}p{0.28\linewidth}p{0.66\linewidth}@{}} \toprule Item & Use and nontransfer boundary\\ \midrule \endhead OpenAI Family 282 & The supplied catalogue records a local enhancement claim under its own framework. It is not an independently verified premise. No machine-checked proof of that claim is supplied here \cite{OpenAICatalogue}.\\[4pt] Oai282-A & Predecessor description of algebraic improvement, finite fitting, conditional affine action, and local/global refusals. Its full source was not supplied as a proof premise \cite{CheungA}.\\[4pt] Landmark II & Methodological comparison for complete parameterization and spanning-family discipline. Its finite certificate does not reconstruct \(L\) \cite{CheungII}.\\[4pt] OpenAI 280 / Oai280-A & A VOA--net correspondence is not this four-dimensional scalar improvement. A Q-system does not determine a metric or a radiation history; chiral circle nets and the present bounded four-dimensional net do not derive each other \cite{OpenAICatalogue,Cheung280A}.\\[4pt] OpenAI 290 / Hawk04-A & A modular-band or bicentralizer record is not a constant complete complement. Hawk04-A's exact-recovery criterion concerns a complete channel dilation, not the present Ward current \cite{OpenAICatalogue,CheungHawk04A}.\\[4pt] OpenAI 265, 362, 376 & An area law is not the entropy history of a Ward record. Smoothness or a forced computation is not an Einstein equation. These are catalogue comparisons only \cite{OpenAICatalogue}.\\[4pt] VII-H & A free-scalar susceptibility benchmark, kept distinct from \(T_{\mu\nu}\), \(L\), and stress-current Ward contacts \cite{CheungVIIH}.\\[4pt] III, V, VI & Classical compact Ward observables, admissible currents, and classical jet descent are comparison interfaces. Their distinct quantum-product or RG obligations are not discharged by this free benchmark \cite{CheungIII,CheungV,CheungVI}.\\[4pt] XIV-H, XIV-I, XIV-J & The internal amplitude construction and supported-sector reference are not the complete characteristic contraction. The stated full-map stop remains open \cite{CheungXIVH,CheungXIVI,CheungXIVJ}.\\[4pt] XV-B, version 2 & A technical continuation proving descent and extension interfaces, not an admitted full graph coefficient and not the downstream quantum admission exit \cite{CheungXVB}.\\[4pt] XVI-N & A conditional prepared-insertion obstruction on a specified massless special family. It supplies no physical \(L\) here and is not a characteristic-product construction \cite{CheungXVIN}.\\[4pt] En03-A / En05-A & Finite schedule-past envelopes on prescribed dynamics. A diameter or schedule bound is not a special-conformal transformation \cite{CheungEn03A,CheungEn05A}.\\[4pt] En06-A & Its algebraic microcausality interface requires eventually commuting bounded generators, norm-convergent comparison maps, and asymptotic preservation of both cross-products. Commuting channels alone are insufficient \cite{CheungEn06A}.\\[4pt] En06-B & Complete payload transport on prescribed routes and speed. Transport identity does not derive the Minkowski metric or integrate these currents \cite{CheungEn06B}.\\ \bottomrule \end{longtable} The statements about the series in this table are documentary scope comparisons based on the supplied records, not independent verification of their proofs or an exhaustive scan of that series. %===================================================================== \section{Exact release and remaining obligations} \label{sec:release} %===================================================================== \begin{theorem}[Oai282-B release] \label{thm:release} For the free massless scalar and the original Weyl net specified in this paper: \begin{enumerate} \item Physical membership of \(L=-Q/2\), including mixed products, a common invariant domain containing the benchmark's old domain, and compact affiliated realizations, is proved. \item Translation, Lorentz, dilation, and special-conformal contacts are explicitly calculated. Their local currents act on the complete ordered Wick-field algebra and obey the stated Lie compatibility. \item A complete stress-tensor classification and two actual separating probes prove coverage in the declared homogeneous polynomial class. \end{enumerate} The following are not constructed: \begin{enumerate} \item an interacting improvement scalar; \item a global integration theorem for the conformal generators; \item a complete arbitrary time-ordered stress-insertion normalization scheme; \item the inherited six-branch characteristic map and its admitted quantum coefficient; \item a gravitational or spacetime-emergence dictionary. \end{enumerate} O.1 remains \NOTCLOSED. \end{theorem} \begin{proof} The positive assertions are Propositions~\ref{prop:domain} and \ref{prop:physicalL}, Proposition~\ref{prop:elementarycontact}, Theorem~\ref{thm:localaction}, Proposition~\ref{prop:brackets}, and Theorems~\ref{thm:classification} and \ref{thm:coverage}. The listed exclusions are outside their objects, domains, or prescription classes. No construction of those objects has been inserted into their proofs. \end{proof} \begin{center} \fbox{\parbox{0.92\linewidth}{ \[ \begin{gathered} L=-\frac12\Wick{\phi^2} \quad\text{is physically constructed in the declared free net},\\ \text{local conformal Ward action: proved on the complete Wick class},\\ \text{finite coverage: proved in the declared stress class},\\ \text{global integration: not attempted},\\ \text{interacting characteristic coefficient: not admitted},\\ \text{O.1: \NOTCLOSED}. \end{gathered} \] }} \end{center} %===================================================================== \appendix \section{Descent and the separate characteristic-product programme} \label{app:descent} %===================================================================== \subsection{An elementary factorization criterion} The supplied XIV-J and XV-B records make reconstruction descent explicit \cite{CheungXIVJ,CheungXVB}. We restate and prove the underlying linear-algebra fact to make the nontransfer boundary self-contained. \begin{proposition}[Descent through reconstruction] \label{prop:descent} Let \[ \Rec:\mathscr P\longrightarrow\mathscr K \] be a surjective linear reconstruction map, and let \(\widetilde M:\mathscr P\to V\) be linear. There exists a unique linear \(M:\mathscr K\to V\) with \[ \widetilde M=M\Rec \] if and only if \[ \ker\Rec\subseteq\ker\widetilde M. \] \end{proposition} \begin{proof} A factorization annihilates every reconstruction-null presentation. Conversely, if two presentations reconstruct to the same kernel, their difference belongs to \(\ker\Rec\). The proposed value \(M(\Rec p)=\widetilde M(p)\) is therefore independent of the presentation. Surjectivity gives uniqueness. \end{proof} This criterion constructs no distributional cross-line product. Its use requires an already specified candidate \(\widetilde M\). The simplest degree-mixing example is \[ q\delta'(q)+\delta(q)=0. \] Indeed, \[ \pair{q\delta'}{\psi} =-\left.\frac{\mathrm d}{\mathrm dq}(q\psi(q))\right|_{q=0} =-\psi(0). \] Under the supplied reconstruction convention, \[ \Rec\bigl((q\,\ii\theta+1)M_{\mathrm R}\bigr)=0. \] The two separate terms reconstruct to opposite nonzero distributions. Thus highest presentation degree is not an intrinsic reconstructed symbol. This does not invalidate estimates on a specified presentation module; it requires descent before promoting a presentation-dependent splitting. \subsection{Why the free Ward construction does not supply the joint map} The retained analytic target in the supplied XIV-I/J record is a contraction of the complete internal sum \[ K_{\mathrm{ret}} = K_{47}+K_5+K_6+K_8+K_9 \] with grouped quantum cross-lines on tests meeting the characteristic locus. The symbol \(L^{\mu,I}\) appearing in that coordinate-response programme is not our improvement scalar \(L=-Q/2\). XIV-I's internal retarded-pole amplitude represents already defined internal lines. XIV-J's reference concerns only the supported sector \[ K_{\mathrm{supp}}=K_6+K_8+K_9 \] on the proved local complement of the characteristic locus. Neither is the required full six-branch contraction \cite{CheungXIVI,CheungXIVJ}. Our elementary insertion \(\phi(x)R(x-y)\) has independent coordinates and a directly defined tensor product. That simple existence argument has not been shown to apply to the different cross-line incidence of the inherited target. Similar words such as ``retarded'', ``contact'', ``scalar'', or ``Ward'' do not identify these operations. \subsection{The unchanged six-step chain} The separate programme still requires, for the same complete coefficient: \begin{enumerate} \item a full joint characteristic map on tests meeting the locus; \item a joint section and controlled complete remainder; \item uniform transverse estimates; \item compatibility with the permitted contacts and singular spectators; \item a complete output wavefront theorem after the actual pushforward; \item identification and admission of the normalized quantum coefficient in the stated output space. \end{enumerate} In its supplied notation, the last step would include the unasserted line \[ Q_HR_1^{(1)}\in\mathscr E_H. \] No such line is proved here. The technical XV-B version 2 proves useful relative-lifting, normal-jet, and extension interfaces. It records the full graph as unconstructed at its inherited B1 boundary \cite{CheungXVB}. This must be distinguished from the downstream admission exit also referred to as XV-B in earlier source contracts. Technical interface results do not open that exit. A normal-order bound alone is not a finite-dimensional numerical ambiguity space: tangential coefficient distributions can vary in infinitely many directions. A conditional extension equivalence is not a selected extension. A point-puncture annihilator correction is not, without additional argument, an extension theorem for a positive-dimensional characteristic locus. None of these interfaces is supplied by the two-probe Ward matrix of Theorem~\ref{thm:coverage}. \scopebox{Standing boundary}{% The full characteristic map remains unconstructed in the supplied record. This is an unresolved construction, not a theorem that every possible construction fails. Oai282-B forms no Ward defect from an absent coefficient, applies no differential to it, and admits no coefficient from that programme.} %===================================================================== \section{Debugging and verification ledger} \label{app:debug} %===================================================================== \begin{longtable}{@{}p{0.30\linewidth}p{0.64\linewidth}@{}} \toprule Potential error & Correction made in this manuscript\\ \midrule \endhead Wrong improvement sign & \(T^\mu{}_\mu=-K\), \(\Box Q=2K\), \(L=-Q/2\), and the coefficient of \((\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q\) is \(+1/6\).\\[4pt] Conservation called implementation & The elementary retarded contact and the compact-current commutator are calculated separately.\\[4pt] Descendants treated as primaries & The full prolonged jet formula \eqref{eq:jetaction} and derivative-of-delta contacts \eqref{eq:jetcontact} are retained.\\[4pt] Lie-bracket sign silently reversed & The positive point-field action \(v_X\) is distinguished from the representation derivation \(\delta_X=-v_X\).\\[4pt] Sharp-time quadratic charge assumed & Only smooth spacetime-smeared charges are operators. Sharp-time expressions compute weak linear-contraction commutators.\\[4pt] Hilbert density called a graph core & For \(Q(f)\), a separate number-bound approximation proves the core statement. No such inference is made for an arbitrary closed field realization.\\[4pt] Physical \(L\) defined by a Green operator & The explicit Wick field and local quadratic approximants construct \(L\) in the original net.\\[4pt] Finite fit called a global functional identity & The admissible stress class is fully enumerated and its complete residual span is proved before probes are applied.\\[4pt] Finite normal order called finite dimension & The Ward class uses constant homogeneous polynomial coefficients. No such reduction is assigned to the different distribution-valued characteristic ambiguity.\\[4pt] Ordered Ward action called a complete time-ordered scheme & Arbitrary renormalized time-ordered stress insertions and curved-background Weyl identities are excluded.\\[4pt] Global free conformality denied & The chart and deficiency examples establish logical separation only; they are not obstructions to the known free-model comparison frameworks.\\[4pt] Technical XV-B confused with admission exit & Version 2's interfaces are distinguished from the downstream exit requiring an admitted coefficient.\\[4pt] Restatement called verification & Supplied Cheung source descriptions remain documentary comparisons. No compilation, proof-assistant check, external referee review, or journal acceptance is asserted.\\ \bottomrule \end{longtable} \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Mathematical scope.} The proved model is the real free massless scalar in the declared vacuum Fock representation. The coverage theorem is exhaustive for Definition~\ref{def:stressclass}, not for all QFT stress tensors or all distributional contact prescriptions. \paragraph{Original net.} Affiliation of \(L\) is proved in the scalar Weyl net \eqref{eq:net}. No net is enlarged merely to make that assertion true. Compatible affiliation of every higher Wick polynomial, or every originally supplied realization in another framework, is not asserted. \paragraph{No fixed-point transfer.} The free benchmark is not marked ``imported from Family 282''. The supplied Family 282 record is not a verification of its own hypotheses in this model. No interacting renormalization-group fixed point is identified. \paragraph{Source provenance.} Cheung references below identify supplied manuscripts or supplied synopses by their identifiers. Where an exact title was not supplied, no bibliographic title is invented. The comparison ledger is not an exhaustive series audit. \paragraph{Compilation and formal verification.} This is a single-file pdfLaTeX source with an internal bibliography and no external figures or bibliography database. Two pdfLaTeX passes are intended to resolve references. No successful execution of those passes, machine checking, independently executed symbolic calculation, referee review, or acceptance is claimed. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \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{Nelson} E.~Nelson, ``Analytic vectors,'' \emph{Annals of Mathematics} \textbf{70} (1959), 572--615. doi:10.2307/1970331. \bibitem{Trotter} H.~F.~Trotter, ``On the product of semi-groups of operators,'' \emph{Proceedings of the American Mathematical Society} \textbf{10} (1959), 545--551. doi:10.1090/S0002-9939-1959-0108732-6. \bibitem{Sanders} K.~Sanders, ``Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes,'' \emph{Journal of Mathematical Physics} \textbf{53} (2012), 042502. arXiv:1010.3978. doi:10.1063/1.3703516. \bibitem{DutschBoas} M.~D\"utsch and F.-M.~Boas, ``The Master Ward Identity,'' \emph{Reviews in Mathematical Physics} \textbf{14} (2002), 977--1049. arXiv:hep-th/0111101. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428. Reference framework for functional products and their domains; no unspecified product theorem is imported. \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. doi:10.1007/BF02096738. \bibitem{DKST} A.~Dymarsky, Z.~Komargodski, A.~Schwimmer, and S.~Theisen, ``On scale and conformal invariance in four dimensions,'' \emph{Journal of High Energy Physics} \textbf{10} (2015), 171. arXiv:1309.2921. doi:10.1007/JHEP10(2015)171. \bibitem{DFKLP} A.~Dymarsky, K.~Farnsworth, Z.~Komargodski, M.~A.~Luty, and V.~Prilepina, ``Scale invariance, conformality, and generalized free fields,'' \emph{Journal of High Energy Physics} \textbf{02} (2016), 099. arXiv:1402.6322. doi:10.1007/JHEP02(2016)099. \bibitem{Baume} F.~Baume, B.~Keren-Zur, R.~Rattazzi, and L.~Vitale, ``The local Callan--Symanzik equation: structure and applications,'' \emph{Journal of High Energy Physics} \textbf{08} (2014), 152. arXiv:1401.5983. doi:10.1007/JHEP08(2014)152. \bibitem{CheungA} N.~Cheung, \emph{Oai282-A}. Supplied predecessor synopsis, n.d. The full source was not supplied for this continuation. Referenced labels include \texttt{thm:ward}, \texttt{prop:finitefit}, \texttt{prop:affine-ward}, \texttt{prop:flow}, \texttt{prop:nonintegrable}, \texttt{prop:coverage}, and \texttt{prop:descent}. Documentary comparison, not a proof premise. \bibitem{CheungII} N.~Cheung, \emph{Landmark II}. Supplied synopsis, n.d.; in particular \texttt{lem:coverage}. Documentary comparison for finite contact certificates and spanning-family discipline. \bibitem{CheungVIIH} N.~Cheung, \emph{VII-H}, version 1. Supplied manuscript excerpts, n.d.; in particular \texttt{thm:main} and its Wick-square susceptibility benchmark. Enter Calculus Companion, \url{https://enter.grok.me}. Documentary comparison, not a Ward prescription used here. \bibitem{CheungIII} N.~Cheung, \emph{Landmark III}. Supplied synopsis, n.d.; \texttt{prop:Ward} and \texttt{thm:Killing}. Classical comparison interface. \bibitem{CheungV} N.~Cheung, \emph{Landmark V}. Supplied synopsis, n.d.; \texttt{thm:rawcurrent}. Admissible-current comparison interface. \bibitem{CheungVI} N.~Cheung, \emph{Landmark VI}. Supplied synopsis, n.d.; \texttt{thm:descent} and \texttt{thm:plateau-obstruction}. Classical jet-descent comparison interface. \bibitem{CheungXIVH} N.~Cheung, \emph{Landmark XIV-H: Internal kernel relations on the operator route}, version 1. Supplied excerpts of \texttt{XIV-H.tex}, n.d.; \texttt{prop:K5exact}, \texttt{thm:H2}, and \texttt{thm:release}. Enter Calculus Companion, \url{https://enter.grok.me}. \bibitem{CheungXIVI} N.~Cheung, \emph{Landmark XIV-I}, version 1. Supplied excerpts of \texttt{XIV-I.tex}, n.d.; \texttt{prop:pole}, \texttt{thm:I2}, \texttt{thm:I3}, and \texttt{thm:release}. Enter Calculus Companion, \url{https://enter.grok.me}. Internal amplitude construction; complete joint-map boundary kept separate. \bibitem{CheungXIVJ} N.~Cheung, \emph{Landmark XIV-J}, version 1. Supplied excerpts of \texttt{XIV-J.tex}, n.d.; \texttt{prop:descent}, \texttt{lem:null}, \texttt{thm:reference}, \texttt{thm:conditionalextension}, \texttt{thm:J2}, and \texttt{cor:noadmission}. Enter Calculus Companion, \url{https://enter.grok.me}. \bibitem{CheungXVB} N.~Cheung, \emph{Landmark XV-B}, version 2. Supplied excerpts of \texttt{XV-B.tex}, n.d.; \texttt{thm:lifting}, \texttt{thm:contact}, \texttt{thm:jetnull}, \texttt{thm:pointcorrection}, and \texttt{thm:release}. Enter Calculus Companion, \url{https://enter.grok.me}. Technical descent and extension interfaces; not admission of the inherited full graph. \bibitem{CheungXVIN} N.~Cheung, \emph{Landmark XVI-N}, version r12. Supplied excerpts of \texttt{XVI-N.tex}, n.d.; \texttt{thm:massless-obstruction} and \texttt{lem:field-only}. Enter Calculus Companion, \url{https://enter.grok.me}. Scoped prepared-insertion comparison, not a physical-\(L\) theorem. \bibitem{CheungEm02E} N.~Cheung, \emph{Em02-E}, version 1. Supplied excerpts of \texttt{Em02-E-v1.tex}, n.d.; \texttt{thm:lapse}. Enter Calculus Companion, \url{https://enter.grok.me}. Classical relational-lapse comparison. \bibitem{CheungEm03A} N.~Cheung, \emph{Em03-A}, version 1. Supplied excerpts of \texttt{Em03-A-v1.tex}, n.d.; \texttt{thm:main}. Enter Calculus Companion, \url{https://enter.grok.me}. Fixed-background scalar-emission comparison. \bibitem{CheungEn03A} N.~Cheung, \emph{En03-A}, version 1. Supplied excerpts of \texttt{En03-A.tex}, n.d.; \texttt{thm:locality}. Enter Calculus Companion, \url{https://enter.grok.me}. Prescribed-schedule locality comparison. \bibitem{CheungEn05A} N.~Cheung, \emph{En05-A}, version 1. Supplied excerpts of \texttt{En05-A.tex}, n.d.; \texttt{thm:cone}. Enter Calculus Companion, \url{https://enter.grok.me}. Schedule-past and metric-envelope comparison. \bibitem{CheungEn06A} N.~Cheung, \emph{En06-A}, version 1. Supplied excerpts of \texttt{En06-A.tex}, n.d.; \texttt{thm:algmicro} and its observable-limit hypotheses. Enter Calculus Companion, \url{https://enter.grok.me}. Conditional algebraic-microcausality comparison. \bibitem{CheungEn06B} N.~Cheung, \emph{En06-B}, version 1. Supplied excerpts of \texttt{En06-B.tex}, n.d.; \texttt{thm:identity}. Enter Calculus Companion, \url{https://enter.grok.me}. Complete payload-transport comparison. \bibitem{CheungHawk04A} N.~Cheung, \emph{Hawk04-A}, version 1. Supplied excerpts of \texttt{Hawk04-A.tex}, n.d.; \texttt{thm:exact}. Enter Calculus Companion, \url{https://enter.grok.me}. Complete-complement recovery comparison. \bibitem{Cheung280A} N.~Cheung, \emph{Oai280-A}. Supplied synopsis, n.d.; \texttt{thm:extensionseparation}. Documentary separation of net extensions from metric and radiation-history claims. \bibitem{OpenAICatalogue} OpenAI, \emph{OpenAI manuscript catalogue and Family 282 source record} (2026). Supplied record dated October 7, 2026; in particular \texttt{rem:oai-p-282-01}, together with the supplied comparison descriptions of Families 280, 290, 265, 362, and 376. Enter Calculus Companion, \url{https://enter.grok.me}. Catalogue records are not independently verified theorem premises. \end{thebibliography} \end{document}