% Em06-B.tex % Self-contained pdfLaTeX source; no external bibliography file is required. % B1--B5 are proved below. The optional finite-mode metric gate B6 is not entered. % Compilation, registration, numerical execution, and machine verification % are not asserted. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Em06-B: Robust classical concentration and quantum fluctuations}, pdfauthor={}, pdfsubject={Scalar-only plateau relational observables} } \setlength{\emergencystretch}{3em} \setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{specification}[theorem]{Specification} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\One}{\mathbf 1} \newcommand{\Hil}{\mathfrak h} \newcommand{\Fock}{\mathcal F_{\mathrm s}} \newcommand{\Dfin}{\mathscr D_{\mathrm{fin}}} \newcommand{\Obs}{\mathcal O} \newcommand{\Sch}{\mathcal S} \newcommand{\cl}{\mathrm{cl}} \newcommand{\mc}{\mathrm{mc}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\abs}[1]{\lvert #1\rvert} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Dom}{Dom} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Rea}{Re} \DeclareMathOperator{\Ima}{Im} \DeclareMathOperator{\WF}{WF} \title{ Em06-B: Robust classical concentration and quantum fluctuations\\ of scalar-only plateau relational observables\\[5pt] \large Finite families, displaced states, and controlled classical inputs } \author{} \date{} \begin{document} \maketitle \begin{abstract} We prove a quantitative extension of the scalar-only first plateau coefficient studied in the supplied Em06-A draft. The metric coefficient remains a prescribed classical input. We independently derive the density variation, zero-coordinate-time Cauchy prescription, compact-test representation, and restricted gauge identity. On common compact supports we prove an explicit, nonoptimal estimate using six derivatives of the observation test and metric coefficient and three derivatives of the switch. For displaced Fock-normal states with finite residual number expectation \(n_\epsilon\), the condition \(\epsilon n_\epsilon\to0\) gives uniform mean, variance, mean-square, and spectral Wasserstein concentration on bounded input families. Observable-specific second moments give the exact necessary and sufficient condition, so the number condition is sufficient but not necessary. Finite commuting families possess joint spectral laws; general families are treated by ordered Weyl expectations, not by an assumed joint probability distribution. Normalized fluctuations retain the declared, possibly degenerate CCR form. Their laws are exactly the laws of the undisplaced fields in the residual states, and concentration alone does not determine a fluctuation limit. We prove input-stability estimates and evaluate nonzero success, bias-failure, amplified-variance, and nonconvergent-fluctuation examples. An explicitly defined additive-noise POVM separates intrinsic from measured concentration. The release comprises the scalar-only gates B1--B5. The optional finite-mode metric gate B6 is not entered. No interacting gravitational product, regulator removal, backreaction, metric reconstruction, Einstein dynamics, holographic identification, or inherited gravitational coefficient admission is obtained. \end{abstract} \section{Question, scope, and source discipline} \label{sec:scope} The question is: \begin{quote} Under which explicit preparation and input-control assumptions do families of scalar-only plateau relational coefficients concentrate on their named classical values, and what survives at the normalized fluctuation scale? \end{quote} The observable is the first relational coefficient, not a full finite-coupling nonlinear observable. Its architecture agrees with the first-order plateau construction described in Cheung's VII-A \texttt{thm:first} \cite{CheungVIIA} and with the supplied Em06-A draft \cite{Em06A}. Every identity needed here is rederived in the conventions printed below. No second-order theorem is imported. The coherent-state mechanism is established mathematics \cite{Glauber,Hepp}. Our contribution is the explicit quantitative package for this named scalar-only coefficient: compact-test estimates, finite families, residual-state assumptions, input control, and the separation between concentration and quantum fluctuations. We do not claim a new general coherent-state or semiclassical-limit principle. The Wigner-measure literature provides a broader bosonic framework \cite{ANphase,ANgeneral}. Its nonlinear propagation results have their own hypotheses \cite{ANpropagation}. They are not premises of the fixed-input estimates proved here. Likewise, dynamical fluctuation theorems such as \cite{BenArous} are comparisons, not substitutes for our exact displacement calculation. \paragraph{The precise weakening is retained.} Only the scalar is quantized. The metric coefficient is fixed before the compact scalar test is passed to the quantum representation. We do not construct a quantized metric-dependent insertion, an interacting BRST/BV complex, or an interacting relational product. This retains the distinction emphasized in V and in the sharp joint insertion problem of VII-B \cite{CheungV,CheungVIIB}. \paragraph{The attached XVII-A is a different project.} The supplied file \texttt{XVII-A.tex} declares a finite-matrix \(\mathrm{AdS}_2\) boundary reconstruction benchmark \cite{XVIIA}. Its geometry, matrices, boundary algebra, and correspondence \(U\) are not inputs here. No identification between its effect and the present Minkowski relational coefficient is asserted. \paragraph{Documentary comparisons are not verified premises.} Cheung citations refer to the supplied versions and excerpts. The added descriptions of VII-I and VII-J include theorem labels without the full corresponding proofs. We therefore use those descriptions only as comparison material \cite{Dossier}, not as independently audited theorems or premises. In particular, a summary calling a landmark ``verified'' does not verify its signs, domains, or compatibility with the present model. \section{Frozen conventions and the classical object} \label{sec:classical} \subsection{Background, supports, and input topologies} \begin{specification}[Common classical data] \label{spec:data} Fix \[ M=\R^{1+3},\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Sigma_0=\{t=0\}, \] a mass \(m>0\), and \(T>0\). Set \[ \Box=\partial_t^2-\Delta,\qquad P_m=\Box+m^2. \] Fix compact sets \[ K,L\subset (0,T)\times B_R \] for some \(R>0\), where \(B_R\) is the spatial ball of radius \(R\). Fix an open set \(U\) compactly contained in \((0,T)\times B_R\). The allowed observation tests and switches are \[ f\in C_K^\infty(M;\R),\qquad \chi\in C_L^\infty(M;\R),\qquad 0\leq\chi\leq1, \] with \(\chi=1\) on a neighborhood of \(\overline U\). We assume this switch class is nonempty. The metric coefficient \(h\) is any real smooth symmetric covariant tensor on \(M\). Its relevant control region is \[ E=[0,T]\times\overline B_{R+T+2}. \] The admitted gauge parameters are exactly \[ \mathfrak g_U=C_c^\infty(U;TM). \] All support sets, the initial surface, \(m\), and \(T\) remain fixed. \end{specification} For spacetime multi-indices, define \begin{align} p_r(f)&=\max_{|\alpha|\leq r}\sup_M|\partial^\alpha f|,\\ c_r(\chi)&=\max_{|\alpha|\leq r}\sup_M|\partial^\alpha\chi|,\\ H_r(h)&=\max_{\mu,\nu,\ |\alpha|\leq r} \sup_E|\partial^\alpha h_{\mu\nu}|. \label{eq:seminorms} \end{align} The input-control topology is \(p_6\) for fixed-support observation tests, \(c_3\) for fixed-support switches, and the local seminorm \(H_6\) for smooth metric coefficients. The preparation amplitude will be controlled in its one-particle Hilbert norm. These are explicit local input controls, not a claim about convergence in a boundary-sensitive kernel topology or a topology of interacting gravitational functionals. Indices are raised and lowered with \(\eta\). For \(\xi\in\mathfrak g_U\), write \[ (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \qquad X_\xi=\xi^\mu\partial_\mu. \] \subsection{Density variation and Green signs} Set \begin{align} \tau(h)&=\eta^{\mu\nu}h_{\mu\nu},\\ a^{\mu\nu}(h)&=\frac12\tau(h)\eta^{\mu\nu}-h^{\mu\nu}, \qquad b(h)=\frac12\tau(h),\\ C(h)^\nu&=\partial_\mu h^{\mu\nu} -\frac12\partial^\nu\tau(h),\\ Q_hv&=\partial_\mu\!\left(a^{\mu\nu}(h)\partial_\nu v\right) +m^2b(h)v. \label{eq:Q} \end{align} The switched scalar action through first order is \begin{equation} \begin{split} S_\chi[\phi;h] ={}&\frac12\int_M \left(\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-m^2\phi^2\right) \dd^4x\\ &+\frac{\kappa}{2}\int_M\chi \left(a^{\mu\nu}(h)\partial_\mu\phi\partial_\nu\phi -m^2b(h)\phi^2\right)\dd^4x. \end{split} \label{eq:action} \end{equation} Its Euler equation is \[ P_m\phi+\kappa Q_{\chi h}\phi=0. \] In particular, the operator is \(Q_{\chi h}\), not \(\chi Q_h\). Derivatives of the switch are retained. The metric-jet parameter \(\kappa\) is independent of the classicality parameter \(\epsilon\). Let \(G_m^{\mathrm R}\) and \(G_m^{\mathrm A}\) be the retarded and advanced Green operators of \(P_m\). With \(D_m=\sqrt{-\Delta+m^2}\), our signs are \begin{align} (G_m^{\mathrm R}F)(t) &=\int_{-\infty}^t \frac{\sin((t-s)D_m)}{D_m}F(s)\dd s,\\ (G_m^{\mathrm A}F)(t) &=\int_t^\infty \frac{\sin((s-t)D_m)}{D_m}F(s)\dd s. \label{eq:Green} \end{align} For compact smooth \(F\), \[ P_mG_m^{\mathrm R}F=P_mG_m^{\mathrm A}F=F, \qquad (G_m^{\mathrm R})^*=G_m^{\mathrm A}. \] We use their standard finite-propagation and Cauchy uniqueness properties \cite{BGP}. The estimates needed for this model are proved in Appendix~\ref{app:PDE}. \subsection{Coordinate prescription and observable} For \(t\geq0\), define the unswitched coordinate coefficient by \begin{equation} \Box w[h]^\mu=C(h)^\mu,\qquad w[h]^\mu|_{\Sigma_0}=0,\qquad \partial_tw[h]^\mu|_{\Sigma_0}=0. \label{eq:w} \end{equation} These are coordinate-time Cauchy data. They are not replaced by a metric-dependent normal-derivative prescription. For a real smooth solution \(P_m\phi_0=0\), define \[ \phi_{1,\chi}[h;\phi_0]=-G_m^{\mathrm R}Q_{\chi h}\phi_0. \] The source is compact and separated from \(\Sigma_0\), and the response has zero initial data. \begin{definition}[Scalar-only first plateau coefficient] \label{def:observable} The named classical functional is \begin{equation} \boxed{ \Obs_{f,h,\chi}[\phi_0] =\int_M f\left( -G_m^{\mathrm R}Q_{\chi h}\phi_0 -w[h]^\mu\partial_\mu\phi_0 \right)\dd^4x. } \label{eq:observable} \end{equation} \end{definition} For the formal metric \(g_\kappa=\eta+\kappa h\), \[ \left.\partial_\kappa\Box_{g_\kappa}v\right|_{\kappa=0} =-h^{\mu\nu}\partial_\mu\partial_\nu v-C(h)^\nu\partial_\nu v. \] Thus \(x^\mu+\kappa w[h]^\mu\) is the harmonic coordinate map through first order, and its inverse has coefficient \(-w[h]\). Equation~\eqref{eq:observable} is precisely the corresponding first scalar coefficient after composition with that inverse map. The theorem concerns this coefficient itself. It does not infer a controlled full finite-\(\kappa\) observable from a formal first-order expansion. \section{Independent classical audit and compact-test estimates} \label{sec:audit} \begin{lemma}[Sign-sensitive identities] \label{lem:identities} The following identities hold: \begin{align} Q_h &=-h^{\mu\nu}\partial_\mu\partial_\nu -C(h)^\nu\partial_\nu+\frac12\tau(h)P_m, \label{eq:Q-expanded}\\ C(K\xi)&=\Box\xi, \label{eq:CK}\\ Q_{K\xi} &=(\partial_\mu\xi^\mu)P_m-[P_m,X_\xi]. \label{eq:QK} \end{align} For real \(h\), \(Q_h\) is formally self-adjoint. \end{lemma} \begin{proof} Expanding the divergence in \eqref{eq:Q} and using \[ \partial_\mu a^{\mu\nu}(h)=-C(h)^\nu \] gives \eqref{eq:Q-expanded}. The trace of \(K\xi\) is \(2\partial_\mu\xi^\mu\), and direct differentiation gives \eqref{eq:CK}. Finally, \[ [P_m,X_\xi] =(\Box\xi^\nu)\partial_\nu +2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu. \] Substituting \(h=K\xi\) into \eqref{eq:Q-expanded} proves \eqref{eq:QK}. Formal self-adjointness follows by integration by parts in the divergence form, including its real zeroth-order term. \end{proof} \begin{proposition}[Exact compact scalar test] \label{prop:test} Put \(u_f=G_m^{\mathrm A}f\) and \begin{equation} \boxed{ q_{f,h,\chi} =-Q_{\chi h}u_f+\partial_\mu(fw[h]^\mu). } \label{eq:q} \end{equation} Then \(q_{f,h,\chi}\) is real, smooth, and compactly supported, with \begin{equation} \supp q_{f,h,\chi} \subset K\cup\bigl(L\cap J^-_\eta(K)\bigr). \label{eq:q-support} \end{equation} For every smooth free scalar solution, \begin{equation} \Obs_{f,h,\chi}[\phi_0] =\int_M q_{f,h,\chi}\phi_0\dd^4x. \label{eq:compact-representation} \end{equation} The dependence support in the classical \(h,\phi_0\) variables is contained in \[ \mathcal D_K=J^-_\eta(K)\cap\{0\leq t\leq T\}, \] which is compact. \end{proposition} \begin{proof} Transposition of the retarded Green operator gives \[ -\int fG_m^{\mathrm R}Q_{\chi h}\phi_0\dd^4x =-\int u_fQ_{\chi h}\phi_0\dd^4x =-\int (Q_{\chi h}u_f)\phi_0\dd^4x. \] The switched coefficient is compactly supported, so the last integration by parts has no Cauchy-surface contribution. The other term obeys \[ -\int fw[h]^\mu\partial_\mu\phi_0\dd^4x =\int\partial_\mu(fw[h]^\mu)\phi_0\dd^4x. \] This proves the formula and its sign. The advanced support property bounds the first term by \(L\cap J^-_\eta(K)\); the second is supported in \(K\). Both sets are separated from \(t=0\). Although \(w[h]\) is defined here for \(t\geq0\), its product with \(f\) extends smoothly by zero near and below \(t=0\). Finite propagation for the scalar response and coordinate Cauchy problem gives dependence only in \(\mathcal D_K\). A compact observation set has bounded spatial causal past on a finite closed time slab, so \(\mathcal D_K\) is compact. \end{proof} \begin{proposition}[Plateau gauge identity, before and after scalar descent] \label{prop:gauge} For \(\xi\in\mathfrak g_U\), \begin{align} w[h+K\xi]-w[h]&=\xi,\\ \phi_{1,\chi}[h+K\xi;\phi_0] -\phi_{1,\chi}[h;\phi_0]&=X_\xi\phi_0,\\ \Obs_{f,h+K\xi,\chi}[\phi_0] &=\Obs_{f,h,\chi}[\phi_0]. \label{eq:plateau} \end{align} At the test level, \begin{equation} \boxed{ q_{f,h+K\xi,\chi}-q_{f,h,\chi} =P_m(X_\xi u_f),\qquad X_\xi u_f\in C_c^\infty(M). } \label{eq:test-gauge} \end{equation} \end{proposition} \begin{proof} Since \(\xi\) and all its derivatives are supported where \(\chi=1\), \(\chi K\xi=K\xi\). On the free scalar shell, \[ Q_{K\xi}\phi_0=-P_m(X_\xi\phi_0). \] Retarded inversion gives \(G_m^{\mathrm R}P_m(X_\xi\phi_0) =X_\xi\phi_0\), because this smooth function is compactly supported away from the initial surface. Equation~\eqref{eq:CK} and Cauchy uniqueness similarly give \(w[K\xi]=\xi\). The two variations in \eqref{eq:observable} cancel. For the stronger compact-test identity, \(P_mu_f=f\) and \eqref{eq:QK} yield \[ -Q_{K\xi}u_f+\partial_\mu(f\xi^\mu) =P_m(X_\xi u_f). \] The multiplier \(\xi\) makes \(X_\xi u_f\) compact. \end{proof} This reproduces the required first-order plateau statement within the present conventions \cite{CheungVIIA}. It proves no unrestricted diffeomorphism theorem. \subsection{One-particle map and classical preparation} Use the unitary spatial Fourier convention \[ \widehat F(p)=(2\pi)^{-3/2}\int_{\R^3}F(x)e^{-\ii p\cdot x}\dd^3x. \] Let \[ \Hil=L^2(\R^3,\dd^3p),\qquad \langle v,z\rangle=\int\overline{v(p)}z(p)\dd^3p, \qquad \omega_p=\sqrt{|p|^2+m^2}. \] The inner product is linear in its second argument. For a real compact smooth spacetime test \(F\), define \begin{equation} k_F(p)=\frac{1}{(2\pi)^{3/2}\sqrt{2\omega_p}} \int_M F(t,x)e^{\ii\omega_pt-\ii p\cdot x}\dd^4x. \label{eq:k} \end{equation} Write \begin{equation} v(f,h,\chi)=k_{q_{f,h,\chi}}. \label{eq:v} \end{equation} For \(z\in\Sch(\R^3;\C)\), define the real smooth free scalar \begin{equation} \phi_z(t,x)=\frac1{(2\pi)^{3/2}}\int_{\R^3} \frac{z(p)e^{-\ii\omega_pt+\ii p\cdot x} +\overline{z(p)}e^{\ii\omega_pt-\ii p\cdot x}} {\sqrt{2\omega_p}}\dd^3p. \label{eq:phi-z} \end{equation} Direct integration gives \begin{equation} \Obs_{f,h,\chi}[\phi_z] =2\Rea\langle v(f,h,\chi),z\rangle =:o(f,h,\chi,z). \label{eq:classical-symbol} \end{equation} The target therefore depends on the scalar preparation as well as the prescribed metric coefficient. Writing \(o[g_{\cl}]\) without naming the preparation would suppress an essential input. \begin{lemma}[Mass-shell estimate and scalar equation descent] \label{lem:mass-shell} On a fixed compact support \(S\subset M\), \[ \|k_F\|_{\Hil}\leq C_{m,S}p_2(F), \qquad k_{P_m\psi}=0\quad(\psi\in C_c^\infty(M)). \] \end{lemma} \begin{proof} Spatial integration by parts gives \[ |k_F(p)| \leq \frac{(2\pi)^{-3/2}}{\sqrt{2\omega_p}(1+|p|^2)} \|(1-\Delta_x)F\|_{L^1(M)}. \] The integral \[ \int_{\R^3}\frac{\dd^3p}{\omega_p(1+|p|^2)^2} \] is finite: \(m>0\) controls the origin, and the radial integrand decays as \(r^{-3}\) at infinity. The \(L^1\) norm is bounded by a fixed-support constant times \(p_2(F)\). The exponential in \eqref{eq:k} is annihilated by \(P_m\). Compact integration by parts proves \(k_{P_m\psi}=0\). \end{proof} \begin{theorem}[Quantitative compact-test construction] \label{thm:estimate} There is a finite constant \(C\), depending only on the fixed support geometry, \(m,T,R\), and a fixed spatial cutoff used in Appendix~\ref{app:PDE}, such that \begin{align} p_2(q_{f,h,\chi}) &\leq C p_6(f) \left(c_3(\chi)H_3(h)+H_6(h)\right),\\ \|v(f,h,\chi)\|_{\Hil} &\leq C p_6(f)H_6(h)\left(1+c_3(\chi)\right). \label{eq:main-estimate} \end{align} The derivative orders are sufficient and are not claimed optimal. Consequently the map to \(v\) is continuous in the stated input controls, uniformly on bounded families. \end{theorem} \begin{proof} The detailed Cauchy and advanced estimates are proved in Appendix~\ref{app:PDE}. They bound \(u_f\) through four derivatives by \(Cp_6(f)\), and \(w[h]\) on \(K\) through three derivatives by \(CH_6(h)\). Applying the product rule to \eqref{eq:q} gives the first estimate. Lemma~\ref{lem:mass-shell}, with common compact support \(K\cup L\), gives the second. \end{proof} By Proposition~\ref{prop:gauge} and Lemma~\ref{lem:mass-shell}, \begin{equation} v(f,h+K\xi,\chi)=v(f,h,\chi). \label{eq:v-gauge} \end{equation} This is exact descent into the free \emph{scalar} representation. It is not a theorem about a joint gravitational quantum ideal or elimination of interacting auxiliary fields. \section{The free scalar carrier and unbounded moments} \label{sec:carrier} On \(\mathcal H=\Fock(\Hil)\), let \(a(v)\) be antilinear and \(a^\dagger(v)\) linear, with \[ [a(v),a^\dagger(w)]=\langle v,w\rangle\One. \] Let \(N\) be the number operator and \[ \Phi(v)=a(v)+a^\dagger(v). \] Initially these operators act on the finite-particle domain \(\Dfin\). \begin{lemma}[Self-adjoint fields and the number estimate] \label{lem:field} For every \(v\in\Hil\), \(\Phi(v)\) is essentially self-adjoint on \(\Dfin\). Its closure, denoted by the same symbol, satisfies \[ \Dom(N^{1/2})\subset\Dom\Phi(v) \] and the quadratic-form inequality \begin{equation} \boxed{ \|\Phi(v)\psi\|^2 \leq \|v\|^2 \left(4\|N^{1/2}\psi\|^2+2\|\psi\|^2\right). } \label{eq:number-bound} \end{equation} \end{lemma} \begin{proof} The creation--annihilation bounds are \[ \|a(v)\psi\|\leq\|v\|\|N^{1/2}\psi\|, \qquad \|a^\dagger(v)\psi\|\leq\|v\|\|(N+1)^{1/2}\psi\|. \] Using \(\|x+y\|^2\leq2\|x\|^2+2\|y\|^2\) proves \eqref{eq:number-bound} on \(\Dfin\). For a vector supported in particle sectors at most \(r\), \[ \|\Phi(v)^j\psi\| \leq (2\|v\|)^j \sqrt{\frac{(r+j)!}{r!}}\|\psi\|. \] Its analytic-vector series converges. The analytic-vector criterion \cite{Nelson} gives essential self-adjointness. Particle truncation converges in the graph norm of \(N^{1/2}\); the displayed bound and closedness then extend the estimate and domain inclusion. \end{proof} We always mean these self-adjoint closures. Commutator identities for unbounded fields are stated on the invariant common core, or as the corresponding quadratic-form identities, not as everywhere-defined products. Define the unitary displacement and scaled Weyl operators by \begin{align} D(z)&=\exp(a^\dagger(z)-a(z)),\\ W_\epsilon(v)&=\exp(\ii\sqrt{\epsilon}\Phi(v)), \qquad 0<\epsilon\leq1. \end{align} Their conventions give \begin{align} D(z)^*\Phi(v)D(z) &=\Phi(v)+2\Rea\langle v,z\rangle\One, \label{eq:displacement}\\ W_\epsilon(v)W_\epsilon(w) &=e^{-\ii\epsilon\Ima\langle v,w\rangle} W_\epsilon(v+w), \label{eq:Weyl}\\ [\Phi(v),\Phi(w)] &=2\ii\Ima\langle v,w\rangle\One \quad\hbox{on }\Dfin. \label{eq:CCR} \end{align} These follow by the scalar commutators of the generators, first on analytic vectors and then for the bounded unitary groups. They are the standard displacement/Weyl identities \cite{Glauber}, with the signs fixed by our inner-product convention. \begin{definition}[Residual and displaced states] \label{def:states} Let \(\rho_\epsilon\) be positive trace-class operators with \[ \Tr\rho_\epsilon=1,\qquad n_\epsilon=\Tr(\rho_\epsilon N)<\infty. \] For a declared \(z\in\Sch(\R^3;\C)\), put \begin{equation} \widetilde\rho_\epsilon =D(z/\sqrt{\epsilon})\rho_\epsilon D(z/\sqrt{\epsilon})^*. \label{eq:rho-tilde} \end{equation} For \(v=v(f,h,\chi)\), the represented relational coefficient is \begin{equation} \widehat O_\epsilon=\sqrt{\epsilon}\Phi(v), \qquad o=2\Rea\langle v,z\rangle. \label{eq:O} \end{equation} \end{definition} The parameter \(\epsilon\) is a dimensionless effective CCR parameter relative to a fixed reference unit of action. It is neither \(\kappa\) nor a smoothing width. The classical symbol is fixed while the representation section in \eqref{eq:O} varies. \paragraph{Meaning of unbounded traces.} If \(\rho=\sum_\ell p_\ell\kb{\psi_\ell}{\psi_\ell}\) has finite number expectation, its positively weighted vectors with \(p_\ell>0\) lie in \(\Dom N^{1/2}\). We define \[ \langle\Phi(v)^2\rangle_\rho :=\sum_\ell p_\ell\|\Phi(v)\psi_\ell\|^2 =\|\Phi(v)\rho^{1/2}\|_{\mathrm{HS}}^2. \] The first moment is the absolutely convergent associated sum. These are spectral moments, independent of the decomposition. Notation such as \(\Tr(\rho\Phi(v)^2)\) below means this quadratic moment; it does not assume that the formal product \(\rho\Phi(v)^2\) is an already-defined trace-class operator. Lemma~\ref{lem:field} implies \begin{equation} \langle\Phi(v)^2\rangle_{\rho_\epsilon} \leq(4n_\epsilon+2)\|v\|^2. \label{eq:state-number} \end{equation} \section{Displaced-state concentration: the exact criterion} \label{sec:concentration} For a self-adjoint operator \(A\) in a state \(\rho\), its spectral law is \(\mu_{\rho,A}(J)=\Tr(\rho E_A(J))\). For probability laws with finite second moments, \(W_2\) denotes the quadratic Wasserstein metric. In particular, \begin{equation} W_2(\mu,\delta_o)^2=\int_\R|x-o|^2\dd\mu(x), \label{eq:pointmass} \end{equation} because every coupling with a point mass has constant second coordinate. \begin{theorem}[Exact identities and concentration criterion] \label{thm:states} For Definition~\ref{def:states}, write \[ b_\epsilon(v)=\langle\Phi(v)\rangle_{\rho_\epsilon}, \qquad M_\epsilon(v)=\langle\Phi(v)^2\rangle_{\rho_\epsilon}, \] and let \(\mu_\epsilon\) be the spectral law of \(\widehat O_\epsilon\) in \(\widetilde\rho_\epsilon\). Then \begin{align} \langle\widehat O_\epsilon\rangle_{\widetilde\rho_\epsilon}-o &=\sqrt{\epsilon}\,b_\epsilon(v), \label{eq:mean-general}\\ \Var_{\widetilde\rho_\epsilon}(\widehat O_\epsilon) &=\epsilon\bigl(M_\epsilon(v)-b_\epsilon(v)^2\bigr), \label{eq:variance-general}\\ \langle(\widehat O_\epsilon-o)^2\rangle_{\widetilde\rho_\epsilon} &=\epsilon M_\epsilon(v) =W_2(\mu_\epsilon,\delta_o)^2. \label{eq:mse} \end{align} For a fixed named \(v,o\), the following are equivalent: \begin{enumerate} \item the mean tends to \(o\) and the variance tends to zero; \item \(\epsilon M_\epsilon(v)\to0\); \item \(W_2(\mu_\epsilon,\delta_o)\to0\). \end{enumerate} A sufficient, not necessary, condition is \begin{equation} \epsilon n_\epsilon\longrightarrow0. \label{eq:number-condition} \end{equation} Under that condition, \begin{align} |\langle\widehat O_\epsilon\rangle_{\widetilde\rho_\epsilon}-o| &\leq \sqrt{\epsilon(4n_\epsilon+2)}\,\|v\|,\\ \Var_{\widetilde\rho_\epsilon}(\widehat O_\epsilon) &\leq\epsilon(4n_\epsilon+2)\|v\|^2,\\ W_2(\mu_\epsilon,\delta_o) &\leq\sqrt{\epsilon(4n_\epsilon+2)}\,\|v\|. \label{eq:concentration-budget} \end{align} \end{theorem} \begin{proof} The exact self-adjoint conjugation is \[ D(z/\sqrt{\epsilon})^*\widehat O_\epsilon D(z/\sqrt{\epsilon}) =o\One+\sqrt{\epsilon}\Phi(v). \] Unitary covariance of spectral measures gives the mean, variance, and second-moment identities. Their moments exist by \eqref{eq:state-number}. Equation~\eqref{eq:pointmass} proves the Wasserstein identity. For every probability law with a finite second moment, \[ \int|x-o|^2\dd\mu_\epsilon =\Var(\mu_\epsilon) +\left(\int x\dd\mu_\epsilon-o\right)^2. \] Both summands are nonnegative, proving the equivalences. Cauchy--Schwarz gives \(b_\epsilon(v)^2\leq M_\epsilon(v)\); \eqref{eq:state-number} then proves the sufficient condition and all three estimates. \end{proof} The general spectral law in this theorem need not be Gaussian. No quasifree-state assumption was used. \begin{corollary}[Uniform concentration on controlled input families] \label{cor:uniform} Let \(\mathcal C\) be any family of data in Specification~\ref{spec:data} with \[ p_6(f)\leq F,\qquad H_6(h)\leq H,\qquad c_3(\chi)\leq X. \] Let \(L_*=CFH(1+X)\), with \(C\) from Theorem~\ref{thm:estimate}. For a common residual state family, all bounds in \eqref{eq:concentration-budget} hold uniformly over \(\mathcal C\) with \(\|v\|\) replaced by \(L_*\). The classical targets are the corresponding \(o(f,h,\chi,z)\). \end{corollary} \begin{proof} Theorem~\ref{thm:estimate} gives \(\sup_{\mathcal C}\|v\|\leq L_*\). Apply Theorem~\ref{thm:states}. \end{proof} This is uniform concentration around the named, possibly varying classical targets. Convergence to a \emph{single fixed target} additionally requires control of their variation; that is proved in Section~\ref{sec:robustness}. \section{Finite families: spectral laws versus ordered Weyl correlations} \label{sec:families} Fix real \(f_1,\ldots,f_J\), with common \(h,\chi,z\), and write \[ v_j=v(f_j,h,\chi),\qquad o_j=2\Rea\langle v_j,z\rangle,\qquad \widehat O_{j,\epsilon}=\sqrt{\epsilon}\Phi(v_j). \] Define the real matrices \begin{equation} S_{ij}=\Rea\langle v_i,v_j\rangle,\qquad A_{ij}=\Ima\langle v_i,v_j\rangle. \label{eq:SA} \end{equation} The matrix \(S\) is positive semidefinite and \(A\) is antisymmetric. \subsection{Second moments and commutators} For a residual state \(\rho\), define its field cross moments by \[ M_{ij}^\rho =\sum_\ell p_\ell \langle\Phi(v_i)\psi_\ell,\Phi(v_j)\psi_\ell\rangle, \qquad b_i^\rho=\langle\Phi(v_i)\rangle_\rho. \] These sums exist by \eqref{eq:state-number}. Set \[ C_{ij}^\rho=\Rea M_{ij}^\rho-b_i^\rho b_j^\rho. \] \begin{proposition}[Two distinct finite-family structures] \label{prop:covariance} For every finite-number residual state, \[ \Ima M_{ij}^\rho=A_{ij}. \] The symmetric covariance in the displaced state is \begin{equation} \operatorname{Cov}_{\mathrm{sym}} (\widehat O_{i,\epsilon},\widehat O_{j,\epsilon}) =\epsilon C_{ij}^{\rho_\epsilon}. \label{eq:cov-general} \end{equation} On the common core, \begin{equation} [\widehat O_{i,\epsilon},\widehat O_{j,\epsilon}] =2\ii\epsilon A_{ij}\One. \label{eq:comm-general} \end{equation} For the displaced vacuum, \[ C_{ij}^{|\Omega\rangle\langle\Omega|}=S_{ij}, \] so the symmetric covariance is exactly \(\epsilon S\). \end{proposition} \begin{proof} On finite-particle vectors the weak commutator gives \[ \langle\Phi(v_i)\psi,\Phi(v_j)\psi\rangle -\langle\Phi(v_j)\psi,\Phi(v_i)\psi\rangle =2\ii A_{ij}\|\psi\|^2. \] Particle truncation converges in \(\Dom N^{1/2}\), and \eqref{eq:number-bound} passes this identity to that form domain. Summing over the state decomposition proves the imaginary-part statement. Displacement shifts each field by its real constant \(o_j\) after scaling. Covariance is unaffected by this shift, proving \eqref{eq:cov-general}. The core commutator is \eqref{eq:CCR} with the scale included. For the vacuum, \[ \Phi(v_j)\Omega=a^\dagger(v_j)\Omega,\qquad \langle\Phi(v_i)\Omega,\Phi(v_j)\Omega\rangle =\langle v_i,v_j\rangle, \] and all first moments vanish. \end{proof} Thus symmetric noise covariance and antisymmetric commutation are not the same quantity. \subsection{The commuting sector} \begin{theorem}[Joint spectral concentration precisely in the commuting sector] \label{thm:commuting} If \(A_{ij}=0\) for every \(i,j\), the self-adjoint fields \(\Phi(v_j)\) strongly commute. Their scaled counterparts have a joint spectral law \(\mu_\epsilon^{\mathrm{joint}}\) in every state considered above. For general residual states, \begin{equation} W_2(\mu_\epsilon^{\mathrm{joint}},\delta_o)^2 =\epsilon\sum_{j=1}^J \langle\Phi(v_j)^2\rangle_{\rho_\epsilon} \leq\epsilon(4n_\epsilon+2)\sum_{j=1}^J\|v_j\|^2, \label{eq:joint-general} \end{equation} where \(o=(o_1,\ldots,o_J)\) and the metric on \(\R^J\) is Euclidean. For the displaced vacuum, \begin{equation} \mu_\epsilon^{\mathrm{joint}}=\mathcal N(o,\epsilon S), \qquad W_2(\mu_\epsilon^{\mathrm{joint}},\delta_o)^2 =\epsilon\operatorname{tr}S. \label{eq:joint-vacuum} \end{equation} Degenerate Gaussian covariance is allowed. If some \(A_{ij}\ne0\), the corresponding operators do not strongly commute. In that case no common joint spectral law of these operators is asserted. \end{theorem} \begin{proof} When \(A_{ij}=0\), \eqref{eq:Weyl} shows that their unitary groups commute for all real parameters. Commuting spectral measures follow from the spectral theorem for commuting unitary groups. The squared Euclidean distance to \(o\) is the sum of the squared coordinate distances. Applying \eqref{eq:mse} to every marginal proves \eqref{eq:joint-general}. Vacuum normal ordering gives \[ \langle\Omega,e^{\ii\Phi(V)}\Omega\rangle =e^{-\|V\|^2/2}. \] For \(V=\sum_jt_jv_j\), displacement therefore gives the joint characteristic function \[ \exp\left(\ii\sum_jt_jo_j -\frac{\epsilon}{2}\sum_{i,j}t_iS_{ij}t_j\right). \] This identifies \eqref{eq:joint-vacuum}. Conversely, strong commutativity would make the two unitary groups commute for every pair of parameters. Their Weyl interchange phase is \(e^{-2\ii stA_{ij}}\); it cannot be one for every \(s,t\) when \(A_{ij}\ne0\). \end{proof} \subsection{General families: ordering is retained} For real \(t=(t_1,\ldots,t_J)\), put \[ V(t)=\sum_jt_jv_j,\qquad a(t)=\sum_{i