\documentclass[11pt]{article} % I-B-VP-v3.tex % Single-file manuscript for pdfLaTeX. % Compile at least twice to resolve references. % No external bibliography, figures, scripts, or data files are required. % % Consolidated from the supplied: % I-B-VP-v2.tex % I-B-VP-AB.tex % I-B-VP-SAE.tex % % This source does not execute the scripts named in the archival appendices. % No successful compilation, numerical execution, or machine verification % is asserted by the consolidation. \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[a4paper,margin=2.5cm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,array,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[ hidelinks, pdftitle={I-B-VP-v3: Vacuum polarization, stress tensor and induced current on a prescribed conical spacetime}, pdfauthor={Ngo Cheung} ]{hyperref} \newcommand{\RR}{\mathbb R} \newcommand{\ZZ}{\mathbb Z} \newcommand{\NN}{\mathbb N} \newcommand{\CC}{\mathbb C} \newcommand{\HH}{\mathcal H} \newcommand{\WW}{\mathcal W} \newcommand{\dd}{\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\id}{\mathrm{id}} \newcommand{\ren}{\mathrm{ren}} \newcommand{\PV}{\operatorname{PV}} \newcommand{\Real}{\operatorname{Re}} \newcommand{\Imag}{\operatorname{Im}} \newcommand{\SourceLabel}[1]{\nolinkurl{#1}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\ket}[1]{\lvert#1\rangle} \newcommand{\bra}[1]{\langle#1\rvert} \newcommand{\proj}[1]{\ket{#1}\bra{#1}} \newcommand{\ANALYTIC}{\texttt{ANALYTIC}} \newcommand{\SUPPLIED}{\texttt{SUPPLIED\_\allowbreak NUMERIC}} \newcommand{\UNCERTIFIED}{\texttt{UNCERTIFIED}} \newcommand{\NOTCLOSED}{\texttt{NOT\_CLOSED}} \newcommand{\bsr}{\texttt{bulk\_\allowbreak symmetry\_\allowbreak restored}} % These marks record the access grades reported in the supplied sources. % They do not assert that every marked paper was read in full during % preparation of this consolidated source. \newcommand{\Mem}{\textsuperscript{\(\dagger\)}} \newcommand{\Meta}{\textsuperscript{\(\ddagger\)}} \newcommand{\Ver}{\textsuperscript{\(\ast\)}} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\spec}{spec} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Rot}{Rot} \newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \setlist{itemsep=3pt,topsep=5pt} \setlength{\LTpre}{8pt} \setlength{\LTpost}{8pt} \setlength{\tabcolsep}{5pt} \Urlmuskip=0mu plus 2mu \emergencystretch=2em \allowdisplaybreaks[2] \title{\textbf{I-B-VP-v3: Closing the spatial-product extension ledger}\\[5pt] \large Vacuum polarization, stress tensor, and induced current\\ on a prescribed conical spacetime with Aharonov--Bohm holonomy} \author{Ngo Cheung} \date{October 2026} \begin{document} \maketitle \begin{abstract} We give a unified construction and extension ledger for a massless scalar field on the regular conical product prescribed in I-B. The spatial operator, its self-adjoint extension, the state prescription, the curvature coupling, and the flat-bundle holonomy are explicit inputs. The extension class consists of transverse self-adjoint extensions tensored with the ordinary longitudinal Laplacian; it includes rotation-breaking finite-dimensional boundary conditions but excludes general non-product extensions. For the Friedrichs extension, an explicit heat-kernel estimate proves infrared finiteness of the massless spectral covariance on compact tests. Direct Euclidean subtraction leaves a jointly real-analytic bisolution at every regular diagonal point, including image-boundary parameters. Analytic continuation gives the local Hadamard form used for point splitting. For one complex field, with \(\Phi=(\phi_1+\ii\phi_2)/\sqrt2\), \(T=T[\phi_1]+T[\phi_2]\), signature \((-,+,+,+)\), and mixed components ordered as \((t,r,\varphi,z)\), define \[ P_\beta=\alpha^{-2}\bigl(1-6\beta(1-\beta)\bigr)-1,\qquad C_\beta=\alpha^{-4}\bigl(1-30\beta^2(1-\beta)^2\bigr)-1. \] The Friedrichs results are \[ \langle\Phi^\dagger\Phi\rangle_\ren =\frac{P_\beta}{48\pi^2r^2},\qquad \langle j_\varphi\rangle_\ren =-\frac{e\,\beta(1-\beta)(1-2\beta)} {12\pi^2\alpha^2r^2}, \] \[ \langle T^\mu{}_\nu\rangle_\ren =\frac{C_\beta}{720\pi^2r^4}\diag(1,1,-3,1) +2\left(\xi-\frac16\right) \frac{P_\beta}{48\pi^2r^4}\diag(4,-2,6,4). \] The neutral real tensor is half the untwisted complex tensor. The current uses the complex first jet of the regular two-point function; replacing that function by its real part would erase it. For every nonnegative extension in the declared product class, a finite-rank Krein formula gives an infrared-finite spectral state and an analytic regular-part correction. We supply the formula also on boundary charts of the extension space. Positivity is equivalent to positivity of the finite boundary relation. Negative transverse eigenvalues produce unstable longitudinal bands; a stationary finite-second-moment state is then excluded by a commutator-growth argument. Two projected alternatives are recorded, together with their nonlocality and Hadamard defects. Scale-invariant irregular modes admit closed forms expressed through analytic branches of the Friedrichs polynomials at shifted arguments. At half holonomy an additional \(\mathbb{CP}^1\) family has explicit angular-dependent observables. Scale-dependent nonnegative extensions have convergent integral representations and proved asymptotics. Generic rotation-breaking extensions with unequal deficient orders can carry a radial current. The familiar Friedrichs coefficients are not claimed as new. Historical formula comparisons and all inherited numerical residuals are retained with their evidence grades. No numerical execution, apex renormalization, backreaction, interacting construction, or downstream programme closure is asserted. \end{abstract} \tableofcontents %====================================================================== \section{Purpose, provenance, and the consolidation} \label{sec:introduction} %====================================================================== Vacuum polarization on an idealized cosmic-string background is established subject matter \cite{HK1986,Linet1987,FS1987,Dowker1987,GL1994}. The separate question of a boundary condition at the removed apex is essential: the regular metric alone does not select a self-adjoint extension \cite{KayStuder,DR2017}. This manuscript consolidates the supplied records \SourceLabel{I-B-VP-v2.tex}, \SourceLabel{I-B-VP-AB.tex}, and \SourceLabel{I-B-VP-SAE.tex} \cite{RecordVP2,RecordAB,RecordSAE}. Those files are retained records; this consolidation neither deletes them nor assumes that their version names form an unambiguous numerical sequence. The first supplied file already identifies itself as version~2, whereas the AB and SAE records refer to an earlier \SourceLabel{I-B-VP.tex} as version~1. We therefore identify predecessor records by filename. \paragraph{How to read this version.} Sections~\ref{sec:host}--\ref{sec:friedrichs-results} contain the Friedrichs construction and all three observables. Sections~\ref{sec:boundary}--\ref{sec:matrix} treat the product extension ledger. Section~\ref{sec:ledger} is the unified status table. The appendices preserve the reported computations and corrections without using them as proof. \subsection{Preserved additions: the AB change record} The AB source's items C1--C7 are retained in substance: \begin{enumerate}[label=\textbf{C\arabic*.}] \item The induced current is added to the Wick square and stress tensor. \item Explicit comparisons with the flux literature are included. \item Reports A1--A5 include character-weighted quotient images and high-precision contour comparisons. \item The Friedrichs formulas hold for every \(0<\alpha<1\), not only quotient angles. Their analytical proof does not use finite sampling. \item The AB source revises its explanation of the earlier \(10^{-17}\)-scale residuals: it reports a cast of image terms to IEEE double complex and an omitted diffraction term at a rounded quotient parameter. These remain reported diagnostics because the scripts and raw logs were not supplied. \item The bibliography retains the source's access grades, with their meaning stated in Section~\ref{sec:evidence}. The Smith chapter's 1989/1990 citation variation is retained as a bibliographic note. \item The deficient twisted modes and their extension families are made explicit. \end{enumerate} \subsection{Preserved additions: the SAE change record} The SAE source's items D1--D6 are likewise retained: \begin{enumerate}[label=\textbf{D\arabic*.}] \item Boundary data, a Krein correction, positivity, unstable extensions, projected constructions, scale-invariant closed forms, and scale-dependent integral representations are included. \item The AKO and Filgueiras--Moraes comparisons and the SAE verification reports are retained. \item Nonnegative non-Friedrichs product extensions move from ``Open'' in the predecessor ledger to constructed spectral states with evaluated observables or convergent evaluation formulas. \item The established Friedrichs formulas are unchanged. Shifted arguments refer to an entire \emph{geometric branch}, not to the globally periodic physical absolute-value mode sum. \item Rotation-breaking \(U(2)\) boundary conditions are admitted only after dropping rotation invariance, while retaining the spatial-product assumption. \item Bibliographic access marks remain provenance statements; they are not a replacement for a formula-level comparison or an execution certificate. \end{enumerate} \subsection{Analytical corrections required for a coherent merge} \label{sec:corrections} A submission cannot preserve a false implication merely because it appeared in a predecessor. The following corrections are incorporated and their affected reports remain visible: \begin{enumerate}[label=\textbf{R\arabic*.}] \item \textbf{Infrared finiteness is proved.} Nonnegativity alone is not used to infer a massless covariance. The Friedrichs heat estimate and the non-Friedrichs Krein bound supply the necessary estimates. \item \textbf{Image boundaries are included.} The total Euclidean remainder is treated by joint ellipticity. No diagonal parameter is discarded as a measure-zero exception. \item \textbf{Current subtraction is stated correctly.} The derivative difference of the direct parametrix does not vanish along arbitrary approaches to coincidence. The current is defined using the subtracted complex kernel. For the time splitting used to compute its azimuthal component, the direct contribution is identically zero. \item \textbf{Boundary charts are genuine boundary relations.} An ``infinite eigenvalue'' is replaced by an explicit subspace and a compressed Hermitian relation. The finite-matrix small-radius limit is not asserted for every boundary chart. \item \textbf{Small-radius remainder orders are corrected.} The first correction for \(0<\nu<1/2\) is retained, but its remainder has additional regimes at \(\nu=1/3\). The logarithmic case \(\nu=1/2\) is supplied. \item \textbf{Asymptotic zeros are respected.} A relative error is not asserted when a displayed leading stress coefficient vanishes. \item \textbf{Transverse and full spectra are separated.} A negative transverse eigenvalue generates a full longitudinal continuous band, not a full-space \(L^2\) eigenfunction. \item \textbf{State selection is not universal state uniqueness.} The distinguished zero-mean spectral Fock state is specified. No classification of all algebraic ground states is claimed. \item \textbf{Projection changes the field algebra.} The two projected constructions are not states with the original local canonical commutator. Their defects are stated on their actual domains. \item \textbf{Computation is not silently promoted to proof.} All inherited execution claims are \SUPPLIED{} and \UNCERTIFIED{} as rigorous total-error bounds. \end{enumerate} These corrections do not alter the Friedrichs polynomials, its current, the scale-invariant correction formulas, or the finite-chart Krein coefficient. %====================================================================== \section{Background, non-extraction, and mathematical dependence} \label{sec:background} %====================================================================== The Enter companion host is prescribed, not derived in this paper. I-B distinguishes a finite residual density matrix from a covariant tensor-valued distribution: the field, observable, state, renormalization prescription, and distributional interpretation must be supplied separately \cite{CheungIB}. We supply those ingredients for a free test field. None is extracted from an Enter readout. The destructive-readout interpretation of Enter \cite{CheungEnter,CheungPhil,CheungEn09} and the reporting discipline of XVI-O1-ST \cite{CheungO1ST} constrain interpretation but do not construct the continuum covariance. The finite instrument and its limitation are reproduced in Section~\ref{sec:finite}. \paragraph{Use of I-B-G.} The supplied Friedrichs foundation identifies \SourceLabel{I-B-G-v2.tex}, with source labels \SourceLabel{ass:host}, \SourceLabel{ass:product}, \SourceLabel{thm:positivity}, \SourceLabel{thm:green}, \SourceLabel{thm:speed}, \SourceLabel{thm:closed}, \SourceLabel{cor:structure}, \SourceLabel{lem:window}, \SourceLabel{thm:transfer}, and \SourceLabel{ass:transfer-contract} \cite{CheungIBG}. The AB and SAE records also refer to an earlier title and theorem numbering. This paper uses source labels when discussing the companion and reproduces the operator, kernel, boundary, and positivity calculations needed for its principal results. Finite propagation and supported transfer are used only in their separately stated scope. \paragraph{Companion comparisons are not imported constructions.} X-A-ST, X-B-ST, and X-C-ST concern supported scalar completion operations \cite{CheungXAST,CheungXBST,CheungXCST}. XI-E separates a named Hadamard product domain from selection of a positive state \cite{CheungXIE}. Oai282-A separates finite tests from continuum coverage \cite{CheungOai282A}. These comparisons do not import a gauge domain, interacting product, contact classification, or dynamics law. \paragraph{Meaning of ``Open''.} An item marked \textbf{Open} is not resolved by this manuscript. This is not a claim that it is unknown in the literature. %====================================================================== \section{Host, bundle, field normalization, and signature} \label{sec:host} %====================================================================== \begin{assumption}[Prescribed background] \label{ass:host} Fix \[ 0<\mu<2\pi,\qquad \alpha=1-\frac{\mu}{2\pi}\in(0,1). \] The regular spacetime is \[ M=\RR_t\times(0,\infty)_r\times (\RR/2\pi\ZZ)_\varphi\times\RR_z, \] with \[ g=-\dd t^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2. \] The metric completion adds \(\WW=\{r=0\}\), but \(\WW\not\subset M\). We use \(\hbar=c=1\), and \(t\) is length-valued time. The massless equation is \[ P\Phi=(-\Box_g+\xi R)\Phi=(\partial_t^2+B)\Phi=0. \] On \(M\), \(R=0\). No distributional apex curvature coupling is included. \end{assumption} Let \[ N=(0,\infty)\times S^1\times\RR,\qquad q=\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2, \qquad \dd V=\alpha r\,\dd r\,\dd\varphi\,\dd z. \] For \(\beta\in[0,1)\), the flat Hermitian line bundle \(L_\beta\) is represented on the angular cover by \[ u(r,\varphi+2\pi,z)=e^{2\pi\ii\beta}u(r,\varphi,z). \] In a lifted parallel frame its connection is ordinary differentiation. Put \[ q_n=n+\beta,\qquad \nu_n=\frac{|q_n|}{\alpha},\qquad \psi_n(\varphi)=\frac{e^{\ii q_n\varphi}}{\sqrt{2\pi\alpha}}. \] The spatial Hilbert space is \(\HH_\beta=L^2_\beta(N,\dd V)\). \begin{assumption}[Extension class] \label{ass:product} The main ledger concerns \[ B=B_\perp\otimes\id+\id\otimes(-\partial_z^2), \] where \(B_\perp\) is a self-adjoint extension of the transverse cone Laplacian in \(L^2_\beta(C_\alpha^\circ,\alpha r\,\dd r\,\dd\varphi)\). It is complex linear. Rotation invariance is imposed for the diagonal subledger but not for the full finite-dimensional boundary relation. General \(z\)-dependent, longitudinal-momentum-dependent, and non-product extensions are outside this class. \end{assumption} The Friedrichs extension is denoted \(B_F\). It is the closure of the nonnegative energy form initially defined on compactly supported smooth sections. \subsection{Real and complex normalization} A neutral real field uses \(\beta=0\). For the charged field, \[ \Phi=\frac{\phi_1+\ii\phi_2}{\sqrt2},\qquad T^{\mathrm{complex}}=T[\phi_1]+T[\phi_2]. \] Thus \(\langle\Phi^\dagger\Phi\rangle\) has the same scalar covariance normalization as one real Wick square, whereas the complex stress tensor has twice the corresponding real differential expression. The real components need not be separate globally trivial bundles. The antiparticle sector uses the conjugate bundle and conjugate extension. A real sign bundle at \(\beta=1/2\) is not what is called one charged complex field here. For \[ D_\mu=\nabla_\mu+\ii e A_\mu, \] a periodic gauge may be chosen locally with \(eA=\beta\,\dd\varphi\), up to an integer gauge shift. The twisted parallel representation satisfies \[ \Phi_{\mathrm{tw}}=e^{\ii\beta\varphi}\Phi_{\mathrm{per}}. \] For a flux \(\mathcal F\), our convention is \(\beta=\operatorname{frac}(e\mathcal F/2\pi)\). \subsection{Signature ledger} On a developing chart, \[ X=r\cos(\alpha\varphi),\qquad Y=r\sin(\alpha\varphi). \] If a companion completion uses \[ \eta=\dd t^2-\dd X^2-\dd Y^2-\dd z^2, \] then \[ \mathrm{dev}^*\eta=-g,\qquad -\Box_g=\mathrm{dev}^*\Box_\eta. \] The equation operator and action Hessian have opposite signs. This distinction matters when comparing inverse conventions. Euclidean continuation uses \[ g_E=\dd\tau^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2, \qquad L_E=-\partial_\tau^2+B. \] With \(t=-\ii\tau\), mixed stress components obey \[ T^t{}_t=T^\tau{}_\tau\big|_E,\qquad T^i{}_j=T^i{}_j\big|_E. \] The static energy density is \[ \varepsilon=T_{tt}=-T^t{}_t. \] \subsection{Angular lifts and distances} \label{sec:angles} For chosen lifts define \[ \vartheta=\alpha(\varphi-\varphi'),\qquad \mathcal K(\vartheta) =\{k\in\ZZ:|\vartheta-2\pi\alpha k|<\pi\}. \] Set \begin{align} R_k^2 &=r^2+r'^2-2rr'\cos(\vartheta-2\pi\alpha k)+\Delta z^2, \nonumber\\ \rho(u)^2 &=r^2+r'^2+2rr'\cosh u+\Delta z^2, \nonumber\\ \rho_0^2&=(r+r')^2+\Delta z^2. \label{eq:distances} \end{align} An image boundary is a locus \[ |\vartheta-2\pi\alpha k|=\pi. \] If \(\vartheta=\psi+2\pi\alpha\ell\), with \(\psi\in(-\pi\alpha,\pi\alpha]\), a lifted kernel satisfies \[ K(\vartheta)=e^{2\pi\ii\beta\ell}K(\psi). \] A reduced-angle expression must retain this bundle factor. All diagonal calculations use one local parallel frame. %====================================================================== \section{Friedrichs heat kernel and the massless state} \label{sec:state} %====================================================================== \begin{proposition}[Heat kernel and large-time estimate] \label{prop:heat} Relative to \(\dd V\), the Friedrichs heat kernel is \begin{align} K_\beta(s;x,x') ={}& \frac{e^{-\Delta z^2/(4s)}}{\sqrt{4\pi s}}\, \frac{e^{-(r^2+r'^2)/(4s)}}{4\pi\alpha s} \sum_{n\in\ZZ} e^{\ii q_n(\varphi-\varphi')} I_{\nu_n}\left(\frac{rr'}{2s}\right). \label{eq:heat} \end{align} For each compact regular spatial set \(K\), \[ \sup_{x,x'\in K}|K_\beta(s;x,x')| \le C_K s^{-3/2-\nu_*},\qquad s\ge1, \qquad \nu_*=\frac{\min(\beta,1-\beta)}{\alpha}. \] At \(\beta=0\), \(\nu_*=0\). \end{proposition} \begin{proof} The Friedrichs radial transform uses \(J_\nu(kr)\) with measure \(k\,\dd k\). Weber's integral is \[ \int_0^\infty e^{-sk^2}J_\nu(kr)J_\nu(kr')k\,\dd k =\frac{e^{-(r^2+r'^2)/(4s)}}{2s} I_\nu\left(\frac{rr'}{2s}\right). \] Combining the normalized angular modes with the longitudinal heat kernel gives \eqref{eq:heat} \cite{Watson,DLMF}. On \(K\), the power series gives, uniformly for \(s\ge1\), \[ I_\nu\left(\frac{rr'}{2s}\right) \le C_K\frac{(C_K/s)^\nu}{\Gamma(1+\nu)}. \] The orders grow linearly in \(|n|\), so their sum is bounded by \(C_Ks^{-\nu_*}\). The remaining prefactor is \(O(s^{-3/2})\). \end{proof} \begin{lemma}[Infrared finiteness] \label{lem:infrared} For every \(v\in C_c^\infty(N;L_\beta)\), \(B_F^{-1/4}v\in\HH_\beta\). On a fixed compact support this map is continuous in the compact smooth test topology. There is no \(L^2\) zero eigenfunction. \end{lemma} \begin{proof} The spectral theorem gives \[ \norm{B_F^{-1/4}v}^2 =\frac1{\sqrt\pi}\int_0^\infty s^{-1/2}\langle v,e^{-sB_F}v\rangle\,\dd s. \] For \(s\le1\), contraction bounds the integrand by \(s^{-1/2}\norm v^2\). For \(s\ge1\), Proposition~\ref{prop:heat} bounds it by \[ C_Ks^{-2-\nu_*}\norm v_{L^1}^2. \] Both integrals converge and give continuity. Zero form energy would imply a locally parallel section; no nonzero parallel section is square integrable on \(N\). \end{proof} \begin{definition}[Distinguished spectral state] \label{def:state} For a nonnegative spatial extension whose compact tests belong to \(\mathrm{dom}(B^{-1/4})\), define \[ {\cal K}_Bf =\int_\RR e^{\ii t\sqrt B}(2\sqrt B)^{-1/2}f(t)\,\dd t, \] and the positive-frequency kernel \[ W_B(t-t';x,x') =\left[\frac{e^{-\ii(t-t')\sqrt B}}{2\sqrt B}\right](x,x'). \] The zero-mean quasifree spectral Fock state has smeared covariance \(\langle{\cal K}_Bf,{\cal K}_Bg\rangle\). For a complex field the particle and antiparticle constructions use \(B\) and its conjugate extension. Charge-changing two-point functions vanish. \end{definition} The algebra is the extension-selected test-field CCR algebra. Its commutator is the causal kernel of that extension. No compact-test exactness assertion \(\ker E=PC_c^\infty(M)\) is needed. That separate question is recorded as open in the supplied I-B-G \SourceLabel{rem:compact-exact}. \begin{proposition}[Friedrichs state and symmetries] \label{prop:state} The Friedrichs prescription is a continuous positive covariance on compact smooth spacetime tests and defines a spectral ground state. With \[ E_\beta(t)=\frac{\sin(t\sqrt{B_F})}{\sqrt{B_F}}, \] the real and charged commutators are respectively \[ [\phi(X),\phi(X')]=-\ii E_0(X,X'),\qquad [\Phi(X),\Phi^\dagger(X')]=-\ii E_\beta(X,X'). \] The state is invariant under \(t,z\) translations, rotations, and proper time-oriented \((t,z)\) boosts. Its covariance has scaling degree two: \[ W_F(\lambda t,\lambda r,\varphi,\lambda z; \lambda t',\lambda r',\varphi',\lambda z') =\lambda^{-2}W_F(X,X'). \] \end{proposition} \begin{proof} Lemma~\ref{lem:infrared} bounds the one-particle vector on fixed compact supports. The Gram form proves positivity and continuity; the spectral Fock construction gives a nonnegative Hamiltonian \cite{Wald1994}. The difference of time orientations is \[ \frac{e^{-\ii t\sqrt B}-e^{\ii t\sqrt B}}{2\sqrt B} =-\ii\frac{\sin(t\sqrt B)}{\sqrt B}. \] Differentiating in the second time variable at equal times gives the canonical \(+\ii\id\) sign. Translations and rotations commute with the operator. For boosts use \begin{align*} W_F={}&\frac1{2\pi\alpha} \sum_n e^{\ii q_n(\varphi-\varphi')} \int_0^\infty k\,\dd k\,J_{\nu_n}(kr)J_{\nu_n}(kr')\\ &\quad\times\int_\RR\frac{\dd p}{2\pi} \frac{e^{\ii p\Delta z-\ii\sqrt{k^2+p^2}\Delta t}} {2\sqrt{k^2+p^2}}. \end{align*} The inner mass-shell measure is the boost-invariant \(1+1\)-dimensional positive-frequency measure of mass \(k\). The infrared estimate justifies compact testing. Rescaling \(k,p\) proves the dilation identity. \end{proof} For Friedrichs, intrinsic finite propagation gives local commutativity for intrinsically spacelike-separated supports \cite[\SourceLabel{thm:speed}]{CheungIBG}. This is not a statement about an arbitrary completion. \begin{remark}[Extension selection is not state uniqueness] \label{rem:uniqueness} The zero-mean spectral state is distinguished by the prescription, not by a theorem classifying every algebraic ground state. Additional zero-frequency covariance terms or appropriate coherent shifts are not ruled out merely by positivity of the extension. \end{remark} %====================================================================== \section{Euclidean kernels, diffraction, and local Hadamard form} \label{sec:hadamard} %====================================================================== For Euclidean points set \[ \cosh\eta= \frac{r^2+r'^2+\Delta\tau^2+\Delta z^2}{2rr'},\qquad \sigma_E^2=r^2+r'^2-2rr'\cos\vartheta+ \Delta\tau^2+\Delta z^2. \] \begin{proposition}[Euclidean mode sum] \label{prop:modes} The heat-integrated kernel \[ G_{F,E}(X,X') =\int_0^\infty \frac{e^{-\Delta\tau^2/(4s)}}{\sqrt{4\pi s}}\, K_\beta(s;x,x')\,\dd s \] is a fundamental distribution for \(L_E\) in each variable. For \(\eta>0\), \begin{align} G_{F,E} &=\frac1{8\pi^2\alpha rr'\sinh\eta} \sum_{n\in\ZZ} e^{-|q_n|\eta/\alpha}e^{\ii q_n(\varphi-\varphi')} \label{eq:modes}\\ &=\frac1{8\pi^2\alpha rr'\sinh\eta} \left[ \frac{e^{-\beta a+\ii\beta b}}{1-e^{-a+\ii b}} +\frac{e^{-(1-\beta)a-\ii(1-\beta)b}}{1-e^{-a-\ii b}} \right], \nonumber \end{align} where \(a=\eta/\alpha\), \(b=\varphi-\varphi'\). For \(\beta=0\), \[ G_{F,E} =\frac{\sinh(\eta/\alpha)} {8\pi^2\alpha rr'\sinh\eta [\cosh(\eta/\alpha)-\cos(\varphi-\varphi')]}. \] At \(\alpha=1,\beta=0\), this is \((4\pi^2\sigma_E^2)^{-1}\). \end{proposition} \begin{proof} Large-time convergence follows from Proposition~\ref{prop:heat}; the additional Euclidean factor gives \(s^{-2-\nu_*}\). Integrating the heat equation proves the fundamental-distribution identities. Substitute \(\zeta=rr'/(2s)\) and use \[ \int_0^\infty e^{-\zeta\cosh\eta}I_\nu(\zeta)\,\dd\zeta =\frac{e^{-\nu\eta}}{\sinh\eta}, \qquad \eta>0,\quad \nu>-1. \] The constants become \((8\pi^2\alpha rr')^{-1}\). Absolute convergence justifies the Friedrichs sum for \(\eta>0\). Splitting \(n\ge0\) and \(n\le-1\) gives the two geometric series \cite{Watson,DLMF}. \end{proof} \subsection{Angular summation} Define \[ \Phi_\beta(u,\vartheta)= \sum_{n\in\ZZ} \sin(\pi\nu_n)e^{-\nu_nu}e^{\ii q_n\vartheta/\alpha}, \qquad u>0. \] \begin{lemma}[Sommerfeld summation and boundary control] \label{lem:angular} For \(\zeta>0\), at generic angles, \begin{align} \sum_n I_{\nu_n}(\zeta)e^{\ii q_n\vartheta/\alpha} ={}& \alpha\sum_{k\in\mathcal K(\vartheta)} e^{2\pi\ii\beta k}e^{\zeta\cos(\vartheta-2\pi\alpha k)} \nonumber\\ &-\frac1\pi\int_0^\infty e^{-\zeta\cosh u}\Phi_\beta(u,\vartheta)\,\dd u . \label{eq:sommerfeld} \end{align} The identity extends after joint distributional testing through image boundaries. Near such a boundary \(\vartheta_j\), \[ |\Phi_\beta(u,\vartheta)| \le C\left(1+ [u^2+(\vartheta-\vartheta_j)^2]^{-1/2}\right). \] The weight decays exponentially at large \(u\). For \(\beta=0\), \[ \Phi_0(u,\vartheta)=\frac12\sum_\pm \frac{\sin((\pi\pm\vartheta)/\alpha)} {\cosh(u/\alpha)-\cos((\pi\pm\vartheta)/\alpha)}. \] \end{lemma} \begin{proof} Set \(A=\pi/\alpha\), \(b=u/\alpha\), \(c=\vartheta/\alpha\), and \[ F(\gamma,w)=\frac1{2\ii} \left[ \frac{e^{\gamma(w+\ii A)}}{1-e^{w+\ii A}} -\frac{e^{\gamma(w-\ii A)}}{1-e^{w-\ii A}} \right]. \] Splitting positive and negative frequencies gives \[ \Phi_\beta=F(\beta,-b+\ii c)+F(1-\beta,-b-\ii c). \] The denominators give the asserted joint local bound. For \(0<\beta<1\), a decay rate is \(\min(\beta,1-\beta)/\alpha\). At \(\beta=0\), the zero-order sine vanishes and rate \(1/\alpha\) suffices. Insert the integral representation \[ I_\nu(\zeta)=\frac1\pi\int_0^\pi e^{\zeta\cos v}\cos(\nu v)\,\dd v -\frac{\sin(\pi\nu)}{\pi} \int_0^\infty e^{-\zeta\cosh u-\nu u}\,\dd u \] \cite[Eq.~10.32.4]{DLMF}. Twisted Fourier summation of the first term gives the image indicator and its phases. An Abel factor \(e^{-\epsilon\nu_n}\) justifies summation first. The joint bound is locally integrable in \((u,\vartheta)\); it and the large-\(u\) decay justify the distributional limit. Combining fractions at \(\beta=0\) gives the last formula. \end{proof} \begin{proposition}[Image--diffraction representation] \label{prop:contour} At generic pairs, \begin{align} G_{F,E} ={}&\sum_{k\in\mathcal K(\vartheta)} \frac{e^{2\pi\ii\beta k}}{4\pi^2(R_k^2+\Delta\tau^2)} \nonumber\\ &-\frac1{\pi\alpha}\int_0^\infty \frac{\Phi_\beta(u,\vartheta)} {4\pi^2(\rho(u)^2+\Delta\tau^2)}\,\dd u . \label{eq:contour} \end{align} Across image boundaries the right-hand side denotes the total joint distribution fixed by the heat kernel. \end{proposition} \begin{proof} Apply Lemma~\ref{lem:angular} inside the four-dimensional heat integral and use \[ \int_0^\infty(4\pi s)^{-2}e^{-D^2/(4s)}\,\dd s =\frac1{4\pi^2D^2}. \] At generic pairs the integrals converge absolutely. The joint local bound gives the tested boundary prescription. \end{proof} \begin{theorem}[Analytic direct subtraction at every regular point] \label{thm:analytic} At every \(X_0\) with \(r_0>0\), choose a sufficiently small Euclidean developing neighbourhood \(V_E\) and parallel frame. Then \[ H_{F,E}(X,X')= G_{F,E}(X,X')-\frac1{4\pi^2\sigma_E^2(X,X')} \] is jointly real-analytic on \(V_E\times V_E\), including its diagonal, Hermitian, and a bisolution of the flat Euclidean Laplacian. No condition such as \(1/(2\alpha)\notin\NN\) is required. \end{theorem} \begin{proof} In developing Cartesian coordinates the two terms are fundamental distributions of the same local operator in each variable. Their delta sources cancel. The difference is therefore annihilated, distributionally, by \(\Delta_X\) and \(\Delta_{X'}\), hence by the elliptic operator \(\Delta_X+\Delta_{X'}\) on the product. Elliptic regularity gives smoothness and analytic elliptic regularity gives joint real analyticity \cite{MorreyNirenberg,Hormander}. Hermiticity follows from the self-adjoint heat kernel. This treats the total distribution and never differentiates a moving image-sector indicator in isolation. \end{proof} Near a generic diagonal pair, \(H_{F,E}\) is precisely \eqref{eq:contour} with the \(k=0\) term removed. At a boundary parameter its analytic representative is the one in Theorem~\ref{thm:analytic}. For \(\alpha=1/N\) and \(N\beta\in\ZZ\), every \(\sin(\pi\nu_n)\) vanishes and the kernel is the full finite character-weighted quotient image kernel. For even \(N\), the strict image set at zero angle omits the two boundary representatives of one quotient image. The full quotient sum or an equivalent limiting prescription must be used; simply dropping them is incorrect. For incompatible characters diffraction need not vanish. \begin{theorem}[Local Lorentzian Hadamard form] \label{thm:hadamard} Every regular Lorentzian point has a sufficiently small developing neighbourhood on which \[ W_F(X,X')= \frac1{4\pi^2[ |\mathbf X-\mathbf X'|^2-(t-t'-\ii0)^2]} +H_F(X,X'), \] where \(H_F\) is smooth and locally analytic. It is obtained by continuing the time difference of \(H_{F,E}\). \end{theorem} \begin{proof} For positive Euclidean separation the spectral kernel is \(e^{-a\sqrt{B_F}}/(2\sqrt{B_F})\). Its boundary value at \(a=\epsilon+\ii(t-t')\) is \(W_F\), by the spectral theorem and Lemma~\ref{lem:infrared}. The analytic remainder has a holomorphic extension near the regular diagonal. After shrinking the neighbourhood it continues smoothly in the time difference. The direct term gives the displayed boundary value. The local geometry and connection are flat and the field is massless, so the local logarithmic Hadamard coefficient vanishes \cite{DecaniniFolacci}. \end{proof} \begin{remark}[Local, not global, microlocal statement] A global Radzikowski characterization on this non-globally-hyperbolic host is not proved \cite{Radzikowski}. Extension-dependent geometric and diffractive fronts require a separate analysis \cite{CheegerTaylor,CheungIBG}. A broken path through the completed apex is not a regular null geodesic of \(M\). No statement is made at \(r=0\). \end{remark} %====================================================================== \section{Renormalization, complex first jets, and Ward identities} \label{sec:renormalization} %====================================================================== \begin{assumption}[Renormalization class] \label{ass:ren} On regular developing neighbourhoods use local covariant Hadamard point splitting, the usual scaling and smooth mass-dependence conditions, and zero Minkowski-vacuum normalization. The flat connection is an external background. No state-dependent or holonomy-dependent nonlocal subtraction, and no apex-supported counterterm, is included. \end{assumption} This is the local framework of \cite{Wald1977,HollandsWald2001,HollandsWald2005,DecaniniFolacci}. Its local formulas are applied on regular charts, not as a globally hyperbolic existence theorem for all of \(M\). Let brackets denote coincidence and let \(H\) be the regular complex two-point kernel in a parallel frame. Put \[ F=[H]. \] For one real field use \(h=\Real H\), and for complex stress use \(h=2\Real H\). For a smooth symmetric real bisolution define \begin{align} [\mathcal D_{\mu\nu}h] ={}&(1-2\xi)[\nabla_{(\mu}\nabla_{\nu)'}h] +\left(2\xi-\frac12\right)g_{\mu\nu} [\nabla_\rho\nabla^{\rho'}h] \nonumber\\ &-\xi[(\nabla_\mu\nabla_\nu+ \nabla_{\mu'}\nabla_{\nu'})h]. \label{eq:pointsplit} \end{align} Indices are parallel transported before coincidence. For the charged field, \begin{equation} \langle j_\mu\rangle_\ren =\ii e[(D_\mu-D_{\mu'}^*)H], \qquad j_\mu=\ii e[ \Phi^\dagger D_\mu\Phi-(D_\mu\Phi)^\dagger\Phi]. \label{eq:current-prescription} \end{equation} The two slots carry opposite connection characters. \begin{lemma}[Regular-part identities and ambiguities] \label{lem:ward} Under Assumption~\ref{ass:ren}: \begin{enumerate}[label=\textup{(\roman*)}] \item The Wick-square ambiguity \(am^2+bR\), the usual stress ambiguities, and local background-gauge-curvature terms vanish on the regular part. The possible current ambiguity proportional to \(\nabla^\nu F_{\nu\mu}\) also vanishes. \item Local conservation corrections and the regular conformal trace anomaly vanish. \item For \(F_h=[h]\), \[ \nabla^\mu T_{\mu\nu}=0,\qquad T^\mu{}_\mu=\frac12(6\xi-1)\Box_gF_h, \] \[ T_{\mu\nu}(\xi)-T_{\mu\nu}(1/6) =\left(\xi-\frac16\right) (g_{\mu\nu}\Box_g-\nabla_\mu\nabla_\nu)F_h. \] For the complex field \(F_h=2F\). \item The current defined by \eqref{eq:current-prescription} is real and conserved. \end{enumerate} These statements do not classify completion-supported distributions or contact terms in higher products. \end{lemma} \begin{proof} All admissible local geometric ambiguity and anomaly expressions vanish when \(m=0\), the Riemann tensor is zero, and the external connection curvature is zero. Holonomy is not a local curvature invariant. In a parallel Cartesian chart put \[ A_{\mu\nu}=[\partial_\mu\partial_{\nu'}h], \qquad K=A^\mu{}_\mu. \] Symmetry, the bisolution equations, and Synge's rule give \[ \Box F_h=2K,\qquad \partial_\mu\partial_\nu F_h =2[\partial_\mu\partial_\nu h]+2A_{\mu\nu}, \qquad \partial^\mu A_{\mu\nu}=\frac12\partial_\nu K. \] Thus \[ T_{\mu\nu} =A_{\mu\nu}-\xi\partial_\mu\partial_\nu F_h +\left(2\xi-\frac12\right)g_{\mu\nu}K. \] Taking a divergence, trace, and coupling difference proves (iii). Hermiticity makes the current real. Applying a divergence to its coincidence expression gives \(\ii e[(\Box_X-\Box_{X'})H]=0\). \end{proof} \begin{remark}[The complex first-jet distinction] The symmetric real part suffices for the stress tensor but not for the current. If \(H\) is replaced by \(\Real H\) before taking the antisymmetric first derivative, the azimuthal current is lost. In the periodic gauge, \[ H_{\mathrm{per}}= e^{-\ii\beta(\varphi-\varphi')}H_{\mathrm{tw}}. \] The covariant derivatives in \eqref{eq:current-prescription} cancel this phase. The prescription is therefore gauge invariant. The parametrix must be transformed with the same parallel factor. \end{remark} \begin{remark}[What ``no subtraction needed'' can mean] For the azimuthal current computed by time splitting at \(\Delta\varphi=0\), the direct parametrix has zero azimuthal derivative difference. No extra subtraction appears in that particular calculation. It is false that the derivative difference of \((4\pi^2\sigma_E^2)^{-1}\) vanishes along every splitting. For example a spatial derivative of that singular function is generally nonzero and divergent. Equation~\eqref{eq:current-prescription}, using the regular kernel, is the splitting-independent definition. \end{remark} %====================================================================== \section{Exact Friedrichs observables and analytical checks} \label{sec:friedrichs-results} %====================================================================== Define \begin{align} P_\beta(\alpha) &=\alpha^{-2}[1-6\beta(1-\beta)]-1, \label{eq:P}\\ C_\beta(\alpha) &=\alpha^{-4}[1-30\beta^2(1-\beta)^2]-1. \label{eq:C} \end{align} \begin{lemma}[Time-split expansion] \label{lem:expansion} For \(r=r'\), \(\Delta\varphi=\Delta z=0\), and \(\Delta\tau=s\), \[ H_{F,E}(s)= \frac{P_\beta}{48\pi^2r^2} -\frac{C_\beta+10P_\beta}{2880\pi^2r^4}s^2 +O(s^4). \] \end{lemma} \begin{proof} The angular sum is \[ S_\beta(a)= \frac{\cosh((1/2-\beta)a)}{\sinh(a/2)} =\frac2a+B_2(\beta)a+\frac{B_4(\beta)}{12}a^3+O(a^5), \] where \[ B_2(\beta)=\beta^2-\beta+\frac16,\qquad B_4(\beta)=\beta^4-2\beta^3+\beta^2-\frac1{30}. \] Also \[ \eta=2\operatorname{arsinh}\frac{s}{2r},\qquad \eta^{-2}=\frac{r^2}{s^2}+\frac1{12} -\frac{s^2}{240r^2}+O(s^4), \] and \[ \frac{S_\beta(\eta/\alpha)}{\alpha\sinh\eta} =\frac2{\eta^2}+\frac{B_2}{\alpha^2}-\frac13 +\eta^2\left[ \frac{B_4}{12\alpha^4} -\frac{B_2}{6\alpha^2}+\frac7{180}\right]+O(\eta^4). \] Multiply by \((8\pi^2r^2)^{-1}\), subtract \((4\pi^2s^2)^{-1}\), and use \(P_\beta=6B_2/\alpha^2-1\), \(C_\beta=-30B_4/\alpha^4-1\). Theorem~\ref{thm:analytic} identifies these with the unique regular coincidence limits. \end{proof} \begin{theorem}[Friedrichs Wick squares and stress tensors] \label{thm:friedrichs} For one neutral real scalar, \[ \langle\phi^2\rangle_\ren =\frac{\alpha^{-2}-1}{48\pi^2r^2}, \] \begin{align} \langle T^\mu{}_\nu\rangle_\ren ={}&\frac{\alpha^{-4}-1}{1440\pi^2r^4}\diag(1,1,-3,1) \nonumber\\ &+\left(\xi-\frac16\right) \frac{\alpha^{-2}-1}{48\pi^2r^4}\diag(4,-2,6,4). \label{eq:T-real} \end{align} For one charged complex scalar, \[ \langle\Phi^\dagger\Phi\rangle_\ren =\frac{P_\beta}{48\pi^2r^2}, \] \begin{align} \langle T^\mu{}_\nu\rangle_\ren ={}&\frac{C_\beta}{720\pi^2r^4}\diag(1,1,-3,1) \nonumber\\ &+2\left(\xi-\frac16\right) \frac{P_\beta}{48\pi^2r^4}\diag(4,-2,6,4). \label{eq:T-complex} \end{align} Their traces are respectively \[ \left(\xi-\frac16\right) \frac{\alpha^{-2}-1}{4\pi^2r^4}, \qquad \left(\xi-\frac16\right) \frac{P_\beta}{2\pi^2r^4}. \] \end{theorem} \begin{proof} The Wick squares follow from Lemma~\ref{lem:expansion}. For stress first apply the real differential expression to \(\Real H_{F,E}\). Reflections, angular evenness of the real part, translations, boosts, and scaling imply \[ T^\mu{}_\nu=r^{-4}\diag(a,b,c,a). \] Radial conservation gives \(c=-3b\). For \(F=P_\beta/(48\pi^2r^2)\), \[ [\partial_\rho\partial^{\rho'}H_{F,E}] =\frac12\Delta_EF=\frac{P_\beta}{24\pi^2r^4}. \] The time-split identities give \[ T^\tau{}_\tau =-H_{F,E}''(0) +\left(2\xi-\frac12\right)\frac{P_\beta}{24\pi^2r^4} =\frac{C_\beta}{1440\pi^2r^4} +\left(\xi-\frac16\right) \frac{4P_\beta}{48\pi^2r^4}. \] At conformal coupling the trace vanishes, hence \(a=b\). For other couplings use \[ (g\Box-\nabla\nabla)^\mu{}_\nu r^{-2} =r^{-4}\diag(4,-2,6,4). \] Take \(\beta=0\) for one real field; multiply by two for one complex stress tensor. The trace formulas follow directly. \end{proof} At conformal coupling the neutral energy density is negative: \[ \varepsilon=-\frac{\alpha^{-4}-1}{1440\pi^2r^4}. \] At \(\alpha=1/2\), the single nontrivial image gives \[ H_E(s)=\frac1{4\pi^2(4r^2+s^2)}. \] Then \[ F=\frac1{16\pi^2r^2},\qquad H_E''(0)=-\frac1{32\pi^2r^4},\qquad \frac12\Delta_EF=\frac1{8\pi^2r^4}, \] so \(T^t{}_t=1/(96\pi^2r^4)\). This is an independent sign check. \begin{theorem}[Friedrichs induced current] \label{thm:current} For one charged complex field, \[ \langle j_\varphi\rangle_\ren =-\frac{e\,\beta(1-\beta)(1-2\beta)} {12\pi^2\alpha^2r^2}, \] \[ \langle j_{\hat\varphi}\rangle_\ren =-\frac{e\,\beta(1-\beta)(1-2\beta)} {12\pi^2\alpha^3r^3}, \qquad j_t=j_r=j_z=0. \] It is conserved and independent of \(\xi\). With the fractional-part convention it is periodic in flux, odd under \(\beta\mapsto1-\beta\), and zero at \(0,1/2\). \end{theorem} \begin{proof} Time split at \(\Delta\varphi=0\). The direct azimuthal term vanishes. Differentiating \eqref{eq:modes}, \[ \partial_{\Delta\varphi}G_{F,E} =\frac{\ii\Sigma(a)}{8\pi^2\alpha r^2\sinh\eta}, \] where \[ \Sigma(a)=\sum_nq_ne^{-|q_n|a} =-\frac{\dd}{\dd a} \frac{\sinh((1/2-\beta)a)}{\sinh(a/2)} =\frac{\beta(1-\beta)(1-2\beta)}3\,a+O(a^3). \] Since \(\partial_\varphi-\partial_{\varphi'}= 2\partial_{\Delta\varphi}\), \eqref{eq:current-prescription} gives the result. The other components vanish by the time and longitudinal evenness and radial endpoint symmetry. Conservation follows from \(j^r=0\) and angular independence. The remaining properties follow from the polynomial and mode relabeling. \end{proof} \subsection{Diffraction identity and quotient sums} \begin{corollary}[Untwisted diffraction identity] \label{cor:diffraction} If \(1/(2\alpha)\notin\NN\), then \begin{align*} &\sum_{0<|k|<1/(2\alpha)}\csc^2(\pi\alpha k)\\ &\quad-\frac1{\pi\alpha}\int_0^\infty \frac{\sin(\pi/\alpha)\,\dd u} {[\cosh(u/\alpha)-\cos(\pi/\alpha)]\cosh^2(u/2)} =\frac{\alpha^{-2}-1}{3}. \end{align*} For \(\alpha>1/2\), the image sum is empty, and on a sufficiently small diagonal neighbourhood the regular remainder is wholly represented by diffraction. \end{corollary} \begin{proof} At coincidence \(R_k^2=4r^2\sin^2(\pi\alpha k)\) and \(\rho(u)^2=4r^2\cosh^2(u/2)\). Multiply the subtracted contour representation by \(16\pi^2r^2\). \end{proof} This representation does not make vacuum polarization a retarded signal emitted from an apex event. \begin{proposition}[Exact quotient identities] \label{prop:quotient} For \(N\ge2\), \[ \sum_{k=1}^{N-1}\csc^2\frac{\pi k}{N}=\frac{N^2-1}{3}, \qquad \sum_{k=1}^{N-1}\csc^4\frac{\pi k}{N} =\frac{(N^2-1)(N^2+11)}{45}. \] For \(j=0,\ldots,N-1\), put \(u=j(N-j)\). Then \[ \sum_{k=1}^{N-1} e^{2\pi\ii jk/N}\csc^2\frac{\pi k}{N} =\frac{N^2-1}{3}-2u, \] \[ \sum_{k=1}^{N-1} e^{2\pi\ii jk/N}\csc^4\frac{\pi k}{N} =\frac{(N^2-1)(N^2+11)}{45} -\frac23u(u+2). \] \end{proposition} \begin{proof} Logarithmic differentiation of the sine product gives \[ \sum_{k=0}^{N-1}\csc^2\left(x+\frac{\pi k}{N}\right) =N^2\csc^2(Nx). \] Compare constant and quadratic terms, using \(\frac12(\csc^2x)''=3\csc^4x-2\csc^2x\). This proves the untwisted identities. For the twisted sums, the discrete second difference of the first sum is \(4\) away from \(j=0\) modulo \(N\), and that of the second is minus four times the first. The displayed polynomials satisfy these equations and the periodic endpoint values; the periodic homogeneous second-difference equation has only constants. The known \(j=0\) values fix those constants. \end{proof} At \(\alpha=1/N\), \(\beta=j/N\), the full quotient kernel and these identities reproduce the Wick square and time-split stress coefficients exactly. This is an all-\(N\) analytical check, distinct from the reported finite symbolic runs. \subsection{Flat and flux limits} At fixed \(r>0\), for \(\beta=0\), \[ P_0=\frac{\mu}{\pi}+O(\mu^2),\qquad C_0=\frac{2\mu}{\pi}+O(\mu^2), \] hence \[ \frac{\langle\phi^2\rangle_\ren}{\mu} \longrightarrow\frac1{48\pi^3r^2},\qquad \frac{\langle T^t{}_t\rangle_\ren}{\mu} \longrightarrow\frac1{720\pi^3r^4} \quad(\xi=1/6). \] These are not uniform apex or wave-front limits. At fixed nonzero \(\beta\), \(\alpha\to1\) retains the punctured flat-space Aharonov--Bohm effect: \[ F\longrightarrow-\frac{\beta(1-\beta)}{8\pi^2r^2}, \] and the complex conformal tensor coefficient tends to \(-\beta^2(1-\beta)^2/(24\pi^2r^4)\). The conformal energy may have either sign as the parameters vary. %====================================================================== \section{Boundary data and the complete transverse product ledger} \label{sec:boundary} %====================================================================== The radial expression in angular mode \(n\) is \[ -\partial_r^2-r^{-1}\partial_r+\nu_n^2r^{-2} \quad\hbox{in }L^2((0,\infty),r\,\dd r). \] The deficient modes are \[ {\cal D}_\beta=\{n:|n+\beta|<\alpha\}. \] For \(0<\beta<1\), these can only be \(0,-1\): \[ 0\in{\cal D}_\beta\iff\beta<\alpha,\qquad -1\in{\cal D}_\beta\iff1-\beta<\alpha. \] There are two precisely when \(1-\alpha<\beta<\alpha\). If there are no deficient modes, Friedrichs is the unique transverse extension. For \(0<\nu_j<1\), boundary data are \[ u_j(r)=a_jr^{-\nu_j}+b_jr^{\nu_j}+o(r^{\nu_j}). \] The integrated Green form is, with one fixed overall sign convention, \[ 2\sum_j\nu_j(\bar a_jb'_j-\bar b_ja'_j). \] Its null Lagrangians classify the extensions. Write \(N=\diag(\nu_j)\). \begin{definition}[Boundary relation, including boundary charts] \label{def:relation} Choose a subspace \(W\subseteq\CC^d\), an isometric inclusion \(V:\CC^{\dim W}\to\CC^d\) with range \(W\), and a Hermitian operator \(H_W\) on \(\CC^{\dim W}\). The boundary condition is \begin{equation} a\in W,\qquad V^\dagger Nb=H_WV^\dagger a. \label{eq:relation} \end{equation} The component of \(Nb\) in \(W^\perp\) is unrestricted. For \(W=\CC^d\), this is \[ b=N^{-1}Ha,\qquad H=H^\dagger. \] For \(W=\{0\}\), it is Friedrichs. The diagonal choice \(H_{jj}=\nu_j\lambda_j\) gives \(b_j=\lambda_ja_j\). The cases \(\lambda_j=\infty\) are represented by removing that direction from \(W\), not by an ordinary finite matrix entry. \end{definition} Every Lagrangian relation has this form. Indeed, let \(W\) be its projection onto the \(a\)-space. The vertical part is \(W^\perp\) in the \(Nb\)-variable, and the remaining graph is Hermitian on \(W\). This is the finite-dimensional relation version of the usual \(U(d)\) parametrization. For \(\beta=0\), only \(n=0\) is deficient and \[ u_0(r)=a\log r+b+o(1). \] Its extensions are Friedrichs or \[ b=-a\log L,\qquad 00, \] as Hermitian matrices. \end{lemma} \begin{proof} The inverse in \eqref{eq:CW} is bounded above by \((V^\dagger\Delta V)^{-1}\). Moreover \[ V(V^\dagger\Delta V)^{-1}V^\dagger\le\Delta^{-1}, \] because conjugation by \(\Delta^{1/2}\) turns the left side into the orthogonal projection onto \(\operatorname{ran}(\Delta^{1/2}V)\). Consequently \[ C_W\le\frac12E^{-1}\Delta^{-1}E^{-1}=S^{-1}. \] \end{proof} \begin{theorem}[Euclidean correction and spectral-state existence] \label{thm:nonnegative-state} If \(H_W\ge0\), the massless spectral covariance is finite and continuous on compact tests. Its Euclidean kernel is \(G_{W,E}=G_{F,E}+\delta G_W\), where \begin{align} \delta G_W(X,X') =\frac1{4\pi^2\alpha}\int_0^\infty p\,\dd p\, J_0\!\left(p\sqrt{\Delta\tau^2+\Delta z^2}\right) \sum_{jk}&e^{\ii q_j\varphi}K_{\nu_j}(pr) \nonumber\\[-2pt] &\times C_{W,jk}(p)K_{\nu_k}(pr') e^{-\ii q_k\varphi'}. \label{eq:dG} \end{align} This correction is jointly real-analytic at every pair of regular Euclidean points. The spectral state has the local Hadamard form of Theorem~\ref{thm:hadamard}. The state remains \(t,z\)-translation invariant and \((t,z)\)-boost invariant. Rotation and dilation invariance hold exactly when the extension has those symmetries. \end{theorem} \begin{proof} Fourier transform the product resolvent in the two Euclidean variables \((\tau,z)\). The radial angular integral in their momentum plane gives \(J_0\), and the normalized transverse modes give the factor \((4\pi^2\alpha)^{-1}\). At small \(p\), Lemma~\ref{lem:Cbound} and \(K_{\nu_j}(pr)=O(p^{-\nu_j})\), locally uniformly in \(r>0\), bound every term by \[ C_Kp^{1-\nu_j-\nu_k}. \] This is integrable because \(\nu_j,\nu_k<1\). Spatial and angular derivatives preserve this integrability; time and longitudinal derivatives improve it. At large \(p\), the bound and the \(K\)-Bessel asymptotic give exponential decay \(e^{-p(r+r')}\) times a polynomial. In particular the equal-time correction is bounded on fixed compact regular spatial pairs. The quadratic form of \(B^{-1/2}\), first defined with a positive spectral or mass cutoff, is the Friedrichs form plus this finite correction. Monotone convergence in the spectral integral removes the cutoff. Lemma~\ref{lem:infrared} therefore supplies the remaining Friedrichs part. This proves compact-test finiteness and continuity rather than inferring them from \(B\ge0\). For analyticity, use the two-dimensional Fourier representation before angular integration. On a small complex neighbourhood of any real regular pair, the imaginary displacement in \((\tau,z)\) is less than the positive real part of \(r+r'\). The exponential large-\(p\) estimate then persists. The small-\(p\) bound is locally uniform in that neighbourhood. Holomorphic parameter integration gives a holomorphic extension, hence real analyticity. The difference is a Euclidean bisolution. Its continuation is smooth locally, so the same direct Hadamard singularity remains. The spectral construction gives positivity, and the product mass-shell argument proves boosts. \end{proof} \begin{remark}[No universal locality assertion for arbitrary relations] The preceding theorem constructs the extension-selected spectral state and its local near-diagonal Hadamard form. It does not automatically extend the Friedrichs intrinsic finite-propagation theorem to every rotation-breaking boundary relation. The commutator is always the one belonging to the chosen extension. \end{remark} %====================================================================== \section{Spectrum, positivity, and scale} \label{sec:spectrum} %====================================================================== \begin{theorem}[Complete product positivity ledger] \label{thm:spectrum} For \(0<\beta<1\): \begin{enumerate}[label=\textup{(\roman*)}] \item The negative transverse eigenvalues are \(-\kappa^2\) satisfying \[ \det[H_W+V^\dagger\Delta(\kappa)V]=0. \] Their number, with multiplicity, is the number of negative eigenvalues of \(H_W\). There are no transverse zero eigenfunctions. Thus the full product operator is nonnegative exactly when \(H_W\ge0\). \item In a full finite chart, rotation invariance is equivalent to diagonality of \(H\). Rotation by \(\gamma\) sends \(H_{0,-1}\) to \(e^{\ii\gamma}H_{0,-1}\). On boundary charts the corresponding condition is invariance of the whole relation. \item If the deficient orders are distinct, the dilation-invariant relations are exactly the \(2^d\) coordinate choices of Friedrichs or \(\mathsf E_0\). If \(\beta=1/2\) and \(\alpha>1/2\), there is in addition \[ a\in\CC v,\qquad b\perp v,\qquad [v]\in\mathbb{CP}^1. \] This is \(\mathsf E_0\) on \(v\) and Friedrichs on \(v^\perp\). \end{enumerate} For \(\beta=0\), every \(\mathsf L_L\) has one transverse eigenvalue \(-\kappa_L^2\). Friedrichs is the only nonnegative massless extension in the untwisted product class. \end{theorem} \begin{proof} A negative transverse eigenfunction is a linear combination of \(K_{\nu_j}(\kappa r)\psi_j\). Its boundary condition is precisely the displayed determinant condition, by the calculation in Theorem~\ref{thm:krein}. The compressed matrix \(V^\dagger\Delta(\kappa)V\) increases strictly in Loewner order from zero to infinity. The ordered eigenvalues of its sum with \(H_W\) therefore increase strictly and continuously from the eigenvalues of \(H_W\) to positive infinity. Exactly its initially negative eigenvalues cross zero, counted with multiplicity. The resolvent difference has finite rank, so the transverse essential spectrum remains \([0,\infty)\). Zero-energy radial solutions \(r^{\pm\nu}\), or \(1,\log r\), cannot be square integrable at infinity. Rotations multiply each boundary component by \(e^{\ii q_j\gamma}\). For two deficient modes their charge difference is one, giving the stated transformation. Modulo a common scalar normalization, dilations act by \(a_j\mapsto s^{\nu_j}a_j\), \(b_j\mapsto s^{-\nu_j}b_j\). With distinct positive orders all weights are distinct, so an invariant Lagrangian is a coordinate sum containing exactly one of each paired \(a_j,b_j\) axes. At equal orders an invariant Lagrangian is \(W_a\oplus W_a^\perp\) in the \(a,b\) weight spaces. The intermediate one-dimensional choices give \(\mathbb{CP}^1\). Finally \(K_0(\kappa r)\) matches \(\log(r/L)\) at \(\kappa=2e^{-\gamma_E}/L\). \end{proof} A transverse eigenvalue does not remain an \(L^2(N)\) eigenvalue after adding the noncompact \(z\)-direction. It produces the branch \[ p_z^2-\kappa^2, \] a continuous band containing negative spectrum. This distinction is essential in the instability argument. In a finite chart, \(H_{jk}\) has length dimension \(\mathrm{length}^{-\nu_j-\nu_k}\). Thus \(H_{0,-1}\) has dimension \(\mathrm{length}^{-1/\alpha}\). Generic finite off-diagonal data introduce a length. The half-holonomy scale-invariant line family is a boundary relation and is not an exception represented by a finite nonzero off-diagonal \(H\). %====================================================================== \section{Unstable extensions and the two projected alternatives} \label{sec:unstable} %====================================================================== \begin{theorem}[No stationary finite-second-moment local state] \label{thm:nogo} Suppose a product extension has a negative transverse eigenvalue \(-\kappa^2\). There is no time-translation-invariant state on its extension-selected compact-test field algebra having finite second moments for every compact smooth field test. In particular, no regular spectral ground state or regular KMS state with that finite-moment property exists. This excludes every massless \(\mathsf L_L\), every diagonal combination with some \(\lambda_j<0\), and every boundary relation with \(H_W\not\ge0\). \end{theorem} \begin{proof} The full spatial operator is bounded below. Choose a compact smooth spatial section \(v\) with nonzero spectral measure on some interval \([-b,-a]\subset(-\infty,0)\). Such a section exists: compact tests are dense, and the negative spectral projection is nonzero. For a charged field use the underlying real phase space. Choose a nonnegative, nonzero \(\chi\in C_c^\infty(\RR)\), supported in \([-A,A]\), and let \(f(t,x)=\chi(t)v(x)\). The commutator pairing of \(f\) and its time translate \(f_T\) is obtained by integrating \[ E(T+s-s')=\frac{\sin((T+s-s')\sqrt B)}{\sqrt B} \] against \(\chi(s)\chi(s')\). For \(\lambda=-\omega^2<0\), this contribution equals \[ \frac{\sinh(\omega T)}{\omega} \left(\int\chi(s)e^{\omega s}\,\dd s\right) \left(\int\chi(s)e^{-\omega s}\,\dd s\right). \] It is positive and grows at least as a positive constant times \(e^{\sqrt a\,T}\) on a smaller negative spectral interval. For nonnegative spectrum, the elementary bound \(|\sin(u\sqrt\lambda)/\sqrt\lambda|\le |u|\) bounds the entire contribution by \(C(T+2A)\norm v^2\). The commutator magnitude is therefore unbounded in \(T\). In an invariant state with finite \(\omega(\Phi(f)^2)\), Cauchy--Schwarz gives \[ |\omega([\Phi(f_T),\Phi(f)])| \le 2\,\omega(\Phi(f)^2), \] independently of \(T\), a contradiction. \end{proof} \begin{remark}[The unstable-data proof retained] The SAE source also gives a phase-space proof. In a transverse bound channel choose longitudinal data supported in \(|p_z|\le\kappa/2\), and put \(\Omega=\sqrt{\kappa^2-p_z^2}\). For \(u_\pm=(f,\pm\Omega f)\), \[ |\sigma(T_tu_+,u_-)| =2\langle f,\Omega e^{t\Omega}f\rangle \ge2e^{\sqrt3\kappa t/2}\langle f,\Omega f\rangle. \] Robertson's inequality contradicts invariance if both variances are finite. That argument concerns an algebra containing these generally noncompact data. Theorem~\ref{thm:nogo} avoids silently identifying that data algebra with the compact-test algebra. \end{remark} \subsection{Principal-value reference} For an unstable extension let \(G_{\mathrm{PV}}\) denote the Euclidean resolvent integral with principal values at its negative transverse poles. It is a useful reference distribution, not a positive spectral ground-state covariance. The finite-rank correction to the Friedrichs kernel is locally analytic at regular diagonal pairs, because its high-momentum part is exponentially decaying and its finite poles are treated as principal-value distributions. \begin{proposition}[Projected constructions and their defects] \label{prop:projections} Suppose for simplicity there is one normalized transverse bound eigenfunction \(b(x_\perp)\) at \(-\kappa^2\). \begin{enumerate}[label=\textup{(\alph*)}] \item Removing the entire transverse bound channel gives \[ G_{\mathrm{cont}} =G_{\mathrm{PV}}-b(x_\perp)\bar b(x'_\perp)Q_\kappa(\Delta), \] where \[ Q_\kappa(\Delta) =\PV\int\frac{\dd^2k}{(2\pi)^2} \frac{e^{\ii k\cdot\Delta}}{k^2-\kappa^2} =-\frac14Y_0(\kappa|\Delta|), \qquad \Delta=(\Delta\tau,\Delta z). \] Since \(Q_\kappa\sim(2\pi)^{-1}\log(1/|\Delta|)\), the subtracted Wick-square limit is \(-\infty\) where \(b\ne0\). The kernel also has extra singularities at \(\Delta\tau=\Delta z=0\) for distinct transverse points. It is not locally Hadamard for the original field. \item Restricting only to the full nonnegative spectral subspace keeps the bound-channel modes with \(|p_z|>\kappa\). Its Euclidean kernel is \[ G_+=G_{\mathrm{PV}}+b\bar b'\,D(\Delta\tau,\Delta z), \] where \[ D(0,z)=0,\qquad D(\tau,0)=\frac14\mathbf H_0(\kappa|\tau|). \] The correction tends to zero at the diagonal, so its Wick-square value agrees with the principal-value reference. It is not a smooth regular remainder: \[ D(\tau,0)=\frac{\kappa|\tau|}{2\pi}+O(|\tau|^3). \] Thus the usual covariant stress point-splitting prescription does not apply. The projected covariance is not boost invariant. \end{enumerate} Both constructions have projected equal-time CCR and are nonlocal modifications of the original field algebra. \end{proposition} \begin{proof} Spectrally, \[ R_\perp(-p^2)=R_{\mathrm{cont}}(-p^2) +\frac{P_b}{p^2-\kappa^2}. \] Fourier transformation in \((\tau,z)\) proves (a). The \(p^{-2}\) bound-channel tail is precisely the logarithm removed from the full reference. The continuum resolvent difference therefore has the compensating algebraic tail; it must not be described as exponentially decaying. For (b), the retained bound contribution is \[ \int_{|p_z|>\kappa}\frac{\dd p_z}{2\pi} e^{\ii p_z z}\frac{e^{-|\tau|\sqrt{p_z^2-\kappa^2}}} {2\sqrt{p_z^2-\kappa^2}}. \] At \(\tau=0\) it is \(-Y_0(\kappa|z|)/4\). At \(z=0\), substituting \(u=\sqrt{p_z^2-\kappa^2}\) gives \[ \frac1{2\pi}\int_0^\infty \frac{e^{-|\tau|u}}{\sqrt{u^2+\kappa^2}}\,\dd u =\frac14[\mathbf H_0-Y_0](\kappa|\tau|). \] Subtract \(Q_\kappa\). The omitted compact momentum interval gives a bounded Fourier contribution tending to zero with \(\tau\), so the coincidence agreement is not restricted to the two axes. The two axis formulas prove failure of Euclidean rotational, hence Lorentzian boost, invariance. Projection replaces the spatial identity in the canonical commutator by the corresponding spectral projection. \end{proof} These special-function identities can also be obtained by Laplace and Hankel transformation \cite{DLMF,GR}. The construction does not repair the original unstable dynamics. \begin{proposition}[Logarithmic principal-value Wick square] \label{prop:logPV} For \(\mathsf L_L\), \begin{align} \delta F_{\mathrm{PV}} &=\frac1{4\pi^2\alpha r^2} \PV\int_0^\infty \frac{xK_0(x)^2}{\log(x/\kappa_Lr)}\,\dd x \label{eq:logPV}\\ &=-\frac1{8\pi^2\alpha r^2\ell} \left[1+\frac{\log2-1-\gamma_E}{\ell} +O(\ell^{-2})\right], \qquad \ell=\log(\kappa_Lr), \nonumber \end{align} as \(\ell\to+\infty\) or \(\ell\to-\infty\). \end{proposition} \begin{proof} Set \(y=\log x\) and \(w(y)=e^{2y}K_0(e^y)^2\). This is a Schwartz function on \(\RR\). The principal-value integral is \(\PV\int w(y)/(y-\ell)\,\dd y\). Split into \(|y|<|\ell|/2\), a neighbourhood of the moving pole, and the remaining tails. On the first region expand the reciprocal. Near the pole subtract \(w(\ell)\) symmetrically and use Schwartz bounds on \(w'\); all tail errors are smaller than any prescribed inverse power after enough moments are kept. Hence \[ \PV\int\frac{w(y)}{y-\ell}\,\dd y =-\frac1\ell\int w -\frac1{\ell^2}\int yw+O(|\ell|^{-3}). \] The Mellin formula in Appendix~\ref{app:mellin} gives \[ \int_0^\infty xK_0^2\,\dd x=\frac12,\qquad \int_0^\infty xK_0^2\log x\,\dd x =\frac12(\log2-1-\gamma_E). \] \end{proof} %====================================================================== \section{Scale-invariant irregular extensions} \label{sec:scale-invariant} %====================================================================== \begin{theorem}[Single-mode corrections and shifted branches] \label{thm:shift} Let \(n\) be deficient with \(q=n+\beta\ne0\) and \(\nu=|q|/\alpha<1\). Take \(\mathsf E_0\) in that mode and Friedrichs in the others. Then \begin{equation} \delta G_n= \frac{\sinh(\nu\eta)} {4\pi^2\alpha rr'\sinh\eta} e^{\ii q(\varphi-\varphi')}. \label{eq:E0kernel} \end{equation} For one complex field, \[ \delta F=\frac{|q|}{4\pi^2\alpha^2r^2},\qquad \delta j_\varphi=-\frac{e\,q|q|}{2\pi^2\alpha^2r^2}, \qquad \delta j_t=\delta j_r=\delta j_z=0, \] \begin{align*} \delta T^t{}_t=\delta T^z{}_z &=\frac{|q|(12\alpha^2\xi-2\alpha^2-q^2)} {6\pi^2\alpha^4r^4},\\ \delta T^r{}_r &=-\frac{|q|(6\alpha^2\xi-\alpha^2+q^2)} {6\pi^2\alpha^4r^4},\qquad \delta T^\varphi{}_\varphi=-3\delta T^r{}_r. \end{align*} Let \(P_X(\beta)\) be any Friedrichs polynomial formula for \(X=F,j_\varphi,T^\mu{}_\nu\), regarded as a polynomial rather than periodically reduced. Then \[ X_{\mathsf E_0^{(0)}}=P_X(\beta+1),\qquad X_{\mathsf E_0^{(-1)}}=P_X(\beta-1). \] If both modes are deficient and both are irregular, \[ X_{\mathsf E_0^{(0)}\mathsf E_0^{(-1)}} =P_X(\beta+1)+P_X(\beta-1)-P_X(\beta). \] \end{theorem} \begin{proof} For \(c=c_0(\nu)\), the radial heat kernel is obtained by replacing \(I_\nu\) by \(I_{-\nu}\). The Laplace transform used in Proposition~\ref{prop:modes} remains valid because \(-\nu>-1\). Their difference gives \eqref{eq:E0kernel}. Let \(c=\cosh\eta\). Near \(c=1\), \[ \frac{\sinh(\nu\eta)}{\sinh\eta} =\nu+\frac{\nu(\nu^2-1)}3(c-1)+O((c-1)^2). \] This proves regularity directly and supplies its second jet. Coincidence gives \(\delta F=\nu/(4\pi^2\alpha r^2)\); the angular derivative gives \(\delta j_\varphi=-2eq\,\delta F\). The time-split second derivative, conservation, and the improvement identity give the displayed stress components. For the branch identity set \[ x=e^{-\eta/\alpha+\ii\Delta\varphi},\qquad y=e^{-\eta/\alpha-\ii\Delta\varphi}, \] and fix their displayed logarithms. Define the entire function of the branch parameter \[ {\cal S}(b)=\frac{x^b}{1-x}+\frac{y^{1-b}}{1-y}. \] On \(01/2\), and \(\nu=1/(2\alpha)\). For \[ v=(\cos\theta,\sin\theta\,e^{\ii\chi}), \] take \(\mathsf E_0\) on \(v\) and Friedrichs on \(v^\perp\). Then \(C=c_0vv^\dagger\), and \[ \delta G= \frac{\sinh(\eta/(2\alpha))} {4\pi^2\alpha rr'\sinh\eta} w(\varphi)\overline{w(\varphi')}, \] \[ w(\varphi)=\cos\theta\,e^{\ii\varphi/2} +\sin\theta\,e^{\ii\chi}e^{-\ii\varphi/2}. \] Put \[ u=\sin2\theta\cos(\varphi-\chi),\qquad \varpi=\sin2\theta\sin(\varphi-\chi),\qquad k_\xi=6\xi-1. \] For one complex field, \[ \delta F=\frac{1+u}{8\pi^2\alpha^2r^2},\qquad \delta j_\varphi=-\frac{e\cos2\theta}{8\pi^2\alpha^2r^2}, \qquad \delta j_r=\delta j_t=\delta j_z=0, \] \begin{align*} \delta T^t{}_t=\delta T^z{}_z &=\frac{8\alpha^2k_\xi(1+u)-2k_\xi u-1} {48\pi^2\alpha^4r^4},\\ \delta T^r{}_r &=\frac{-4\alpha^2k_\xi(1+u)-2k_\xi u-1} {48\pi^2\alpha^4r^4},\\ \delta T^\varphi{}_\varphi &=\frac{4\alpha^2k_\xi(1+u)+1} {16\pi^2\alpha^4r^4},\\ \delta T^r{}_\varphi &=\alpha^2r^2\delta T^\varphi{}_r =-\frac{k_\xi\varpi}{8\pi^2\alpha^2r^3}. \end{align*} They satisfy the local conservation and trace identities. At conformal coupling the correction is independent of \((\theta,\chi)\) and equals the one-mode conformal correction. \end{theorem} \begin{proof} In \eqref{eq:CW}, take \(V=v\), \(H_W=0\), and use equal orders. This gives \(C=c_0vv^\dagger\). The kernel is therefore \eqref{eq:E0kernel} with its angular mode replaced by \(w\). The necessary angular identities are \[ |w|^2=1+u,\qquad |w'|^2=\frac{1-u}{4},\qquad w''=-\frac14w,\qquad w'\bar w-w\bar w'=\ii\cos2\theta. \] Together with the radial second jet in the proof of Theorem~\ref{thm:shift}, they give the displayed coincidence derivatives. At \(\xi=1/6\), substitution in \eqref{eq:pointsplit} gives \(-[48\pi^2\alpha^4r^4]^{-1}\diag(1,1,-3,1)\). For general \(\xi\), apply twice the improvement operator to \(\delta F\). In particular \[ \nabla_r\nabla_\varphi(r^{-2}(1+u)) =3r^{-3}\varpi, \] which gives the off-diagonal entry. The Ward identities also follow from the smooth bisolution property, independently of these component formulas. \end{proof} %====================================================================== \section{Scale-dependent diagonal nonnegative extensions} \label{sec:diagonal} %====================================================================== For \(0<\nu<1\), \(\lambda>0\), define \(M=M_\lambda\) by \[ (M/2)^{2\nu} =\lambda\frac{\Gamma(1+\nu)}{\Gamma(1-\nu)}. \] Then \[ c_\lambda(p)=c_0(\nu)\frac{p^{2\nu}}{p^{2\nu}+M^{2\nu}}. \] \begin{theorem}[Convergent observable integrals] \label{thm:diagonal} For one irregular deficient mode of charge \(q\), put \(\varepsilon=Mr\), \(K=K_\nu(x)\). Then \[ \delta F= \frac{\nu}{4\pi^2\alpha r^2}\Phi_\nu(\varepsilon), \qquad \Phi_\nu(\varepsilon)=\frac{c_0}{\nu} \int_0^\infty \frac{x^{1+2\nu}K_\nu(x)^2}{x^{2\nu}+\varepsilon^{2\nu}} \,\dd x, \] \[ \delta j_\varphi=-2eq\,\delta F,\qquad \delta j_t=\delta j_r=\delta j_z=0, \] and \[ \delta T^a{}_b= \frac1{4\pi^2\alpha r^4} \int_0^\infty x\,\dd x\, \frac{c_0x^{2\nu}}{x^{2\nu}+\varepsilon^{2\nu}} t^a{}_b(x), \] where \begin{align*} t^t{}_t=t^z{}_z &=x^2K^2+(4\xi-1) (\nu^2K^2+x^2K'^2+x^2K^2),\\ t^r{}_r &=x^2K'^2-(x^2+\nu^2)K^2+4\xi xKK',\\ t^\varphi{}_\varphi &=2\nu^2K^2-4\xi xKK' +(4\xi-1)(\nu^2K^2+x^2K'^2+x^2K^2). \end{align*} All off-diagonal entries vanish. Corrections from independent diagonal modes add. \end{theorem} \begin{proof} Differentiate \eqref{eq:dG} before coincidence. The Euclidean momentum angular average gives \([\partial_\tau\partial_{\tau'}]\mapsto p^2/2\). Radial differentiation and \[ K_\nu''=-K_\nu'/x+(1+\nu^2/x^2)K_\nu \] reduce the point-split tensor to the displayed expressions. The exponential and small-momentum bounds of Theorem~\ref{thm:nonnegative-state} justify differentiation. At fixed \(p\), the relevant stress integrand is proportional to \(r^{-2}t(pr)\). The Bessel equation gives \[ x(t^r{}_r)' -t^r{}_r=t^\varphi{}_\varphi, \] which is radial conservation before momentum integration. The trace follows from Lemma~\ref{lem:ward}. \end{proof} \begin{theorem}[Small-radius asymptotics with corrected remainders] \label{thm:small} As \(\varepsilon\downarrow0\), \(\Phi_\nu\to1\). For \(0<\nu<1/2\), \[ \Phi_\nu=1-A_\nu\varepsilon^{2\nu}+R_\nu(\varepsilon), \] where \[ A_\nu=\frac{c_0}{\nu} \frac{\sqrt\pi\,\Gamma(1-2\nu)\Gamma(1-\nu)} {4\Gamma(3/2-\nu)} \] and \[ R_\nu(\varepsilon)= \begin{cases} O(\varepsilon^{4\nu}),&0<\nu<1/3,\\ O(\varepsilon^{4/3}\log(1/\varepsilon)),&\nu=1/3,\\ O(\varepsilon^{2-2\nu}),&1/3<\nu<1/2. \end{cases} \] At \(\nu=1/2\), \[ \Phi_{1/2}(\varepsilon) =1-2\varepsilon e^{2\varepsilon}E_1(2\varepsilon) =1-2\varepsilon\log(1/\varepsilon)+O(\varepsilon). \] For \(1/2<\nu<1\), \[ \Phi_\nu=1-B_\nu\varepsilon^{2-2\nu} +O(\varepsilon^{2\nu}), \] where \[ B_\nu=\frac{c_0}{\nu} \frac{4^\nu\Gamma(\nu)^2}{4} \frac{\pi}{2\nu\sin(\pi(1-\nu)/\nu)}. \] The appropriately rescaled current and stress converge to their \(\mathsf E_0\) values. \end{theorem} \begin{proof} The bound by one in the coefficient and the integrability of \(xK_\nu^2\) give dominated convergence. For \(\nu<1/2\), expand once: \[ \frac1{1+(\varepsilon/x)^{2\nu}} =1-(\varepsilon/x)^{2\nu} +\frac{(\varepsilon/x)^{4\nu}} {1+(\varepsilon/x)^{2\nu}}. \] The first correction is the convergent Mellin integral defining \(A_\nu\). In the last term use \(K_\nu(x)^2=O(x^{-2\nu})\) near zero and split at \(x=\varepsilon,1\). The remaining power integral is governed by \(x^{1-6\nu}\), giving the three remainder regimes. For \(\nu>1/2\), subtract the leading small-\(x\) expression \[ K_\nu(x)^2=a_\nu x^{-2\nu}+O(1), \qquad a_\nu=2^{2\nu-2}\Gamma(\nu)^2. \] The leading deficit is obtained by \(x=\varepsilon y\) and \[ \int_0^\infty\frac{y^{1-2\nu}}{1+y^{2\nu}}\,\dd y =\frac{\pi}{2\nu\sin(\pi(1-\nu)/\nu)}. \] After this subtraction the remainder divided by \(\varepsilon^{2\nu}\) is integrable both at zero and infinity, which proves the stated bound. For \(\nu=1/2\), use \(K_{1/2}(x)^2=\pi e^{-2x}/(2x)\) and integrate directly. The stress limits follow by the same domination applied to the derivative integrands. \end{proof} \begin{theorem}[Long-range tail] \label{thm:large} As \(r\to\infty\) at fixed \(\lambda>0\), \[ \delta F= \frac{\nu}{4\pi^2\alpha(1+2\nu)\lambda\,r^{2+2\nu}} +O(r^{-2-4\nu}). \] The leading stress coefficients are \begin{align*} \delta T^r{}_r &= \frac{\nu[(1+2\nu)-2\xi(1+\nu)(3+2\nu)]} {2\pi^2\alpha\lambda(1+2\nu)(3+2\nu)\,r^{4+2\nu}} +O(r^{-4-4\nu}),\\ \delta T^t{}_t=\delta T^z{}_z &= \frac{\nu(1+\nu) [4\xi(1+\nu)(3+2\nu)-(2\nu^2+3\nu+2)]} {2\pi^2\alpha\lambda(1+2\nu)(3+2\nu)\,r^{4+2\nu}} +O(r^{-4-4\nu}),\\ \delta T^\varphi{}_\varphi &=-(3+2\nu)\,[\delta T^r{}_r]_{\mathrm{leading}} +O(r^{-4-4\nu}). \end{align*} Where a leading coefficient is nonzero, its relative correction is \(O((Mr)^{-2\nu})\). No relative assertion is made at a zero coefficient. \end{theorem} \begin{proof} For \(\varepsilon\to\infty\), \[ \frac{x^{2\nu}}{x^{2\nu}+\varepsilon^{2\nu}} =\varepsilon^{-2\nu}x^{2\nu} +O(\varepsilon^{-4\nu}x^{4\nu}). \] All resulting \(K\)-Bessel moments, including the derivative moments, converge. Evaluate them with Appendix~\ref{app:mellin} and the defining relation between \(M\) and \(\lambda\). This gives \(\delta F\) and \(T^r{}_r\). Conservation at power \(r^{-4-2\nu}\) gives the angular coefficient; the trace identity gives the time coefficient. The same moment estimate supplies the absolute remainder. \end{proof} The approach to Friedrichs is slow when \(\nu\) is small: relative to \(r^{-2}\) or \(r^{-4}\), the correction is of order \(r^{-2\nu}\). This is the power-law analogue of the logarithmic tail in \eqref{eq:logPV}. At fixed regular points the limits \(\lambda\downarrow0\) and \(\lambda\to\infty\) give \(\mathsf E_0\) and Friedrichs, respectively, by dominated convergence. %====================================================================== \section{Rotation-breaking nonnegative relations} \label{sec:matrix} %====================================================================== Every nonnegative boundary relation already has the convergent kernel \eqref{eq:dG}. This section makes its observable content and asymptotic qualifications explicit. \begin{proposition}[Matrix observables and radial current] \label{prop:matrix} For \(d=2\), let \(C=C_W\) and \(K_j=K_{\nu_j}(pr)\). Then \[ \delta F= \frac1{4\pi^2\alpha}\int_0^\infty p\,\dd p \sum_{jk}C_{jk}K_jK_k e^{\ii(q_j-q_k)\varphi}, \] \[ \delta j_\varphi= -\frac e{4\pi^2\alpha}\int_0^\infty p\,\dd p \sum_{jk}(q_j+q_k)C_{jk}K_jK_k e^{\ii(q_j-q_k)\varphi}. \] Writing the two labels as \(0,-1\), \[ \delta j_r= -\frac e{2\pi^2\alpha}\int_0^\infty p^2\,\dd p\, \Imag(C_{0,-1}e^{\ii\varphi}) [K_{\nu_0}'(pr)K_{\nu_{-1}}(pr) -K_{\nu_0}(pr)K_{\nu_{-1}}'(pr)]. \] Also \(j_t=j_z=0\). Stress is obtained by applying \eqref{eq:pointsplit} to \(2\Real\delta G_W\). All these operations converge locally and satisfy \[ \nabla_\mu T^\mu{}_\nu=0,\qquad T^\mu{}_\mu=(6\xi-1)\Box F, \] \[ \frac1r\partial_r(rj^r)+\partial_\varphi j^\varphi=0. \] Generically, unequal deficient orders and off-diagonal boundary data give a nonzero radial current and a nonzero shear component. At \(\beta=1/2\), equal orders force the displayed radial current to vanish for every such relation. Other components may still have angular dependence. \end{proposition} \begin{proof} Apply the derivative difference to \eqref{eq:dG}. An angular derivative gives \(\ii(q_j+q_k)\). For a radial derivative combine the \(j,k\) and \(k,j\) terms using Hermiticity of \(C\); this gives the imaginary part shown. At equal orders the radial Wronskian factor is zero. All interchanges are justified by Theorem~\ref{thm:nonnegative-state}. The Ward identities are Lemma~\ref{lem:ward}. \end{proof} Changing the finite-chart entry \(H_{0,-1}\) to \(e^{\ii\gamma}H_{0,-1}\) changes the same phase in \(C_{0,-1}\); it is equivalent to \(\varphi\mapsto\varphi+\gamma\). Statements that every component must depend on angle, or that every non-diagonal relation must carry radial current, would ignore the equal-order exceptions. \begin{proposition}[Finite positive matrices: large and small radius] \label{prop:matrix-asymptotics} Suppose \(W=\CC^2\) and \(H>0\). Then \[ \delta F\sim\frac1{4\pi^2\alpha} \sum_{jk}(H^{-1})_{jk} \frac{\nu_j\nu_k}{1+\nu_j+\nu_k} \frac{e^{\ii(q_j-q_k)\varphi}} {r^{2+\nu_j+\nu_k}} \quad(r\to\infty), \] with a termwise asymptotic expansion. The leading azimuthal-current terms are obtained by weighting the summands by \(-e(q_j+q_k)\). The off-diagonal power is \(r^{-2-1/\alpha}\). For every full finite \(H\ge0\), the small-radius leading rescaled observables are those of \((\mathsf E_0,\mathsf E_0)\): \(r^2\delta F\), \(r^2\delta j_\varphi\), \(r^3\delta j_r\), and \(r^4\delta T\) in an orthonormal frame have the corresponding limits. \end{proposition} \begin{proof} At small \(p\), positive invertibility gives \[ C(p)=\frac12E^{-1}H^{-1}E^{-1} +\hbox{higher powers of }p. \] Set \(x=pr\) and use \[ \int_0^\infty x^{1+\nu_j+\nu_k} K_{\nu_j}(x)K_{\nu_k}(x)\,\dd x = \frac{2^{\nu_j+\nu_k-1} \Gamma(1+\nu_j+\nu_k) \Gamma(1+\nu_j)\Gamma(1+\nu_k)} {\Gamma(2+\nu_j+\nu_k)}. \] Gamma reflection and recurrence give the displayed coefficient. A low/high momentum split, the Neumann expansion at small momentum, and exponential decay at high momentum justify the asymptotic integration and the derivative expansions. At large \(p\), \[ C(p)=(S+2EHE)^{-1}\longrightarrow S^{-1} \] for every full finite \(H\). After \(p=x/r\), Lemma~\ref{lem:Cbound} supplies an integrable dominating function for each rescaled observable, including its required derivatives. Dominated convergence gives the two-mode \(\mathsf E_0\) limits. \end{proof} \begin{remark}[Boundary-chart and rank qualifications] The large-radius formula uses \(H^{-1}\) and is not asserted for singular \(H\). The integral representation remains valid there. The small-radius two-mode limit is a statement for a full \emph{finite} chart. It is not valid for every boundary chart: the scale-invariant line family already provides a counterexample, because it is irregular in only one boundary direction. For arbitrary boundary relations use \eqref{eq:CW}; no unsupported universal endpoint interpolation is imposed. The limits just stated are rescaled limits. They do not assert that divergent unscaled quantities converge to finite numbers. \end{remark} %====================================================================== \section{Formula comparisons with the literature} \label{sec:literature} %====================================================================== The literature comparisons below concern stated formulas and translated conventions. They do not assert an exhaustive priority search or verification of an entire paper. The source access grades are defined in Section~\ref{sec:evidence}. \subsection{Friedrichs flux formulas} Sriramkumar's equations (2.12)--(2.16) \cite{Sriramkumar2001} use \((+,-,-,-)\), radius \(\rho\), and fractional flux \(\gamma\). Set \(\rho=r,\gamma=\beta\). Its lower-index current is our Theorem~\ref{thm:current}. For its minimally coupled complex tensor, the coefficients are \[ {\cal A}=-C_\beta,\qquad {\cal B}=-20P_\beta, \] in \[ \frac1{720\pi^2r^4} \left\{{\cal A}\diag(1,1,-3,1) -{\cal B}\diag(1,-1/2,3/2,1)\right\}. \] With the covariant stress normalization held fixed, changing the metric sign reverses mixed components. This reproduces \eqref{eq:T-complex} at \(\xi=0\). Its periodic-gauge two-point expression, equation (2.4), is twice our equal-time kernel after the gauge phase is included. Spinelly--Bezerra de Mello \cite{SBdM2003}, equations (30) and (36), agree with \(F\) after the flux-even identification \(\gamma=\beta\). Their equation (37), as printed in the inspected version, is \[ \frac{1-\alpha^2-6\gamma(1-\gamma)} {48\pi^2\alpha^4r^2}. \] That display carries an extra \(\alpha^{-2}\) relative to (30) and (36). It does not agree with \(F\). The earlier sentence in this ledger, which copied (37) with denominator \(48\pi^2\alpha^2r^2\) and called it an agreement, was our transcription error. I-B-VP-4 records the same printed discrepancy as SB2. Their equations (54)--(56), in their displayed stress normalization, have half our complex multiplicity; their coupling term agrees after that translation. \paragraph{Printed discrepancy retained.} Their equation (55) prints \(\gamma(1-\gamma^2)\), rather than \(\gamma^2(1-\gamma)^2\), in the flux-dependent conformal term. The former is not symmetric under \(\gamma\mapsto1-\gamma\): it is \(0.273\) at \(0.3\) and \(0.357\) at \(0.7\). The latter is the polynomial obtained here and in Sriramkumar's equation (2.15). We identify an apparent typographical discrepancy in that displayed formula, not a demonstrated error in every result of the paper. No published erratum is asserted. Bezerra de Mello--Bezerra--Saharian--Harutyunyan \cite{BdMBSH2015}, equation (4.6), gives the zero-thickness massless current as an image sum plus an integral. For \(D=3\), \(q=1/\alpha\), their physical component \(-j_2/r\), with \(j_2=j_\varphi/\alpha\), equals \[ \frac{e q^3\beta(1-\beta)(1-2\beta)}{12\pi^2r^3} \] in their signature convention. Their equation (4.7) is the \(q=1\) limit. The reported numerical comparisons are retained in Appendix~\ref{app:ABreports}; they are not needed to prove the polynomial. Guimar\~aes \cite{Guimaraes1995} gives another explicit zero-temperature comparison. With \(B=\alpha,\gamma=\beta,\rho=r\), \[ \omega_2=\frac{P_\beta}{24\pi^2},\qquad \omega_4=\frac{C_\beta+10P_\beta}{720\pi^2}. \] Its equations (19) and (30) give \(F=\omega_2/(2r^2)\) and, in order \((t,z,r,\varphi)\), \[ \frac{\omega_4-\omega_2/3}{r^4}\diag(1,1,1,-3) +\frac{4(\xi-1/6)\omega_2}{r^4} \diag(1,1,-1/2,3/2). \] Reordering gives \eqref{eq:T-complex}. This comparison uses the stated overlap of its integral presentation, \(B>1/2\); our all-angle result is proved independently. \subsection{Logarithmic boundary conditions and AKO} In Allen--Kay--Ottewill \cite{AKO1996}, their scattering length \(R\) and pole \(q\) map to our \(L\) and \(\kappa_L\): \[ q=2e^{-\gamma_E}/R. \] Their equations (3.17), (4.3), and Appendix A give the \(\nu=0\) Krein coefficient and principal-value integral. Equation (4.8) agrees with the leading term of \eqref{eq:logPV}. Their equations (4.13)--(4.15) are the one-real-field \(\nu=0\) versions of our diagonal stress integrands. The complex tensor here is twice those expressions. Their equation (4.17) is the same radial conservation identity. AKO fit idealized exterior formulas to a smooth core. The literal zero-thickness logarithmic extension has an unstable band and no stationary finite-moment state by Theorem~\ref{thm:nogo}. The principal-value value is therefore not silently called its ground-state expectation. \subsection{Filgueiras--Moraes and other contexts} For \(\beta=1/2\), their von Neumann boundary parameter \(\eta_0\) maps through the small-radius coefficients to \[ E_\perp= -\kappa_{\mathrm{ref}} \left[ \frac{\cos(\eta_0/2+\pi/(8\alpha))} {\cos(\eta_0/2-\pi/(8\alpha))} \right]^{2\alpha}, \] in the energy units used in their equation (10) \cite{FM2007}. That equation also includes the free longitudinal kinetic term. The ratio must select the real bound-state branch; in our boundary convention this is \(\lambda<0\). The reported eight-point comparison is retained below. The inverse-square analysis of \cite{DR2017} supplies broader operator context. The fermionic references \cite{SousaGerbert1989,BordagVoropaev1993,BdFS1999} concern different fields and boundary structures, not confirmations of the scalar tensor or current computed here. The SAE source reports finding no direct scalar computation of all three observables for these zero-thickness \(\mathsf E_0,\mathsf E_\lambda,U(2)\) choices. That is retained as a limited search report, not a theorem of novelty or an exhaustive literature claim. %====================================================================== \section{Supported transfer does not transfer a state} \label{sec:transfer} %====================================================================== For an angularly small chart \[ U\subset\{r>r_U,\ \varphi\in I\},\qquad r_U>0,\quad |I|<\pi, \] with a fixed lift, define \[ T_{\mathrm{arr}}(x,x') =\min\left( \{\rho_0\}\cup \{R_k:k\ne0,\ |\vartheta-2\pi\alpha k|\le\pi\} \right). \] The supplied Friedrichs transfer theorem gives \[ T_{\mathrm{arr}}(x,x')\ge L_U=2r_U\sin(\pi\alpha/2)>0 \] \cite[\SourceLabel{lem:window}, \SourceLabel{thm:transfer}]{CheungIBG}. \begin{assumption}[Conjunctive supported-transfer contract] A comparison with a completion Green operation requires: \begin{enumerate}[label=\textup{(T\arabic*)}] \item the angularly small chart, signature ledger, and parallel frame; \item a declared compact regular reserve containing all retained shadows, initial-data regions, and intermediate source supports; \item for each sign \(\sigma=\pm1\), the strict inequality \[ \sigma(t-t')0\) & Excluded as stationary finite-moment state & One negative transverse eigenvalue and an unstable full band; Theorems~\ref{thm:spectrum} and \ref{thm:nogo}. \\ Entire transverse bound channel removed & Projected construction; not local Hadamard & Extra logarithmic singularity and divergent subtracted square; Proposition~\ref{prop:projections}(a). Projected CCR. \\ Only negative full spectrum removed & Projected Wick square evaluated; stress prescription not supplied & Principal-value Wick square; nonsmooth direction-dependent remainder and boost breaking; Proposition~\ref{prop:projections}(b). Not the original local field. \\ \(\mathsf E_0\) in one deficient mode & Constructed; closed forms & Theorem~\ref{thm:shift}, with the legitimate shifted branch. Replaces predecessor ``Open'' status. \\ \((\mathsf E_0,\mathsf E_0)\) & Constructed; closed forms & Additive formula in Theorem~\ref{thm:shift}. \\ Diagonal \(\lambda_j\in[0,\infty]\) & Constructed; convergent integrals & Theorems~\ref{thm:nonnegative-state}, \ref{thm:diagonal}--\ref{thm:large}. Finite-\(\lambda\) endpoint and asymptotic qualifications apply. \\ Any diagonal combination with a negative \(\lambda_j\) & Excluded as stationary finite-moment state & Theorems~\ref{thm:spectrum} and \ref{thm:nogo}. \\ Half-holonomy \(\mathbb{CP}^1\) line family & Constructed; closed forms & Theorem~\ref{thm:line}. Rotation breaking is allowed; radial current vanishes. \\ Full finite \(U(2)\) chart, \(H\ge0\) & Constructed; integral evaluation & Theorems~\ref{thm:krein}, \ref{thm:nonnegative-state}; Propositions~\ref{prop:matrix}, \ref{prop:matrix-asymptotics}. Large-radius formula needs \(H>0\). \\ Nonnegative boundary charts & Constructed; integral evaluation & Compressed coefficient \eqref{eq:CW}. The full finite-chart two-mode ultraviolet limit is not imposed on these relations. \\ Any relation with \(H_W\not\ge0\) & Excluded as stationary finite-moment state & Theorems~\ref{thm:spectrum} and \ref{thm:nogo}. \\ Massive extensions, including \(\mathsf L_L\) with \(m\ge\kappa_L\) & \textbf{Open} here & A massive baseline and covariance analysis are not supplied. At equality, nonnegativity is insufficient because of the longitudinal threshold infrared divergence. \\ Non-product or longitudinal-momentum-dependent extensions & \textbf{Open} & Boundary data are infinite dimensional and require an operator-valued relation and Weyl function. Generic choices need not preserve longitudinal boosts. \\ Geometrically twisted tensor sectors, spinors, gauge fields & \textbf{Open} & No spin structure, constrained gauge domain, or multiplet state is constructed. \\ Thermal and other non-ground states & \textbf{Open} & Not constructed. Their infrared conditions are separate from the ground-state estimates. \\ Apex-supported extensions of the observables & \textbf{Open} & No distributional completion or apex counterterm class is chosen. \\ Global microlocal spectrum analysis & \textbf{Open} & Local Hadamard form is proved; global diffractive propagation is not classified here. \\ \end{longtable} \endgroup \begin{remark}[Massive threshold warning] If a transverse bound eigenvalue is \(-\kappa^2\) and \(m=\kappa\), its longitudinal dispersion is \(|p_z|\). A test with nonzero zero-momentum projection then gives \[ \int_{|p_z|<1} \frac{|\widehat v(p_z)|^2}{2|p_z|}\,\dd p_z=\infty. \] Thus nonnegativity at a massive threshold does not guarantee the ordinary covariance on every compact test. \end{remark} \paragraph{Remaining mathematical questions.} The unclosed items are: \begin{enumerate}[label=\textbf{V\arabic*.}] \item completion-supported Wick-square, current, and stress extensions and their counterterms; \item global microlocal propagation for the selected states; \item massive states and observables, including threshold restrictions; \item induced Maxwell fields and broader source-by-source literature audits; \item a smoothed-core or distributional apex curvature-coupling model selecting an effective extension; \item general non-product extensions; \item spinors and constrained gauge systems; \item thermal and other non-ground states; \item simpler closed stress formulas for generic matrix relations. \end{enumerate} The first and fifth questions are not answered by the vanishing regular-part curvature anomaly. On the declared regular manifold, \(\xi\) changes the stress expression but not the fixed scalar equation or chosen extension. %====================================================================== \section{Non-identifications and conclusion} \label{sec:conclusion} %====================================================================== The cone, deficit, extension, spectral state prescription, \(\xi\), and \(\beta\) are declared inputs. None is derived from an Enter record. No physical preference among the nonnegative extensions is claimed, and no zero-thickness relation is asserted to model a particular real string core. A symmetric, conserved, traceless tensor need not source the prescribed background. For example, in a regular flat chart \[ t_{\mu\nu}=\diag(3e_0,e_0,e_0,e_0),\qquad e_0>0, \] is constant, conserved, and traceless, while the local Einstein tensor is zero. Equivalently a constant null covector gives the witness \(k_\mu k_\nu\). This is the I-B \SourceLabel{prop:not-source} separation \cite{CheungIB,CheungXAST}. We do not insert the expectation values into Einstein or Maxwell equations, compute a sourced magnetic field, solve backreaction, or claim spacetime emergence. Negative energy is a property of the stated field, state, and normalization, not a certification of a programme direction. The observables are smooth tensor or scalar fields, hence distributions, on \(M\). Their generic \(r^{-2}\), \(r^{-3}\), and \(r^{-4}\) behaviours are not ordinary locally integrable completion data with transverse measure \(r\,\dd r\). Distributional extensions may exist, but require a prescription and may have apex-supported ambiguities. For finite nonnegative charts the leading diagonal ultraviolet powers are the scale-invariant ones; singular one-endpoint boundary data must not be confused with a universal stronger simultaneous diagonal power. Local point splitting does not establish an interacting algebra, time-ordered product, quantum Ward identity, BRST closure, graviton or ghost sector, or apex gauge condition. A finite reporting instrument is not a continuum covariance, and finite numerical tests are not a coverage theorem. No companion status record is updated. In particular: \begin{itemize} \item no new closure of O.1 is asserted outside any independently declared scoped result; \item any \NOTCLOSED{} status outside such a scope remains unchanged; \item the frozen XVI-N insertion is not discharged; \item XV-B's downstream quantum-admission exit is not opened; \item \bsr{} is not promoted; \item no XVI-R or frozen-tree release is asserted. \end{itemize} These are non-transfer boundaries, not premises of the scalar proofs \cite{CheungO1ST,CheungXVIO,CheungXVIN,DocumentaryBoundary}. The supplied XIV-H/I/J and XV-B excerpts distinguish constructed internal kernels or off-characteristic references from an unconstructed complete characteristic quantum map. Nothing here identifies our scalar extension problem with that graph problem or supplies its missing admission. A registered technical manuscript named XV-B is not declared nonexistent; ``not opened'' refers only to the stated downstream admission gate. \paragraph{Conclusion.} Within the declared massless spatial-product class, every extension has a precise positivity and state status. Nonnegative relations have infrared-finite spectral covariances, local Hadamard form, and convergent regular-part observable formulas. Scale-invariant cases admit explicit closed forms. Unstable relations are excluded by a local compact-test stationarity contradiction, while their projected substitutes are identified as different, nonlocal constructions. The familiar Friedrichs results, the complex first-jet current, and all retained computational reports are presented with separate analytical and evidential status. No completion, dynamics, or programme closure follows from them. %====================================================================== \appendix \section{Integral identities and asymptotic calculations} \label{app:mellin} %====================================================================== For real \(\mu,\rho\) and \(s>|\mu|+|\rho|\), \begin{equation} \int_0^\infty x^{s-1}K_\mu(x)K_\rho(x)\,\dd x = \frac{2^{s-3}}{\Gamma(s)} \Gamma\left(\frac{s+\mu+\rho}{2}\right) \Gamma\left(\frac{s+\mu-\rho}{2}\right) \Gamma\left(\frac{s-\mu+\rho}{2}\right) \Gamma\left(\frac{s-\mu-\rho}{2}\right). \label{eq:mellin} \end{equation} One derivation inserts the Schwinger representations for the two \(K\)-functions, integrates the Gaussian power of \(x\), then sets the two positive Schwinger parameters equal to \(tv,t(1-v)\). The \(t\)-integral is a gamma integral and the \(v\)-integral is a beta integral. The stated condition justifies absolute convergence. This also provides a direct proof of the identity used in the asymptotic sections; see \cite{Watson,GR,DLMF} for the standard special-function framework. In particular, \[ \int_0^\infty xK_\nu(x)^2\,\dd x =\frac{\pi\nu}{2\sin\pi\nu},\qquad 0<\nu<1, \] and at \(\nu=0\) the continuous limit is \(1/2\). Differentiation in \(s\) at \(s=2,\mu=\rho=0\) gives \[ \int_0^\infty xK_0(x)^2\log x\,\dd x =\frac12(\log2-1-\gamma_E). \] For derivative moments, integration by parts gives, whenever the endpoint terms vanish, \[ \int_0^\infty x^{a+1}K_\nu K_\nu'\,\dd x =-\frac{a+1}{2}\int_0^\infty x^aK_\nu^2\,\dd x. \] The Bessel equation then reduces \(K'^2\) moments to the same Mellin family. These operations yield the coefficients of Theorem~\ref{thm:large} without an appeal to a symbolic run. \paragraph{Boundary-data normalization.} The identities \[ d_\nu e_\nu=\frac1{2\nu},\qquad \Delta_\nu=\nu s_\nu d_\nu/e_\nu,\qquad 2e_\nu^2\Delta_\nu=s_\nu \] are the normalization checks behind \eqref{eq:Cfinite}. They prevent a second angular normalization factor from being inserted into \eqref{eq:dG}. That equation uses unnormalized exponentials in its displayed sum; \eqref{eq:resolvent} uses the normalized \(\psi_j\)'s. %====================================================================== \section{Retained Friedrichs computational reports} \label{app:FReports} %====================================================================== Every execution statement in this and the following appendices is \SUPPLIED{} and \UNCERTIFIED{}. The original scripts and raw logs were not included in the supplied text. No run is asserted here. The named files are \[ \SourceLabel{code/symbolic.py},\quad \SourceLabel{code/numerics.py},\quad \SourceLabel{code/numerics_v2.py}, \] \[ \SourceLabel{logs/symbolic.json},\quad \SourceLabel{logs/numerics.json},\quad \SourceLabel{logs/numerics.log},\quad \SourceLabel{logs/numerics_v2.json}. \] The sources report SymPy~1.14 and mpmath~1.3, initially 30-decimal-digit work and subsequently 40 digits, with parameters and coordinates entered as IEEE doubles. These are inherited provenance descriptions, not available execution artifacts. To distinguish duplicated source labels, this appendix calls the Friedrichs symbolic suite F-S1--F-S4. The SAE suite retains SAE-S1--SAE-S4 below. \subsection{Symbolic reports F-S1--F-S4} \begin{description}[style=nextline] \item[F-S1] The geometric mode sum and time-split expansions reportedly matched the displayed Wick-square and stress formulas. The log keys \SourceLabel{phi2_beta0_matches_lit}, \SourceLabel{Ttt_beta0_xi16_matches_lit}, and \SourceLabel{Ttt_beta0_general_xi_matches_FS} were reported \texttt{true}. The corresponding identities have the independent proof in Lemma~\ref{lem:expansion}. \item[F-S2] For \(N=2,\ldots,6\), \(\alpha=1/N\), and \(\xi=0,1/6,1/4\), the full Cartesian image tensor reportedly matched exactly in 15 cases: 60 diagonal and 180 off-diagonal entries. All off-diagonal entries and the coincidence on-shell residual were reported zero. As \(1440\pi^2r^4(T^t{}_t,T^r{}_r, T^\varphi{}_\varphi,T^z{}_z)\), the retained examples are \[ \begin{array}{c|c|c} N&\xi&\text{reported vector}\\ \hline 2&1/6&(15,15,-45,15)\\ 3&1/6&(80,80,-240,80)\\ 5&0&(144,864,-2592,144)\\ 5&1/4&(864,504,-1512,864) \end{array} \] They also follow directly from \eqref{eq:T-real}. \item[F-S3] The \(\csc^2\) and \(\csc^4\) identities were reportedly checked exactly for \(N=2,\ldots,8\). Proposition~\ref{prop:quotient} proves the stronger all-\(N\) identities. \item[F-S4] Conservation, trace, and \((g\Box-\nabla\nabla)r^{-2}\) were reportedly checked symbolically. They are proved analytically in Lemma~\ref{lem:ward} and Theorem~\ref{thm:friedrichs}. \end{description} \subsection{N1: off-diagonal contour comparisons} For each of \[ \alpha\in\left\{ 1-\frac1{2\pi},\ 1-\frac{\sqrt2}{2\pi},\ \frac34,\ \frac1{\sqrt2}\right\},\qquad\beta=0, \] and for \[ (\alpha,\beta)= \left(1-\frac1{2\pi},0.3\right),\quad(0.4,0.3), \] the source reports three pairs and a maximum absolute difference \(9.5\times10^{-19}\) between contour and mode values of order \(10^{-2}\) to \(1\). The source attributed this scale to double-precision chart inputs. Without the raw paired values and error propagation this is a reported diagnostic, not a verified cause. \subsection{N2: coincidence Wick square and revised diagnostics} The source reports relative differences at most \(1.1\times10^{-31}\) for a set containing \[ \alpha= 1-\frac1{2\pi},\ 1-\frac{\sqrt2}{2\pi},\ \frac1{\sqrt3},\ \frac34 \] and two twisted cases with \(\beta=0.3\). For a second set containing \[ \alpha=\frac12,\frac13,\frac14,0.3,\qquad (\alpha,\beta)=(0.4,0.7), \] the retained earlier residual is \(8.6\times10^{-17}\). The AB record reports that image terms had been cast to double complex. It further reports an omitted diffraction correction at a double approximation to \(1/3\). After removing the cast, it reports N2 discrepancies at most \(4\times10^{-41}\), except \(\alpha=\texttt{double}(1/3)\), for which the omitted term reportedly leaves \(5.8\times10^{-17}\). It reports N2/N3 reproduction at at most \(6\times10^{-39}\). All of these numbers and explanations are retained; none replaces a raw log or an error enclosure. Two supplied values of \(r^2F_0\) are \[ 8.7471068907447577\times10^{-4}\quad(\mu=1), \] \[ 1.6886863940389629\times10^{-2}\quad(\alpha=1/3). \] The exact second expression is \(8/(48\pi^2)\). Pure diffraction at \(\alpha>1/2\) and compatible finite images at exact quotient parameters are analytical conclusions, not consequences of those decimals. \subsection{N3: full tensor residuals} The source describes symbolic second differentiation followed by quadrature, for \[ \alpha\in\left\{ 1-\frac1{2\pi},\ 1-\frac{\sqrt2}{2\pi},\ \frac1{\sqrt3}\right\},\quad\beta=0, \] and \((\alpha,\beta)=(1-1/(2\pi),0.3)\), with \(\xi=0,1/6,1/4\) and \(r=1,2\), giving 24 runs. \begin{center} \scriptsize \setlength{\tabcolsep}{3pt} \begin{tabular}{@{}llL{.43\linewidth}ll@{}} \toprule \(\alpha,\beta\)&\(\xi\)& \(1440\pi^2r^4(T^t{}_t,T^r{}_r,T^\varphi{}_\varphi,T^z{}_z)\)& rel.\ diff.&\(\mathrm{tr}\,T\,r^4\)\\ \midrule \(1-1/(2\pi),0\)&\(1/6\)& \((1.000488680294,\cdot,-3.001466040882,\cdot)\)& \(4.6\,10^{-31}\)&\(-2.5\,10^{-34}\)\\ \(1-1/(2\pi),0\)&\(0\)& \((-7.28723785,5.14435194,-15.4330558,\cdot)\)& \(1.8\,10^{-31}\)&\(-1.74942\,10^{-3}\)\\ \(1-\sqrt2/(2\pi),0\)&\(1/6\)& \((1.773131450488,\cdot,-5.319394351463,\cdot)\)& \(1.9\,10^{-31}\)&\(-1.9\,10^{-34}\)\\ \(1/\sqrt3,0\)&\(1/6\)& \((8,8,-24,8)\)&\(2.3\,10^{-31}\)&\(-3.9\,10^{-34}\)\\ \(1/\sqrt3,0\)&\(1/4\)& \((28,-2,6,28)\)&\(9.8\,10^{-32}\)&\(4.22172\,10^{-3}\)\\ \(1-1/(2\pi),0.3\)&\(1/6\)& \((-3.292315687470,\cdot,9.876947062410,\cdot)\)& \(6.2\,10^{-31}\)&\(1.9\,10^{-34}\)\\ \bottomrule \end{tabular} \end{center} A dot means equality to the time component. The reported maxima are: \[ \begin{array}{l|l} \text{relative tensor difference}&6.2\times10^{-31}\\ \text{off-diagonal magnitude}&1.4\times10^{-41}\\ \text{dimensionless radial conservation residual}&1.5\times10^{-33}\\ \text{on-shell coincidence residual}&7.7\times10^{-34} \end{array} \] The conformal trace was reported of order \(10^{-34}\), and the \(r=1,2\) values reportedly obeyed \(r^{-4}\) scaling. The original prose stated the untwisted real trace for every nonconformal run. That is not the correct comparator for a twisted complex run. The latter must use \[ \left(\xi-\frac16\right)\frac{P_\beta}{2\pi^2r^4}. \] The retained untwisted example \(-1.74942137814895\times10^{-3}\) at \(\alpha=1-1/(2\pi)\), \(\xi=0\), is compatible with the real trace formula. \subsection{N4: small-deficit slopes} At \(r=1,\beta=0\), with stress at \(\xi=1/6\): \begin{center} \small \begin{tabular}{@{}ccc@{}} \toprule \(\mu\)&\(F_0/\mu\)&\(T^t{}_t/\mu\)\\ \midrule \(10^{-1}\)&\(6.88294860341\times10^{-4}\)& \(4.66344397439\times10^{-5}\)\\ \(10^{-2}\)&\(6.73514437791\times10^{-4}\)& \(4.49725955867\times10^{-5}\)\\ \(10^{-3}\)&\(6.72067407377\times10^{-4}\)& \(4.48116263845\times10^{-5}\)\\ \midrule exact limit&\(1/(48\pi^3)\)&\(1/(720\pi^3)\)\\ reported decimal&\(6.71906967358\times10^{-4}\)& \(4.47937978239\times10^{-5}\)\\ \bottomrule \end{tabular} \end{center} This is compatible with the proved pointwise slopes; it is not a uniform-limit proof. %====================================================================== \section{Retained AB checks A1--A5} \label{app:ABreports} %====================================================================== The named additional files are \[ \SourceLabel{code/ab.py},\quad \SourceLabel{code/ab_common.py},\quad \SourceLabel{logs/ab.json}. \] The source reports 40-digit work at shared IEEE-double inputs. \subsection{A1: exact phased images} For \(N=2,\ldots,6\), \(\alpha=1/N\), and \(\beta=j/N\), \(j=1,\ldots,N-1\), the source reports 15 exact symbolic cases. The reported quantities are the Wick square, all current components, and conformal time stress. The retained scaled examples are \[ \left(48\pi^2r^2F,\ 12\pi^2r^2j_\varphi/e,\ 720\pi^2r^4T^t{}_t\right): \] \[ \begin{array}{c|c|c} N&\beta&\text{reported triple}\\ \hline 3&1/3&(-4,-2/3,-40)\\ 4&1/4&(-3,-3/2,-15)\\ 5&1/5&(0,-12/5,144)\\ 6&1/6&(5,-10/3,545)\\ 6&1/2&(-19,0,-1135) \end{array} \] The conjugate pairs reportedly have equal Wick squares and stress and opposite current. The quotient identity and the closed formulas provide independent analytical checks. \subsection{A2: irrational-angle contour checks} The reported parameter pairs are \begin{itemize} \item \((1-1/(2\pi),0.3)\), \((1-1/(2\pi),0.8)\); \item \((1-\sqrt2/(2\pi),5^{-1/2})\); \item \((3^{-1/2},0.3)\), \((3^{-1/2},0.45)\), with reported \(\mu=2.656\); \item \((0.3,0.37)\), \((\pi^{-1},0.7)\), where nontrivial images occur. \end{itemize} The reported maximum relative differences are \(3.1\times10^{-41}\) for \(F\) and \(2.1\times10^{-40}\) for \(j_\varphi\). The other current components are reported zero. Two retained current values are \[ r^2j_\varphi/e =-1.003151077025388291759762\times10^{-3} \] at \((1-1/(2\pi),0.3)\), and \[ r^2j_\varphi/e=+7.000000000000000\times10^{-3} \] at \((\pi^{-1},0.7)\). At the exact latter parameters the polynomial value is \(0.007\). The full twisted tensor is the N3 case above; the AB source reports its revised reproduction at \(6\times10^{-39}\). \subsection{A3: symmetries} The source reports \[ |F(\beta)-F(1-\beta)|,\quad |j(\beta)+j(1-\beta)|\le7\times10^{-19}, \] and exact zero at \(\beta=1/2\). It attributes the residual to forming \(1-\beta\) in double precision. It also reports off-diagonal mode periodicity to \(3.6\times10^{-17}\), attributed to the double rounding of \(1.3\). These attributions remain unverified diagnostics. \subsection{A4--A5: external formula comparisons} The source reports current comparisons with \cite{BdMBSH2015} at eight pairs whose approximate \(q=1/\alpha\) values are \[ 1.189,\ 1.290,\ 1.732,\ 1.732,\ 3.333,\ \pi,\ 3.846,\ 1.923. \] It reports relative differences at most \(1.2\times10^{-39}\), except \(2.9\times10^{-31}\) at \(q=1.923\), attributed to quadrature near a small denominator. The \(q=1\) limit at \(\beta=0.1,0.3,0.45,0.7\) reportedly agrees to \(5\times10^{-38}\). Against \cite{Sriramkumar2001}, it reports \(2\times10^{-43}\) for the signature-translated minimal tensor and \(1.2\times10^{-44}\) for the current. It also reports exact agreement with the coupling term of \cite{SBdM2003} and agreement with its conformal term only after replacing the printed flux factor as discussed in Section~\ref{sec:literature}. That reported agreement does not cover the printed equation (37), which carries the extra \(\alpha^{-2}\) corrected above. The source states that full texts were stored in \SourceLabel{lit/}. That directory was not supplied here. %====================================================================== \section{Retained SAE checks} \label{app:SAEreports} %====================================================================== The named files are \[ \SourceLabel{code/sae_common.py},\ \SourceLabel{code/sae_sym.py},\ \SourceLabel{code/sae_num.py},\ \SourceLabel{code/sae_lead.py},\ \SourceLabel{code/sae_fm.py}, \] with \SourceLabel{logs/sae_*.json}, including \SourceLabel{logs/sae_lead.json} and \SourceLabel{logs/sae_fm.json}. The source reports 30-digit work unless stated otherwise and a double-exponential quadrature rule tested on \(\int xK_\nu^2\) to \(4\times10^{-29}\). The source numbering skips C6. No missing C6 result is invented. \begin{description}[style=nextline] \item[SAE-S1] The exact \(\mathsf E_0\) kernel reportedly gives the shifted Wick square, current, and four diagonal stress formulas with residual exactly zero in both modes. Conservation, trace, current conservation, and diagonality were reported exact. Theorem~\ref{thm:shift} supplies the independent proof. \item[SAE-S2] For the half-holonomy line family, conservation, trace, and current conservation were reported exactly zero. The compact forms reportedly agreed at rational sample points to \(8\times10^{-174}\), and \(\theta=0\) reproduced the single-mode formulas. Sampled high precision is not an exact identity certificate; Theorem~\ref{thm:line} gives the analytical calculation. \item[SAE-S3] The Krein stress integrands were reported to satisfy pointwise conservation numerically below \(10^{-166}\) (the abstract also reported \(10^{-167}\)). Their \(\nu=0\) translation to AKO (4.13)--(4.15) was reported with symbolic residual zero. The Bessel identity in Theorem~\ref{thm:diagonal} proves the conservation relation. \item[SAE-S4] A truncated comparison of the branch series \({\cal S}(\beta\pm1)\) was reported with residual \(10^{-63}\). It is a truncation report, separate from the exact branch identity in Theorem~\ref{thm:shift}. \item[C1] Ten Krein-integral versus \(\mathsf E_0\) closed-form cases reportedly included irrational \(\alpha,\beta\), mode \(0\), mode \(-1\), and both, with \(\xi=0,1/6,0.25,-0.1\). The maximum relative deviation was \(5.9\times10^{-30}\). Reported off-diagonal stress was at most \(2\times10^{-36}\), radial current at most \(10^{-65}\), and differentiated conservation, trace, and current residuals at most \(7\times10^{-31}\). \item[C2] An eigenfunction expansion \[ \int_0^\infty k\,\dd k\, \frac{f_k(r)f_k(r')-J_\nu(kr)J_\nu(kr')}{k^2+p^2}, \qquad f_k\propto t(k)J_\nu+J_{-\nu}, \] was reportedly compared with the Krein term at \(r\ne r'\). Four \(\mathsf E_0\) or \(\lambda>0\) cases gave \(1.1\times10^{-15}\) at 20 digits, described as limited by an oscillatory tail. \item[C3] For matrix relations the reports give: Hermiticity residual \(3\times10^{-47}\); boundary-condition residual \(10^{-42}\) from the \(I_{\pm\nu}\) split and \(6\times10^{-16}\) from a direct small-radius fit; and positive \(C\) in the tested nonnegative cases. For 12 random Hermitian matrices, the reported determinant-root count agreed with the number of negative matrix eigenvalues, including zero, one, and two transverse bound states. Theorem~\ref{thm:spectrum} proves the count without random tests. \item[C4] At \(\alpha=0.8,\beta=0.3\), the tested orders were \(\nu=0.375,0.875\). The source reports small-\(\varepsilon\) asymptotic relative errors \(4.5\times10^{-3}\) and \(2.6\times10^{-9}\) at \(\varepsilon=10^{-6}\). For large \(\varepsilon=10^6\), it reports \(2.8\times10^{-5}\) and \(5.8\times10^{-11}\). It reports \(j/F=-2eq\), agreement of leading stress coefficients to 12 printed digits, structural relations to \(10^{-29}\), and conservation and trace residuals at most \(10^{-30}\). The conjugate-mode swap reportedly agrees in square and stress to \(3.5\times10^{-30}\), with opposite current. The small-radius tests lie in regimes compatible with the corrected Theorem~\ref{thm:small}; they do not establish the incorrect predecessor remainder uniformly for all \(0<\nu<1/2\). \item[C5] For \(\alpha=0.83,\beta=0.45\), and a positive matrix with reported eigenvalues \(0.49,2.11\), the source reports conservation \(1.5\times10^{-31}\), trace \(7\times10^{-31}\), current conservation \(1.3\times10^{-32}\), and \(j_r=-1.63\times10^{-5}\ne0\). The rotation rule was reported exact. The off-diagonal-to-zero limit reportedly agrees with the diagonal result to \(4\times10^{-31}\), and the line family integral with its closed form to \(6\times10^{-30}\). The SAE text additionally reports large-radius relative error \(5\times10^{-4}\) at \(r=10^3\), described as decreasing like \(r^{-2\nu_0}\), and a small-radius scaled error \(2\times10^{-4}\) at \(r=10^{-6}\). These tests are not extended to arbitrary boundary charts. \item[C7] The reported bound-state diagnostics are: \begin{itemize} \item after pole subtraction, values \(-0.04893705\) on both sides of \(p=\kappa(1\pm10^{-8})\), differing by \(1.6\times10^{-9}\); \item an algebraic \(p^{-2}\) tail coefficient \(-\kappa^2K_0K_0/(\pi\alpha)\), agreeing to \(3\times10^{-5}\) at \(p=200\); \item the Struve identity to \(3\times10^{-26}\), the longitudinal split identity to \(5\times10^{-15}\), and the principal-value Hankel identity below \(10^{-25}\); \item the \(\mathsf E_\lambda\) bound-state residue matching \(\norm{K_\nu}^{-2}\) to 12 digits; \item logarithmic asymptotic tests at \(\kappa_Lr=10^{\pm3},10^{\pm6},10^{\pm12}\), with two-term relative error \(2.2\times10^{-3}\) at \(10^{12}\), described as decreasing like \(\log^{-2}\). \end{itemize} \item[Filgueiras--Moraes report] Eight \((\alpha,\eta_0)\) comparisons were reported at most \(6\times10^{-19}\). The energy scale and longitudinal term must be restored when comparing the dimensionless expression with their equation (10). \end{description} \paragraph{What would strengthen these reports.} Supply the actual scripts, exact parameter encodings, dependency versions, raw values, precision schedules, quadrature tolerances, oscillatory truncations, pole prescriptions, certified tails, and image-boundary handling. Until then the retained numbers are comparisons, not rigorous error enclosures or proof of a universal identity. %====================================================================== \section{Technical and documentary clarifications} \label{app:clarifications} %====================================================================== \begin{enumerate}[label=\textup{(\roman*)}] \item \textbf{Product completeness.} The finite ledger is complete for Assumption~\ref{ass:product}, not for every self-adjoint extension of the full three-dimensional minimal operator. Non-product boundary data are infinite dimensional. \item \textbf{Positivity and infrared behaviour.} The nonnegative extension condition, the compact-test covariance estimate, and thermal infrared behaviour are separate questions. \item \textbf{Boundary relations.} A finite Hermitian matrix is one chart, not the entire extension space. Compression in \eqref{eq:CW} supplies the remaining charts. \item \textbf{Transverse versus longitudinal spectrum.} A transverse bound eigenfunction gives a longitudinal continuous branch. The instability proof uses that branch rather than an invented full-space eigenvector. \item \textbf{Analytic branch versus physical flux periodicity.} The shifted formulas refer to a fixed entire geometric branch. They do not eliminate the extension change by integer flux relabeling. \item \textbf{Complex first jet.} Taking a real part is appropriate for the stress prescription but destroys the current information if done before its first jet. \item \textbf{All diagonal cone parameters.} Joint ellipticity proves the regular representative even where individual image sectors jump. \item \textbf{Projection versus original CCR.} The principal-value reference is not positive; the projected positive covariances have projected commutators and do not solve the original local-state problem. \item \textbf{Completion terms.} Vanishing local curvature ambiguities on \(M\) does not classify worldsheet-supported terms or stress-correlation contacts. \item \textbf{Transfer versus covariance.} Supported equality of Green operations does not identify their symmetric states. \item \textbf{Finite arrays versus continuum normalization.} A finite report, finite normal-jet array, or small numerical residual is not automatically a finite-dimensional exhaustive normalization class or a continuum identity. \item \textbf{External programme boundaries.} The supplied XIV-H/I/J and XV-B excerpts describe different objects, fields, domains, contacts, and quantum gates. Internal retarded-kernel reconstruction, characteristic composition, output wavefront control, and quantum admission remain distinct there. No theorem here supplies a missing graph product or promotes a classical master-equation result to quantum closure. \item \textbf{Verification status.} No successful compilation, independent execution, machine-checked proof, referee acceptance, or journal acceptance is claimed. The mathematical assertions released here are the displayed proofs with their explicit hypotheses. \end{enumerate} %====================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} %====================================================================== \bibitem{HK1986} T.~M.~Helliwell and D.~A.~Konkowski, Vacuum fluctuations outside cosmic strings, \emph{Physical Review D} \textbf{34} (1986), 1918--1920. DOI: \nolinkurl{10.1103/PhysRevD.34.1918}. \Meta \bibitem{Linet1987} B.~Linet, Quantum field theory in the space-time of a cosmic string, \emph{Physical Review D} \textbf{35} (1987), 536. DOI: \nolinkurl{10.1103/PhysRevD.35.536}. The supplied records disagree on the last page; only the first page is used here. \Meta \bibitem{FS1987} V.~P.~Frolov and E.~M.~Serebriany, Vacuum polarization in the gravitational field of a cosmic string, \emph{Physical Review D} \textbf{35} (1987), 3779--3782. DOI: \nolinkurl{10.1103/PhysRevD.35.3779}. \Meta \bibitem{Dowker1987} J.~S.~Dowker, Casimir effect around a cone, \emph{Physical Review D} \textbf{36} (1987), 3095--3101. DOI: \nolinkurl{10.1103/PhysRevD.36.3095}. \Meta \bibitem{Dowker1987b} J.~S.~Dowker, Vacuum averages for arbitrary spin around a cosmic string, \emph{Physical Review D} \textbf{36} (1987), 3742--3746. \Meta \bibitem{GL1994} M.~E.~X.~Guimar\~aes and B.~Linet, Scalar Green's functions in an Euclidean space with a conical-type line singularity, \emph{Communications in Mathematical Physics} \textbf{165} (1994), 297--310. DOI: \nolinkurl{10.1007/BF02099773}. \Meta \bibitem{Guimaraes1995} M.~E.~X.~Guimar\~aes, Vacuum polarization at finite temperature around a magnetic flux cosmic string, \emph{Classical and Quantum Gravity} \textbf{12} (1995), 1705--1714. DOI: \nolinkurl{10.1088/0264-9381/12/7/012}. \url{https://arxiv.org/abs/gr-qc/9502003}. Used: the zero-temperature formulas (19), (30), and the displayed \(\omega_2,\omega_4\) coefficients. \bibitem{Sriramkumar2001} L.~Sriramkumar, Fluctuations in the current and energy densities around a magnetic-flux-carrying cosmic string, \emph{Classical and Quantum Gravity} \textbf{18} (2001), 1015. DOI: \nolinkurl{10.1088/0264-9381/18/6/304}. \url{https://arxiv.org/abs/gr-qc/0011074}. \Ver Used: (2.4), (2.10)--(2.16). \bibitem{SBdM2003} J.~Spinelly and E.~R.~Bezerra de Mello, Vacuum polarization of a charged massless scalar field on cosmic string spacetime in the presence of a magnetic field, \emph{Classical and Quantum Gravity} \textbf{20} (2003), 873--887. \url{https://arxiv.org/abs/hep-th/0301169}. \Ver Used: (22), (30), (36), (37), (54)--(56), (68). Equations (30) and (36) agree with \(F\). The printed equation (37) has denominator \(48\pi^2\alpha^4r^2\), an extra \(\alpha^{-2}\), and does not agree. The earlier agreement claim for (37) was our transcription error. The printed discrepancy in (55) is discussed explicitly and is retained. \bibitem{BdMBSH2015} E.~R.~Bezerra de Mello, V.~B.~Bezerra, A.~A.~Saharian, and H.~H.~Harutyunyan, Vacuum currents induced by a magnetic flux around a cosmic string with finite core, \emph{Physical Review D} \textbf{91} (2015), 064034. \url{https://arxiv.org/abs/1411.1258}. \Ver Used: (2.8)--(2.10), (3.4), and (4.2)--(4.7). \bibitem{KayStuder} B.~S.~Kay and U.~M.~Studer, Boundary conditions for quantum mechanics on cones and fields around cosmic strings, \emph{Communications in Mathematical Physics} \textbf{139} (1991), 103--139. DOI: \nolinkurl{10.1007/BF02102731}. \Meta \bibitem{DR2017} J.~Derezi\'nski and S.~Richard, On Schr\"odinger operators with inverse square potentials on the half-line, \emph{Annales Henri Poincar\'e} \textbf{18} (2017), 869--928. DOI: \nolinkurl{10.1007/s00023-016-0520-7}. \url{https://arxiv.org/abs/1604.03340}. \bibitem{AKO1996} B.~Allen, B.~S.~Kay, and A.~C.~Ottewill, Long-range effects of cosmic string structure, \emph{Physical Review D} \textbf{53} (1996), 6829--6841. DOI: \nolinkurl{10.1103/PhysRevD.53.6829}. \url{https://arxiv.org/abs/gr-qc/9510058}. \Ver Used: (3.7), (3.17), (4.3), (4.8), (4.13)--(4.17), and Appendix A. \bibitem{FM2007} C.~Filgueiras and F.~Moraes, The bound-state Aharonov--Bohm effect around a cosmic string revisited, \emph{Physics Letters A} \textbf{361} (2007), 13--15. DOI: \nolinkurl{10.1016/j.physleta.2006.09.030}. \url{https://arxiv.org/abs/gr-qc/0509100}. \Ver Used: (8), (10). \bibitem{AllenOttewill1990} B.~Allen and A.~C.~Ottewill, Effects of curvature couplings for quantum fields on cosmic-string space-times, \emph{Physical Review D} \textbf{42} (1990), 2669--2677. \Meta \bibitem{SouradeepSahni} T.~Souradeep and V.~Sahni, Quantum effects near a point mass in \(2+1\)-dimensional gravity, \emph{Physical Review D} \textbf{46} (1992), 1616--1633. \Mem \bibitem{Smith1990} A.~G.~Smith, Gravitational effects of an infinite straight cosmic string on classical and quantum fields: self-force and vacuum fluctuations, in G.~W.~Gibbons, S.~W.~Hawking, and T.~Vachaspati, eds., \emph{The Formation and Evolution of Cosmic Strings}, Cambridge University Press, 1990, pp.~263--292. \Meta The supplied AB/SAE records note a 1989 citation in \cite{SBdM2003}; this variation is not resolved here. \bibitem{SousaGerbert1989} Ph.~de Sousa Gerbert, Fermions in an Aharonov--Bohm field and cosmic strings, \emph{Physical Review D} \textbf{40} (1989), 1346--1349. \Meta \bibitem{BordagVoropaev1993} M.~Bordag and S.~Voropaev, Charged particle with magnetic moment in the Aharonov--Bohm potential, \emph{Journal of Physics A} \textbf{26} (1993), 7637--7649. \url{https://arxiv.org/abs/hep-th/9304017}. \Meta \bibitem{BdFS1999} C.~G.~Beneventano, M.~De Francia, and E.~M.~Santangelo, Dirac fields in the background of a magnetic flux string and spectral boundary conditions, \emph{International Journal of Modern Physics A} \textbf{14} (1999), 4749--4762. \url{https://arxiv.org/abs/hep-th/9809081}. \Meta \bibitem{CheegerTaylor} J.~Cheeger and M.~Taylor, On the diffraction of waves by conical singularities. I, II, \emph{Communications on Pure and Applied Mathematics} \textbf{35} (1982), 275--331 and 487--529. \bibitem{Watson} G.~N.~Watson, \emph{A Treatise on the Theory of Bessel Functions}, 2nd edition, Cambridge University Press, 1944. \bibitem{GR} I.~S.~Gradshteyn and I.~M.~Ryzhik, \emph{Table of Integrals, Series, and Products}, 7th edition, Academic Press, 2007. \Mem \bibitem{DLMF} F.~W.~J.~Olver et al., eds., \emph{NIST Digital Library of Mathematical Functions}. Used: Bessel representations, transforms, and Struve integrals, especially Sections 10.22, 10.32, 10.43, and 11.5. \url{https://dlmf.nist.gov/}. \bibitem{MorreyNirenberg} C.~B.~Morrey, Jr., and L.~Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations, \emph{Communications on Pure and Applied Mathematics} \textbf{10} (1957), 271--290. DOI: \nolinkurl{10.1002/cpa.3160100204}. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, 2nd edition, Springer, 1990. \bibitem{ReedSimonII} M.~Reed and B.~Simon, \emph{Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness}, Academic Press, 1975. \Mem \bibitem{Wald1977} R.~M.~Wald, The back reaction effect in particle creation in curved spacetime, \emph{Communications in Mathematical Physics} \textbf{54} (1977), 1--19. \bibitem{Wald1978} R.~M.~Wald, Trace anomaly of a conformally invariant quantum field in curved spacetime, \emph{Physical Review D} \textbf{17} (1978), 1477--1484. \Mem \bibitem{Wald1994} R.~M.~Wald, \emph{Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics}, University of Chicago Press, 1994. \bibitem{HollandsWald2001} S.~Hollands and R.~M.~Wald, Local Wick polynomials and time ordered products of quantum fields in curved spacetime, \emph{Communications in Mathematical Physics} \textbf{223} (2001), 289--326. DOI: \nolinkurl{10.1007/s002200100540}. \url{https://arxiv.org/abs/gr-qc/0103074}. \bibitem{HollandsWald2005} S.~Hollands and R.~M.~Wald, Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes, \emph{Reviews in Mathematical Physics} \textbf{17} (2005), 227--312. \url{https://arxiv.org/abs/gr-qc/0404074}. \bibitem{DecaniniFolacci} Y.~D\'ecanini and A.~Folacci, Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension, \emph{Physical Review D} \textbf{78} (2008), 044025. DOI: \nolinkurl{10.1103/PhysRevD.78.044025}. \url{https://arxiv.org/abs/gr-qc/0512118}. \bibitem{Radzikowski} M.~J.~Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, \emph{Communications in Mathematical Physics} \textbf{179} (1996), 529--553. \bibitem{KayWald} B.~S.~Kay and R.~M.~Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon, \emph{Physics Reports} \textbf{207} (1991), 49--136. \Mem \medskip \noindent\textit{Retained manuscript records and scoped companion references.} These entries identify the supplied records or documentary interfaces. They do not assert independent verification of entire companion manuscripts or transfer of their mathematical constructions. \medskip \bibitem{RecordVP2} N.~Cheung, \emph{I-B-VP: Vacuum polarization of the Friedrichs ground state on a prescribed conical spacetime}, supplied \SourceLabel{I-B-VP-v2.tex}, October 2026. Retained source for the heat estimate, all-diagonal analytic subtraction, local renormalization, transfer separation, finite reporting, and integrity discipline. \bibitem{RecordAB} N.~Cheung, \emph{I-B-VP-AB: Vacuum polarization and induced current of the Friedrichs ground state on the conical Enter spacetime}, supplied \SourceLabel{I-B-VP-AB.tex}, October 2026. Retained AB successor record, called version 2 in that source. \bibitem{RecordSAE} N.~Cheung, \emph{I-B-VP-SAE: Closing the self-adjoint extension ledger}, supplied \SourceLabel{I-B-VP-SAE.tex}, October 2026. Retained extension-ledger record, called version 3 in that source. Its qualifications and corrections are stated in the present text. \bibitem{CheungEnter} N.~Cheung, The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement, supplied bibliographic record: \emph{Frontiers in Physics} \textbf{14} (2026), article 1954943. DOI: \nolinkurl{10.3389/fphy.2026.1954943}. The supplied source describes a publisher accepted-article record. Current publication status is not independently asserted here. Cited only for finite operational semantics. \bibitem{CheungPhil} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}, supplied source identifier \SourceLabel{EntPhil-v2.tex}. Documentary orientation, not a continuum theorem premise. \bibitem{CheungIB} N.~Cheung, \emph{I-B-v1: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}, version 1, Enter Calculus Companion. Used: prescribed host, \SourceLabel{prop:not-source}, and \SourceLabel{thm:instrument}. \url{https://enter.grok.me/browse/I-B}. \bibitem{CheungIBG} N.~Cheung, \emph{I-B-G: Intrinsic and holonomy-twisted Green operators on a prescribed conical spacetime: Self-adjoint extensions, diffraction, finite propagation, and supported transfer from flat-completion laboratories}, supplied \SourceLabel{I-B-G-v2.tex}, October 2026. The earlier AB/SAE records use a different title and theorem numbering for \SourceLabel{I-B-G.tex}. Relevant source labels are identified in Section~\ref{sec:background}. \bibitem{CheungXAST} N.~Cheung, \emph{X-A-ST}, version 1, Enter Calculus Companion. Used for the signature and non-sourcing comparison. \url{https://enter.grok.me/browse/X-A-ST}. \bibitem{CheungXBST} N.~Cheung, \emph{X-B-ST: One scalar retarded map on the X-A-ST host}, version 1, Enter Calculus Companion. Supported retained-shadow and equation/Hessian conventions. \url{https://enter.grok.me/browse/X-B-ST}. \bibitem{CheungXCST} N.~Cheung, \emph{X-C-ST: Second-order scalar completion on the inherited X-A-ST/X-B-ST host}, version 1, Enter Calculus Companion. Support contract only; no higher product is imported. \url{https://enter.grok.me/browse/X-C-ST}. \bibitem{CheungXIE} N.~Cheung, \emph{XI-E}, version 1, Enter Calculus Companion. Supplied excerpts \SourceLabel{def:domain}, \SourceLabel{thm:star}. Product-domain versus state-selection distinction only. \url{https://enter.grok.me/browse/XI-E}. \bibitem{CheungO1ST} N.~Cheung, \emph{XVI-O1-ST}, version 5, supplied identifier \SourceLabel{XVI-O1-ST-v5.tex}. Used for finite reporting and integrity-aware residual retention. No loop, phase reconciliation, or quantum closure is imported. \url{https://enter.grok.me/browse/XVI-O1-ST}. \bibitem{CheungOai282A} N.~Cheung, \emph{Oai282-A}, version 1, Enter Calculus Companion. Supplied \SourceLabel{prop:coverage}, \SourceLabel{prop:invisible}. \url{https://enter.grok.me/browse/Oai282-A}. \bibitem{CheungEn09} N.~Cheung, \emph{En09-A}, version 1, Enter Calculus Companion. Supplied destructive finite reporting and record--residual separation excerpts. \url{https://enter.grok.me/browse/En09-A}. \bibitem{CheungXVIO} N.~Cheung, \emph{Landmark XVI-O}, supplied series status record. Read-only reference; no O.1 update is asserted. \url{https://enter.grok.me/browse/XVI-O}. \bibitem{CheungXVIN} N.~Cheung, \emph{Landmark XVI-N}, supplied frozen prepared-source record. Not discharged by this scalar calculation. \url{https://enter.grok.me/browse/XVI-N}. \bibitem{DocumentaryBoundary} N.~Cheung, supplied excerpts from \emph{XIV-H}, version 1; \emph{XIV-I}, version 1; \emph{XIV-J}, version 1; and \emph{XV-B}, version 2. Used only to preserve the documentary non-transfer boundary: internal kernels and partial reference constructions do not constitute complete characteristic quantum admission. No graph-specific theorem is imported into the present scalar extension problem. \end{thebibliography} \end{document}