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\title{\textbf{I-B-VP-2: The exact global wavefront relation\\
of the conical Friedrichs ground state}\\[5pt]
\large Geometric propagation, apex diffraction, and positive frequency\\
on the prescribed regular Enter spacetime}
\author{Ngo Cheung}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
We determine the ordinary global wavefront set of the massless
holonomy-twisted Friedrichs spectral ground-state covariance on
the regular conical spacetime prescribed in I-B. The result is an
equality, valid for every opening \(0<\alpha<1\) and fractional
holonomy \(0\leq\beta<1\), including the angular transition where
a geometric front meets the apex-diffracted front.

Let \(\Gc^+\) be the closure of the positive-frequency relation of
regular null geodesics, including its diagonal limit, and let
\(\Dc^+\) be the positive conormal relation of
\[
 (t-t')^2=(r+r')^2+(z-z')^2.
\]
Then
\[
 \WF(W_\beta)=
 \begin{cases}
 \Gc^+,&\alpha=1/N,\ \beta=p/N,\\
 \Gc^+\cup\Dc^+,&\text{otherwise},
 \end{cases}
\]
where \(N\geq2\) and \(p=0,\ldots,N-1\).
Thus the full strictly diffractive relation disappears exactly in
the finite-character quotient sectors. In other sectors it remains
in the joint wavefront set even at isolated angular zeros of its
leading coefficient.

The proof sums the Euclidean angular modes before continuation,
uses a two-variable meromorphic expression to control every
angular transition, and combines its reduced-variable structure
with the bisolution equations. It also gives the explicit
inverse-square-root diffractive coefficient and its
half-order regularity improvement away from transition angles.

The contribution is a convention-complete global analysis of the
specified covariance within the companion framework, not a claim
of priority for conical diffraction. It distinguishes local
Hadamard subtraction from a global geodesic-only spectrum
condition. In particular, supported Green-operator agreement does
not identify vacuum covariances, and the charged commutator is not
the pointwise imaginary part of one complex bundle kernel.
The result resolves V2 for this state and extension at the level
of the ordinary two-point wavefront set. It neither restores global
hyperbolicity nor constructs an apex field algebra, an interacting
product, a measurement acquisition of wavefront data, or a
gravitational source law.
\end{abstract}

\tableofcontents

%======================================================================
\section{Purpose, background, and corrections to the proposed route}
\label{sec:introduction}
%======================================================================

Cheung's I-B prescribes a regular conical spacetime together with
a separate finite operational interface \cite{CheungIB}.
I-B-G constructs intrinsic scalar propagation after a spatial
self-adjoint extension has been supplied \cite{CheungIBG}.
I-B-VP-v2 constructs the distinguished massless Friedrichs
spectral ground state and establishes its local Hadamard form
\cite{CheungIBVP}. I-B-VP-0 evaluates its charged current
\cite{CheungIBVPZero}; I-B-VP-1 constructs prescribed extensions
of selected one-point distributions across the apex in an
additional smooth completion \cite{CheungIBVPOne}.

The present object is different: the global covariance on the
\emph{regular} spacetime \(M\), with both endpoints satisfying
\(r,r'>0\). No smooth structure at the apex and no apex-supported
one-point counterterm enters the proof.

\paragraph{Established analytical background.}
Sommerfeld-type cone kernels and cosmic-string Euclidean Green
functions have a substantial history
\cite{Dowker,Linet}. Geometric and diffractive propagation on
cones and edges are treated by Cheeger--Taylor,
Melrose--Wunsch, and Melrose--Vasy--Wunsch
\cite{CheegerTaylorI,CheegerTaylorII,MelroseWunsch,MelroseVasyWunsch}.
The magnetic Aharonov--Bohm diffraction problem provides another
relevant model \cite{Yang}.
We use these works for context, not as substitutes for checking
the exact holonomy, state, and transition formulas below.

The global statement is proved directly from an explicitly summed
kernel. The standard tools needed are distributional boundary
values, the wavefront pullback and product theorems, microlocal
elliptic regularity, and propagation of singularities for a smooth
real-principal-type operator on the regular host
\cite{HormanderI,HormanderIV}.

\subsection{Four necessary corrections}

The proposed V2 route requires the following corrections.

\begin{enumerate}[label=\textup{(\roman*)}]
\item \textbf{A retarded transfer theorem is not a state-transfer theorem.}
Equality of supported retarded or advanced operators in a laboratory
window does not imply equality of positive-frequency covariances.
The appropriate comparison is equality of the local singular
structure, or smoothness of a specified covariance difference,
not equality of states.

\item \textbf{The global relation must include limiting geometric rays.}
A regular null geodesic can approach a ray through the removed
apex. Its limiting covector pair belongs to a closed wavefront
set, even though the limiting ray is not a geodesic in \(M\).
We therefore use the closure \(\Gc^+\), rather than only the
relation of actual regular null geodesics.

\item \textbf{A diffraction bound must allow exact disappearance.}
Some cone--character pairs admit a finite image formula with
no strictly diffractive relation. Conversely, a zero of one
leading coefficient at one angular value does not remove a
covector from the \emph{joint} wavefront set if nearby angular
values retain that covector in the limit.

\item \textbf{The charged commutator is not obtained by taking a
pointwise imaginary part.}
The spatial bundle kernel can be complex before any Lorentzian
time prescription is introduced. The correct commutator uses
opposite operator ordering and the conjugate charge sector.
The operator-valued time-odd part, not the pointwise imaginary
part of \(W_\beta\), is fixed by the causal propagator.
\end{enumerate}

These are corrections to the logical route, not changes to the
prescribed geometry, Friedrichs extension, or spectral state.

\subsection{Enter and the non-extraction boundary}

\emph{The Philosophy of Enter} distinguishes destructive readout
from quantum recovery, a classical record from a decoder, and an
analogy from an identification \cite{CheungPhilosophy}.
I-B and En09-A implement a finite destructive instrument returning
a classical code and a surviving residual, not the measured
quantum clock register \cite{CheungIB,CheungEn09}.
That interface does not select the continuum field, apex extension,
vacuum, or wavefront relation.

The background, worldsheet defect, extension, and state remain
inputs. A finite conditional density matrix is not the
bidistribution analyzed here. A wavefront condition is not an
Einstein equation or a derivation of the metric
\cite[\SourceLabel{prop:not-source}]{CheungIB}
\cite[\SourceLabel{prop:noeinstein}]{CheungXAST}.
The philosophical discussion fixes interpretation; it is not a
premise of the continuum proof.

\paragraph{Scope of the new result.}
We prove an exact ordinary \(C^\infty\)-wavefront theorem on
\(M\times M\), not a theorem about an unspecified boundary
wavefront calculus. We do not claim a complete symbol calculus at
the geometric--diffractive transition, a classification of all
ground states, or an interacting microlocal algebra.
The exact wavefront equality does not require any of those stronger
claims.

%======================================================================
\section{The regular host and the spectral covariance}
\label{sec:state}
%======================================================================

\subsection{Geometry, bundle, and non-global-hyperbolicity}

Fix
\[
 0<\alpha<1,\qquad 0\leq\beta<1.
\]
The spacetime and metric are
\begin{equation}
 M=\RR_t\times(0,\infty)_r
       \times(\RR/2\pi\ZZ)_\varphi\times\RR_z,
 \qquad
 g=-\dd t^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2.
 \label{eq:host}
\end{equation}
The future orientation is that of \(\partial_t\).
The completion adds the worldsheet
\[
 \WW=\RR_t\times\{o\}\times\RR_z.
\]
It is not part of \(M\).

A lifted parallel representation of the charged line bundle is fixed by
\begin{equation}
 \Phi(\varphi+2\pi)=e^{2\pi\ii\beta}\Phi(\varphi).
 \label{eq:character}
\end{equation}
Changing a local parallel frame multiplies the covariance by a
smooth nowhere-zero phase, and therefore does not change its
wavefront set.

Write
\[
 q_N=\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2,\qquad
 \dd V=\alpha r\,\dd r\,\dd\varphi\,\dd z.
\]
Let \(B_\beta\) be the nonnegative Friedrichs extension of
\(-\Delta_{q_N}\) on the twisted spatial Hilbert space.
The extension is an input, as in I-B-G
\cite{CheungIBG,KayStuder}.
No distributional curvature-coupling interaction at \(\WW\) is
introduced.

\begin{proposition}[The regular host is not globally hyperbolic]
\label{prop:nonGH}
The function \(t\) is temporal, but \(M\) is not globally
hyperbolic.
\end{proposition}

\begin{proof}
The inverse metric gives \(g^{-1}(\dd t,\dd t)=-1\).
Choose \(R>0\), \(T_0>R\), and fixed \(\varphi_0,z_0\).
The points
\[
 p=(-T_0,R,\varphi_0,z_0),\qquad
 q=(T_0,R,\varphi_0,z_0)
\]
have a causal diamond containing
\[
 x_m=(0,R/m,\varphi_0,z_0),\qquad m\geq2:
\]
the radial spatial travel length from either endpoint is less
than \(R<T_0\).
This sequence has no convergent subsequence in \(M\).
Thus the diamond is not compact.
\end{proof}

This reproduces the relevant geometric obstruction of
I-B-G \SourceLabel{prop:nonGH} without importing a global
Cauchy theorem. A self-adjoint extension supplies the evolution
used here; it does not make the regular slices Cauchy surfaces
for all causal curves.

\subsection{Definition and compact-test finiteness}

Put \(H_\beta=\sqrt{B_\beta}\).
The covariance is
\begin{equation}
 W_\beta(T;x,x')
 =\left[\frac{e^{-\ii T H_\beta}}{2H_\beta}\right](x,x'),
 \qquad T=t-t'.
 \label{eq:W}
\end{equation}
For compact spacetime sections,
\[
 \mathcal K_\beta f
 =\int_\RR e^{\ii tH_\beta}(2H_\beta)^{-1/2}f(t)\,\dd t,
 \qquad
 W_\beta(f,g)=
 \langle\mathcal K_\beta f,\mathcal K_\beta g\rangle.
\]
For a real neutral scalar take \(\beta=0\).
For a charged scalar use the particle and antiparticle sectors
on the bundle and its conjugate, as in
I-B-VP-v2 \SourceLabel{def:state} \cite{CheungIBVP}.

For completeness, the estimate needed at zero energy is recalled.

\begin{lemma}[Infrared control]
\label{lem:IR}
Every compact smooth spatial section belongs continuously to
\(\operatorname{dom}(B_\beta^{-1/4})\) in its fixed-support test
topology. Consequently \eqref{eq:W} defines a continuous positive
covariance on compact spacetime tests.
\end{lemma}

\begin{proof}
With
\[
 \nu_n=\frac{|n+\beta|}{\alpha},
\]
the heat kernel is
\begin{align}
 K_\beta(s;x,x')
 ={}&
 \frac{e^{-(z-z')^2/(4s)}}{\sqrt{4\pi s}}\,
 \frac{e^{-(r^2+r'^2)/(4s)}}{4\pi\alpha s}
 \nonumber\\
 &\times
 \sum_{n\in\ZZ}e^{\ii(n+\beta)(\varphi-\varphi')}
              I_{\nu_n}\left(\frac{rr'}{2s}\right).
 \label{eq:heat}
\end{align}
It follows from the Friedrichs Bessel transform and Weber's
integral \cite{Watson,CheungIBG}.
On a compact regular spatial set and for \(s\geq1\), the Bessel
power series gives
\[
 |K_\beta(s;x,x')|
 \leq C s^{-3/2-\nu_*},
 \qquad
 \nu_*=\frac{\min(\beta,1-\beta)}{\alpha}.
\]
For a compact spatial section \(v\),
\[
 \norm{B_\beta^{-1/4}v}^2
 =\frac1{\sqrt\pi}\int_0^\infty
 s^{-1/2}\langle v,e^{-sB_\beta}v\rangle\,\dd s.
\]
Use the contraction bound for \(0<s<1\), and the displayed
compact heat bound with \(\norm{v}_{L^1}^2\) for \(s\geq1\).
The latter integrand is \(O(s^{-2-\nu_*})\).
The bounds are continuous in fixed-support test seminorms.
Time integration then defines \(\mathcal K_\beta\), and its Gram
form proves positivity.
\end{proof}

The spectral construction is quasifree and has nonnegative
one-particle Hamiltonian \(H_\beta\). It is the specified
distinguished spectral state, not a uniqueness theorem among all
algebraic ground states.

On regular charts,
\begin{equation}
 P_XW_\beta=P_{X'}W_\beta=0,\qquad
 P=\partial_t^2-\Delta_{q_N}=-\Box_g.
 \label{eq:bisolution}
\end{equation}
These equations follow by spectral calculus and integration by
parts on compact tests. Their principal symbol is
\[
 p(X,\kappa)=-\kappa_t^2+\kappa_r^2
        +\frac{\kappa_\varphi^2}{\alpha^2r^2}+\kappa_z^2,
\]
up to an irrelevant overall sign.

\subsection{Wavefront and frequency conventions}

All wavefront sets below are ordinary, unprimed wavefront sets:
\[
 (X,\kappa;X',\kappa')\in\WF(W_\beta)
\]
means the covector of the bidistribution on \(M\times M\).
A conventional canonical-relation notation would replace
\(\kappa'\) by \(-\kappa'\).

We use the local Fourier convention
\[
 \widehat u(\xi)=\int e^{-\ii x\cdot\xi}u(x)\,\dd x.
\]
The factor \(e^{-\ii\omega(t-t')}\), \(\omega>0\), therefore
has first temporal covector \(\kappa_t=-\omega<0\).
For signature \((-+++)\), such a null covector is future directed
after raising its index. This is our positive-frequency
orientation.

%======================================================================
\section{An exact summed kernel and its boundary values}
\label{sec:closed}
%======================================================================

Use the differences
\[
 T=t-t',\qquad Z=z-z',\qquad
 \delta=\varphi-\varphi'
\]
in a chosen angular lift. For \(\varepsilon>0\), define
\begin{equation}
 Q_\varepsilon
 =\frac{r^2+r'^2+Z^2-(T-\ii\varepsilon)^2}{2rr'},
 \qquad
 \eta_\varepsilon=\arcosh Q_\varepsilon,
 \label{eq:Qeta}
\end{equation}
where the branch has \(\operatorname{Re}\eta_\varepsilon>0\).
For real \(T\neq0\), \(Q_\varepsilon\) approaches the real axis
from the side \(\sgn T\).

\subsection{Summation before continuation}

Define
\begin{equation}
 S_\beta(\eta,\delta)
 =\sum_{n\in\ZZ}
 e^{-|n+\beta|\eta/\alpha}e^{\ii(n+\beta)\delta},
 \qquad \operatorname{Re}\eta>0.
 \label{eq:Sseries}
\end{equation}

\begin{lemma}[Closed angular sum]
\label{lem:sum}
For \(a=\eta/\alpha\),
\begin{align}
 S_\beta(\eta,\delta)
 ={}&
 \frac{e^{-\beta a+\ii\beta\delta}}
             {1-e^{-a+\ii\delta}}
 +\frac{e^{-(1-\beta)a-\ii(1-\beta)\delta}}
             {1-e^{-a-\ii\delta}}
 \label{eq:Sgeom}\\
 ={}&
 \frac{
 e^{\ii\beta\delta}\sinh((1-\beta)a)
 +e^{-\ii(1-\beta)\delta}\sinh(\beta a)}
 {\cosh a-\cos\delta}.
 \label{eq:Sclosed}
\end{align}
The formulas include \(\beta=0\).
In particular, the meromorphic continuation of \(S_\beta\) is odd
in \(\eta\).
\end{lemma}

\begin{proof}
Separate \(n\geq0\) and \(n\leq-1\) and sum the two convergent
geometric series. Putting their terms over a common denominator
gives \eqref{eq:Sclosed}. Its numerator is odd in \(a\) and its
denominator is even.
\end{proof}

\begin{proposition}[Global regularized covariance]
\label{prop:closed}
For \(\varepsilon>0\),
\begin{equation}
 W_{\beta,\varepsilon}(X,X')
 =
 \frac{S_\beta(\eta_\varepsilon,\delta)}
      {8\pi^2\alpha rr'\sinh\eta_\varepsilon}
 =
 \left[
 \frac{e^{-(\varepsilon+\ii T)H_\beta}}{2H_\beta}
 \right](x,x').
 \label{eq:Wclosed}
\end{equation}
Its distributional boundary value as \(\varepsilon\downarrow0\)
is \(W_\beta\).
\end{proposition}

\begin{proof}
For a real Euclidean separation \(a_0>0\), the spectral identity
\[
 \frac{e^{-a_0\sqrt B}}{2\sqrt B}
 =\int_0^\infty
 (4\pi s)^{-1/2}e^{-a_0^2/(4s)}e^{-sB}\,\dd s
\]
and \eqref{eq:heat} give
\[
 G_{E,\beta}
 =\frac{S_\beta(\eta,\delta)}
       {8\pi^2\alpha rr'\sinh\eta},
 \qquad
 \cosh\eta=
 \frac{r^2+r'^2+Z^2+a_0^2}{2rr'}.
\]
The heat integral is justified at large \(s\) by
Lemma~\ref{lem:IR}. The change of variable
\(\zeta=rr'/(2s)\) uses
\[
 \int_0^\infty e^{-\zeta\cosh\eta}I_\nu(\zeta)\,\dd\zeta
 =\frac{e^{-\nu\eta}}{\sinh\eta}
\]
\cite{Watson}. This is also the Euclidean mode kernel of
I-B-VP and I-B-VP-0 \cite{CheungIBVP,CheungIBVPZero}.

Both sides continue holomorphically in \(a_0\) to
\(\operatorname{Re}a_0>0\).
The denominators in \eqref{eq:Sgeom} are nonzero there under
\eqref{eq:Qeta}, since \(\operatorname{Re}\eta>0\).
Set \(a_0=\varepsilon+\ii T\).
Spectral dominated convergence on the compact-test form of
Lemma~\ref{lem:IR} proves the boundary assertion.
\end{proof}

\subsection{A boundary-value fact used below}

\begin{lemma}[One complex variable with smooth parameters]
\label{lem:boundaryvalue}
Let \(F(z,y)\) be holomorphic for \(\operatorname{Im}z>0\),
smooth in real parameters \(y\), and locally bounded by a power of
\((\operatorname{Im}z)^{-1}\), together with each fixed finite
number of \(y\)-derivatives. Then it has a distributional boundary
value on compact real coordinate patches, and
\begin{equation}
 \WF(F(s+\ii0,y))
 \subset\{(s,y;\lambda,\nu):\lambda\geq0\}.
 \label{eq:BVorientation}
\end{equation}
For a lower-half-plane boundary value, \(\lambda\leq0\).

The assertion is local and remains valid after a real-analytic
change of the complex transverse coordinate with the corresponding
half-plane orientation.
\end{lemma}

\begin{proof}
This is the distributional boundary-value theorem with parameters
\cite{HormanderI}. We recall the two ingredients needed here.
Sufficiently many holomorphic primitives in \(z\) have controlled
boundary limits; differentiating those limits defines the original
distributional boundary value. The estimates with parameters make
this construction continuous on joint tests.

For the wavefront assertion, localize and Fourier transform.
In a cone where \(\lambda<-c|(\lambda,\nu)|\), shift the \(z\)
contour upward. An almost-analytic extension of the real cutoff
makes its \(\bar\partial\)-error vanish to arbitrarily high order
at the boundary. That order can be chosen larger than the fixed
polynomial growth order of \(F\). The shifted term decays
exponentially in the negative \(\lambda\)-cone; the cutoff errors
decay to every inverse power. This proves
\eqref{eq:BVorientation}. Reflection proves the lower-half-plane
case. The same contour argument in an analytic transverse
coordinate proves the last assertion.
\end{proof}

For the function
\[
 \mathcal F_\beta(Q,\delta)
 =\frac{S_\beta(\arcosh Q,\delta)}{\sinh(\arcosh Q)},
 \label{eq:reducedF}
\]
the required polynomial bounds hold near \(Q=-1\), uniformly on
compact real angular intervals. Indeed,
\(\operatorname{Re}\arcosh Q\geq c\,\operatorname{Im}Q\) on a
sufficiently small upper-half-plane compact patch, the geometric
denominators in \eqref{eq:Sgeom} are bounded below by a constant
times \(\operatorname{Re}\arcosh Q\), and
\(\sinh(\arcosh Q)\) has only a square-root zero.
Angular differentiation changes the polynomial bound, but not
its existence.

Consequently, near a point with \(T\neq0\), the covariance has
the reduced form
\begin{equation}
 W_\beta
 =\frac1{8\pi^2\alpha rr'}\,
 \mathcal F_\beta(Q+\ii0\,\sgn T,\delta),
 \qquad
 Q=\frac{r^2+r'^2+Z^2-T^2}{2rr'}.
 \label{eq:reducedboundary}
\end{equation}
The analytic change from \(T\) to \(Q\) is regular there, since
\(\partial_TQ=-T/(rr')\neq0\).
This observation is particularly useful at an angular transition:
no moving image indicator is differentiated.

\subsection{The local direct term}

\begin{lemma}[Local Hadamard form from the closed expression]
\label{lem:local}
On a sufficiently small developing neighbourhood of each regular
point, in a local parallel frame,
\begin{equation}
 W_\beta(X,X')
 =\frac1{4\pi^2\bigl(
 |\mathbf X-\mathbf X'|^2-(T-\ii0)^2\bigr)}
     +H_\beta(X,X'),
 \label{eq:localHadamard}
\end{equation}
with \(H_\beta\) smooth; in fact the local regular remainder is
real-analytic in the corresponding real variables.
\end{lemma}

\begin{proof}
Take a lift with \(\delta\) near zero and \(Q\) near \(1\).
The quotient in \eqref{eq:reducedF} is meromorphic in
\((Q,\delta)\), because both its numerator divided by
\(\sinh\eta\) and its denominator are even analytic functions of
\(\eta\) near \(\eta=0\).
Its denominator factors as
\[
 \cosh(\arcosh Q/\alpha)-\cos\delta
 =(Q-\cos(\alpha\delta))h(Q,\delta),
 \qquad h(1,0)=\alpha^{-2}.
\]
For \(\delta\neq0\) sufficiently small, either term in
\eqref{eq:Sgeom} computes the pole residue. It gives
\[
 \mathcal F_\beta(Q,\delta)
 -\frac{\alpha}{Q-\cos(\alpha\delta)}
\]
analytic across the pole. Analytic division extends the same
statement through \((Q,\delta)=(1,0)\).
Multiplication by the prefactor in \eqref{eq:Wclosed} gives
the direct kernel in \eqref{eq:localHadamard}, because
\[
 2rr'(Q-\cos(\alpha\delta))
 =r^2+r'^2-2rr'\cos(\alpha\delta)+Z^2-T^2.
\]
The remaining analytic function has a smooth Lorentzian boundary
value. The direct term is the ordinary Minkowski
positive-frequency distribution in the developing chart.
\end{proof}

This independently recovers the local statement used in
I-B-VP and I-B-VP-0. The global proof below is not an application
of a globally hyperbolic local-to-global theorem.

%======================================================================
\section{The geometric and apex relations}
\label{sec:relations}
%======================================================================

For a chosen lift define
\begin{equation}
 \theta_k=\alpha(\delta-2\pi k),\qquad k\in\ZZ,
 \label{eq:thetak}
\end{equation}
\begin{equation}
 R_k^2=r^2+r'^2-2rr'\cos\theta_k+Z^2,\qquad
 Q_k=R_k^2-T^2.
 \label{eq:Qk}
\end{equation}
Only finitely many \(k\) satisfy \(|\theta_k|\leq\pi\) on a
fixed compact lifted angular patch.

\begin{definition}[Closed geometric positive relation]
\label{def:geometric}
Let \(\Gc_{\mathrm{reg}}^+\) be the ordinary positive-frequency
covector relation of null geodesic segments entirely in \(M\).
It includes
\[
 (X,\kappa;X',-\kappa')
\]
when \(\kappa'\) is parallel transported to \(\kappa\) along that
segment and \(\kappa\) is future directed.
Define
\begin{equation}
 \Gc^+=\overline{\Gc_{\mathrm{reg}}^+}
 \subset T^*(M\times M)\setminus0.
 \label{eq:Gclosed}
\end{equation}
The closure includes the standard diagonal null relation.
\end{definition}

Away from the diagonal, the actual regular geodesic relation is
locally described by
\begin{equation}
 Q_k=0,\qquad |\theta_k|<\pi,\qquad
 (\kappa,\kappa')=\lambda\,\dd_{X,X'}Q_k,\qquad
 \lambda T>0.
 \label{eq:Gcoordinates}
\end{equation}
Its closure allows \(|\theta_k|=\pi\).

\begin{lemma}[Geometric interpretation]
\label{lem:geometric}
Formula \eqref{eq:Gcoordinates} describes all regular null
geodesic segments. The additional off-diagonal limits in
\(\Gc^+\) occur at \(|\theta_k|=\pi\).
\end{lemma}

\begin{proof}
A geodesic away from the apex develops into a straight line in
the transverse plane, together with an affine longitudinal
coordinate. A straight transverse segment avoiding the origin has
a continuous angular change strictly between \(-\pi\) and \(\pi\).
Its length is \(R_k\) for the corresponding lift.
The null condition is \(|T|=R_k\).
Differentiating its squared separation gives the parallel
endpoint covectors in \eqref{eq:Gcoordinates}.

A limit with angular change \(\pi\) or \(-\pi\) passes through
the transverse origin. Conversely, those segments are limits
of straight segments avoiding the origin.
The local finiteness of the allowable lifts excludes other
off-diagonal limits.
\end{proof}

Put
\begin{equation}
 L_{\mathrm{ap}}=\sqrt{(r+r')^2+Z^2},
 \qquad
 F_{\mathrm{ap}}=(r+r')^2+Z^2-T^2.
 \label{eq:Fap}
\end{equation}

\begin{definition}[Apex-diffracted positive relation]
\label{def:diffraction}
Define
\begin{equation}
 \Dc^+
 =\left\{
 (X,\kappa;X',\kappa'):
 \begin{array}{l}
 F_{\mathrm{ap}}=0,\\
 (\kappa,\kappa')=\lambda\,\dd_{X,X'}F_{\mathrm{ap}},\\
 \lambda T>0
 \end{array}
 \right\}.
 \label{eq:Drelation}
\end{equation}
No angular covector occurs in this relation.
The angular transition set is
\begin{equation}
 \Bc=
 \{\delta\pmod{2\pi}:
          \cos\delta=\cos(\pi/\alpha)\}.
 \label{eq:Bangles}
\end{equation}
\end{definition}

Both endpoint covectors in \eqref{eq:Drelation} are nonzero and
null. For example,
\[
 \dd_XF_{\mathrm{ap}}
 =-2T\,\dd t+2(r+r')\,\dd r+2Z\,\dd z.
\]
The first raised covector is future directed precisely when
\(\lambda T>0\).

\begin{proposition}[The broken ray and longitudinal matching]
\label{prop:broken}
For \(T=L_{\mathrm{ap}}>0\), the relation \(\Dc^+\) is the
endpoint relation of an incoming radial null segment and an
outgoing radial null segment meeting the removed worldsheet at
\[
 z_*=\frac{rz'+r'z}{r+r'},\qquad
 t_*=t'+\frac{r'}{r+r'}T.
 \label{eq:scatterpoint}
\]
The energy and longitudinal momentum agree across that meeting.
The incoming and outgoing angular positions need not agree.
Moreover,
\[
 \Dc^+\cap\Gc^+
\]
consists exactly of these radial covector pairs at the angular
transition \(\delta\in\Bc\).
\end{proposition}

\begin{proof}
Minimize the length of a two-segment spatial path through the
worldsheet:
\[
 \sqrt{r'^2+(z_*-z')^2}
       +\sqrt{r^2+(z-z_*)^2}.
\]
Equality of the two longitudinal slopes gives
\eqref{eq:scatterpoint} and total length \(L_{\mathrm{ap}}\).
The corresponding null segments have common longitudinal
velocity \(Z/T\) and transverse speed \((r+r')/T\).
The covectors in \eqref{eq:Drelation} record the same energy and
longitudinal momentum.

For a limiting geometric ray to have this length,
\[
 R_k^2=L_{\mathrm{ap}}^2
 \quad\Longleftrightarrow\quad
 \cos\theta_k=-1.
\]
With \(|\theta_k|\leq\pi\), this means
\(\theta_k=\pi\) or \(-\pi\), equivalently \(\delta\in\Bc\).
At that point \(\dd Q_k=\dd F_{\mathrm{ap}}\).
\end{proof}

The meeting point is not an added field-theoretic degree of
freedom. It is a description of the endpoint relation selected
by the Friedrichs kernel.

%======================================================================
\section{Global wavefront containment, including every transition}
\label{sec:upper}
%======================================================================

\begin{proposition}[Global positive-frequency upper bound]
\label{prop:upper}
For every \(0<\alpha<1\) and \(0\leq\beta<1\),
\begin{equation}
 \WF(W_\beta)\subset\Gc^+\cup\Dc^+.
 \label{eq:upper}
\end{equation}
In particular, there are no one-zero-endpoint covectors and
no additional pure angular wavefronts at an image boundary.
\end{proposition}

\begin{proof}
The proof separates the real values of \(Q\) in
\eqref{eq:reducedboundary}.

\paragraph{1. The regions \(Q>1\) and \(Q<-1\).}
For \(Q>1\), \(\eta=\arcosh Q\) is real and positive, and the
summed expression is smooth.
For \(Q<-1\), its two boundary values have
\[
 \eta=v+\ii\pi\,\sgn T,\qquad v>0.
\]
The geometric denominators in \eqref{eq:Sgeom} are nonzero
because \(e^{-v/\alpha}<1\). The factor \(\sinh\eta\) is also
nonzero. Thus each boundary value is smooth.

\paragraph{2. The interval \(-1<Q<1\).}
Write
\[
 \eta=\ii\gamma\,\sgn T,\qquad 0<\gamma<\pi.
\]
The only poles in \eqref{eq:Sgeom} occur when
\[
 \gamma=|\alpha(\delta-2\pi k)|
       =|\theta_k|,\qquad |\theta_k|<\pi.
\]
They are precisely \(Q_k=0\).
Each of the two geometric fractions has a simple pole in a smooth
defining function for the corresponding front.
Even where two fronts cross in the base, the two fractions
provide the separate conormal descriptions.
Consequently the wavefront is contained in their conormals.

The half-plane boundary-value orientation in
Lemma~\ref{lem:boundaryvalue} gives a nonnegative coefficient
of \(\dd Q\) for \(T>0\), and a nonpositive one for \(T<0\).
Since
\[
 \partial_tQ=-\frac{T}{rr'},
\]
the first temporal covector is negative.
This is exactly the orientation \(\lambda T>0\) in
\eqref{eq:Gcoordinates}.

\paragraph{3. The branch point \(Q=1\).}
At an angular value not congruent to zero, the evenness in
\(\eta\) makes \eqref{eq:reducedF} analytic in \(Q\).
Near \(\delta=0\pmod{2\pi}\), the factorization in the proof of
Lemma~\ref{lem:local} gives the direct geometric conormal.
At a diagonal point the complete local wavefront is the usual
positive-frequency diagonal null relation by
\eqref{eq:localHadamard}.

\paragraph{4. The branch point \(Q=-1\), including \(\delta\in\Bc\).}
Here \(T\neq0\). The map
\[
 (X,X')\longmapsto(Q,\delta)
\]
is a submersion, and \eqref{eq:reducedboundary} expresses the
kernel as the pullback of a two-variable distribution, multiplied
by the smooth factor \(1/(8\pi^2\alpha rr')\).
The pullback theorem therefore gives
\[
 (\kappa,\kappa')
 =\lambda\,\dd Q+\nu\,\dd\delta
 \label{eq:span}
\]
for every possible wavefront covector at this branch point.
There is no contribution from passive coordinates.

On \(F_{\mathrm{ap}}=0\),
\[
 \dd Q=\frac1{2rr'}\dd F_{\mathrm{ap}}.
\]
The first endpoint part of \(\dd Q\) is null and has no angular
component. Hence
\[
 p(X,\kappa)=\frac{\nu^2}{\alpha^2r^2}.
\]
The bisolution equation \eqref{eq:bisolution} and microlocal
elliptic regularity force \(p(X,\kappa)=0\), so \(\nu=0\).
The total covector is nonzero, therefore \(\lambda\neq0\);
both endpoint covectors are nonzero.
Lemma~\ref{lem:boundaryvalue} fixes its positive-frequency
orientation. This is precisely \(\Dc^+\).

This last argument applies to the \emph{total summed kernel}.
It does not assign or differentiate separate values of moving
image-sector indicators.

\paragraph{5. Exhaustion.}
At \(T=0\), the only possible singular points are on the diagonal:
indeed \(Q\geq1\), with equality only when \(r=r'\) and \(Z=0\),
and a singular angular value then requires
\(\delta=0\pmod{2\pi}\).
The preceding cases cover all of \(M\times M\).
\end{proof}

\begin{remark}[Why support or a remainder estimate would not suffice]
The proof uses a reduced-variable pullback theorem, an oriented
boundary-value theorem, and the characteristic equations.
Neither finite propagation nor a finite-order estimate alone
would prove \eqref{eq:upper}. This is the distinction between
distributional control and a wavefront theorem also emphasized
in the supplied Em01-D material \cite{CheungEmOneD}.
\end{remark}

\subsection{The geometric lower bound}

\begin{proposition}[Every closed geometric covector occurs]
\label{prop:geolower}
For all parameters,
\begin{equation}
 \Gc^+\subset\WF(W_\beta).
 \label{eq:geolower}
\end{equation}
\end{proposition}

\begin{proof}
Lemma~\ref{lem:local} gives every positive diagonal null covector.
Fix a regular null geodesic segment starting at \(X'\).
Propagation of singularities for \(P_X\), with the primed
variables retained as parameters, transports the first covector
along that entire segment \cite{HormanderIV}.
The path remains in the smooth regular host; no passage through
the apex is made in this argument.
Thus every element of \(\Gc_{\mathrm{reg}}^+\) occurs.
The wavefront set is closed in the punctured cotangent bundle,
so it contains \(\Gc^+\).
\end{proof}

One can also read the nonzero coefficient directly from
\eqref{eq:Sgeom}: a geometric branch has principal term
\[
 \frac{e^{2\pi\ii\beta k}}
            {4\pi^2\bigl(R_k^2-(T-\ii0)^2\bigr)}.
\]
Its character factor is never zero. Propagation avoids any
need to select a leading term at a crossing of geometric fronts.

%======================================================================
\section{The diffracted coefficient and its nonvanishing criterion}
\label{sec:diffraction}
%======================================================================

Set
\[
 c=\frac{\pi}{\alpha}.
\]
On the complement of the angular transition set, define
\begin{equation}
 \Acoef(\delta)=
 \frac{
 e^{\ii\beta\delta}\sin((1-\beta)c)
 +e^{-\ii(1-\beta)\delta}\sin(\beta c)}
 {\cos c-\cos\delta}.
 \label{eq:A}
\end{equation}
This is a complex, real-analytic angular function in each
transition-free angular interval.

\begin{lemma}[Exact local form at a strictly diffracted front]
\label{lem:diffnormal}
Near a point with \(F_{\mathrm{ap}}=0\) and
\(\delta\notin\Bc\), there are smooth functions \(A_*\) and
\(B_*\), analytic in the local real variables, such that
\begin{equation}
 W_\beta
 =A_*
   \bigl(F_{\mathrm{ap}}+\ii0\,\sgn T\bigr)^{-1/2}
       +B_*,
 \label{eq:diffnormal}
\end{equation}
and
\begin{equation}
 A_*\big|_{F_{\mathrm{ap}}=0}
 =\frac{\Acoef(\delta)}
             {8\pi^2\alpha\sqrt{rr'}}.
 \label{eq:diffcoefficient}
\end{equation}
The square root is the principal boundary value.
\end{lemma}

\begin{proof}
Take first \(T>0\). Near \(Q=-1\), write
\(\eta=\ii\pi+w\). Since
\[
 Q=-\cosh w,
 \qquad
 Q+1=\frac{F_{\mathrm{ap}}}{2rr'},
\]
one has
\[
 w=-\ii\sqrt{\frac{F_{\mathrm{ap}}+\ii0}{rr'}}
                \bigl(1+O(F_{\mathrm{ap}})\bigr),
\]
\[
 \sinh\eta
 =\ii\sqrt{\frac{F_{\mathrm{ap}}+\ii0}{rr'}}
                \bigl(1+O(F_{\mathrm{ap}})\bigr).
\]
The angular denominator in \eqref{eq:Sclosed} is nonzero at
\(w=0\), and
\[
 S_\beta(\ii\pi,\delta)=\ii\Acoef(\delta).
\]
Expand \(S_\beta(\ii\pi+w,\delta)/\sinh(\ii\pi+w)\) in its
convergent Laurent series in \(w\).
Its even and odd powers separate into an analytic function of
\(F_{\mathrm{ap}}\) plus
\((F_{\mathrm{ap}}+\ii0)^{-1/2}\) times another analytic function.
Multiplication by the prefactor in \eqref{eq:Wclosed} gives
\eqref{eq:diffcoefficient}.

For \(T<0\), expand at \(\eta=-\ii\pi\).
Oddness of \(S_\beta\) and of \(\sinh\eta\) gives the same
coefficient and the lower boundary value.
\end{proof}

\begin{lemma}[Wavefront of the boundary square root]
\label{lem:sqrtWF}
At \(s=0\),
\[
 \WF((s+\ii0)^{-1/2})
 =\{(0,\lambda):\lambda>0\},
\]
and the lower boundary value has \(\lambda<0\).
Multiplication by a smooth coefficient nonzero on \(s=0\)
preserves that conormal wavefront.
\end{lemma}

\begin{proof}
The representation
\[
 (s+\ii0)^{-1/2}
 =\frac{e^{-\ii\pi/4}}{\sqrt\pi}
       \int_0^\infty e^{\ii\lambda s}\lambda^{-1/2}\,\dd\lambda
\]
is understood as a damped oscillatory integral.
It gives the one-sided wavefront and an elliptic conormal
amplitude at every positive \(\lambda\).
The lower boundary value is obtained by conjugating the scalar
model. Smooth elliptic multiplication preserves the wavefront
\cite{HormanderI}.
\end{proof}

\begin{definition}[Finite-character quotient parameters]
\label{def:exceptional}
Define
\begin{equation}
 \Exc=
 \left\{
 (\alpha,\beta):
 \alpha=\frac1N,\quad
 \beta=\frac pN,\quad
 N\in\NN,\ N\geq2,\quad 0\leq p<N
 \right\}.
 \label{eq:exceptional}
\end{equation}
\end{definition}

\begin{proposition}[Exact criterion for absence of a strict
diffractive coefficient]
\label{prop:criterion}
The function \(\Acoef\) vanishes identically on a nonempty
transition-free angular interval if and only if
\((\alpha,\beta)\in\Exc\).

If \((\alpha,\beta)\notin\Exc\), its nonzero angular values are
dense in the complement of \(\Bc\), and
\begin{equation}
 \Dc^+\subset\WF(W_\beta).
 \label{eq:difflower}
\end{equation}
\end{proposition}

\begin{proof}
Multiplying the numerator in \eqref{eq:A} by
\(e^{-\ii\beta\delta}\) gives
\[
 \sin((1-\beta)c)+e^{-\ii\delta}\sin(\beta c).
\]
It vanishes on an interval precisely when both constants vanish:
\[
 \frac{\beta}{\alpha}\in\ZZ,\qquad
 \frac{1-\beta}{\alpha}\in\ZZ.
\]
Their sum gives \(1/\alpha=N\in\NN\), \(N\geq2\), and then
\(\beta=p/N\) with \(0\leq p<N\).
The converse is immediate.

Outside this exceptional case, the analytic numerator has no
open zero interval. At every nonzero coefficient value,
Lemmas~\ref{lem:diffnormal} and \ref{lem:sqrtWF} give all of
\(\Dc^+\) over that angular value.
Closedness of the joint wavefront set supplies the same radial
covectors at its angular zeros and at the transition angles.
This proves \eqref{eq:difflower}.
\end{proof}

\begin{remark}[A zero on an angular slice is not joint smoothness]
At a special fixed value of \(\delta\), a restricted scalar
coefficient can vanish. That is not a neighbourhood on
\(M\times M\). The lower bound above is a joint statement in
the angular and remaining variables. No inference from a
single angular slice is used.
\end{remark}

\subsection{Comparison with the image--diffraction representation}

For orientation and normalization, define
\[
 \rho(u)^2=r^2+r'^2+2rr'\cosh u+Z^2,
\]
\[
 \Phi_\beta(u,\delta)
 =\sum_{n\in\ZZ}
 \sin(\pi\nu_n)e^{-\nu_nu}e^{\ii(n+\beta)\delta},
 \qquad u>0.
\]
The notation here uses \(\delta\), rather than the developing
angle \(\alpha\delta\), as its second argument.

The Bessel angular identity gives, off the transition angles,
\begin{align}
 W_\beta
 ={}&
 \sum_{\substack{k\in\ZZ\\|\theta_k|<\pi}}
 \frac{e^{2\pi\ii\beta k}}
       {4\pi^2\bigl(R_k^2-(T-\ii0)^2\bigr)}
 \nonumber\\
 &-\frac1{4\pi^3\alpha}
 \int_0^\infty
 \frac{\Phi_\beta(u,\delta)}
      {\rho(u)^2-(T-\ii0)^2}\,\dd u.
 \label{eq:contourW}
\end{align}
One obtains this formula by applying the heat-kernel angular
summation before integrating the Euclidean time heat factor,
and then continuing its \emph{total} kernel.
It is the covariance counterpart of the
Sommerfeld--Carslaw--Dowker formulas in
I-B-G and I-B-VP \cite{CheungIBG,CheungIBVP}.
At transition angles the joint distribution, not separate
pointwise sector values, is the object.

The coefficient in \eqref{eq:A} can be checked without assigning
a value to a divergent angular series:
\[
 \Phi_\beta(u,\delta)
 =\frac{S_\beta(u-\ii\pi,\delta)
             -S_\beta(u+\ii\pi,\delta)}{2\ii},
 \qquad u>0.
\]
Its transition-free limit is therefore
\[
 \Phi_\beta(0,\delta)=-\Acoef(\delta).
\]
Since
\[
 \rho(u)^2-(T-\ii0)^2
 =F_{\mathrm{ap}}+rr'u^2+O(u^4)+\ii0\,\sgn T,
\]
the endpoint model
\[
 \int_0^\infty
       \frac{\dd u}{s+rr'u^2}
 =\frac{\pi}{2\sqrt{rr'}}\,s^{-1/2}
\]
recovers \eqref{eq:diffcoefficient}.
The full wavefront proof at the angular transition was instead
given by the reduced-variable argument of
Proposition~\ref{prop:upper}; this endpoint check is not used to
differentiate a singular moving image sector.

\begin{corollary}[Relative strength away from transitions]
\label{cor:strength}
At a nonzero diffractive coefficient and away from geometric
fronts, the transverse Fourier amplitude of \(W_\beta\) has order
\(-1/2\). At a single noncoincident geometric front away from
other fronts it has order \(0\).
Thus the strictly diffracted covariance is one-half derivative
smoother in its normal variable.

Equivalently, at such elliptic points the local normal Sobolev
thresholds are \(s<0\) for the diffracted front and \(s<-1/2\)
for the geometric front.
No uniform half-order assertion at their transition is made.
\end{corollary}

\begin{proof}
The square-root model has Fourier amplitude
\(\lambda^{-1/2}\) on a half-line.
The simple-pole geometric model has amplitude of order \(0\).
After localization, the corresponding one-dimensional Sobolev
integrals have thresholds
\[
 \int_1^\infty \lambda^{2s-1}\,\dd\lambda<\infty
 \quad\Longleftrightarrow\quad s<0,
\]
and
\[
 \int_1^\infty \lambda^{2s}\,\dd\lambda<\infty
 \quad\Longleftrightarrow\quad s<-1/2.
\]
The nonzero leading coefficients make these thresholds sharp.
\end{proof}

%======================================================================
\section{The quotient sectors and the exact global theorem}
\label{sec:main}
%======================================================================

\subsection{An independent finite-image construction}

\begin{proposition}[Finite-character covariance]
\label{prop:quotient}
If \(\alpha=1/N\) and \(\beta=p/N\), then
\begin{equation}
 W_{p/N}(X,X')
 =\sum_{k=0}^{N-1}e^{2\pi\ii pk/N}
       W_{\mathrm{flat}}\bigl(\widetilde X,\mathcal R_k\widetilde X'\bigr),
 \label{eq:finiteimages}
\end{equation}
where
\[
 \widetilde X=(t,r\cos(\varphi/N),r\sin(\varphi/N),z),
\]
\(\mathcal R_k\) rotates the covering transverse plane by
\(2\pi k/N\), and
\[
 W_{\mathrm{flat}}(Y,Y')
 =\frac1{4\pi^2\bigl(
 |\mathbf Y-\mathbf Y'|^2-(t-t'-\ii0)^2\bigr)}.
\]
Consequently,
\begin{equation}
 \WF(W_{p/N})=\Gc^+.
 \label{eq:quotientWF}
\end{equation}
\end{proposition}

\begin{proof}
Apply the character-weighted image sum first to the flat heat
kernel. Its angular Fourier sum projects covering modes
\(m\in\ZZ\) to
\[
 m=Nn+p,
\]
because
\[
 \sum_{k=0}^{N-1}e^{2\pi\ii(p-m)k/N}
 =
 \begin{cases}
 N,&m\equiv p\pmod N,\\
 0,&\text{otherwise}.
 \end{cases}
\]
The projected angular factor is
\(e^{\ii(n+p/N)(\varphi-\varphi')}\), and the Bessel order is
\(|Nn+p|=|n+\beta|/\alpha\).
The factor \(N\) supplies precisely the quotient normalization
\(1/\alpha\) relative to the conical volume.
Thus the result is the Friedrichs heat kernel \eqref{eq:heat}.
Euclidean integration and spectral continuation give
\eqref{eq:finiteimages}.

Every image has the ordinary flat null wavefront relation.
If its straight segment avoids the origin, it gives a regular
geodesic on the quotient. If it crosses the origin, its endpoint
covector is a limit of such geodesics.
Thus the finite sum has wavefront contained in \(\Gc^+\).
Proposition~\ref{prop:geolower} gives the reverse inclusion.
This argument uses the full finite image sum, including boundary
representatives when \(N\) is even.
\end{proof}

\subsection{Exact global wavefront equality}

\begin{theorem}[Global Friedrichs wavefront relation]
\label{thm:main}
For the massless Friedrichs spectral covariance
\eqref{eq:W} on the regular host \eqref{eq:host},
\begin{equation}
 \boxed{
 \WF(W_\beta)=
 \begin{cases}
 \Gc^+,&(\alpha,\beta)\in\Exc,\\[2pt]
 \Gc^+\cup\Dc^+,&(\alpha,\beta)\notin\Exc.
 \end{cases}}
 \label{eq:main}
\end{equation}
The equality holds on all of \(M\times M\), including the
diagonal, crossings of geometric fronts, and every
geometric--diffractive angular transition.

Every first endpoint covector is future null, every second
endpoint covector is past null, and neither endpoint covector
is zero.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:upper} gives the global upper bound.
Proposition~\ref{prop:geolower} gives the geometric lower bound.
Outside \(\Exc\), Proposition~\ref{prop:criterion} gives the
entire diffracted lower bound.
Inside \(\Exc\), Proposition~\ref{prop:quotient} removes every
strictly diffractive covector outside \(\Gc^+\).
The orientation and nonzero-endpoint statements were established
in Proposition~\ref{prop:upper} and the definitions of the two
relations.
\end{proof}

\begin{corollary}[Singular support and the smooth tail]
\label{cor:singsupp}
The singular support is the projection to the base of the
relation in \eqref{eq:main}. In particular:
\begin{enumerate}[label=\textup{(\roman*)}]
\item off the diagonal, geometric singularities occur at the
regular or limiting geometric travel times \(T=\pm R_k\);
\item in nonquotient sectors, the additional singular front is
\(T=\pm L_{\mathrm{ap}}\);
\item the open region \(T^2>L_{\mathrm{ap}}^2\) is smooth.
\end{enumerate}
The third statement does not say that the covariance or causal
kernel vanishes there.
\end{corollary}

\begin{proof}
The first two statements follow from the exact wavefront equality.
For the third, \(Q<-1\), where the summed boundary expression is
smooth by the first step of Proposition~\ref{prop:upper}.
\end{proof}

\begin{remark}[What the result contributes]
The exact equality, the treatment of all angular transitions,
and the holonomy criterion for disappearance of the strictly
diffractive relation are the mathematical release of this
companion.
Cone diffraction and quotient image methods are established
subjects \cite{CheegerTaylorI,CheegerTaylorII,MelroseWunsch,Yang};
the present paper does not claim that these phenomena originate
in the Enter framework.
\end{remark}

%======================================================================
\section{Positive frequency, the commutator, and the Hadamard distinction}
\label{sec:spectrum}
%======================================================================

\subsection{The correct charged ordering identity}

Define
\[
 E_\beta(T)=\frac{\sin(TH_\beta)}{H_\beta}.
\]
The bundle parameter \(-\beta\) below is understood modulo one.

\begin{proposition}[Spectral splitting and charged commutator]
\label{prop:CCR}
At the operator-kernel level,
\begin{align}
 \frac{W_\beta(T)+W_\beta(-T)}2
 &=\frac{\cos(TH_\beta)}{2H_\beta},
 \label{eq:even}\\
 W_\beta(T)-W_\beta(-T)&=-\ii E_\beta(T).
 \label{eq:odd}
\end{align}
For the charged spacetime kernels, the second term in
\eqref{eq:odd} is the reversed opposite-charge ordering:
\begin{equation}
 W_\beta(X,X')-W_{-\beta}(X',X)
 =-\ii E_\beta(X,X').
 \label{eq:chargedCCR}
\end{equation}
Thus the state has the extension-selected canonical commutator.
The pointwise real and imaginary parts of the scalar expression
for \(W_\beta\) in a parallel frame are not, in general, the two
operators in \eqref{eq:even}--\eqref{eq:odd}.
\end{proposition}

\begin{proof}
The first two identities are the elementary spectral cosine
and sine identities for \eqref{eq:W}.
Complex conjugation of the spatial bundle character, followed
by endpoint reversal, identifies
\[
 W_{-\beta}(-T;x',x)=W_\beta(-T;x,x').
\]
This can be checked directly by reindexing the angular modes.
The antiparticle Fock construction gives precisely this reversed
operator ordering, proving \eqref{eq:chargedCCR}.

In a twisted parallel representation, the angular factors
\(e^{\ii(n+\beta)\delta}\) can make the time-even spatial kernel
complex. Therefore scalar complex conjugation at the same
ordered pair is not the operation \(T\mapsto-T\) with spatial
endpoints retained.
\end{proof}

For the neutral real scalar, this reduces to
\[
 [\phi(X),\phi(X')]=-\ii E_0(X,X').
\]
The sign agrees with the spectral convention in the supplied
I-B-G and I-B-VP records \cite{CheungIBG,CheungIBVP}.

Let \(\Gc^-=-\Gc^+\) and \(\Dc^-=-\Dc^+\), where the minus sign
negates cotangent fibers without changing base points.

\begin{corollary}[Wavefront of the causal kernel]
\label{cor:EWF}
The wavefront of \(E_\beta\) is the union of both frequency
orientations of \eqref{eq:main}:
\[
 \WF(E_\beta)=
 \begin{cases}
 \Gc^+\cup\Gc^-,
       &(\alpha,\beta)\in\Exc,\\
 \Gc^+\cup\Gc^-\cup\Dc^+\cup\Dc^-,
       &(\alpha,\beta)\notin\Exc.
 \end{cases}
\]
\end{corollary}

\begin{proof}
Equation~\eqref{eq:odd} writes \(E_\beta\) as the difference
of two kernels with opposite, disjoint temporal orientations.
Theorem~\ref{thm:main}, applied to time reversal, identifies the
two sets. A microlocal singularity in one temporal orientation
cannot be cancelled by a kernel smooth in that orientation.
\end{proof}

Finite propagation of the Friedrichs evolution gives causal
commutator support for the intrinsic spatial length metric, as in
I-B-G \SourceLabel{thm:speed} \cite{CheungIBG}.
This support statement is compatible with the diffracted
relation; it is not a support statement for \(W_\beta\) itself.

\subsection{Local Hadamard form is not a geodesic-only global equality}

Radzikowski's theorem relates the Hadamard condition and an
oriented geodesic wavefront condition under its globally
hyperbolic free-field hypotheses \cite{Radzikowski}.
Here one can still \emph{define} the relation of ordinary regular
null geodesics, but the hypotheses of that global theorem do not
hold.

\begin{proposition}[The additional global relations are genuine]
\label{prop:notordinary}
The covariance is locally of Hadamard form at every regular
point. Nevertheless:
\begin{enumerate}[label=\textup{(\roman*)}]
\item outside \(\Exc\), its global wavefront contains all strictly
diffractive pairs in
\(\Dc^+\setminus\Gc^+\), which are not linked by a regular null
geodesic with the displayed endpoint covectors;
\item even in a finite-character quotient sector, the closed
geometric relation can contain limiting rays through the removed
apex that are not actual regular geodesics.
\end{enumerate}
\end{proposition}

\begin{proof}
Local Hadamard form is Lemma~\ref{lem:local}.
For (i), choose \(\delta\notin\Bc\) on the apex front.
Proposition~\ref{prop:broken} shows that its radial covector pair
is outside \(\Gc^+\), while Theorem~\ref{thm:main} places it in
\(\WF(W_\beta)\).
A regular geodesic carrying that radial covector cannot avoid
the apex and connect the two endpoints.

For (ii), the finite image kernel includes a flat null segment
whose transverse projection crosses the covering origin whenever
the corresponding image endpoints are antipodal.
It is the limit of regular quotient geodesics and hence belongs
to \(\Gc^+\), but its limiting radial segment is not in \(M\).
\end{proof}

The exact result is therefore a positive-frequency spectrum
condition for a specified geometric-plus-apex relation.
Calling it a geodesic-only globally Hadamard theorem would
discard part of the actual wavefront.
Conversely, its additional global singularities do not invalidate
local Hadamard point splitting.

%======================================================================
\section{Products, restrictions, and the one-point companions}
\label{sec:calculus}
%======================================================================

\subsection{Same-orientation products}

\begin{proposition}[Products justified by the global orientation]
\label{prop:products}
Let \(W_{\beta_1},\ldots,W_{\beta_m}\) be covariances of the
form considered here, with the same time orientation.
Their product as kernels on \(M\times M\), with the appropriate
tensor-product bundle, is defined by the ordinary wavefront
product theorem. The same holds after any fixed finite
differentiations of the factors.

This assertion does not supply a product between arbitrary
oppositely oriented kernels or an interacting algebra.
\end{proposition}

\begin{proof}
Every nonzero wavefront covector of every factor has a
future-null first component.
A finite sum of nonzero future-causal covectors cannot vanish.
Thus no selection of wavefront covectors from the factors can
sum to the zero covector on \(M\times M\).
The product criterion applies \cite{HormanderI}.
Differentiation does not enlarge a wavefront set.
\end{proof}

For example, the ordinary sufficient criterion fails for
\(W_\beta\overline{W_\beta}\) at an elliptic geometric null
covector: the two factors have opposite covectors there.
That criterion failure is not a universal nonexistence theorem
for every renormalized prescription.

The examples in VII-H and the declared polynomial domains in
XI-E are methodological comparisons
\cite{CheungVIIH,CheungXIE}. Their observables and products are
not identified with the present conical bundle kernels.

\subsection{Equal-time and timelike restrictions}

\begin{proposition}[Two useful pullbacks]
\label{prop:pullbacks}
The equal-time restriction of \(W_\beta\) exists and equals the
spatial covariance \(1/(2H_\beta)\).
For any smooth timelike curve \(\gamma\) in \(M\), the pullback
\[
 (\gamma\times\gamma)^*W_\beta
\]
also exists as a bidistribution on its parameter domain.
\end{proposition}

\begin{proof}
A covector normal to an equal-time embedding has only temporal
components at its endpoints.
A nonzero null covector cannot be purely temporal, so the
normal set is disjoint from \(\WF(W_\beta)\).
The spectral restriction is therefore legitimate and gives
\(1/(2H_\beta)\).

A nonzero null covector cannot annihilate a timelike tangent.
Hence neither endpoint of a wavefront pair can lie in the
annihilator required to obstruct the timelike pullback.
The wavefront pullback theorem applies \cite{HormanderI}.
\end{proof}

\subsection{The diagonal is different}

\begin{proposition}[Point splitting still requires subtraction]
\label{prop:diagonal}
The unrenormalized diagonal pullback of \(W_\beta\) fails the
ordinary wavefront criterion. In contrast, the local remainder
\(H_\beta\) in \eqref{eq:localHadamard} has well-defined diagonal
jets of every finite order.
Consequently the Wick square, stress tensor, and current
prescriptions of the regular-part companions remain
well-defined at every point of \(M\).
\end{proposition}

\begin{proof}
The diagonal wavefront contains
\[
 (X,\kappa;X,-\kappa),
 \qquad \kappa\neq0,\quad \kappa\text{ future null}.
\]
These covectors are normal to the diagonal embedding.
Thus one cannot justify the unrenormalized pullback by the
ordinary criterion.
The subtracted remainder is smooth in a neighbourhood of each
regular diagonal point, so all its required local diagonal
derivatives exist.

The apex front cannot meet the spacetime diagonal:
at \(X=X'\),
\(F_{\mathrm{ap}}=4r^2>0\).
The additional global fronts therefore do not alter this local
subtraction argument.
\end{proof}

For the charged current, one must retain the complex first jet
of \(H_\beta\), as in I-B-VP-0
\cite[\SourceLabel{rem:imaginary},
      \SourceLabel{def:current}]{CheungIBVPZero}.
The global theorem does not turn that prescription into the
real-part stress prescription.

The apex finite parts of I-B-VP-1 concern one-point
distributions in a separately declared completion
\cite{CheungIBVPOne}. Their contact parameters neither change
\(W_\beta\) on \(M\times M\) nor provide an extension of its
two-point kernel to \(\WW\).

%======================================================================
\section{A regular timelike diffraction echo}
\label{sec:echo}
%======================================================================

The global relation has a direct consequence for an already
well-defined restriction, without constructing a detector model.

Fix a static worldline
\[
 \gamma(u)=(u,r_0,\varphi_0,z_0),\qquad r_0>0.
\]
Its proper-time parameter is \(u\).

\begin{corollary}[A nonzero delayed singularity on a static restriction]
\label{cor:echo}
Suppose
\[
 \cos(\pi/\alpha)\neq1,\qquad
 \Acoef(0)\neq0.
 \label{eq:echoconditions}
\]
Then, near \(u-u'=\pm2r_0\),
\begin{align}
 (\gamma\times\gamma)^*W_\beta
 ={}&
 \frac{\Acoef(0)}{8\pi^2\alpha r_0}
 \bigl(4r_0^2-(u-u'-\ii0)^2\bigr)^{-1/2}
 \nonumber\\
 &+\text{terms of lower singular order and smooth terms}.
 \label{eq:echo}
\end{align}
The restricted covariance therefore has a genuine delayed
singularity at each of those nonzero proper-time separations.

For example, \(\alpha=2/3,\ \beta=0\) gives
\(\Acoef(0)=1\), and the hypotheses hold.
\end{corollary}

\begin{proof}
The pullback exists by Proposition~\ref{prop:pullbacks}.
On this worldline, \(\delta=Z=0\) and
\(F_{\mathrm{ap}}=4r_0^2-(u-u')^2\).
The first condition in \eqref{eq:echoconditions} excludes an
angular transition at \(\delta=0\).
Every geometric travel time at this fixed spatial point is
strictly below \(2r_0\) unless it is a transition, so no other
front meets a sufficiently small neighbourhood of either
displayed separation.
Lemma~\ref{lem:diffnormal} and
\eqref{eq:diffcoefficient} now give \eqref{eq:echo}.
Nonvanishing of the coefficient and
Lemma~\ref{lem:sqrtWF} prove the singularity.

For the example,
\(c=3\pi/2\), so
\[
 \Acoef(0)=\frac{\sin(3\pi/2)}{\cos(3\pi/2)-1}=1.
\]
\end{proof}

This is a statement about a free-field covariance restricted to
a prescribed worldline. It does not assert a detector response
rate, an acquisition channel, or the performance of an Enter
readout. At angular or holonomy values excluded by
\eqref{eq:echoconditions}, this corollary makes no conclusion
from a vanishing leading coefficient.

%======================================================================
\section{Supported laboratories, reporting, and non-transfer}
\label{sec:interfaces}
%======================================================================

\subsection{What supported Green agreement actually implies}

I-B-G's supported-transfer result concerns retarded and advanced
scalar operations in a signed arrival window, with independent
retained-shadow and compact-reserve conditions
\cite[\SourceLabel{thm:transfer},
      \SourceLabel{ass:transfer-contract}]{CheungIBG}.
The corresponding ST sheets retain those support obligations
\cite{CheungXAST,CheungIXAST}.

It is not a theorem that
\(W_\beta=W_{\mathrm{flat}}\) in that window.
Already for \(\beta=0\), the regular diagonal difference is
\[
 [H_0]=\frac{\alpha^{-2}-1}{48\pi^2r^2},
\]
which is nonzero for \(0<\alpha<1\)
\cite[\SourceLabel{thm:phi2}]{CheungIBVP}.
Thus exact covariance equality would contradict the
regular-part Wick square.

The correct local statement is
\[
 W_\beta-W_{\mathrm{flat}}\in C^\infty
\]
on a sufficiently small developing product neighbourhood,
and on any further specified pair region where the direct
singularity has been subtracted and no additional front is met.
The explicit kernel and Theorem~\ref{thm:main}, not a
state-transfer inference, justify that assertion.

\subsection{Finite records do not determine a wavefront}

The finite supplied-report interface can install words describing
\[
 (\alpha,\beta,\text{bundle convention},\text{formula},
   \text{microlocal localization},\text{status}).
\]
For a calibrated clock basis, put
\[
 D_j=\bra{\tau_j}_C\otimes I_A,\qquad
 \mathcal M(X)=\sum_j
       \proj{j,w_j}_R\otimes D_jXD_j^\dagger.
\]
Then
\[
 \sum_jD_j^\dagger D_j=I_{CA},
 \qquad
 \Tr_R\mathcal M(X)=\Tr_CX,
\]
also with every untouched finite reference
\cite{CheungIB,CheungEn09,CheungOOneST}.
The output contains \(A\), not a quantum return of \(C\).
It reports installed descriptions; it does not acquire an
unknown continuum wavefront set.

Finite linear tests do not determine a general distribution,
and therefore do not determine its wavefront set.
For example, for any fixed finite collection of tests one can
choose more point-supported distributions than tests, at
distinct regular points of a coordinate patch. Their evaluation
map has a nonzero kernel. A nonzero combination in that kernel
is singular but invisible to all the chosen tests.
This is a coverage limitation in an unrestricted distribution
class, not a construction of another Friedrichs covariance.
The class-relative distinction is also emphasized in
Oai282-A \cite{CheungOaiTwoEightTwo}.

\subsection{Evidence and unrelated analytic problems}

All wavefront conclusions here are \ANALYTIC{} conclusions from
the displayed representation and microlocal arguments.
No numerical sample is promoted to an equality of wavefront
sets.

A future numerical probe must retain its angular lifts, bundle
phases, boundary-value side, test localizations, frequency
directions, precision, cutoffs, raw residuals, and total error
allowances. We retain the distinctions
\ANALYTIC{}, \SUPPLIED{}, and \UNCERTIFIED{} used in
XVI-O1-ST \cite{CheungOOneST}.
No numerical execution, compiler execution, independent rebuild,
or proof-assistant check is claimed.

The apex-diffracted relation of this paper is not the
aligned-null characteristic contraction of the separate
XIV/XV programme. No joint characteristic graph map is
constructed. The supplied XIV-I, XIV-J, and XV-B records retain
their own domains and analytic obligations
\cite{CheungXIVI,CheungXIVJ,CheungXVB}.
A global theorem for this free two-point kernel cannot discharge
those different products.

%======================================================================
\section{Ledger update and conclusion}
\label{sec:conclusion}
%======================================================================

\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}L{.27\linewidth}L{.22\linewidth}L{.44\linewidth}@{}}
\toprule
Object & Status here & Exact boundary\\
\midrule
\endfirsthead
\toprule
Object & Status here & Exact boundary\\
\midrule
\endhead
\bottomrule
\endfoot

Massless Friedrichs spectral covariance on \(M\)
&
Retained and globally analyzed
&
The kernel and state prescription are \eqref{eq:W};
compact-test infrared control is retained.
\\

Local Hadamard form
&
Constructed
&
Lemma~\ref{lem:local}; each regular developing neighbourhood.
\\

Global ordinary two-point wavefront set
&
Exactly determined
&
Theorem~\ref{thm:main}, including all angular transitions.
This is the V2 result for the specified state.
\\

Strict diffraction
&
Classified
&
Absent exactly for \((\alpha,\beta)\in\Exc\);
otherwise the complete joint relation \(\Dc^+\) occurs.
\\

Positive frequency and causal-kernel orientation
&
Constructed
&
Theorem~\ref{thm:main} and
Corollary~\ref{cor:EWF}, with the charged ordering
\eqref{eq:chargedCCR}.
\\

Selected products and restrictions
&
Constructed
&
Same-orientation kernel products, equal-time pullback,
and timelike-curve pullback; not arbitrary interacting products.
\\

Global hyperbolicity or state uniqueness
&
Not established
&
Non-global-hyperbolicity remains.
No uniqueness among all algebraic ground states is claimed.
\\

Quantum state or field algebra across \(\WW\)
&
\textbf{Open here}
&
Neither the present wavefront theorem nor the one-point
finite parts of I-B-VP-1 construct it.
\\

Massive, non-Friedrichs, thermal, spinor, constrained, interacting cases
&
\textbf{Open here}
&
No transfer from the present exact massless covariance is asserted.
\\
\end{longtable}
\endgroup

\paragraph{The precise V2 update.}
For the fixed massless Friedrichs spectral ground state,
V2 is resolved at the level of the exact ordinary wavefront
set of the global two-point covariance on the regular host.
This is stronger than a generic-pair containment and includes
transition angles and exceptional quotient sectors.
It is not a classification of every state or every extension.

V1 retains the prescribed-completion qualifications of
I-B-VP-1 \cite{CheungIBVPOne}.
The other V-items retain their own extension, mass, thermal,
apex-curvature, spinor, and interacting scopes.
Here ``Open'' means not resolved in the declared companion
scope, not unknown throughout the literature.

\paragraph{No sourcing or programme-wide status change.}
The cone, character, Friedrichs condition, and spectral state
are prescribed inputs. The theorem does not derive the metric,
impose Einstein or Maxwell dynamics, solve backreaction, or
establish spacetime emergence
\cite{CheungIB,CheungXAST}.
No OpenAI catalogue claim is a mathematical premise.

No standing version-specific status elsewhere is updated.
Any O.1=\NOTCLOSED{} entry remains so unless independently
closed in its own stated scope; an independently reported
subsector result is neither verified nor revoked here.
XVI-N is not discharged, the downstream XV-B quantum-admission
exit is not opened, and \bsr{} is not promoted by this paper
\cite{CheungOOneST,CheungEn09}.
These are non-transfers, not corpus-wide impossibility claims.

\paragraph{Conclusion.}
Summing before continuation gives a global kernel whose only
possible nongeometric branch point is the apex travel front.
At its angular transition, reduced-variable pullback and the
bisolution equations remove the spurious angular covectors
that a moving-sector differentiation could create.
The exact diffraction coefficient then supplies the lower
bound, while finite-character projection identifies every
nondiffractive sector.
The resulting equality \eqref{eq:main} separates three facts:
the covariance is locally Hadamard, its frequency orientation
is globally controlled, and its global propagation relation
need not be the relation of regular null geodesics alone.

%======================================================================
\appendix
\section{Two checks on the transition argument}
\label{app:checks}
%======================================================================

\subsection{Why a null-set convention cannot control a wavefront}

At \(\delta\in\Bc\), the geometric image set changes and the
diffraction weight has its corresponding angular transition.
Choosing a pointwise value on that angular set does not determine
a distributional extension: distributions supported there are
not ruled out by agreement off the set.

The proof instead uses \eqref{eq:Wclosed}, already identified with
the spectral covariance before taking a boundary value.
At \(Q=-1\), every possible wavefront covector is constrained to
\[
 \lambda\,\dd Q+\nu\,\dd\delta.
\]
Since
\[
 Q+1=\frac{F_{\mathrm{ap}}}{2rr'},
\]
its endpoint components are
\[
 \kappa=
 -\frac{\lambda T}{rr'}\,\dd t
 +\frac{\lambda(r+r')}{rr'}\,\dd r
 +\nu\,\dd\varphi
 +\frac{\lambda Z}{rr'}\,\dd z,
\]
\[
 \kappa'=
 \frac{\lambda T}{rr'}\,\dd t'
 +\frac{\lambda(r+r')}{rr'}\,\dd r'
 -\nu\,\dd\varphi'
 -\frac{\lambda Z}{rr'}\,\dd z'.
\]
Their characteristic equations are
\[
 p(X,\kappa)=\frac{\nu^2}{\alpha^2r^2},\qquad
 p(X',\kappa')=\frac{\nu^2}{\alpha^2r'^2}.
\]
Thus both force \(\nu=0\).
This calculation works when one or two angular transition
representatives coincide, and in particular when
\(1/\alpha\) is an integer.
No separate endpoint prescription is needed.

\subsection{Why the exceptional criterion is joint}

The two conditions
\[
 \sin(\pi\beta/\alpha)=0,\qquad
 \sin(\pi(1-\beta)/\alpha)=0
\]
are simultaneous conditions on the entire angular coefficient.
They are stronger than cancellation at \(\delta=0\), or at any
other chosen angular slice.

When they hold, every Bessel order is an integer:
\[
 \nu_n=|Nn+p|.
\]
The finite-character heat projection then proves the exact
finite-image covariance, not merely cancellation of a first
diffractive asymptotic coefficient.
When they do not hold, analyticity and closedness of the
wavefront set prove the full \(\Dc^+\) lower bound.
These two different arguments are what make the exceptional
classification exhaustive.

%======================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}
%======================================================================

\paragraph{Mathematical domain.}
All two-point distributions and all wavefront equalities are on
the smooth regular spacetime \(M\), with \(r,r'>0\).
No value of a quantum field on \(\WW\) is defined.

\paragraph{Dependency scope.}
The Friedrichs operator, bundle convention, and spectral
identification are those stated explicitly in
Section~\ref{sec:state}.
The kernel summation, all-angle wavefront upper bound,
diffraction coefficient, quotient criterion, and equality
theorem are proved in this paper.
Other companion citations are identified as analytical inputs
or scoped comparisons; unquoted corpus material is not an
additional premise.

\paragraph{Novelty scope.}
The paper claims a completed analysis for the stated companion
problem. It does not claim invention of conical diffraction,
Aharonov--Bohm diffraction, the Hadamard condition, spectral
ground states, or microlocal product calculus.

\paragraph{Verification and submission scope.}
This is a single-file pdfLaTeX manuscript with an internal
bibliography. No successful build, numerical experiment,
independent referee verification, journal acceptance, or
proof-assistant certification is asserted.

%======================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
%======================================================================

\bibitem{Dowker}
J.~S.~Dowker,
Quantum field theory on a cone,
\emph{Journal of Physics A: Mathematical and General}
\textbf{10} (1977), 115--124.
\newblock DOI: \nolinkurl{10.1088/0305-4470/10/1/023}.
\newblock Historical cone-kernel context; the present wavefront
proof uses the displayed summed covariance.

\bibitem{Linet}
B.~Linet,
Quantum field theory in the space-time of a cosmic string,
\emph{Physical Review D} \textbf{35} (1987), 536--539.
\newblock DOI: \nolinkurl{10.1103/PhysRevD.35.536}.
\newblock Cited for cosmic-string Euclidean Green functions and
local subtraction, not for the global theorem proved here.

\bibitem{CheegerTaylorI}
J.~Cheeger and M.~Taylor,
On the diffraction of waves by conical singularities. I,
\emph{Communications on Pure and Applied Mathematics}
\textbf{35} (1982), 275--331.
\newblock DOI: \nolinkurl{10.1002/cpa.3160350302}.

\bibitem{CheegerTaylorII}
J.~Cheeger and M.~Taylor,
On the diffraction of waves by conical singularities. II,
\emph{Communications on Pure and Applied Mathematics}
\textbf{35} (1982), 487--529.
\newblock DOI: \nolinkurl{10.1002/cpa.3160350403}.

\bibitem{MelroseWunsch}
R.~Melrose and J.~Wunsch,
Propagation of singularities for the wave equation on conic manifolds,
\emph{Inventiones Mathematicae}
\textbf{156} (2004), 235--299.
\newblock \url{https://arxiv.org/abs/math/0111255}.
\newblock General conic propagation context; no unverified
application of a compact-cone theorem to the present edge
product is used.

\bibitem{MelroseVasyWunsch}
R.~Melrose, A.~Vasy, and J.~Wunsch,
\emph{Propagation of singularities for the wave equation on edge manifolds}.
\newblock \url{https://arxiv.org/abs/math/0612750}.
\newblock Cited for the broader geometric/diffractive edge
framework; the exact product-cone calculation here is explicit.

\bibitem{Yang}
M.~Yang,
\emph{Diffraction of the Aharonov--Bohm Hamiltonian}.
\newblock \url{https://arxiv.org/abs/2008.04129}.
\newblock Cited for magnetic diffraction and its diffraction
coefficient; its two-dimensional Hamiltonian is not identified
without calculation with the four-dimensional covariance.

\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.
\newblock DOI: \nolinkurl{10.1007/BF02100096}.
\newblock The globally hyperbolic equivalence theorem is a
comparison, not a local-to-global theorem applied to this
non-globally-hyperbolic host.

\bibitem{HormanderI}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I:
Distribution Theory and Fourier Analysis},
2nd edition, Springer, 1990.
\newblock Used for distributional boundary values, wavefront
sets, pullbacks, products, and microlocal elliptic regularity.

\bibitem{HormanderIV}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators IV:
Fourier Integral Operators},
Springer, 1985.
\newblock Used for propagation of singularities on the smooth
regular host; no propagation theorem through an unspecified
apex domain is invoked.

\bibitem{Watson}
G.~N.~Watson,
\emph{A Treatise on the Theory of Bessel Functions},
2nd edition, Cambridge University Press, 1944.
\newblock Used for the Bessel heat integral and its Laplace
transform.

\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.
\newblock DOI: \nolinkurl{10.1007/BF02102731}.
\newblock Cited for the independent role of the apex
self-adjoint extension.

\medskip
\noindent\textit{Cheung companion records.}
The following references are unpublished manuscripts or supplied
source excerpts, identified by version and source label.
Citation does not assert an audit of an entire record, a
successful compiler run, or formal verification.
\medskip

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter:
Destructive Readout, Quantum Recovery, and the Refusal of Identification},
version~2, \SourceLabel{EntPhil-v2.tex},
Enter Calculus Companion.
\newblock Introductory philosophical map, not a theorem premise.
\newblock \url{https://enter.grok.me/enter}.

\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.
\newblock Used: the prescribed regular host and worldsheet,
\SourceLabel{thm:instrument}, the Einstein comparator, and
\SourceLabel{prop:not-source}.
\newblock \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},
version~2, October 2026.
\newblock Supplied source \SourceLabel{I-B-G-v2.tex}
and quoted kernel passages, Enter Calculus Companion.
\newblock Relevant labels:
\SourceLabel{ass:host},
\SourceLabel{prop:nonGH},
\SourceLabel{thm:closed},
\SourceLabel{thm:speed},
\SourceLabel{cor:structure},
\SourceLabel{thm:transfer}, and
\SourceLabel{ass:transfer-contract}.
\newblock Supported propagation is not covariance or state transfer.

\bibitem{CheungIBVP}
N.~Cheung,
\emph{I-B-VP: Vacuum polarization of the Friedrichs ground state
on a prescribed conical spacetime:
Local Hadamard subtraction, holonomy, and the renormalized stress tensor},
version~2, October 2026.
\newblock Supplied source \SourceLabel{I-B-VP-v2.tex},
Enter Calculus Companion.
\newblock Relevant labels:
\SourceLabel{def:state},
\SourceLabel{lem:infrared},
\SourceLabel{prop:state},
\SourceLabel{prop:euclidean},
\SourceLabel{thm:analytic},
\SourceLabel{thm:hadamard},
\SourceLabel{thm:phi2}, and
\SourceLabel{prop:contour}.
\newblock Its V2 is the specific global two-point target
addressed here.

\bibitem{CheungIBVPZero}
N.~Cheung,
\emph{I-B-VP-0: The holonomy-odd vacuum current of the charged
Friedrichs ground state:
A local observable on the prescribed conical Enter spacetime},
version~1, October 2026.
\newblock Supplied source \SourceLabel{I-B-VP-0-v1.tex},
Enter Calculus Companion.
\newblock Relevant labels:
\SourceLabel{prop:kernel},
\SourceLabel{def:current},
\SourceLabel{rem:imaginary},
\SourceLabel{thm:current}, and the differentiated
image--diffraction appendix.
\newblock The current is a one-point observable, not the
two-point wavefront relation proved here.

\bibitem{CheungIBVPOne}
N.~Cheung,
\emph{I-B-VP-1: Prescribed apex finite parts on a conical Enter spacetime:
Wick square, stress tensor, and charged current:
strong extension, Ward identities, and classified contact freedom},
version~1, October 2026.
\newblock Supplied source \SourceLabel{I-B-VP-1-v1.tex},
Enter Calculus Companion.
\newblock Cited only to distinguish prescribed one-point
extensions in an auxiliary smooth completion from a two-point
state or field algebra across the apex.
\newblock Its finite-part and contact-space theorems are not
premises of the present wavefront proof.

\bibitem{CheungXAST}
N.~Cheung,
\emph{X-A-ST}, version~1, Enter Calculus Companion.
\newblock Used for the non-sourcing and supported-completion
distinctions, including
\SourceLabel{prop:noeinstein}.
\newblock \url{https://enter.grok.me/browse/X-A-ST}.

\bibitem{CheungIXAST}
N.~Cheung,
\emph{IX-A-ST}, version~1, Enter Calculus Companion.
\newblock Supplied excerpts concerning finite destructive
clock reporting and its distinction from a global continuum
field theory.
\newblock \url{https://enter.grok.me/browse/IX-A-ST}.

\bibitem{CheungVIIH}
N.~Cheung,
\emph{VII-H}, version~1,
\SourceLabel{VII-H.tex}, Enter Calculus Companion.
\newblock Supplied free-field response and release excerpts.
\newblock Methodological comparison for oriented products and
the restriction of an ambiguity class, not a transferred
conical product theorem.

\bibitem{CheungXIE}
N.~Cheung,
\emph{XI-E}, version~1,
\SourceLabel{XI-E.tex}, Enter Calculus Companion.
\newblock Supplied excerpts concerning its declared polynomial
Hadamard domain and separate classical calculations.
\newblock No conical interacting algebra or state selection is
imported.
\newblock \url{https://enter.grok.me/browse/XI-E}.

\bibitem{CheungEmOneD}
N.~Cheung,
\emph{Em01-D}, version~2,
\SourceLabel{Em01-D-v2.tex}, Enter Calculus Companion.
\newblock Supplied scope and Fourier-certificate descriptions.
\newblock Used only for the distinction between a finite-order
bound and an actual wavefront theorem.

\bibitem{CheungOOneST}
N.~Cheung,
\emph{XVI-O1-ST}, version~5,
\SourceLabel{XVI-O1-ST-v5.tex},
Enter Calculus Companion.
\newblock Used for the complete supplied-report instrument,
evidence classes, retained residuals, and execution boundaries.
\newblock \url{https://enter.grok.me/browse/XVI-O1-ST}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{En09-A}, version~1,
Enter Calculus Companion.
\newblock Supplied destructive-clock and standing-boundary
excerpts.
\newblock \url{https://enter.grok.me/browse/En09-A}.

\bibitem{CheungOaiTwoEightTwo}
N.~Cheung,
\emph{Oai282-A}, version~1,
Enter Calculus Companion.
\newblock Cited for the class-relative finite-test coverage
distinction in the supplied companion material.
\newblock No external catalogue theorem is a premise.
\newblock \url{https://enter.grok.me/browse/Oai282-A}.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{Landmark XIV-I}, version~1,
\SourceLabel{XIV-I.tex}, Enter Calculus Companion.
\newblock Supplied excerpts:
\SourceLabel{thm:I2},
\SourceLabel{thm:I3}, and
\SourceLabel{thm:release}.
\newblock Internal amplitude construction and the separate
unconstructed characteristic composition are recorded as
non-transfer boundaries only.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Landmark XIV-J}, version~1,
\SourceLabel{XIV-J.tex}, Enter Calculus Companion.
\newblock Supplied excerpts:
\SourceLabel{prop:descent},
\SourceLabel{thm:reference},
\SourceLabel{thm:J2}, and
\SourceLabel{cor:noadmission}.
\newblock Its supported-sector reference is not identified with
the present apex-diffracted covariance.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B}, version~2,
\SourceLabel{XV-B.tex}, Enter Calculus Companion.
\newblock Supplied relative-lifting, normal-jet, and release
excerpts.
\newblock Its technical existence and its distinct downstream
quantum-admission exit remain separate; neither is discharged by
the present free two-point theorem.

\end{thebibliography}

\end{document}