% ix-e-2.tex
% Intended engine: pdflatex.
% Run pdflatex twice to resolve cross-references.
%
% This manuscript evaluates the locked first-stage commutator.
% It does not construct a time-ordered product.
% Its release outcome is explicitly Unresolved: the strongly separated
% witness class is settled, but complete two-way support coverage is not.
%
% Internal source statements are imported only in the form supplied.
% No unprovided source manuscript, support theorem, or old product is
% reconstructed.

\documentclass[11pt]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathrsfs}
\usepackage[margin=1in]{geometry}
\usepackage{array,booktabs,longtable}
\usepackage{hyperref}

\hypersetup{
  colorlinks=true,
  linkcolor=blue,
  citecolor=blue,
  urlcolor=blue,
  pdftitle={Landmark IX-E-2: Exact imaginary part of the locked Cauchy kernel},
  pdfauthor={}
}

\setlength{\emergencystretch}{3em}

\newcommand{\dd}{\mathrm d}
\newcommand{\id}{\mathrm{id}}
\newcommand{\Span}{\operatorname{span}_{\mathbb C}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\fsupp}{\operatorname{supp}_{\mathrm{fun}}}
\newcommand{\Ker}{\operatorname{ker}}
\newcommand{\Imag}{\operatorname{Im}}
\newcommand{\Real}{\operatorname{Re}}
\newcommand{\R}{\mathbb R}
\newcommand{\Cplx}{\mathbb C}
\newcommand{\Mplus}{M_+}
\newcommand{\Mint}{M_+^\circ}
\newcommand{\Padm}{\mathscr P_{\mathrm{adm}}}
\newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}}
\newcommand{\Dfirst}{D_{\mathrm{first}}}
\newcommand{\Ddecl}{D_{\mathrm{decl}}}
\newcommand{\Dold}{D_{\mathrm{old}}}
\newcommand{\Dcmp}{D_{\mathrm{cmp}}}
\newcommand{\Dtwo}{\mathcal D_2}
\newcommand{\Otwo}{\mathcal O_2}
\newcommand{\Oact}{\mathcal O_{\mathrm{act}}}
\newcommand{\Ttwo}{\mathsf T_2}
\newcommand{\Athree}{\Cplx[\hbar]/(\hbar^4)}
\newcommand{\Vthree}{\mathscr V^{[3]}}
\newcommand{\Vplus}{\hbar\Vthree}
\newcommand{\Qfg}{\mathcal Q_{f,g}}
\newcommand{\QRgf}{\mathcal Q^R_{g,f}}
\newcommand{\Cfg}{\mathcal C_{f,g}}
\newcommand{\DK}{K-\overline K}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{hypothesis}[theorem]{Hypothesis}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\title{\textbf{Landmark IX-E-2}\\[0.4em]
Exact imaginary part of the locked Cauchy kernel\\
and the first-stage commutator\\
on the retained two-way sector}
\author{}
\date{}

\usepackage{tikz}
\begin{document}
\maketitle

\begin{abstract}
We evaluate the imaginary part of the locked mixed Cauchy kernel
and apply it to the first-stage commutator on the fixed comparison
domain of IX-E. A common Abel regulator gives an exact
distributional decomposition into a direct sine contribution and
two cosine contributions. Away from the spatial apex, the
imaginary part consists of a direct null-cone delta distribution
and a locally integrable contribution in the observation-spacelike
region between the direct and reflected cones. The direct and
reflected cosine tails cancel outside the reflected cone.

Consequently, the commutator vanishes identically on IX-E's entire
strongly separated two-way witness class. An explicit
observation-spacelike control has nonzero commutator, but is not
certified as a two-way pair of actual functional supports.
The remaining domain-wide obligation is therefore an actual
functional-support bridge, not another evaluation of the kernel
on the settled witness class.

The release remains \emph{Unresolved}. No time-ordered replacement,
contact classification, normalized retarded product, or
fixed-domain obstruction is claimed. The inherited free-sector
results and external non-identification boundaries are unchanged.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Commutator coverage};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Time-ordered construction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {This paper does not propose a time-ordered product.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Commutator coverage is not time ordering. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-IX-E-2}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited target};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Time-ordered construction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {This paper does not propose a time-ordered product.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Commutator coverage is not time ordering. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-IX-E-2}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited target};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Commutator coverage};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Time-ordered construction};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {This paper does not propose a time-ordered product.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-IX-E-2}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited target};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {Time-ordered construction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Commutator coverage. This paper does not propose a time-ordered product.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Inherited target is not Time-ordered construction. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-IX-E-2}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Inherited target};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Commutator coverage};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Time-ordered construction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {This paper does not propose a time-ordered product.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-IX-E-2}
\end{figure}

Figure~\ref{fig:split-IX-E-2} records one view of the same separation. Figure~\ref{fig:chain-IX-E-2} records one view of the same separation. Figure~\ref{fig:pair-IX-E-2} records one view of the same separation. Figure~\ref{fig:gate-IX-E-2} records one view of the same separation. The drawings add no hypothesis.


Figure~\ref{fig:scope-IX-E-2} records that separation.
The drawing adds no hypothesis.



\section{Release contract and fixed target}

\subsection{Scope and unchanged legal outcomes}

This manuscript is a commutator-coverage paper. It does not propose
a time-ordered construction.

The target inherited from IX-E is a complex-linear actual tensor map
\[
 \Ttwo:\Dtwo\longrightarrow\Vthree,
 \qquad
 \Vthree=\Padm\otimes_{\Cplx}\Athree.
\]
Here ``actual'' means that tensors are formed from actual
functionals, not from unquotiented test-function presentations.
The formal parameter is fixed by conjugation.

We retain IX-E's three legal outcomes unchanged:
\begin{enumerate}
\item \textbf{Constructed:} all coefficients exist and every declared
interface clause holds on the fixed domain.
\item \textbf{Proved obstructed:} a contradiction excludes every map
satisfying that interface on the fixed domain.
\item \textbf{Unresolved:} at least one required consistency,
extension, admission, contact, continuity, or overlap obligation
remains undischarged.
\end{enumerate}

The Constructed outcome is retained for series consistency but is
unavailable within the deliberately restricted scope of this paper.
A vanishing commutator is not a construction.

\subsection{Frozen domain}

Let \(\sigma\) be complex-linear flip and \(\gamma\) componentwise
antilinear conjugation:
\[
 \sigma(F\otimes G)=G\otimes F,
 \qquad
 \gamma(F\otimes G)=F^*\otimes G^*.
\]
Thus
\[
 d^\dagger=\sigma\gamma d,
 \qquad
 \sigma^2=\gamma^2=\id,
 \qquad
 \sigma\gamma=\gamma\sigma.
\]

The inherited first-stage space and frozen target are
\begin{align}
 \Dfirst
 &= (\mathscr B\otimes\mathscr W)
    +(\mathscr W\otimes\mathscr B),\\
 \Dtwo
 &=\Span\left\{
     \sigma^a\gamma^b d:
     d\in\Dfirst,\quad a,b\in\{0,1\}
   \right\},
 \qquad \Otwo=\{0\}.
 \label{eq:frozen-domain}
\end{align}
The spaces \(\mathscr B\) and \(\mathscr W\) are the inherited
spaces of actual first-stage functionals. In particular, every
\(B_f\otimes W_g\) under consideration is retained.

IX-E proves the stability and containment statements
\[
 \Dtwo\subset\Dcmp,
 \qquad
 \Dcmp=\Ddecl\cap\sigma(\Ddecl),
 \qquad
 \sigma\Dtwo=\gamma\Dtwo=\Dtwo^\dagger=\Dtwo.
 \label{eq:domain-stability}
\]
These are imported results, not new domain claims.

The zero old seed does not evaluate the actual old intersection:
\[
 \Oact=\Dtwo\cap\Dold,
 \qquad
 \Dold=\Pmc\otimes\Pmc.
 \label{eq:old-intersection}
\]
No nonzero old time-ordered map is supplied here. Conjugation
saturation cannot be ignored when auditing~\eqref{eq:old-intersection}.

\begin{center}
\fbox{\begin{minipage}{0.92\textwidth}
\textbf{Release box.}
The outcome is \textbf{Unresolved} on the fixed
domain~\eqref{eq:frozen-domain}.

\smallskip
\textbf{Witness-class result:}
the commutator vanishes exactly on every pair in IX-E's strongly
separated witness class.

\smallskip
\textbf{Fixed-domain result:}
no nonzero commutator on a certified retained two-way pair is
established. Complete two-way coverage remains unfinished at the
actual functional-support bridge.

\smallskip
No retained pair is deleted. No time-ordered replacement is
introduced. No implication from witness-class vanishing to a
constructed causal product is asserted.
\end{minipage}}
\end{center}

\section{Explicit inherited inputs and their limits}

The internal prerequisites below are the statements supplied for
this successor. Their attribution records the supplied provenance;
it does not assert independent verification of unprovided source
manuscripts. Every imported property used in a proof is stated
explicitly. References to other landmarks are not substitute
existence theorems~\cite{Inputs,IXE}.

\subsection{Sharp insertion, locked covariance, and free product}

Work in the flat half-space
\[
 \Mplus=[0,\infty)\times\R^3,
 \qquad
 \Sigma=\{0\}\times\R^3,
 \qquad
 \Mint=(0,\infty)\times\R^3.
\]
The configuration space is the unrestricted smooth off-shell
scalar and metric configuration space inherited from IX-A.
Neither an on-shell quotient nor a gauge-fixed quotient is used.

The sharp insertion is
\begin{equation}
 L=\mathsf E_0C,
 \qquad
 B_f[h,\phi]
 =-\int_{\Mplus}
 f(x)(Lh)^\mu(x)\partial_\mu\phi(x)\,\dd^4x.
 \label{eq:sharp-insertion}
\end{equation}
For \(x=(t,\mathbf x)\), \(y=(s,\mathbf y)\), the locked mixed
covariance supplied from VII-C and restated on the VII-D convention
sheet is
\begin{equation}
\begin{split}
 K(x,y)
 =\int_{\R^3}\frac{\dd^3p}{(2\pi)^3}
 e^{ip\cdot(\mathbf x-\mathbf y)}
 \Bigg[
 &\left(-\frac{t}{4|p|}+\frac{i}{8|p|^2}\right)
 e^{-i|p|(t-s)}
 \\
 &-\frac{i}{8|p|^2}e^{i|p|(t+s)}
 \Bigg].
\end{split}
\label{eq:locked-kernel}
\end{equation}
No graviton positivity is assumed. The inherited statements
\(H_{ww}=0\) and the characteristic-cone failure of \(B_f\) remain
unchanged.

The declared free product \(\star_c\) and its first-stage
coefficients are inputs:
\begin{align}
 B_f\star_cW_g
 &=B_fW_g+\hbar\Qfg,
 &
 \Qfg
 &=-2\pair{f(x)g(y)K(x,y)}
                 {\partial_t\phi(x)h_{00}(y)},\\
 W_g\star_cB_f
 &=B_fW_g+\hbar\QRgf,
 &
 \QRgf
 &=-2\pair{g(y)f(x)\overline{K(x,y)}}
                 {h_{00}(y)\partial_t\phi(x)}.
 \label{eq:ordered-coefficients}
\end{align}
These are the inherited exact first-stage formulas in the truncated
coefficient space. Their reversal identity is
\begin{equation}
 \star_c(d)^*=\star_c(d^\dagger).
 \label{eq:free-reversal}
\end{equation}

The released degree-two and degree-four descent, the word through
\(\hbar^3\), the evaluations of
\(\Ker p_L,\Ker p_R,\mathcal C_{\mathrm{desc}},
\mathcal C_{\mathrm{glue}}\), and F9a fixed-support continuity
are not reopened. F9a is not promoted to hypocontinuity.

Full support of \(K\), where inherited, is not used to determine its
imaginary part. A real distribution may have full support.

\subsection{The off-shell adjoint and support bounds}

For the adjoint variable \(b\) associated with a test covector \(a\),
the supplied VII-B identity is
\begin{align}
 q_a
 &=\theta C^*b+\delta(t)\mathsf B(b|_\Sigma),
 \label{eq:adjoint}\\
 \mathsf B^{\alpha\beta}(v)
 &=-n^{(\alpha}v^{\beta)}
   +\frac12\eta^{\alpha\beta}n^\mu v_\mu.
 \notag
\end{align}
Equivalently, in the inherited convention,
\[
 C^*(\theta b)
 =\theta C^*b+\delta(t)\mathsf B(b|_\Sigma).
 \label{eq:sharp-adjoint}
\]
There is no \(\delta'(t)\) term. The supplied on-shell smooth
representative does not replace this off-shell derivative.

For \(F=\supp f\) and \(G=\supp g\), the available support bounds are
\begin{equation}
 \fsupp B_f\subset P_F\cup F,
 \qquad
 P_F=J^-(F)\cap\{t\ge0\},
 \qquad
 \fsupp W_g\subset G.
 \label{eq:support-upper}
\end{equation}
These are upper inclusions. They are not lower support theorems.

\subsection{VII-C and the finite part are not repaired}

Under VII-C's joint hypotheses (E1)--(E4), namely the old Wick
product on the polynomial domain, off-shell admission of the sharp
insertion, sequential normal-topology admission, and hypocontinuous
multiplication with a continuous \(\hbar^3\) coefficient map, the
mandatory word has no sequentially continuous extension. Its
supplied separated-support residue is
\begin{equation}
 \mathcal C(f,g,f')
 =\frac{1}{64(2\pi)^3}
 \int g(y)\,\dd^4y
 \int_{S^2}
 A_f(y,\mathbf n)A_{f'}(y,\mathbf n)\,\dd\Omega(\mathbf n),
 \label{eq:old-residue}
\end{equation}
where \(A_f\) denotes the inherited reflected light-ray transform.
This residue is not subtracted here.

For the supplied VII-D regularization family,
\begin{equation}
 u_\mu^\rho
 =U_m+(\kappa_\rho+\log\mu)r,
 \qquad
 r=c_0\Psi_*\nu,
 \qquad
 c_0=\frac{1}{64(2\pi)^3},
 \label{eq:numerical-finite-part}
\end{equation}
is a strong distributional limit, and admissible mollifiers differ
by \(c_{ab}r\). This is numerical finite-part data, not an algebra
construction. Neither \(u_\mu^\rho+cr\) nor the triangle scalar is
renamed as \(\star_c\) or \(\Ttwo\).

\subsection{Other inherited boundaries}

V's externally retarded object
\[
 R_{1,1}=i(T_2-V\phi),
 \qquad
 u_f=-G_Af,
 \qquad
 q_{g,h,f}=\mathsf Q_{gh}u_f
\]
belongs to its compactly switched minimal vertex problem. Its
linear coefficient, finite-mode oscillator coefficient, and cubic
Newton--wave Wick form remain different objects. In particular,
V does not supply \(T_2(V,V)\) or the sharp map sought here.

VI's first-order compact-switch defect and VII-A's restricted
plateau intertwining statement are retained. The expression
\[
 \mathcal O^{(2)}
 +R_1(V_1;\mathcal O^{(1)})
 +R_1(V_2;\mathcal O^{(0)})
 +\frac12R_2(V_1,V_1;\mathcal O^{(0)})
\]
is an obligation list, not a construction imported into this paper.

VII-E/F and VIII-A--C supply their stated observation, exchange,
particle-link, affine-matching, packet, and scheduling certificates.
No supplied map identifies those outputs with \(B_f\), \(W_g\),
a triangle finite part, or an element of \(\Dtwo\).
VIII-C's token policy supplies a semantic promise, not an acquired
expectation.

\section{Pairing conventions and the commutator sign audit}

Set
\begin{equation}
 \tau=t-s,\qquad
 u=t+s,\qquad
 \mathbf z=\mathbf x-\mathbf y,\qquad
 R=|\mathbf z|.
 \label{eq:variables}
\end{equation}

Distributional pairing is complex-bilinear: no conjugation of the
test argument is implicit. Distributional conjugation is defined by
\[
 \pair{\overline T}{\chi}
 =\overline{\pair{T}{\overline\chi}},
 \qquad
 \Imag T=\frac{T-\overline T}{2i}.
\]
Kernel conjugation is at fixed \((x,y)\). It is not the operation
\(K(x,y)\mapsto K(y,x)\).

\begin{proposition}[Sign audit]
The inherited coefficients~\eqref{eq:ordered-coefficients} give
\begin{align}
 B_f\star_cW_g-W_g\star_cB_f
 &=\hbar\Cfg,\\
 \Cfg
 &=-2\pair{f(x)g(y)(K-\overline K)(x,y)}
                 {\partial_t\phi(x)h_{00}(y)}
 \label{eq:commutator}\\
 &=-4i\pair{f(x)g(y)\Imag K(x,y)}
                  {\partial_t\phi(x)h_{00}(y)}.
 \notag
\end{align}
\end{proposition}

\begin{proof}
The classical terms cancel. The scalar tests and smooth
configuration factors in \(\QRgf\) commute, so their order may be
changed without exchanging singular distributions. Subtracting
the two order-\(\hbar\) coefficients gives the factor \(-2\)
multiplying \(K-\overline K\). The definition of the imaginary part
then gives the factor \(-4i\).
\end{proof}

\subsection{Existence of the Fourier-defined distribution}

For \(\varepsilon>0\), insert the same factor
\(e^{-\varepsilon|p|}\) into every term
of~\eqref{eq:locked-kernel}; call the result \(K_\varepsilon\).
This Abel parameter is not the VII-C mollifier.

\begin{lemma}[Fourier definition and regulator convergence]
The locked formula defines a tempered distribution in
\((t,s,\mathbf z)\). Moreover,
\[
 K_\varepsilon\longrightarrow K
 \quad\text{in }\mathcal S'(\R^2\times\R^3).
\]
The distribution is smooth in \((t,s)\) with values in
\(\mathcal S'(\R^3)\).
\end{lemma}

\begin{proof}
Let \(\Xi(t,s,\mathbf z)\) be a compactly supported smooth test
function and take its partial spatial Fourier transform. On its
time support the absolute value of the momentum coefficient is
bounded by
\[
 C\bigl(|p|^{-1}+|p|^{-2}\bigr),
\]
with \(C\) independent of \(0<\varepsilon\le1\).
The partial Fourier transform decreases faster than any inverse
power of \(|p|\), uniformly on that support.

Near \(p=0\), spherical measure gives
\[
 \int_{|p|<1}
 \bigl(|p|^{-1}+|p|^{-2}\bigr)\,\dd^3p
 =4\pi\int_0^1(q+1)\,\dd q<\infty.
\]
At infinity, the Fourier decay supplies an integrable majorant.
Dominated convergence therefore defines the unregulated pairing
and proves convergence on compactly supported tests.

For Schwartz tests, the same estimate is combined with temporal
Schwartz weights sufficient to absorb the explicit factor \(t\).
This gives the tempered estimate and convergence in
\(\mathcal S'\). Temporal differentiation of the momentum
coefficients introduces only additional nonnegative powers of
\(|p|\) and polynomial temporal factors. Spatial Fourier decay
again dominates them. Differentiation under the paired integral
therefore proves smooth dependence on the time parameters.
\end{proof}

The kernel in \((x,y)\) is obtained by the invertible coordinate
change to \((t,s,\mathbf z,\mathbf y)\), with constant dependence on
the last variable. Consequently the compactly supported pairings
in~\eqref{eq:commutator} are meaningful for arbitrary smooth
configuration factors.

All distribution identities below are first proved with ambient
time variables in \(\R\). Interior restrictions require no boundary
choice. For inherited tests meeting \(\Sigma\), smooth dependence
on the time parameters supplies the corresponding half-space
pairing. No new boundary contact is introduced into \(K\).

\section{Exact direct-plus-reflected subtraction}

Define the Fourier distributions
\begin{align}
 D(v,\mathbf z)
 &=\int_{\R^3}\frac{\dd^3p}{(2\pi)^3}
 e^{ip\cdot\mathbf z}\frac{\sin(|p|v)}{|p|},
 \label{eq:D-fourier}\\
 H(v,\mathbf z)
 &=\int_{\R^3}\frac{\dd^3p}{(2\pi)^3}
 e^{ip\cdot\mathbf z}\frac{\cos(|p|v)}{|p|^2}.
 \label{eq:H-fourier}
\end{align}
The same argument as above defines these distributions and their
Abel-regulated versions. The letter \(H\) here denotes only this
cosine distribution; it is not a new Wick covariance.

\begin{proposition}[Exact subtraction]
With the convention~\eqref{eq:variables},
\begin{equation}
\boxed{
 \DK
 =\frac{it}{2}D(\tau,\mathbf z)
  +\frac{i}{4}H(\tau,\mathbf z)
  -\frac{i}{4}H(u,\mathbf z).
}
\label{eq:exact-subtraction}
\end{equation}
Equivalently,
\begin{equation}
\boxed{
 \Imag K
 =\frac{t}{4}D(\tau,\mathbf z)
  +\frac18H(\tau,\mathbf z)
  -\frac18H(u,\mathbf z).
}
\label{eq:imag-decomposition}
\end{equation}
\end{proposition}

\begin{proof}
Conjugate~\eqref{eq:locked-kernel} at fixed observation arguments.
The spatial phase becomes \(e^{-ip\cdot\mathbf z}\). Replacing
\(p\) by \(-p\) restores the original spatial phase without changing
\(|p|\).

The direct difference is
\[
 -\frac{t}{4|p|}
 \bigl(e^{-i|p|\tau}-e^{i|p|\tau}\bigr)
 +\frac{i}{8|p|^2}
 \bigl(e^{-i|p|\tau}+e^{i|p|\tau}\bigr)
 =
 \frac{it}{2|p|}\sin(|p|\tau)
 +\frac{i}{4|p|^2}\cos(|p|\tau).
\]
The reflected difference is
\[
 -\frac{i}{8|p|^2}
 \bigl(e^{i|p|u}+e^{-i|p|u}\bigr)
 =
 -\frac{i}{4|p|^2}\cos(|p|u).
\]
These identities hold with the common Abel regulator, and hence
after its distributional limit. Division by \(2i\) gives
\eqref{eq:imag-decomposition}.
\end{proof}

Thus the correctly named pieces are
\[
 (\DK)_{\mathrm{dir}}
 =\frac{it}{2}D(\tau,\mathbf z)
  +\frac{i}{4}H(\tau,\mathbf z),
 \qquad
 (\DK)_{\mathrm{ref}}
 =-\frac{i}{4}H(u,\mathbf z).
\]
The direct contribution is not a pure function of \(\tau\), and
is not a pure Pauli--Jordan contribution. Distinct Fourier phases
do not prevent their integrated distributions from cancelling on
an open region.

The substitutions \(v=\tau\) and \(v=u\) are distributionally
unambiguous. For example, \((t,s)\mapsto(\tau,s)\) is an invertible
linear coordinate change, after which \(D(\tau,\mathbf z)\) is
\(D\) tensored with the constant distribution in \(s\). The same
argument applies to \(u\). No time-ordering prescription or
wavefront extension is hidden in these substitutions.

\section{Common-regulator evaluation}

\subsection{The cosine distribution}

For \(R>0\), angular integration gives
\begin{equation}
 H_\varepsilon(v,R)
 =\frac{1}{2\pi^2R}
 \int_0^\infty
 e^{-\varepsilon q}
 \frac{\sin(qR)\cos(qv)}{q}\,\dd q.
 \label{eq:H-radial-regulated}
\end{equation}

\begin{lemma}[Abel integral]
For \(\varepsilon>0\) and \(a\in\R\),
\[
 \int_0^\infty e^{-\varepsilon q}
 \frac{\sin(aq)}{q}\,\dd q
 =\arctan\frac a\varepsilon.
\]
\end{lemma}

\begin{proof}
The integral vanishes at \(a=0\). Differentiation with respect to
\(a\) is justified by the exponential regulator and gives
\[
 \int_0^\infty e^{-\varepsilon q}\cos(aq)\,\dd q
 =\Real\frac{1}{\varepsilon-ia}
 =\frac{\varepsilon}{\varepsilon^2+a^2}.
\]
Integration from \(0\) to \(a\) proves the assertion.
\end{proof}

Using the sine-cosine product identity in
\eqref{eq:H-radial-regulated} yields
\begin{equation}
 H_\varepsilon(v,R)
 =\frac{1}{4\pi^2R}
 \left[
  \arctan\frac{R+v}{\varepsilon}
  +\arctan\frac{R-v}{\varepsilon}
 \right].
 \label{eq:H-arctan}
\end{equation}

\begin{proposition}[Cosine limit]
On \(R>0\),
\begin{equation}
 \boxed{
 H(v,R)=\frac{1}{4\pi R}\Theta(R-|v|).
 }
 \label{eq:H-evaluated}
\end{equation}
Here \(\Theta\) is the locally integrable Heaviside distribution;
its assigned point value at zero is irrelevant.
\end{proposition}

\begin{proof}
For \(R>|v|\), both arctangent arguments tend to positive infinity,
so their sum tends to \(\pi\). For \(R<|v|\), one tends to positive
infinity and the other to negative infinity, so the sum tends to
zero.

For all \(R>0\),
\[
 |H_\varepsilon(v,R)|\le\frac{1}{4\pi R}.
\]
The majorant is locally integrable in
\((v,\mathbf z)\in\R\times\R^3\), since spatial polar measure is
\(R^2\,\dd R\,\dd\Omega\). Thus dominated convergence proves local
distributional convergence, including neighbourhoods of the
spatial apex. Its limit agrees with the independently defined
Fourier distribution.

The radial display is used on \(R>0\). At the apex the Fourier
definition remains authoritative. The global convergence argument
does not leave freedom to add an apex contact.
\end{proof}

\subsection{The sine distribution}

The companion calculation is
\begin{align}
 D_\varepsilon(v,R)
 &=\frac{1}{2\pi^2R}
 \int_0^\infty e^{-\varepsilon q}
 \sin(qR)\sin(qv)\,\dd q\\
 &=\frac{1}{4\pi^2R}
 \left[
  \frac{\varepsilon}{\varepsilon^2+(R-v)^2}
  -\frac{\varepsilon}{\varepsilon^2+(R+v)^2}
 \right].
 \label{eq:D-regulated}
\end{align}

\begin{proposition}[Sine limit and derivative check]
On \(R>0\),
\begin{equation}
 \boxed{
 D(v,R)=\frac{1}{4\pi R}
 \bigl[\delta(v-R)-\delta(v+R)\bigr].
 }
 \label{eq:D-evaluated}
\end{equation}
Globally, with the Fourier definitions,
\begin{equation}
 D=-\partial_vH.
 \label{eq:D-H-check}
\end{equation}
\end{proposition}

\begin{proof}
For a one-dimensional compactly supported smooth test function
\(\zeta\),
\[
 \int_\R
 \frac{\varepsilon}{\varepsilon^2+w^2}\zeta(w)\,\dd w
 =
 \int_\R\frac{\zeta(\varepsilon a)}{1+a^2}\,\dd a
 \longrightarrow \pi\zeta(0).
\]
The integrable majorant
\(\|\zeta\|_\infty/(1+a^2)\) proves this delta-sequence limit.
It is the standard distributional interpretation, not a pointwise
limit on the cone~\cite{NIST}.

Applying it to the two translated terms
of~\eqref{eq:D-regulated} proves~\eqref{eq:D-evaluated}. The argument
also holds under joint pairing in \((v,\mathbf z)\) on compact
subsets of \(R>0\), where \(1/R\) is smooth and bounded.

At finite regulator,
\[
 D_\varepsilon=-\partial_vH_\varepsilon
\]
both by Fourier differentiation and by
\eqref{eq:H-arctan}. Distributional differentiation is continuous,
so the global convergence of \(H_\varepsilon\) proves
\eqref{eq:D-H-check}, including the apex in the Fourier sense.
\end{proof}

The momentum singularities have therefore required no infrared
finite part. The common regulator evaluates the locked
distribution; it does not select a new normalization.

\section{Evaluated imaginary part and its support}

\begin{theorem}[Exact imaginary part]
On \(t,s>0\) and \(R>0\),
\begin{equation}
\boxed{
\begin{aligned}
 \Imag K(x,y)
 ={}&
 \frac{t}{16\pi R}
 \bigl[\delta(\tau-R)-\delta(\tau+R)\bigr]
 \\
 &+\frac{1}{32\pi R}
 \left[
  \Theta(R-|\tau|)-\Theta(R-u)
 \right].
\end{aligned}
}
\label{eq:imag-evaluated}
\end{equation}
Away from the direct and reflected cone boundaries,
\begin{equation}
 \Imag K=
 \begin{cases}
  0,&R<|t-s|,\\[1mm]
  \dfrac{1}{32\pi R},&|t-s|<R<t+s,\\[2mm]
  0,&R>t+s.
 \end{cases}
 \label{eq:imag-regions}
\end{equation}
\end{theorem}

\begin{proof}
Substitute~\eqref{eq:H-evaluated}
and~\eqref{eq:D-evaluated} into
\eqref{eq:imag-decomposition}. Since \(t,s>0\),
\[
 u=t+s>|\tau|.
\]
The Heaviside difference is therefore one precisely in the open
region \(|\tau|<R<u\), and zero in the other two open regions.
The direct-cone delta contribution is kept separately.
\end{proof}

\begin{corollary}[Support away from the apex]
In the open set \(t,s>0,\ R>0\), the support of \(\Imag K\) is
exactly
\[
 \{(x,y):|t-s|\le R\le t+s\}.
 \label{eq:imag-support}
\]
The direct cone carries the displayed delta contribution.
The reflected cone is a jump locus of the locally integrable
contribution, not an additional delta distribution.
\end{corollary}

\begin{proof}
The formula vanishes in the complementary open regions.
It is a strictly positive smooth function on
\(|t-s|<R<t+s\). The closure of that region within
\(t,s>0,\ R>0\) is the displayed set, so every point of that set
lies in the support. The formula itself contains no reflected
delta term.
\end{proof}

Thus observation-spacelike pairs can have nonzero imaginary part.
The relevant spacelike region is the region between the two
cones, not the region beyond the reflected cone.

Two checks follow directly from the Fourier identity. The
functions \(H\) and \(D\) are respectively even and odd in \(v\).
At \(t=0\), the locked kernel vanishes and
\eqref{eq:imag-decomposition} gives zero because
\(H(-s)=H(s)\). At \(s=0\), the two cosine terms cancel and
\[
 \Imag K(t,0,\mathbf z)=\frac t4D(t,\mathbf z).
\]
These are identities in the spatial distribution variable, not
assignments of pointwise cone values.

\section{The IX-E witness class: exact zero}

\subsection{Actual support ordering}

For actual functionals, retain IX-E's definition
\begin{equation}
 F\gtrsim G
 \quad\Longleftrightarrow\quad
 \fsupp F\cap J^-(\fsupp G)=\varnothing.
 \label{eq:ordering}
\end{equation}
Causal relations include null relation and coincidence.

Both orderings hold exactly when no point of one actual functional
support is causally related in either direction to a point of the
other. Indeed, the first ordering excludes one direction and the
second excludes the reverse. In particular, double ordering
excludes intersection of the actual supports.

\begin{lemma}[Strongly separated witness class]
Suppose \(f,g\) have time supports in \([0,T]\) and spatial
supports in
\[
 \overline B(x_0,r),
 \qquad
 \overline B(y_0,s_0),
\]
respectively. If
\begin{equation}
 |x_0-y_0|>2T+r+s_0,
 \label{eq:strong-separation}
\end{equation}
then both \(B_f\gtrsim W_g\) and \(W_g\gtrsim B_f\) hold.
\end{lemma}

\begin{proof}
A point in \(P_F\cup F\) has time in \([0,T]\) and spatial
distance at most \(T+r\) from \(x_0\). A point in \(G\) has
time in \([0,T]\) and spatial distance at most \(s_0\) from
\(y_0\). Their spatial distance is greater than \(T\), whereas
their time difference is at most \(T\). The upper support sets
are causally disjoint. The actual supports are subsets of them
by~\eqref{eq:support-upper}.
\end{proof}

\begin{theorem}[Vanishing on the entire witness class]
For every \(f,g\) satisfying~\eqref{eq:strong-separation},
\begin{equation}
 \boxed{
 f(x)g(y)\Imag K(x,y)=0,
 \qquad
 \Cfg=0.
 }
 \label{eq:witness-zero}
\end{equation}
The first equality is distributional and the second is an identity
of actual functionals on every allowed configuration.
\end{theorem}

\begin{proof}
On \(\supp f\times\supp g\),
\[
 R\ge |x_0-y_0|-r-s_0>2T\ge t+s.
\]
Hence neither direct-cone delta meets the observation support.
Moreover,
\[
 H(t-s,R)=\frac{1}{4\pi R}
 =H(t+s,R)
\]
there, so the cosine contributions cancel.

The inequalities have a positive uniform margin on the compact
observation support. Even if a time support meets \(\Sigma\),
there is an ambient open neighbourhood on which
\[
 R>\max\{|t-s|,|t+s|\}.
\]
On that neighbourhood the global Fourier identity gives
\(\Imag K=0\). Multiplication by \(f(x)g(y)\) therefore gives
the zero distribution, with no boundary ambiguity.
Equation~\eqref{eq:commutator} then vanishes for every smooth
configuration.
\end{proof}

This result needs no detection lemma: there is no nonzero
distribution to detect. The strong separation is precisely why
this witness class does not meet the reflected cone.

\subsection{An explicit representative}

Define smooth nonnegative bumps
\[
 \beta(a)=
 \begin{cases}
  \exp\!\bigl(-1/(1-a^2)\bigr),&|a|<1,\\
  0,&|a|\ge1,
 \end{cases}
 \qquad
 \chi(\mathbf v)=
 \begin{cases}
  \exp\!\bigl(-1/(1-|\mathbf v|^2)\bigr),&|\mathbf v|<1,\\
  0,&|\mathbf v|\ge1.
 \end{cases}
 \label{eq:bumps}
\]
Let \(\mathbf e_1=(1,0,0)\), and take
\begin{align}
 f(t,\mathbf x)
 &=\beta\bigl(8(t-\tfrac12)\bigr)\chi(10\mathbf x),\\
 g(s,\mathbf y)
 &=\beta\bigl(8(s-\tfrac12)\bigr)
   \chi\bigl(10(\mathbf y-3\mathbf e_1)\bigr).
 \label{eq:explicit-zero-tests}
\end{align}
Their time supports lie in
\([3/8,5/8]\subset(1/4,3/4)\), their spatial radii are \(1/10\),
and their center distance is \(3\). Taking \(T=1\) gives
\[
 3>2+\frac1{10}+\frac1{10}.
\]
This is a retained two-way pair with exact zero commutator.
No numerical integration is involved.

\section{A nonzero observation-spacelike control}

A zero on the settled witness class must not be confused with zero
on all observation-spacelike pairs.

\begin{proposition}[Explicit nonzero control]
There are nonnegative compactly supported tests \(f,g\) whose
observation supports are strictly spacelike separated and for
which \(\Cfg\neq0\). The calculation alone does not certify
double ordering of their actual functional supports.
\end{proposition}

\begin{proof}
Use the bumps~\eqref{eq:bumps}, set \(\eta=1/20\), and choose
\begin{align}
 f(t,\mathbf x)
 &=\beta\bigl((t-1)/\eta\bigr)\chi(\mathbf x/\eta),\\
 g(s,\mathbf y)
 &=\beta\bigl((s-1)/\eta\bigr)
   \chi\bigl((\mathbf y-\mathbf e_1)/\eta\bigr).
 \label{eq:control-tests}
\end{align}
On their product support,
\[
 |t-s|\le\frac1{10},
 \qquad
 \frac9{10}\le R\le\frac{11}{10},
 \qquad
 t+s\ge\frac{19}{10}.
\]
Thus the whole product support lies strictly inside
\[
 |t-s|<R<t+s.
\]
No cone distribution meets it.

Choose the unrestricted smooth off-shell configuration
\[
 \phi(t,\mathbf x)=t,
 \qquad
 h_{00}(s,\mathbf y)=\alpha>0,
 \qquad
 h_{\mu\nu}=0\quad\text{for the other components}.
\]
Then~\eqref{eq:commutator} and~\eqref{eq:imag-regions} give
\begin{equation}
 \boxed{
 \Cfg[h,\phi]
 =-\frac{i\alpha}{8\pi}
 \int_{\Mplus^2}
 \frac{f(x)g(y)}{|\mathbf x-\mathbf y|}
 \,\dd^4x\,\dd^4y\neq0.
 }
 \label{eq:control-pairing}
\end{equation}
The integral is finite and strictly positive before multiplication
by the nonzero prefactor: the denominator is bounded away from
zero and both bumps are nonnegative and nonzero. More explicitly,
writing \(I\) for the integral,
\[
 \frac{10}{11}
 \left(\int f\right)\left(\int g\right)
 \le I\le
 \frac{10}{9}
 \left(\int f\right)\left(\int g\right).
\]
This is an exact pairing with explicit tests, not numerical
sampling.
\end{proof}

\begin{remark}[Status of the control]
The tests~\eqref{eq:control-tests} are observation-spacelike.
The support of \(B_f\), however, can extend into \(P_F\).
No lower support calculation establishing both actual orderings
has been supplied for this pair. Accordingly,
\eqref{eq:control-pairing} is a nonzero observation-spacelike
control, not an obstruction witness.
\end{remark}

\section{Actual support coverage and the unfinished bridge}

\subsection{Detection on the unrestricted configuration space}

The following elementary fact makes the remaining bridge a precise
commutator obligation.

\begin{lemma}[Compact detection]
Let
\[
 T_{f,g}=f(x)g(y)\Imag K(x,y)
\]
be a nonzero compactly supported distribution with support in
\(\Mint^2\). On the unrestricted smooth off-shell configuration
space, \(\Cfg\neq0\) as an actual functional.
\end{lemma}

\begin{proof}
There exist compactly supported smooth \(a(x),b(y)\) such that
\[
 \pair{T_{f,g}}{a\otimes b}\neq0.
\]
To justify this reduction, choose a compact product rectangle
containing the relevant supports. A compactly supported smooth
test on the rectangle can be periodized inside a larger rectangle
and expanded in its smooth Fourier series. Multiplication by
separate compact cutoffs in the two variables gives approximating
finite sums of tensor-product tests, converging with every
derivative on a fixed compact set. If a distribution vanished on
all tensor-product tests, continuity would therefore make it
vanish on every test.

Choose a smooth scalar configuration satisfying
\[
 \partial_t\phi=a,
 \qquad
 \phi(t,\mathbf x)=\int_{t_*}^{t}a(v,\mathbf x)\,\dd v,
\]
and choose \(h_{00}=b\), with the other metric components zero.
Equation~\eqref{eq:commutator} detects \(T_{f,g}\).
If real configurations are required, decomposing \(a,b\) into
real and imaginary parts gives a real tensor-product detector.
Compactness is required of the kernel, not of \(\phi\).
\end{proof}

For interior tests, distributional vanishing of \(T_{f,g}\) and
vanishing of the actual commutator are consequently equivalent.
This uses unrestricted off-shell configurations and cannot be
transferred without proof to a constrained quotient.

\subsection{The fixed-domain obstruction}

\begin{theorem}[Retained-pair obstruction]
Suppose \(B_f\otimes W_g\in\Dtwo\), both actual support orderings
hold, and \(\Cfg\neq0\). Then no symmetric causally factorizing
map \(\Ttwo\) exists on \(\Dtwo\).
\label{thm:obstruction}
\end{theorem}

\begin{proof}
Factorization and symmetry would require
\[
 B_f\star_cW_g
 =\Ttwo(B_f,W_g)
 =\Ttwo(W_g,B_f)
 =W_g\star_cB_f.
\]
This contradicts~\eqref{eq:commutator}. The nonzero
order-\(\hbar\) coefficient remains nonzero modulo \(\hbar^4\).
No higher coefficient or contact can make the two prescribed free
values equal.
\end{proof}

This is IX-E's fixed-domain obstruction criterion, now equipped
with the evaluated kernel~\cite[Theorem 4.6]{IXE}. Its missing
premise is still certification of a nonzero two-way pair.

\subsection{Statement of the bridge}

The remaining coverage obligation is
\begin{equation}
\boxed{
 \begin{gathered}
 B_f\gtrsim W_g,\qquad W_g\gtrsim B_f\\
 \Longrightarrow\quad
 f(x)g(y)\Imag K(x,y)=0.
 \end{gathered}
}
\tag{SB}
\label{eq:SB}
\end{equation}
It concerns every retained first-stage pair, not just the strongly
separated class.

A sufficient route is to prove that two-way functional separation
excludes the observation region
\[
 |t-s|\le R\le t+s,
\]
including the direct-cone contribution, with a separate check of
any apex or boundary cases used in that argument. This exclusion
would imply~\eqref{eq:SB}. It is sufficient, not asserted necessary:
multiplication by tests can also annihilate a distribution through
vanishing or cancellation.

\subsection{Why upper support bounds do not prove the bridge}

For \(R\le t+s\), the time-zero balls
\[
 \overline B(\mathbf x,t),
 \qquad
 \overline B(\mathbf y,s)
\]
intersect. Indeed, along the segment joining the centers one can
choose distances \(a\le t\) and \(R-a\le s\) precisely when
\(R\le t+s\). The resulting point
\[
 z=(0,\mathbf z_0)
\]
lies in \(J^-(x)\cap J^-(y)\).

This explains the reflected geometry. It does not show that
\(z\in\fsupp B_f\). Membership in the upper set \(P_F\) is not
membership in the actual support. Likewise, a use of
\(y\in\fsupp W_g\) requires more than \(y\in G\).

\begin{lemma}[Support through first derivatives]
For a differentiable functional on the unrestricted affine smooth
configuration space, its functional support is the closure of the
union of the supports of its first-derivative distributions over
all configurations and all field components.
\label{lem:derivative-support}
\end{lemma}

\begin{proof}
If the functional is independent of perturbations supported in
an open set \(U\), every first derivative annihilates such
perturbations. Conversely, if every first derivative annihilates
them at every configuration, then for a perturbation \(\eta\)
supported in \(U\),
\[
 F(\psi+\eta)-F(\psi)
 =\int_0^1
   \mathrm DF(\psi+\lambda\eta)[\eta]\,\dd\lambda
 =0.
\]
This is exactly absence of functional dependence in \(U\).
Taking the complement of the union of such open sets proves the
support characterization.
\end{proof}

For the sharp insertion, the metric variation is
\[
 \mathrm D_hB_f[h,\phi][\dot h]
 =-\int f(x)(L\dot h)^\mu(x)\partial_\mu\phi(x)\,\dd^4x.
\]
Set \(a_\mu=-f\,\partial_\mu\phi\) and
\(b=\mathsf E_0^*a\), in the inherited adjoint convention. Its
metric derivative is represented by the full distribution
\[
 q_a=L^*a
 =\theta C^*b+\delta(t)\mathsf B(b|_\Sigma).
\]
The scalar variation remains
\[
 \mathrm D_\phi B_f[h,\phi][\dot\phi]
 =-\int f(x)(Lh)^\mu(x)\partial_\mu\dot\phi(x)\,\dd^4x.
\]

Lemma~\ref{lem:derivative-support} therefore identifies what a
lower support proof must examine. It does not evaluate those
supports. Such a proof must control the actual bulk distribution,
its surface trace, all allowed scalar gradients, and possible
cancellations. A nonzero candidate cone, or the support of one
formal summand, does not by itself establish nonzero dependence
of the complete functional.

The surface term cannot be discarded, and the on-shell smoothing
cannot replace it. Actual-presentation cancellations must also
be respected: support belongs to the functional, not to a chosen
test presentation.

\subsection{Conjugate-sector coverage}

Conjugation preserves functional support on the unrestricted real
configuration space, since vanishing of a dependence is equivalent
to vanishing of its complex conjugate. The same statement holds
for its usual complexification.

Let \(S(d)=\star_c(d)\) and
\[
 J(d)=S(d)-S(\sigma d).
\]
The inherited reversal identity gives
\[
 J(\gamma d)=-J(d)^*.
\]
Indeed,
\[
 S(\gamma d)^*=S(\sigma d),
 \qquad
 S(\sigma\gamma d)^*=S(d).
\]
Thus a complete commutator-vanishing proof on the original
first-stage pairs transfers to their declared conjugates.
This transfer is explicit; it does not supply the missing proof
on the original pairs. Nor does generatorwise vanishing evaluate
all actual linear relations among prescribed tensors.

\subsection{Precise unfinished status}

No proof of~\eqref{eq:SB} is supplied here. No retained counterexample
with both actual support orderings is supplied either.

The kernel identity is closed. The strongly separated witness
calculation is closed. The nonzero control is closed as an
observation-spacelike calculation. The domain-wide stop remains
the actual functional-support bridge.

It would be incorrect to move that stop to a replacement
distribution, or to describe the control as an evaluated
fixed-domain obstruction.

\section{Remaining consistency obligations and release decision}

\subsection{Vanishing is not construction}

Even a future proof of~\eqref{eq:SB} would clear only the
first-stage double-ordering commutator test. IX-E's actual-tensor
relation test is separate.

For clarity, its algebraic structure can be recorded without
constructing a new map. With
\[
 A(d)=\frac12\bigl(S(d)+S(\sigma d)\bigr),
\]
a candidate \(\Ttwo=A+X\) must satisfy
\[
 X(\sigma d)=X(d),
 \qquad
 X(\gamma d)=-X(d)^*.
\]
For prescribed tensors \(d_i\) with prescribed values \(t_i\), put
\(x_i=t_i-A(d_i)\). In the flip quotient
\[
 Q=\Dtwo/\operatorname{im}(\id-\sigma),
\]
the inherited finite-relation condition is
\[
 \sum_i a_i[d_i]+\sum_i b_i[\gamma d_i]=0
 \quad\Longrightarrow\quad
 \sum_i a_ix_i-\sum_i b_ix_i^*=0,
\]
where each sum is finite. This condition is not evaluated here.
IX-E's algebraic sufficiency proof uses a Hamel complement and
does not establish continuity~\cite[Theorem 5.1]{IXE}.

The ordered slab also remains relevant: its nonzero
\(\QRgf\) excludes classical multiplication as the time-ordered
map. A later slab proves an ordering, not necessarily its
uniqueness, and is not a substitute for a two-way calculation.

If the commutator gate is eventually clear, the inherited E2--E6
obligations remain. They include a specified replacement
distribution, admission of coefficients and descendants, named
fixed-support continuity, exact overlap, contact and involution
compatibility, and the hypotheses of conditional subtraction.
This successor begins none of those constructions.

\subsection{Decision table}

\begin{longtable}{
 >{\raggedright\arraybackslash}p{0.43\textwidth}
 >{\raggedright\arraybackslash}p{0.47\textwidth}}
\toprule
Evidence & Legal consequence\\
\midrule
\endhead
Nonzero commutator on a certified retained two-way pair
&
Proved obstructed on the fixed \(\Dtwo\); stop.\\
\addlinespace
Exact zero on the strongly separated witness class only
&
Unresolved; that class is settled, but complete two-way coverage
is unfinished.\\
\addlinespace
Vanishing on every retained two-way first-stage pair
&
The double-ordering commutator gate is clear; the complete
causal-product release remains Unresolved.\\
\addlinespace
Unclosed kernel or conjugation identity
&
Unresolved at that identity. This is not the present stop.\\
\bottomrule
\end{longtable}

\begin{longtable}{
 >{\raggedright\arraybackslash}p{0.23\textwidth}
 >{\raggedright\arraybackslash}p{0.67\textwidth}}
\toprule
Gate & Status established here\\
\midrule
\endhead
Fixed domain
&
Inherited target retained without deletion; actual old intersection
unevaluated.\\
Kernel evaluation
&
Closed by the common-regulator proof and exact imaginary-part
formula.\\
Witness class
&
Closed: exact zero for every strongly separated pair.\\
Spacelike control
&
Closed: explicit nonzero pairing, without two-way certification.\\
Complete two-way coverage
&
Open at the actual functional-support bridge.\\
Actual tensor relations
&
The inherited finite-relation test remains unevaluated.\\
E2--E5
&
No replacement, admission, continuity, contact, overlap, or
analytic involution construction is attempted.\\
E6
&
IX-E's subtraction theorem remains conditional; no normalized
retarded output is released.\\
\bottomrule
\end{longtable}

\begin{theorem}[Release decision]
Relative to the stated inherited inputs and the results proved in
this manuscript, the release on~\eqref{eq:frozen-domain} is
\textbf{Unresolved}. Neither a Constructed outcome nor a
Proved obstructed outcome is established.
\end{theorem}

\begin{proof}
No time-ordered map satisfying the complete interface is
constructed. The explicit certified two-way class has zero
commutator. The explicit nonzero control has not been certified
as two-way in actual functional support, and no violation of the
actual-tensor relation test is evaluated. The support bridge
remains unfinished. These establish precisely the third release
status, not nonexistence of every possible future construction.
\end{proof}

\section{Non-identification ledger and retained exclusions}

\begin{longtable}{
 >{\raggedright\arraybackslash}p{0.21\textwidth}
 >{\raggedright\arraybackslash}p{0.69\textwidth}}
\toprule
Source or object & Retained boundary\\
\midrule
\endhead
I and IV
&
Measurement semantics and represented records do not identify
their outputs with the present kernel or commutator.\\
II and III
&
A finite contact-span certificate and classical descent are not
sharp time-ordered existence theorems.\\
V
&
Its external \(R_{1,1}\), oscillator coefficient, and Newton--wave
cubic are not \(K\), a Cauchy-reflection triangle, or \(\Ttwo\).\\
VI and VII-A
&
The compact-switch defect and restricted plateau intertwining
statements do not construct the missing higher retarded products.\\
VII-C
&
Its conditional normal-topology obstruction remains in force
under its joint hypotheses (E1)--(E4).\\
VII-D and triangle scalar
&
A numerical finite part and mollifier conversion are not a
binary product or a binary contact classification.\\
VII-E/F and VIII-A--C
&
The certificates and token promise supply no map into the
present actual tensor sector and no conical-current
identification of \(\Qfg\).\\
IX-Free-2
&
The released free product, descent, word coherence, and F9a are
not time ordering and are not strengthened here.\\
\bottomrule
\end{longtable}

All supplied free coefficients and the triangle parameter remain
fixed. No contact is added to repair unequal prescribed values;
no \(u_\mu^\rho\) is subtracted and renamed as a product. Removing
a retained pair would change the target, not solve it.

G2 remains unopened. In particular, the inherited compact-switch
witness
\[
 B_f[h+\mathcal K\xi,\phi]-B_f[h,\phi]
 =-\int f\,X_\xi\phi,
 \qquad
 g\mathcal K\xi=0,
\]
retains its nonzero first-order dressing variation under its stated
hypotheses. A correction beginning at second order cannot cancel
a nonzero first-order coefficient while that prescription is
retained.

No Feynman replacement, scaling-degree extension, normalized
\(R_1\), \(R_2\), or \(\mathsf T_3\) is begun. No Einstein-equation
or compilation claim is made. No repair of VII-C and no
normal-topology algebra claim follows from the present calculation.

\section{Conclusion}

The locked kernel has an explicitly evaluated imaginary part,
supported away from the spatial apex on the direct null cone and
the observation-spacelike region between the direct and reflected
cones. Both the direct cosine contribution and the reflected
cosine contribution are essential: their tails cancel beyond the
reflected cone.

On IX-E's strongly separated two-way witness class the resulting
commutator is exactly zero, as a distribution and as an actual
functional on every allowed configuration. This closes that
class without qualification. The explicit nonzero spacelike
control shows why observation separation alone cannot replace
actual functional-support ordering.

No fixed-domain obstruction is established. A domain-wide
conclusion requires the actual functional-support bridge, with
the sharp adjoint surface term retained and actual-presentation
cancellations controlled. Even universal commutator vanishing
would not construct a time-ordered extension or discharge the
remaining actual-relation and analytic obligations.

The release is therefore Unresolved at complete two-way
commutator coverage. The evaluated kernel, the settled witness
class, and the remaining support question are now separately
visible without reconstructing the series.

\begin{thebibliography}{9}

\bibitem{IXE}
\emph{Landmark IX-E: Binary time ordering and causal subtraction
on the declared sharp comparison sector}.
Supplied manuscript \texttt{IX-E.tex}.
Used for the fixed domain, support ordering, first-stage
coefficients, retained-pair obstruction, algebraic relation test,
and conditional subtraction boundary.

\bibitem{Inputs}
\emph{Internal prerequisite statements supplied for Landmark
IX-E-2}.
The quoted statements from I--V, VI, VII-A--F, VIII-A--C,
IX-A, IX-B, IX-D, IX-Free, and IX-Free-2 are imported only in the
explicit forms recorded in this manuscript.
No unprovided full-source theorem or nonzero old time-ordered map
is reconstructed.

\bibitem{NIST}
National Institute of Standards and Technology,
\emph{Digital Library of Mathematical Functions},
Sections 1.16 and 1.17(i).
Background for distributional differentiation and delta-sequence
interpretation. The regulator identities and pairing limits used
here are proved directly.

\end{thebibliography}

\end{document}