==== TABLE OF CONTENTS ====
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//main.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/01-introduction.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/02-foundations.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/03-null-current.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/04-dispersion.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/05-charges-cones.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/06-extraction.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/07-short-products.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/08-reconstruction.tex
Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//references.bib
===========================

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//main.tex ====
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\hypersetup{pdftitle={Scale and conformal symmetry in four-dimensional operational quantum field theory},pdfauthor={OpenAI}}
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\title{Scale and conformal symmetry in four-dimensional\\operational quantum field theory}
\author{OpenAI}
\date{September 26, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that scale symmetry implies local conformal symmetry for a class
of four-dimensional unitary, positive-energy quantum field theories with a
Poincar\'e- and scale-invariant vacuum. The class has a discrete,
bounded-below spectrum of scaling dimensions with finite multiplicities,
and each original field has finite scaling support. The original fields
and their adjoints have compatible affiliated realizations in a causally
commuting bounded local net, and a physical local dilatation current
generates scale transformations through its local Ward identities. The
operational framework admits sharply localized reconstructed fields after
their joint products and compatible affiliated realizations in the same
net have been established. Under these hypotheses, the stress tensor admits
a symmetric, conserved, traceless improvement with unchanged translation
and Lorentz charges. The corresponding conformal currents satisfy unbroken
local Ward identities on the physical field algebra. The conclusion is
local; integration to a global conformal action on the whole bounded net
remains a separate question.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-foundations}
\input{sections/03-null-current}
\input{sections/04-dispersion}
\input{sections/05-charges-cones}
\input{sections/06-extraction}
\input{sections/07-short-products}
\input{sections/08-reconstruction}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//main.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/01-introduction.tex ====
\section{Introduction}
\label{sec:introduction}

Scale symmetry constrains correlation functions under uniform changes of length. Conformal symmetry also includes special conformal transformations and their local Ward identities. The implication from the former to the latter is therefore a statement about the local operator content of a theory, not only about the action of a global charge.

For a symmetric conserved stress tensor in four dimensions, a conserved dilatation current has the form
\begin{equation}\label{eq:intro-current}
 D_\mu=x^\nu T_{\mu\nu}-V_\mu,
 \qquad T^\mu{}_{\mu}=\partial^\mu V_\mu.
\end{equation}
The problem is to improve $T$ to a traceless tensor without changing its affine charges or leaving the original physical field algebra. A sufficient condition is a local scalar $L$ with $T^\mu{}_{\mu}=\Box L$, but the general improvement problem allows tensor potentials. Accordingly, a proof must derive the scalar condition if it uses one; it cannot impose it as part of the desired conclusion.

Stress-tensor improvements that preserve the Poincar\'e generators were a central feature of the work of Callan, Coleman and Jackiw~\cite{CallanColemanJackiw1970}. The issue here is to construct the required scalar within the original physical field algebra.

\subsection{The result and its scope}

We work in a vacuum representation of a local, unitary, positive-energy
theory on $\R^{1,3}$. The vacuum is invariant under Poincar\'e
transformations and continuous dilatations. The original point-field
labels form an algebraic direct sum
\[
 \mathcal F=\bigoplus_{\Delta\in\Sigma}\mathcal F_\Delta,
\]
where $\mathcal F_\Delta$ consists of homogeneous dilation eigenfields
of dimension $\Delta$. The real dimension spectrum $\Sigma$ is discrete,
with no finite accumulation, bounded below, and has finite multiplicities
in the Lorentz-component convention of Assumption~\ref{ass:framework}.
Thus each original field is a finite linear combination of homogeneous
fields: it has \emph{finite scaling support}. Both $T$ and the virial field
$V$ in \eqref{eq:intro-current} are physical local fields, with the
corresponding compact-flux Ward identities.

The operational data also include a fixed covariant, causally commuting
net of bounded local algebras, with compatible affiliated realizations
of the original fields and their adjoints. A further finite homogeneous
field family belongs to this physical theory only after its localization
in arbitrarily small regions, all finite mixed products, common invariant
domain and original-net affiliation have been established. The required
domain contains the original vacuum-word domain
$\mathcal D_{\mathrm{old}}$, constructed in
Proposition~\ref{F:old-domain}. Thus membership is a conclusion to be
proved for the improvement scalar. Finite scaling support restricts
field labels; arbitrary finite word lengths and joint Schwartz test
kernels remain available.

We use \emph{physical field algebra} in this joint sense: polynomial
insertions from the original fields and any finite family admitted
together with them under Assumption~\ref{ass:framework}. Products of
separately constructed families require their mutual reconstruction.
Proposition~\ref{R:compatible-families} will extend the same improvement
and Ward action to each such admitted family. The framework requires
neither Haag duality nor a split property, a phase-space estimate, or
field-operator bounds in terms of energy.

\Needspace{14\baselineskip}
\begin{theorem}\label{thm:main}
Suppose Assumption~\ref{ass:framework} holds, including the algebraic
direct-sum decomposition with finite scaling support for every original
field. Then the physical stress tensor admits a local symmetric conserved
improvement with the same translation and Lorentz Ward charges and with
zero trace. More precisely, let $T^{(4)}$ and $V^{(3)}$
be the Hermitian dimension-four and dimension-three components selected
by the affine Ward identities. There is a Hermitian physical scalar
field $L$ of dimension two such that
\begin{equation}\label{eq:intro-improvement}
 \partial^\mu V^{(3)}_\mu=\Box L,
 \qquad
 t_{\mu\nu}=T^{(4)}_{\mu\nu}
       +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L
\end{equation}
is symmetric, conserved and traceless. The replacement $T\mapsto T^{(4)}$ also preserves the affine charges. The currents $X^\nu t_{\mu\nu}$ for conformal Killing vector fields $X$ have unbroken local Ward identities on the physical field algebra, extending its Poincar\'e and dilation actions.
\end{theorem}

The preserved physical data are the Hilbert space, bounded local net,
original polynomial fields on $\mathcal D_{\mathrm{old}}$, and their
translation and Lorentz charges. The common domain is enlarged by the
reconstructed vacuum words. This does not identify new affiliated
extensions with previously chosen closed operators on every vector of
their larger domains. The conclusion is a local conformal Ward action;
integration to a global conformal action on the whole bounded net is a
separate question.

The field-space and bounded-net hypotheses delimit the class in
Theorem~\ref{thm:main}. In particular, weak commutation of unbounded
fields alone does not supply the bounded locality used by the proof.
In a gauge description, the construction takes place in the chosen
gauge-invariant local algebras.

\paragraph{The full free scalar theory.}
The massless real scalar in its four-dimensional vacuum Fock
representation gives a concrete example. Use its full bounded local
net and the Wick monomials in the scalar and its derivatives as
original fields. Their scaling grades are finite-dimensional and
integer-valued. The joint-domain and affiliation verification is given
in the example of Section~\ref{F:free-scalar-example}.

With $Q=:\!\phi^2\!:$, the canonical stress tensor and virial field are
\[
 T^{\mathrm{can}}_{\mu\nu}
 =:\!\partial_\mu\phi\,\partial_\nu\phi
       -\tfrac12\eta_{\mu\nu}(\partial\phi)^2\!:,
 \qquad V_\mu=-\tfrac12\partial_\mu Q .
\]
The free equation gives
$(T^{\mathrm{can}})^\mu{}_\mu
=-:\!(\partial\phi)^2\!:
=\partial^\mu V_\mu=\Box(-Q/2)$.
The canonical commutation relations give the Poincar\'e and dilation
Ward actions on compact Cauchy slabs. Here the scalar is already
present, $L=-Q/2$, and \eqref{eq:intro-improvement} is the familiar
improvement
$t_{\mu\nu}=T^{\mathrm{can}}_{\mu\nu}
+(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)Q/6$.

The local-current hypothesis is substantive. Restricting instead to
shift-invariant Wick polynomials removes $Q$ and
$V_\mu=-:\!\phi\partial_\mu\phi\!:$; the available dimension-three
linear fields $\partial_\mu\partial_\nu\phi$ contain no Lorentz
vector, and products of differentiated fields start at dimension
four. This is the field-algebra issue in the free-scalar and dual
noncompact two-form examples discussed by
\citet{DymarskyZhiboedov2015,Nakayama2015}. A different operational
theory must also specify its bounded net and intrinsic field
membership: deleting a name from the original list alone does not
exclude a field reconstructible in the same net.

\subsection{History and the local-field question}

The two-dimensional history runs through Zamolodchikov's $c$-theorem
and Polchinski's enhancement result under suitable hypotheses
\cite{Zamolodchikov1986,Polchinski1988}. In four dimensions, local
renormalization-group consistency conditions give a substantial
perturbative result. The flow relevant to the trace includes
flavor-current contributions, and positivity in the perturbative regime
forces that flow to vanish at a scale-invariant theory. Perturbative
limit cycles of ordinary beta functions can therefore describe conformal
theories \cite{FortinGrinsteinStergiou2013,JackOsborn2014}.
The required positivity is not established by those consistency
identities at arbitrary coupling. Nakayama reviews the distinctions
among scale currents, improvements, unitarity and the hypotheses of
enhancement arguments \cite{Nakayama2015}.

The four-dimensional $a$-theorem of Komargodski and Schwimmer uses
anomaly matching and a positive dilaton dispersion relation for flows
between conformal endpoints \cite{KomargodskiSchwimmer2011}.
Luty, Polchinski and Rattazzi developed this method to establish
conformal asymptotics in perturbatively controlled regimes and to
constrain nonperturbative scale-invariant theories
\cite{LutyPolchinskiRattazzi2013}. Their general argument leaves a
passage from on-shell dilaton constraints to an off-shell local trace
identity. Dymarsky, Komargodski, Schwimmer and Theisen sharpened the
vanishing-amplitude argument, retaining infrared-regulation hypotheses
and the need to justify the inference from trivial probe scattering to
a removable local interaction \cite{DKST2015}.

That inference is sensitive to contact and semi-local terms.
Bzowski and Skenderis showed how semi-local contributions can carry
the scale anomaly even when on-shell dilaton amplitudes vanish
\cite{BzowskiSkenderis2014}. Dymarsky, Farnsworth, Komargodski, Luty
and Prilepina analyzed the generalized-free-trace possibility:
without a suitable dimension-two scalar the anomaly rules it out;
when such a scalar exists, either an improvement is conformal or an
improved trace can be chosen that is not generalized free
\cite{DFKLP2016}. The latter alternative retains an improvement
problem.

Earlier arguments also aim directly at a local scalar.
Yonekura proposes the inverse-wave-operator construction
$\Box^{-1}T^\mu{}_\mu$ and states the crossing and momentum-space
analyticity assumptions used to establish its locality
\cite{Yonekura2014}. Sachs gives a Weyl-gauging and consistency
argument for a restricted class of metric-couplable Lagrangian theories
with a gauge-invariant virial current \cite{Sachs2015}.
Here the required limited momentum continuation is derived inside
primitive causal tubes, and the improvement is reconstructed in the
prescribed physical theory. No asymptotic scattering states or
independently assumed crossing relation enter the argument.

Null-cut modular methods have a substantial earlier development. Faulkner, Leigh, Parrikar and Wang related first-order deformations of half-space modular Hamiltonians to horizon stress-tensor integrals~\cite{FaulknerLeighParrikarWang2016}. Casini, Test\'e and Torroba studied null-plane modular Hamiltonians and their half-sided modular relations~\cite{CasiniTesteTorroba2017}. The bounded-cut argument below establishes the flux limits, domains and commuting transverse measure needed under Assumption~\ref{ass:framework}.

The remaining issue is an operator one. Intrinsic recovery of point
fields from local algebras under polynomial energy bounds was studied
by Fredenhagen and Hertel~\cite{FredenhagenHertel1981}; those bounds
are not assumed here. Dividing the trace's vacuum
polarization by invariant mass squared produces a scalar two-point wave.
Even if that wave has the right dimension and localization, it is not yet
a physical local operator. One must construct its mixed products at every
finite length, establish their positivity and a compatible common domain,
and prove affiliation in the original algebras.
Section~\ref{sec:reconstruction} is devoted to this passage.

\Needspace{7\baselineskip}
\subsection{Structure of the proof}

\begin{enumerate}
\item \emph{Bounded null cuts.} Wedge modular data and half-sided modular inclusions give positive generators for null cuts. A bounded-profile Ward identity identifies their linear response with the physical stress tensor. It produces a commuting positive transverse measure and its quadratic moment. The spectral comparison used here is stronger than form order; its proof uses modular strip analyticity.
\item \emph{Removal of the critical scalar tail.} The available scalar comparison is sharpened by ruling out its sole possible obstruction, a critical tail. A causal stress/virial dispersion comparison between the two null horizons isolates that tail as an order-zero Hankel leakage coefficient. A logarithmic test leaves a positive multiple of the scalar leakage norm as the leading term, forcing the tail to vanish.
\item \emph{From charge forms to actual modular action.} The improved
scalar comparison first gives local forms with one scalar insertion.
Their quadratic charges are identified with positive self-adjoint null
moments on a specified domain. Future and past cone comparisons force
the residual dilation charge to vanish. A KMS argument then identifies
the actual cone modular groups, and half-sided inclusions give the
actual time-M\"obius action on line diamonds.
\item \emph{Extraction and identification of the scalar wave.} Actual
diamond modular transformations extract a dimension-two scalar wave
$S$ from the physical virial field, with real localization and bounded
approximants in the original net. The formal trace wave $l$ need not
initially equal $S$. Hilbert products through length four give a
conformal representation on pairs. Its radial positivity and normalized
current fluxes force the residual $l-S$ to vanish.
\item \emph{Reconstruction and improvement.} Bounded approximants give
analytic products at every finite length. Ambient differential identities
control pure scalar products; a causal wave equation and characteristic
Gaussian tests propagate temperedness and spectral support through mixed
orderings with homogeneous original slots. Finite multilinearity then
extends the scalar and Hilbert-valued boundaries to every fixed list of
original fields, with the same joint test kernels. The resulting products
give a common invariant domain, and exchange with the original bounded
commutants gives affiliated realizations. Only after this reconstruction
is the improvement in \eqref{eq:intro-improvement} an operator identity in
the physical algebra.
\end{enumerate}

The intermediate results separate tools that may be useful beyond the present theorem: a bounded-cut modular Ward formula, a two-horizon critical-tail test, and a reconstruction argument that distinguishes a local vacuum wave from a field of the original theory. Their hypotheses and domain restrictions are stated at the points of use.

Three group actions serve different purposes in this argument. Their
relation is summarized here to distinguish the two-point comparison
from the actual local operations used in the construction. The space
$\Pi$ is the closed sum of the scalar $l$, conserved-vector $J$, and
traceless-tensor $t$ vacuum-wave channels. The pair space
$\mathcal H_{\leq2}$ is spanned by the vacuum and the one- and
two-insertion tube vectors for $L,S,J,t$.
\begin{center}
\small
\begin{tabular}{@{}p{0.24\textwidth}p{0.23\textwidth}p{0.45\textwidth}@{}}
\toprule
Action & Space & Role\\
\midrule
Auxiliary conformal action & One-field space $\Pi$ &
Fixed by the primary two-point kernels; supplies localization and
spectral comparisons. The null moments are later shown to act this
way on $\Pi$ (Theorem~\ref{C:canonical-Pi}).\\[4pt]
Actual time-M\"obius action & Original $\cH$, for each time direction &
Uses cone and diamond modular groups. It transports bounded
operations in line diamonds and extracts $S$
(Theorem~\ref{C:time-mobius}).\\[4pt]
Conformal action on pairs & $\mathcal H_{\leq2}$ &
Constructed from short-product Ward identities and four-point
positivity. It supplies the radial estimates proving $l=S$
(Proposition~\ref{P:pair-representation}).\\
\bottomrule
\end{tabular}
\end{center}

Section~\ref{sec:foundations} fixes the field-space conventions and
modular inputs. Sections~\ref{sec:null-current} and~\ref{sec:dispersion}
establish the null-cut and dispersion arguments.
Sections~\ref{sec:charges-cones} and~\ref{sec:extraction} identify actual
modular symmetries and extract the scalar vacuum wave.
Section~\ref{sec:short-products} proves its equality with the trace
polarization. Section~\ref{sec:reconstruction} constructs the field and
completes the local Ward identities.

\Needspace{12\baselineskip}
The following notation records the successive constructions.
\begin{center}
\small
\begin{tabular}{@{}p{0.10\textwidth}p{0.84\textwidth}@{}}
\toprule
Symbol & Role in the proof\\
\midrule
$l$ & Trace scalar vacuum wave, defined in \eqref{F:wave-decomposition}.\\
$S$ & Scalar vacuum wave extracted from the virial field, with physical real localization (Proposition~\ref{E:accessible-scalar}) and later bounded approximants (Proposition~\ref{E:bounded-approximation}).\\
$\ell$ & Residual wave $l-S$, written $L-S$ inside short products; Theorem~\ref{P:return} proves $\ell=0$.\\
$L$ & Initially a scalar insertion symbol, then an entry in analytic products. Its physical field realization is obtained only after Theorem~\ref{R:mixed-products}, the common-domain construction, and Proposition~\ref{R:affiliation}.\\
\bottomrule
\end{tabular}
\end{center}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/01-introduction.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/02-foundations.tex ====
\section{Operational framework, vacuum waves, and null cuts}
\label{sec:foundations}

We use the metric $\eta=\operatorname{diag}(1,-1,-1,-1)$ and
$U(x)=e^{iP\cdot x}$. Scalar products are antilinear in their first
argument. Our dilation convention is
\[
 U_D(s)P_\mu U_D(-s)=e^sP_\mu,\qquad
 U_D(s)O(x)U_D(-s)=e^{sd}O(e^sx)
\]
for a field of dimension $d$. The distinction between a field, its
vacuum wave, and a bounded local operation will be maintained throughout.

\subsection{Original physical data and the vacuum-generated domain}

The hypotheses below specify the original fields, their bounded local
net, and their Ward currents. We also fix the domain required when a
new field is admitted to that theory. Proposition~\ref{F:old-domain}
constructs this domain from the original vacuum distributions and
proves its compatibility with the given affiliated realizations.

\Needspace{6\baselineskip}
\begin{assumption}[Operational physical theory]\label{ass:framework}
The physical Hilbert space $\cH$ is generated from a vacuum $\Omega$ by
the original physical fields and their adjoints. The fields, in finite-component
Lorentz multiplets, and their ordered products are operator-valued
Wightman distributions on a common invariant polynomial domain, which we
call the initial realization domain $\mathcal E$. They have positive translation spectrum,
Poincar\'e covariance, and locality,
with the usual grading when fermions occur. The vacuum is invariant under
Poincar\'e transformations and continuous dilations; uniqueness of the
invariant vector is not assumed.

Let $\mathcal D_{\mathrm{fact}}\subset\mathcal E$ be the algebraic span of
$\Omega$ and all finite original vacuum words with compactly supported
factor tests. To specify the vacuum-generated domain, for every finite ordered list $\mathbf A=(A_1,\ldots,A_n)$ of original fields and adjoints,
write $\Psi^{\mathrm{old}}_{\mathbf A}(h)$ for the continuous extension of
its original factor-test vacuum map to joint kernels
$h\in\mathcal S((\R^4)^n)$. These kernels
include $C_c^\infty((\R^4)^n)$ and finite sums of factorized tests. Set
\[
 \mathcal D_{\mathrm{old}}
 =\Span\{\Omega,\Psi^{\mathrm{old}}_{\mathbf A}(h):
          n=1,2,\ldots,\ h\in\mathcal S((\R^4)^n)\}.
\]
The existence of these extensions and their insertion actions is proved
in Proposition~\ref{F:old-domain}; it is not an additional assumption.
This is an algebraic span: it has no
Hilbert or graph completion and no bound on word length or tensor rank.
The domain whose containment is required below is precisely
$\mathcal D_{\mathrm{old}}$, not an independently larger choice of
$\mathcal E$.

Let $\mathcal F$ be the complex vector space of original point-field
labels, modulo equality of their operator-valued distributions on
$\mathcal D_{\mathrm{fact}}$; Proposition~\ref{F:old-domain} makes this
equivalently equality on $\mathcal D_{\mathrm{old}}$. It is the algebraic direct sum
\[
 \mathcal F=\bigoplus_{\Delta\in\Sigma}\mathcal F_\Delta,
 \qquad
 A=\sum_{\Delta\in S_A}A_\Delta,
 \quad S_A\subset\Sigma\ \text{finite},
\]
where $\Sigma\subset\R$ is a discrete set of real dimensions, bounded
below and locally finite: $\Sigma\cap[a,b]$ is finite for every finite
interval $[a,b]$. Each $\mathcal F_\Delta$ is a finite-dimensional
Lorentz-stable space of finite-component fields satisfying
\[
 U_D(s)A_\Delta(x)U_D(-s)
       =e^{s\Delta}A_\Delta(e^s x)
\]
as operator-valued distributions on $\mathcal D_{\mathrm{fact}}$, and hence
on $\mathcal D_{\mathrm{old}}$ by that proposition.
Multiplicity counts independent component fields after including all
Lorentz component indices and imposing linear field identities;
equivalently, each grade contains finitely many finite-component Lorentz
multiplets. Homogeneous components are themselves original physical
fields. Adjunction preserves each real grade, with conjugate Lorentz
components, and covariant distributional differentiation sends
$\mathcal F_\Delta$ into $\mathcal F_{\Delta+1}$, with a missing grade
interpreted as zero. Lorentz transformations preserve the grades and
statistics; translations act on spacetime arguments. No completion by
infinitely many scaling components is taken.

There is a fixed isotone net $O\mapsto\cM(O)$ of weakly closed
algebras of physical bounded operations, covariant under the same
physical Poincar\'e and dilation representations. Algebras at spacelike separation
commute, with the fixed statistics twist in the graded case. Each smeared
original physical field has a closed affiliated realization in its localization
algebra, extending its realization on $\mathcal E$, whose adjoint also
extends the field adjoint there. The
corresponding extensions on $\mathcal D_{\mathrm{old}}$ are derived in
Proposition~\ref{F:old-domain}.
For any other open region $R$,
\[
 \cM(R)=\bigvee_{\overline O\subset R}\cM(O),
\]
where $O$ ranges over double cones. Physical point-field membership in
$\Aphys$ is intrinsic to this fixed net, not to an enumeration of symbols.
An operationally reconstructed finite family of homogeneous
finite-component point fields of real scaling dimensions is admitted,
together with its finite linear span, once its localization supports can
shrink inside arbitrarily small double cones; all joint field distributions
of the family with itself and with every fixed finite list of original
fields and adjoints have been established at every finite length on a
compatible common invariant domain containing $\mathcal D_{\mathrm{old}}$,
with the required adjoints and the preceding covariance, spectrum, and
locality/statistics properties; and compatible smeared realizations
affiliated with the original localization algebras have been established,
with their adjoints extending the field adjoints on that domain. Separately
constructed fields are not combined into a common polynomial field algebra
without their mutual joint distributions, common invariant domain with the
required adjoints, and original-net affiliations having been established.
The criterion asserts no candidate's existence or analytic properties and
admits no point field merely by convergence of an infinite nonhomogeneous
scaling sum.

The original physical fields include a symmetric conserved stress tensor
$T_{\mu\nu}$ with the translation and orbital Lorentz Ward identities,
and a local virial field $V_\mu$ such that
\[
 D_\mu=x^\nu T_{\mu\nu}-V_\mu,\qquad
 \partial^\mu D_\mu=0,\qquad
 T^\mu{}_{\mu}=\partial^\mu V_\mu.
\]
The Ward identities of $D$ generate the stated dilations. Charge Ward
identities are understood through compact Cauchy slabs surrounding an
insertion and extend to polynomial products by the derivation rule.
The identities displayed above are flat-space field identities; the
usual contact terms in time-ordered distributions are allowed.
\end{assumption}

Finite scaling support concerns field labels. It leaves all finite product
lengths and all joint Schwartz and compact smooth kernels available,
without a finite tensor-rank requirement. It restricts neither Hilbert
completions and generator spectra nor weak closures of bounded algebras.
The bounded-locality and original-net affiliation clauses are separate
hypotheses: neither weak commutation of field restrictions nor affiliation
in a larger relatively local net supplies them. No Haag duality, split
property, phase-space estimate, scattering theory, or conformal symmetry
is assumed.

\Needspace{6\baselineskip}
\begin{proposition}[Original joint-test domain]\label{F:old-domain}
For every fixed finite ordered list of original fields and adjoints,
the vacuum distribution has a unique tempered extension and its vacuum
map has a continuous Hilbert-valued extension to all joint Schwartz
kernels. The space $\mathcal D_{\mathrm{old}}$ is Hilbert-dense and
invariant under old field insertions and continuous joint polynomial
insertions, with their prescribed adjoints. Finite field-label
identities, covariance, locality, and the compact-slab Ward identities
pass to this domain with their stated support conditions.

For a compactly smeared original field $B(f)$, each retained closed
affiliated realization and its Hilbert adjoint agree on
$\mathcal D_{\mathrm{old}}$ with these insertion actions. This assertion
also holds for an originally supplied Schwartz realization. It does
not require $\mathcal D_{\mathrm{old}}$ to be a graph core for that
realization.
\end{proposition}

\begin{proof}
\emph{Tempered distributions and finite field-label expansions.}
For every fixed ordered list of homogeneous original fields
$A_i\in\mathcal F_{\Delta_i}$, its already given vacuum distribution is
tempered. To see this, remove the common translation coordinate. In the
remaining relative coordinates $\xi$, dilation invariance of the vacuum
gives
\[
 W(\lambda\xi)=\lambda^{-\sum_i\Delta_i}W(\xi),\qquad \lambda>0,
\]
in the distributional pullback sense. A compact neighborhood of the
origin is controlled by the local finite order of the given distribution.
On a fixed compact annulus it is bounded by finitely many test-function
seminorms. Dilating that annulus transfers the bound to every outer dyadic
annulus with a fixed power determined by the dimensions and the seminorm
order. Summation against a sufficiently high Schwartz seminorm proves
temperedness. The zero- and one-point cases are immediate.

Adjoints are homogeneous at the same real grade. For any fixed list of
original fields $A_i=\sum_{\Delta\in S_i}A_{i,\Delta}$, multilinearity on
$\mathcal D_{\mathrm{fact}}$ first gives the component expansion on compact
factor tests. Compact tensor-test density and continuity of the given joint
distributions then give, for the same compact joint kernel $\varphi$,
\begin{equation}\label{F:finite-label-extension}
 W(A_1,\ldots,A_n)[\varphi]
 =\sum_{(\Delta_1,\ldots,\Delta_n)\in S_1\times\cdots\times S_n}
 W(A_{1,\Delta_1},\ldots,A_{n,\Delta_n})[\varphi].
\end{equation}
The finite sum of homogeneous tempered extensions defines the unique
tempered extension of the left side, so the same identity holds for every
joint Schwartz kernel. It agrees with any originally specified Schwartz
distribution by its stated continuity and compact-test density. Its constants
and seminorm orders may depend on the fixed list; no single total scaling
dimension is assigned to a nonhomogeneous sum.

\emph{Hilbert-valued words and insertion actions.}
For a fixed old list $\mathbf A$, write $\Psi^0_{\mathbf A}(u)$ for the
original vacuum vector of a finite sum $u$ of compact factor tests. The
doubled tempered distribution gives a continuous Schwartz seminorm $p$
and a constant $C$ such that
\[
 \|\Psi^0_{\mathbf A}(u)-\Psi^0_{\mathbf A}(v)\|^2
 =W(\mathbf A^*,\mathbf A)[(u-v)^*\otimes(u-v)]
 \le C p(u-v)^2.
\]
Here the star reverses and adjoints the field list and conjugates and
reverses the kernel. Compact finite tensor tests are dense in the joint
Schwartz space. The estimate therefore constructs the unique continuous
Hilbert-valued extension $\Psi^{\mathrm{old}}_{\mathbf A}$ used in the
definition of $\mathcal D_{\mathrm{old}}$; polarization passes all its Gram
identities to the limit. Any originally specified Schwartz vacuum-word map
agrees with this extension by its stated continuity and the same density.
Consequently the original vacuum-generation assumption also makes
$\mathcal D_{\mathrm{fact}}$ Hilbert-dense.
This argument concerns only original products whose existence is assumed;
it supplies no product containing a newly constructed field.

For a fixed old word
$\mathbf A$ and a fixed finite old prefix $\mathbf B$, the same doubled
vacuum-distribution estimate applies to the longer word. Consequently, if
$h_j\to h$ in the joint Schwartz topology, then for a fixed prefix kernel
$g$,
\[
 \Psi^{\mathrm{old}}_{\mathbf A}(h_j)\longrightarrow
 \Psi^{\mathrm{old}}_{\mathbf A}(h),\qquad
 \Psi^{\mathrm{old}}_{\mathbf B,\mathbf A}(g\otimes h_j)
 \longrightarrow
 \Psi^{\mathrm{old}}_{\mathbf B,\mathbf A}(g\otimes h)
\]
in Hilbert norm. The estimates are continuous jointly in $g,h$ on their
Schwartz spaces. Reversed adjoint kernels obey the same bounds. Compact
joint kernels can be approximated, after a finite partition into product
patches, by finite sums of factor tests with the prescribed local supports;
Schwartz kernels admit finite tensor approximations in the Schwartz
topology. Concatenation, reversal, and differentiation are continuous in
these test spaces.

Left concatenation of old test words is well-defined on their vacuum
vectors. Indeed, if a finite old test polynomial $P$ represents zero,
positivity and Cauchy--Schwarz applied to $P$ and $Q^*QP$ give
$W(P^*Q^*QP)=0$ for every finite old test polynomial $Q$. This is the
squared norm of the proposed vector for $QP$. Thus the null vectors form
a left ideal, including cancellations between different words, and the
same vacuum pairing places adjoint insertion inside the Hilbert adjoint.
Concatenation therefore makes $\mathcal D_{\mathrm{old}}$ invariant
under the old field and polynomial insertions. It is Hilbert-dense because
it contains the original factorized vacuum-polynomial vectors that generate
$\cH$.

\emph{Agreement with the retained realizations.}
Let $T_B(f)$
be a chosen closed affiliated realization of an original field with compact
test $f$, and choose finite sums of compact tensor tests $h_j\to h$ as above.
On the vectors in $\mathcal D_{\mathrm{fact}}\subset\mathcal E$ the original extension clause gives
\[
 T_B(f)\Psi^{\mathrm{old}}_{\mathbf A}(h_j)
   =\Psi^{\mathrm{old}}_{B,\mathbf A}(f\otimes h_j).
\]
Both sides converge in Hilbert norm by the two displayed continuity
statements. Closedness of $T_B(f)$ therefore proves the same equality for
$h$. Applying the argument to the closed Hilbert adjoint $T_B(f)^*$ gives
its equality with the insertion of $B^\dagger(\bar f)$ there. Iterating
this one-field statement proves the corresponding actions of every finite
factorized prefix and its reversed adjoint on
$\mathcal D_{\mathrm{old}}$. Joint polynomial insertions are the continuous
concatenations already defined; this argument does not assign an affiliated
closure to an arbitrary joint smear.

If an original Schwartz smearing $B(f)$ and its closed affiliated
realization were already supplied, the same conclusion holds for that
realization. To identify its action on a fixed vector of
$\mathcal D_{\mathrm{fact}}$, approximate $f$ by compact tests in Schwartz
topology. The longer doubled correlations give Hilbert convergence of the
inserted vectors, while the original Schwartz distributional continuity
gives the same limiting matrix elements against
$\mathcal D_{\mathrm{fact}}$. Density identifies the vectors, also for the
adjoint. The preceding $h_j$ closed-graph argument then applies unchanged.
This uses only an originally supplied Schwartz action and its stated
continuity; it asserts no pre-existing realization for a smearing that was
not originally supplied.

\emph{Identities and local Ward actions.}
The same paired vector and inserted-vector limits extend every finite
field-label identity from $\mathcal D_{\mathrm{fact}}$ to
$\mathcal D_{\mathrm{old}}$. The converse follows from
$\mathcal D_{\mathrm{fact}}\subset\mathcal D_{\mathrm{old}}$, so the two
equality conventions for $\mathcal F$ agree. The homogeneous dilation law
and the adjoint and derivative grade laws extend in the same way, because
the transformed, reversed, and differentiated kernels remain in the stated
test spaces. Thus none of these identities on joint vectors was used to
construct those vectors.

The original covariance and flat field identities pass from the factorized
vectors to joint vectors by the same longer-word continuity, with transformed
or differentiated kernels. Locality and compact-slab Ward identities pass
under their stated support conditions. For a general Schwartz spectator
vector, keep the localized slab and insertion tests fixed and approximate
only the spectator kernel in Schwartz topology. If a localized insertion
polynomial itself has a compact joint kernel, use the support-preserving
compact approximants.
Thus the operational identities used below hold on the dense invariant
vacuum-generated domain $\mathcal D_{\mathrm{old}}$, with the original
vacuum correlations, Hilbert space, and bounded net unchanged. The
factor approximations just used converge in the graph of each specified
$T_B(f)$ on these joint vectors; Hilbert density alone was not used for
that conclusion. No claim is made that $\mathcal D_{\mathrm{old}}$ is a
graph core for a chosen realization, that the closure of its field
restriction equals that realization, or that a newly constructed field acts on arbitrary additional
vectors of $\mathcal E$.
\end{proof}

We will use \eqref{F:finite-label-extension} as a finite-label permanence
principle for later distributions. Every term uses the same joint kernel;
continuity, common spectral cones, and linear distributional identities
pass through the finite sum. Positivity and descent through null
polynomial vectors require their full summed forms and will be checked
where they arise.

The dimension set is countable, and each grade has a finite component
basis. Countable dense test families for the resulting countable
collection of finite words, together with the Hilbert-valued continuity
just proved, show that $\cH$ is separable. This supplies the separability
used in the direct-integral descriptions below.

The spectrum condition now gives tube analyticity, uniqueness from an open
boundary patch, and Reeh--Schlieder for these original distributions. In
particular, a separated-point field identity valid on a mutually spacelike
open patch extends to ordered vacuum distributions by tube uniqueness.
This assertion does not assign differentiated Ward identities to a
particular choice of time ordering.

\subsection{Example: the full free scalar theory}
\label{F:free-scalar-example}

The massless real scalar in its four-dimensional vacuum Fock
representation gives a concrete example. Take the full scalar net
$\cM(O)=\{e^{i\overline{\phi(f)}}:
f\in C_c^\infty(O,\R)\}^{\prime\prime}$ and, as original fields, finite
linear combinations of Wick monomials
$:\!\prod_{j=1}^n\partial^{\alpha_j}\phi\!:$.
Their dimensions are the nonnegative integers
$n+\sum_j|\alpha_j|$, with finitely many component monomials at
each grade. The Weyl relations give bounded locality, and Wick's
construction supplies all joint products and their invariant domain
\cite[Theorem~5.7 and its following remark]{BrunettiFredenhagenKoehler1996}.
Affiliation has a concrete justification here. On the common domain
invariant under Wick fields and Weyl operators
\cite[Section~2.2, equation~(11)]{BrunettiFredenhagen2000},
a Wick smearing commutes with exterior Weyl operators. Its closed
graph retains this commutation and domain invariance. Double-cone
duality of the free massless scalar
\cite[Section~3, Corollary~2]{HislopLongo1982} identifies the generated
exterior algebra with $\cM(O)'$, proving affiliation. The formal
adjoint gives the required Hilbert-adjoint extension. This is a
property of the example, not a duality hypothesis of the theorem.

\subsection{Selecting the current weights}
\label{F:weights}

We may, and henceforth do, replace $T$ and $V$ by Hermitian homogeneous
fields of dimensions four and three. We justify both the selection and
preservation of the charges. First take the even components of the
spacetime currents. The prescribed unitary affine derivations preserve
parity, so the odd-current Ward contributions vanish by parity separation
on parity-homogeneous insertions; the flat field identities split by parity as
well. For an even current $J$ with a real slab test, polynomial adjunction
on $\mathcal D_{\mathrm{old}}$ gives
\[
 \bigl(i[J,A]\bigr)^\dagger=i[J^\dagger,A^\dagger].
\]
The prescribed unitary derivation preserves adjoints, and its Ward identity
also applies to $A^\dagger$ with the same support. Hence $J^\dagger$ and
$(J+J^\dagger)/2$ have the same Ward action as $J$. Applying this to the
translation and orbital currents of $T$ and to $D=xT-V$ shows that the
Hermitian parts preserve all required charges. The real metric, coordinate,
and derivative coefficients also preserve the flat field identities under
adjunction. Relabel these even Hermitian representatives as $T,V$.
The adjoint grade law preserves their decompositions. Write the finite sums
\[
 T=\sum_{d\in S_T}T^{(d)},\qquad
 V=\sum_{e\in S_V}V^{(e)},
\]
with every missing component interpreted as zero. Directness of the grading,
Lorentz covariance, and the derivative grade law split symmetry,
Hermiticity, conservation, and the trace identity. Thus, for every
$d\in S_T\cup\{e+1:e\in S_V\}$,
\[
 T^{(d)}_{\mu\nu}=T^{(d)}_{\nu\mu},\qquad
 \partial^\mu T^{(d)}_{\mu\nu}=0,\qquad
 (T^{(d)})^\mu{}_{\mu}=\partial^\mu V^{(d-1)}_\mu,
\]
and the components can be taken Hermitian. In particular each geometric
current
\[
 D^{(d)}_\mu=x^\nu T^{(d)}_{\mu\nu}-V^{(d-1)}_\mu
\]
is conserved.

Fix a homogeneous insertion $O$ of dimension $\alpha$ and homogeneous
spectators. A compact-slab translation commutator made from $T^{(d)}$ has
weight $\alpha+d-3$: the slab integral removes three powers of length.
An orbital Lorentz commutator from $xT^{(d)}$ has weight $\alpha+d-4$,
as does a dilation commutator from the pair $D^{(d)}$. Conservation of
these component currents permits the slabs to be rescaled while keeping
the insertion inside. After testing the homogeneous spectators, these are
identities between ordinary scalar distributions. The prescribed
translation, Lorentz, and dilation actions have weights $\alpha+1$,
$\alpha$, and $\alpha$, respectively. After testing the remaining variables,
the Ward identities are finite scalar sums of real exponentials in the
logarithmic scale. Their linear independence
selects the $d=4$ translation and orbital contributions and the $d=4$
dilation pair $xT^{(4)}-V^{(3)}$; the other corresponding Ward
contributions vanish. This is a statement about charges, not the vanishing
of the nonselected fields.

Finite multilinearity now extends these identities to every fixed list of
original fields and adjoints under the same test kernels, and the retained
derivation rule extends them to every finite polynomial word. The selected
components retain conservation, the trace identity, and all required Ward
actions. Replacing the original stress tensor by $T^{(4)}$ adds a local
symmetric conserved tensor with zero translation and Lorentz charges.
This is an admissible charge-preserving change of stress tensor; it does
not presume a scalar improvement. The field decompositions here are
identities on $\mathcal D_{\mathrm{old}}$; no equality between a chosen
closed realization of a sum and a sum of chosen closed realizations is
needed. Each selected component has its own affiliated realization by
Assumption~\ref{ass:framework}.

\subsection{Modular input and a spectral comparison lemma}
\label{F:modular}

For a standard real subspace $X\subset\cH$, write
\[
 \mathfrak s_X(x+iy)=x-iy\quad(x,y\in X),\qquad
 \mathfrak s_X=J_X\Delta_X^{1/2},\qquad
 S_X=-\frac{1}{2\pi}\log\Delta_X.
\]
Here standard means $\overline{X+iX}=\cH$ and $X\cap iX=\{0\}$.
The symplectic complement is
$X'=\{\xi:\operatorname{Im}\ip{\xi}{x}=0\text{ for }x\in X\}$.
For a standard algebra $(\cM,\Omega)$ its associated real space is
$X_\cM=\overline{\{A\Omega:A=A^*\in\cM\}}$.
Statistics twists in complementary spaces and algebras are understood
in all statements involving odd fields.

Every original polynomial word in this subsection has the finite component
expansion described above, with its joint test kernel unchanged. Thus the
preceding temperedness and finite-component covariance apply to every word
used in the polynomial argument.

The Wightman Bisognano--Wichmann theorem identifies the wedge polynomial
Tomita graph with $J_W e^{-\pi B}$, where $B$ generates the wedge boost;
hence $\Delta_W^{it}=e^{-2\pi itB}$ and the opposite wedge has the
complementary real space \cite{BW1976}. The published theorem assumes a
unique translation-invariant vacuum. We explain its boundary calculation
and core argument in the version needed here, followed by the passage
to bounded algebras. First, uniqueness of the vacuum is unnecessary for
this graph identification. The reflection itself must be constructed before the
core argument. Extend the covariance of the vacuum distributions along
the connected complex Lorentz group. On a real Jost patch, locality
reverses the order of the fields with the prescribed statistics signs;
analytic uniqueness extends the resulting distributional equality.
Combine it with the Hermitian-conjugation identity for vacuum
distributions. If $F$ and $G$ are smeared polynomial words and
$\xi_F=F\Omega$, it says
\[
 \ip{\xi_F^{\natural}}{\xi_G^{\natural}}
       =\ip{\xi_G}{\xi_F}.
\]
Here $\xi_F^{\natural}$ is the reflected-adjoint word: conjugate its
test functions, reflect their arguments by
$j_W(u,v,y)=(-u,-v,y)$, and replace the components by their adjoints
with the finite-component matrices of the imaginary wedge boost at
rapidity $i\pi$ and the fixed statistics twist. Apply this operation
to each factor in the \emph{original word order}; the order reversal
above occurs in the vacuum distributions used to compute scalar
products. This is precisely the finite-component reflection boundary
calculation in the polynomial BW proof. It uses covariance, the spectrum
condition, locality, and Hermitian conjugation, with no cluster limit
or condition on the dimension of the invariant subspace.

The displayed identity shows that the conjugate-linear word map
preserves norms and descends through all zero-norm polynomial relations.
Reflected test functions again range over the full test space, so its
range is dense. It therefore extends to an antiunitary $J_W$;
reflection twice, with the normal-statistics phases, gives $J_W^2=1$.
Now use polynomial vacuum vectors whose first point and successive gaps
lie in an open wedge-ordered patch. Imaginary boost
through rapidity $\pi$ takes their ordered distributions through the
positive-spectrum tubes, and locality at the spacelike boundary reverses
the fields. Tube uniqueness makes their complex span Hilbert-dense.
Adjoin the adjoint vectors and all real boost translates. The resulting
span is boost-invariant and its adjunction agrees with
$J_W e^{-\pi B}$. To check that it is a graph core, let $\mathcal G$ be
its closed graph and smooth by $e^{-\epsilon B^2}$. This smoothing
preserves $\mathcal G$, and both
$e^{-\epsilon B^2}$ and $e^{-\pi B}e^{-\epsilon B^2}$ are bounded.
Hilbert approximation by the indicated span consequently approximates
every smoothed vector in graph norm. Finally
$e^{-\epsilon B^2}\xi\to\xi$ in the graph norm of $e^{-\pi B}$ for
$\xi$ in its domain. This proves graph totality without a clustering
argument.

Second, the polynomial and bounded-algebra graphs coincide under
Assumption~\ref{ass:framework}. Start with finite sums of factorized
smeared words. If $A$ is a closed affiliated realization of such a
polynomial with $\Omega\in\Dom A\cap\Dom A^*$, its bounded polar
cutoffs $A_m=A\mathbf1_{[0,m]}(|A|)$ satisfy
\[
 A_m\Omega\longrightarrow A\Omega,\qquad
 A_m^*\Omega\longrightarrow A^*\Omega.
\]
For the second convergence use
$A_m^*\Omega=\mathbf1_{[0,m]}(|A|)A^*\Omega$.
Use natural-domain products and sums of the chosen closed affiliated
single-field realizations. Their domains contain $\mathcal D_{\mathrm{old}}$,
and their adjoints contain the corresponding adjoint polynomials there,
so they are closable. Their natural domains and graphs are invariant
under commutant unitaries; the closures remain affiliated. These are
the extensions to which the same polar-cutoff argument applies.
For an admitted compact joint kernel, first use a finite partition into
product patches inside the localization region, then approximate each part
by finite sums of tensor tests in its smooth test-function topology. The
reversed adjoint kernels of these approximants converge to the reversed
adjoint kernel in the same topology. The old Hilbert-valued
joint-distribution bounds proved above pass both the vacuum and
adjoint-vacuum vectors to the limit. Closedness of the bounded-algebra
Tomita graph therefore gives the same graph inclusion for the joint word.
This vector-level limit does not assert a closed affiliated realization for
an arbitrary joint smear.
Let $X_{\rm pol}(R)$ be the closed real span of vacuum vectors of finite
original polynomial expressions that are Hermitian on
$\mathcal D_{\mathrm{old}}$ and compactly localized in $R$, with
admitted compact joint kernels included by the graph limits just proved.
This is an auxiliary vacuum-word graph space; it neither restricts the
original test-function spaces nor redefines $\mathcal D_{\mathrm{old}}$.
Consequently,
\begin{equation}\label{F:BW-sandwich}
 X_{\rm pol}(W)\subset X_{\cM(W)}
 \subset X_{\cM(W')}^{\prime}
 \subset X_{\rm pol}(W')^{\prime}=X_{\rm pol}(W).
\end{equation}
This proves equality and the bounded wedge modular formula. The
inclusion in the middle is bounded locality, not a claim about commuting
unbounded restrictions. To pass from graph equality to algebra wedge
duality, use the modular-invariant-subalgebra theorem: a subalgebra
invariant under the modular group of a faithful normal state admits a
state-preserving conditional expectation \cite{Takesaki1972}; see also
\cite[Theorems~3.5 and~5.2]{AccardiCecchini1982}. If the subalgebra is
$\mathcal N$ and $E$ is this expectation, its GNS implementation is the
vector identity
\[
 E(x)\Omega=P_{\overline{\mathcal N\Omega}}x\Omega.
\]
Apply it to the local inclusion of the
opposite, twist-corrected wedge algebra in $\cM(W)'$. The larger
algebra's modular group is the geometric opposite boost and preserves
the smaller algebra. The latter has cyclic vacuum, so the implementing
projection is the identity. The expectation is therefore the identity
by separation, proving equality of these algebras. Double-cone duality
is not used.

We use the following exact forms of the modular translation theorems
\cite{Borchers1992,ArakiZsido2005}; for the positive-translation
statement we use the explicit formulation and proof in
\cite[Theorem~II.5.2]{Borchers2000}. If $e^{icb}$ fixes $\Omega$, $b\ge0$,
and $\Ad(e^{icb})\cM\subset\cM$ for $c\ge0$, where $\Omega$ is cyclic
and separating for $\cM$, then
\begin{equation}\label{F:Borchers}
 e^{isS_\cM}b e^{-isS_\cM}=e^s b,\qquad J_\cM bJ_\cM=b.
\end{equation}
If $\mathcal N\subset\cM$ have the same cyclic separating vector and
$\Ad(e^{isS_\cM})\mathcal N\subset\mathcal N$ for $s\ge0$, the
half-sided modular inclusion theorem gives
\begin{equation}\label{F:HSMI}
 \begin{split}
 H&=\overline{S_\cM-S_\mathcal N}\ge0,\qquad H\Omega=0,\qquad
 e^{isS_\cM}He^{-isS_\cM}=e^sH,\\
 J_\cM HJ_\cM&=H,\qquad
 \mathcal N=\Ad(e^{iH})\cM,\qquad
 e^{isS_\mathcal N}=e^{i(1-e^s)H}e^{isS_\cM}.
 \end{split}
\end{equation}
The closure in the first line is the self-adjoint closure on the common
logarithm domain; subsequent sums of affine-group generators will have
this meaning. The applicable inclusion result is
\cite[Theorem~2.1]{ArakiZsido2005}, specialized to a vector state. It
requires no cyclicity for $\mathcal N'\cap\cM$. Since $J_\cM$ is
antiunitary, its unitary-group formula is
$J_\cM e^{itH}J_\cM=e^{-itH}$, consistently with the generator formula
above. These results also hold for standard real subspaces; one may
apply the algebra result to their CCR Weyl algebras and restrict the
vacuum modular objects to the one-particle space.

For later form comparisons, $X\subset X_1$ implies
$\mathfrak s_X\subset\mathfrak s_{X_1}$. Thus the closed form of
$\Delta_X$ restricts that of $\Delta_{X_1}$ and
$\Delta_X\ge\Delta_{X_1}$ in form order. The resolvent integral for the
operator-monotone logarithm gives $S_X\le S_{X_1}$ on common operator
domains. This statement by itself is weaker than the next lemma. Koot's
spectral characterization of inclusions of standard pairs uses the
modular-strip and half-plane mechanism
\cite[Theorem~3.1, Lemma~3.4 and Theorem~3.5]{Koot2025}.
We prove the contraction version between two possibly different
standard spaces that is needed for the embedded comparison channels.

\Needspace{6\baselineskip}
\begin{lemma}[Strip comparison]\label{F:strip-comparison}
Let $X_i$ be standard real spaces with normalized modular generators
$B_i$, and let $K_i\ge0$ be self-adjoint with
$e^{itB_i}K_i e^{-itB_i}=e^tK_i$. Suppose $j:\cH_1\to\cH_2$ is a
contraction intertwining the boost groups and
\[
 j e^{iK_1}X_1\subset e^{iK_2}X_2.
\]
Then, for every $R\ge0$,
\begin{equation}\label{F:strip-bands}
 \mathbf1_{(R,\infty)}(K_2)j\mathbf1_{[0,R]}(K_1)=0.
\end{equation}
In particular, for $s>0$ and $\xi\in\Dom K_1^s$,
$j\xi\in\Dom K_2^s$ and
$\norm{K_2^s j\xi}\le\norm{K_1^s\xi}$.
\end{lemma}

\begin{proof}
We first prove that, for every $r>0$,
\begin{equation}\label{F:strip-exponential}
 e^{rK_2}j e^{-rK_1}\ \text{is everywhere defined and contractive}.
\end{equation}
Its exponential growth on a high target band will then exclude that band
from the image of a low source band.

Set $A=e^{-iK_2}je^{iK_1}$. It maps $X_1$ into $X_2$, so
$\mathfrak s_2A\xi=A\mathfrak s_1\xi$ on
$\Dom\mathfrak s_1=\Dom\Delta_1^{1/2}$. Equivalently,
\[
 \Delta_2^{1/2}A\xi
 =J_2AJ_1\Delta_1^{1/2}\xi
 \qquad(\xi\in\Dom\Delta_1^{1/2}).
\]
Thus $\Delta_2^{1/2}A\Delta_1^{-1/2}$, initially on
$\Dom\Delta_1^{-1/2}$, has the contractive bounded extension
$J_2AJ_1$. The operators
\[
 F(z)=\Delta_2^{iz}A\Delta_1^{-iz},\qquad
 -\tfrac12<\operatorname{Im}z<0,
\]
have contractive boundary values: the real boundary multiplies $A$
on either side by a unitary, and the other boundary is
$\Delta_2^{it}J_2AJ_1\Delta_1^{-it}$. The analytic-extension theorem
\cite[Theorem~2.3]{ArakiZsido2005}, followed by three lines, gives the
contractive analytic continuation to the strip. Equivalently, sandwich
between bounded spectral intervals of the modular logarithms, apply
three lines to their scalar matrix coefficients, and remove the
intervals. The displayed Tomita relation supplies the full source domain
needed at the second boundary.
Write $\widetilde F(\zeta)=F(z)$ for $\zeta=e^{-2\pi z}$, which
maps the strip bijectively onto the upper half-plane. Boost covariance
and the intertwining property of $j$ identify its positive real boundary
value as $e^{-i\zeta K_2}j e^{i\zeta K_1}$. To justify the interior
expression despite its potentially unbounded first factor, take
$\xi$ and $\eta$ in bounded spectral bands of $K_1$ and $K_2$,
respectively. The scalar function
\[
 \ip{e^{i\overline\zeta K_2}\eta}{j e^{i\zeta K_1}\xi}
\]
is entire and agrees with
$\ip{\eta}{\widetilde F(\zeta)\xi}$ on a positive real interval.
Boundary uniqueness, or reflection of their zero difference across
that interval, identifies them throughout the upper half-plane. At
$\zeta=ir$, $r>0$, contractivity therefore gives
\[
 \left|\ip{e^{rK_2}\eta}{j e^{-rK_1}\xi}\right|
 \le\norm{\eta}\,\norm{\xi}.
\]
Bounded spectral vectors are a core for $e^{rK_2}$. The adjoint-domain
criterion implies $j e^{-rK_1}\xi\in\Dom e^{rK_2}$ and identifies its
image with $\widetilde F(ir)\xi$. Finally approximate any source vector
by its $K_1$ spectral truncations. Boundedness of $e^{-rK_1}$ and
$\widetilde F(ir)$, followed by closedness of $e^{rK_2}$, proves that
$e^{rK_2}je^{-rK_1}$ is everywhere defined and contractive.
If $\xi=\mathbf1_{[0,R]}(K_1)\xi$, apply this extension to
$e^{rK_1}\xi$ to obtain
\[
 \norm{e^{rK_2}j\xi}\le e^{rR}\norm{\xi}.
\]
Projection onto $K_2\ge R+\epsilon$ and $r\to\infty$ shows that this
projection annihilates $j\xi$. Taking $\epsilon\downarrow0$ proves
\eqref{F:strip-bands}. The resulting tail inequality is
\[
 \norm{\mathbf1_{(a,\infty)}(K_2)j\xi}
 \le\norm{\mathbf1_{(a,\infty)}(K_1)\xi}.
\]
Integrate its square against $2s a^{2s-1}\dd a$. Truncation followed by
monotone convergence proves both the domain assertion and the norm
bound.
\end{proof}

\subsection{Timelike vacuum channels}
\label{F:waves}

Fix future null vectors $n,\bar n$ with $n\cdot\bar n=1$ and orthonormal
transverse axes $e_i$, $i=1,2$, with $e_i\cdot e_j=-\delta_{ij}$. Write
\begin{equation}\label{F:null-coordinates}
 \begin{gathered}
 x=vn+u\bar n+y,\qquad p=an+b\bar n+k,\qquad
 p\cdot x=au+bv-\boldsymbol k\cdot\boldsymbol y,\\
 2ab=M^2+|\boldsymbol k|^2.
 \end{gathered}
\end{equation}
The variables $a,b,\boldsymbol k$ also denote their commuting momentum
operators. Coordinate subscripts on covariant tensors mean contraction
with the coordinate basis vectors. The boost $B$ acts on spacetime by
$v\partial_v-u\partial_u$, and null rotations $G_i$ by
$u\partial_i+y_i\partial_v$.

Disintegrate the positive two-point measure of a dimension-$d$ multiplet
on the open timelike cone. Lorentz transformations and dilations act
transitively there, so relative to $\dd^4p$ its vacuum amplitude at rest
has norm proportional to $M^{d-2}$. The proportionality is a positive
finite-dimensional rotation-invariant matrix; transport to other
momenta uses the finite Lorentz matrices and unitary fiber transport.
There is no independent positive singular measure on an interior mass
shell, since dilation moves every positive mass shell and fixes the
homogeneous measure class on their union.

A possible contribution on the nonzero light cone is normalized instead
by $\dd^4p\,\theta(p^0)\delta(p^2)$. Its amplitude on a fixed null ray
would grow as energy to the power $d-1$. A boost along that ray has only
the boost weights of the finite Lorentz multiplet; positivity therefore
excludes the contribution if $d-1$ exceeds their maximum. The maxima
are one for a vector and two for a symmetric rank-two tensor, whereas
$d-1$ is respectively two and three for $V$ and $T$. A zero-momentum
atom for a positive-dimensional field is also impossible: its norm is
invariant under unitary dilation but would be multiplied by $e^{sd}$.
Thus both $V\Omega$ and $T\Omega$ have purely timelike spectrum.

Let $\Theta=T^\mu{}_{\mu}$. At this stage define only vacuum waves,
\begin{equation}\label{F:wave-decomposition}
 l=-\Theta/M^2,\qquad
 J_\mu=V_\mu-ip_\mu l,\qquad
 t_{\mu\nu}=T_{\mu\nu}
       -\tfrac13(-\eta_{\mu\nu}M^2+p_\mu p_\nu)l.
\end{equation}
The local Ward identities imply
$p^\mu J_\mu=0$, $p^\mu t_{\mu\nu}=0$, and
$\eta^{\mu\nu}t_{\mu\nu}=0$. At rest their rotation types are respectively
spin zero, one, and two, so they are orthogonal. Their rest norms are
constant, proportional to $M$, and proportional to $M^2$. In particular
$l$ is a well-defined dimension-two wave; division by $M^2$ has introduced
no infrared singularity in its two-point measure.

These channels have the two-point data of a scalar primary of dimension
two, a conserved vector primary of dimension three, and a conserved
symmetric traceless primary of dimension four. This follows directly
from the uniqueness of a positive rotation-invariant metric on each of
the three rest representations and homogeneous Lorentz transport.
Equivalently their position-space kernels are the positive-energy
boundary values proportional to
\[
 \frac1{(x^2)^2},\qquad
 \frac{I_{\mu\nu}(x)}{(x^2)^3},\qquad
 \frac{I_{\mu\rho}I_{\nu\sigma}+I_{\mu\sigma}I_{\nu\rho}
                 -\eta_{\mu\nu}\eta_{\rho\sigma}/2}{2(x^2)^4},
 \quad I_{\mu\nu}=\eta_{\mu\nu}-\frac{2x_\mu x_\nu}{x^2}.
\]
Their overall constants include the prescribed spectral boundary
normalization. Let $\Pi\subset\cH$ be the Hilbert sum of these wave
channels, omitting zero-norm channels. Kernel covariance defines an
auxiliary unitary conformal action on $\Pi$: transform tube point vectors
with their indicated tensor matrices and weights, observe preservation
of their scalar products, and extend from their dense span. This is a
comparison representation, not yet a symmetry of the physical theory.
Its real localization spaces can equivalently be constructed as those
of the corresponding positive generalized free fields.

The following rest axes are adapted to the fixed null direction:
\begin{equation}\label{F:tetrad}
 e_z^{(n)}=\frac{M}{b}n-\frac pM,\qquad
 e_i^{(n)}=e_i+\frac{k_i}{b}n.
\end{equation}
They are orthogonal to $p$, have squared norm $-1$, and are mutually
orthogonal. Define the constant spin labels $j_A$ and $s_{AB}$ by the
components of $J/(iM)$ and $t/M^2$ in this tetrad, and put $s_z=s_{zz}$.
Then
\begin{equation}\label{F:longitudinal-labels}
 V_v=ibc_V,\qquad T_{vv}=b^2c_T,\qquad
 c_V=l+j_z,\qquad c_T=l/3+s_z.
\end{equation}
The phases refer to real derivatives of Hermitian fields. Expansion of
\eqref{F:tetrad} shows that every stress component is a linear combination
of constant labels with polynomial coefficients in $a,b,k,M$, allowing
only powers of $b$ in the denominators. Its norm is bounded by a constant
times $w_\alpha w_\beta$, where
\begin{equation}\label{F:component-weights}
 w_u=a,\qquad w_i=\sqrt{ab},\qquad w_v=b.
\end{equation}
For example $M,|k|\le\sqrt{2ab}$, and the apparent $1/b$ factors in the
rest axes acquire a compensating $M$ or $k$ from the rest amplitude.

\subsection{Scalar comparison copies}
\label{F:scalar-comparisons}

For each label in the following table, some tangential derivative of
$M^\nu d_*$ is the vacuum wave of a physical differential field:
\begin{equation}\label{F:comparison-table}
 \begin{array}{c|c}
 d_*&\nu\\\hline
 c_V,c_T&0\\
 l,j_z,s_z&2\\
 j_i,s_{zi}&1\\
 (s_{ij})_{\perp\mathrm{TF}}&2
 \end{array}
\end{equation}
Here the derivative has multiplier $b^m$ for some integer $m\ge0$,
up to a fixed phase. To see the physical content of this claim, set
$q_i=be_i+k_i n$, a vector whose contractions are differential
combinations with polynomial coefficients. The identities
\[
 M^2l=-\Theta,\qquad
 bM^2j_z=M^2V_v/i+b\Theta,\qquad
 b^2M^2s_z=M^2T_{vv}+b^2\Theta/3
\]
prove the scalar and longitudinal entries. Since $p\cdot q_i=0$,
\[
 V(q_i)=ibMj_i,\qquad T(n,q_i)=b^2Ms_{zi},\qquad
 \bigl(T(q_i,q_j)\bigr)_{\perp\mathrm{TF}}
       =b^2M^2(s_{ij})_{\perp\mathrm{TF}}.
\]
The scalar improvement in \eqref{F:wave-decomposition} drops from the
first mixed tensor contraction and from the transverse traceless part.
Finally $c_V,c_T$ come directly from \eqref{F:longitudinal-labels}.
All factors $M^2,b,k_i$ in these identities are polynomial momentum
multipliers; no inverse differential operator has been declared local.
The scalar reality phase may be chosen as $-i$ on the $j_i,s_{zi}$
copies and as $1$ on the other entries. This accounts for the factor
$i$ in a real tangential derivative.

Suppose a region $R$ is invariant under positive
$v$ translations, as are the cuts used below. Derivatives in $v$ can be
removed in the real closure of its wave space. Choose real
$\rho\in C_c^\infty(0,1)$ with integral one, and set
$\rho_R(v)=R^{-1}\rho(v/R)$. For a test function $f$ supported in the
region, the wave of
$(1-\rho_R*_{v})^m f$ has multiplier
$(1-\widehat\rho(Rb))^m$ and converges to that of $f$ by dominated
convergence, since $b>0$ almost everywhere. Each such test has its first
$m$ $v$-moments zero. Its $m$-fold primitive is therefore compactly
supported; choosing the primitive to vanish at $v=-\infty$ keeps its
support in the positive-$v$ hull of the original support and its positive
translates. Thus it is the $m$th derivative of a real test in the same
region. With the fixed derivative phase, the scalar generalized free
wave space of dimension $2+\nu$ on each label embeds in $X(R)$.

For $W=\{u<0,v>0\}$ this copy has precisely its own scalar wedge
Tomita data. Indeed, its embedding intertwines boosts and the wedge
reflection, including the derivative phase; hence its projection reduces
$J_We^{-\pi B}$, and the fixed real space is the scalar one. This does
not say that its curved-cut real space exhausts physical localization.

In light-front variables the measure is, up to one common fixed
normalization,
\begin{equation}\label{F:lightfront-measure}
 \dd^4p=\frac{\dd b}{b}\,\dd^2k\,M\dd M,
 \qquad b>0, M>0.
\end{equation}
For a scalar copy of dimension $2+\nu$, put $Y=i\partial_k$ and define
\begin{equation}\label{F:Hankel-generator}
 H_\nu=b(|Y|^2+h_\nu),\qquad
 h_\nu=-\partial_M^2-M^{-1}\partial_M+\nu^2/M^2.
\end{equation}
The operator $h_\nu$ is the regular nonnegative Hankel realization on
$L^2(\R_+,M\dd M)$: the Hankel transform with kernel $J_\nu(Mr)$
conjugates it to multiplication by $r^2$. In particular, the logarithmic
solution at zero is excluded when $\nu=0$.

The associated null conformal map is
\begin{equation}\label{F:null-conformal-map}
 (u,y,v)\longmapsto
 (\gamma u,\gamma y,v+c\gamma|y|^2),\qquad
 \gamma=(1-2cu)^{-1},\qquad c>0.
\end{equation}
It maps $W$ diffeomorphically to
\[
 -\frac1{2c}<u<0,\qquad
 v>\frac{|y|^2}{2u+1/c}.
\]
Two-point conformal covariance therefore maps the scalar wedge real
space onto the scalar real space of this region. Its unitary implementer
is $e^{icH_\nu}$. For verification, apply \eqref{F:Hankel-generator} to
$M^\nu e^{ip\cdot x}$. The direct expression is
\[
 H_\nu(M^\nu e^{ip\cdot x})
 =\left(2u^2a-2u k\cdot y+b|y|^2-2iu(\nu+2)\right)
       M^\nu e^{ip\cdot x}.
\]
For complex tube coordinates, the squares in this formula are analytic
bilinear squares. Multiplication by $i$ gives the infinitesimal
coordinate action
\[
 2u^2\partial_u+2uy_i\partial_i+|y|^2\partial_v
       +2u(2+\nu),
\]
which integrates to \eqref{F:null-conformal-map} with the scalar weight.
This calculation is valid on tube point vectors: their imaginary
future displacement gives exponential damping in $b$ and Gaussian
damping in $k,M$ at fixed $b$; at $M=0$ their radial behavior is
$M^\nu$ times an even smooth function, the regular Hankel boundary
condition. Applying any power of $b(|Y|^2+h_\nu)$ retains an integrable
polynomial times this damping. The factors of $b$ cancel the inverse
Gaussian variances as $b\downarrow0$. Consequently differentiation
occurs in the operator domain. Uniqueness of the unitary evolution
identifies this realization with the kernel-covariance implementer.

\subsection{The physical null-cut generators}
\label{F:cuts}

For a real transverse profile $f$, define
\begin{equation}\label{F:cut-definition}
 W_f=\operatorname{int}\left\{u<0:
 v+\frac{|y-z|^2}{-2u}\ge f(z)\text{ for every }z\in\R^2\right\}.
\end{equation}
Equivalently, put
\[
 \mathcal L_d f(y)=\sup_z\left(f(z)-\frac{|y-z|^2}{2d}\right);
 \qquad W_f=\{u<0, v>\mathcal L_{-u}f(y)\}^{\circ}.
\]
The allowed profiles initially are nonnegative bounded smooth functions
and nonnegative circular parabolas of positive curvature. We subsequently
allow constants of either sign, and sums of a bounded smooth function
with a positive-curvature circular parabola. All bounded smooth profiles
used in estimates have the stated derivatives bounded. Write
$p_d(y)=|y-d|^2$ and $W=W_0$.

For a nonnegative profile, $W_f\subset W$ has nonempty open interior and
an open spacelike complement. Reeh--Schlieder and the test-space density
above make the vacuum vectors of finite factorized polynomials from one
small double cone dense. Their affiliated realizations have bounded polar
cutoffs in the algebra, so $\Omega$ is cyclic for $\cM(W_f)$. Locality and
cyclicity in the complement make it separating.
The boost sends $W_f$ to $W_{e^sf}$, which is contained in $W_f$ for
$s\ge0$. Every hypothesis of \eqref{F:HSMI} is therefore met, and it
defines $H_f\ge0$ with
\begin{equation}\label{F:cut-modular}
 \begin{gathered}
 S(W_f)=B-H_f,\qquad
 \cM(W_f)=\Ad(e^{iH_f})\cM(W),\\
 e^{isS(W_f)}=e^{i(1-e^s)H_f}e^{isB},\qquad
 H_{\lambda f}=\lambda H_f\quad(\lambda>0).
 \end{gathered}
\end{equation}
The last assertion follows by conjugating the modular difference by a
boost and applying its affine covariance.

Translations $e^{icb}$ act inward on both $W$ and $W_f$ for $c\ge0$.
Equation~\eqref{F:Borchers} says that both modular groups scale $b$ by
the same factor; their quotient in \eqref{F:cut-modular} consequently
commutes with the whole $b$-group. Hence $H_f$ and $b$ strongly commute.
A constant translation gives
\begin{equation}\label{F:constant-shifts}
 H_{f+c}=H_f+cb.
\end{equation}
Use this identity to define signed profiles bounded below; it agrees
with their modular difference and is independent of the positive shift
chosen. Inclusion and modular-log order then give, for bounded $g$,
\begin{equation}\label{F:bounded-cut-bound}
 -\norm{g}_\infty b\le H_g\le\norm{g}_\infty b.
\end{equation}
The inequalities can first be tested on vectors with $b$ in a compact
band, smooth for the relevant affine-group representations. On such a
band they assert a bounded self-adjoint extension, and exhaustion of
the bands gives the displayed decomposable-operator inequality.

\Needspace{6\baselineskip}
\begin{proposition}[Parabolic generators]\label{F:parabolic-generator}
On $\{b>0\}$ there is a boost-independent internal Hilbert space
$\mathcal K$ and a nonnegative self-adjoint operator $h$ on it such that
\begin{equation}\label{F:parabolic-form}
 \cH_{b>0}\simeq
 L^2(\R_+,\dd b/b)\otimes L^2(\R^2,\dd^2k)\otimes\mathcal K,
 \qquad H_{p_d}=b\bigl(|Y-d|^2+h\bigr),\quad Y=i\partial_k.
\end{equation}
Here $G=bY$, $P_i=-k_i$, and
$(e^{isB}\psi)(b,k)=\psi(e^sb,k)$.
The operators $H_{p_d}$ vanish on $\ker b$. Moreover
$J_WbJ_W=b$, $J_WYJ_W=Y$, and $J_WhJ_W=h$.
\end{proposition}

\begin{proof}
Put $H=H_{p_0}$. Completion of the square in
\eqref{F:cut-definition} shows that $W_{p_0/(2R)}$ is the portion,
with $u<0$, of the future cone with tip $(u,v,y)=(-R,0,0)$.
The null rotations about this tip have generators $G_i+RP_i$.
They preserve the algebra and the vacuum, so commute with its modular
group
$e^{i(1-e^s)H/(2R)}e^{isB}$.
Set $\tau=(1-e^s)/(2R)$. Boost conjugation of the rotation group then
gives, at the level of Weyl elements, the transformation
\[
 ((1-2R\tau)\beta,R\beta)\longmapsto(\beta,R\beta)
\]
for coefficients of $(G,P_\perp)$; its central coefficient $b$ is fixed.
For a fixed sufficiently small nonzero $\tau$, vary $R$ over an open
positive interval. Two distinct values give linearly independent
coefficient pairs. Multiplying the corresponding Weyl elements, with
the same central Weyl phase on both sides, yields the full shear
\begin{equation}\label{F:parabolic-shear}
 \Ad(e^{i\tau H})G_i=G_i,\qquad
 \Ad(e^{i\tau H})P_i=P_i+2\tau G_i.
\end{equation}
The group law extends it to every real $\tau$.

Positive energy confines the nonzero momentum spectrum on $\ker b$ to
one null ray. A small spatial rotation takes this ray to a disjoint
ray; a vector supported there would be orthogonal to its arbitrarily
small rotations, contradicting strong continuity. Thus $\ker b$ is
precisely the zero-momentum subspace. On it $P_i=0$, and
\eqref{F:parabolic-shear} forces $G_i=0$. Repeating in all null directions
makes every connected Lorentz generator vanish there. In particular
$B=0$, and $e^{isB}He^{-isB}=e^sH$ forces $H=0$ on this subspace.

For $b>0$, the null rotations and transverse translations form the Weyl
pair $G_i=bY_i$, $P_i=-k_i$. Their regular representation is the
Schr\"odinger representation with a multiplicity space; the boost action
on $\log b$ trivializes that multiplicity over $b$. This gives the
representation in \eqref{F:parabolic-form}. The operator $b|Y|^2$
implements exactly the shear \eqref{F:parabolic-shear}. Consequently
$e^{-itb|Y|^2}e^{itH}$ commutes with the transverse Weyl representation
and with $b$. It is a one-parameter unitary group on the multiplicity
space, fiber by fiber. Boost homogeneity identifies its generator as
$bh$ with $h$ independent of $b$. Since $H\ge0$, its expression
$b(|Y|^2+h)$ implies $h\ge0$: localize the $Y$ variable near zero and
test an arbitrary spectral interval of $h$.

Transverse translation covariance replaces $Y$ by $Y-d$, proving the
formula for $p_d$. The modular-conjugation relation for $H$ and $b$,
together with geometric conjugation of the null rotations, gives the
three statements involving $J_W$.
\end{proof}

The use of a regular Weyl representation in this proof is the
multiplicity form of the Stone--von Neumann theorem, or equivalently
translation imprimitivity. No assertion that the internal space consists
only of one-field waves is made. In the channels of $\Pi$, $M$ and the
spin labels are internal variables.

\Needspace{6\baselineskip}
\begin{proposition}[Bounded cuts commute with the parabolic data]
\label{F:bounded-commutation}
For bounded smooth $g$,
\[
 H_g=bh_g,\qquad \norm{h_g}\le\norm{g}_\infty,
\]
where $h_g$ is independent of $b$ and strongly commutes with $Y,h$.
\end{proposition}

\begin{proof}
The formula \eqref{F:parabolic-form} and constant shifts make all
parabolic generators commute. Consequently
$\Ad(e^{itH_{p_d}})\cM(W_q)=\cM(W_{q+tp_d})$ for every
positive-curvature circular parabola $q$ and $t\ge0$.
To pass to a bounded $g$ we need an algebra-generation argument.
For $x_0=(u_0,v_0,y_0)$, $u_0<0$, its null shadow is
\[
 q_{x_0}(z)=v_0+\frac{|z-y_0|^2}{-2u_0},\qquad
 W_{q_{x_0}}=I^+(x_0)\cap\{u<0\}.
\]
Given a double cone compactly contained in $W_g$, choose $x_0$ slightly
timelike before its lower tip, still inside $W_g$. Its shadow strictly
dominates $g$ and its future cone contains the double cone. Conversely,
$q\ge g$ implies $W_q\subset W_g$. Inner generation therefore proves
\begin{equation}\label{F:parabolic-generation}
 \cM(W_g)=\bigvee_{q>g}\cM(W_q).
\end{equation}
For $W_{g+tp_d}$ the interior-shadow construction supplies a generating
subfamily of dominating parabolas of curvature strictly greater than
$t$. Indeed lower curvature cannot dominate $g+tp_d$ at transverse
infinity, while equality corresponds to the boundary value
$u_0=-1/(2t)$ and cannot be the shadow of an interior point. Thus the
join defining $\cM(W_{g+tp_d})$ may be restricted to all dominating
parabolas of curvature greater than $t$. Subtraction of $tp_d$ bijects
this restricted family with the positive-curvature parabolas dominating
$g$. Arbitrary dominating parabolas may have curvature exactly $t$;
they are not needed for the generating family. Taking joins yields
\[
 \Ad(e^{itH_{p_d}})\cM(W_g)=\cM(W_{g+tp_d}).
\]
Conjugate the modular group and use
$\Ad(e^{itH_{p_d}})B=B-tH_{p_d}$. Applying the same identity to
$\lambda g$ and $\lambda t$, then dividing by $\lambda$ by boost
homogeneity, gives
\begin{equation}\label{F:parabola-addition}
 H_{g+tp_d}=tH_{p_d}
    +e^{i\lambda tH_{p_d}}H_g e^{-i\lambda tH_{p_d}}
 \quad(\lambda>0).
\end{equation}
There is no need to differentiate an unspecified unbounded sum. On a
band $\epsilon\le b\le L$, $H_g$ is bounded by
$L\norm{g}_\infty$. The second term in \eqref{F:parabola-addition} is
independent of $\lambda$, and its strong limit as $\lambda\downarrow0$
is $H_g$. It follows that $H_g$ commutes with the full group of every
$H_{p_d}$. Exhausting the $b$ bands proves strong commutation globally.
In the joint calculus of the commuting parabolic family,
\[
 H_{p_d}-H_{p_0}=b(-2d\cdot Y+|d|^2).
\]
These differences recover $Y$ on $b>0$, and $H_{p_0}/b-|Y|^2$ recovers
$h$. Hence $H_g$ commutes with $Y,h$. Finally
\eqref{F:bounded-cut-bound} gives the bounded operator $h_g=H_g/b$,
and boost homogeneity makes it independent of $b$.
\end{proof}

We next justify the reflected localization needed for lower bounds.
The wedge conjugation reflects $(u,v,y)$ to $(-u,-v,y)$. Even if its
action on an arbitrarily chosen bounded double-cone algebra has not
been postulated, it produces a net compatible with the original one
in spacelike separation. Indeed every bounded operation in a double
cone belongs to all containing wedge algebras; their images under
$J_W$ are the corresponding reflected, twist-corrected wedge algebras
by \eqref{F:BW-sandwich} and Poincar\'e covariance. Two compact
spacelike separated double cones admit separating complementary
wedges. To see this, write their causal tips as $a_i,b_i$. A future
null linear functional separating $b_1$ from $a_2$, and a second one
separating $b_2$ from $a_1$, define the two wedge boundaries; their
strict inequalities place the cones in opposite wedges. Locality of
these wedge algebras gives the claim.

In particular, the reflected region $J_WW_{-g}$ is spacelike to $W_g$.
This also follows directly by minimizing the sum of the two quadratic
penalties in \eqref{F:cut-definition}; for $s,t>0$ its minimum is
$|y_--y_+|^2/(2(s+t))$. Thus
\begin{equation}\label{F:reflected-order}
 J_WX(W_{-g})\subset X(W_g)',\qquad H_g\ge-H_{-g}.
\end{equation}
For the second implication use modular-log order,
$J_WBJ_W=-B$, and $J_WH_gJ_W=H_g$. The latter follows from the
positive-cut conjugation relation and the constant-shift formula.

\Needspace{6\baselineskip}
\begin{lemma}[Taylor comparison]\label{F:taylor-bound}
If $g$ is bounded and smooth with bounded Hessian, then
\begin{equation}\label{F:taylor-equation}
 |h_g-g(Y)|\le C h,\qquad
 C=\tfrac12\sup_y\norm{g''(y)}.
\end{equation}
In particular
$\norm{(h_g-g(Y))\psi}\le C\norm{h\psi}$ for $\psi\in\Dom h$.
\end{lemma}

\begin{proof}
For every transverse center $z$,
\[
 g(y)\le g(z)+\nabla g(z)\cdot(y-z)+C|y-z|^2.
\]
The right side is a signed positive-curvature parabola if $C>0$.
Inclusion, modular-log order, and \eqref{F:parabolic-form} give
\[
 h_g\le g(z)+\nabla g(z)\cdot(Y-z)+C|Y-z|^2+Ch.
\]
Applying the same argument to $-g$ and then \eqref{F:reflected-order}
gives the lower inequality with both occurrences of $C$ negated.
For this fixed $g$, all operators involved commute strongly by
Proposition~\ref{F:bounded-commutation}. Impose the inequalities first
for a countable dense set of $z$ and then, on each joint spectral fiber,
let $z$ tend to the value of $Y$. They become
$-Ch\le h_g-g(Y)\le Ch$ in a commuting spectral calculus, which proves
both assertions. If $C=0$, boundedness makes $g$ constant and the
result is \eqref{F:constant-shifts}.
\end{proof}

\Needspace{6\baselineskip}
\begin{proposition}[Scalar spectral comparison]\label{F:spectral-comparison}
On any scalar copy from \eqref{F:comparison-table}, with its embedding
into the ambient internal space understood, one has
\begin{equation}\label{F:spectral-equation}
 \mathbf1_{(\lambda,\infty)}(h)
       \mathbf1_{[0,\lambda]}(h_\nu)=0,\qquad
 \norm{h^s\psi}\le\norm{h_\nu^s\psi}\quad(s>0).
\end{equation}
The norm assertion includes the corresponding inclusion of domains.
It does not assert that $h$ leaves the scalar copy invariant.
For bounded profiles $f\le g$ one also has
\begin{equation}\label{F:bounded-spectral-order}
 \mathbf1_{(\lambda,\infty)}(h_f)
       \mathbf1_{(-\infty,\lambda]}(h_g)=0.
\end{equation}
\end{proposition}

\begin{proof}
Let $j_\nu$ embed the scalar comparison copy. The derivative-removal
argument in Section~\ref{F:scalar-comparisons} gives its real wave
localization inside the physical cut space. By
\eqref{F:null-conformal-map} and \eqref{F:cut-modular},
\[
 j_\nu e^{icH_\nu}X_\nu(W)
       \subset e^{icH_{p_0}}X(W)\qquad(c>0).
\]
The embedding intertwines the flat wedge boosts, so
Lemma~\ref{F:strip-comparison} applies. It also intertwines the
external variables $b,Y$. To see the latter point independently of
the internal operator $h$, a null rotation $N_\beta$ fixes $n,b,M$
and sends $k$ to $k+b\beta$ and $e_i$ to $e_i+\beta_i n$.
The adapted axes in \eqref{F:tetrad} therefore satisfy
$N_\beta e_A^{(n)}(p)=e_A^{(n)}(N_\beta p)$ for $A=z,1,2$.
The constant labels and their fixed derivative phases transform
accordingly. Thus $j_\nu$ intertwines the null-rotation groups
generated by $G=bY$, as well as $b$; disintegration on $b>0$
gives the claimed $Y$ intertwining. This uses the Poincar\'e action
on the comparison waves and does not require $h$ to preserve their
individual channels. Both sides are decomposable in
$b,Y$, and $j_\nu$ intertwines these variables. In the common spectral
fibers their generators are
$b(|Y|^2+h_\nu)$ and $b(|Y|^2+h)$. Subtracting the common scalar
$b|Y|^2$ and dividing by $b>0$ in the band-inclusion conclusion gives
the first assertion in \eqref{F:spectral-equation}. More formally,
apply the whole-generator conclusion for rational band endpoints,
localize $(b,Y)$ to shrinking measurable rectangles, and use monotone
convergence of spectral projections. The power bound follows by
integrating spectral tails as in Lemma~\ref{F:strip-comparison}.

If $f\le g$, the cut inclusion is $W_g\subset W_f$. Apply the same
lemma with source $g$, target $f$, and the identity embedding. If needed
add the same constant to make both profiles nonnegative. After
disintegrating $b$, removing this common constant gives
\eqref{F:bounded-spectral-order}. This is spectral order obtained from
an analytic strip, rather than from ordinary quadratic-form order.
\end{proof}

\Needspace{8\baselineskip}
\subsection{Normal displacement and the required form domain}

\begin{lemma}[Normal form bound]\label{F:lipschitz-form}
Let $g$ be bounded and smooth with Lipschitz constant $C_1$, and write
$H_g(-\delta)=e^{-i\delta a}H_g e^{i\delta a}$. For $\delta,\lambda>0$,
\begin{equation}\label{F:normal-displacement}
 \pm\bigl(H_g-H_g(-\delta)\bigr)
 \le\delta\left(\frac a\lambda+\frac{\lambda C_1^2}2 b\right)
\end{equation}
as forms on a joint momentum-smooth core. The commutator form with
$a$ extends continuously to $\Dom\sqrt{1+a+b}$. If
$\psi\in\bigcap_{m\ge1}\Dom(1+a+b)^m$, then
$H_g\psi\in\Dom\sqrt a$.
\end{lemma}

\begin{proof}
The quadratic sup-convolution obeys
$\mathcal L_{d+\delta}f=\mathcal L_d(\mathcal L_\delta f)$.
Completing the square in the Lipschitz estimate gives
\[
 \mathcal L_\delta(\lambda g)(y)
 \le\lambda g(y)+\frac{\delta\lambda^2 C_1^2}{2}.
\]
Hence the cut with profile
$\lambda g+\delta\lambda^2C_1^2/2$, translated by $u=-\delta$, lies
inside $W_{\lambda g}$. The translated boost is $B-\delta a$.
Modular-log order thus gives
\[
 B-\delta a-\lambda H_g(-\delta)
     -\frac{\delta\lambda^2C_1^2}{2}b
 \le B-\lambda H_g.
\]
This is the upper sign of \eqref{F:normal-displacement}. Apply $J_W$,
which fixes $a,b,H_g$ and reverses the translation parameter by
antiunitarity, and then translate by $-\delta$; this gives the lower
sign. One may initially use vectors smooth for translations and the
boost, then remove the boost smoothing in the resulting form bound.

Put $E=1+a+b$ and
$Q_\lambda=a/\lambda+\lambda C_1^2 b/2$. Differentiating at
$\delta=0$ on momentum-smooth vectors defines the Hermitian form
\[
 c_g(\phi,\psi)=i\bigl(
  \ip{a\phi}{H_g\psi}-\ip{H_g\phi}{a\psi}\bigr).
\]
Differentiation is legitimate using $H_g=bh_g$, bounded $h_g$, and
translation smoothness of both vectors; it does not presume that
$H_g$ preserves $\Dom a$. The two inequalities give
$|c_g(\psi,\psi)|\le\ip{\psi}{Q_\lambda\psi}$. Polarization, or
representation of this Hermitian form after $Q_\lambda^{1/2}$, gives
\begin{equation}\label{F:commutator-form}
 |c_g(\phi,\psi)|\le
 \norm{Q_\lambda^{1/2}\phi}\,
 \norm{Q_\lambda^{1/2}\psi}.
\end{equation}
For fixed $\lambda$ this proves continuity in the $E^{1/2}$ form norm.

For the domain assertion set $\chi=H_g\psi$. The $b$ bound and strong
$b$ commutation imply $\chi\in\Dom\sqrt{1+b}$. On a joint spectral
core for $a,b$,
\[
 \ip{a\phi}{\chi}
   =\ip{H_g\phi}{a\psi}-i c_g(\phi,\psi),\qquad
 |\ip{H_g\phi}{a\psi}|
 \le\norm{g}_\infty\norm{\sqrt b\,\phi}
                           \norm{\sqrt b\,a\psi}.
\]
Both terms are continuous in $\norm{\sqrt E\,\phi}$. Adding
$\ip{(1+b)\phi}{\chi}$ consequently gives
\[
 |\ip{E\phi}{\chi}|\le C_\psi\norm{\sqrt E\,\phi}.
\]
The spectral theorem characterizes this inequality as
$\chi\in\Dom\sqrt E$: test with spectral truncations of $\chi$ and
let their endpoints tend to infinity. Thus $H_g\psi\in\Dom\sqrt a$,
as required.
\end{proof}

When opposite localization of conjugated bounded operations is needed,
one may therefore use the reflected, twist-corrected net just described.
It is spacelike compatible with the original net. Conjugating a chosen
original affiliated realization by $J_W$ gives an affiliated realization
in the reflected algebra, with its polynomial restriction on
$J_W\mathcal D_{\mathrm{old}}$. On generated factorized vacuum words,
its vacuum and adjoint-vacuum data are the reflected-adjoint word data
from the wedge boundary calculation; admitted compact joint word data follow by the graph limits
above. No equality of $J_W\mathcal D_{\mathrm{old}}$ with
$\mathcal D_{\mathrm{old}}$ is used. The construction can also be repeated with past cuts
of the original net. Section~\ref{sec:null-current} will identify their
bounded-profile generators from the physical current and thereby compare
the resulting moments.
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/02-foundations.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/03-null-current.tex ====
\section{The bounded null-cut Ward identity}
\label{sec:null-current}

The distinction between a modular generator and a stress-tensor flux is
essential here.  The preceding construction supplies the generators
$H_g$ without identifying their action on physical fields.  We now make
that identification for bounded profiles.  Throughout this section $g$
is real and belongs to $C_b^\infty(\R^2)$, meaning that every derivative
is bounded.  All estimates involve only finitely many of these bounds.
We write $h_{\mathbf1}=\id$ for the constant profile $1$; a numerical
subscript $\nu$ in $h_\nu$ always denotes the radial Hankel operator of
Section~\ref{sec:foundations}.

The proof has two distinct outputs. First we identify matrix elements of
$H_g$ with a stress-tensor flux and extend that identity to physical
polynomial tests. Then we use its linear dependence on $g$, together
with the spectral order already proved, to construct a commuting
transverse measure. Its quadratic moment is the input to
Section~\ref{sec:dispersion}. The identification concerns bounded
profiles throughout; the unbounded parabolic generator retains its
separate role as a comparison operator.

Put
\begin{equation}\label{N:boost-current}
 j_\alpha=vT_{\alpha v}-uT_{\alpha u},\qquad
 \partial^\alpha j_\alpha=0,
 \qquad
 \mathcal C_\epsilon(\phi,Z)
   =\ip{\phi}{Z}-\epsilon\ip{Z}{\phi}.
\end{equation}
The last expression is a real bilinear form, rather than a complex
bilinear pairing.  Complex versions below are obtained by polarization.
The stress tensor is even.  Thus only the case in which the two local
tests have equal total parity contributes; $\epsilon$ is their
interchange sign.

\subsection{Bounded local tests and differentiation}

Let $A$ and $C$ be bounded Hermitian operators, localized in compact
double cones strictly inside $W$ and its opposite, respectively.  We
first smooth them by compactly supported convolution under the
Poincar\'e group.  A slightly larger double cone still contains each
support.  Such smoothing gives bounded commutators of every prescribed
finite order with the infinitesimal Poincar\'e action, and the vacuum
vectors and their products are smooth translation vectors.  On the
opposite side one may use the twist-corrected reflected net described
in Section~\ref{sec:foundations}.

The first task is to keep both tests separated while varying them with
$H_g$. This requires containment of the conjugated algebras for both
signs of the flow parameter. The bound $|H_g|\leq\norm{g}_\infty b$
alone supplies a domain estimate and does not supply that containment.

There are $\alpha,c>0$ such that the support of $A$ lies strictly inside
the truncated cut with profile $\alpha|y|^2+c$; the reflected assertion
holds for $C$.  Write $d=\norm{g}_\infty$.  For $t>0$, the positive-profile
half-sided inclusion and the constant-shift identity give
\[
 \Ad(e^{itH_g})\cM(W)=\cM(W_{tg})
       \supset\cM(W_{td})=\Ad(e^{itdb})\cM(W).
\]
Conjugating this inclusion by $e^{-itH_g}$, and using commutation
with $b$, yields
\[
 \Ad(e^{-itH_g})\cM(W)\subset\cM(W_{-td}).
\]
For positive $t$ the same upper containment follows from $tg\geq-td$.
These are algebra inclusions obtained from the cuts; they do not
follow merely from an operator-form bound.  Since $H_g$ commutes
with the parabolic generators and $b$, conjugation of
$\cM(W_{\alpha p_0+c})$ therefore lies in
$\cM(W_{\alpha p_0+c-|s|d})$.  Choosing $|s|d<c/2$ shows that
\[
 A_s=\Ad(e^{isH_g})(A),\qquad C_s=\Ad(e^{isH_g})(C)
\]
remain in fixed, strictly separated truncated regions.  Reflection
gives the assertion for $C_s$.  No identity $H_{-g}=-H_g$ is used.
Hence $A_s$ and $C_t$ graded commute for all sufficiently small $s,t$.

Set $Z=CA\Omega$.  Since $|H_g|\leq \norm{g}_\infty b$, these smooth
vectors lie in $\Dom H_g$.  Differentiating vectors, rather than
unbounded operator products, gives
\begin{equation}\label{N:product-derivative}
 Z_{C'}=\epsilon A(iH_gC\Omega),\qquad
 Z_{A'}=C(iH_gA\Omega),\qquad
 Z_{C'}+Z_{A'}=iH_gZ.
\end{equation}
For example, $C_sA\Omega=\epsilon AC_s\Omega$ in a whole interval
about zero, so the first derivative follows from strong vector
differentiation.  The product rule then follows by subtracting and
adding $C_sA\Omega$ in $C_sA_s\Omega$.  The normal form estimate of
Lemma~\ref{F:lipschitz-form} and the bounded translation commutators give
\begin{equation}\label{N:normal-domain}
 Z_{A'},Z_{C'}\in\Dom(1+a)^{1/2}.
\end{equation}
For example, Lemma~\ref{F:lipschitz-form} puts $H_gA\Omega$ in
$\Dom(1+a)^{1/2}$. The bounded commutator of $C$ with the normal
translation generator makes $C$ bounded on $\Dom(1+a)$ with its graph
norm; interpolation with its Hilbert-space bound gives the same
property on $\Dom(1+a)^{1/2}$. This proves the assertion for $Z_{A'}$,
and the roles of $A,C$ give the other one. In particular no invariance
of $\Dom a$ under $H_g$ is being used.

These are the only normal-energy domains needed for the narrow-strip
term below.  The boost calculation uses only
$\Dom B\cap\Dom H_g$. Indeed the affine unitary relation
$e^{itB}e^{isH_g}=e^{ie^t sH_g}e^{itB}$, differentiated in $t$ on
this intersection, gives
\[
 B e^{isH_g}\xi=e^{isH_g}(B+sH_g)\xi.
\]
Thus the $H_g$ unitary group preserves this intersection and is continuous
in its $B$ graph norm. Differentiating the corresponding matrix identity,
and moving $B$ onto the fixed first vector in one difference quotient,
gives
\begin{equation}\label{N:affine-form}
 i\bigl(\ip{B\xi}{H_g\eta}-\ip{H_g\xi}{B\eta}\bigr)
   =\ip{\xi}{H_g\eta},
 \qquad \xi,\eta\in\Dom B\cap\Dom H_g.
\end{equation}
Our smoothed vacuum vectors belong to this domain. No domain for
$BH_g\eta$ is required.

Decrease $c$ and relabel it so that the fixed containing truncations
for all sufficiently small $s$ have profiles $\alpha p_0+c$.
We choose a time step whose two half-slabs remain separated from the
opposite truncated regions. The geometry can be read directly from the
cut formula. Put $R_0=1/(2\alpha)$. Then
\[
 W_{\alpha p_0+c}
 =\left\{-R_0<u<0:\quad
       v>c+\frac{|y|^2}{2(R_0+u)}\right\}.
\]
For $u=-d<0$, the reflected-cut separation inequality says that
\[
 v>-c-\frac{|y|^2}{2(R_0+d)}
\]
is spacelike to the reflected opposite truncation. The margin satisfies
\begin{equation}\label{N:slab-margin}
 d+c+\frac{|y|^2}{2(R_0+d)}
 \ge c_0\sqrt{1+d^2+|y|^2}\qquad(d\ge0)
\end{equation}
for some $c_0>0$: $d+c$ controls $R_0+d$, and the
arithmetic--geometric mean controls $|y|$. Reflection gives the
corresponding inequality for $u>0$.

Choose a positive smooth width $w(u,y)$ comparable to
$\sqrt{1+u^2+|y|^2}$, with bounded first derivatives and homogeneous of
degree one outside a compact set. If $\sigma$ is a fixed smooth step,
zero below $-1$ and one above $1$, set
$\rho=\sigma((u+v)/(\varepsilon w(u,y)))$.
For sufficiently small $\varepsilon$, its transition slab is spacelike,
and \eqref{N:slab-margin} makes its negative and positive halves
spacelike to the respective opposite truncated regions. Near $u=0$ we
also require separation from both fixed compact tests $A,C$.
Their compact normal-coordinate margins give this gap on bounded
transverse sets; at large $|y|$, their quadratic spacelike separation
dominates the slab width. Shrinking $\varepsilon$ supplies the required
central gap. This last assertion concerns the compact tests, not both
entire truncated regions.

Take $\chi\in C_c^\infty(\R\times\R^2)$, real and equal to one near
zero, and write
\[
 \chi_R(u,y)=\chi(u/R,y/R),\qquad F_R=\chi_R\,\dd\rho,
 \qquad j_F\Omega=\int F\mathbin{\cdot}j(x)\Omega\,\dd^4x.
\]
The full flux vectors satisfy
\begin{equation}\label{N:full-flux}
 \sup_{R\geq R_0}\norm{j_{F_R}\Omega}<\infty,
 \qquad j_{F_R}\Omega\rightharpoonup0\quad(R\longrightarrow\infty).
\end{equation}
Indeed conservation allows replacement of $\rho$ by an entirely scaled
step agreeing with it near $\supp\dd\chi_R$.  The difference of the
fluxes is the integral of $\dd\chi_R(\rho-\rho_R)$ against $j$ and
vanishes.  The resulting compact flux has scale degree zero, so its
vector is a unitary dilation of one fixed vector.  Its spectral mass
is driven to zero under this dilation.  The stress-tensor vacuum
channel has no zero-momentum part, proving weak convergence; the norm
is unchanged.  This argument uses neither clustering nor a norm bound
on the field.

The physical Ward identities are assumed for original polynomials.
The following two mixed functionals extend them in exactly the form
needed for differentiation. In each, the boost acts on the fixed
smoothed test.

Let $\vartheta$ be smooth, equal to one on $(-\infty,-2]$ and zero on
$[-1,\infty)$, and put
\[
 \theta_{-,\delta}(u)=\vartheta(u/\delta),\qquad
 \theta_{+,\delta}(u)=\theta_{-,\delta}(-u),\qquad
 j_\pm=j_{\theta_{\pm,\delta}F_R}.
\]

\Needspace{7\baselineskip}
\begin{lemma}[Mixed boost Ward identities]\label{N:mixed-Ward}
For the bounded smoothed opposite tests above, for sufficiently large
$R$, sufficiently small $\delta>0$, and $s$ in a neighborhood of zero,
\begin{align}\label{N:undifferentiated-Ward}
 \ip{C_s\Omega}{BA\Omega}
   &=\mathcal C_\epsilon(j_-\Omega,C_sA\Omega),\notag\\
 \ip{BC\Omega}{A_s\Omega}
   &=-\mathcal C_\epsilon(j_+\Omega,CA_s\Omega).
\end{align}
\end{lemma}

\begin{proof}
Start with a Hermitian finite algebraic sum $C_0$ of factorized original
words compactly localized in the opposite wedge. Its closed affiliated
realization was constructed in Section~\ref{sec:foundations}. The boost
Ward identity applies to its original polynomial restriction; affiliation
and opposite locality allow the compact current flux around $C_0$ to
pass the bounded operator $A$.

Choose the partition for the flux around $A$ once. Its time-zero causal
shadow is compactly contained in the positive spatial halfspace, so a
small slab neighborhood of that shadow lies compactly inside $W$ and
is spacelike to every opposite-wedge polynomial. Denote the resulting
fixed compact current vacuum vector by $\eta_A$. For each $C_0$, its
own time-zero causal shadow is a compact subset of the negative
halfspace. Thin or deform the slab only near that shadow, leaving the
$A$ partition and the homogeneous shape at infinity unchanged.
Conservation and \eqref{N:full-flux}, applied to each deformation,
transfer the opposite flux to this same $\eta_A$. The signs of the
commutator give
\[
 \ip{BC_0\Omega}{A\Omega}
 =\ip{C_0\Omega}{BA\Omega}
 =\mathcal C_\epsilon(\eta_A,C_0A\Omega).
\]
In particular the current side obeys
\[
 |\mathcal C_\epsilon(\eta_A,C_0A\Omega)|
 \le2\norm{\eta_A}\norm A\norm{C_0\Omega},
\]
since $C_0A\Omega=\epsilon AC_0\Omega$. No uniform bound on the
changing flux near $C_0$ is needed. The boost has been moved to the
fixed vector $A\Omega$ before taking any Hilbert limit.

Hermitian factorized polynomial vacuum vectors are real dense in the
opposite wedge standard subspace by the Tomita-graph argument in
Section~\ref{sec:foundations}. The displayed bounded functional therefore
extends to $C_s\Omega$. For the fixed family of truncated localizations,
the compact spatial partition can be replaced by the negative half-slab:
$C_s$ is spacelike there, and the commutator vanishes outside the fixed
causal intersection of $A$. This gives the first identity in
\eqref{N:undifferentiated-Ward}. Interchanging the fixed side gives the
second. Its minus sign comes from placing the boost on the bra; the
opposite flux contributes
$\epsilon\ip{CA_s\Omega}{j_+\Omega}-\ip{j_+\Omega}{CA_s\Omega}$.
The operators $A_s,C_s$ need only lie in their fixed truncated algebras;
compact localization of these varied operators is not required.
\end{proof}

Differentiate \eqref{N:undifferentiated-Ward} in the real parameter $s$
and add. Equation~\eqref{N:affine-form} identifies the left side as
\[
 i\bigl(\ip{BC\Omega}{H_gA\Omega}
        -\ip{H_gC\Omega}{BA\Omega}\bigr)
 =\ip{C\Omega}{H_gA\Omega}.
\]
The right side is
$\mathcal C_\epsilon(j_-\Omega,Z_{C'})
-\mathcal C_\epsilon(j_+\Omega,Z_{A'})$.
Substituting \eqref{N:product-derivative} and writing the two half-slabs
as the full slab minus its central strip gives
\begin{align}\label{N:finite-variation}
 \ip{C\Omega}{H_gA\Omega}
 &=\mathcal C_\epsilon(j_{\theta_{-,\delta}F_R}\Omega,iH_gZ)
   -\mathcal C_\epsilon(j_{F_R}\Omega,Z_{A'})\notag\\
 &\quad+
 \mathcal C_\epsilon
  (j_{(1-\theta_{-,\delta}-\theta_{+,\delta})F_R}\Omega,Z_{A'}).
\end{align}
All current entries here are actual smeared vacuum vectors. This
calculation extends a bounded vector functional; it does not postulate
a stress-tensor operator commutator with an arbitrary bounded operation.

\subsection{The three cutoff estimates}

There are three different error mechanisms in
\eqref{N:finite-variation}. The narrow strip disappears weakly after
one inverse half-power of normal energy. The normal face is compared
in norm with the multiplication operator $bg(Y)$, which yields the
physical plane flux. The distant cap is controlled after dilation by
a mass-dyad sum. Keeping these mechanisms separate specifies which
vector domains are needed and which bounds must be uniform in the
normal width.

Here and below $a=(M^2+|k|^2)/(2b)$, $b>0$.  Constants in the
one-field norm can be suppressed, since there are only finitely many
polarizations.  The two useful versions of the measure are
\begin{equation}\label{N:measure-coordinates}
 \dd a\,\dd b\,\dd^2k\quad(2ab\geq |k|^2),
 \qquad
 \frac{\dd b}{2b}\,\dd^2k\,\dd m,\qquad m=M^2\geq0.
\end{equation}
All inequalities below are componentwise inequalities in the finite
rest-spin spaces, summed at the end.

We make the dependence on test functions precise.  For a compact
smooth function $q(u,y,v)$ put
\[
 p_m(q)=\sum_{|\beta|\leq m}
  \norm{(1+|u|+|y|+|v|)^m\partial^\beta q}_{L^1}.
\]
In estimates involving a step, this seminorm is applied to its $v$
derivative and to its products with the indicated compact functions
of $(u,y)$, not to the step itself.  At a fixed $R$, all constants
denoted $C_{R,N}$ are bounded by a polynomial in the corresponding
$p_{4N+20}$ seminorms, the derivatives of $\chi_R$ on their compact
support, and the derivatives of the fixed step $\vartheta$.  After
rescaling a cap by $R$, use the same seminorms of its fixed smooth
shapes; the constants are then independent of $R$ and of the sharp
normal width.  Factors $u$, $v$, or a coordinate derivative appearing
in the current are included in these seminorms.  This convention
allows uniformity for compact families of local tests and for all
the smoothing operations used below.

First consider the last term in \eqref{N:finite-variation}.  Its
normal support has width $O(\delta)$.  Fourier transformation gives
a factor $\delta$ times a rapidly decreasing function of
$(\delta a,b,k)$; the part containing $uT_{\alpha u}$ has an additional
$\delta$.  The polarization weights for indices $u,i,v$ are
$a,\sqrt{ab},b$.  Consequently, after applying $(1+a)^{-1/2}$,
the square norm is bounded by a constant times
\begin{equation}\label{N:strip-bound}
 \int_0^\infty
  \bigl\{\delta^2(1+a)+\delta^4(1+a)^3\bigr\}
     (1+\delta a)^{-6}\,\dd a\leq C_R.
\end{equation}
The omitted $(b,k)$ integrals are convergent rapid-decay integrals;
extending their domain only enlarges this bound.  On every compact
momentum set the weighted Fourier coefficients tend to zero.
Uniform boundedness and compact-momentum density therefore give
weak convergence to zero.  By \eqref{N:normal-domain},
\begin{equation}\label{N:strip-vanishing}
 \lim_{\delta\downarrow0}
 \mathcal C_\epsilon
 (j_{(1-\theta_{-,\delta}-\theta_{+,\delta})F_R}\Omega,Z_{A'})=0.
\end{equation}

Thus the strip estimate uses exactly the half-energy domain established
before the differentiation. We turn to the first term, whose sharp
normal boundary will survive as the Ward flux.

Next integrate the first term of \eqref{N:finite-variation} by parts.
Moving $iH_g$ to the first entry means applying $-iH_g$ there.  Before
that application, conservation replaces the flux by
\begin{equation}\label{N:boundary-cap}
 B_{R,\delta}-E_{R,\delta}
 =\int(-\theta_{-,\delta}')\chi_R\rho\,j_v\Omega\,\dd^4x
 -\int\theta_{-,\delta}\rho
   \bigl[(\partial_u\chi_R)j_v-(\partial_i\chi_R)j_i\bigr]
       \Omega\,\dd^4x.
\end{equation}
For this integration first introduce a smooth $v$ cutoff of length
$S$.  At fixed $R,\delta$, let $S\to\infty$ before making any other
limit.  At $b>0$, the transforms of $\rho$ and $v\rho$ have respectively
the bounds $b^{-1}$ and $b^{-2}$ times rapid large-$b$ symbols.  This
follows by differentiating the compact function $\partial_v\rho$;
the possible distributions supported at $b=0$ are killed by the
displayed stress-tensor powers of $b$.  The error containing the
derivative of the $v$ cutoff, after multiplication by $b$, is bounded
by
\[
 C_{R,\delta,N}\,b(abS+a^2)(1+Sb+a+|k|)^{-N}.
\]
Its square integral in \eqref{N:measure-coordinates} is $O(S^{-3})$
for $N$ large: set $B=Sb$, and enlarge the cone to the full positive
$(a,b)$ quadrant.  Since $\norm{H_g\xi}\leq
\norm{g}_\infty\norm{b\xi}$, this removes the cutoff in the required
matrix element.  The remaining terms converge after multiplication
by $b$, using normal Fourier decay at fixed $\delta$.

We now estimate the normal face $B_{R,\delta}$.  Its decomposition
requires only the scalar comparison channels with $\nu=0,1$:
\[
 T_{vv}=b^2c_T,\qquad
 T_{iv}/b=M s_{zi}-k_i c_T,
\]
while conservation expresses $T_{uv}$ as a linear combination of
$(ab,|k|^2)c_T$ and $k_iM s_{zi}$.  Before a radial derivative, the
coefficient of the $vT_{vv}$ term has order one.  That of the
$uT_{uv}$ term has order $\delta(M^2+|k|^2)/b$; the mixed term
$\delta|k|M/b$ is bounded by the same expression.  Differentiating
in $m=M^2$ costs $\delta/b$, and
\begin{equation}\label{N:radial-identity}
 h_\nu\bigl(M^\nu f(m)\bigr)
 =-4M^\nu\bigl((1+\nu)f'(m)+mf''(m)\bigr).
\end{equation}
For example, a mixed $\nu=1$ coefficient has the form
$\delta k_iM f_\delta(m,b,k)/b$, up to fixed constants and finite sums.
Rescaling its normal test by $u=\delta s$ gives
\[
 |\partial_m^j f_\delta|
 \le C_{R,N}(\delta/b)^j
       (1+b+|k|+\delta a)^{-N},\qquad j=0,1,2.
\]
Its radial image is
\[
 b h_1\left(\frac{\delta k_iM}{b}f_\delta\right)
 =-4\delta k_iM\left(2\partial_m f_\delta
                              +m\partial_m^2 f_\delta\right).
\]
The potentially singular factor is now
$\delta |k|M/b\le\delta a$; also $\delta m/b\le2\delta a$.
These factors are absorbed by the rapid decay, leaving an overall
factor $\delta$. The $\nu=0$ terms follow from the same radial identity.
Consequently, for every $N$,
each coefficient of $bh_\nu B_{R,\delta}$ obeys
\begin{equation}\label{N:face-symbol}
 \abs{bh_\nu B_{R,\delta}(b,k,M)}
 \leq C_{R,N}\delta
      (1+b+|k|+\delta a)^{-N}.
\end{equation}
The spectral comparison \eqref{F:spectral-equation}, applied to
these channels, and the Taylor bound \eqref{F:taylor-equation} imply
\begin{equation}\label{N:face-error}
 \norm{(H_g-bg(Y))B_{R,\delta}}
 \leq C_R\norm{g''}_\infty\sqrt\delta.
\end{equation}
In particular the estimate has no unrecorded lower cutoff in $b$.
Indeed the square integral of the right side of
\eqref{N:face-symbol} is bounded by
\begin{align*}
 C\delta^2\int\frac{\dd b}{b}\,\dd^2k
  \int_0^\infty
   \left(1+b+|k|+\frac{\delta(m+|k|^2)}{2b}\right)^{-2N}\dd m
 &\leq C_N\delta
       \int\dd b\,\dd^2k(1+b+|k|)^{1-2N}\\
 &\leq C_N'\delta.
\end{align*}

For the principal part use $w=1+b+M^2/b$, which commutes with
$g(Y)$.  For any fixed $q>1/2$, the vectors
$w^{-q}bB_{R,\delta}$ are norm bounded and converge to their plane
limits.  To see this, the coefficients after multiplication by $b$
are bounded by
\[
 C_{R,N}b(1+\delta a)(1+b+|k|+\delta a)^{-N};
\]
the $m$ integral against $w^{-2q}$ is finite, and the remaining
$b,k$ integrals converge.  Dominated convergence applies, and the
term with $u$ tends to zero.  Since a translation-smooth $Z$ belongs
to $\Dom w^q$ for every fixed $q$, this proves convergence in the
pairing.  Fourier inversion in $k$ inserts $g(y)$.  Finally $ib$
acts as $\partial_v$ on $T_{vv}\Omega$; integration by parts leaves
\begin{equation}\label{N:plane-before-gap}
 \int\chi_R(0,y)g(y)
   (\rho(0,y,v)+v\partial_v\rho(0,y,v))
   \mathcal C_\epsilon(T_{vv}(0,y,v)\Omega,Z)\,\dd v\,\dd^2y.
\end{equation}
The term with $\partial_v\rho$ vanishes by the spacelike gap.

The face comparison has now removed the internal operator $h$ from
the surviving flux. Its error tends to zero with $\delta$ at each
fixed $R$, including the region $b\downarrow0$. To complete the
argument we need a different estimate for the distant cap: its
bound must remain valid after the normal step has become sharp.

It remains to control the cap $E_{R,\delta}$ uniformly as the normal
step becomes sharp.  Dilate its variables by $R$.  Its smooth shapes
then range over a fixed bounded set of the seminorms specified above,
and its current smearing has scale degree zero.  In these coordinates
the generator becomes
\begin{equation}\label{N:rescaled-generator}
 R^{-1}b h_{g_R},\qquad g_R(y)=g(Ry).
\end{equation}
Normal integration by parts gives a factor $(1+a)^{-1}$ from a
half-space step, and $(1+a)^{-2}$ if its coefficient contains $u$.
These bounds, as well as their logarithmic $a$ derivatives up to
order two, are uniform in the smoothing width.  For example, split
the normal integral into $|u|\leq(1+a)^{-1}$ and its complement.
The first part has the claimed length, or its square with an extra
$u$; on the second part integrate by parts.  The derivative of the
step has uniformly bounded total variation, and its support has
$|u|=O(\delta/R)$.  Repeating this argument proves the stated symbol
bounds.  Tangential integrations by parts give arbitrary rapid
decay in $b,k$, with the same $b^{-2},b^{-1}$ factors from $v\rho,\rho$.

Let $F_v,F_i$ denote the resulting cap coefficients before
multiplication by $b$.  The polarization weights give, for any $N$,
\begin{equation}\label{N:cap-symbols}
 |F_v|\leq C_N(1+a)^{-1}\tau_N(b,k),\qquad
 |F_i|\leq C_N\frac{\sqrt{a/b}}{1+a}\tau_N(b,k),\qquad
 \tau_N=(1+b+|k|)^{-N}.
\end{equation}
The same inequalities hold after up to two logarithmic $M$
derivatives, with a different $C_N$.  This last assertion can also
be checked directly in each tetrad coefficient: every occurrence
of $a$ is $(M^2+|k|^2)/(2b)$, so $M\partial_Ma\leq2a$.

Use a smooth dyadic partition $\eta_{\mathrm{lo}}(M)+\sum_{j\geq0}\eta_j(M)=1$
with $\eta_{\mathrm{lo}}$ supported in $M\leq2$ and
$\eta_j$ supported in $N_j/2\leq M\leq2N_j$, $N_j=2^j\geq1$,
and uniformly bounded logarithmic derivatives.  If $F_j=\eta_jF$
and $F_{\mathrm{lo}}=\eta_{\mathrm{lo}}F$, then
\begin{equation}\label{N:cap-dyads}
 \norm{bF_{\mathrm{lo}}}\leq C,\qquad
 \norm{bF_j}\leq C,\qquad
 \norm{bhF_j}\leq C N_j^{-2}\quad(j\geq0).
\end{equation}
Here the last expression uses the high-mass pieces only.  For clarity,
the potentially worst integral in the first two bounds is
\[
 \int_{M\asymp N}\frac{\dd m}{2b}\,
       \frac{ab}{(1+a)^2}
 =\frac12\int_{M\asymp N}\frac{a}{(1+a)^2}\,\dd m
 \leq Cb\quad(N\geq1),
\]
followed by the convergent $(b,k)$ integral with $\tau_N^2$.
For $M\leq2$ the integrand is simply bounded and the $m$ interval
has bounded length.  Thus the factor $b^{-1/2}$ in
\eqref{N:cap-symbols} is harmless.  To prove the $h$ bound, decompose
into the finite pure-label channels of
\eqref{F:comparison-table}.  On $M\asymp N_j$, the identity
\eqref{N:radial-identity} and the logarithmic derivative bounds give
a factor $N_j^{-2}$, including when derivatives hit $\eta_j$.
Apply \eqref{F:spectral-equation} to each channel and sum their
norms.  No commutation of $h$ with invariant mass is asserted or used.

Write $h_{g_R}=g_R(Y)+D_R$.  The bounded-profile estimate and
\eqref{F:taylor-equation} give simultaneously
\[
 \norm{D_R\xi}\leq2\norm{g}_\infty\norm{\xi},\qquad
 \norm{D_R\xi}\leq\tfrac12R^2\norm{g''}_\infty\norm{h\xi}.
\]
The second is an operator-norm estimate on $\Dom h$, since $D_R$
strongly commutes with $h$.  Thus the high dyads of the remainder
in \eqref{N:rescaled-generator} have norms at most
\[
 \frac{C_g}{R}\min(1,R^2/N_j^2),
\]
and their sum, including the low-mass part, is
\begin{equation}\label{N:cap-remainder}
 O_g\bigl((1+\log R)/R\bigr).
\end{equation}
Indeed, the dyads with $N_j\leq R$ contribute at most
$C_g/R$ each, and there are $O(1+\log R)$ of them. The remaining
bound is a geometric series in $R^2/N_j^2$, whose sum is bounded
independently of $R$. This gives the stated rate without any
orthogonality assumption on the images of the dyads.

This estimate allows $h$ to mix masses: the summation is a triangle
inequality after applying $D_R$, not an orthogonal decomposition
of its image.  The principal part $R^{-1}bg_R(Y)$ does preserve
mass.  Pair its $j$th dyad against the correspondingly rescaled
$Z_R$.  Translation smoothness gives, for every fixed integer $r$,
\[
 \norm{1_{M\asymp N_j}Z_R}
 \leq C_r(Z)(1+N_j/R)^{-r}.
\]
Consequently its pairing is bounded by
\begin{equation}\label{N:cap-principal}
 \frac{C_gC_r(Z)}R
 \left(1+\sum_{j\geq0}(1+N_j/R)^{-r}\right)
 \leq C_{g,r,Z}\frac{1+\log R}{R},\qquad r\geq1.
\end{equation}
Equations~\eqref{N:cap-remainder}--\eqref{N:cap-principal} prove
cap vanishing in precisely the order required by
\eqref{N:finite-variation}.

\subsection{The Ward formula and polynomial tests}

The order of limits is
\begin{equation}\label{N:limit-order}
 S\longrightarrow\infty,\qquad
 \delta\downarrow0\text{ at fixed }R,\qquad
 R\longrightarrow\infty.
\end{equation}
The strip term vanishes by \eqref{N:strip-vanishing}, the cap by
\eqref{N:cap-remainder}--\eqref{N:cap-principal}, and the full-flux
term by \eqref{N:full-flux}.  The normal face gives
\eqref{N:plane-before-gap}.  In particular its dependence on $g$
is linear before the last limit.  This proves linearity of $H_g$
as a form on these local vectors for all profiles under consideration.
The estimate
\[
 |\ip{\xi}{H_g\eta}|
 \leq\norm{g}_\infty\norm{b^{1/2}\xi}\norm{b^{1/2}\eta}
\]
extends this equality to the whole form domain.  The needed graph
density follows by applying a small compact translation convolution
to Reeh--Schlieder approximants in any fixed double cone: convolution
is bounded from Hilbert norm to the graph norm of $b^{1/2}$, and its
approximate identity tends to the identity in that graph norm.

\Needspace{6\baselineskip}
\begin{proposition}[Bounded-profile Ward formula]\label{N:Ward-proposition}
The map $g\mapsto H_g=bh_g$ is real linear on $C_b^\infty(\R^2)$.
For $g$ compactly supported or Schwartz, and the bounded smooth
opposite tests above,
\begin{equation}\label{N:bounded-Ward}
 \ip{C\Omega}{H_gA\Omega}
 =\int_{\R^2}\int_\R g(y)\rho(v)
    \mathcal C_\epsilon(T_{vv}(0,y,v)\Omega,CA\Omega)
        \,\dd v\,\dd^2y.
\end{equation}
Here $\rho$ is any smooth step whose derivative lies in the $v$ gap
between the two causal intersections with the plane.  The expression
is a vacuum-vector distribution paired with the indicated smooth
vector, not an unsmeared operator integral.
\end{proposition}

To replace the original plane step by this $\rho(v)$, interpolate
inside the spacelike gap and use locality.  The half-step vacuum
wave is continuous after an inverse polynomial energy weight:
$T_{vv}$ supplies $b^2$, the step costs $b^{-1}$, and tangential
smearing supplies rapid Fourier decay.  This is also the weighted
convergence proved in the normal-face calculation.  It justifies
the plane restriction and the removal of $\chi_R$ in
\eqref{N:bounded-Ward}.

\Needspace{6\baselineskip}
\begin{proposition}[Polynomial extension]\label{N:polynomial-extension}
Equation~\eqref{N:bounded-Ward} holds for compactly smeared physical
polynomial tests, including their adjoints, on the common field
domain $\mathcal D_{\mathrm{old}}$.  The convergence to it is uniform when the smearing tests
vary over a compact smooth parameter family.
\end{proposition}

The extension has three separate passages, with uniformity for fixed word
labels and lengths and a compact smooth family of their tests. For finite sums of factorized
polynomials, polar truncation first replaces each polynomial by a bounded
operation in its original localization algebra. Group convolution then
gives the translation derivatives needed in the Ward formula. The rapid
truncation error absorbs the polynomial growth of the norms of the other
bounded factors; only after that limit do we remove the convolution.
Finally, test-space density and the original joint-distribution bounds
extend the identity to admitted compact joint kernels.

\begin{proof}
First let $A$ be a Hermitian finite algebraic sum of factorized physical
polynomial words on the invariant field domain $\mathcal D_{\mathrm{old}}$.
Sums and products of its affiliated single-field realizations,
followed by closure, give a closed affiliated operator $T$ such that
$T|_{\mathcal D_{\mathrm{old}}}=A$ and
$T^*|_{\mathcal D_{\mathrm{old}}}=A$.  The product is
closable because its adjoint contains the reversed adjoint product
on the dense domain $\mathcal D_{\mathrm{old}}$.  Write $T=U|T|$, let
$E_L=1_{[0,L]}(|T|)$, and set
\[
 T_L=TE_L,\qquad A_L=\tfrac12(T_L+T_L^*).
\]
Then $A_L$ is bounded Hermitian in the same local algebra, with
$\norm{A_L}\leq L$.  For $\psi\in\mathcal D_{\mathrm{old}}$ and every nonnegative integer $m$,
the spectral-tail bounds give
\begin{align}\label{N:polar-tails}
 \norm{(T-T_L)\psi}
 &\leq L^{-2m}\norm{|T|^{2m+1}\psi},\notag\\
 \norm{(T^*-T_L^*)\psi}
 &\leq L^{-2m}\norm{|T|^{2m}T^*\psi}.
\end{align}
Both right sides are finite polynomial-domain norms.  Indeed
$\mathcal D_{\mathrm{old}}$ is invariant under $T,T^*$, and powers of $T^*T$ and
$TT^*$ there agree with the corresponding iterated physical
polynomials.  Hence $A_L\psi-A\psi=O(L^{-N})$ for every prescribed
$N$.  This proves the same assertion for any fixed finite family
of polynomial states and moments needed in the argument.  No
essential self-adjointness is required.

One must not differentiate the sharp spectral projection $E_L$.
Instead first convolve $A_L$ under the spacetime group, with a fixed
compact smooth kernel $f$.  Group derivatives then fall on $f$.
The estimate \eqref{N:polar-tails} is uniform for the transformed
polynomial tests on the compact support of $f$, since its right
sides are continuous norms of smeared polynomial vectors.  Thus
the convolved vacuum vectors converge in every fixed translation
Sobolev norm.  Apply the same construction to $C$ separately.
Their localization regions remain spacelike.  For example,
\[
 C_LA_L\Omega-CA\Omega
 =C_L(A_L-A)\Omega+(C_L-C)A\Omega,
\]
whose first term is bounded by $L$ times a faster-than-any-power
tail and whose second is another polynomial-state tail.  The same
argument after convolution gives convergence of products in the
finite Sobolev norms used above.  To make the product convergence explicit, let $\alpha_r$ denote the
Poincar\'e action and write $A_L(r)=\alpha_r(A_L)$,
$A(r)=\alpha_r(A)$, with analogous notation for $C$. Uniformly for
$r,s$ in fixed compact group sets,
\[
 \norm{C_L(s)A_L(r)\Omega-C(s)A(r)\Omega}
 \leq L C_NL^{-N}+C_N'L^{-N}.
\]
For the second term apply the polar-tail estimate to the polynomial
state $A(r)\Omega$; its iterated $C(s)$-polynomial norms are uniformly
bounded on these compact sets. For a product of the two convolved
operators, a common translation is left translation of both group
variables. Integration by parts consequently places each prescribed
translation derivative on the convolution kernels. The resulting
finite sum of compact kernel integrals has the same tail estimate,
with new constants. This proves the required Sobolev convergence
without differentiating either polar projection.

For these factorized polynomials, the left side of
\eqref{N:bounded-Ward} converges by its $b^{1/2}$ form bound, and
the right side by the inverse-energy estimates for the half-step
wave.  Finally remove the convolution on the original smooth field
tests.  All constants are finite test seminorms, so this last limit
and compact-parameter uniformity follow from distributional
continuity.

It remains to pass to the admitted compact joint kernels. For an old
word $A$, write $\Psi_A(\varphi_A)$ for its vacuum vector smeared with a
joint kernel $\varphi_A$. First partition a compact support into finitely
many product patches inside the required localization region. On each
patch, finite sums of tensor tests are dense in the smooth test-function
topology. Choose such kernels $\varphi_{A,j}\to\varphi_A$ with their supports
in those patches; their reversed adjoint kernels converge as well. Do the
same for the kernel $\varphi_C$ of $C$. Polarization gives the factorized
identity for arbitrary complex word tests, so it applies to these
approximants. This is a test-space approximation, not a finite-rank
requirement on the given kernels.

For every fixed nonnegative integer $k$, the original joint-distribution
bounds imply
\[
 \bigl\|(1+P^0)^k\bigl(\Psi_A(\varphi_{A,j})-\Psi_A(\varphi_A)\bigr)\bigr\|
 \longrightarrow0,
\]
and analogously for $C$ and for the concatenated old word with kernel
$\varphi_{C,j}\otimes\varphi_{A,j}\to\varphi_C\otimes\varphi_A$. Indeed, a collective
time-translation derivative of a vacuum word is the same word with the
corresponding derivative of its joint test. The doubled original vacuum
distributions therefore bound each displayed graph norm by finitely many
joint test seminorms. Positive translation spectrum makes these norms
control the $b^{1/2}$ graph norm and every fixed translation-Sobolev norm
used above. The left side of \eqref{N:bounded-Ward} is continuous by its
$b^{1/2}$ form bound; the right side is continuous by the inverse-energy
half-step bound applied to the concatenated old vacuum word. Taking this
kernel limit after the polar and convolution limits proves the stated Ward
identity for the joint kernels.

On a compact smooth parameter family the same finitely many seminorms are
uniformly bounded. The tensor approximations can be chosen uniformly in
those seminorms, using a finite cover of the compact family and a smooth
partition in its parameter. The preceding graph and flux bounds then give
the asserted compact-parameter uniformity. This final passage does not
assert a closed affiliated operator for an arbitrary joint smear.
\end{proof}

\subsection{The commuting measure and its second moment}

The analytic part of the argument has established the bounded Ward
identity and its polynomial extension. We now use the resulting
linearity to organize all bounded profiles simultaneously. Positivity
will give a measure, but normalization requires a separate tightness
argument, and its quadratic moment requires comparison with the
actual parabolic cuts.

Linearity upgrades the spectral order of bounded profiles to
commutativity.  To record the argument, put $A=h_g$, $B=h_f$ and
$\varepsilon=\norm{f}_\infty/s$.  The profile comparison of
Section~\ref{sec:foundations} gives the spectral brackets
\[
 A-\varepsilon\id\ \leq_{\rm sp}\ A+B/s
       \ \leq_{\rm sp}\ A+\varepsilon\id.
\]
If $r<t$ and $2\varepsilon<t-r$, the low $A$ subspace
$1_{(-\infty,r]}(A)\cH$ is contained in the spectral subspace of
$A+B/s$ below $r+\varepsilon$, while the high $A$ subspace
$1_{[t,\infty)}(A)\cH$ is contained in its spectral subspace above
$t-\varepsilon$.  Thus
\[
 1_{(-\infty,r]}(A)(A+B/s)1_{[t,\infty)}(A)=0.
\]
The same matrix of $A$ is zero, so the matrix of $B$ is zero.
Varying $r,t$ proves that $B$ commutes with every spectral projection
of $A$.  This argument uses the spectral-support comparison
\eqref{F:bounded-spectral-order}, not merely ordinary operator order.

\Needspace{6\baselineskip}
\begin{proposition}[Internal measure and quadratic moment]
\label{N:measure-proposition}
On the $b>0$ representation there is a normalized positive
operator-valued Borel measure $\mu$
on $\R^2$, with mutually commuting values on the internal
multiplicity space, such that
\begin{equation}\label{N:measure-representation}
 H_g=b\int_{\R^2}g(Y+z)\,\mu(\dd z),
 \qquad \mu(\R^2)=\id.
\end{equation}
Its values commute with $h$ and with the external variables $b,Y$.
Its first moment is zero as a polarized integral on
$\Dom h^{1/2}$. Its trace second moment is the increasing closed-form
limit of compactly truncated moments and defines a positive
self-adjoint operator $\kappa$ satisfying
\begin{equation}\label{N:moment-comparison}
 \int z\,\mu(\dd z)=0,\qquad
 \kappa=\int |z|^2\,\mu(\dd z)\leq h,
 \qquad K_n=b(|Y|^2+\kappa).
\end{equation}
The second-moment inequality is a quadratic-form inequality and,
since the operators strongly commute, has its usual joint
spectral meaning.  No assertion that $\mu$ is projection-valued,
or that $\kappa=h$, is made.
\end{proposition}

\begin{proof}
The map $g\mapsto h_g$ is positive and contractive in the supremum
norm, and $h_{\mathbf1}=\id$. It commutes strongly with $Y,h,b$ and is
boost independent. In the $Y$ representation write $h_g=M_{F_g}$,
where $F_g$ is an essentially bounded internal-operator-valued function.
For $\tau_tg(z)=g(z+t)$, transverse covariance gives
\[
 F_{\tau_tg}(y)=F_g(y+t)\quad\text{almost everywhere},\qquad
 \|F_g(\cdot+t)-F_g\|_{L^\infty}
 \le\|\tau_tg-g\|_\infty.
\]
The right side tends to zero. Weak-operator convolution with a smooth
approximate identity therefore converges to $F_g$ in essential-supremum
norm. The convolutions are norm continuous; their pairwise
essential-supremum bounds are consequently actual supremum bounds.
Their uniform limit selects a unique norm-continuous representative
$\widetilde F_g$. Covariance now holds pointwise, so
\[
 \mathcal L(g)=\widetilde F_g(0),\qquad
 \widetilde F_g(y)=\mathcal L(\tau_yg)
\]
defines one positive contractive internal functional. Linearity,
positivity, and the commuting relations pass to these representatives.
For inequalities involving the unbounded $h$, first restrict to a
bounded spectral band of $h$, evaluate the resulting continuous
inequality at zero, and then exhaust the bands.

The Riesz representation theorem applied to scalar matrix elements
of $\mathcal L$ on compact tests, and polarized, gives a positive
operator-valued measure $\mu$. Commutativity of all $h_g$ and
commutation with $h$ pass to its Borel values by bounded monotone
approximation. This proves \eqref{N:measure-representation} first for
compactly supported smooth profiles.

It remains to rule out mass at infinity, which is not automatic
from a representation on compact tests.  Choose a radial
$\chi_0\in C_c^\infty(\R^2)$, $0\leq\chi_0\leq1$, equal to one
near zero and nonincreasing along rays.  Set
$\chi_{0,R}(z)=\chi_0(z/R)$.  The Taylor estimate at the $Y=0$
fiber gives, on $\Dom h^{1/2}$,
\begin{equation}\label{N:tightness}
 0\leq\id-\mu(\chi_{0,R})
 \leq C R^{-2}h.
\end{equation}
For $\mathcal E_R=\id-\mu(\chi_{0,R})$, positivity gives
$0\leq \mathcal E_R\leq\id$, hence
$\norm{\mathcal E_R\psi}^2\leq\ip{\psi}{\mathcal E_R\psi}$. Equation~\eqref{N:tightness}
therefore gives strong convergence to zero first on
$\Dom h^{1/2}$ and then, by density and the contraction bound,
on the whole internal space.
Monotone convergence and density imply
$\mu(\chi_{0,R})\uparrow\id$ strongly.  This proves normalization
and excludes escaping mass.  It also extends the representation
from compact tests to bounded smooth profiles.  Indeed positivity
and linearity at the $Y=0$ fiber give
\[
 |\mathcal L(g)-\mu(g\chi_{0,R})|
 \leq\norm{g}_\infty\bigl(\id-\mu(\chi_{0,R})\bigr).
\]
The right side tends strongly to zero; it controls the omitted
scalar matrix elements, and polarization completes the extension.

For the precise second-moment constant, put
$q_R(z)=|z|^2\chi_{0,R}(z)$.  These bounded smooth profiles increase
to $|z|^2$ and satisfy $q_R\leq p_0$.  Direct cut inclusion and log
order give $h_{q_R}\leq |Y|^2+h$.  At $Y=0$ this is
\[
 0\leq\mu(q_R)\leq h.
\]
Monotone convergence defines the densely defined closed second
moment form $\kappa$, dominated by $h$.  All its bounded spectral
approximants commute with $h$, so $\kappa$ is affiliated with the
same commutative algebra.  This proves $\kappa\leq h$ with coefficient
one, rather than a Hessian-dependent constant.

Finally let $e\in\R^2$ and
$a_R(z)=(e\cdot z)\chi_{0,R}(z)$.  Then $a_R(0)=0$ and
$\norm{a_R''}_\infty\leq C|e|/R$, so
\[
 |\mu(a_R)|\leq C|e|R^{-1}h.
\]
The second-moment bound makes the first moment integrable on the
$h$ form domain: for $\psi\in\Dom h^{1/2}$,
\[
 \int|z|\,\ip{\psi}{\mu(\dd z)\psi}
 \leq\norm{\psi}\,\norm{h^{1/2}\psi}.
\]
Dominated convergence and polarization now give
$\int e\cdot z\,\mu(\dd z)=0$.  This proves
\eqref{N:moment-comparison}.  Covariance of the construction under
changes of transverse axes and dilations follows from covariance
of the original cut generators; boost independence has already
removed the choice of boost coordinate.
\end{proof}

The measure and its moment have been constructed using one horizon.
For the comparison between two horizons we also need them to be
independent of whether the bounded tests were taken in the original
or the reflected net. The physical Ward formula supplies precisely
that identification.

\Needspace{6\baselineskip}
\begin{corollary}[Reflected agreement]\label{N:reflected-agreement}
Repeat the construction with past cuts of the original physical
net, or with the twist-corrected reflected net, and reflect back
to the same future orientation.  The bounded profile functionals,
the measure $\mu$, and the quadratic moment $K_n$ agree.
\end{corollary}

\begin{proof}
Wightman wedge reflection gives the same polynomial vacuum data
and the same physical stress-tensor flux on the right side of
\eqref{N:bounded-Ward}.  Proposition~\ref{N:polynomial-extension}
therefore identifies the two bounded-profile forms on opposite
polynomial vacuum vectors.  Their $b^{1/2}$ graph density and the
uniform form bound identify the operators.  The compact-profile
measure and its monotone moments are consequently identical.
This argument identifies the bounded functionals and their
quadratic moment; it does not identify the full parabolic modular
generator with $K_n$ at this stage.
\end{proof}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/03-null-current.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/04-dispersion.tex ====
\section{Two horizons and the scalar spectral comparison}
\label{sec:dispersion}

The comparison needed for the trace polarization is stronger than the
order-two comparison furnished by its physical differential field.
We prove that the quadratic moment of the bounded-profile generators
satisfies an order-zero comparison on the vacuum wave $l$.
No local field with vacuum wave $l$ is used in this Section.

The argument has two parts.  First, the same causal stress-tensor
commutator is evaluated from two null directions.  This relates their
bounded-profile matrices after all local subtraction terms have been
removed.  Second, the difference between Hankel orders two and zero is
shown to have only one possible tail.  A logarithmically long radial
test makes the two-horizon identity detect the squared norm of that
tail, and positivity forces it to vanish.

We recall precisely the inputs from Sections~\ref{sec:foundations}
and~\ref{sec:null-current}.  On the internal Hilbert space at fixed
light-front coordinates, $h\geq0$ is the parabolic operator and
$h_\nu=-\partial_M^2-M^{-1}\partial_M+\nu^2M^{-2}$ has its regular
Hankel realization in $L^2(\R_+,M\,\dd M)$.  The comparison
\eqref{F:spectral-equation} applies with
\begin{equation}\label{D:orders}
\begin{array}{c|c}
 \text{constant polarization label}&\nu\\ \hline
 c_V=l+j_z,\quad c_T=l/3+s_z&0\\
 l,\ j_z,\ s_z&2\\
 j_i,\ s_{zi}&1.
\end{array}
\end{equation}
The bounded Ward formula \eqref{N:bounded-Ward}, including its polynomial
extension, and the moment representation give
\begin{equation}\label{D:measure-input}
 H_g=b\int g(Y+z')\,\mu(\dd z'),\qquad
 \mu(1)=\id,\qquad \mu(z'_i)=0,\qquad
 \kappa:=\mu(|z'|^2)\leq h.
\end{equation}
Here $\mu$ is a positive commutative operator-valued measure; its values
commute with $h$, $Y$ and $b$.  It need not be projection-valued.
The construction is covariant under dilatations and rotations, independent
of the boost coordinate, and agrees with the reflected construction.
Consequently, on every copy in \eqref{D:orders},
\begin{equation}\label{D:inherited-comparison}
 \mathbf1_{(s,\infty)}(\kappa)\mathbf1_{[0,s]}(h_\nu)=0,
 \qquad \norm{\kappa^t\psi}\leq\norm{h_\nu^t\psi}
 \quad(t>0).
\end{equation}
Indeed $\kappa$ and $h$ commute, so $\kappa\leq h$ combines with the
spectral comparison for $h$.  All internal formulas below are Hilbert
space formulas or distributions tested on compact output mass intervals;
they do not presume integral kernels for $\mu$.

In particular, the comparison in \eqref{D:inherited-comparison}
retains the embedding of each scalar copy in the ambient internal
space.  It bounds the spectrum of its image; it does not say that
$\kappa$ preserves that copy.  Reduction will be proved only after
the charge identities are available.

\subsection{The plane matrix and its causal continuation}

The bounded Ward identity initially supplies a localized plane matrix.
To compare the two horizons we need its causal continuation and a
fixed-normal restriction in the particular three-point channel used
here.  We establish these two operations before carrying out any
Cauchy subtraction.

Fix future null vectors $n,\bar n$ with $n\cdot\bar n=1$ and use
the transverse frame of Section~\ref{sec:foundations}.  Write
\begin{equation}\label{D:kinematics}
\begin{aligned}
 p&=a n+b\bar n+w, &p^2&=M_o^2>0,\\
 k&=a_x n+q, &p-k&=(a-a_x)n+b\bar n+r,\qquad w=q+r,\\
 X&=-k^2=|q|^2, &z&=-(p-k)^2=|r|^2-2b(a-a_x).
\end{aligned}
\end{equation}
The comparison point is $a_x=a$, where both virtualities are strictly
spacelike: $X=|q|^2>0$ and $z=|r|^2>0$.

\Needspace{6\baselineskip}
\begin{lemma}[The full-plane matrix]\label{D:plane-matrix}
On the spacelike-transfer part of $n\cdot k=0$, the full $T_{nn}$
matrix integrated along $n$ is the matrix of
$b\int g(Y+z')\mu(\dd z')$.  Initially the assertion is a joint
distributional identity, averaged over the null direction $n$ and tested
in the normal displacement $u$ of the plane. For the stress-tensor
three-point matrix with timelike spin-one output used below, the angular
average can be removed: $n$ may be fixed. The displacement $u$ remains
a distributional variable, whose Fourier transform supplies $a_x$.
\end{lemma}

\begin{proof}
Translate the plane normally by $u$, test $u$ with a compact smooth
function, and choose $\widehat g$ smooth with compact support in a
transverse annulus.  Average $n$ over a small smooth patch, taking a
smooth transverse frame and a sphere normalization of $n$.  Opposite
wedge tests can be chosen uniformly for these small changes.  Insert
a cutoff in the longitudinal coordinate $v$ and split its matrix by
the separating step used in \eqref{N:bounded-Ward}.  One term is that
Ward formula; the remaining term is an all-plane vacuum matrix.
The latter vanishes at spacelike transfer by the vacuum spectrum.
The operator on the Ward side at normal coordinate $u$ is
$U_a(u)H_gU_a(-u)$.

The removal of the $v$ cutoff has an independent distributional
meaning.  In momentum space it inserts $2\pi\delta(n\cdot k)$ after
averaging over $n$. On its support, $k=a_xn+q$ with $q$ in the
chosen annulus.  The derivative of $n\cdot k$ in an angular direction
of $n$ has the nonzero transverse component $q$.  The coarea formula
therefore turns the averaged delta insertion into a smooth multiplier.
The Fourier transform of the normal test is Schwartz.  Derivatives of
the moving coordinates cost polynomial factors, absorbed by this
decay and the compact annular test.  Thus the cutoff multipliers
converge in the Schwartz seminorms needed to pair with any of the
tempered external field distributions.  For an arbitrary fixed list
of original fields, these are the finite sums of homogeneous-list
distributions established in Section~\ref{sec:foundations}; no single
scaling dimension is assigned to the whole list.

The identity on opposite wedge tests extends to arbitrary external
translations by positive-energy tube uniqueness.  On the operator
side, $|H_g|\leq\norm{g}_\infty b$ supplies the polynomial energy
bound: insert positive-energy exponential damping in the two
external vectors, obtain the holomorphic matrix elements, and remove
the damping as distributional boundary values.  Equality on the open
wedge patch then determines the boundary distribution everywhere.
This establishes the identity jointly with the normal parameters.
The fixed-normal pullback needed here is justified in
Lemma~\ref{D:causal-continuation}.
\end{proof}

\Needspace{6\baselineskip}
\begin{lemma}[Fixed-spacelike-virtuality dispersion]
\label{D:causal-continuation}
Test $[T_{\alpha\beta}(x),T_{\rho\sigma}(0)]\Omega$ against the
spin-one output wave, with output measure $\dd^4p$ on the open
timelike cone.  At fixed $X>0$, the contracted matrix in
\eqref{D:kinematics} is the jump of polynomially bounded functions
holomorphic in the upper and lower $z$ half-planes.  The jump is the
first Wightman order and is supported in $z\leq0$.
At fixed $z>0$, the corresponding $X$ jump is the reversed Wightman
order with the same sign.  The two gap values come from one causal
splitting of this commutator.
\end{lemma}

\begin{proof}
Retain compact timelike output tests throughout the construction.
Testing against a $j$ label amounts to testing with $V$ and smooth
transverse projectors on that compact set. Lorentz and scale covariance
trivialize the open timelike orbit, leaving only the finite rest-rotation
representation. They identify the resulting distributions in simultaneous
moving coordinates. Fixed-$p$ formulas below refer to this covariant
disintegration; their estimates are uniform on compact positive output
mass ranges.

Local commutativity supports the relative-position commutator in
$\overline V_+\cup\overline V_-$.  Off the origin choose future and
past cutoffs constant near these two cones on the unit sphere.  Finite
distribution order allows their products to be extended through the
origin by subtracting a sufficiently long Taylor jet of the test
function. This is the cone-supported distribution extension used in
causal splitting; see \cite[Section~5.1]{EpsteinGlaser1973}.
Restore the total commutator by a distribution supported
at the origin.  This gives tempered cone-supported distributions
whose difference is the commutator.  Average the splitting over rest
rotations and transport it along the timelike Lorentz and scale
orbit.  Its local ambiguity is a finite sum of derivatives of a
delta function.  With the Fourier convention
$D(p,x)=\int e^{ikx}D(p,k)\,\dd k$, the resulting boundary values
$D^+,D^-$ are holomorphic in the upper and lower primitive tubes,
respectively, with polynomial bounds in momentum and inverse
distance to the tube boundary
\cite[Section~1, pp.~155--156]{BrosIagolnitzer1973}.

Here is a direct path from those tubes to the required half-planes.
In the rest frame of $p$, set
\begin{equation}\label{D:virtuality-path}
 k^0=\frac{M_o^2-X+z}{2M_o},\qquad
 R^2=(k^0)^2+X,\qquad \boldsymbol k=R\omega,
 \quad\omega\in S^2\text{ real}.
\end{equation}
Write $k^0=u+iv$, $R=s+it$.  Direct calculation gives
\begin{equation}\label{D:tube-distance}
 v^2-t^2=
 \frac{2Xv^2}{u^2+v^2+X+|(u+iv)^2+X|}>0,
 \qquad
 |v|-|t|\geq\frac{X|v|}{2(u^2+v^2+X)}.
\end{equation}
Thus \eqref{D:virtuality-path} lies strictly in the relevant tube
whenever $\operatorname{Im}z\ne0$.  On compact positive $X,M_o$
ranges the distance is bounded below by a constant times
$|\operatorname{Im}z|/(1+|z|)^2$.  The tube bounds therefore give
polynomial bounds in $|z|+|\operatorname{Im}z|^{-1}$.

The square root in \eqref{D:virtuality-path} introduces no extra
branch.  A tensor with the fixed finite collection of external
indices has only finitely many rotation harmonics.  A harmonic of
degree $s$ is $R^s$ times a Cartesian harmonic polynomial in
$\omega$; its remaining scalar coefficient is even and analytic in
$R$.  At $R=0$, where the path is still inside a tube, analyticity
in the Cartesian momentum components proves regularity of that
coefficient.  Divide by $R^s$ and reconstruct with Cartesian harmonic
polynomials.  The scalar coefficients are single-valued functions of
the two invariants.  They can consequently be evaluated at the
actual moving light-front momentum in \eqref{D:kinematics}, even
though that complex momentum itself has null imaginary part.
The branch points $k^0=\pm i\sqrt X$ are nonreal; none belongs
to the real locus on which a distributional restriction is taken.

We also verify that the real boundary obtained this way is the
required restriction.  For fixed real $\omega$,
$\dd R/\dd k^0=k^0/\sqrt{(k^0)^2+X}$ has absolute value less than
one.  The real tangent is strictly timelike.  Integration by parts
against a compact test interval therefore gives the uniform decay
on the dual cone needed for the Fourier--Laplace boundary limit.
The harmonic reconstruction gives the moving-direction restriction.
There is a useful equivalent noncharacteristic check.  The tube
boundary distributions have ordinary wavefront covectors in the
corresponding causal cones: compact localization in momentum
convolves the cone-supported tempered inverse Fourier transform
with a Schwartz function, giving rapid decay in directions outside
that cone.  For the full finite tensor family, rest-rotation
covariance gives equations $L_{ij}D=\rho_{ij}D$, where $L_{ij}$
is the angular rotation vector field and $\rho_{ij}$ a constant
matrix on the tensor indices.  The principal symbols of these
equations force every singular covector to annihilate the angular
orbit.  Since real spacelike $k$ has nonzero spatial part, its
annihilator is exactly $\Span\{p,k\}$.  If
$\xi=A p+B k$ annihilates the tangent $n$ in
\eqref{D:kinematics}, then $A b=0$, so $\xi=B k$.  As $k^2=-X<0$,
no nonzero such covector is causal. The pullback along
$a_x\mapsto a_xn+q$ is therefore noncharacteristic. On compact
positive $X$ and timelike output ranges the separation of its conormal
bundle from these wavefront cones persists in a small angular patch
of $n$. The restricted distribution thus depends smoothly on that
patch, removing the angular average in Lemma~\ref{D:plane-matrix}.
The normal displacement $u$ is still tested, or Fourier transformed;
no point evaluation of an arbitrary distribution in $u$ is needed.
This is a restriction
statement for this covariant three-point matrix, not for arbitrary
null-plane field distributions.

In the first Wightman order the vacuum leg has momentum $p-k$;
in the reverse order it has momentum $k$.  At $X>0$ the reverse
order vanishes by the spectrum condition, and the first can occur
only for $z\leq0$.  The two holomorphic boundary values glue in
the positive $z$ gap.  If instead $z>0$ is fixed and $X$ varies,
upper-half-plane continuation corresponds to the lower original
tube; its jump is minus the commutator, hence the reverse Wightman
order with a positive sign.  Both gap values are values of the
same split commutator at two spacelike virtualities.  This proves
the comparison directly from causal tubes, without a crossing
assumption.
\end{proof}

\subsection{Subtractions and the two-horizon identity}

Causal continuation identifies the two spacelike gap values, but a
dispersion formula determines each only up to a polynomial.  The next
step tracks the dependence of these polynomials on the transverse
variables.  The useful conclusion is their separated form: each
ambiguity is polynomial in at least one of the two variables.

Write $[\psi]_{j_A}$ for the output distribution tested against the
constant spin-one label in the indicated horizon frame, and put
$\mu_n(q)=\int e^{iq\cdot z'}\mu(\dd z')$ in that frame.
Application of Lemma~\ref{D:plane-matrix} to the first jump gives,
after testing it with $\phi(z)$, a fixed nonzero common constant times
\begin{equation}\label{D:jump}
 \left[\mu_n(q)(M^2+|r|^2)^2
 c_T^{(\bar n)}(M,r)\,\phi(-M^2)\right]_{j_A}(M_o).
\end{equation}
The normalization is worth checking.  The plane insertion is
$b\mu_n(q)e^{iq\cdot Y}$, conjugated by $U_a(u)$, and the input
polarization is $T_{\bar n\bar n}=a_i^2c_T^{(\bar n)}$ with
$a_i=(M^2+|r|^2)/(2b)$.  The normal Fourier measure is
$\dd a_x=\dd z/(2b)$.  Thus the jump with respect to $\dd z$
contains $2b^2a_i^2=(M^2+|r|^2)^2/2$; no boost factor remains.
The opposite channel has exactly the same constant.  These
identities can first be tested on external wave packets, so
\eqref{D:jump} means bounded operator application to an internal
$M\,\dd M$ wave and testing on compact output masses.

\Needspace{6\baselineskip}
\begin{lemma}[The subtraction ambiguities]\label{D:subtractions}
There are continuous $L^2(M\,\dd M)$-valued regularizations
$\chi^n_{\rm reg}(r,M)$ of
\begin{equation}\label{D:chi}
 \chi^n(r,M)=(M^2+|r|^2)c_T^{(\bar n)}(M,r)
\end{equation}
such that, with $w=q+r$ and
\begin{equation}\label{D:AB}
A_A=[\mu_n(q)\chi^n_{\rm reg}(r)]_{j_A^{(n)}},\qquad
B_A=[\mu_{\bar n}(r)\chi^{\bar n}_{\rm reg}(q)]_{j_A^{(\bar n)}},
\end{equation}
one has
\begin{equation}\label{D:two-horizon}
 (M_o^2+|w|^2)A_z-(|w|^2-M_o^2)B_z+2M_ow_iB_i
 =P_q+P_r.
\end{equation}
The degree of each polynomial is uniformly finite.  The coefficients
of $P_q$, polynomial in $q$, may depend on $r$; those of $P_r$,
polynomial in $r$, may depend on $q$.
\end{lemma}

\begin{proof}
Choose a positive subtraction point $z_0$ and retain the full finite
covariant tensor family.  Use one integer $d$ larger than all its
polynomial growth orders, perform the Cauchy subtraction on that
family, and then contract with $nn,\bar n\bar n$ and the output
frame.  The subtracted Cauchy kernel is
\[
 \frac{(z-z_0)^d}{(z'-z_0)^d(z'-z)}.
\]
Using \eqref{D:jump}, its value at $z=|r|^2$ is proportional to
\begin{equation}\label{D:cauchy-wave}
 \left[\mu_n(q)\chi^n(r,M)
 \left(\frac{|r|^2-z_0}{-M^2-z_0}\right)^d\right]_{j_A}.
\end{equation}
Subtracting this Cauchy transform leaves zero jump.  Its boundary
Fourier transform is supported in both closed half-lines, hence
at zero; the remainder is a polynomial in $z$.

Its dependence on the transverse parameter must also be controlled.
A null rotation with parameter $\beta$, fixing $n$, changes
\[
 r\longmapsto r+b\beta,\quad
 a\longmapsto a+\beta\cdot(q+r)+\tfrac12b|\beta|^2,\quad
 a_x\longmapsto a_x+\beta\cdot q.
\]
It fixes $b,M_o,q,X,z$ and transports the output adapted frame.
The remaining tensor contractions transform polynomially in
$\beta$, by a matrix independent of $z$.  This matrix commutes
with the common-order Cauchy subtraction.  Covariance of the split
commutator therefore makes the
coefficients of its subtraction polynomial polynomial in $r$ at
fixed $q$.  Evaluation at $z=|r|^2$ preserves this property.  The
opposite calculation gives a polynomial in $q$ at fixed $r$.
The degrees are uniform because the distribution order in a rest
frame and the degrees of the finite tensor representation are
finite.  The opposite calculation may hold its own boost coordinate
$a$ fixed; the evaluated contracted matrix has net boost weight zero,
so it can then be written at the same $b$ as the first calculation.

For completeness the wave regularization can be made explicitly.
Set $R_2=|r|^2$, $D=M^2+R_2$.  The change of spatial frame gives
\begin{equation}\label{D:polarization}
 c_T^{(\bar n)}=l/3+d_Ad_Bs_{AB},\qquad
 d_z=\frac{R_2-M^2}{D},\qquad d_i=-\frac{2Mr_i}{D}.
\end{equation}
Write $s_{ij}^{\rm TF}=s_{ij}+\delta_{ij}s_z/2$.  Expansion of
\eqref{D:polarization} gives
\begin{equation}\label{D:chi-expansion}
 \chi^n=\frac D3l+
 \left(R_2-5M^2+\frac{6M^4}{D}\right)s_z
 +\left(-4Mr_i+\frac{8M^3r_i}{D}\right)s_{zi}
 +\frac{4M^2r_ir_j}{D}s_{ij}^{\rm TF}.
\end{equation}
In particular the radial coefficient follows from
$s_{11}+s_{22}=-s_z$:
$((R_2-M^2)^2-2M^2R_2)/D=R_2-5M^2+6M^4/D$.
The vector coefficient is $2Dd_zd_i$.

Choose a smooth mass cutoff $\zeta$ equal to zero below $1$ and
one above $2$, and subtract $\zeta(M)$ times
\[
 \frac{M^2+R_2}{3}l+(M^2-5R_2)s_z
 +(4Mr_i-8r_iR_2/M)s_{zi}+4r_ir_js_{ij}^{\rm TF}.
\]
The remaining large-mass terms are $O(M^{-2})$ or better on
compact $r$ sets, and hence belong to $L^2(M\,\dd M)$.
At small mass, $|d|=1$ in \eqref{D:polarization}, so the norm is
bounded by a constant times $M^2+|r|^2$.  Dominated convergence
now proves continuity also at $r=0$; the same formulas give
polynomial growth in $r$.

Replacing \eqref{D:cauchy-wave} by this regularization changes
only an $L^2$-valued polynomial in $r$.  Indeed
\[
 (R_2-z_0)^d-(-M^2-z_0)^d
\]
is divisible by $D$, canceling the sole denominator in
\eqref{D:chi-expansion}.  Both regulated expressions are $L^2$
waves; polynomial interpolation then shows that every coefficient
of their polynomial difference is an $L^2$ wave.  Choose the same
mass cutoff and subtraction prescription in the other horizon.

Finally the output frame rotation is
\[
 j_z^{(n)}=\frac{|w|^2-M_o^2}{M_o^2+|w|^2}j_z^{(\bar n)}
 -\frac{2M_ow_i}{M_o^2+|w|^2}j_i^{(\bar n)}.
\]
The common gap value from Lemma~\ref{D:causal-continuation} and the
common jump normalization therefore give \eqref{D:two-horizon}
off $q=0$ and $r=0$.

There is no extra distribution supported on those axes.  The left
side is continuous in $q,r$, with values on compact mass tests.
Take polynomial interpolation projections $\mathcal P_q,
\mathcal P_r$ of degrees larger than the two degree bounds, using
nodes away from zero.  On the off-axis patch,
$(1-\mathcal P_q)(1-\mathcal P_r)$ annihilates the left side.
By continuity it does so everywhere.  Decomposing with these two
projections extends its separated-polynomial form to all $q,r$.
\end{proof}

Fix the transverse Fourier convention
\[
 \widehat g(r)=(2\pi)^{-2}\int e^{-ir\cdot z'}g(z')\,\dd^2z',
 \qquad g(z')=\int e^{ir\cdot z'}\widehat g(r)\,\dd^2r.
\]
If both position tests vanish near the origin, every term $P_q+P_r$
in \eqref{D:two-horizon} is annihilated.  For a real radial
$g\in C_c^\infty(\R^2\setminus\{0\})$, put
\begin{equation}\label{D:Phi}
 \Phi(g)=\int\widehat g(r)\chi^n(r,M)\,\dd^2r,
 \qquad \mu(g)=\int g(z')\mu(\dd z').
\end{equation}
The subtraction makes no difference in this formula.

Indeed, integrating a polynomial in $r$ against $\widehat g(r)$
gives a finite linear combination of derivatives of $g$ at zero,
all of which vanish.  With one such test in each transverse variable,
this removes $P_q$ and $P_r$ separately, even though their remaining
coefficients can depend on the other variable.  The regularized
formula supplies the meaning of this operation on the original wave.
To perform the horizon exchange explicitly, let $\mathsf R$ be the spatial
half-turn about $e_2$. It sends $n$ to $\bar n$, $\bar n$ to $n$, and acts
on transverse coordinates by $\mathsf S=\operatorname{diag}(-1,1)$.
The adapted tetrads obey
\[
 \mathsf R e_z^{(n)}(p)=e_z^{(\bar n)}(\mathsf Rp),\qquad
 \mathsf R e_i^{(n)}(p)=\mathsf S_{ij}e_j^{(\bar n)}(\mathsf Rp).
\]
Thus the longitudinal label has no extra sign. In the covariantly
identified output frames, with the same regularization on both horizons,
\[
 A_z(q,r)=B_z(\mathsf Sr,\mathsf Sq),\qquad
 A_i(q,r)=\mathsf S_{ij}B_j(\mathsf Sr,\mathsf Sq).
\]
Write $\langle F\rangle_g=\int\widehat g(q)\widehat g(r)F(q,r)
\,\dd^2q\,\dd^2r$. Radiality makes the test invariant under
$(q,r)\mapsto(\mathsf Sr,\mathsf Sq)$, including the weight
$|q+r|^2$. Consequently
\[
 \langle A_z\rangle_g=\langle B_z\rangle_g=:C_g,\qquad
 \langle |q+r|^2A_z\rangle_g=\langle |q+r|^2B_z\rangle_g,
\]
and \eqref{D:two-horizon} reduces to
\[
 2M_o^2C_g+2M_o\sum_i\langle(q_i+r_i)B_i\rangle_g=0.
\]
Our Fourier convention gives
$q_i\widehat g(q)=-i\widehat{\partial_i g}(q)$. Each of the two
summands differentiates one test, with the same sign. On returning
their vector contraction to the $n$ frame, the transverse matrix
$\mathsf S$ occurs twice and cancels. Division by $2M_o$ gives
\begin{equation}\label{D:radial-identity}
 M_o[\mu(g)\Phi(g)]_{j_z}
 =i\sum_i
 [\mu(\partial_i g)\Phi(g)+\mu(g)\Phi(\partial_i g)]_{j_i}.
\end{equation}
Reversing the Fourier orientation would change the common phase,
which will play no role; the relative coefficients are fixed.
In particular, \eqref{D:chi-expansion} shows that the radial and
vector kernels surviving away from the position origin are
$6M^4/(|r|^2+M^2)$ and $8M^3r_i/(|r|^2+M^2)$.
The two-dimensional massive Green transform gives respectively
$M^4K_0(M|z'|)$ and a fixed imaginary multiple of
$M^4\widehat z'_iK_1(M|z'|)$.  The scalar and transverse trace-free
terms do not contribute to a radial test, nor to its first angular
harmonic derivative, as appropriate.

\subsection{Exact Hankel leakage}

We now isolate the only obstruction to improving the order-two
comparison on $l$ to order zero.  A test with zero first radial moment
has no leakage beyond its Hankel support.  It follows that every
possible exterior tail depends on that single moment; dilation
covariance will determine its exact shape.

Let $\mathcal H_\nu$ be the unitary Hankel transform
\[
 (\mathcal H_\nu F)(M)=\int_0^\infty J_\nu(Mv)F(v)v\,\dd v.
\]
It diagonalizes $h_\nu$ as multiplication by $v^2$.
Put $\lambda=\sqrt\kappa$.  On its positive spectral part,
dilatation covariance permits a homogeneous realization
$L^2(\R_+,\lambda\,\dd\lambda;\mathcal K)$ with constant
multiplicity space $\mathcal K$.  Its zero spectral subspace is
kept separately.  Write $\mathcal U$ for this spectral
realization, and $f c$ for the input radial wave with label $c$.

\Needspace{6\baselineskip}
\begin{lemma}[Exact tail]\label{D:exact-tail}
For $c=l,j_z,s_z$ there is $d_c\in\mathcal K$ such that, if
$F\in C_c^\infty(0,R)$, then
\begin{equation}\label{D:tail}
 \mathcal U\bigl(c\mathcal H_0F\bigr)(\lambda)
 =\left(\int_0^\infty vF(v)\,\dd v\right)\lambda^{-2}d_c
 \quad\text{for almost every }\lambda>R.
\end{equation}
The label map is linear, and
\begin{equation}\label{D:tail-relations}
 d_{j_z}=-d_l,\qquad d_{s_z}=-d_l/3.
\end{equation}
\end{lemma}

\begin{proof}
The Bessel recurrence $J_2(x)=2J_1(x)/x-J_0(x)$ and
$\int_0^v rJ_0(Mr)\,\dd r=vJ_1(Mv)/M$ give
\begin{equation}\label{D:hankel-transfer}
 (\mathcal H_2\mathcal H_0F)(v)
 =\frac2{v^2}\int_0^v rF(r)\,\dd r-F(v).
\end{equation}
The identity first follows on smooth compact tests by the unitary
Hankel inversion and then in $L^2(v\,\dd v)$.  If $F$ has zero
first radial moment and support below $R$, the right side also
has support below $R$.  The order-two comparison in
\eqref{D:inherited-comparison} therefore puts
$c\mathcal H_0F$ in the $\lambda\leq R$ subspace.

Consequently the restriction of the transfer to any interval
strictly above $R$ factors through the single functional
$F\mapsto\int vF(v)\,\dd v$.  Choose one annular test with this
moment equal to one.  Overlapping support bounds identify the
resulting vector-valued coefficient at every $\lambda>0$.
In these coordinates internal dilatations act as
$f(M)\mapsto e^sf(e^sM)$ on the input and
$\psi(\lambda)\mapsto e^{-s}\psi(e^{-s}\lambda)$ on the output.
Since the moment of the corresponding Hankel-side test scales
by $e^s$, its coefficient must be $\lambda^{-2}d_c$.
A reference test and its square-integrable tail ensure
$d_c\in\mathcal K$.  This proves an exact tail, not only its
asymptotic order.

The order-zero comparisons on $c_V$ and $c_T$ say that
$c_V\mathcal H_0F$ and $c_T\mathcal H_0F$ have no spectrum above
the support radius of $F$.  Linearity of the tail gives
$d_l+d_{j_z}=0$ and $d_l/3+d_{s_z}=0$.
\end{proof}

\subsection{The logarithmic pairing}

The tail relations alone allow a nonzero vector $d_l$.  We will insert
a radial test covering many scales into \eqref{D:radial-identity}.
Its two scalar-channel tails produce a term proportional to
$\log L\,\|d_l\|^2$, whereas the first angular harmonics give smaller
terms.  The estimates below keep this leading coefficient and the
errors separate.

Choose $0\leq\eta_L\leq1$, smooth and supported in
$(0,\log L)$, equal to one outside fixed-width neighborhoods
of the endpoints, with uniformly bounded derivatives.  Set
\begin{equation}\label{D:plateau}
 g_L(z')=|z'|^2\eta_L(\log|z'|).
\end{equation}
We retain the nonzero normalization constant $c_*$ throughout
the estimates; it is determined by the radial Fourier kernel.

\Needspace{6\baselineskip}
\begin{lemma}[Plateau estimates]\label{D:plateau-estimates}
There are scalar functions $f_L,u_L$ such that
\begin{equation}\label{D:plateau-waves}
 \Phi(g_L)=s_zf_L(M),\qquad
 \Phi(\partial_i g_L)=M s_{zi}u_L(M),
\end{equation}
and, writing $\mathscr D=M\partial_M$,
\begin{align}
 &\norm{f_L-c_*\mathbf1_{(1/L,1)}}_{L^2(\dd M/M)}=O(1),
 &&c_*\ne0,\label{D:plateau-main}\\
 &\norm{\mathscr D^a f_L}_{L^2(\dd M/M)}+
 \norm{\mathscr D^a u_L}_{L^2(\dd M/M)}=O(1)
 &&(a=1,2),\label{D:plateau-derivatives}\\
 &\norm{\kappa\Phi(\partial_i g_L)}=O(1),\qquad
 \norm{f_L/M}_{L^2(M\,\dd M)}=O(\sqrt{\log L}).
 \label{D:plateau-norms}
\end{align}
For $c=s_z,j_z$, in the positive $\lambda$ representation,
\begin{equation}\label{D:plateau-tail}
 \lambda\mathcal U(c f_L)
 =c_*\mathbf1_{(1,L)}(\lambda)\lambda^{-1}d_c+E_{c,L},
 \qquad \norm{E_{c,L}}_{L^2(\lambda\,\dd\lambda;\mathcal K)}=O(1).
\end{equation}
\end{lemma}

\begin{proof}
The kernels computed after \eqref{D:radial-identity}, with
$t=M|z'|$, give explicitly in our transverse Fourier convention
\begin{align}
 f_L(M)&=6\int_0^\infty t^3K_0(t)
     \eta_L(\log t-\log M)\,\dd t,\label{D:f-convolution}\\
 u_L(M)&=-4i\int_0^\infty t^2K_1(t)
     (2\eta_L+\eta_L')(\log t-\log M)\,\dd t.
 \label{D:u-convolution}
\end{align}
Changing the Fourier orientation changes the fixed phase in the
second formula only.  The first has
\[
 c_*=6\int_0^\infty t^3K_0(t)\,\dd t=24
\]
with the convention fixed above.  In particular it cannot vanish.
On the logarithmic variable these are convolutions with integrable
kernels having finite first moment.  At $t=0$ their densities
decay exponentially in $|\log t|$, and at infinity they decay
faster; the relevant logarithmic derivatives have the same
integrability.  Convolution of the interval plateau differs from
its kernel integral times that plateau by a uniformly bounded
$L^2$ error: the difference is supported near the two edges up
to integrable tails.  Positive derivatives act on these two
fixed-width edges.  This proves
\eqref{D:plateau-main}--\eqref{D:plateau-derivatives}.
The functions themselves and their derivatives have integrable
tails at zero and infinity for each fixed $L$.

Direct differentiation gives
\[
 h_1(Mu)=-M^{-1}(\mathscr D^2+2\mathscr D)u,
 \qquad
 \norm{h_1(Mu)}^2=
 \int_0^\infty|\mathscr D^2u+2\mathscr Du|^2\frac{\dd M}{M}.
\]
The order-one comparison proves the first estimate in
\eqref{D:plateau-norms}; the second is
\eqref{D:plateau-main}.

To prove \eqref{D:plateau-tail} without assigning a divergent
form norm to an individual nonzero-moment test, fix a smooth
annular $F$ with $\int vF(v)\,\dd v=1$ and put $G=\mathcal H_0F$.
Replace $f_L$ by
\[
 c_*\mathcal H_0\bigl(F-L^{-2}F(\cdot/L)\bigr)(M)
 =c_*\bigl(G(M)-G(LM)\bigr).
\]
Since $G(0)=1$ and $G$ decreases rapidly at infinity, this
difference has the same plateau and the same uniform edge
estimates as $f_L$.  Its error $e_L$ satisfies
\[
 \norm{\sqrt{h_2}e_L}^2
 =\int_0^\infty
 (|\mathscr De_L|^2+4|e_L|^2)\frac{\dd M}{M}=O(1).
\]
The comparison \eqref{D:inherited-comparison} gives an $O(1)$
error after multiplication by $\lambda$.  Above a fixed radius,
the $G$ term has the exact tail of Lemma~\ref{D:exact-tail}.
The norm of $G(LM)$ in $L^2(M\,\dd M)$ is $L^{-1}\norm G$;
hence its $\lambda$-weighted norm below $CL$ is $O(1)$ for
any fixed $C$.  Above both Hankel support radii the two exact
tails cancel, because their moments are both one.  The remaining
fixed lower interval and fixed-ratio upper interval contribute
$O(1)$ to the weighted norm.  This proves
\eqref{D:plateau-tail}.  The $\lambda=0$ spectral subspace
contributes nothing to that estimate.

Using the difference of the two dilated tests is essential here:
their individual $\lambda^{-2}$ tails need not have finite
$\lambda$-weighted norm, but the exact cancellation above the larger
support radius makes their difference admissible.  No estimate has
been assigned to either divergent weighted tail separately.
\end{proof}

\Needspace{6\baselineskip}
\begin{proposition}[Order-zero scalar comparison]
\label{D:critical-comparison}
The quadratic moment $\kappa$ of the physical bounded-profile
generators satisfies
\begin{equation}\label{D:order-zero}
 \mathbf1_{(s,\infty)}(\kappa)
 \mathbf1_{[0,s]}(h_0)\big|_l=0\qquad(s\geq0).
\end{equation}
\end{proposition}

\begin{proof}
On the positive $\lambda$ spectrum define
\[
 A_L(\lambda)=\lambda^{-2}\mu(g_L),\qquad
 B_{i,L}(\lambda)=\lambda^{-1}\mu(\partial_i g_L).
\]
The moments in \eqref{D:measure-input} imply
$0\leq A_L\leq\id$ and $\norm{B_{i,L}}\leq C$ uniformly:
$|\partial_i g_L(z')|\leq C|z'|$, and positive-measure
Cauchy--Schwarz bounds its integral by $C\sqrt\kappa$.
Scale covariance identifies the law of $z'/\lambda$ in each
fiber with one fixed positive measure on $\mathcal K$.  Its
mean is zero and its trace-second moment is $\id$.  Hence,
on every fixed multiplicity vector,
\begin{equation}\label{D:bulk}
 A_L(\lambda)\longrightarrow\id,\qquad
 B_{i,L}(\lambda)\longrightarrow0
 \quad\text{if }\lambda\longrightarrow\infty,
 \quad L/\lambda\longrightarrow\infty.
\end{equation}
For the first limit use the integrable weight $|z'/\lambda|^2$;
for the second use $|z'/\lambda|$ and the zero first moment.
The limiting derivative integrand is $2z'_i/\lambda$.
Bounded convergence for the positive measure proves strong
convergence; no uniform rate is required.

These are statements about the fixed multiplicity vectors that enter
the tail formula.  Positivity and the second moment control the
truncated integrands; projection-valuedness of $\mu$ is unnecessary.
On $\ker\kappa$ the second moment is zero, so $\mu$ is concentrated at
$z'=0$ there.  The profiles $g_L$ and $\partial_i g_L$ vanish near that
point, and hence this spectral subspace makes no contribution to the
pairing.

Pair \eqref{D:radial-identity} with $f_L(M_o)/M_o$ and use
\eqref{D:plateau-waves}.  This is allowed although the identity
was first tested on compact mass intervals.  Cut off this test
smoothly at zero and infinity.  The approximants converge both
in its $L^2(M_o\,\dd M_o)$ norm and after multiplication by
$M_o$.  For fixed $L$, the profile operators are bounded, and
\eqref{D:plateau-norms} controls the derivative wave, so every
pairing has a limit.

The left side becomes
\begin{align*}
 \ip{j_zf_L}{\mu(g_L)s_zf_L}
 &=\ip{\lambda\mathcal U(j_zf_L)}
 {A_L\lambda\mathcal U(s_zf_L)}\\
 &=|c_*|^2\int_1^L
 \ip{d_{j_z}}{A_L(\lambda)d_{s_z}}\frac{\dd\lambda}{\lambda}
 +O(\sqrt{\log L}).
\end{align*}
The exact normalization $|c_*|^2$ is common to the bra and ket.
Equation~\eqref{D:bulk} and boundedness imply, by averaging on
the logarithmic interval, that this is
\begin{equation}\label{D:leading-pairing}
 |c_*|^2\log L\,\ip{d_{j_z}}{d_{s_z}}+o(\log L).
\end{equation}
For example, discard two logarithmic edge intervals of fixed
width, apply the strong bulk convergence on the remaining
interval, and then let the discarded width grow.  This proves
the asserted Cesaro limit directly.

For the first term on the right of \eqref{D:radial-identity},
\eqref{D:bulk}, \eqref{D:plateau-tail} and uniform boundedness give
\[
 \norm{\mu(\partial_i g_L)s_zf_L}
 =\norm{B_{i,L}\lambda\mathcal U(s_zf_L)}
 =o(\sqrt{\log L}).
\]
The output test $j_if_L/M_o$ has norm $O(\sqrt{\log L})$.
Their pairing is therefore $o(\log L)$.  For the last term,
$0\leq\mu(g_L)\leq\kappa$ and commutativity give
\[
 \norm{\mu(g_L)\Phi(\partial_i g_L)}
 \leq\norm{\kappa\Phi(\partial_i g_L)}=O(1).
\]
Its pairing is $O(\sqrt{\log L})$.
Comparison with \eqref{D:leading-pairing} forces
\[
 0=\ip{d_{j_z}}{d_{s_z}}=\tfrac13\norm{d_l}^2,
\]
by \eqref{D:tail-relations}.  Thus $d_l=0$.
If either of the spin labels used above is zero, the same
conclusion already follows from \eqref{D:tail-relations}.

Lemma~\ref{D:exact-tail} now says that $l\mathcal H_0F$ has no
$\lambda$ spectrum above $R$ whenever $F$ is supported below
$R$.  Smooth compact tests in $(0,R)$ are dense in
$L^2((0,R),v\,\dd v)$.  Hankel unitarity and closedness of
spectral subspaces give \eqref{D:order-zero} with $s=R^2$;
at $s=0$ the regular $h_0$ copy has no zero eigenvector.
The result concerns the moment $\kappa$.  It does not identify
the full parabolic operator $h$ with that moment.
\end{proof}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/04-dispersion.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/05-charges-cones.tex ====
\section{Single insertions, quadratic charges, and time cones}
\label{sec:charges-cones}

Throughout this Section, $l$ is the dimension-two vacuum wave constructed in
Section~\ref{sec:foundations}. The letter $L$ will initially denote only an
insertion in a sesquilinear Wightman form. Neither an operator $L(f)$ nor a
product containing two copies of $L$ is being asserted. This distinction is
essential when conserved currents are used below.

The section proceeds through three distinct constructions.  We first
extend the extreme vacuum values of $l$ to local forms with one
insertion.  Their conserved currents then define charge forms, whose
quadratic part is identified with the positive moment $K_n$ on a
specified domain.  Finally the vanishing residual dilation charge
identifies the actual cone modular groups and produces the actual
time-M\"obius action used for scalar extraction.

\subsection{From scalar localization to one local insertion}

The order-zero spectral comparison has a localization interpretation.
There are two transports: from the scalar comparison to the moment
$\kappa$, and from that moment to a physical bounded cut. We state both,
together with the reverse containment for actual parabolic cuts that
will be needed later.

\begin{lemma}[Localization transport]\label{C:localization-transport}
Let $X_0(W)$ be the scalar wedge real space in the $l$ channel,
$\iota$ its embedding into the physical Hilbert space, and $X(W)$ the
physical wedge real space. For $p(z)=c|z-d|^2+c_0$, $c>0$, put
\[
 K_{0,p}=b\{c(|Y-d|^2+h_0)+c_0\},\qquad
 K_p=b\{c(|Y-d|^2+\kappa)+c_0\}.
\]
Then
\begin{equation}\label{C:moment-localization}
 \iota e^{iK_{0,p}}X_0(W)\subset e^{iK_p}X(W).
\end{equation}
If $g$ is bounded and smooth and $g\le p$, the left side also lies in
$X(W_g)=e^{iH_g}X(W)$. Conversely, the actual parabolic real space
satisfies
\begin{equation}\label{C:actual-parabolic-localization}
 X(W_p)\subset e^{iK_p}X(W).
\end{equation}
These statements hold after translations of the null plane and reflection.
\end{lemma}

\begin{proof}
The scalar wedge conjugation is the real scalar conjugation; it is
identified already on the dense waves generated by $M^2l$. To prove
\eqref{C:moment-localization}, cancel the common external factors and
consider $e^{-icb\kappa}\iota e^{icbh_0}$. Let $E_i$ be the
actual $\kappa$ step projections and $P_j$ the reference $h_0$ step
projections. Clip both spectral variables at one common finite bound,
then approximate the clipped
$\kappa$ from below and the clipped $h_0$ from above.  If their step
values are $a_i,c_j$, spectral inclusion gives
$E_i\iota P_j=0$ whenever $a_i>c_j$.  Each surviving ordered term is
therefore $E_i\iota P_j e^{icb(c_j-a_i)}$, with nonnegative
translation parameter; no joint spectral resolution is used.
The coefficients intertwine the boost and wedge conjugation, so
they map the reference real wedge space into the actual one.  The
positive translations preserve that actual space.  At the common
clipping endpoint both step values coincide, which preserves the
same sign.  Strong convergence follows from the separate spectral
calculi as the mesh tends to zero and the clipping bound tends to
infinity. Closedness of $X(W)$ proves the inclusion. This uses the
ordered products of two separate spectral resolutions, not a joint
resolution of $\kappa$ and $h_0$.

For the second transport, positivity, normalization, and the zero first
moment of $\mu$ give
\[
 H_g=b\int g(Y+z)\mu(\dd z)
 \le b\{c(|Y-d|^2+\kappa)+c_0\}=K_p.
\]
This inequality is in the common commuting calculus. Its difference
is $b$ times a nonnegative boost-independent operator commuting with
wedge conjugation. Thus $e^{i(K_p-H_g)}$ is a direct integral of
inward null translations and preserves $X(W)$. Apply it to
\eqref{C:moment-localization} to obtain membership in $X(W_g)$.

Finally the actual parabolic generator is
$H_p=b\{c(|Y-d|^2+h)+c_0\}$. Since $h$ and $\kappa$ commute and
$\kappa\le h$, the same argument applies to
$H_p-K_p=bc(h-\kappa)\ge0$ and proves
\eqref{C:actual-parabolic-localization}. The operators vanish on
$b=0$. Covariance and reflected agreement give the remaining placements.
No equality of $h$ with $\kappa$ is used.
\end{proof}

Here is the resulting extreme-ordering statement. First let
$A_i(y_i)$ be homogeneous original physical fields. The two
expressions with $L(x)$ at the extreme left or
extreme right are defined by pairing the wave $l$ with a physical
polynomial vector. They agree whenever $x$ is spacelike to all $y_i$ and,
for some future null $n$,
\begin{equation}\label{C:extreme-patch}
 n\cdot x<\min_i n\cdot y_i
 \quad\hbox{or}\quad
 n\cdot x>\max_i n\cdot y_i.
\end{equation}
For the first alternative, place a null plane between $x$ and the
$y_i$ and use the future shadow of a tip just before $x$ as a dominating
parabola. The past shadows of the $y_i$ lie strictly below it by
spacelikeness. Their differences from it tend quadratically to
$-\infty$ at transverse infinity, so a bounded smooth truncation from
below still separates the supports. These strict inequalities persist
in small neighborhoods. Locality of bounded operations in that cut,
followed by Tomita adjunction of the whole physical product, proves the
equality. The opposite alternative uses the reflected construction.
The scalar insertion is even, so no additional graded sign occurs.
For a permitted compact joint kernel, the same statement follows by
the tensor-test approximation and closed Tomita-graph limit in the
discussion of \eqref{F:BW-sandwich}; it does not require an affiliated
realization for the joint-smeared polynomial itself.

The remaining task is to place this insertion between arbitrary
physical factors while retaining their ordered spectra. Causal solutions
at adjacent interchanges will have the required wave source, but their
compatibility leaves a homogeneous-wave defect. The next lemma recovers
such a defect from its support and spectator analyticity. We then apply
it to assemble the insertion with all its ordering cones.

\Needspace{7\baselineskip}
\begin{lemma}[Recovery of a homogeneous wave defect]
\label{C:homogeneous-defect}
Let $H(x,y_1,\ldots,y_m)$, $m\ge1$, be a tempered, jointly
translation-invariant distribution with $\Box_xH=0$ and
\begin{equation}\label{C:H-support}
 \supp_x H\subset C_y\cup K_y,\qquad
 C_y=\bigcup_iJ(y_i),\qquad
 K_y=\bigcap_{n\ {\mathrm{future\ null}}}
 \{x:\min_i n\cdot y_i\le n\cdot x\le\max_i n\cdot y_i\}.
\end{equation}
For a compact transfer-momentum test $\varphi(k)$, let $H_\varphi(y)$
be its pairing with the partial Fourier transform in $x$, using
$e^{-ikx}$. Suppose these pairings are holomorphic in the ordered
spectator tube and, for $\supp\varphi\subset\{|k|\le K\}$, $0<K\le1$,
obey
\begin{equation}\label{C:shifted-test-bound}
 |H_\varphi(y)|\le C p_N(\varphi)
 (1+|y|+\delta(y)^{-1})^N
 \exp(CK|\operatorname{Im}y|).
\end{equation}
Here $p_N$ is one fixed finite Schwartz seminorm and $\delta(y)$ is the minimum
distance of the successive imaginary gaps from the cone boundary
($\delta=1$ for $m=1$). Then $H$ is a finite sum
\[
 H(x,y)=\sum_{i,P}C_{iP}(y)P(\partial_x)\Delta_0(x-y_i),
\]
where the coefficients have tempered boundaries with the ordinary
ordered spectator spectra. They are invariant under common spectator
translation. The polynomials $P$ may be chosen independently modulo
$k^2$. We normalize the massless commutator solution by
$\Delta_0(0,\boldsymbol r)=0$ and
$\partial_0\Delta_0(0,\boldsymbol r)=\delta(\boldsymbol r)$.
\end{lemma}

\begin{proof}
Normalizing $n=(1,\omega)$ and taking $\omega=\pm e_a$ shows that
$K_y$ is compact, uniformly on compact spectator sets. We first remove
its part outside $C_y$, then recover the coefficients of the remaining
wave solutions with bounds that preserve the unshifted spectator spectra.

We give the support argument explicitly. Choose an open spectator
patch around distinct equal-time spatial extremal vertices, so that
each $y_i$ is a strict maximum and a strict minimum for open sets of
future null directions. If $x\in K_y\cap J^+(y_i)$, the maximizing
directions give $0\le n\cdot(x-y_i)\le0$. A causal vector orthogonal
to an open set of null directions is zero. The minimizing directions
treat $J^-(y_i)$. Hence
\begin{equation}\label{C:vertex-intersection}
 K_y\cap C_y=\{y_1,\ldots,y_m\}
\end{equation}
on a smaller patch. Fix a test neighborhood $U$ spacelike to all the
vertices, and choose small vertex neighborhoods $N_i$ whose closures
remain spacelike to $U$. There is a compactly supported smooth
$\chi(x)$ equal to one near $K_y$ whose derivatives meet $C_y$ only
in $N=\bigcup_iN_i$. Positive separation margins make one such cutoff
work uniformly after shrinking the spectator patch. A compact
spectator cutoff causes no change to $\Box_x$. Thus $v=\chi H$ has
compact $x$-support and
\[
 \Box_xv=[\Box_x,\chi]H=:f,\qquad \supp_xf\subset N.
\]
For compactly supported distributions the exact identity
$v=E_{\rm ret}*\Box_xv$ follows by moving $\Box$ through convolution.
Its right side vanishes on $U$, since $J^+(N)\cap U=\varnothing$.
Near $K_y$ one has $v=H$; elsewhere the original support statement
already suffices. This removes $K_y\setminus C_y$ from
\eqref{C:H-support}, without any decomposition of distributions on
touching closed sets.

In a common thin time slab the cone components are disjoint.
A spatial partition separates them with derivatives off their
supports. Each component still solves the wave equation there.
In $r=x-y_i$ coordinates its displacement and normal-derivative data
on $r^0=0$ are supported at $\boldsymbol r=0$. Distributional
solutions of the wave equation have distributional Cauchy data on
this non-characteristic slice. The point-support structure theorem
therefore writes both data as finite sums of spatial delta
derivatives with spectator-distribution coefficients. On a smaller
relatively compact patch their orders are uniformly bounded.
With $\Delta_0(0,\boldsymbol r)=0$ and
$\partial_0\Delta_0(0,\boldsymbol r)=\delta(\boldsymbol r)$,
their evolutions are derivatives of $\Delta_0$. Cauchy uniqueness
first in the slab and then globally in $x$ gives
\begin{equation}\label{C:vertex-wave-expansion}
 H(x,y)=\sum_{i,P}C_{iP}(y)P(\partial_x)\Delta_0(x-y_i).
\end{equation}
Choose the finitely many polynomials $P$ independently modulo $k^2$.
Enlarging the finite family if necessary, use homogeneous bases
$P_{q\alpha}$ of the quotient by $k^2$ in each degree
$q=0,\ldots,N$.

We next justify continuation of the coefficients, including its
growth bound. This is needed to recover the unshifted spectator
spectra. Smear the Fourier transform of
$P(\partial_x)\Delta_0(x-y_i)$ against compact $k$-tests. Its column
is, up to constant factors,
\[
 \big\langle P(k)\widehat\Delta_0(k)e^{ik y_i},\varphi(k)\big\rangle.
\]
All augmented minors expressing that the corresponding tests of
$H$ lie in the span of these columns vanish on the initial patch.
They are holomorphic in the ordered spectator tube; boundary
uniqueness makes them vanish throughout that tube. The columns have
full rank at every interior point, with a quantitative bound as
follows. Choose finitely many fixed positive null vectors
$n_r=(1,\omega_r)$ such that, for every $q\le N$, the evaluation
matrix
\[
 (E_q)_{r\alpha}=P_{q\alpha}(n_r)
\]
is injective. Such a choice exists: a homogeneous polynomial
vanishing on every positive null ray is zero modulo $k^2$, and
finite dimensionality permits a finite separating set of
evaluations. Taking the union of these sets for $q\le N$ gives
one collection of rays. Fix a left inverse for each $E_q$.

On the $r$th ray the polynomial-exponential column sum is
\[
 \sum_{i=1}^m\sum_{q=0}^N
 a_{iqr}s^qe^{is n_r\cdot y_i},
 \qquad
 a_{iqr}=\sum_\alpha C_{i,q\alpha}(y)P_{q\alpha}(n_r).
\]
Let $d=m(N+1)$, $\varepsilon=c/(1+|y|)$, and use the
$d$ samples $s=t\varepsilon$, $t=1,\ldots,d$. The matrix
from the coefficients $a_{iqr}$ to these samples is
\[
 (V_r)_{t,(i,q)}=(t\varepsilon)^qz_{ir}^{\,t},
 \qquad z_{ir}=e^{i\varepsilon n_r\cdot y_i}.
\]
Its determinant is the confluent Vandermonde determinant
\begin{equation}\label{C:Vandermonde}
 \det V_r=c_{m,N}\varepsilon^{mN(N+1)/2}
 \prod_i z_{ir}^{(N+1)(N+2)/2}
 \prod_{i<j}(z_{jr}-z_{ir})^{(N+1)^2},
 \qquad c_{m,N}=\Bigl(\prod_{q=0}^Nq!\Bigr)^m.
\end{equation}
For example this follows by replacing powers of $t$ by falling
factorials and using the derivative columns of the ordinary
Vandermonde matrix. Thus the ray inversion distinguishes the
degree $q$ and center $i$ before the fixed left inverses of
$E_q$ recover each coefficient $C_{i,q\alpha}$. This describes
a left inverse for the full rectangular multi-ray matrix;
independence on each ray alone would not suffice.

The scale of the samples is also significant.  Keeping them at fixed
nonzero momenta would preserve the exponential type of the shifted
spectator cones.  Taking $s=O((1+|y|)^{-1})$ instead keeps that
exponential bounded.  The determinant estimate ensures that this
shrinking choice costs only polynomial powers, which are compatible
with tempered spectral boundary values.

For the tube distance $\delta(y)$ in the statement, the moduli
of the $z_{ir}$ are bounded
above and below, and for $i\ne j$
\[
 |z_{ir}-z_{jr}|\ge\big||z_{ir}|-|z_{jr}|\big|
 \ge c'\varepsilon\delta(y)\quad(i\ne j),
\]
because the imaginary gaps are strictly ordered and the directions
are fixed. The determinant formula and cofactor bounds therefore
give the full left inverse polynomial norm in
$\varepsilon^{-1}$ and $1+\delta(y)^{-1}$. To turn these point
samples of the known columns into legitimate compact $k$-tests,
replace each sample $t\varepsilon n_r$ by a smooth bump of width $w$,
normalized to have integral one on the positive null shell. The centers
are a distance comparable to $\varepsilon$ from its vertex. For
$w\le c_1\varepsilon$, its shell Jacobian and the prescribed test
derivatives therefore give
\[
 p_N(\varphi)\le C\varepsilon^{-a}w^{-b}
\]
for fixed finite $a,b$. Derivatives of the column functions
$P(k)e^{ik\cdot y}$ cost polynomial powers of $1+|y|$ and
$\varepsilon^{-1}$, while their exponential is bounded on these
supports. If $A_0$ is the point-sample matrix and $L_0A_0=1$,
choose
\[
 w=c_2\varepsilon(1+|y|+\delta(y)^{-1})^{-J}
\]
with fixed $J$ sufficiently large and $c_2$ sufficiently small. The
resulting test-column matrix $A$ then satisfies
$\|L_0(A-A_0)\|<1/2$. Its left inverse
$(L_0A)^{-1}L_0$ has the same polynomial bound, and the displayed
seminorm cost remains polynomial.

This proves full rank using genuine test columns. On a neighborhood
of any fixed interior point, keep a full-rank finite set of tests
fixed; its inverse and the vanishing augmented minors define
holomorphic coefficients. Uniqueness identifies them on overlaps.
The parameter-dependent test choices above serve only to estimate
these already defined holomorphic functions. Their supports satisfy
$|k|=O(\varepsilon)$ and their seminorms have polynomial cost.
Applying \eqref{C:shifted-test-bound} with $K=O(\varepsilon)$
therefore costs only polynomial powers: the remaining exponential
is bounded by $\exp(O(\varepsilon|\operatorname{Im}y|))=O(1)$.
The left inverse consequently gives
\begin{equation}\label{C:coefficient-bound}
 |C_{iP}(y)|\le C(1+|y|)^N(1+\delta(y)^{-1})^N
\end{equation}
for some $C,N$ on the tube. The coefficients have ordinary tempered
ordered spectral boundaries. Testing all compact $k$-sets, including
a neighborhood of zero, proves \eqref{C:vertex-wave-expansion}
globally. Translation covariance also shows that the coefficients
are invariant under common spectator translation.

\end{proof}

The construction is first carried out for a fixed list of homogeneous
original entries.  We will then extend it by finite multilinearity in
their scaling components.  Descent to polynomial vectors will be
checked for the whole summed bra or ket, so cancellations between
different homogeneous terms remain allowed.

\Needspace{6\baselineskip}
\begin{proposition}[One-insertion forms]\label{C:single-insertion}
There are unique tempered, Hermitian, Poincar\'e- and scale-covariant
Wightman forms with exactly one insertion $L$ and arbitrarily many
physical insertions, whose extreme vacuum values are those of $l$ and
which satisfy $\Box L=\Theta$. They have the ordinary ordered spectra,
are local under every allowed adjacent interchange, and depend only on
the physical polynomial vectors on the two sides of $L$.
\end{proposition}

\begin{proof}
\emph{Causal source solutions.}
The empty physical product has only the specified zero vacuum
expectation, since $l$ has no zero-momentum component. Suppose
$m\ge1$ and fix an order of homogeneous original entries
$A_i\in\mathcal F_{\Delta_i}$, $i=1,\ldots,m$.
Use the Fourier convention
$e^{-ikx-i\sum_i p_i y_i}$, and put $q_h=\sum_{i\le h}p_i$, with
$q_0=0$. For an insertion just after $A_j$, the required spectral
conditions are
\begin{equation}\label{C:insertion-spectra}
 q_h\in\overline V_+\quad(1\le h\le j),
 \qquad k+q_h\in\overline V_+\quad(j\le h\le m-1),
 \qquad k+q_m=0.
\end{equation}
Redundant zero partial sums at the endpoints may be included. Write
$E_R,E_L$ for the known extreme values. For each $i$, take the
difference of the physical distributions with $\Theta$ immediately
before and immediately after $A_i$. Its support, in $r=x-y_i$, is
causal. Split it into forward and backward parts by a finite-order
Taylor subtraction in $r$ alone, and apply respectively the retarded
and advanced wave fundamental solutions. This gives a tempered $B_i$
with the same causal support and the prescribed wave source. The
convolutions are proper on the respective cones: two future causal
vectors, or two past causal vectors, are bounded on the inverse image
of a bounded set under addition. The same cone geometry gives the
polynomial bounds needed for tempered convolution.

This construction preserves precisely the common spectral cones
\begin{equation}\label{C:merged-spectra}
 q_h\in\overline V_+\ (h<i),\qquad
 k+q_h\in\overline V_+\ (h\ge i).
\end{equation}
Indeed in the coordinates $(r,y_1,\ldots,y_m)$ the spectator momenta
are $p_j$ for $j\ne i$ and $k+p_i$ for $j=i$.
All the conditions in \eqref{C:merged-spectra} involve only those
spectator momenta. Multiplication, subtraction, and convolution in $r$
do not alter them. This is why solving the wave equation separately
in the external momentum before doing causal splitting would be
insufficient.

\emph{Removing the homogeneous defect.}
Set
\[
 R_j=E_R+\sum_{i>j}B_i,\qquad
 L_j=E_L-\sum_{i\le j}B_i.
\]
Both have the wave source appropriate to the $j$th position. The first
has all the initial cones in \eqref{C:insertion-spectra}; the second
has all its final cones. Their difference
$H_*=R_j-L_j$ is independent of $j$ and satisfies $\Box_xH_*=0$.
After a compact Fourier test in $k$, its spectator partial sums lie in
$\overline V_+$ up to bounded shifts of size $O(\sup|k|)$.
It consequently has an analytic continuation to the ordinary ordered
spectator tube, with polynomial bounds multiplied by an exponential
of type $O(\sup|k|)$. Equation~\eqref{C:extreme-patch} and the causal
support of the $B_i$ give
the support condition $\supp_xH_*\subset C_y\cup K_y$ of
\eqref{C:H-support}.

Its shifted spectator cones also give
\eqref{C:shifted-test-bound}: for a transfer test supported in
$|k|\le K$, every partial cone is shifted by at most a fixed multiple
of $K$. Fourier--Laplace damping contributes
$\exp(CK|\operatorname{Im}y|)$; the finite-order tempered bound on the
joint distribution controls the remaining terms by one fixed seminorm
$p_N(\varphi)$ and polynomial powers of $|y|$ and the inverse tube
distance. This estimate is uniform in the test, including when its
support shrinks toward $k=0$. Lemma~\ref{C:homogeneous-defect} therefore
gives the expansion \eqref{C:vertex-wave-expansion} for $H_*$.

Let $H_i$ denote the summand centered at $y_i$. In Fourier variables
it shifts $p_i$ by $k$, and no other spectator momentum. Consequently
it satisfies exactly the common cones
\eqref{C:merged-spectra}. The required distributions are
\begin{equation}\label{C:Wj}
 W_j=R_j-\sum_{i>j}H_i
     =L_j+\sum_{i\le j}H_i.
\end{equation}
The first expression supplies the initial cones and the second the
final cones; $W_m=E_R$, $W_0=E_L$, and
$W_j-W_{j-1}=-B_j+H_j$ is supported in $J(y_j)$.

\emph{Locality and descent to polynomial vectors.}
For an adjacent interchange of physical fields in the suffix,
smear all suffix variables with any compact test supported where
that pair is spacelike. Keep this test fixed. The two orderings
have the same partial spectral cones in the prefix and $x$:
$q_h\in\overline V_+$ for $h\le j$ and
$q_j+k\in\overline V_+$. Smearing the suffix does not change
those cones. Their difference is therefore the boundary of a
holomorphic function in the ordered tube of the $j+1$ variables
$y_1,\ldots,y_j,x$, with the last imaginary difference measured
from the real boundary. On the nonempty open set where $x$ is
spacelike to every prefix point, the already proved adjacent
$L$ interchanges move it to the extreme left. At that extreme,
physical locality makes the fixed suffix swap zero, regardless
of the prefix. Boundary uniqueness in this partial tube makes
the difference zero for all prefix and $x$ variables. Since the
fixed suffix test was arbitrary on the spacelike-pair patch,
this proves the required distributional locality. For a swap
in the prefix, hold its test fixed instead and move $L$ to the
extreme right. The common final cones, equivalently the negative
reversed partial sums by total momentum conservation, give the
opposite ordered tube in $x$ and the suffix variables. The same
argument applies.

For a fixed homogeneous list, locality moves $L$ to an extreme
position on a spacelike open set.  Ordered tube uniqueness therefore
shows that the specified extreme values determine the form uniquely.
In particular, the result is multilinear in the homogeneous labels
and is covariant, independently of the auxiliary causal splittings.
For arbitrary original entries write
\[
 A_i=\sum_{\Delta\in S_i}A_{i,\Delta},\qquad |S_i|<\infty,
\]
and take the finite sum of the constructed forms over
$S_1\times\cdots\times S_m$, using the same joint test kernel in
every term.  The common ordered spectral cones, locality, wave
equation, and extreme pairings survive this finite sum.  Its dilation
covariance is the termwise action on the labels, not homogeneity of
one total degree.  The distribution orders and constants may depend
on this fixed finite list.

The vector-dependence assertion uses exactly the preceding choice of
which variables to hold fixed, now for the whole finite summed form.
If a fixed ket polynomial $A\Omega$ is zero, even by cancellation
between component terms or word lengths, vary only the opposite
prefix and $x$.  Once the entire ket has been smeared and held fixed,
the finite sum has the same prefix-and-$x$ tube, regardless of the
word lengths on that fixed side.
On the open set where the $L$ interchanges through that prefix
are spacelike, the extreme-left value pairs $l$ with
$C^*A\Omega=0$. The partial-tube argument just given makes
the insertion zero everywhere. It never requires a null
polynomial vector to remain zero under independent translations
of its individual factors. If the bra polynomial is zero,
hold it fixed and use the extreme right and the opposite tube.
For a permitted joint kernel, distributional continuity first
supplies the summed form and its common partial-tube boundary for
that fixed whole kernel.  The zero extreme pairing is then used for
the whole kernel, not for its approximants.  Hermiticity and the
stated covariance follow from uniqueness of the whole form.
This proves
a statement about forms with one $L$, without a positivity
assertion for repeated insertions.

In particular, the construction may now be differentiated in the
$L$ slot and paired with arbitrary physical polynomial vectors.
These are precisely the operations needed to form the conserved
currents below; none requires composing two $L$ insertions.
\end{proof}

\subsection{Conserved charge forms and their vacuum flux}

Use Proposition~\ref{C:single-insertion} to define the conserved
one-insertion forms
\begin{equation}\label{C:formal-currents}
 t_{\mu\nu}=T_{\mu\nu}
 +\frac13(\partial_\mu\partial_\nu-\eta_{\mu\nu}\Box)L,
 \qquad J_\mu=V_\mu-\partial_\mu L.
\end{equation}
Thus $t$ is symmetric and traceless and $\partial^\mu J_\mu=0$.
For a conformal Killing vector $X$, $X^\nu t_{\mu\nu}$ is conserved.
Its commutator flux through a compact Cauchy slab around a compact
physical polynomial $A$, paired with $C\Omega$, defines a charge
form $q_X(C,A)$. The cutoff equals a time step on the relevant causal
support. Locality and conservation permit compact deformations of
this slab. To move the charge from ket to bra requires an additional
statement: a complete vacuum flux through a physical product must
vanish even when $X$ is quadratic.

The compact commutator flux and this complete vacuum flux have
different roles.  Conservation and locality make the former
independent of a compact deformation.  The latter is a limit over
increasing scales and requires an infrared estimate.  We establish
that estimate uniformly in the mass band so that the bands can
subsequently be removed.

\Needspace{6\baselineskip}
\begin{lemma}[Low-energy estimate]\label{C:low-energy}
Let $Z$ be an even Hermitian compact polynomial vacuum vector, smooth
under translations. For an invariant-mass spectral band
$I\Subset(0,\infty)$ and $\Lambda>0$,
\begin{equation}\label{C:low-energy-bound}
 \|1_{P^0<\Lambda}E_I Z\|
 \le C_Z\Lambda\,\langle Z,P^0E_I Z\rangle^{1/2},
\end{equation}
where the constant is independent of $I$ and $\Lambda$.
\end{lemma}

\begin{proof}
The mass projection commutes with the wedge modular data and hence
preserves all wedge real spaces containing $Z$. Wedge separation
therefore makes the self-commutator function of $E_IZ$ vanish at
bounded times and sufficiently large spatial separation, with the
spatial bound independent of $I$. Its time derivative at zero has
compact spatial support and is bounded in absolute value by
$2\langle Z,P^0E_IZ\rangle$. If $\mu_I$ is the positive translation
spectral measure of $E_IZ$, the spatial Fourier transform of this
derivative, after division by its fixed imaginary factor, is the
sum of the positive marginal measures of $p^0d\mu_I(p)$ at
$\boldsymbol p$ and $-\boldsymbol p$. Fourier transformation of the
compactly supported bounded function bounds each of these positive
measures by
$C_Z\langle Z,P^0E_IZ\rangle d^3\boldsymbol p$.
On the positive-energy cone $p^0\ge|\boldsymbol p|$. Hence
\[
 \int_{p^0<\Lambda}d\mu_I
 \le C_Z\langle Z,P^0E_IZ\rangle
 \int_{|\boldsymbol p|<\Lambda}\frac{d^3\boldsymbol p}{|\boldsymbol p|}
 \le C'_Z\Lambda^2\langle Z,P^0E_IZ\rangle.
\]
The integrable singularity at $\boldsymbol p=0$ causes no problem;
the positive mass band has no atom there. Taking square roots proves
the estimate.
\end{proof}

A fully scaled smooth vacuum flux for a quadratic $X$ is of the
form $R U_D(\log R)F$, where $F$ is a fixed purely timelike,
translation-smooth wave. For a compact mass band of $F$, the paired
part of $Z$ lies in $I_R=[c/R,C/R]$. Equation
\eqref{C:low-energy-bound}, with $\Lambda=N/R$, gives a bound
proportional to
$N\langle Z,P^0E_{I_R}Z\rangle^{1/2}$, which tends to zero.
To remove the compact-energy and compact-mass restrictions, write
$F_N=1_{N<P^0<2N}F$ for dyadic $N>0$. The same argument without
a mass restriction bounds the pairing on this dyad by
$C_ZN\|F_N\|\langle Z,P^0Z\rangle^{1/2}$, uniformly in $R$.
The stress-tensor vacuum amplitude is $O((p^0)^2)$ and the
four-momentum measure of a dyadic cone segment is $O(N^4)$.
Thus $\|F_N\|=O(N^4)$ at zero. At infinity the smooth flux
test gives arbitrarily rapid decrease. Hence
$\sum_N N\|F_N\|<\infty$. On a fixed dyad, compact timelike
mass approximation has arbitrarily small norm, and its complement
is controlled by the same uniform bound. Dominated convergence
now proves the full flux assertion. Lower-degree $X$ have a smaller scaling
factor. Polarization into even Hermitian tests proves that every
complete vacuum flux under consideration vanishes.

The left and right charge prescriptions consequently agree and
$q_X$ is Hermitian. For an affine Killing field the improvement
changes the current by the superpotential
\begin{equation}\label{C:affine-superpotential}
 \frac13\partial^\lambda
 \bigl(X_\lambda\partial_\mu L-X_\mu\partial_\lambda L
             +(\partial_\lambda X_\mu)L\bigr).
\end{equation}
Its compact commutator flux is zero. Thus the translation and
Lorentz charge forms are the physical ones. Integration by parts
for dilatations gives
\begin{equation}\label{C:Dprime}
 D'=D+Q,\qquad Q=q_J.
\end{equation}
At this point $D'$ and $Q$ are charge forms, not asserted
self-adjoint operators. Their dependence on two separately
translated polynomial tests is tempered and has each test's ordinary
vacuum translation spectrum. For the bra spectrum one lets the
compact flux follow the ket, translating the polynomial vector
field $X$ as well; Hermiticity supplies the ket spectrum. In
particular strictly energy-damped collective translations are
legitimate arguments of these forms.

\subsection{Identification of the quadratic charge and its domains}

The comparison with the physical current will first be made on
opposite wedge polynomials, smoothed under translations and rotations.
Compact smooth field tests have both kinds of derivatives; the angular
derivatives will control the distant caps. We keep
the large longitudinal cutoff and the growing transverse cutoff
separate: the first is removed at fixed transverse radius, and only
then is that radius allowed to grow.

Write
\[
 k_n(x)=2(n\cdot x)x-x^2n
       =|y|^2n+2uy+2u^2\bar n.
\]
We shall identify its charge with the already constructed positive
self-adjoint moment
$K_n=b(|Y|^2+\kappa)$. First take opposite compact Hermitian wedge
polynomials, as in the bounded-profile Ward identity of
Section~\ref{sec:null-current}. Locality identifies the commutator
flux through $u<0$ with the extreme vacuum commutator of
$(t\cdot k_n)\Omega$ against their product. Translation smoothing
makes these matrix elements smooth and supported in the union of
the corresponding causal supports. Conservation gives a plane
term $|y|^2t_{vv}$ and the following cap terms.

On the lower cap, the nonzero part with positive step function is to
the future of the right wedge polynomial. It has bounded $u$,
$|y|\asymp R$, and $v\gtrsim R^2$. Product cutoffs can therefore be
chosen with $\partial_u\chi_R=0$ there. On a dyad $v\asymp S$,
$S\gtrsim R^2$, the transverse cap amplitudes obey
\begin{equation}\label{C:cap-amplitude}
 C_NRS\bigl(R^2b\sqrt{ab}+Rab+a\sqrt{ab}\bigr)(1+bS)^{-N}.
\end{equation}
The factors follow respectively from the three terms of $k_n$,
the stress-tensor component bounds, the transverse cutoff
derivative, and the cap volume. Integration in $v$ gives the last
factor. With sufficiently many inverse energy weights these
amplitudes have uniformly bounded Hilbert norms. More explicitly,
$p^0$ is comparable to $a+b$, and the allowed transverse momenta
have area at most $Cab$. In the $d^4p$ realization the squared
norms of the three terms of \eqref{C:cap-amplitude}, weighted
by $(1+p^0)^{-m}$ with $m>5/2$, are bounded respectively by
\begin{equation}\label{C:cap-norms}
 C\bigl(R^6S^{-3},\ R^4S^{-2},\ R^2S^{-1}\bigr).
\end{equation}
For example the first squared integrand, after transverse
integration, is bounded by
$CR^6S^2a^2b^4(1+a+b)^{-2m}(1+bS)^{-2N}$;
its $b$ integral is $O(S^{-5})$ and its $a$ integral is finite.
The other two have powers $a^3b^3$ and $a^4b^2$.
Since $S\gtrsim R^2$, all three bounds are uniform.
There is also a positive power gain when paired with a test vector
smooth under rotations. Fix a sufficiently small $\varepsilon>0$.
The region $a<S^{-\varepsilon}$ gains at least
$S^{-3\varepsilon/2}$ in norm by the same integral;
$b>S^{-1+\varepsilon}$ is negligible by increasing $N$.
In the remaining region
$b/p^0\le C S^{-1+2\varepsilon}$, so the momentum direction lies
in a spherical cap of power-small area. Rotation Sobolev estimates
bound the spectral projection of the fixed test vector onto this
cap by a positive power of its area. This follows by averaging
the squared projected norms over rotations and then applying
Sobolev evaluation on the rotation group; the same estimate holds
after each of the finitely many required rotation derivatives and
energy weights. The angular cap has area
$O(S^{-1+2\varepsilon})$, so its projection norm is
$O(S^{-1/2+\varepsilon})$. For example any
$0<\varepsilon<1/2$ gives
$\gamma=\min(3\varepsilon/2,1/2-\varepsilon)>0$.
The paired dyads are bounded by $CS^{-\gamma}$, and summing
$S\gtrsim R^2$ gives $O(R^{-2\gamma})$.

A regulator at large $v$ can first be removed at fixed $R$.
Its derivative has size $S^{-1}$, its volume is $O_R(S)$, and the
orbital coefficients are then bounded independently of $S$.
The rapid $bS$ decay gives a vanishing energy-weighted norm.
On the plane, locality restricts the $|v|\asymp S$ tails to
$|y|\lesssim\sqrt S$. Their amplitudes are bounded by
\begin{equation}\label{C:plane-amplitude}
 C_NS^3b^2(1+bS)^{-N}.
\end{equation}
Here transverse integration bounds the squared weighted norm by
$CS^6\int a b^5(1+a+b)^{-2m}(1+bS)^{-2N}\,\dd a\dd b$,
which is uniform. Restriction to $a<S^{-\varepsilon}$ gives
$O(S^{-\varepsilon})$ in norm. The preceding angular argument
therefore again gives summable
paired tails, uniformly when the growing quadratic spatial
profile is truncated by a bounded-factor smooth cutoff.
The term $\partial_v^2L$ integrates to zero: the derivative of
the separating step is in a spacelike gap. Moving both derivatives
onto a distant tail cutoff gives the scalar-wave amplitude
$C_NS(1+bS)^{-N}$. Its squared weighted norm is bounded by
$CS^2\int ab(1+a+b)^{-2m}(1+bS)^{-2N}\,\dd a\dd b$,
and has the same low-$a$ and angular gains. The surviving plane integral is
therefore the growing second moment of the bounded-profile Ward
identity. It equals the form of $K_n$.

This identity must be extended on a genuine domain. We record the
domain argument rather than treating an unbounded charge form as
an operator.

Its two inputs are geometric Tomita localization and smoothness under
the boost group.  They first yield power domains for vectors with
strictly one-sided supports.  Positive-energy analyticity then
transports these estimates to damped translates, where the charge
identity can be compared on one common dense domain.

\Needspace{6\baselineskip}
\begin{lemma}[An affine Hardy estimate]\label{C:affine-Hardy}
Let $B=-i\partial_\tau$ on $L^2(\R,\dd\tau;\mathcal K)$, where
$\mathcal K$ is any Hilbert space. Suppose
$f\in\bigcap_m H^m(\R;\mathcal K)$ and
$g(\tau)=e^{-ice^\tau}f(\tau)\in\Dom e^{-\pi B}$, with $c>0$.
For every $r>0$ there is a finite integer $m$ such that
\begin{equation}\label{C:Hardy-power-bound}
 \|e^{r\tau}f\|\le C_{r,c}
 \bigl(\|f\|_{H^m}+\|e^{-\pi B}g\|\bigr).
\end{equation}
The constants are uniform when $c$ varies in a compact subinterval
of $(0,\infty)$.
\end{lemma}

\begin{proof}
Choose $\chi\in C_c^\infty(\R)$ equal to one on $[0,1]$, put
$\chi_j(\tau)=\chi(\tau-j)$, and write $L=e^j$, $j\ge0$.
At Fourier frequencies $\zeta<-\delta L$, split the convolution
$\widehat\chi_j*\widehat g$ at source frequency $\nu=-\delta L/2$.
The Tomita-domain assumption gives
\[
 \|\mathbf1_{\nu<-\delta L/2}\widehat g\|
 \le e^{-\pi\delta L/2}\|e^{-\pi B}g\|.
\]
For the remaining source frequencies, $|\zeta-\nu|>\delta L/2$.
The Schwartz $L^1$ tail of $\widehat\chi_j$ is $O_N(L^{-N})$,
uniformly in $j$, since translation changes only its phase. Young's
inequality therefore gives
\[
 \|\mathbf1_{\zeta<-\delta L}\widehat{\chi_jg}\|
 \le C_NL^{-N}\bigl(\|g\|+\|e^{-\pi B}g\|\bigr).
\]
On $\zeta\ge-\delta L$, take $\delta$ smaller than half the minimum
of $ce^{\tau-j}$ on $\supp\chi_j$. The phase
$-ce^\tau-\zeta\tau$ then has derivative of absolute value at least
$C(L+|\zeta|)$. Repeated integration by parts in the Hilbert-valued
integral gives
\[
 \|\widehat{\chi_jg}(\zeta)\|
 \le C_N(L+|\zeta|)^{-N}\sum_{k\le N}\|f^{(k)}\|_2.
\]
Its squared integral on this half-line is $O(L^{-2N+1})$ times the
displayed Sobolev bound squared. Since $\|\chi_jf\|=\|\chi_jg\|$,
these two estimates give arbitrary inverse powers of $e^j$ for the
localized norm. Choose $N>r+1$ and sum over $j$; on $\tau<0$ the
weight is at most one. This proves \eqref{C:Hardy-power-bound} and
the stated uniformity.
\end{proof}

\Needspace{6\baselineskip}
\begin{lemma}[Domains for the null moment]\label{C:K-domains}
A smooth compact physical polynomial vacuum vector whose supports
lie strictly on one side of a null plane is in $\Dom K_n^r$ for
every $r>0$, locally uniformly in such placements. Every strictly
energy-damped translate of a compact physical polynomial vector
also has this property. Applying $K_n^r$ to the latter vectors
preserves strong tube holomorphy and the indicated one-sided
boundary continuity.
\end{lemma}

\begin{proof}
On the lower side choose $c>0,d\in\R$ so that all supports lie
in $W_{cp_0+d}$. Tomita affiliation, after undoing
$U_H(c)U_b(d)$, gives the domain condition for $e^{-\pi B}$.
For a compact joint kernel, the same condition is the closed
Tomita-graph limit of factorized tests from
Section~\ref{sec:foundations}.  Collective boost derivatives act on
the smooth joint kernel, so their norms are controlled by the old
joint-distribution seminorms.  No affiliated joint-smear operator is
needed for this vector statement.
Remove the fixed constant translation $U_b(d)$ first; it commutes
with $H$ and preserves boost smoothness of these field vectors.
On the positive part of the affine representation use
$H=e^\tau$, $B=-i\partial_\tau$. The resulting vector $f$ satisfies
\[
 e^{-ice^\tau}f\in\Dom e^{-\pi B},
\]
where the original vector $f$ has Sobolev derivatives of every
order in $\tau$. Lemma~\ref{C:affine-Hardy} gives
$e^{r\tau}f\in L^2$ for every $r>0$. The $H=0$ part is
harmless. Since $K_n\le H$ strongly and the two commute, this
proves the assertion for the lower placement. Reflection and
equality of the reflected moments prove the upper placement.
The choices have positive geometric margins, giving locally
uniform bounds.

For $U(x+i\eta)A\Omega$, $\eta\in V_+$, consider the bounded
Hilbert-valued holomorphic function
\[
 1_{(L,\infty)}(K_n)U(x+s\eta)A\Omega,
 \qquad \operatorname{Im}s>0.
\]
A real interval of $s$ places the entire translated support on
one side of the plane. On that interval its norm is $O(L^{-N})$
for every $N$, locally uniformly; elsewhere it is bounded by
the original vector norm. The subharmonic logarithmic norm
estimate with harmonic measure of that interval yields any
prescribed inverse power at every fixed interior point, by
starting with a sufficiently large $N$. Spectral-tail
integration gives all the power domains, locally uniformly.
Closedness of $K_n^r$ and uniform convergence of spectral
truncations give strong holomorphy and the stated boundaries.
\end{proof}

Let $\mathscr D$ be the linear span of all vectors obtained from these
strictly energy-damped compact polynomial translates by finitely many
real translations, dilatations, and compact smooth dilation convolutions
\[
 \int \rho(s)U_D(s)\xi\,\dd s,\qquad
 \rho\in C_c^\infty(\R).
\]
The locally uniform bounds of Lemma~\ref{C:K-domains} and closedness
allow each such integral in the graph of every required $K_n^r$, also
after translating the origin; iteration gives the same conclusion for
every finite sequence of these operations. The space is dense and
invariant under real translations and dilatations by construction.
Its defining smooth kernels also
place it in $\Dom D$; the compact dilation convolutions form a graph
core for $D$, so $\mathscr D$ is such a core.
The preceding charge identity extends to $\mathscr D$ by
tube uniqueness. Thus
\begin{equation}\label{C:charge-identity}
 q_{k_n}(C,A)=\langle C\Omega,K_nA\Omega\rangle
\end{equation}
on this common domain, for every origin and future null normal.
The linear relations among conformal vector fields give vector
relations among the $K_n$ there: test them against the dense
domain and use \eqref{C:charge-identity}. In particular
\begin{equation}\label{C:quadratic-charge-translation}
 K_n(s):=U_a(s)K_nU_a(-s)
       =K_n-2s(D'-B)+2s^2a\quad\hbox{on }\mathscr D.
\end{equation}
This equation realizes $D'$ and $Q=D'-D$ as symmetric vector
operators on $\mathscr D$, by a difference of the indicated
operators. Only each $K_n(s)$ is being used as self-adjoint;
no essential self-adjointness of $Q$ is required.

\subsection{The canonical action on the one-field space}

The charge identity supplies more information than the spectral
comparison alone.  Its translated quadratic dependence first kills
the nonnegative comparison defect in the scalar channels.  The
spectral support statement then upgrades equality of compression
forms to a reducing-subspace identity.  Only after that step do we
treat the remaining helicities.

\Needspace{6\baselineskip}
\begin{theorem}\label{C:canonical-Pi}
The one-field space $\Pi$ reduces each $K_n$. On $\Pi$, $K_n$
equals the null special-conformal generator of the auxiliary
positive conformal representation carried by $l,J,t$.
Moreover $\langle Q\phi,\psi\rangle=0$ for
$\phi\in\mathscr D$, $\psi\in\Pi$.
\end{theorem}

\begin{proof}
We first treat the scalar labels $l,j_z,s_z$. The comparison
\eqref{D:order-zero}, together with $c_V=l+j_z$ and
$c_T=l/3+s_z$, gives order-zero comparison on each of them.
For smooth compact waves with $b,M>0$ compare the compression
of $K_n$ with $K_{0,n}=b(|Y|^2+h_0)$.
Their nonnegative expectation difference, evaluated on
$U_a(-s)\psi$, has zero second derivative in $s$.
Indeed the second-difference identity from
\eqref{C:quadratic-charge-translation} transfers to these waves
by symmetry against $\mathscr D$; the domains follow from the
spectral comparison. Direct differentiation gives the same
second difference for $K_{0,n}$. A nonnegative affine function
on the whole real line is constant.

Testing with separated compact multipliers in $b,k$ now shows
that the positive radial form
$h_0-\kappa_{\rm compr}$ is invariant under multiplication by
$e^{itM^2}$. Its distribution kernel is annihilated by
$M^2-M'^2$. On $M,M'>0$ this forces a diagonal kernel; positivity
makes it a positive measure $\nu(dM)$. It is bounded by the
$h_0$ form. For any compact interval $I\Subset(0,\infty)$,
choose logarithmic cutoffs $f_L=1$ on $I$, vanishing outside
successively larger compact intervals, with
\[
 \int_0^\infty M|f_L'(M)|^2\,dM\longrightarrow0.
\]
For example piecewise linear ramps in $\log M$ have energy
$O(L^{-1})$, and smoothing preserves that bound. Consequently
$\nu(I)\le\int|f_L|^2d\nu\to0$. The compression forms agree.

This is also equality of operators on a reducing copy. The
spectral comparison gives
\[
 \|1_{(s,\infty)}(\kappa)\psi\|
 \le\|1_{(s,\infty)}(h_0)\psi\|.
\]
Equality of the first moments makes the squared tail norms
equal almost everywhere. The support inclusion underlying this
inequality and equality in the projection norm inequality give
equality of the projected vectors. Spectral continuity gives
it at all continuity points, hence equality of all spectral
measures and reduction. Comparing the odd translation
differences with the canonical scalar formula shows that
$Q\mathscr D$ is orthogonal to these labels. Varying the null
direction spans all polarizations of $\Pi$, proving the last
assertion of the Theorem.

For clarity, the projection argument above does not require the two
spectral measures to commute.  Write $\iota$ for the scalar-copy
embedding, $E_s=\mathbf1_{(s,\infty)}(\kappa)$ and
$P_s=\mathbf1_{(s,\infty)}(h_0)$.  Spectral comparison gives
$E_s\iota=E_s\iota P_s$.  Equality of the first moments is equality
of the integrals of $\|E_s\iota\psi\|^2$ and $\|P_s\psi\|^2$;
their pointwise difference is nonnegative, so it vanishes almost
everywhere.  Equality in the contraction estimate for the orthogonal
projection $E_s$ therefore gives $E_s\iota\psi=\iota P_s\psi$.
One first uses a countable dense form core and then spectral
continuity to obtain the operator identity for every threshold.
This proves reduction rather than merely equality of expectations.

We recall explicitly the reference generator on the remaining
polarizations. On transverse helicity of absolute weight $d$
it is
\begin{equation}\label{C:canonical-hankel}
 K_{c,n}=b(|Y|^2+h_d).
\end{equation}
For a rank-$r$ primary of dimension $r+2$, $r=1,2$, contract
$r-d$ slots with $n$ and use
$\partial_vO_i-\partial_iO_v$ in each of the other slots,
discarding joint transverse traces. The wave is a constant
multiple of $b^rM^d$ times its helicity label. Under the null
conformal map of Section~\ref{sec:foundations}, the transverse
Jacobian is $\gamma(e_i+2cy_i n)$. The curl cancels the
$y_i n$ terms; derivatives of those terms contribute only
the discarded traces. The transformed wave thus has the
effective scalar weight $2+d$, giving
\eqref{C:canonical-hankel} by the scalar computation. This
uses only the positive primary two-point kernels and also
acts on their descendant combinations $V,T$. Its origin
translation formula contains $D$, without $Q$.

On smooth compact timelike $\Pi$ waves the actual domains follow
from \eqref{F:spectral-equation}. Put
$X_n=K_n-K_{c,n}$ there. Symmetry-testing
\eqref{C:quadratic-charge-translation} on $\mathscr D$, using
the orthogonality just proved, shows that $X_n$ intertwines
translations and is linear in $n$. The same is true on the
energy-damped compact $V,T$ waves in $\mathscr D$: symmetry
now gives $Q=0$ as a vector on these waves, and their reference
domains follow from the primary and descendant tube action.
The translation formula gives polynomial graph bounds.
Integrating translates against Schwartz functions therefore
commutes $X_n$ through compact timelike momentum multipliers.
Such multiples of the damped $V,T$ waves span the smooth
compact timelike tests of $\Pi$.

It follows that $X_n$ is a measurable fiber coefficient map.
Linearity in $n$ supplies a covector index $\lambda$. The
coefficient tensors
\[
 X_\lambda V_\mu(p)=X_\lambda J_\mu(p),\qquad
 X_\lambda T_{\mu\nu}(p)=X_\lambda t_{\mu\nu}(p)
\]
are locally square integrable up to the nonzero light cone
from inside: this follows first with the damping and smooth
test multipliers, and then locally without them by choosing
those multipliers nonvanishing. Their dimensions are
respectively $d'=2,3$, and they are covariant tensors.
A rest-frame boost component of weight $\omega$ has norm
proportional to $M^{d'-2-\omega}$ when boosted to a fixed
nonzero energy and $M\downarrow0$. The transverse mass
measure is proportional to $M\,dM$. Local integrability
therefore excludes every weight $\omega\ge d'-1$.

For the first tensor, contracting the $\mu$ slot with a null
vector leaves the $\lambda$ covector parallel to its lowered
null vector: its transverse and same-null contractions have
the forbidden weights. At rest conservation makes the
$\mu$ slot purely spatial. Reversing its spatial direction
changes the contracted tensor by a sign, but would require
it to be parallel to a different null covector. It is zero.
For the second tensor, contract both $\mu,\nu$ slots with
the same null vector. The forbidden weights again leave
only that parallel covector. Both slots are spatial at rest,
so reversing direction leaves the contraction unchanged and
again forces it to be zero. Varying the direction and using
symmetry recovers every spatial tensor coefficient. Hence
both tensors vanish and $K_n=K_{c,n}$ on compact timelike
tests.

For $d\ge1$, conjugation by $M^{1/2}$ turns the radial
operator into
$-\partial_M^2+(d^2-1/4)M^{-2}$, which is limit-point at
both endpoints. Compact smooth radial tests, combined with
smooth transverse cores and compact $b$ cutoffs, are thus
graph cores for \eqref{C:canonical-hankel}. The $d=0$ part
was already reducing with its specified Hankel extension.
The closed graph of the actual self-adjoint $K_n$ contains
the graph of the reference self-adjoint operator on each
copy. Resolvent uniqueness gives reduction of their sum
$\Pi$ and equality there, as claimed.
\end{proof}

\subsection{The residual dilation charge vanishes}

The remaining charge $Q$ is not assigned an independent
self-adjoint realization.  Instead, future and past cone comparisons
provide two families of genuine self-adjoint generators whose first
moments contain the same $Q$.  Opposite exponential bounds on their
limiting measures will force both families to concentrate on the
physical dilation spectrum.

Fix $n,\bar n$ and write $K=K_n$. Truncate the future time
cone by $u<R$ and the past cone by $u>-R$. Their actual
real spaces are contained in comparison real spaces
constructed with $K$ in place of the full parabolic
generator. This is \eqref{C:actual-parabolic-localization} for the
translated parabolic cut; reflection treats the past cone.
Let $T_R^+$ be the comparison future modular generator
and let $-T_R^-$ be the comparison past modular generator.
The affine product formulas and
\eqref{C:quadratic-charge-translation} give self-adjoint
operators satisfying
\begin{equation}\label{C:cone-comparison-generators}
 T_R^\pm=D+Q\mp\frac{K}{2R}\quad\hbox{on }\mathscr D.
\end{equation}
For example the future expression is
$B(R)-K(R)/(2R)$; the past modular expression is
$-B(-R)-K(-R)/(2R)$. Their spectral measures have uniformly
bounded second moments on $\mathscr D$ as $R\to\infty$.
The same is true on a smooth single-field vector supported
compactly in either time cone: the relevant planes avoid
its support, so Lemma~\ref{C:K-domains} and the affine
product domains apply. Testing against $\mathscr D$ shows
that
\[
 T_R^\pm\xi\pm K\xi/(2R)
\]
is a Hilbert vector independent of $R$ and of the sign.

\Needspace{6\baselineskip}
\begin{proposition}\label{C:Q-zero}
The charge form $Q$ vanishes on all physical polynomial
vectors. In particular $Q=0$ on $\mathscr D$.
\end{proposition}

\begin{proof}
Average the spectral measures of $T_R^\pm$ over
$S<\log R<2S$, with normalized Lebesgue measure, and take
common weak limit points as $S\to\infty$. One may first
test continuous functions on the compactified real line.
The uniform second moments on dense vectors remove mass
at infinity. The limits are normalized positive
operator-valued measures, not assumed projection-valued.
Dilatations shift $\log R$ by a fixed amount, whose
boundary contribution in the averages is $O(S^{-1})$.
Thus the limits commute with $D$. They have joint
positive measures $E^\pm(dd,ds)$ with the spectral
measure of $D$. Uniform second moments also give uniform
integrability of first moments, including polarized
cross moments on the vectors just described.

Only these first moments will be compared with $D+Q$.  The
logarithmic averaging is used to obtain commutation with the known
self-adjoint operator $D$, while the uniform second moments prevent
loss of first moments in the weak limit.  Neither operation turns
the limiting positive operator-valued measures into spectral measures.

Let $O$ be a Hermitian scaling component of real dimension
$\alpha$ and $\xi=O(f)\Omega$, with $f$ real, smooth and compact
in the future cone. The scaling orbit continues for
$0<\operatorname{Im}z<\pi$ because $e^zx$ has forward
timelike imaginary part there. Compact interior-cone
support and integration by parts give Hilbert boundary
values at both ends; real scale covariance gives uniform
norm bounds along the strip. Consequently
\begin{equation}\label{C:single-scale-strip}
 e^{-\pi D}\xi=e^{i\pi\alpha}O(f^-)\Omega,
 \qquad f^-(x)=f(-x).
\end{equation}
We need this strip identity also for complex linear combinations of
real dilation translates. Write $\xi_s=U_D(s)\xi$ and let
$\xi_s^-$ be its reflected-test wave. Hermitian conjugation of the
two-point function, with both test coordinates reflected, gives
\[
 \langle\xi_s^-,\xi_t^-\rangle=\langle\xi_t,\xi_s\rangle.
\]
For $p(D)=\sum_jc_jU_D(s_j)$ put
$p^\#(D)=\sum_j\overline{c_j}U_D(s_j)$; the real parameters $s_j$
are unchanged. With $\omega_\alpha=e^{i\pi\alpha}$ and
$\eta=e^{-\pi D}\xi=\omega_\alpha\xi^-$, the reflected Gram identity yields
\begin{equation}\label{C:filtered-adjoint-norms}
 \|p(D)e^{-\pi D}\xi\|=\|p^\#(D)\xi\|,
 \qquad \|p^\#(D)\eta\|=\|p(D)\xi\|.
\end{equation}
For all sufficiently large truncations, the comparison Tomita partner
of $p(D)\xi$ is $p^\#(D)\xi$, and that of $p(D)\eta$ on the past
side is $\omega_\alpha^{-2}p^\#(D)\eta$. Their norms are therefore exactly
the two norms in \eqref{C:filtered-adjoint-norms}. The finite set of
dilated supports remains compactly in the respective cone. In
particular, we do not require a complex filter to preserve its real space.

Real-space containment and these Tomita norm bounds, followed by the
spectral-measure limits in the future and past cones, give
\begin{align}
 \int |p(d)|^2e^{-2\pi s}\,\dd\langle E^+\rangle_\xi
 &\le\|p(D)e^{-\pi D}\xi\|^2,\label{C:Jensen-plus}\\
 \int |p(d)|^2e^{-2\pi d}e^{2\pi s}
                  \,\dd\langle E^-\rangle_\xi
 &\le\|p(D)\xi\|^2.\label{C:Jensen-minus}
\end{align}
Truncation followed by monotone convergence justifies
the unbounded exponential tests in the limiting measures.
The first moments in $s$, with every such filter, agree
for the two signs by
\eqref{C:cone-comparison-generators} and the uniform
moment bounds. Exponential polynomials determine finite
measures: polarization recovers every exponential
$e^{itd}$, and Fourier uniqueness then recovers the
measure. Thus these equalities and inequalities hold
conditionally on the common $d$ marginal $\mu_\xi$.

To spell out the first-moment assertion, the finite filters $p(D)$
preserve the class of smooth cone-supported single-field vectors;
they also commute with $e^{-\pi D}$.  Thus both signs admit every
same filter.  Their limiting first-moment measures in $d$ are finite
by the uniform second-moment bounds.  Taking
$p(d)=1+z e^{itd}$ and polarizing in $z\in\mathbb C$ shows that
these two measures have identical Fourier transforms, hence agree.
This is equality of conditional means, rather than just equality
of their integrals over $d$.

Disintegrate the two scalar joint measures as
$\mu_\xi(dd)\nu_d^\pm(ds)$. Their conditional first
moments $m_\pm(d)$ agree almost everywhere, while
\eqref{C:Jensen-plus}--\eqref{C:Jensen-minus} say
\[
 \int e^{-2\pi s}\nu_d^+(ds)\le e^{-2\pi d},
 \qquad
 \int e^{2\pi s}\nu_d^-(ds)\le e^{2\pi d}.
\]
Jensen gives $m_+(d)\ge d\ge m_-(d)$.
Equality of the means forces both to equal $d$; strict
convexity then forces both conditional laws to be
$\delta_d$. This argument requires only positivity and
normalization of the measures, not multiplicativity of
the operator-valued limits. Positivity also gives
Cauchy--Schwarz for each measurable set, so cross
measures against $\xi$ are supported on $s=d$.
Their first moments, paired with $\phi\in\mathscr D$,
now give
the following identity.  Indeed, if a measurable set is disjoint
from $s=d$, its positive operator has zero quadratic value on $\xi$,
and Cauchy--Schwarz makes every mixed value against $\xi$ zero as
well.  The mixed measure is therefore concentrated on the diagonal;
its $d$ marginal is the mixed spectral measure of $D$.  Its first
moment is consequently that of $D$, while the uniformly integrable
first-moment limits from \eqref{C:cone-comparison-generators}
give the form of $D+Q$.  Subtracting the two gives
\begin{equation}\label{C:Q-single-orthogonality}
 \langle Q\phi,\xi\rangle=0.
\end{equation}

The strip and Jensen argument just completed used a Hermitian
homogeneous field of dimension $\alpha$ with a real test. Splitting a
complex test into its real and imaginary parts extends
\eqref{C:Q-single-orthogonality} to all tests. A complex homogeneous
field is the sum of its Hermitian real and imaginary parts,
\[
 O_\Delta=\frac{O_\Delta+O_\Delta^\dagger}{2}
       +i\,\frac{O_\Delta-O_\Delta^\dagger}{2i},
\]
which have the same grade because adjunction preserves real scaling
dimensions.  Linearity therefore extends
\eqref{C:Q-single-orthogonality} to every homogeneous original field.
For $O=\sum_{\Delta\in S_O}O_\Delta$, its finite scaling support then
extends the conclusion to $O$ with the same smearing test.  This
step sums conclusions; it does not assign the nonhomogeneous field
one strip phase or apply Jensen's inequality to cross-grade terms.

It remains to pass from single fields to products.
Define the insertion $\delta_QO(y)$ by the commutator
with $J(y+r)$ integrated against a compact flux cutoff
in $r$. Conservation and locality shrink the flux as
needed. This insertion is relatively local to physical
fields and is tempered with the ordered spectra: in
this cutoff operation the adjacent momenta are first
combined, so the spectator spectral cones are unchanged.
Its extreme vacuum values against every physical
polynomial vanish by
\eqref{C:Q-single-orthogonality} and the charge identity.
To justify that passage, damp both collective
translations first, remove the ket damping on the
smooth future-cone single-field test, and remove the
bra damping distributionally. Translation covariance
removes the future-cone support restriction. Move the
insertion to the extreme on a spacelike open set;
ordered tube uniqueness makes the zero equality global.
Finally the commutator flux obeys the derivation rule
over every finite list of product slots.  This rule is the
telescoping of the adjacent single-insertion commutators just
constructed, not a new operator axiom for $J$ or $L$.  For a joint
test kernel the telescoping identity is first an identity of joint
distributions and is then tested with that same kernel.  Each
summand is zero, proving the
Proposition.
\end{proof}

\subsection{Modularity of time cones and the actual time action}

We now return from comparison real spaces to algebras of actual
bounded operations.  The equality $Q=0$ determines the limiting
comparison generator, but identifying the physical cone modular
group also requires a KMS identity and separation of the vacuum.
Both will be obtained for the original cone algebra before using
half-sided modular inclusion for line diamonds.

\Needspace{6\baselineskip}
\begin{theorem}\label{C:cone-modularity}
The vacuum is cyclic and separating for the actual
future and past cone algebras. Their modular generators,
with the normalization of Section~\ref{sec:foundations},
are respectively $D$ and $-D$.
\end{theorem}

\begin{proof}
By Proposition~\ref{C:Q-zero},
$T_R^\pm\psi\to D\psi$ on $\mathscr D$. This is a
core for $D$: it is dense and invariant under its
unitary group, and smoothing by compactly supported
smooth functions of the group parameter supplies a
graph core. The self-adjoint core convergence criterion
therefore gives $T_R^\pm\to D$ in strong resolvent
sense, and their real unitary groups converge strongly.

Take two bounded operations in a fixed compact
subregion of the future cone. For all sufficiently
large $R$, their vacuum and adjoint-vacuum vectors
belong to the comparison standard space and its
complexification with the required Tomita domains.
Writing the two operations as $A,B$, the standard-space
two-vector identity gives the scalar holomorphic continuation
of $\langle B^*\Omega,e^{itT_R^+}A\Omega\rangle$ to
$0\le\operatorname{Im}z\le2\pi$. Its values are defined by
\[
 F_R(t+iy)=
 \big\langle e^{-yT_R^+/2}B^*\Omega,
 e^{itT_R^+}e^{-yT_R^+/2}A\Omega\big\rangle,
 \qquad 0\le y\le2\pi,
\]
with the opposite boundary
\[
 F_R(t+2\pi i)
 =\langle e^{itT_R^+}A^*\Omega,B\Omega\rangle.
\]
One obtains the width $2\pi$ by assigning up to $\pi$
of imaginary evolution to each vector and using
$e^{-\pi T_R^+}=J_RS_R$; the comparison conjugation
disappears from the displayed boundary formula. The split-vector
pairing uses only $\Dom e^{-\pi T_R^+}$; no vector
$e^{-2\pi T_R^+}A\Omega$ is required.
The bounds depend only on the four vector norms.
Normal-family convergence
on the strip, together with the strong convergence on
the real edge, gives the vacuum KMS identity for $D$
on these actual operations. The identical argument
in the past gives the identity for $-D$. Bounded
strong-star approximation extends the identity from
compactly localized operations to the cone algebras.

Scaling already acts as an automorphism of each actual
cone algebra. The vacuum is cyclic by Reeh--Schlieder.
For clarity, faithfulness is part of the conclusion:
the KMS identity implies that a bounded operation
annihilating the vacuum has adjoint annihilating it.
Apply this to $BA$ for arbitrary $B$ if $A\Omega=0$.
Then $A^*B^*\Omega=0$; cyclicity gives $A^*=0$.
Thus the vacuum is separating. Uniqueness of the
modular automorphism group of this faithful vector
state identifies the generators as claimed.
\end{proof}

\Needspace{6\baselineskip}
\begin{theorem}[Actual time-M\"obius action]\label{C:time-mobius}
For every future unit timelike vector $e$, the actual
Hilbert space carries a positive-energy representation
of the universal covering group of the time-M\"obius
group. Locally it maps physical diamond algebras with
tips on the line $\R e$ to the corresponding diamond
algebras. Its affine subgroup is the physical time
translation and dilation subgroup. The canonical lift of
the connected stabilizer of each finite interval is its
actual diamond modular group.
\end{theorem}

\begin{proof}
Choose two line parameters $a<b$. Let $A,B_t$ be the
dilation generators centered at $ae,be$, and let $C$
be the modular generator of the physical diamond with
those tips. Inclusion of the diamond into the future
cone at $ae$ and the past cone at $be$ is half-sided
modular by Theorem~\ref{C:cone-modularity} and isotony.
The common vacuum is cyclic and separating for both
inclusions; no cyclicity of a relative commutant is
needed. The half-sided inclusion theorem gives the
affine relations corresponding to
\begin{equation}\label{C:three-mobius-fields}
 A=(z-a)\partial_z,\qquad
 B_t=(z-b)\partial_z,\qquad
 C=\frac{(z-a)(b-z)}{b-a}\partial_z.
\end{equation}
More specifically, $A-C\ge0$ scales with weight $-1$
under $A$, with the negative inward inclusion convention
$N=U_{A-C}(-1)MU_{A-C}(1)$; and $-B_t-C\ge0$ scales
with weight $+1$ under $-B_t$. The pair $A,B_t$
already has the physical affine relation with
$P_e\ge0$.

These are relations of unitary groups, not just
commutators on a tentative common domain. Ordered
products of the three classical one-parameter groups
are coordinates near the identity. The three affine
relations give exact pair-switching identities, so
reordering products proves a local unitary
homomorphism. It integrates uniquely to the simply
connected cover. This is the three-subgroup
integration criterion of
\cite[Lemma~1.1]{GuidoLongoWiesbrock1998}; see also
\cite[Lemmas~2.3--2.4]{MorinelliTanimoto2019}.
In its circle formulation the intervals are $(a,b)$, $(b,\infty)$,
and $(\infty,a)$, with stabilizer fields $C,B_t,-A$. Their three
pair relations are precisely the two cone--diamond relations above
and the physical affine relation between the two cone groups.
Its translation subgroup is the physical one, so
the representation has positive energy.

Affine covariance transports $C$ to the modular
generator of every line diamond. A sufficiently small
M\"obius transformation relative to a fixed finite
interval factors into an affine map of its tips and
an element of its connected stabilizer, with the lift
near the identity. Both factors have the
asserted action on its actual algebra. This proves
the local geometric action, without replacing the
physical net by a larger net.

The construction is made for each chosen time direction using its
actual cone and diamond algebras.  Its local geometric statement is
the one needed below: the connected interval stabilizer acts by that
diamond's modular group.  Compatibility sufficient to integrate all
spacetime conformal transformations on the full net has not been
used in this argument.
\end{proof}

We finish by recording the representation data needed for extraction.
Separate the trivial representation. The nontrivial irreducible
positive-energy representations of the simply connected cover are
the lowest-weight modules with $h>0$
\cite[Theorem~1.5.3 and the following direct-integral discussion]{LongoLectures2008}.
The general representation may contain a direct integral of these
modules, rather than a discrete sum. To obtain the decomposition,
disintegrate the central unitary of a full rotation. Within each
central character, compact energy lies in one integer translate;
positivity and the raising and lowering norm relations decompose it
into orthogonal lowest-weight ladders. The real ladder coefficients
give measurable multiplicity spaces $\mathcal K_h$.

Each nontrivial irreducible module restricts to the unique strictly
positive irreducible affine representation
\cite[Corollary~2.33 and equation~(6)]{NeebOlafsson2017}.
Multiplication by $E^{1/2}$ converts its $L^2(\dd E)$ realization
to the common affine realization
\begin{equation}\label{C:time-primary-waves}
 \mathcal H_h=L^2(\R_+,\dd E/E),\qquad
 E=P_e,\qquad
 (U_D(s)\psi)(E)=\psi(e^sE),\qquad
 \psi_{h,z}(E)=E^he^{izE}.
\end{equation}
The last wave, for $\operatorname{Im}z>0$, transforms as a primary of
weight $h$: its inner product kernel is
\[
 \int_0^\infty E^{2h-1}e^{iE(z-\bar w)}\dd E
 =\frac{\Gamma(2h)}{[-i(z-\bar w)]^{2h}},
\]
with the branch fixed in the upper half-plane. Kernel covariance on
the dense span of these waves fixes the primary realization and its
lift to the cover.
Here $h$ denotes a lowest-weight parameter, distinct from the internal
parabolic operator used in the null-cut comparison.
On the nontrivial summand, put
\[
 \mathcal K=\int_{h>0}^{\oplus}\mathcal K_h\,\dd\nu(h),\qquad
 \cH_{\mathrm{nontriv}}=L^2(\R_+,\dd E/E)\otimes\mathcal K,
 \qquad \mathsf h v_h=h v_h.
\]
Thus a measurable weight-band projection is
$\id\otimes\mathbf1_I(\mathsf h)$. No discreteness of this weight
operator follows from, or is required by, the field-label spectrum.

\Needspace{6\baselineskip}
\begin{lemma}\label{C:mobius-real-spaces}
On any bounded weight band contained in $h>0$, the
real standard space of a line diamond is generated
by the primary waves \eqref{C:time-primary-waves}
smeared with real smooth compact interval tests and
multiplicity vectors real for the appropriate phased
conjugation. Weight-band projections commute with
the diamond Tomita operator.
\end{lemma}

\begin{proof}
Let $J$ be the actual future-cone modular conjugation. The
positive-translation relation gives $JP_eJ=P_e$. The half-sided
cone--diamond relation gives the same formula for its positive
special-conformal partner, while $JDJ=-D$. These three generating
groups show that $J$ implements time reflection on the integrated
M\"obius representation.

In each lowest-weight ladder choose the canonical reflection
conjugation, measurably in $h$; its existence and multiplicity form
are given in \cite[Theorem~1.6.3 and its universal-cover extension]{LongoLectures2008}.
The product of this conjugation with $J$ commutes with the full
representation. Inequivalent lowest weights cannot be mixed by that
commutant, so $J$ preserves every measurable weight band. In the
common affine realization it is pointwise energy conjugation together
with an energy-independent multiplicity conjugation $j$, preserving
the spectral projections of $\mathsf h$. This argument distinguishes
weights $h$ and $1-h$ even when their Casimir values coincide.

For a real compact half-line test $f$, the strip continuation gives
\[
 e^{-\pi D}\bigl(E^h\widehat f(E)v\bigr)
 =e^{i\pi h}E^h\widehat f(-E)v.
\]
Hence its Tomita fixed-point condition is
$v=e^{-i\pi h}jv$, the phased real structure in the statement.
These tests are complex cyclic by
Fourier uniqueness. Their real closure is invariant
under dilatations and contained in the half-line
real space. Gaussian smoothing in $D$ gives graph
density for Tomita, hence equality of the two real
spaces. On compact positive weight bands these operations have uniform
bounds, and exhaustion treats a bounded band approaching zero.
The half-sided unitary sends the half-line
space onto the finite-interval space and transports
the tests by primary covariance. All these operations
commute with the weight-band projections. Each such projection
therefore commutes with the diamond conjugation, strongly commutes
with its modular operator, and preserves the domain and action of
its Tomita operator.
\end{proof}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/05-charges-cones.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/06-extraction.tex ====
\section{Extraction of a localizable scalar wave and accessible products}
\label{sec:extraction}

Throughout this Section, $l$ remains the scalar \emph{vacuum wave}
constructed from the trace.  In particular, division by $M^2$ on that
vacuum wave has not defined an operator on other vectors.  We first extract
from the physical virial field a scalar wave $S$ with actual local
approximants.  Only subsequently do we construct products of such accessible
waves.  The notation $L$ in a formal insertion still refers to the
single-insertion forms of Proposition~\ref{C:single-insertion}; a physical field
$L$ will be reconstructed in Section~\ref{sec:reconstruction}.

The construction has two distinct localization steps.  We first use
the canonical two-point spaces to select the part of the virial vacuum
wave with actual time weight two.  The actual time-M\"obius unitaries
then express that part through operations in physical diamonds.  After
proving independence of the chosen time direction, we quantify these
operations by bounded approximants and construct their analytic
products.  These products have real boundaries on spacelike patches;
the remaining real-axis reconstruction is a separate task.

\subsection{Projection into canonical diamond localization}

Write $X_{\mathrm{phys}}(O)$ for the closed real vacuum space of the
physical algebra in $O$, and $X_c(O)$ for the additive canonical real
space on the one-field representation $\Pi$.  The latter is determined by
the positive scalar, conserved-vector and traceless-tensor two-point
kernels of Section~\ref{sec:foundations}.

\Needspace{6\baselineskip}
\begin{lemma}\label{E:projection-localization}
For every bounded diamond $O$,
\begin{equation}\label{E:diamond-projection}
 P_\Pi X_{\mathrm{phys}}(O)\subset X_c(O).
\end{equation}
This assertion uses conformal covariance only of the auxiliary two-point
representations.
\end{lemma}

\begin{proof}
The actual parabolic containment
\eqref{C:actual-parabolic-localization} of
Lemma~\ref{C:localization-transport}, followed by the reduction in
Theorem~\ref{C:canonical-Pi}, gives the desired projected membership
for every parabolic diamond $N$ containing $\overline O$: actual
localization is contained in the comparison localization, and the
comparison generator acts canonically on $\Pi$.
The conjugation on $\Pi$ is the primary reality operation by wedge
Bisognano--Wichmann; the same statement holds in the reverse orientation.
We explain why these containments determine the bounded diamond.

The point of the geometry is to recover a bounded diamond from the
larger parabolic regions already controlled by the null-charge
comparison.  We carry out this recovery in the canonical real spaces,
where the complementary diamond can be described explicitly.

Use the Einstein cylinder, with coordinates $(\tau,x)\in\R\times S^3$
and round distance $d$.  By a Poincar\'e transformation and a dilation,
the physical patch and the chosen diamond can be written
\[
 |\tau|+d(x,N_0)<\pi,
 \qquad O=\{\,|\tau|+d(x,N_0)<\alpha\,\},\quad 0<\alpha<\pi.
\]
On the canonical representations,
\begin{equation}\label{E:canonical-complement}
 X_c(O)'=X_c(O^\perp),\qquad
 O^\perp=\{d(x,N_0)>\alpha+|\tau|\},
\end{equation}
where prime denotes symplectic complement.  Indeed, a connected conformal
transformation carries the wedge and its opposite to this diamond pair.
Transporting wedge modular data proves the equality.  There is no
assumption here about a physical algebra's duality.

For completeness, these transports are well-defined across the required
charts.  The two-point kernels are rational functions with the finite
Jacobian and weight factors of their integral bosonic primaries.  On
strict complex point waves in sufficiently small boxes the rational
covariance identity preserves every inner product.  Their span is dense,
so it defines local unitaries; the same identity gives local composition
and hence integration to the connected cover.  On nonsingular real
patches, the differentials preserve the future cone, so smaller strict
imaginary approach cones have distributional boundary values after
transport.  On overlaps, both boundary prescriptions are continuations
of the same rational complex kernel.  In strict spacelike separation
the two prescriptions agree.  Additivity therefore transports the real
smear spaces, even when the opposite diamond meets several charts.

Here is an explicit covering of the complement needed in
\eqref{E:canonical-complement}.  Let a target have $\tau\geq0$ and
$\theta=d(x,N_0)>\alpha+\tau$.  Choose
$\alpha<\beta<\theta-\tau$ and take the past tip of $N$ to be
$(-\beta,N_0)$.  Its future tip is $(\pi-r,q)$, where
$d(q,N_0)=r<(\pi-\alpha)/2$; it lies on the future null boundary of the
physical patch.  This parabolic diamond contains $\overline O$ with
strict margins.  The target is outside the causal future of its past tip
because $\theta>\beta+\tau$.  To put it outside the causal past of the
future tip, it suffices to choose $q$ so that
\[
 d(x,q)+r>\pi-\tau.
\]
At fixed $r$, the maximum left side is
$\min(\theta+2r,2\pi-\theta)$.  As $r$ increases to
$(\pi-\alpha)/2$, this tends to
$\min(\theta+\pi-\alpha,2\pi-\theta)$, which exceeds
$\pi-\tau$, except when $(\tau,x)=(0,-N_0)$.  Time reversal treats
$\tau<0$.  Consequently the spacelike shadows of these $N$ cover
$O^\perp$ except possibly its central antipode.

If a vector symplectically annihilates all these shadows, its pairing
with canonical real point waves vanishes on their union.  The resulting
distribution cannot be supported only at the omitted point: locally its
singular covectors are causal, from the positive-energy tube and its
conjugate, whereas a nonzero point-supported distribution has all
nonzero covector directions in its wavefront set.  It therefore
annihilates $X_c(O^\perp)$.  Taking symplectic complements in
\eqref{E:canonical-complement} proves
\eqref{E:diamond-projection}.
\end{proof}

\subsection{Timelike restrictions of the original fields}

The time-weight argument uses point waves restricted to a timelike
line.  Later, bounded approximation will also require uniform moments
of the original source field as a transverse smearing shrinks to that
line.  Both facts follow from the same ordered-spectrum estimate.

\Needspace{6\baselineskip}
\begin{lemma}\label{E:timelike-restriction}
Every fixed finite original vacuum distribution has a restriction to
parallel timelike lines, continuous in joint Schwartz temporal tests.
The restrictions of its fixed transverse derivatives exist as well.
Transverse mollification converges in these distributions, and in the
one-field vacuum and adjoint-vacuum vectors, with error $O(\delta)$
at width $\delta$.  Each bound uses finitely many temporal test
seminorms, uniformly on compact families of line positions.  In
particular every fixed source-field polynomial moment is uniformly
bounded under this mollification.  For a Hermitian field, a real line
test has its vacuum vector in $X_{\mathrm{phys}}(O)$ whenever the
compact line support is contained in the double cone $O$.
\end{lemma}

\begin{proof}
Choose the line direction as the time axis and write the individual
Fourier variables as $k=(\omega,\boldsymbol k)$.  On the ordered
spectral support $\Gamma_n$, positive energy of the partial sums gives
$\sum_i|\boldsymbol k_i|\leq C_n\sum_i|\omega_i|$.
Choose a smooth cutoff $\chi=1$ near $\Gamma_n$, supported where
$|\boldsymbol k|\leq C'_n(1+|\omega|)$, with polynomially bounded
derivatives.  For a joint Schwartz temporal test $h$,
$\chi(k)\widehat h(\omega)$ is therefore Schwartz in all variables,
with each Schwartz seminorm bounded by finitely many temporal ones.
Compact families of spatial translations and fixed transverse
derivatives introduce only polynomial factors and obey the same bound.

For the tempered Fourier distribution $\widehat W_n$, fix a finite
Schwartz seminorm $p_m$ controlling its action.  Transverse mollifier
multipliers are uniformly bounded in the required weighted derivative
seminorms, and
\[
 |\partial_{\boldsymbol k}^{\gamma}
       (\widehat\rho(\delta\boldsymbol k)-1)|
       \leq C_\gamma\delta(1+|\boldsymbol k|),\qquad 0<\delta\leq1.
\]
After multiplication by $\chi\widehat h$, this gives an $O(\delta)$
bound in $p_m$, with finitely many additional temporal seminorms.
It constructs the restriction and proves the stated distributional
convergence.  For the squared one-field vacuum error, the doubled
correlation has two mollifier differences; its seminorm is
$O(\delta^2)$, giving vector error $O(\delta)$.  The adjoint and all
fixed alternating source-field words are covered by the same original
vacuum distributions.  Thus every fixed moment required below is
uniformly bounded.  This argument tests tempered distributions; it
assumes no ordinary Fourier density for an $n$-point function.

For a real line test of a Hermitian field, sufficiently thin compact
transverse mollifications are supported in $O$.  Their vacuum vectors
belong to its closed real space by affiliation.  Their Hilbert limit
has the same membership.  Translation and Lorentz covariance give the
assertion for all the lines under consideration.
\end{proof}

\subsection{Actual and auxiliary time weights}

Fix a future timelike unit $e$ and put $E=P_e>0$.  Use the common affine
realization $L^2(\R_+,\dd E/E;\mathcal K)$, where translations multiply
by $e^{itE}$ and dilations rescale $E$.  In a lowest-weight module of
weight $h>0$, the point wave is $E^h e^{izE}v$.  Here $h$ is a
scalar time lowest weight, unrelated to the internal parabolic operator
$h$ of Proposition~\ref{F:parabolic-generator} or the radial comparison
operators $h_\nu$.  Theorem~\ref{C:time-mobius} and
Lemma~\ref{C:mobius-real-spaces} constructed the \emph{actual}
time-M\"obius action and its interval real spaces.  The auxiliary
canonical representation has the same affine convention, with its own
primary weights $s$.  Write $H$ for the actual lowest-weight operator
on the affine multiplicity space.  We will compare interval real
spaces as well as affine actions: the affine actions alone do not
distinguish lowest weights.

\Needspace{6\baselineskip}
\begin{lemma}[Interval Paley regularity and weight selection]
\label{E:paley-selection}
Let $X_s(I)$ be the closed real interval space of a primary of weight
$s>0$.  Dividing a wave in $X_s(I)$ by $E^s$ gives an entire Fourier
transform; evaluation and each derivative on a compact complex-frequency
set are continuous in its real-space Hilbert norm.  The bounds are
uniform on compact positive weight bands.  In particular:
\begin{enumerate}
\item Suppose a bounded affine intertwiner $j_s$ from the actual
representation to a weight-$s$ primary representation satisfies
$j_sX_{\mathrm{act}}(I)\subset X_s(I)$ for bounded intervals $I$.
In the common affine realization write $j_s=\id\otimes T_s$.
Then
\[
 T_s\mathbf1_{\R\setminus(s+\mathbb Z_{\geq0})}(H)=0.
\]
\item A physically real localized vacuum point wave of scale dimension
$d>0$ has actual weights only among $h=d-j>0$,
$j\in\mathbb Z_{\geq0}$.
\end{enumerate}
\end{lemma}

\begin{proof}
For real $f\in C_c^\infty(I)$ the primary norm, up to a positive
constant, is
\[
 \int_0^\infty E^{2s-1}|\widehat f(E)|^2\,\dd E.
\]
Reality determines negative frequencies.  To evaluate
$\partial_z^k\widehat f(z)$, choose a compactly supported smooth test
which agrees on $I$ with $(it)^k e^{izt}$.  Modify it outside $I$ to
annihilate its first $m$ Fourier jets at zero, with $m>s-1$.
Such a modification exists: finitely many disjoint small bumps outside
$I$, with distinct centers, have a moment matrix arbitrarily close to
a nonsingular Vandermonde matrix.  Cauchy--Schwarz with dual weight
$|E|^{1-2s}$ now bounds the pairing.  Its high-frequency integral
converges by smoothness, and its low-frequency integral converges by
the vanished jets.  The construction and all bounds are uniform on
compact sets of $z$ and compact bands of $s$.  They extend to the real
Hilbert closure.  The extended evaluations satisfy the
Cauchy--Riemann equations and give the asserted entire function.

The regularity at zero frequency now detects the allowed weights.
In the common affine realization, changing the primary weight changes
only a power of $E$.  Entire continuation permits precisely the
nonnegative integral powers of that variable.

An intertwiner commuting with $E$ and its rescalings is an
energy-independent bounded multiplicity map $T_s$.  For a real
multiplicity section $v$ in a compact $H$-band, projection of the actual
localized wave has quotient
\begin{equation}\label{E:weight-multiplier}
 \widehat f(E)T_s E^{H-s}v.
\end{equation}
The assumed real-space inclusion and the preceding continuity make
this holomorphic at $E=0$.  Choose $\widehat f(0)\ne0$.  On a compact
band $T_s\exp((H-s)z)v$ is an entire function of $z=\log E$.
Single-valued continuation in $E$ implies
\[
 T_s\bigl(e^{2\pi i(H-s)}-\id\bigr)v=0.
\]
Complexifying real sections and applying the spectral theorem shows
that $T_s$ is supported on $h-s\in\mathbb Z$.  Only finitely many
such values occur on the band.  Their finite Laurent expansion in
\eqref{E:weight-multiplier} has no pole, so its negative powers vanish.
Exhausting the positive weight axis proves the first assertion.

For the second, write a dimension-$d$ point wave as $E^d u$ in the
actual affine representation.  Project onto compact $H$-bands, which
commute with interval Tomita data.  In each fiber division by its
primary power gives $\widehat f(E)E^{d-h}u_h$.  The same argument forces
$d-h\in\mathbb Z_{\geq0}$.  There is no translation-invariant component
of a positive-dimension point wave: its zero-momentum vacuum amplitude
would have to transform by a nonunit-modulus dilation character.
For original fields, the required timelike restrictions and real
localization are supplied by Lemma~\ref{E:timelike-restriction}.
The same assertion for a subsequently constructed wave uses its stated
Hilbert-valued restriction and physical real localization.
\end{proof}

The norm-continuous evaluations above concern the closed real space.
For a complex localized operator vector, use its Hermitian real and
imaginary parts; the bound then involves the sum of the vacuum and
adjoint-vacuum norms.  No Hilbert-norm evaluation bound on an arbitrary
completion of $X_s(I)+iX_s(I)$ is being used.

Here the intertwiners are specific canonical projections.  Project by
$P_\Pi$, then onto an auxiliary time-primary summand of weight $s$.
Lemma~\ref{E:projection-localization} and the canonical interval
Tomita data give the required real-space inclusion for this map $j_s$.
The same construction applies when the canonical comparison space is
enlarged below.  Taking adjoints of the selection identity gives
\[
 \mathbf1_{\R\setminus(s+\mathbb Z_{\geq0})}(H)T_s^*=0.
\]
Write $l(0)=E^2u_l$.  Its auxiliary temporal primary has weight two,
so its actual affine label is $u_l=T_2^*v_l$ for its canonical label
$v_l$; it has actual weights in $2+\mathbb Z_{\geq0}$.
Likewise $J_e(0)=E^3u_{J_e}$ has
$u_{J_e}=T_3^*v_{J_e}$ and actual weights in
$3+\mathbb Z_{\geq0}$.  These are adjoint support statements, not
commutation of actual and auxiliary weight projections.
The temporal virial wave has $E^3$-label $iu_l+u_{J_e}$.
Its physical real localization and dimension three bound its actual
weights above by three.  It therefore has only weights two and three.
The weight-two label is $iu_{S_e}$, where
\begin{equation}\label{E:definition-Se}
 u_{S_e}=P^{\mathrm{act}}_{h=2}u_l,
 \qquad S_e(0)=E^2u_{S_e}.
\end{equation}
This is a vacuum-wave definition.

The derivative of a primary of weight two has an inhomogeneous
transformation term.  Differentiating its primary transformation, and
adding the weight-three part of $V_e$, gives for a small real
non-affine time-M\"obius map $g$
\begin{equation}\label{E:finite-difference}
 U_e(g)V_e(ze)\Omega
  =(g'(z))^3V_e(g(z)e)\Omega
     +2g'(z)g''(z)S_e(g(z)e).
\end{equation}
Choose an interval on which $g'g''\ne0$.  After changing variables and
multiplying by smooth real coefficients, this equation expresses any
smearing of $S_e$ there as a difference of a physical virial smearing
and an actual modular transform of one.  Consequently, for a real test
supported in a line interval whose diamond lies in $O$,
$S_e(f)\in X_{\mathrm{phys}}(O)$.  Complex linear extension gives its
adjoint-wave reality relation; this is the meaning of a Hermitian wave
here.  It is even and physically real localized on all translated lines.
Translation
covariance defines it as a spacetime Hilbert-valued distribution.  For
example $\|e^{-\epsilon P_e}S_e(0)\|=O(\epsilon^{-2})$,
and $|\boldsymbol P|\leq P_e$ supplies all required spacetime test
bounds.  The family is Lorentz covariant in $e$, a rotational scalar
about $e$, and homogeneous of dimension two.

\Needspace{9\baselineskip}
\subsection{Independence of the time direction}

\begin{proposition}\label{E:accessible-scalar}
The waves $S_e$ agree for all future timelike unit vectors $e$.
Their common value $S$ is a purely timelike Lorentz scalar of dimension
two, with physical real localization.  It has the canonical null
special-conformal action.  It can be included in the auxiliary
representation and in Lemma~\ref{E:projection-localization}.
\end{proposition}

\begin{proof}
There are three issues to settle: the timelike spectral densities,
the possible massless component, and the action of the actual null
generators on the resulting scalar.  The projection identity
\eqref{E:definition-Se} will link the first two issues: comparison on
one energy slice fixes both the normalization and the dependence on
the time direction.

We first prove a two-point claim: the timelike entries of the joint
spectral measure of the family $S_e$ and $l$ are constant multiples of
the ordinary measure $\dd^4p$ on the future timelike cone.  No density
is assumed in this claim.  Write $\mu_{ee'}$ for a joint spectral
entry, including $l$ as an additional label.  The corresponding
commutator distribution is
\[
 C_{ee'}(x)=\int e^{ip\cdot x}\,\dd\mu_{ee'}(p)
              -\int e^{-ip\cdot x}\,\dd\mu_{e'e}(p).
\]
Every invariant-mass projection commutes with wedge Tomita data.
Wedge separation and reality therefore give causal support for this
commutator after projection to any compact mass interval.  For $l$,
the real wedge data are the canonical scalar data already identified
in Section~\ref{sec:foundations}; physical point-field membership is
not needed.

Homogeneity disintegrates the timelike measure as
$M^3\dd M\,\dd\nu(p/M)$, with a polynomially bounded measure on the
unit hyperboloid.  Testing arbitrary mass intervals shows that the
shell commutator has causal support for almost every mass; rescaling
then gives the same statement for the reference shell.  This is a
disintegration of distributions, not an assumed field on a fixed
mass shell.  The shell commutator solves the Klein--Gordon equation.
Its displacement and normal-derivative Cauchy data at zero time are
supported at the spatial origin, and hence are finite sums of delta
derivatives.  Fourier solution of this Cauchy problem expresses its
shell measure as the ordinary commutator measure times a polynomial
in $p/M$, modulo $p^2=M^2$.  In particular this proves absolute
continuity rather than presupposing it.

For a self-pair the shell polynomial is even under $p\mapsto-p$.
Take its representative of least degree modulo the shell relation.
If its highest degree is $k$, that homogeneous term is nonzero on an
open set of null directions; otherwise divisibility by $p^2$ would
reduce the degree.  At fixed nonzero energy and $M\downarrow0$ it
contributes order $M^{-k}$ to the density with respect to $\dd^4p$.
The transverse mass integration is $M\dd M$.  Positivity and local
finiteness therefore require $k<2$.  Evenness leaves only a constant.
Positivity of the matrix measure bounds each cross density by the
geometric mean of the self densities.  A polynomial on the unit
hyperboloid with such a global bound is constant by the same highest
homogeneous term argument.  All timelike density entries are therefore
constant, proving the two-point claim.  Covariance makes the self
constant of $S_e$ independent of $e$.

There is no zero-momentum part.  We must also exclude a massless part;
positivity alone would allow one.  Let $\rho=\sqrt{-W_\mu W^\mu}$ be
the massless Pauli--Lubanski length and project the entire Lorentz family
onto a compact band $0<a\leq\rho\leq b<\infty$.  This projection
commutes with wedge modular data.  Its commutators are causal massless
solutions.  The preceding point-supported Cauchy-data argument makes
them derivatives of the massless commutator, hence supported on the
light cone in four dimensions.

On the complex cyclic span of this projected family, let $X_+$ and
$X_-$ be the real spaces generated in the future and past cones.
Translation-spectrum uniqueness makes both spaces complex cyclic:
a vector orthogonal to smearings on an open cone is orthogonal to all
translates by tube uniqueness.  Huygens support gives
$X_-\subset X_+'$, so $X_+$ is standard.  Lorentz transformations
preserve it and hence commute with its modular group.  Every future
translation maps it into itself.  The positive-generator modular
translation relation therefore scales all $P_\mu$ by one common
nontrivial exponential under that modular group.  Because the Lorentz
generators commute with the group, $\rho$ scales by the same factor.
No nonzero spectrum contained in $[a,b]$ is invariant under all these
rescalings.  The projected space is zero.  Exhausting positive bands
excludes nonzero Pauli--Lubanski length.

The remaining massless representations have trivial little-group
translations and decompose into helicities.  A rotational scalar about
$e$ has only helicity zero, since a spatial rotation fixing both $e$ and
a massless momentum acts trivially on its wave.  Use the scalar
massless invariant measure.  The possible internal part of dilations
is a momentum-independent unitary group $W$ on scalar multiplicities.
Lorentz covariance and dimension two give the amplitude
\begin{equation}\label{E:massless-amplitude}
 S_e(p)=(p\cdot e)W(\log(p\cdot e))\beta.
\end{equation}
For two directions the commutator argument makes the cross density a
homogeneous quadratic shell polynomial.  A quadratic scalar invariant
under rotations fixing $e,e'$ is, on the null shell, a linear
combination of $(p\cdot e)^2$, $(p\cdot e)(p\cdot e')$ and
$(p\cdot e')^2$.  It follows that
\[
 \varphi(u)=\ip{\beta}{W(\log u)\beta},\qquad
 u=\frac{p\cdot e'}{p\cdot e},
\]
is a linear combination of $u^{-1},1,u$ on the attainable interval.
As the relative rapidity grows these intervals exhaust $(0,\infty)$;
the coefficients agree on overlaps.  Since
$|\varphi(u)|\leq\|\beta\|^2$, both nonconstant coefficients vanish.
Thus the massless cross density is
$c(p\cdot e)(p\cdot e')$, with $c=\|\beta\|^2\geq0$.

The only massless possibility left is the scalar density just
displayed.  Its growth with the relative rapidity of $e$ and $e'$ is
incompatible with the bounded timelike cross densities.  The next
energy-slice calculation makes this comparison with its exact
normalization.

Now apply the defining projection in \eqref{E:definition-Se} to
$F=S_{e'}$.  A dimension-two wave with physical real localization has actual weights at most
two; $u_l$ has actual weights $2,3,\ldots$.  In the multiplicity fiber
at $P_e=1$,
\begin{equation}\label{E:projection-pairing}
 \ip{u_F}{u_{S_e}}=\ip{u_F}{u_l}.
\end{equation}
Let $B(e',e)$ and $D(e')$ denote the constant timelike density
entries for $(S_{e'},S_e)$ and $(S_{e'},l)$, respectively.  Write
$b=B(e,e)$, which is independent of $e$ by covariance, and let
$\rho_{ll}$ be the self density of $l$.  With timelike measure
$\dd^4p$ and massless measure $\theta(p^0)\delta(p^2)\dd^4p$,
\eqref{E:projection-pairing} reads explicitly
\begin{equation}\label{E:energy-slice-normalization}
 \frac{4\pi}{3}B(e',e)+2\pi c(e\cdot e')
   =\frac{4\pi}{3}D(e').
\end{equation}
Indeed the timelike slice $p\cdot e=1$ is the unit spatial ball
in the rest frame of $e$, of volume $4\pi/3$.  Lorentz invariance
of $\dd^4p$ makes this normalization independent of the slicing
direction.  On the massless slice the angular integral is
$\frac c2\int_{S^2}(e'^0-\boldsymbol e'\cdot\boldsymbol n)\dd\Omega
=2\pi c(e\cdot e')$.  Positivity gives
$|B(e',e)|\leq b$ and $|D(e')|\leq\sqrt{b\rho_{ll}}$ uniformly.
Arbitrarily large relative rapidity therefore forces $c=0$.
Setting $e=e'$ in \eqref{E:energy-slice-normalization} now gives
$D(e')=b$, and hence $B(e',e)=b$ for all pairs.  Thus
$\|S_e(f)-S_{e'}(f)\|^2=0$ for every test $f$.
It also shows that the constant two-point density of $S$ with $l-S$
is zero.  This is orthogonality of the scalar wave channels, including
all their spacetime translates; it does not assume that the actual
time-weight projection commutes with spatial translations.

Finally, physical real localization supplies the scalar order-zero
parabolic comparison for $S$.  We can also apply the proof of
Lemma~\ref{C:K-domains} to its smooth compact vacuum smearings
in the same strictly one-sided placements, and then to all their
strictly energy-damped translates.
That proof requires real localization, giving the Tomita bound after
parabolic transport, and boost smoothness of every order.  Both are
available here: localization was obtained from
\eqref{E:finite-difference}, and the scalar covariance just proved
identifies every iterated boost derivative with another compact
smooth $S$-smearing.  Its Tomita and Hardy estimates therefore give
the asserted $K_n$ domains without treating $S$ as an operator.
Symmetry against the same dense core transfers the even second
difference of \eqref{C:quadratic-charge-translation},
$K_n(s)+K_n(-s)-2K_n=4s^2a$, to these vectors as a form identity.
The $D'$ term cancels, so no $D'$ operator domain for $S$ is assumed.
The second-difference and zero-capacity argument of
Theorem~\ref{C:canonical-Pi} therefore applies: the
nonnegative comparison defect is translation-diagonal in $M^2$, and
is bounded by $\int_0^\infty M|f'(M)|^2\dd M$.  Logarithmic cutoffs
equal to one on a prescribed compact mass interval make this bound
tend to zero.  Spectral comparison then gives reduction and the exact
canonical null action.  Its wedge reality and two-point kernel are
scalar, so the proof of Lemma~\ref{E:projection-localization} includes
this enlarged canonical space as well.
\end{proof}

\Needspace{6\baselineskip}
\begin{corollary}\label{E:infinity-vanishing}
Put $\ell=l-S$.  Its insertion at imaginary infinity annihilates every
compact physical vacuum polynomial:
\begin{equation}\label{E:residual-infinity}
 \lim_{R\to\infty}R^4\ip{\ell(iRe)}{A\Omega}=0.
\end{equation}
\end{corollary}

\begin{proof}
The multiplicity label $u_\ell$ has actual weights $2+j$, $j\geq1$.
First take a Hermitian polynomial localized in a diamond on the line.
Its vacuum vector $\psi$ is in the corresponding real space.  The
scalar quotient
\[
 F(E)=E^{-2}\ip{u_\ell}{\psi(E)}
\]
is regular at zero, with continuous evaluation, by
Lemmas~\ref{E:projection-localization} and~\ref{E:paley-selection}.
For compact actual weight-band projections of $\psi$, its overlap with
$u_\ell$ has only positive integral powers $E^{h-2}$, so $F(0)=0$.
These projections preserve the interval real space and converge in
Hilbert norm; continuity of evaluation proves the same equality
without the band cutoff.  With the common $\dd E/E$ normalization,
\[
 \ip{\ell(iRe)}{\psi}=\int_0^\infty E^3e^{-RE}F(E)\,\dd E.
\]
The contribution near zero is $o(R^{-4})$, and the complementary
contribution is exponentially small by Cauchy--Schwarz.  Every compact
polynomial is a complex linear combination of Hermitian ones, and its
support fits in a line diamond.  For a compact joint polynomial
kernel, the same real-space statement is the closed Tomita-graph
limit of factorized tests, using the old joint-distribution
continuity from Section~\ref{sec:foundations}.  This proves the claim.
\end{proof}

At this point $S$ is a direction-independent scalar vacuum wave
with physical real localization, and $l-S$ has the stated decay against
physical polynomial vectors.  To multiply $S$ with other entries we
need quantitative bounded operations, including control of both their
vacuum and adjoint-vacuum errors.

\subsection{Bounded approximants for accessible waves}

Call a vacuum wave \emph{accessible} when it has the approximation
property below, including the corresponding adjoint wave.  Physical
fields and $S$ have this property.  This definition does not assert
that an accessible wave already acts on a common field domain.
In the following norm estimates, $a(f)$ denotes a vector: for an
original field it is its smeared vacuum vector, and for $S$ it is
its Hilbert-valued smearing.  The notation
$a^\dagger(\bar f)$ denotes the corresponding adjoint-vacuum vector.

\Needspace{6\baselineskip}
\begin{proposition}\label{E:bounded-approximation}
Let $a$ be a physical field or $S$, and let $f$ have support in a small
real source patch.  Given a double-cone neighborhood of its support,
there are bounded operations $A_R(f)$ in that neighborhood, for dyadic
$R\to\infty$, such that for one fixed exponent $q$
\begin{align}
 \|A_R(f)\|&\leq C R^q,\label{E:approx-norm}\\
 \|A_R(f)\Omega-a(f)\|+
 \|A_R(f)^*\Omega-a^\dagger(\bar f)\|
   &\leq C_N R^{-N}\quad(N=1,2,\ldots).
 \label{E:approx-error}
\end{align}
For each prescribed $N$, constants are uniform on balls of finitely
many test seminorms.  When patches and localization margins shrink to
size $\varepsilon$ in a fixed compact set, these constants cost at
most fixed inverse powers of $\varepsilon$.  The approximants need
not depend linearly on $f$.
\end{proposition}

\begin{proof}
For an ordinary field smearing with closed affiliated realization
$A=v|A|$, take its polar cutoff $A_R=v|A|\mathbf1_{[0,R]}(|A|)$.
It has norm at most $R$.  The tails on the vacuum and adjoint vacuum
satisfy, for every positive integer $m$,
\begin{align*}
 \|(A-A_R)\Omega\|&\leq R^{-m}\||A|^{m+1}\Omega\|,\\
 \|(A^*-A_R^*)\Omega\|&\leq R^{-m}\||A^*|^{m+1}\Omega\|.
\end{align*}
Here the absolute-value moments follow from the original realization
hypothesis together with the generated-domain action check in
Section~\ref{sec:foundations}. Both $A$ and $A^*$ preserve the dense
invariant vacuum-generated domain $\mathcal D_{\mathrm{old}}$ and agree
there with the physical field and its polynomial adjoint. Induction
therefore gives
\[
 \mathcal D_{\mathrm{old}}\subset
 \Dom (A^*A)^m\cap\Dom(AA^*)^m
 \qquad(m=1,2,\ldots),
\]
and these powers on that domain are the corresponding alternating
polynomial words.  Their vacuum norms are finite and are bounded by
finitely many seminorms of the source tests, by the joint Wightman
distributions.  Since $|A|^{2m}=(A^*A)^m$ and
$|A^*|^{2m}=(AA^*)^m$, spectral calculus also gives every lower
integer absolute-value moment used in the tail bounds.  This uses
the adjoint-extension clause of Assumption~\ref{ass:framework} and
does not require essential self-adjointness.

The invariant polynomial domain supplies these moments, uniformly
on finite test-seminorm balls; subdivision supplies the requested
local neighborhoods.  Use a Hermitian basis or adjoint-paired
cutoffs to preserve adjunction and grading.

For a nonhomogeneous original source, any comparison of shrinking
patches by dilation is made on its finitely many homogeneous
components and then summed.  For this fixed field and requested
accuracy, a maximum of the finitely many seminorm orders and powers
is sufficient.  This does not assign the source a single scaling
dimension or require a bound uniform in the set of dimensions, and
it does not change the fixed polynomial norm exponent in
\eqref{E:approx-norm}.

For $S$ the order of construction matters.  We truncate the
ordinary source field before applying the actual modular unitary,
and then add the resulting bounded operations.  We will estimate
the sum by its absolute quadrature weights, so no domain or moment
estimate for an unbounded sum of modularly transformed fields enters.

For $S$, use \eqref{E:finite-difference} on parallel lines.  After
undoing $g$ in the test, a line smearing of $S$ is a fixed linear
combination of a line virial smearing and a modularly transformed
line virial smearing, with smooth real coefficients.  By
Lemma~\ref{E:timelike-restriction}, transverse approximation at width
$\delta_R\leq e^{-R}$ has vacuum error $O(\delta_R)$, and every
fixed source-field moment is bounded uniformly as
$\delta_R\downarrow0$.  Its adjoint data obey the same estimates.

Polar truncate each such source field at $R$ and apply the actual
M\"obius unitary.  It fixes the vacuum and transports the bounded
operation into the destination diamond, so both estimates in
\eqref{E:approx-error} survive.  A transverse quadrature of mesh
at most $e^{-R}$ approximates a full spacetime smearing.  Although
its number of terms grows, the sum of absolute quadrature weights
is bounded.  Hence the sum of the operator norms is $O(R)$, and the
sum of the vacuum errors remains rapidly small.  No moment bound
for a sum of noncommuting transformed fields is being assumed.

All the operations just described use finitely many derivatives for
each requested accuracy.  Rescaling tests and using diamonds with
margins proportional to $\varepsilon$ therefore introduces only
inverse powers of $\varepsilon$.  The fixed non-affine map can be
chosen nonsingular on the entire compact parameter patch, with
$g'g''$ bounded away from zero.  Take
$\delta_R\leq c\varepsilon e^{-R}$ and sufficiently fine quadrature
to retain these margins.  Subdivision completes the construction
for an arbitrary compact source support.
\end{proof}

The norm exponent in \eqref{E:approx-norm} is fixed before an error
order $N$ is requested.  Increasing $N$ changes the finite collection
of moments and test seminorms used to estimate the error, but does not
change that exponent.  This distinction allows rapid vacuum errors
to dominate the finitely many polynomially growing factors in a
product.

\subsection{Holomorphic products, ties, and unsmearing}

For $p$ accessible entries, use increasing imaginary translations:
\begin{equation}\label{E:ordered-tube}
 \operatorname{Im}z_1\in V_+,
 \qquad\operatorname{Im}(z_{j+1}-z_j)\in V_+
 \quad(1\leq j<p).
\end{equation}
The first inequality is needed for vector products.  With tests fixed
in sufficiently small patches, form actual bounded vector products
\begin{equation}\label{E:bounded-product}
 F_R(z)=U(z_1)A_{1R}U(z_2-z_1)A_{2R}\cdots
           U(z_p-z_{p-1})A_{pR}\Omega.
\end{equation}
They are holomorphic in \eqref{E:ordered-tube}, with norm at most a
fixed power of $R$.
Their scalar vacuum expectations have the larger tube determined only
by the successive imaginary gaps: vacuum invariance removes the first
factor $U(z_1)$.  Equivalently, add a common future imaginary translation
to put all the arguments in the vector tube, then pair with $\Omega$.
The scalar answer is independent of this common translation.  We use
$W$ for its limit and retain $\Psi$ for the vector limit.

\Needspace{6\baselineskip}
\begin{proposition}\label{E:accessible-products}
The vectors \eqref{E:bounded-product} have a unique, locally uniform
holomorphic limit.  It is multilinear in the accessible waves and has
Hilbert-valued distributional boundaries at mutually strictly
spacelike real placements.  These boundaries are graded symmetric.
The limits extend to common holomorphic neighborhoods of imaginary
ties: successive blocks may have equal imaginary translation when
real placements within each block are pairwise spacelike and all
inequalities between blocks remain strict.  The limits unsmear to
holomorphic point vectors, denoted $\Psi$, depending only on the total
complex insertion arguments.  For physical entries alone they agree
with the usual Wightman vectors.
For any two finite accessible words, with each word in its strict
vector tube, their scalar functions satisfy
\begin{equation}\label{E:accessible-splitting}
 \begin{split}
 &\big\langle\Psi[A_1(z_1)\cdots A_p(z_p)],
                  \Psi[B_1(w_1)\cdots B_q(w_q)]\big\rangle\\
 &\quad=W\bigl(A_p^\dagger(\bar z_p),\ldots,
               A_1^\dagger(\bar z_1),B_1(w_1),\ldots,B_q(w_q)\bigr).
 \end{split}
\end{equation}
An empty word denotes $\Omega$.  Source tests on the bra word are
conjugated and reversed, together with their field labels.  The identity
holds for distinct words and their finite linear combinations at every
finite total length.  It asserts analytic Hilbert splittings and their
spacelike real boundaries, without yet asserting a distributional
boundary at every real configuration.
\end{proposition}

\begin{proof}
We first prove convergence where locality lets us estimate the
whole product by vacuum errors.  Holomorphic propagation then carries
that convergence into the ordered tube.  Finally, bounded-product
continuity at a spacelike edge gives the common neighborhoods needed
for ties; translation compatibility removes the source smearing.

At real translations making all approximation patches mutually
spacelike, telescope two bounded products, whether at $R,2R$ or
from different approximation choices.  Strong graded locality moves
each difference to the vacuum.  Its vacuum error is rapidly small
by \eqref{E:approx-error}, while all other factors have polynomial
norm growth by \eqref{E:approx-norm}.  The product difference is
therefore $O_N(R^{-N})$ for every $N$, uniformly on smaller placement
patches and appropriate test-seminorm balls.  This proves independence,
graded symmetry and multilinearity at these real placements.

We record the propagation estimate, since vacuum convergence alone
would not justify the products elsewhere.  For a holomorphic vector
difference $G_R$ bounded by $CR^q$, restrict to a complex line
$z=x+\zeta\eta$ with $\eta$ in the ordered cone.  A real interval $I$
can be chosen inside a good spacelike-placement patch.  The
subharmonic logarithm of its norm and harmonic measure give
\begin{equation}\label{E:harmonic-propagation}
 \log\|G_R(x+\zeta\eta)\|
 \leq\omega(\zeta,I)(\log C_N-N\log R)
 +(1-\omega(\zeta,I))(\log C+q\log R).
\end{equation}
On compact subsets reached this way $\omega$ is bounded below by a
positive number.  Interior three-circle chains propagate the same
conclusion to every fixed compact subset of the connected tube.
As $N$ is arbitrary, convergence is rapid there.  Above a smaller
good real patch it is rapid uniformly up to the boundary, because
$\omega$ has a uniform positive lower bound for bounded approach
directions.  Summing the dyadic differences proves existence.

For ties, first keep the common translation of each block and its
spectators strictly inside their allowed complex domains.  The
relative translations within a block have wedges of opposite
imaginary-gap cones for its order and its reversal.  At their common
real edge the bounded products agree, with the graded sign, by
strong locality.  Each bounded product has a norm-continuous
extension to that edge: the real translation groups and the
positive-energy contraction semigroups are strongly continuous,
and only finitely many bounded factors occur.  On fixed smaller
wedges and their common edge its norm is $O(R^q)$.
The continuous-edge form of the local edge-of-the-wedge theorem
therefore applies directly; its continuation neighborhood depends
only on the wedges and the edge patch, and its interior bounds are
controlled by the supremum bounds on those wedges and their edge;
see \cite[Theorems~1$'$ and~2]{Browder1963}.

To make uniformity explicit, apply the theorem after scalarization
to the Hilbert-valued family $(R^{-q-1}F_R)_R$ in the
square-summable direct sum, with dyadic $R$.  Its common edge is
norm continuous: finitely many components are continuous and the
remaining squared norms are bounded uniformly by a convergent tail
of $\sum_R R^{-2}$.  Scalarization gives a bounded holomorphic
Hilbert-valued continuation on one common neighborhood, and each
unweighted component there has a bound $O(R^{q+1})$.  The same
supremum-bound argument is uniform on the prescribed test-seminorm
balls; no estimate from boundary distribution seminorms alone is
being asserted.  Interior
propagation from the strict tube now gives rapid convergence on
compact subsets of these neighborhoods.  Adjacent permutations
within a block are treated on overlapping smaller neighborhoods,
where uniqueness glues the continuations.

By uniqueness from the spacelike real patch, multilinearity persists
throughout these domains.  The seminorm bounds and the test-function
kernel theorem give joint distributions in the source variables,
holomorphic in the translation variables.  Translate a source test
within its patch.  At real spacelike placements this is the same as
translating its insertion argument; the equality involves only the
vacuum wave and hence survives all approximation choices.  Analytic
uniqueness extends this equality.  If $x$ denotes source variables,
the kernel in coordinates
$(x,\operatorname{Re}z+x,\operatorname{Im}z)$ is therefore independent
of its first variables in the distributional sense.  It satisfies
the Cauchy--Riemann equations in the remaining variables.  Thus it is
a holomorphic Hilbert-valued function of $x+z$, rather than merely a
distribution in $x$.  This proves unsmearing.  The same identity glues
source patches and their translations.

For the split identity first keep all source tests smeared, and use
the same bounded approximants on the two sides, taking the adjoints of
the bra approximants in the scalar word.  This is an admissible choice
because both vacuum and adjoint-vacuum errors were controlled.  Bounded
adjunction gives \eqref{E:accessible-splitting} before taking limits.
The displayed scalar word lies in its ordered scalar tube: its reversed
bra gaps and ket gaps are future directed, and its central gap has
imaginary part $\operatorname{Im}w_1+\operatorname{Im}z_1\in V_+$.
Local uniform convergence of the two vector factors and of the scalar
word therefore proves the identity there.  The preceding unsmearing
then gives the point formula.  Conversely, at any cut between two
successive entries of an ordered scalar tuple, choose a common
imaginary shift strictly between their imaginary parts.  Subtracting
that shift puts the reversed-adjoint prefix and the suffix in the two
strict vector tubes of \eqref{E:accessible-splitting}.  Thus the formula
supplies the analytic Hilbert splitting of every ordered scalar tuple
at every cut.  Finite linear combinations retain all
cross terms.  At mutually spacelike real placements, the same identity
follows from the Hilbert-norm limits already proved on those patches;
no unrestricted real-axis limit is needed.

For physical fields, polar truncations converge on the polynomial
domain as well as on the vacuum.  At mutually spacelike real
placements their product limit is consequently the ordinary smeared
product.  For a joint source kernel, finite tensor-test approximation
and the joint distribution estimates extend this equality; the
kernel itself need not have finite tensor rank.  Tube uniqueness
identifies their point vectors everywhere.
Finally the estimates above give distributional boundary convergence
locally uniformly on the good real patches.  In any closed strict
subcone of approach, point bounds near such an edge are polynomial
in inverse distance: test the real variables on balls at that scale
and use the holomorphic mean-value estimate on smaller complex balls.
\end{proof}

\subsection{A boundary pullback rule}

The preceding products will be composed with analytic coordinate
changes before their real limits are taken.  The following formulation
specifies the required boundary operation.

\Needspace{6\baselineskip}
\begin{lemma}\label{E:boundary-calculus}
Let a Hilbert-valued holomorphic function have a local distributional
boundary $u$ and polynomial distance bounds in closed strict approach
subcones.  Let $\Phi$ be a real analytic local diffeomorphism whose
complexification takes a smaller wedge strictly into such approach
directions.  Then its composition with $\Phi$ has boundary
$\Phi_{\R}^{*}u$ on that smaller wedge.
\end{lemma}

\begin{proof}
Shrink the real patch and the approach cone so that the image of a
small imaginary translate of size $\delta$ is a graph at distance
comparable to $\delta$ inside one fixed strict convex subcone.
Interpolate that graph to a straight imaginary translate there.
Suppose the holomorphic function is $O(\delta^{-m})$ on this
interpolation.  Extend a compact smooth test almost analytically to
order $N$: its $\bar\partial$ is $O(\delta^N)$, uniformly with the
finitely many derivatives required on the patch.  Stokes' formula
for the holomorphic volume form compares the two integrals.  Their
error is bounded by a constant times $\delta^{N+1-m}$; take
$N>m+1$.

There is a minor test-function issue in this comparison.  The desired
coefficient on the moving graph is a fixed smooth real test, whereas
an almost-analytic extension evaluated on that graph has nonzero
imaginary Taylor terms.  Include the holomorphic Jacobian, and choose
the real test on the straight surface by inverting these Taylor terms
through order $N$.  More explicitly, the graph evaluation operator
on test jets is $\id+\delta B_1+\cdots+\delta^N B_N$, where each
$B_j$ is a differential operator with smooth coefficients.  It has a
unique inverse modulo $\delta^{N+1}$, obtained successively in each
power of $\delta$.  This is the negative-imaginary-displacement
Taylor correction.  Its corrected real tests converge in every
smooth seminorm to the ordinary real change-of-variables test; the
remaining graph error is $O(\delta^{N+1-m})$.  The boundary limit on
the straight surface is therefore precisely the distributional real
pullback.  Pairing with arbitrary Hilbert vectors proves the weak
statement, and the uniform estimates prove the asserted
Hilbert-valued distributional convergence.
\end{proof}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/06-extraction.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/07-short-products.tex ====
\section{Short products and the return to the accessible scalar}
\label{sec:short-products}

The scalar vacuum wave $l$ has not yet been realized as a physical field.
In this Section, $L$ is an insertion symbol whose one-vacuum wave is $l$;
it is not an operator on the polynomial field domain.  We construct only
products of total length at most four, including derivatives, and prove
exactly the factorization statements needed below.  The symbols
\begin{equation}
 J=V-\partial L,\qquad
 t=T-\frac13(\eta\Box-\partial\partial)L,\qquad
 \ell=L-S
 \label{P:formal-fields}
\end{equation}
initially have this same meaning.  At the end we prove $l=S$ as vacuum
waves.  Construction of mixed products at every finite length and of the physical
operator $L$ is deferred to Section~\ref{sec:reconstruction}.

We use the ordered-vector-tube convention of
Section~\ref{sec:extraction}: the vacuum is on the right, and
\begin{equation}
 \Im z_1\in V_+,\qquad \Im(z_{j+1}-z_j)\in V_+.
 \label{P:vector-tube}
\end{equation}
The strict first inequality is necessary for vector products.  Scalar
vacuum functions depend only on differences and require only the remaining
inequalities.  A derivative always acts on its own insertion.  All entries used in this Section are even.  Tensor
indices and source smearings in the following constructions are retained,
even when suppressed in the notation.

The length bound is chosen to support a specific Hilbert-space
argument.  Actual bounded products and finite interpolation will
first construct the short vectors and their split identities.  Their
four-point scalar functions then determine inner products of pairs,
which give a conformal representation on the pair space.  Its radial
positivity and the normalized current fluxes will force the residual
wave $l-S$ to vanish.

\subsection{Finite interpolation on spacelike patches}

\Needspace{6\baselineskip}
\begin{lemma}[Common wedge for small null flows]
\label{P:common-wedge}
Fix any finite family of sufficiently small, mutually spacelike real
source patches.  There are a point $o$ strictly to their future, an open set of
future null directions $n$, and, for each finite set of sufficiently small
null special-conformal parameters in that set, an opposite open cone
$O_-$ with the following property.  Bounded operations on the source
patches, or on their inverse coordinate images, after conjugation by the
actual centered null-charge unitaries lie in wedges spacelike to $O_-$.
The vectors $C\Omega$, with $C\in\mathcal M(O_-)$, form a total set.
These choices persist under sufficiently small changes of the source
patches and Lorentz transformations.
\end{lemma}

\begin{proof}
Choose a future null direction $m$ separated from the directions $n$ and
use $m$ as the conjugate wedge axis.  Work strictly below the null cut
plane centered at $o$.  In the resulting coordinates put $p(y)=|y|^2$.
Each source patch fits, with a strict margin, inside a cut
$W_{\alpha p+c}$ with $\alpha>0$ sufficiently small and $c$ sufficiently
negative.  The same containment holds for the finitely many inverse
coordinate images when the flow parameters are small enough.

Here is a two-sign bound uniform in quadratic truncation.  Choose smooth
bounded profiles $0\leq q_R\leq p$, increasing to $p$, and write
$U_R(t)=e^{itH_{q_R}}$.  For $0<t<\alpha$, cut inclusion and the
constant-shift identity give
\[
 \cM(W_{\alpha p+c})\subset\cM(W_{tq_R+c})
       =\Ad(U_R(t))\cM(W_c).
\]
Thus conjugation by $U_R(-t)$ sends the source algebra into the fixed
wedge algebra $\cM(W_c)$.  For positive $t$, the source algebra is
already contained in $\cM(W_c)$, and
\[
 \Ad(U_R(t))\cM(W_c)=\cM(W_{tq_R+c})\subset\cM(W_c).
\]
Both signs therefore have the same containing wedge for
$|t|\leq t_0<\alpha$, independently of $R$.  This argument uses
$q_R\leq p$, not a bound on the diverging norms $\|q_R\|_\infty$.
The positive moment construction of Section~\ref{sec:null-current}
gives $U_R(t)\to e^{itK_n}$ strongly.  Weak closedness of the fixed
wedge algebra now preserves the containment for each bounded conjugated
operation in this limit.  Translations give the centered version used
here.

For finitely many choices, take $m\cdot x$ sufficiently negative, then
add a sufficiently large positive multiple of $m$.  This leaves
$m\cdot x$ unchanged and makes all the finitely many $n\cdot x$
sufficiently positive.  A small open double cone about such an $x$ lies
in the required opposite region.  All inequalities are strict, hence
stable under the small changes asserted in the statement.  Local
vacuum cyclicity in this double cone proves totality.  No cyclicity of a
relative commutant is used.
\end{proof}

Let $g$ be the standard null special-conformal coordinate map centered at
$o$, with small parameter vector $b_g=sn$, and let $U_g$ denote the actual
unitary generated by the corresponding centered $K_n$.  On the
one-field space these actions agree canonically by
Theorem~\ref{C:canonical-Pi}.  This does not assert covariance on the
full physical Hilbert space.  Reparametrize an input at $g^{-1}x$ and
multiply its test and indices by the inverse primary factors at $x$.
The transformed one-vacuum data from accessible inputs $S,V,T$ are then
\begin{equation}
 S(x),\qquad
 V(e_j;x)+\frac{4b_g\cdot e_j}{1+2b_g\cdot(x-o)}L(x),\qquad
 t(x)+\mathcal P_g L(x).
 \label{P:transformed-data}
\end{equation}
Here $e_j$ is a fixed target polarization, and $\mathcal P_g$ is a
tensor-valued differential operator of order at most two with analytic
coefficients on the patch.  It is obtained by transforming
$(\eta\Box-\partial\partial)L/3$ as derivatives of a scalar primary of
weight two.  In particular
$\mathcal P_0=(\eta\Box-\partial\partial)/3$.
The coefficient in the vector formula is the derivative of the scalar
primary factor $(1+2b_g\cdot(x-o))^2$; this fixes its normalization.
All these identities also hold after differentiation and source
smearing.

\Needspace{6\baselineskip}
\begin{lemma}[Finite null interpolation]
\label{P:interpolation}
For $q\leq4$ and generic independent polarizations $e_1,\ldots,e_q$,
finite evaluations at arbitrarily small future null $b$ distinguish the
$2^q$ functions
\begin{equation}
 b\longmapsto\prod_{j\in I}
       \frac{4b\cdot e_j}{1+2b\cdot(x_j-o)},
 \qquad I\subset\{1,\ldots,q\},
 \label{P:interpolation-functions}
\end{equation}
uniformly for $x_j$ in sufficiently small compact source patches.
The inverse matrix is analytic in the positions, also in a complex
neighborhood where its determinant stays nonzero.
\end{lemma}

\begin{proof}
Complete the polarizations to a generic basis and put $z_j=b\cdot e_j$.
A polynomial in the square-free monomials $z_I$ that vanishes on an open
part of the null cone vanishes on the full complex quadric and is
divisible by its nondegenerate quadratic equation $Q(z)$.  This last
assertion follows, for example, by solving the quadric for one variable
on a nonsingular chart and applying polynomial division; the remainder
vanishes on an open set and is zero.  In a generic basis $Q$ has a nonzero
$z_j^2$ coefficient for every $j$.  If $Q R$ were nonzero and
multi-affine, then
$\deg_{z_j}(QR)=2+\deg_{z_j}R\geq2$, a contradiction.  Thus the
square-free monomials are independent on every such cone patch.
There is consequently a finite nonsingular evaluation matrix.  Varying
ray lengths separates its homogeneous degrees.

Scale all its sample vectors by $\varepsilon>0$ and divide column $I$
by $(4\varepsilon)^{|I|}$.  As $\varepsilon\downarrow0$, the matrix in
\eqref{P:interpolation-functions} converges uniformly on compact source
patches to the nonsingular monomial matrix.  Its determinant remains
nonzero for sufficiently small $\varepsilon$.  Matrix inversion gives
the last assertion.  Thus the interpolation involves multiplication of
distributions by smooth functions, never pointwise inversion of a
distribution.
\end{proof}

The limit $\varepsilon\downarrow0$ in the preceding proof is used
only to find a nonsingular evaluation matrix.  In the construction
below we fix one sufficiently small positive scale for the compact
placement or continuation geometry under consideration.  We then
shrink the position neighborhoods so that its determinant stays
bounded away from zero, and only afterward take the bounded-
approximation and boundary limits.  The inverse constants may depend
on this fixed scale; a uniform inverse as the scale tends to zero is
not needed.

Generic polarizations also suffice for exact tensor recovery.  The
decomposable tensors coming from admissible generic polarization
tuples span the full tensor product: a linear functional vanishing on
that open set is a multilinear polynomial vanishing identically.
Choose a finite spanning subset once.  Every requested tensor
component is then a fixed linear combination of those evaluations,
without inverting an evaluation matrix at a nongeneric tuple.

\Needspace{6\baselineskip}
\begin{proposition}[Hilbert products through length four]
\label{P:short-products}
For entries from $t,S,V,L$, and hence $J,\ell$ and their derivatives,
there are Hilbert-valued product distributions on mutually spacelike
real patches through total length four.  They are multilinear,
permutation-symmetric, and covariant under translations, Lorentz
transformations and dilations.  They satisfy every leg-local linear
identity of their vacuum waves.  With $C$ in a common opposite cone,
they obey
\begin{equation}
 \ip{C\Omega}{\Psi[O_1\cdots O_p]}
 =\ip{O_1^\dagger\Omega}{C^*\Psi[O_2\cdots O_p]}.
 \label{P:recursive-product}
\end{equation}
Smearings on the left side of the inner product are conjugated.
The scalar functions factor at every split of total length at most
four.  Explicitly, for $p+q\leq4$ and both words in their vector tubes,
\begin{equation}\label{P:short-splitting}
 \begin{split}
 &\big\langle\Psi[A_1(z_1)\cdots A_p(z_p)],
                  \Psi[B_1(w_1)\cdots B_q(w_q)]\big\rangle\\
 &\quad=W\bigl(A_p^\dagger(\bar z_p),\ldots,
               A_1^\dagger(\bar z_1),B_1(w_1),\ldots,B_q(w_q)\bigr).
 \end{split}
\end{equation}
Here an empty word is $\Omega$, and bra source tests are conjugated
and reversed.  The reversed bra gaps, the ket gaps, and the central
gap $\operatorname{Im}w_1+\operatorname{Im}z_1$ are all future
directed.  Thus the scalar word is in its gap-only tube.  In particular
the case $p=q=2$ gives the pair--pair Gram identities.  All these
products extend holomorphically to
\eqref{P:vector-tube} and the spacelike tie neighborhoods of
Proposition~\ref{E:accessible-products}; the split identities persist
wherever both vector factors are defined.
\end{proposition}

\begin{proof}
Proceed by induction on length, and at a fixed length by increasing
number of $t$ slots.  Length one is the given one-vacuum construction.
For accessible inputs use the localized bounded approximants of
Proposition~\ref{E:bounded-approximation}.  Conjugate every factor of an
approximating real product by the same $U_g$.  Lemma~\ref{P:common-wedge}
allows the first factor to commute past $C$.  Vacuum-vector and
adjoint-vacuum convergence for this factor, and convergence of the
shorter tail, therefore yield the right side of
\eqref{P:recursive-product}.  By the shorter induction assertions this
is the multilinear expansion of \eqref{P:transformed-data}, initially
as a functional of $C\Omega$.

Use $T$ in positions intended eventually to contain $t$.  Terms with
fewer $t$ slots have representing vectors already and can be subtracted:
replacing a transformed $T$ by $\mathcal P_gL$ lowers this induction
index.  Interpolation is then needed only for the remaining $V/L$
choices.
At the remaining $V$ positions the unknown functionals are indexed by
the subsets replaced by $L$, with the coefficients
\eqref{P:interpolation-functions}.  Lemma~\ref{P:interpolation} expresses
each unknown as a finite smooth linear combination of already existing
Hilbert-valued distributions.  They therefore have representing
vectors.  Generic polarization choices suffice: multilinearity,
followed by a finite choice of generic bases, recovers all components.
This proves \eqref{P:recursive-product} at the new length.

The same argument with any other sufficiently small $g$ proves the
expansion formula, because it proves equality against the total family
$C\Omega$ in a common opposite cone.  Subdivision of patches or a
change in all auxiliary choices gives the same vectors recursively by
\eqref{P:recursive-product}.  Comparing a construction with a small
translated, Lorentz-transformed or rescaled construction proves the
similarity laws.  Interpolating the symmetry of transformed accessible
products proves spacelike permutation symmetry, again first subtracting
terms with fewer $t$ slots.  Inserting a zero vacuum wave into
\eqref{P:recursive-product}, after permuting it to the first position,
gives zero.  Hence, in particular,
\begin{equation}
 \Box L=\partial\cdot V=\Theta,\qquad
 \partial\cdot J=0,\qquad
 \partial^\mu t_{\mu\nu}=0,\qquad t^\mu{}_{\mu}=0
 \label{P:leg-identities}
\end{equation}
hold in these products.  This argument concerns vector distributions,
not operator-domain equalities.

For bounded approximants the vacuum expectation of a product splits at
any position by Hilbert-space adjunction.  The same is true for its
common unitary conjugate.  Expand the full scalar product and both
factors by \eqref{P:transformed-data}; real transformations respect
adjunction.  Subtract the terms already split by induction in the
number of $t$ slots.  The identical finite interpolation now proves
\eqref{P:short-splitting} for every remaining entry list, with the
same adjoint and coordinate convention as
\eqref{E:accessible-splitting}.  Thus short-length
associativity and positivity come from actual bounded products.

It remains to continue the vectors, rather than merely their matrix
coefficients.  Fix a compact continuation path from a spacelike
boundary patch to an interior tube point or one of the allowed ties.
Choose the parameter scale in Lemma~\ref{P:interpolation} small enough
that every inverse image $g^{-1}z_j$ stays in the accessible product
domain along that path.  Near a tie use its open holomorphic
neighborhood.  The analytic inverse matrices remain nonsingular on a
small neighborhood of the compact path.  Applying $U_g$ to the
accessible analytic vectors and performing the same successive
subtractions and inversions constructs the desired holomorphic
vectors.  The boundary calculus of
Lemma~\ref{E:boundary-calculus} identifies their distributional
boundary with the real construction: the coordinate maps take a
smaller approach wedge strictly into the original wedge.

There is no choice of branch.  On overlapping connected compact
continuation neighborhoods choose parameters sufficiently small for
both constructions; their boundaries agree on a smaller original
patch, and distributional edge-of-the-wedge uniqueness makes the
holomorphic vectors agree.  Exhaustion gives the full ordered tube.
At a tie the accessible germs and previously constructed terms agree
across the relevant permutations, so the interpolated germs agree as
well.  All leg identities and similarity laws continue by the same
uniqueness.  Finally, the two sides of each split identity have equal
boundary distributions when bra and ket are separately smeared on the
original spacelike patches and approached in strict subcones.
Uniqueness in their common split tube proves the asserted continuation
of the identity.  Only a local distributional boundary and an open
convex approach subcone are needed here; no global boundary estimate
for arbitrary $L$ products has been assumed.
\end{proof}

\Needspace{6\baselineskip}
\begin{lemma}[Comparison with the single-insertion forms]
\label{P:single-comparison}
The analytic pairs $LV,VL,JV,VJ,tV,Vt$ agree, when paired with an
ordinary physical polynomial bra, with the corresponding insertions of the
tempered single-$L$ forms of Proposition~\ref{C:single-insertion}.
\end{lemma}

\begin{proof}
First take a finite sum of factorized compact physical bra words and
translate it into a common opposite cone.  Its affiliated bounded
truncations can be used as $C$ in
\eqref{P:recursive-product}.  With $L$ first, the adjoints of these
truncations acting on an ordinary smeared $V\Omega$ converge on the
polynomial domain.  Locality of the single-insertion forms therefore
gives equality on an open real spacelike patch.  For a compact joint
bra kernel, approximate by factor tests in the permitted smooth
topology, preserving the opposite-cone margin.  Continuity of the
vacuum and adjoint-vacuum graph vectors and of the single-insertion
distributions extends this equality to that kernel.  Give bra and pair
strictly separated ordered imaginary translations.  The known
single-insertion spectral cones give analyticity in the two pair
arguments and the common translation of the bra, with its internal
spectator tests fixed.  Analytic uniqueness extends the equality, and
one may remove the separate bra damping while keeping the pair damped.
The other pairs follow from their linear differential definitions.
\end{proof}

\subsection{The Ward rows and the alternating exception}

The vector construction supplies finite transformation identities
for products formed from the accessible entries.  To obtain the
canonical Ward identity for each interpolated entry separately, we
must recover every row of the triangular system mixing $V$ and $L$.
The possible four-slot alternating term is the one case not removed
by the square-free interpolation argument; its elimination uses the
purely timelike spectrum of the last scalar wave.

\Needspace{6\baselineskip}
\begin{proposition}[Conformal covariance of the short scalar functions]
\label{P:short-covariance}
All scalar vacuum functions $W$ of total length at most four obey the
infinitesimal conformal Ward identities, with $L,S,J,t$ primary of
weights $2,2,3,4$ and spins $0,0,1,2$, respectively.  The laws for $V,T$
are their derivative laws from \eqref{P:formal-fields}.
\end{proposition}

\begin{proof}
Fix the $t$ and $S$ positions.  At the remaining $m_0\leq4$ positions
start with polarized vectors $V(e_j)$, and let $W_I$ be the function
obtained by replacing the positions in $I$ by $L$.  The canonical
special-conformal Ward operator $\widehat K_a$ acts diagonally by the
primary differential actions and has off-diagonal part
\begin{equation}
 (\widehat K_a W)_I\big|_{\mathrm{off}}
       =4\sum_{j\notin I}(a\cdot e_j)W_{I\cup\{j\}}.
 \label{P:ward-off-diagonal}
\end{equation}
These operators commute: they are the canonical commuting special
transformations of primaries and their derivatives.  They may be
centered at $o$, since the difference from another center is an affine
Ward operator, already zero on every row.

Unitarity and the expansion in Proposition~\ref{P:short-products} give
\begin{equation}
 (e^{s\widehat K_n}W)_\varnothing=W_\varnothing
 \label{P:null-row}
\end{equation}
for small $s$ and the open family of null $n$.  Initially the tensor
positions contain $T$.  Induct in their number, substitute
$T=t+\mathcal P_0L$, and subtract terms with fewer $t$ positions, whose
covariance is already established.  This gives
\eqref{P:null-row} with the indicated $t$ positions.

Write $X_I(a)=(\widehat K_aW)_I$.  Differentiating
\eqref{P:null-row} once and using the linear span of the open null family
gives $X_\varnothing(a)=0$ for every $a$.  Suppose that $X_I=0$ when
$|I|<k$.  Commuting two Ward operators on a row $H$ of size $k-1$ and
differentiating \eqref{P:null-row} $k+1$ times give, respectively,
\begin{align}
 \sum_{j\notin H}z_j(a)X_{H\cup\{j\}}(b)
   &=\sum_{j\notin H}z_j(b)X_{H\cup\{j\}}(a),
       \label{P:row-symmetry}\\
 \sum_{|I|=k}z_I(n)X_I(n)&=0,
       \label{P:row-null}
\end{align}
where $z_j(a)=a\cdot e_j$ and $z_I=\prod_{j\in I}z_j$.
Indeed all diagonal differentiations of already zero rows vanish; the
first surviving paths through the triangular off-diagonal array are
exactly the $k!$ orders of adding the same $k$ indices.  The common
nonzero factors $4$ and $k!$ have been canceled.

Complete a generic set of polarizations to a basis and write
$X_I(a)=\sum_{h=1}^4z_h(a)x_{Ih}$.  Equation~\eqref{P:row-symmetry}
says that, for each $H$, the matrix with rows outside $H$ equal to
$x_{H\cup\{j\},h}$ and its other rows zero is symmetric.  If $k\geq2$,
choose $H=I\setminus\{j\}$.  For $h\in I$ choose $j\ne h$; its zero
row kills $x_{Ih}$.  A completed-basis row $h>m_0$ likewise kills that
entry.  For $h\notin I$, symmetry identifies all the surviving entries
with a coefficient $c_{I\cup\{h\}}$, depending only on the union.
Thus \eqref{P:row-null} becomes
\begin{equation}
 (k+1)\sum_{|B|=k+1}c_Bz_B(n)=0.
 \label{P:square-free-ward}
\end{equation}
The square-free independence proved in
Lemma~\ref{P:interpolation} kills every coefficient, including the case
where there is no such union.

For $k=1$, \eqref{P:row-symmetry} instead gives a symmetric matrix.
Its quadratic form vanishes on the null cone, so it equals a scalar
multiple of the metric quadratic.  If $m_0<4$, its zero completed-basis
rows force that multiple to vanish.  If $m_0=4$, let
$G_{ij}=e_i\cdot e_j$.  The coefficient matrix is $\beta G^{-1}$,
which says
\begin{equation}
 X_{\{i\}}(e_j)=\beta\delta_{ij}.
 \label{P:exception-evaluation}
\end{equation}
The function $\beta$ is multilinear in all four polarizations: row $i$
is independent of $e_i$, and evaluation on $e_i$ restores its linear
factor.  It vanishes when two polarizations coincide, by
\eqref{P:exception-evaluation} and continuity from generic choices.
It is therefore alternating.  The sole possible exception is
\begin{equation}
 X_{\{i\}}(a)=
 \lambda\,\epsilon(e_1,\ldots,e_{i-1},a,e_{i+1},\ldots,e_4).
 \label{P:alternating-exception}
\end{equation}
Here there are exactly four $V/L$ positions and no other entries.
Divergence in position $i$ of $X_\varnothing=0$ also gives
\begin{equation}
 \Box_iX_{\{i\}}=0.
 \label{P:exception-harmonic}
\end{equation}
For clarity, the canonical array preserves the relation
$\partial\cdot V=\Box L$: its difference is the divergence of the
primary dimension-three vector $J$, whose divergence transforms
homogeneously.  This is the intertwining identity used in deriving
\eqref{P:exception-harmonic}.

We eliminate \eqref{P:alternating-exception} using its last position,
$i=4$.  Smear the first three positions on a real mutually spacelike
patch, keeping the last argument in the future tube.  Split the scalar
function into the triple bra and the one-wave ket.  Its boundary in the
first three positions is obtained from negative ordered bra dampings.
Move the center of the generator sufficiently far to the future of
these compact supports; affine covariance leaves $X$ unchanged.
The last-position canonical differential action on $L$ is exactly the
canonical generator applied to $l(z_4)$.  The ordinary physical triple
$VVV$ bra belongs to the domains of these centered actual $K_n$ by
Lemma~\ref{C:K-domains}.  Moreover $\Pi$ reduces these
generators and their actions on $\Pi$ are canonical by
Theorem~\ref{C:canonical-Pi}.  Projecting the bra to $\Pi$ thus
allows the last-position generator to be moved to the fixed bra.
Linear combinations of the available null parameters span arbitrary
$a$.

Every other term of the smeared $X_{\{4\}}(a)$ pairs the unchanged
last wave with a fixed constructed triple bra, possibly differentiated
or with replacements.  Consequently the whole expression is
$\ip{\eta}{l(z_4)}$ for a fixed Hilbert vector $\eta$, with no
coordinate multiplier left on the last wave.  Its translation
spectrum is purely timelike, since that of $l$ is.  Equation
\eqref{P:exception-harmonic} says that this spectrum is supported on
$p^2=0$; these disjoint supports force it to vanish.  More explicitly,
after any fixed future damping it is a finite spectral pairing, and
multiplication by $p^2$ kills it only if that pairing is zero off the
null shell.  There is no shell or zero-momentum component of $l$.
Boundary uniqueness in the first three arguments and generic
polarization then give $\lambda=0$.

The induction on $k$ is now complete, followed by the induction in the
number of $t$ positions.  This proves every row of the canonical Ward
system.
\end{proof}

\subsection{Euclidean continuation and the residual at infinity}

Wick coordinates are $(\tau,\boldsymbol r)\mapsto i\tau e+\boldsymbol r$.
Sort points by increasing $\tau$ and use the tie neighborhoods when
heights coincide.  A distinct Euclidean configuration has only
spacelike equal-height ties, so the resulting scalar functions are
real analytic on the full distinct-point configuration space.  Use
complex-linear tensor pullback with basis $C_0=ie$ and real spatial
bases $C_j$.  The identities become
\begin{equation}
 V=J+\partial L,\qquad
 T=t+\frac13(\delta\partial^2-\partial\partial)L.
 \label{P:euclidean-identities}
\end{equation}
Here $\partial^2$ is the Euclidean Laplacian.  Integrating the Ward
identities along nonsingular local conformal flows gives their finite
covariance, with reordering through tie neighborhoods as required.

For a scalar primary $A$ of dimension two define
\begin{equation}
 W(A_\infty\,\cdots)=\lim_{R\to\infty}R^4W(A(Ru)\,\cdots),
 \qquad |u|=1.
 \label{P:infinity-definition}
\end{equation}
The limit exists and has the primary transport law.  To verify this,
choose a Euclidean special-conformal transformation carrying infinity
to a finite point disjoint from the spectators.  Join it to the
identity by a flow whose pole ray avoids the finite set of points,
including the moving large point.  At the final transformation every
argument is distinct and finite, so ordinary local analyticity and the
weight-two factor give the limit.  This argument is uniform on small
complex neighborhoods of strictly time-ordered spectator
configurations and proves locally holomorphic convergence there.  It
also proves independence of the direction in
\eqref{P:infinity-definition}.

\Needspace{6\baselineskip}
\begin{proposition}[Residual infinity decoupling]
\label{P:residual-infinity}
For the residual scalar $\ell=L-S$,
\begin{equation}
 W(\ell_\infty O_1\cdots O_j)=0,
 \qquad O_i\in\{S,V,T\},\qquad j\leq3,
 \label{P:accessible-decoupling}
\end{equation}
at all distinct Euclidean points, also after differentiating any
spectator insertion.
\end{proposition}

\begin{proof}
We justify passage from the localized-vector result of
Corollary~\ref{E:infinity-vanishing} to these products; unrestricted
timelike real boundaries for accessible products are unnecessary.
For a real vector localized in a fixed line diamond, the projected
Paley multiplier in that Proposition has zero value at energy zero.
The same local Paley bound controls its first derivative there.  Thus
it is $O(E)\norm{\xi}$ at low energy.  Integrating with the
weight-two energy powers against $e^{-RE}$ in the measure $\dd E/E$,
and using exponential damping beyond a fixed energy, gives
\begin{equation}
 R^4\abs{\ip{\ell(iRe)}{\xi}}
       \leq C R^{-1}\norm{\xi},\qquad R\geq1.
 \label{P:infinity-rate}
\end{equation}
For complex localized operator data the right side is bounded by the
sum of the vacuum and adjoint-vacuum norms.  Consequently the estimate
passes to mutually spacelike accessible products in that diamond by
the two bounded-approximation limits of
Proposition~\ref{E:accessible-products}.

Choose tests on small patches about equal-time mutually spacelike
points and continue their translations a short distance into strictly
increasing imaginary time.  Before using the improved real-edge
estimate, the corresponding smeared pairing multiplied by $R^4$ is
$O(R^2)$ uniformly up to a smaller good real patch: the one-wave norm
is $O(R^{-2})$ and the smeared product bounds are uniform there.  Apply
the subharmonic logarithmic estimate on a half-disk with this real
edge.  If its harmonic measure at an interior point is $\omega$, the
bound becomes
$O(R^{-\omega+2(1-\omega)})=O(R^{2-3\omega})$.
There is an interior open set with $\omega>2/3$, and hence convergence
to zero on that set.  The positive imaginary times can be fixed before
further shrinking the source tests.  On its strict Euclidean
configurations \eqref{P:infinity-definition} already converges locally
holomorphically.  Small real source offsets therefore commute with
that limit.  Unsmeared uniqueness gives
\eqref{P:accessible-decoupling} on an open set, and real analyticity
continues it to every distinct configuration.  Derivatives follow
from the same locally holomorphic identity.
\end{proof}

\subsection{The pair representation and normalized fluxes}

The four-point split identities now supply a Hilbert-space conformal
action on pairs. Its normalized current fluxes will control the possible
singular terms in the radial argument below.

\Needspace{6\baselineskip}
\begin{proposition}[Conformal representation on the pair space]
\label{P:pair-representation}
Let $\mathcal H_{\leq2}$ be the closed span of $\Omega$ and the one- and
two-insertion tube vectors for $L,S,J,t$.  There is a strongly
continuous unitary representation of the universal cover of the
connected conformal group on this space, with the canonical local
primary transformations.  Translations coincide with the actual
positive-energy translations.  Its timelike special-conformal
generator $K_e^{\mathrm{rep}}$ is nonnegative.
\end{proposition}

\begin{proof}
For each length restrict its arguments to a relatively compact open
box in its tube.  These vectors span a dense subspace: a vector
orthogonal to them gives a holomorphic scalar function zero on an
open set, and hence zero throughout the tube.  For all sufficiently
small real conformal transformations, map these vectors by their
primary transformation laws.  The resulting inner products are
unchanged by the split identities of
Proposition~\ref{P:short-products} and the integrated scalar Ward
identities of Proposition~\ref{P:short-covariance}.  Total length four
is exactly what is needed for pair--pair inner products.  The image
is dense by the same open-box argument; thus the map extends to a
unitary.  On smaller boxes the local composition identity is exact.
Holomorphic point dependence gives strong continuity.  Continuing
these local unitaries along paths integrates the action to the
universal cover.

Uniqueness extends the local laws to all tube points and, for
sufficiently small transformations, to Euclidean configurations in
the positive-time half-space.  The point vectors there are smooth
group vectors.  On translations the construction agrees with the
actual translation action, whose spectrum is future directed.  The
time-M\"obius subgroup conjugates its positive time-translation
generator to $K_e^{\mathrm{rep}}$, proving the final assertion.  No
conformal action on the full physical Hilbert space is asserted here.
\end{proof}

For a target $A(y)$ in the positive Euclidean time half-space let
$\Sigma$ be a small outward-oriented sphere about $y$, wholly within
that half-space.  Define Hilbert-valued fluxes from the analytic pair
vectors by
\begin{equation}
 F_J(A;y)=\int_\Sigma\dd\Sigma^a\,\Psi[J_a(x)A(y)],\qquad
 F_X^t(A;y)=\int_\Sigma\dd\Sigma^a\,X^b(x)\Psi[t_{ab}(x)A(y)].
 \label{P:flux-definition}
\end{equation}
At each point of the sphere use the Euclidean order, with the tie
germs providing the transitions.  There is no factor of the sphere
area in \eqref{P:flux-definition}.  Conservation and tracelessness
make the corresponding holomorphic flux forms closed off the target,
so the contour may be deformed, or held fixed when differentiating
the target nearby.  If one instead rescales both currents' fluxes by
a common conventional constant, the constant $c$ below rescales with
them.

\Needspace{6\baselineskip}
\begin{proposition}[Short-length Ward fluxes]
\label{P:flux-wards}
With one consistent nonzero Ward normalization $c$,
\begin{align}
 F_J(A)&=0,& A&=V,L,J,\label{P:J-flux}\\
 F^t_{\partial_b}(A)&=c\,\partial_b A\Omega,& A&=V,L,J.
 \label{P:translation-flux}
\end{align}
For primary targets $L,J$ and every conformal Killing field $X$,
\begin{equation}
 F_X^t(A)=c\,\delta_X A\Omega.
 \label{P:conformal-flux}
\end{equation}
Here $\delta_X$ is the canonical differential action, including spin
and weight.  In particular, a special transformation centered at the
target has zero flux, while centered dilation about $L$ gives
$2cL\Omega$.  With outward orientation in
\eqref{P:flux-definition}, Wick time basis $C_0=ie$, the convention
$i[P_\nu,A]=\partial_\nu A$, and upper-minus-lower order $At-tA$ in
the cap difference, one has $c=-1$.
\end{proposition}

\begin{proof}
First take $A=V$ and pair with an arbitrary ordinary physical
polynomial bra.  Lemma~\ref{P:single-comparison} identifies the
analytic pairs with tempered single-insertion forms.  Deform
$\Sigma$ to a cylinder whose relative Euclidean time caps are
$\pm\delta$ and whose spatial disk has radius $a$.  Give the whole
relative time a small real Minkowski shift $s$, with $|s|<a$.
The shifted contour remains in the holomorphic domains; where the
imaginary relative time crosses zero on the side, strict spatial
separation and tie gluing provide a common neighborhood.

Average smoothly and with integral one over $a$ and $s$, keeping the
shift support strictly inside $|s|<a$.  The side contribution tends to
zero as $\delta\downarrow0$ because it lies on a compact nonsingular
spacelike set and its time thickness tends to zero.  The two caps now
have smooth Cauchy-slab tests and converge to the opposite Lorentzian
order boundaries at a common target damping.  This use of the
single-insertion boundary is legitimate even at a point in the common
complex target translation: its total pair momentum is exponentially
damped, while the smooth relative test decays rapidly in the relative
momentum.  The difference is the commutator charge with the time
average and a spatial cutoff equal to one on its causal support.
The $J$ charge is zero by Proposition~\ref{C:Q-zero}; the translation
charge of $t$ is the original translation charge by
Section~\ref{sec:charges-cones}.  They give
\eqref{P:J-flux}--\eqref{P:translation-flux} for $V$.

The Wick time component contributes $i$, and the tensor translation
index contributes the pullback $C_b^\nu$.  With the order and charge
convention in the statement,
$i[V,P_b]=-\partial_bV$, hence $c=-1$.  This calculation fixes one
common normalization for all components and all subsequently
transported Killing fields.  Pairing with the dense polynomial bras
proves equality of the Hilbert vectors.

A constant-current flux around a primary target extends to a
holomorphic single-wave function of its Minkowski argument throughout
the future tube: choose a relative sphere smaller than its common
damping.  Translation covariance makes it a translation wave;
dilations give its flux dimension.  Small Lorentz transformations act
on the pair and its contour; homotopy of the transformed contour back
to a small sphere gives the tensor transformation.  Differentiate the
identities for $V$ in its target index and use
$\partial\cdot V=\Box L$ while keeping the enclosing contour fixed.
One obtains the homogeneous wave equation for both
\begin{equation}
 F_J(L),\qquad
 F^t_{\partial_b}(L)-c\,\partial_bL\Omega.
 \label{P:harmonic-flux-defects}
\end{equation}
These are, respectively, a scalar wave of dimension two and a vector
wave of dimension three.  Translation covariance and positive-energy
holomorphy realize their values after any fixed future damping as
vacuum spectral amplitudes; further damping agrees by uniqueness.
Their wave equation confines their spectrum to $p^2=0$.
Scale covariance excludes a zero-momentum amplitude, and the
light-cone boost-weight test of Section~\ref{sec:foundations} excludes
the nonzero shell: in these cases $d-1>\max\omega$ for the finite
Lorentz multiplet.  Only local finiteness after exponential damping is
needed for that test.  Thus both defects vanish.  Subtraction of
$\partial L$ from $V$ proves the identities also for $J$.

For $A=L,J$, conjugate the translation identity by any sufficiently
small real conformal transformation in
Proposition~\ref{P:pair-representation}.  Since $t$ is a conserved
traceless primary of dimension four, its flux transforms exactly to
the flux for the pushed-forward vector field: three surface powers,
one vector-field power, and the two orthogonal index factors cancel
its weight.  Deform the transformed sphere back locally.  The target
primary factors are invertible, and the differential action on the
right transforms by the same Lie algebra adjoint action.  Translations
and all their such adjoints span the conformal Lie algebra (successive
conformal brackets give rotations, dilations and special
transformations).  Linearity and complexification give every
Euclidean conformal Killing field and prove
\eqref{P:conformal-flux}.
\end{proof}

The flux laws remain valid inside scalar Euclidean functions up to the
available split length, with exterior spectators, including
$\ell_\infty$.  First place those spectators earlier than the whole
pair by a Euclidean time plane and use the split identity, representing
infinity first by a large negative-time insertion.  Then continue in
the distinct spectator and target positions, choosing a smaller
contour where necessary.  One immediate consequence is
\begin{equation}
 W(\ell_\infty J_a(x)L(y))=0.
 \label{P:ell-J-L}
\end{equation}
Indeed rotations, translations and dilations give the only possible
form $d(x-y)_a/|x-y|^4$.  Its $J$ flux about $L$ is $d|S^3|$, and
\eqref{P:J-flux} makes $d=0$.  Primary covariance transports this
vanishing to any placement of the infinity insertion.

We now have the pair representation and current flux identities
with one fixed normalization.  The remaining argument turns radial
singularities of short functions into spectral statements on that
pair space.  Only the representation classifications at energies
zero, one and two will be needed for this step, so the analysis does
not require a discrete expansion of
all radial states.

\subsection{Radial Hilbert calculus and its low energies}

The function to be tested is
$W(\ell_\infty\ell(x)V_\mu(y)V_\nu(z))$.
Residual infinity decoupling will make it harmonic in $x$.  We will
identify each singular radial power with an actual low-energy spectral
projection on the pair space.  The low-energy bounds and a Maxwell
test leave only dipoles; a tangential reflection then removes those
dipoles from a positive pair norm.  The resulting norm bound eliminates
the remaining low-energy overlaps, and the normalized current fluxes
will finally test the residual scalar's own norm.  The following
calculus supplies these spectral and positivity statements without a
complete expansion into radial eigenstates.

Write a Euclidean point as $y=(y_0,\boldsymbol y)$.  The conformal map
\begin{equation}
 c_{\mathrm{ball}}(y)=
 \frac{(1-|y|^2,\,2\boldsymbol y)}{1-2y_0+|y|^2}
 \label{P:ball-map}
\end{equation}
takes the unit ball to the positive-time half-space.  Define radial
one- and two-insertion kets by primary pullback of the corresponding
half-space vectors; obtain descendants by differentiation.  Radial
adjunction reflects $y$ to $Iy=y/|y|^2$ with its full tensor and
weight factors.  Indeed direct substitution in
\eqref{P:ball-map} gives
$c_{\mathrm{ball}}(Iy)=(-c_{\mathrm{ball}}(y)_0,
 c_{\mathrm{ball}}(y)_{\mathrm{spatial}})$, exactly Wick reflection.
Scalar covariance therefore identifies radial splitting with Hilbert
adjunction.  To transport arbitrary separated interior and exterior
groups, each of size at most two, join the required proper Euclidean
conformal transformations by paths avoiding poles on the finite point
set.  A pole condition has codimension four, so such paths can be
chosen after a small perturbation.  Points at infinity are reached by
\eqref{P:infinity-definition}.  In particular the radial ket
$|\ell\rangle=|\ell(0)\rangle$ has bra $\ell_\infty$.

On the pair representation put
\begin{equation}
 H=\frac12(P_e+K_e^{\mathrm{rep}}).
 \label{P:radial-H}
\end{equation}
This is the self-adjoint compact-time generator on the covering group.
Its derived-generator identity holds on group-smooth vectors, a core
for this generator.  Both terms on the right are positive, so $H\geq0$.
The real Minkowski vector field of $H$, in Wick half-space coordinates,
is
$i((\tau^2-|\boldsymbol r|^2-1)/2,\tau\boldsymbol r)$.
The derivative of \eqref{P:ball-map} takes $iy$ to this field.  Thus
positivity and analytic continuation of the local group law give
\begin{equation}
 r^H|A_1(y_1)\cdots A_p(y_p)\rangle
   =r^{d_1+\cdots+d_p}
       |A_1(ry_1)\cdots A_p(ry_p)\rangle,
 \quad 0<r\leq1,\quad p\leq2.
 \label{P:radial-contraction}
\end{equation}
One first continues through a short contraction, where both analytic
laws apply, and then iterates through interior configurations.
Radial rotations act unitarily with the usual $SO(4)$ tensor law.

The real-form involution on complex radial transformations is complex
conjugation followed by conjugation with $I$.  Write $p_a^+,p_a^-$,
$H,M_{ab}$ for the derived actions corresponding, respectively, to
radial translation, $2y_a y-|y|^2e_a$, dilation, and
$y_a e_b-y_b e_a$.  On smooth vectors,
\begin{equation}
 (p_a^+)^\dagger=p_a^-,\qquad
 [H,p_a^\pm]=\pm p_a^\pm,\qquad
 [p_a^-,p_b^+]=2\delta_{ab}H+2M_{ab}.
 \label{P:radial-algebra}
\end{equation}
These are active-point conventions.  For example,
$M_{ab}|A_d\rangle=\delta_{ad}|A_b\rangle-\delta_{bd}|A_a\rangle$
at the origin.  Inversion takes translations to negative special
transformations, giving the displayed matrix adjunction.

The positivity calculation below is the low-energy part of the
four-dimensional conformal unitarity-bound method of
Mack~\cite{Mack1977}; see also the level-one Gram-matrix derivation
in \cite[Section~2.5]{Minwalla1998}.  We give the required finite-array
calculation on the present pair representation, including its
smooth-density justification.

\Needspace{6\baselineskip}
\begin{lemma}[The required low radial energies]
\label{P:radial-low-energy}
The $H=0$ eigenspace is a trivial representation.  The $H=1$ eigenspace
has only scalar rotation types.  At $H=2$, among the types in a tensor
product of two vectors, only scalars and antisymmetric two-forms can
occur.  Every smooth antisymmetric energy-two array obeys
\begin{equation}
 \sum_a p_a^+|F_{ab}\rangle=0,\qquad
 \sum_a p_a^+|(*F)_{ab}\rangle=0.
 \label{P:Maxwell-null-vectors}
\end{equation}
No discreteness of the full radial spectrum is required.
\end{lemma}

\begin{proof}
We specify the smooth-density issue first.  The element
$Z=e^{2\pi iH}$ is central, since its adjoint action on the conformal
Lie algebra is the identity by \eqref{P:radial-algebra}.  Its fixed
subrepresentation contains every integer $H$ eigenspace.  The
projection onto that subrepresentation commutes with the group,
preserves smooth vectors and has dense smooth image.  There the
spectrum of $H$ is supported on the integers, and
\begin{equation}
 E_H(\{h\})v=\frac1{2\pi}\int_0^{2\pi}e^{it(H-h)}v\,\dd t
 \label{P:integer-projection}
\end{equation}
preserves smoothness.  Averaging also over the compact rotations gives
dense smooth vectors in each rotation type and dense smooth
finite tensor-equivariant arrays.  Rotations descend to $SO(4)$,
since their action on the dense radial kets is the ordinary tensor
action.  The computations below therefore extend by density to the
stated eigenspaces.

At zero energy the expectations of both positive terms in
\eqref{P:radial-H} vanish.  They annihilate a smooth zero-energy vector.
The translation spectrum cone then makes every translation generator
annihilate it, and the conjugate cone makes every special partner
annihilate it.  Their brackets annihilate it as well, proving
triviality.  If $Hv=v$, lowering has energy zero and
$\ip{w}{p_a^-v}=\ip{p_a^+w}{v}=0$ for every zero-energy $w$;
hence $p_a^-v=0$.

For any smooth lowest array of energy $h$, raising norms give
\begin{equation}
 \left\|\sum_a p_a^+v_a\right\|^2
   =2\sum_{a,b}\ip{v_a}{(h\delta_{ab}+M_{ab})v_b}\geq0.
 \label{P:raising-Gram}
\end{equation}
A two-by-two plane-rotation block gives $|m|\leq h$ for each plane
weight.  An $SO(4)$ tensor type has two $SU(2)$ spins $j_+,j_-$, their
sum an integer, with maximal plane weight $j_++j_-$.  At $h=1$ only
scalars, vectors and the two chiral two-form types survive this test.
The divergence arrays exclude the last two.  In detail, the tensor
index rotation rule gives
\begin{align}
 \sum_bM_{ab}|A_b\rangle&=-3|A_a\rangle,\label{P:vector-Gram}\\
 \sum_cM_{ac}|F_{cb}\rangle
   &=-3|F_{ab}\rangle-|F_{ba}\rangle
                  +\delta_{ab}\sum_c|F_{cc}\rangle.
 \label{P:tensor-Gram}
\end{align}
The matrix eigenvalues in \eqref{P:raising-Gram} are therefore $-3$
for the vector divergence, $-4$ for a symmetric traceless rank-two
divergence, and $-2$ for an antisymmetric rank-two divergence.
Their positivity bounds are $h\geq3,4,2$, respectively.  At $h=1$
this leaves only scalars.

At $h=2$ the types in two vectors are even under the central radial
rotation $y\mapsto-y$.  Their lowered arrays are odd but would have to belong to
the scalar-only energy-one space, so they are lowest arrays.  The
$-4$ bound excludes the symmetric traceless type.  The antisymmetric
bound is saturated and makes the first norm in
\eqref{P:Maxwell-null-vectors} zero.  Apply the same argument to the
Hodge dual, which intertwines rotations, to obtain its second
identity.  This proves exactly the needed low-energy statements
without an expansion in a complete family of local operators.
\end{proof}

\Needspace{6\baselineskip}
\begin{lemma}[The exterior Maxwell test]
\label{P:Maxwell-exterior}
An antisymmetric energy-two radial array has zero exterior overlap
with the pair $\ell_\infty,V_\nu(z)$, with $z$ outside the radial ball.
\end{lemma}

\begin{proof}
Let $B_\nu(z)$ be the reflected ket of this exterior data; it is a
smooth pair vector.  Its exterior translation and similarity laws,
including the full derivative reflection of $V$, are
\begin{equation}
 \partial_{z^a}B_\nu=-p_a^-B_\nu,\qquad
 (z\cdot\partial+1)B_\nu=-HB_\nu.
 \label{P:exterior-laws}
\end{equation}
Translation fixes the insertion at infinity; reflection changes its
generator to the lowering generator with the indicated sign.  For
the second formula the exterior vector contributes weight $-6$ and
the reflected interior pair dimension five, giving total degree
$-1-H$.  These statements need only similarity covariance of $V$.

For a smooth Maxwell array, set
$Q_{\nu ab}(z)=\ip{B_\nu(z)}{F_{ab}}$.
Equations~\eqref{P:Maxwell-null-vectors} and
\eqref{P:exterior-laws} make $Q$ homogeneous of degree $-3$ and
make it and its Hodge dual in $a,b$ divergence-free.  Rotation
covariance and antisymmetry leave precisely
\begin{equation}
 Q_{\nu ab}=A\,|z|^{-4}(\delta_{\nu a}z_b-\delta_{\nu b}z_a)
            +B\,|z|^{-4}\epsilon_{\nu abd}z_d.
 \label{P:Maxwell-tensors}
\end{equation}
For $n=z/|z|$, the divergence of the first tensor is
$|z|^{-4}(\delta_{\nu b}-4n_\nu n_b)$, while that of the second is
zero.  Hence $A=0$.  Hodge duality interchanges these tensors up to
orientation sign, so dual divergence gives $B=0$.  Dense smooth
arrays in the energy-two rotation types and continuity of the fixed
exterior pairing prove the general assertion.
\end{proof}

\subsection{Harmonic removal of the residual}

Consider the short Euclidean scalar function
\begin{equation}
 G_{\mu\nu}(x;y,z)
   =W(\ell_\infty\ell(x)V_\mu(y)V_\nu(z)),
 \qquad x,y,z\text{ distinct}.
 \label{P:G-definition}
\end{equation}
By \eqref{P:accessible-decoupling} and the leg identities,
\begin{equation}
 \partial_x^2G=0,\qquad
 \partial_{x^a}G_{\mu\nu}
       =-W(\ell_\infty J_a(x)V_\mu(y)V_\nu(z)).
 \label{P:G-harmonic}
\end{equation}
For the second equation use
$\partial\ell=V-J-\partial S$; the accessible terms vanish.
For the first, $\partial^2\ell=\Theta-\partial^2S$, whose two
terms likewise vanish in this correlation.

For fixed $y,z$, $G$ is bounded as $|x|\to\infty$.
To see the quantitative bound, transport the two scalars colliding
at infinity by a proper spherical inversion, adjusted by a reflection,
to a finite radial ball, placing the vector pair exterior.  The
separation $r'$ of the inner scalar pair is comparable to $|x|^{-1}$.
Its norm is at most $C(r')^{-4}$ by
\eqref{P:radial-contraction} and $H\geq0$, starting from a fixed
inner radius uniformly in angle.  The scalar inversion factor of the
moving point is $O(|x|^{-4})$, and cancels this bound.  The fixed
exterior pair has finite norm, proving boundedness.

We next determine the possible singularities at the two finite
points $y$ and $z$.  Harmonicity gives two radial powers in each
spherical harmonic.  Pair-space positivity will identify a singular
power with an actual low-energy spectral projection; the preceding
representation and flux results can then test that projection.

Project the harmonic function onto a spherical harmonic of degree
$q$ about $y$.  With $r=|x-y|$, its radial coefficient has the form
\begin{equation}
 g_q(r)=A_qr^q+D_qr^{-q-2}.
 \label{P:harmonic-coefficient}
\end{equation}
Choose a small radial ball centered at $y$, let $v_q$ be the harmonic
projection of the inner pair ket
$|\ell(y+r_0n)V_\mu(y)\rangle$ at fixed $r_0$, and let $B$ be the
fixed exterior ket.  The contraction identity gives
\begin{equation}
 r^5g_q(r)=r_0^5\ip{B}{(r/r_0)^Hv_q}.
 \label{P:harmonic-spectral}
\end{equation}
The spectral pairing
$\mu(E)=\ip{B}{E_H(E)v_q}$ is a complex measure of finite total
variation, at most $\norm{B}\norm{v_q}$.  Thus the right side is a
Laplace transform of a finite measure on $[0,\infty)$.
The two powers on the left have exponents $q+5$ and $3-q$.
If $q>3$, boundedness of this transform at $r=0$ kills $D_q$.
Otherwise uniqueness of finite-measure Laplace transforms identifies
$D_q$ with the actual point spectral projection at energy $3-q$.
One can verify uniqueness by setting $u=e^{-h}$: all integer
Laplace parameters give the polynomial moments on $[0,1]$, which
determine the finite measure.  In particular, a continuous spectral
component cannot imitate the singular power in
\eqref{P:harmonic-coefficient}.

The projected array transforms in $\mathcal Y_q\otimes\mathbf4$,
where $\mathcal Y_q$ is the degree-$q$ harmonic representation.
At $q=3$ the possible energy is zero, and at $q=2$ it is one.  Only
scalars occur there by Lemma~\ref{P:radial-low-energy}, whereas
$\mathcal Y_q\otimes\mathbf4$ contains a scalar only for $q=1$.
Both coefficients therefore vanish.  At $q=1$ the energy is two;
the possible types are the scalar and Maxwell two-forms.  The latter
have zero exterior overlap by Lemma~\ref{P:Maxwell-exterior}.  The
unique scalar intertwiner leaves precisely a dipole parallel to the
vector index.  The remaining $q=0$ monopole is killed by integrating
\eqref{P:G-harmonic} over a small sphere about $y$:
$F_J(V)=0$, whereas the radial derivative of $D_0r^{-2}$ has flux
$-2|S^3|D_0$.  Repeat at $z$.

Convergent harmonic expansions on punctured-ball annuli now show
that subtraction of these two dipoles removes both singularities.
The remainder is entire and bounded at infinity, hence constant in
$x$.  We have proved
\begin{equation}
 G_{\mu\nu}(x;y,z)=
 D_\nu(y,z)\frac{(x-y)_\mu}{|x-y|^4}
 +\widetilde D_\mu(y,z)\frac{(x-z)_\nu}{|x-z|^4}
 +C_{\mu\nu}(y,z).
 \label{P:G-decomposition}
\end{equation}
The constant is the limit at $x$ infinity, so its similarity laws are
immediate: it depends only on $y-z$ and is homogeneous of degree $-6$.

The only possible finite singularities are now the two displayed
dipoles.  A tangential polarization kills them simultaneously after
placing the two vector points at inverse radial positions.  This
choice turns the bounded remainder into a bound on a genuine pair
norm, which excludes its low radial energies.

Take radial center $x=0$, choose $|u|=1$, put $y=u/r$, $z=ru$, and
contract both vector indices with a fixed $w\perp u$.  Both dipoles
in \eqref{P:G-decomposition} vanish, and its last term is $O(r^6)$.
For clarity, the full radial reflection law for $V=J+\partial L$ is
\begin{equation}
 V_\mu(x)^\dagger
 =|x|^{-6}R_{\mu\nu}(x)V_\nu(Ix)
      -4x_\mu|x|^{-6}L(Ix),\qquad
 R_{\mu\nu}=\delta_{\mu\nu}-2\widehat x_\mu\widehat x_\nu.
 \label{P:V-reflection}
\end{equation}
The second term is the derivative of $|x|^{-4}$ in the scalar
reflection formula.  Tangential contraction removes it and leaves
$Rw=w$.  Radial splitting therefore gives
\begin{equation}
 \big\|\,|\ell(0)V_w(ru)\rangle\,\big\|^2
     =r^{-6}G_{ww}(0;u/r,ru)=O(1).
 \label{P:tangential-norm}
\end{equation}
This bounds an actual Hilbert norm.  If $\psi$ is the corresponding
ket at a fixed radius $r_0$ and $s=r/r_0$, contraction gives
\begin{equation}
 \big\|\,|\ell(0)V_w(ru)\rangle\,\big\|^2
    =s^{-10}\int_{[0,\infty)}s^{2h}\,
                   \dd\ip{\psi}{E_H(h)\psi}.
 \label{P:no-low-energy}
\end{equation}
A nonzero spectral mass on $[0,5-\varepsilon]$ would make this at
least a positive constant times $s^{-2\varepsilon}$, contrary to
\eqref{P:tangential-norm}.  Taking all $\varepsilon>0$ excludes the
whole spectrum below five, in particular overlap with
$|J_\rho\rangle$ at energy three.

Use \eqref{P:ell-J-L} and its conformal transports to remove the
$\partial L$ part of $V$ in that overlap.  The remaining three-point
function has the similarity form
\begin{equation}
 \langle J_\rho|\ell(0)J_\mu(z)\rangle
      =|z|^{-2}(a\delta_{\rho\mu}
                      +b\widehat z_\rho\widehat z_\mu).
 \label{P:ell-JJ-form}
\end{equation}
Tangential testing forces $a=0$.  Its divergence is
$|z|^{-3}(b-2a)\widehat z_\rho$, so conservation forces $b=0$.
Primary conformal transport yields
\begin{equation}
 W(\ell_\infty JJ)=0.
 \label{P:ell-JJ}
\end{equation}
Expand the already zero $W(\ell_\infty VV)$ using $V=J+\partial L$.
Equations~\eqref{P:ell-J-L} and \eqref{P:ell-JJ} leave the mixed
second derivative of $W(\ell_\infty L(y)L(z))$ equal to zero.
Translation covariance makes this a function of $y-z$ with zero
Hessian, and its degree $-2$ excludes a nonzero affine function.
Consequently
\begin{equation}
 W(\ell_\infty LL)=0.
 \label{P:ell-LL}
\end{equation}

The residual already decouples from the $JL$, $JJ$ and $LL$
three-point functions.  To test its own scalar norm, we need one more
identity involving the stress tensor.  We first remove the allowed
$Jt$ tensor structure with a centered special flux, and then use a
centered dilation flux on the remaining $Lt$ function.

It remains to use the stress flux.  Write $R=|z-y|$ and
$n=(z-y)/R$.  Rotation and scale covariance, symmetry in
$\rho,\sigma$, and the absence of a nonzero parity-odd symmetric
tensor give the general form
\begin{align}
 W(\ell_\infty J_\mu(y)t_{\rho\sigma}(z))
  =R^{-5}\big[&A n_\mu\delta_{\rho\sigma}
       +B(n_\rho\delta_{\sigma\mu}+n_\sigma\delta_{\rho\mu})
       +D n_\mu n_\rho n_\sigma\big].
 \label{P:ell-Jt-general}
\end{align}
Trace gives $4A+2B+D=0$.  Vector divergence gives the coefficients
$-2A+2B$ and $-2D-12B$ of, respectively,
$R^{-6}\delta_{\rho\sigma}$ and $R^{-6}n_\rho n_\sigma$.
Thus $A=B=\gamma$ and $D=-6\gamma$, giving
\begin{equation}
 W(\ell_\infty J_\mu(y)t_{\rho\sigma}(z))
 =\gamma R^{-5}
 (n_\mu\delta_{\rho\sigma}+n_\rho\delta_{\sigma\mu}
       +n_\sigma\delta_{\rho\mu}-6n_\mu n_\rho n_\sigma).
 \label{P:ell-Jt}
\end{equation}
This also obeys tensor divergence.  Take the special vector field
centered at $y$,
$X_\alpha^\sigma=2(z-y)_\alpha(z-y)^\sigma-R^2\delta_\alpha{}^\sigma$.
The contraction of its angular part with the tensor in
\eqref{P:ell-Jt} is
\begin{equation}
 n^\rho(2n_\alpha n^\sigma-\delta_\alpha{}^\sigma)
 (n_\mu\delta_{\rho\sigma}+n_\rho\delta_{\sigma\mu}
       +n_\sigma\delta_{\rho\mu}-6n_\mu n_\rho n_\sigma)
 =-\delta_{\alpha\mu}-2n_\alpha n_\mu.
 \label{P:special-angular}
\end{equation}
Its sphere average is $-3\delta_{\alpha\mu}/2$, since
$\langle n_\alpha n_\mu\rangle=\delta_{\alpha\mu}/4$.
The field contributes $R^2$ and the area $R^3$, canceling $R^{-5}$.
The flux is therefore
$-3\gamma|S^3|\delta_{\alpha\mu}/2$.  It is zero by
\eqref{P:conformal-flux}, because a centered special transformation
annihilates the primary $J$.  Hence $\gamma=0$.

\Needspace{6\baselineskip}
\begin{theorem}[Return of the scalar wave]
\label{P:return}
The formal scalar vacuum wave equals the accessible scalar:
\begin{equation}
 l=S,\qquad \ell=0.
 \label{P:return-equation}
\end{equation}
\end{theorem}

\begin{proof}
Expand $W(\ell_\infty V(y)T(z))=0$, which is a case of
\eqref{P:accessible-decoupling}, using
\eqref{P:euclidean-identities}.  The $Jt$ term has just vanished;
the terms with derivatives of $L$ vanish by
\eqref{P:ell-J-L} and \eqref{P:ell-LL}.  What remains is
$\partial_yW(\ell_\infty L(y)t(z))=0$.
Translation covariance and homogeneity of degree $-4$ force that
function to be zero.  Its centered dilation flux around $L$,
using \eqref{P:conformal-flux}, now gives
\begin{equation}
 0=2c\,W(\ell_\infty L).
 \label{P:final-flux}
\end{equation}
The energy-slice calculation in Proposition~\ref{E:accessible-scalar}
gave $D(e)=B(e,e)=b$ after excluding the massless contribution.  Thus
the constant two-point density of $S$ with $l-S$ vanishes: the scalar
wave channels are orthogonal, not merely their labels on one time
slice.  It follows that $W(\ell_\infty L)$ equals the
scalar two-point coefficient $W(\ell_\infty\ell)$, which is the
nonnegative radial norm $\norm{|\ell\rangle}^2$ and is strictly
positive for a nonzero scalar wave.  Since $c\ne0$,
\eqref{P:final-flux} makes that norm zero.  Therefore $l=S$.
\end{proof}
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/07-short-products.tex ====

==== BEGIN Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/08-reconstruction.tex ====
\section{Real-axis reconstruction and physical conformal currents}
\label{sec:reconstruction}

By Theorem~\ref{P:return}, the extracted scalar wave is the entire
trace polarization: $S=l$.  In this Section we write $L$ for that accessible
entry.  Until the joint domain is constructed below, this notation denotes
its vacuum wave and its analytic products, rather than an operator on
arbitrary states.  In particular, the vacuum-wave division by $M^2$ is
never extended to another leg.  The ingredients from Section~\ref{sec:extraction} are
bounded local approximations, analytic vector products of every finite
length, their reflected Hilbert splittings, and gluing at spacelike ties.
The leg equation is
\begin{equation}\label{R:leg-equation}
 \Box_i W(\ldots,L(x_i),\ldots)
       =W(\ldots,\Theta(x_i),\ldots).
\end{equation}
The remaining task is to establish joint tempered boundaries, their
ordered spectra, and affiliation with the original local algebras.

The construction has three stages.  First, we obtain real-axis
boundaries for pure scalar products and then for every mixed ordering.
Second, their Hilbert splittings provide a common invariant domain,
and bounded approximants give affiliation in the fixed net of
Assumption~\ref{ass:framework}.  Only after these steps do we regard
$L$ as a physical field and use its improvement to define the local
conformal Ward derivations.

The first stage proves the following joint boundary statement.  Its
scalar and vector conclusions are both needed: scalar positivity
defines insertion operators, while vector continuity identifies their
domain with actual vectors in the original Hilbert space.

\Needspace{6\baselineskip}
\begin{theorem}\label{R:mixed-products}
For every finite ordered list of $L$ and original physical fields,
the mixed analytic scalar product is the Fourier--Laplace continuation
of a tempered Wightman distribution, with the ordered spectrum,
adjunction, locality, and
similarity covariance, with dilations acting on the original field
labels.
The analytic vector products have tempered Hilbert-valued boundaries
with the vector spectrum.  Their reflected splittings hold for joint
tests and give all joint Wightman positivity inequalities in the
original Hilbert space.
\end{theorem}

We first prove this for homogeneous original slots.  Pure scalar
vectors give the initial mixed orderings; an adjacent-interchange
argument then reaches every ordering.  After constructing the vector
boundaries and their splittings, finite multilinearity restores the
arbitrary original field labels under the same joint test kernels.

\subsection{Uniformity conventions and pure scalar products}

All assertions below concern a fixed finite list of fields and compact
real coordinate sets.  For a test supported in a compact set $K$, put
\begin{equation}\label{R:seminorm}
 p_{K,m}(f)=\max_{|\alpha|\leq m}\sup_K|\partial^\alpha f|.
\end{equation}
A distribution estimate is uniform on the unit ball of one such seminorm;
the finite integer $m$ is allowed to increase when a prescribed accuracy
increases.  For boxes of diameter $\varepsilon$, the same convention
uses rescaled tests and hence introduces only powers of
$\varepsilon^{-1}$.  Imaginary directions range over compact subsets of
open ordered cones.  Constants can depend on those compact sets, the
field list, and the requested derivative order, but never on the dyadic
approximation parameter or on $0<\varepsilon\leq\varepsilon_0$.

For clarity we record the quantitative content of the bounded
approximation result.  For the product of a fixed number of approximants,
there are fixed $A,Q<\infty$ such that its holomorphic norm is at most
$C\varepsilon^{-A}R^Q$.  On a separated real placement patch its dyadic
difference satisfies, for every $k$,
\begin{equation}\label{R:approx-data}
 \|F_{2R}-F_R\|\leq C_k\varepsilon^{-B_k}R^{-k},
 \qquad
 \|F_R\|\leq C\varepsilon^{-A}R^Q.
\end{equation}
The first estimate follows by telescoping and moving the differing
bounded factor to the vacuum; the finite seminorms for it depend on $k$.
The exponent $Q$ in the second estimate does not.  Both estimates include
the adjoint waves.  These are precisely the shrinking-box estimates of
Proposition~\ref{E:bounded-approximation}; they impose no energy bound on
an unknown field operator.

The two exponents in \eqref{R:approx-data} play different roles.
Increasing the desired accuracy changes $k$ and the required test
seminorm, while the polynomial norm exponent $Q$ remains fixed.
This is what permits a summable dyadic estimate after a positive
fraction of the accuracy has been lost in analytic continuation.

We will obtain spectral support from scaling and local boundary
regularity.  The exact homogeneous case suffices for pure products.
Later we will also need the version in which a scaling error already
has the required spectral support.

\Needspace{6\baselineskip}
\begin{lemma}[Scaling and tube spectrum]\label{R:scaling-spectrum}
Let $C\subset\R^N$ be an open convex cone and set
\[
 K=C^*=\{\xi:\xi\cdot y\geq0\ \hbox{for every }y\in C\}.
\]
Use Fourier waves $e^{ix\cdot\xi}$.  Let $u$ be a tempered distribution
with values in a Hilbert space, locally near the origin the
distributional boundary of a holomorphic function on a wedge with
imaginary directions in $C$.  Suppose that, for a real $d$ and
unitaries $V(\lambda)$, $\lambda>0$, every scaling defect
\[
 \lambda^d u(\lambda x)-V(\lambda)u(x)
\]
has Fourier support in $K$.  Then $\supp\widehat u\subset K$.
In particular this holds for exact scaling covariance, with zero
defect.  For scalar distributions one may take $V(\lambda)=1$.
The wedge-boundary hypothesis may be replaced by rapid decay of
localized Fourier transforms near the origin in every direction
outside $K$.
\end{lemma}

\begin{proof}
A distributional wedge boundary has no analytic, hence no smooth,
wavefront at the origin in a direction outside $K$.  This follows
by shifting a localized Fourier test a small distance in a strict
direction of $C$ with negative frequency pairing.  The usual
distributional boundary bounds on smaller approach wedges justify
the shift.  The argument applies in Hilbert norm, or after
scalarization uniformly on the unit ball.

Fix a smooth Fourier test supported in a compact subset of
$\R^N\setminus K$.  Its pairing with $\widehat u$ after dilation
to frequency $\Lambda$ decreases faster than every power of
$\Lambda$.  To pass from the local wavefront estimate to this
unlocalized pairing, insert a cutoff equal to one near the origin.
The local part has the asserted rapid decay.  Outside the cutoff,
the rescaled Schwartz inverse Fourier test and all its derivatives
decrease rapidly in every Schwartz seminorm, so temperedness
controls the remainder.  A finite covering treats all directions
in the fixed frequency support.

The scaling defect has zero pairing with that test and with every
one of its positive dilates, since $K$ is a cone.  The remaining
scaling identity makes the norm of the dilated pairing a fixed
power of $\Lambda$ times the norm of the original pairing;
$V(\lambda)$ does not change that norm.  Rapid decay forces the
original pairing to vanish.  Every compact Fourier test outside
$K$ is allowed, proving the assertion.
\end{proof}

\Needspace{6\baselineskip}
\begin{proposition}\label{R:pure-products}
Every pure $L$ analytic vector product has a tempered Hilbert-valued
boundary with the ordinary ordered vector spectrum.  Its Fourier--Laplace
continuation is the accessible analytic product.
\end{proposition}

\begin{proof}
First, the product has scalar-primary covariance under sufficiently
small simultaneous special transformations.  The geometry in
Lemma~\ref{P:common-wedge} applies to every fixed finite family of
source and destination patches: its proof uses only finitely many
strict wedge inequalities.  Choose the conjugate null axis $m$ separated from the
finitely many flow directions, take $m\cdot x$ sufficiently negative,
and then add a sufficiently large positive multiple of $m$ to make
all the required opposite inequalities strict.  A small double cone
$O_-$ about that point is spacelike to every containing wedge and to
the destination patches.  Local vacuum cyclicity makes
$\{C\Omega:C\in\cM(O_-)\}$ total.

For a fixed finite product length, reparametrize the source tests by
a small null flow and its inverse primary factors.  Write $F_g^{(n)}$
for the limit of the resulting simultaneously conjugated bounded
products and $F^{(n)}$ for the accessible destination product.  The
former limit exists because the source product converges and the
common conjugating unitary is bounded.  For $C\in\cM(O_-)$, move
its first bounded factor past $C$ and take the vacuum and
adjoint-vacuum limits.  The canonical one-wave action and induction
on the length give
\begin{align}\label{R:all-length-recursion}
 \langle C\Omega,F_g^{(n)}\rangle
   &=\langle L(\bar f_1)\Omega,C^*F_g^{(n-1)}\rangle\\
   &=\langle L(\bar f_1)\Omega,C^*F^{(n-1)}\rangle
    =\langle C\Omega,F^{(n)}\rangle.\notag
\end{align}
Every transformed one-vacuum wave is already the accessible $L$
wave, so this induction uses no finite-length interpolation.  Totality
proves the covariance identity on the real spacelike patch, and
boundary uniqueness proves it in the tube.  Compositions of small
null flows give small special transformations.  For any fixed real
compact set and strict approach subcone, choose the transformations
small both there and on a finite interior path to the initial patch.
Their unitary factors have norm one.  Thus the identity is uniform
near that compact set; localization under arbitrary large conformal
transformations is not required.

Thus covariance has been obtained separately for every finite pure
product length.  The next step uses it to transport the known
boundary estimates for products of $\Theta$ into enough differential
estimates to recover the original pure $L$ product.

Here is an algebraic device that converts this covariance into a
boundary estimate.  On the ambient null cone use
\[
 X=(X^+,X^-,X^\mu),\qquad q(X)=X^\mu X_\mu-X^+X^-,
 \qquad X=\lambda(1,x^2,x).
\]
Here $F$ is the Hilbert-valued analytic vector product.  Its lift is
Hilbert-valued and homogeneous of degree $w=-2$ in each slot;
the differential identities below may be tested against arbitrary
Hilbert vectors, with estimates uniform on their unit ball.  On degree
$w$ homogeneous functions, define the flat Thomas derivative
\begin{equation}\label{R:Thomas}
 \mathscr D_A(w)=(2+2w)\partial_A-X_A\partial^2_{\mathrm{amb}}.
\end{equation}
Successive derivatives use the new degree in their slot.  The product
rule, $\partial^2_{\mathrm{amb}}q=12$, and Euler's identity show that
these derivatives preserve the quadratic ideal, commute successively,
and have zero successive trace.  They therefore act intrinsically on
the cone.  The same calculation gives
\begin{equation}\label{R:Thomas-contraction}
 X^A\mathscr D_A(w)=w(2+2w)\quad\pmod{q}.
\end{equation}
These are the flat ambient identities underlying the Thomas operator;
see also \cite{GoverWaldron2009}.  None of the factors in
\eqref{R:Thomas-contraction} vanishes at $w=-2,-3,\ldots$.

In particular, contraction can undo each successive derivative at
the degrees encountered here.  The weight in
\eqref{R:Thomas} must be lowered after each differentiation in that
slot; using a fixed weight throughout would not give these intrinsic
commuting operators.

Take the null direction $a_\infty$ representing chart infinity, and
extend the lifted function independently of $X^-$.  Then
$a_\infty\cdot\mathscr D$ is a nonzero smooth chart factor times
$\Box_x$.  Consequently
\begin{equation}\label{R:all-slot-derivative}
 G_aF:=\prod_{i=1}^n(a\cdot\mathscr D_i)F
\end{equation}
has polynomial boundary growth for $a=a_\infty$, by the ordinary
all-$\Theta$ product and \eqref{R:leg-equation}.  Primary transport
proves the same statement for $a$ in an open real null-cone patch about
$a_\infty$.  Finite derivatives preserve these bounds after shrinking
the approach subcone, by Cauchy's inequalities.

For this transport, write $\widetilde F$ for the homogeneous lift.
For each fixed sufficiently small transformation $g$, its local
covariance is $\widetilde F(gX)=U_g\widetilde F(X)$, with the
primary chart factors absorbed in the lift.  The flat ambient
chain rule gives $G_{ga}\widetilde F(gX)=U_gG_a\widetilde F(X)$.
The unitary $U_g$ is fixed as the insertion coordinates vary, so
the coordinate derivatives act only on $\widetilde F$.  Taking
$a=a_\infty$ transfers the known bound to the null direction
$ga_\infty$.  The shifted homogeneous weights supply smooth
chart factors bounded on the chosen compact set.  Scalarization
replaces a testing vector by its unitary image and therefore keeps
the Hilbert norm estimate uniform.

We give the finite algebraic argument needed to recover $F$.  In
constant complex variables $\xi_{iA}$, let $I$ be the ideal generated by
$\xi_i^2$ and by $\prod_i(a\cdot\xi_i)$ for all $a$ in that patch.
If a common zero had every $\xi_i\ne0$, the patch would be covered by
the finitely many proper hyperplane sections $a\cdot\xi_i=0$.
The real null patch is Zariski dense in the irreducible complex null
cone and spans the ambient space, so this is impossible.  Thus every
common zero has $\xi_i=0$ for some $i$.

For each of the finitely many monomials
$m=\prod_i\xi_{iA_i}$, the Nullstellensatz gives $m^{e_m}\in I$.
Let $e$ dominate their exponents.  Every monomial of degree at least
$6(e-1)+1$ in each slot is divisible by one of the $m^e$ and belongs
to $I$.  Noetherianity gives finite generator representations, and
separate homogeneity in each slot gives multihomogeneous coefficient
representations.  Substitute the successively commuting operators
\eqref{R:Thomas}.  Trace terms vanish; the remaining terms are finite
derivatives of \eqref{R:all-slot-derivative}.  Hence sufficiently high
derivative tensors of $F$ have polynomial growth.  Contracting their
indices with the $X_i$ and using \eqref{R:Thomas-contraction} repeatedly
recovers $F$ with a nonzero constant factor.  The chart factors are
bounded on the compact set.  This proves moderate growth of $F$.

The algebraic recovery used only a finite number of null directions
and derivatives for this fixed product length.  Its constants may
depend on that length; no estimate uniform over all lengths is needed
for constructing the polynomial field domain.

The moderate-growth tube boundary theorem, applied weakly and then
uniformly in unit Hilbert vectors, gives a Hilbert-valued distribution
locally.  Its exact similarity covariance globalizes this boundary
and makes it tempered: rescaling dyadic annuli to a fixed compact set
changes a finite distribution seminorm by only a power of the radius,
and the dilation unitary leaves its norm unchanged.  Lemma~\ref{R:scaling-spectrum}
gives its ordered vector spectrum.  Fourier--Laplace uniqueness
then identifies its continuation with the original product.
\end{proof}

\subsection{Boundary values and a distribution criterion}

We specify the analytic facts used in the mixed-product argument so
that no growth assertion is concealed in a boundary-value notation.
The point of the distinction is that holomorphy alone will initially
give boundary hyperfunctions for the coefficients of a causal wave
ambiguity.  We will prove a quantitative Gaussian estimate before
using any of those coefficients as an ordinary distribution.
The boundary $b f$ of a holomorphic function on a local convex wedge
exists as a hyperfunction without a growth assumption.  Restriction
to a smaller approach cone is compatible with this boundary map; the
map is injective for the local germ on a connected single wedge and
agrees with ordinary distributional boundary values whenever those
exist.  We fix the Euclidean coordinate volume form when identifying
scalar hyperfunctions with the corresponding top-form convention.  Compactly
supported hyperfunctions are analytic functionals carried by their
support.  Thus, for each complex neighborhood $K_\gamma$ of the
carrier, a representative $u$ satisfies
\begin{equation}\label{R:carrier}
 |\langle u,h\rangle|\leq C_\gamma\sup_{K_\gamma}|h|,
 \qquad h\text{ holomorphic near }K_\gamma.
\end{equation}
Komatsu constructs the injective boundary map on a Stein complex
neighborhood with an open convex imaginary cone
\cite[Section~1]{Komatsu1973}; equations (1.1) and (1.5) on
pp.~253--254 give injectivity and compatibility with smaller cones.
Small complex boxes are Stein, so restriction to a smaller real patch
gives the local germ statement used here.  The comparison on p.~257
identifies polynomial-growth boundaries with distributional ones.
Martineau's Theorems~1--2 give the analytic-functional realization
\cite{Martineau1961}; the carrier estimate is in Section~1(a)--(b)
and Proposition~1 of that paper.  These constructions alone give no
polynomial growth estimate.  In the analytic-functional notation
used below, the canonical distribution embedding is described in
\cite[Section~4, equation~(4.3)]{HoepfnerRagognette2021}.

Use real Euclidean coordinates for the Gaussian, with their
complex-bilinear extension, and the Fourier wave convention
$e^{ix\cdot\xi}$.  For a compact analytic-functional representative put
\begin{equation}\label{R:FBI}
 \mathcal T_\Lambda u(w,\eta)
   =\left\langle u(z),
       e^{i\Lambda(w-z)\cdot\eta-\Lambda(w-z)^2/2}\right\rangle.
\end{equation}
Only real centers $w$ and real frequencies $\eta$ are used for the
estimates below.

\Needspace{6\baselineskip}
\begin{lemma}\label{R:FBI-criterion}
Suppose that, near a real point $w_0$, a hyperfunction $u$ has a compact
analytic-functional representative such that
\begin{equation}\label{R:FBI-polynomial}
 |\mathcal T_\Lambda u(w,\eta)|\leq C(1+\Lambda)^N
 \quad(w\in U, |\eta|=r_*,\ \Lambda\geq1)
\end{equation}
for some neighborhood $U$ of $w_0$ and some $r_*>0$.
Then $u$ is a distribution near $w_0$.

If $u=b f$ from a wedge, its transform near $w_0$ can be computed by
integrating $f$ on a sufficiently small fixed interior translate of
a local real box, with an exponentially small error.  The box and
height can be chosen uniformly for $w$ in a smaller box and $\eta$
in a fixed compact set.
\end{lemma}

\begin{proof}
The first assertion is the distribution case of the generalized FBI
criterion of Hoepfner--Ragognette
\cite[Theorem~6.3 and Section~8.1]{HoepfnerRagognette2021}.
In their notation take the positive quadratic polynomial
$p(v)=|v|^2/(2r_*)$ and $\xi=\Lambda\eta$, $|\eta|=r_*$.
Then $|\xi|p(w-z)=\Lambda(w-z)^2/2$, so their transform is
\eqref{R:FBI} up to an irrelevant fixed normalization.  Their local
inverse decomposition separates a distribution, obtained from the
polynomially bounded part, from an analytic remainder.  Finite
covering of the frequency sphere makes the polynomial estimate
uniform.  The compact representative is immaterial because changing
it away from $w_0$ changes this estimate by an exponentially small
term.

For the second assertion, use the fixed-height integration
representation of a wedge boundary value
\cite[Remark~4.1]{HoepfnerRagognette2021}, using a rounded box with
smooth boundary.  The difference is an analytic
functional carried in an arbitrarily thin complex neighborhood of
the boundary of the chosen real box.  This is also obtained from the
product Cauchy representation: entries that extend across one
coordinate cancel under contour displacement.  Axes strictly inside
the wedge give the requisite product half-domain description.
Let the distance between the smaller box of centers and that boundary
be $a>0$.  For $z=x+iy$, $|\eta|\leq M$, the real exponent in the
Gaussian is
\begin{equation}\label{R:Gaussian-exponent}
 \Lambda\left(y\cdot\eta+\frac{|y|^2}{2}
                    -\frac{|w-x|^2}{2}\right).
\end{equation}
Choose all side contours with $|y|\leq t$ and then the carrier
neighborhood with width $\gamma$ so small that
$(t+\gamma)M+(t+\gamma)^2/2<a^2/4$.
Equation~\eqref{R:carrier} bounds their contributions by $C e^{-c\Lambda}$.
This is the localization estimate of
\cite[Proposition~4.5]{HoepfnerRagognette2021}.  Heights and carrier
neighborhoods are fixed before $\Lambda\to\infty$; no integrability
of $f$ down to the real edge has been assumed.
\end{proof}

A related fact for a tempered distribution will be useful.  Rapid
Gaussian decay near the origin in a frequency cone implies rapid
decay when its Fourier transform is tested at large frequencies
strictly inside that cone.  One can use the smooth wavefront
characterization, or directly integrate the Gaussian inversion
against that frequency test.  Away from the origin, integrations by
parts in the frequency variable gain arbitrary powers; a derivative
of the Gaussian costs at most a square-root power beyond the fixed
distribution order.  This gives the alternative Fourier-decay
hypothesis in Lemma~\ref{R:scaling-spectrum}.

\subsection{A quantitative boundary estimate at one merged face}

Until the finite-multilinearity passage below, every original-field
slot in the mixed-product argument is homogeneous, including every
spectator slot.  Original derivative slots carry their shifted
dimensions; derivatives of $L$ will follow by differentiating its
reconstructed products.
We fix an arbitrary finite list and an arbitrary ordering; neither its
length nor the set of lists is bounded.  The constants and distribution
orders may depend on this list.  We shall first complete both scalar
and Hilbert-valued reconstruction for these homogeneous lists.

The scalar induction is simultaneous over every finite homogeneous
old-field list and is indexed only by the number of $L$ entries.
Its zero stage is the original Wightman family.  At the next stage,
orderings with every $L$ on the right are tempered: the analytic
splitting \eqref{E:accessible-splitting} pairs an ordinary
homogeneous old bra with the pure vector boundary of
Proposition~\ref{R:pure-products}.  This continuous bilinear pairing
on the conjugate bra Hilbert space and the ket Hilbert space gives a
joint Schwartz distribution by the kernel theorem.  The reversed
bra vector cone, the ket vector cone, and translation invariance
supply its scalar ordered cones.

It remains to move $L$ entries through old entries.  In each adjacent
interchange, applying $\Box$ in the $L$ slot replaces it by the
homogeneous old field $\Theta$.  Both source orderings therefore
belong to the preceding induction stage, however many old spectators
are present.  We prove the interchange step below before iterating it.

Let $W_1$ be a mixed scalar ordering already known to be tempered,
and obtain $W_2$ by interchanging an adjacent $L(x)$ and homogeneous
original physical field
$A(y)$.  Write $r=x-y$.  The variable $z$ consists of $y$ and all
spectator coordinates in their merged order.  Holding $z$ in a compact
subset of its strict tube keeps every gap except the new relative
gap bounded away from the boundary.  Replacing this $L$ by $\Theta$
gives two already known tempered sources, with the homogeneous
dimension-four field $\Theta$ in that slot,
$S_i=\Box_rW_i$.

In these coordinates the momentum at the merged $y$ slot is the sum
of the two adjacent momenta; all other spectator momenta are unchanged.
Write $C_z$ for the strict merged approach cone and $K_z=C_z^*$ for
its closed spectral cone, in the Fourier convention of
Lemma~\ref{R:scaling-spectrum}.  A strict relative direction $v_r$
of the target order, sufficiently small compared with a fixed merged
direction $v_z$, makes $(v_r,v_z)$ a strict full target direction.
These are different cones: operations in $r$ preserve $K_z$, while
the missing relative spectral condition will require a further argument.

Here the relative gap tends to the boundary while the merged gaps
remain strict.  The estimate must therefore keep a positive analytic
propagation exponent as the relative gap shrinks.  Once this face
estimate is available, the wave equation will carry it from a thin
real-time strip to every compact relative-coordinate set.

\Needspace{6\baselineskip}
\begin{lemma}\label{R:face-bound}
For compact real $r$-sets, compact strict merged $z$-sets, and compact
sets of relative imaginary directions $v$ of the target order, there
are $C,N<\infty$ such that
\begin{equation}\label{R:face-power}
 |W_2(r+i\varepsilon v;z)|\leq C\varepsilon^{-N},
 \qquad 0<\varepsilon\leq\varepsilon_0.
\end{equation}
The same statement holds for every fixed finite derivative order.
Consequently $W_2$ has an ordinary distributional $r$-boundary
$F_0(r;z)$, holomorphic as a distribution-valued function on the strict
merged tube.
\end{lemma}

\begin{proof}
We first work with $|r^0|\leq C_0\varepsilon$ and bounded
$\mathbf r$.  Shift $\mathbf r$ outward by at most $C_1\varepsilon$
so that the resulting $r_*$ has spacelike clearance at least
$4c\varepsilon$.  Put the real merged placements $z_*$ sufficiently
far apart on a common time slice.  Their spacelike margins, including
separation from both members of the pair, can be chosen of order one.
The tests for the two nearby slots have diameter $c\varepsilon$;
all tests may be taken of that size.  The bounds
\eqref{R:approx-data} hold on the resulting separated patch.

Here $F_R$ denotes the scalar vacuum expectation of the bounded
product \eqref{E:bounded-product}.  On the vector tube, vacuum
invariance removes its initial factor $U(z_1)$.  The resulting
expression depends only on consecutive gaps and extends holomorphically,
with the same polynomial norm bound, to the scalar tube where those
gaps are strict.  Thus this scalar face estimate does not require the
first-gap condition that remains necessary for vector products.

Write $\eta=\operatorname{Im}z$.  Consider the complex line
\begin{equation}\label{R:anisotropic-line}
 \zeta\longmapsto
     (r_*+\varepsilon\zeta v,\ z_*+\zeta\eta),
       \qquad \operatorname{Im}\zeta>0.
\end{equation}
For sufficiently small $\varepsilon$ its direction is in the full
target cone.  There is a real interval $[-a,a]$ on which the
approximant supports remain spacelike, with $a>0$ independent of
$\varepsilon$: the relative displacement is of size $\varepsilon$
and the corresponding clearance is also of that size, whereas all
merged displacements and margins are of order one.  The harmonic
measure of this interval at $\zeta=i$ is
$\alpha=2\arctan(a)/\pi>0$.  The two-constants theorem applied to
$\log|F_{2R}-F_R|$ gives
\[
 |F_{2R}-F_R|
 \leq C_k\varepsilon^{-\alpha B_k-(1-\alpha)A}
           R^{-\alpha k+(1-\alpha)Q}.
\]
Slightly reducing the clearances gives this estimate on a complex
$r$-ball of radius $c_1\varepsilon$ and a $z$-ball of fixed radius.

Returning the spatial shift takes a bounded number of
$\varepsilon$-balls.  At fixed complex $r$, moving the merged
variables to any prescribed merged compact takes a bounded number
of fixed-radius $z$-balls.  Moving $y$ moves $x=y+r$ with it, so the
small internal gap does not restrict those $z$-radii.  More precisely,
take the compact convex hull, inside the strict merged tube, of the
initial placements and the prescribed compact set.  Its gaps have a fixed positive
margin; the two exterior gaps adjacent to the pair keep that margin
when the internal displacement is $O(\varepsilon)$.  A finite cover
therefore supplies radii and a number of balls independent of
$\varepsilon$.  Take overlapping
balls with radii $\rho,2\rho,4\rho$ inside the tube.  Each three-circle
step retains exponent $1/2$.  Thus a fixed $\sigma>0$, independent of
$\varepsilon$, gives
\begin{equation}\label{R:dyadic-face}
 |F_{2R}-F_R|
   \leq C_k\varepsilon^{-E_k}
             R^{-\sigma k+(1-\sigma)Q},
 \qquad E_k=\sigma\max(B_k,A)+(1-\sigma)A.
\end{equation}
Choose one finite $k$ with $\sigma k-(1-\sigma)Q>1$ and then fix its
finite test seminorm.  Summing dyadic differences gives a fixed
inverse power of $\varepsilon$.  If localization starts at a dyadic
$R_0(\varepsilon)$ bounded by a power of $\varepsilon^{-1}$, its
initial norm contributes another finite power.  The construction of
Proposition~\ref{E:bounded-approximation} permits this choice.  The
accuracy order is never increased while $\varepsilon$ tends to zero.

This order of choices is essential: the continuation geometry fixes
$\sigma$, then one chooses $k$, and only then lets the relative gap
shrink.  The resulting $E_k$ is a fixed finite exponent in the face
bound.

For completeness, unsmearing this estimate does not involve point
evaluation of a bounded approximant.  The limit already has a
holomorphic kernel by Proposition~\ref{E:accessible-products}.  Choose
a smooth radial bump $\rho$ on $\C^4$ of integral one, supported in
a sufficiently small ball.  Holomorphic mean value in each of $p$
slots gives, writing $Z$ for the full slot coordinates,
\begin{align}\label{R:unsmear}
 W_2(Z)
 &=\int_{(\C^4)^p}\prod_i\varepsilon^{-8}
               \rho(w_i/\varepsilon)W_2(Z+w)\,\dd w\\
 &=\int \dd b\;
     W_2(f_{\varepsilon,b_1},\ldots,f_{\varepsilon,b_p};
                     Z+i\varepsilon b).\notag
\end{align}
Here $f_{\varepsilon,b}(u)=\varepsilon^{-4}\rho(u/\varepsilon+ib)$,
and $b$ ranges over a fixed compact set and
$p_{K,m}(f_{\varepsilon,b})\leq C_m\varepsilon^{-4-m}$.
The finite seminorm estimate just proved therefore bounds the last
integral by a power.  Cauchy's inequalities in smaller
$\varepsilon$-balls also bound all fixed derivatives.  This proves
\eqref{R:face-power} initially in the thin real-time strip.

Finally fix $z$ and the imaginary displacement $i\varepsilon v$.
The function
$u_\varepsilon(t,\mathbf r)=W_2((t,\mathbf r)+i\varepsilon v;z)$
satisfies the ordinary inhomogeneous wave equation in the real
variables.  Its initial value, spatial derivatives, and first time
derivative have the bounds just obtained.  Its source is the
Fourier--Laplace continuation of the inductively tempered $S_2$,
so it has a power bound of its own at this relative gap.  Kirchhoff's
formula on $|t|\leq T$, $|\mathbf r|\leq B$ gives
\begin{align}\label{R:wave-estimate}
 \|u_\varepsilon\|_{T,B}
 &\leq \|u_\varepsilon(0)\|_{B+T}
   +T\|\nabla u_\varepsilon(0)\|_{B+T}
   +T\|\partial_tu_\varepsilon(0)\|_{B+T}
   +\tfrac12T^2\|S_2\|_{T,B+T},
\end{align}
with an immaterial fixed norm constant.  This proves the bound on
every real compact set.  Derivatives obey the same estimate.
Moderate-growth boundary values now give $F_0$; the estimates uniform
on $z$-compacts give holomorphy and locally uniform finite distribution
order in $r$.
\end{proof}

\subsection{Causal ambiguity and recovery on the wrong massless branch}

The source difference $S_2-S_1$ vanishes for real spacelike $r$ by
inductive locality.  Split it into distributions supported in
$J^+(0)$ and $J^-(0)$ by a partition away from $r=0$ and finite Taylor
subtraction at that point.  Subtraction acts only in $r$, so all
merged $z$-spectra are preserved.  Convolve these two parts with the
retarded and advanced fundamental solutions, respectively.  The
cone-supported convolutions are proper on bounded sets and define
tempered distributions; their Schwartz seminorms involve only
finitely many seminorms of the sources.  Adding their sum to $W_1$
gives a tempered distribution $T_0$ satisfying
\begin{equation}\label{R:causal-solution}
 \Box_r T_0=S_2,\qquad
 \supp_r(T_0-W_1)\subset J(0),
\end{equation}
with the merged spectra.  Let the same symbol denote its
Fourier--Laplace continuation in $z$, valued in $r$-distributions.

At fixed strict $z$, the difference $H_*=F_0-T_0$ solves the
homogeneous wave equation and is supported in $J(0)$; spacelike tie
gluing proves the support assertion.  A homogeneous wave distribution
has distributional Cauchy data.  At time zero those data are supported
at the spatial origin, so each is a finite sum of derivatives of a
delta function.  Cauchy uniqueness consequently gives
\begin{equation}\label{R:wave-ambiguity}
 H_*(r;z)=\sum_{j=1}^m C_j(z)P_j(\partial_r)\Delta_0(r).
\end{equation}
The $P_j$ may be chosen homogeneous and linearly independent modulo
the wave quadratic, with $\Delta_0$ the massless commutator function.
One fixed finite list works on the connected merged tube.  Indeed,
on a relatively compact open subset the distribution order is
uniform, so sufficiently high spatial monomials annihilate both
Cauchy data.  These annihilators continue holomorphically to the
whole tube.  Cauchy restriction is continuous on homogeneous wave
solutions, and fixed dual tests recover the finitely many coefficients;
hence the $C_j$ are holomorphic.

The unknown part of the boundary has now been reduced to finitely
many scalar coefficient functions on the merged tube.  The next
lemma proves their distribution regularity by testing on the
massless branch excluded by the target ordering.  The causal
solution $T_0$ is already tempered and will supply the polynomial
bound in that test.

\Needspace{6\baselineskip}
\begin{lemma}\label{R:coefficient-distributions}
Every coefficient $C_j$ in \eqref{R:wave-ambiguity} has a local ordinary
distribution boundary in the merged variables.
\end{lemma}

\begin{proof}
Fix a real merged center.  Initially $bC_j$ is only a hyperfunction.
Choose a small real box about the center, a smaller box of centers
$w$, and a strict merged direction $v_z$.  Lemma~\ref{R:FBI-criterion}
represents the Gaussian testing by a fixed translate $it_0v_z$,
up to exponentially small side errors.

Choose a strict relative direction $v_r$ for the target ordering,
small enough compared with the merged gaps that $(v_r,v_z)$ is a
strict full target direction.  Choose finitely many nonzero null
covectors $k_a$ on the wrong branch, so $k_a\cdot v_r<0$.
The sample points range over a radially open patch of that branch;
they are not constrained to a fixed energy shell.
The radius variation will be used when separating coefficient
polynomials of different degrees; choosing directions only on one
fixed shell would not give that independence argument.
Then choose
$r_*>0$ sufficiently small that, uniformly for all samples and
$|\eta|=r_*$,
\begin{equation}\label{R:negative-phase}
 k_a\cdot v_r+\eta\cdot v_z\leq-c_0<0.
\end{equation}
Test \eqref{R:wave-ambiguity} in $r$ with
$\chi(r)e^{-i\Lambda r\cdot k_a-\Lambda r^2/2}$, where $\chi=1$
near zero, and in $z$ with the Gaussian
\eqref{R:FBI} on the fixed translated box.

For the $F_0$ term, deform the relative contour from zero to
$it_0v_r$, leaving $z$ at height $it_0v_z$.
Lemma~\ref{R:face-bound} justifies the initial endpoint.  On the final
central contour the real exponent, apart from negative spatial
Gaussians, is at most
\[
 \Lambda\left[-t_0c_0+
                  \tfrac12t_0^2(|v_r|^2+|v_z|^2)\right].
\]
Choose $t_0$ so small that this is $\leq-c\Lambda$.  The continued
$W_2$ is bounded on that fixed interior compact, so this term is
exponentially small.

We specify the cutoff error, since the relative contour starts at a
distributional face.  Let $N$ be an inverse-height growth exponent
from \eqref{R:face-power}, increased to cover the derivatives used
below.  Replace $\chi(x)$ on the moving contour by its almost-analytic
Taylor lift
\[
 \chi^{[M]}(x+iy)=
     \sum_{|\alpha|\leq M}(iy)^\alpha\partial^\alpha\chi(x)/\alpha!,
 \qquad M>N+1.
\]
Its antiholomorphic derivative is $O(|y|^M)$, supported in the
transition region.  There $|x|$ is separated from zero.  By decreasing
$t_0$ before taking the large-parameter limit,
\eqref{R:Gaussian-exponent} is $\leq-c'\Lambda$ throughout the
transition, even after all linear phase costs.  Stokes errors are
bounded by a polynomial in $\Lambda$ times
$e^{-c'\Lambda}\int_0^{t_0}t^{M-N}\,\dd t$.
The same construction applies to any smooth central cutoff in $z$.
For the merely hyperfunctional coefficient boundary we instead use
the fixed-height analytic-functional remainder of
Lemma~\ref{R:FBI-criterion}; we do not multiply that boundary by a
smooth cutoff.

The $T_0$ term is polynomially bounded in $\Lambda$: move its merged
contour back to the real boundary and use its tempered seminorm.
Derivatives of the Gaussian and cutoffs cost only powers of
$\Lambda$.  Its contour errors obey the same estimate, now with the
known finite distribution order of $T_0$.

It remains to check that these tests recover every coefficient.  Put
$d_j=\deg P_j$ and
\[
 M_{aj}(\Lambda)=
  \left\langle P_j(\partial)\Delta_0,
       \chi(r)e^{-i\Lambda r\cdot k_a-\Lambda r^2/2}\right\rangle.
\]
The Fourier transform of $\Delta_0$ is a nonzero normalization times
$\operatorname{sgn}(p^0)\delta(p^2)$.  Gaussian Fourier integration,
with $p=\Lambda k_a+\sqrt\Lambda q$, gives
\begin{equation}\label{R:shell-asymptotic}
 \Lambda^{3/2-d_j}M_{aj}(\Lambda)
       \longrightarrow c(k_a)P_j(ik_a),\qquad c(k_a)\ne0.
\end{equation}
Indeed $\dd p$ contributes $\Lambda^2$, the delta function
$\Lambda^{-3/2}$, and the four-dimensional Gaussian Fourier factor
$\Lambda^{-2}$.  The remaining integral is a nonzero Gaussian
integral on $2k_a\cdot q=0$, with the fixed sign of the branch.
The cutoff error is exponentially small by the temperedness of
$P_j(\partial)\Delta_0$.  The nonzero factor $c(k_a)$ can depend on
the row.

The evaluation matrix $P_j(ik_a)$ has full column rank for a suitable
finite sample set.  Otherwise a nonzero linear combination would
vanish on an open real patch of one null branch, hence on the
irreducible complex quadratic cone, contradicting independence
modulo the wave quadratic.  Radial openness matters for polynomials
of different degrees.  Selecting an invertible square submatrix,
\eqref{R:shell-asymptotic} bounds its inverse by powers of $\Lambda$.
The tested identity therefore gives
\[
 |\mathcal T_\Lambda(bC_j)(w,\eta)|\leq C(1+\Lambda)^N
      \qquad(|\eta|=r_*),
\]
uniformly on the smaller box.  Lemma~\ref{R:FBI-criterion} proves
that every $bC_j$ is an ordinary distribution there.
This conclusion is local in the merged real variables.  Growth at
infinity and the merged spectral support still require the scaling
argument below.
\end{proof}

\subsection{Global growth, spectra, and completion of mixed products}

Local distribution boundaries do not by themselves imply global
temperateness.  Here exact similarities provide the missing control.
The comparison solution $T_0$ need not share the exact homogeneity
of $W_2$.  Its scaling defect will instead appear as an explicitly
tempered error, which is enough both for dyadic growth estimates and
for excluding frequencies outside the merged spectral cone.
If $W_2$ has $n_L$ entries $L$ and homogeneous original entries of
dimensions $\Delta_1,\ldots,\Delta_k$, put
\[
 d_W=2n_L+\sum_{i=1}^k\Delta_i.
\]
Here the old dimensions already include any derivative shifts.
The accessible analytic product has exact similarity covariance for
this homogeneous list, by covariance of the one-wave data and uniqueness
of the multilinear accessible products.  Its distributional
$r$-boundary $F_0$ has the same exact scaling.  Put
$\delta_j=2+\deg P_j$, so
$P_j(\partial)\Delta_0(\lambda r)=\lambda^{-\delta_j}
P_j(\partial)\Delta_0(r)$.
Choose fixed compact $r$-tests $\varphi_j$ dual to the finite list
$P_j(\partial)\Delta_0$.  Exact scaling of $F_0$ and
\eqref{R:wave-ambiguity} show that
\begin{equation}\label{R:scaling-defect}
 \lambda^{d_W-\delta_j}C_j(\lambda z)-C_j(z)
 =\left\langle
      T_0(r;z)-\lambda^{d_W}T_0(\lambda r;\lambda z),
                   \varphi_j(r)\right\rangle_r.
\end{equation}
The right side has a tempered boundary with merged spectral support.
For $\lambda\geq1$ its Schwartz seminorm constants grow at most
polynomially in $\lambda$: scaling and the fixed compact extraction
tests act continuously on Schwartz space with such bounds.

Cover the complement of a compact $z$-ball by dyadic annuli and
rescale each annulus to a fixed compact set.  There the local
coefficient distribution has a fixed finite order.  Formula
\eqref{R:scaling-defect} bounds its value on every outer annulus by
a polynomial in the annulus radius and finitely many Schwartz
seminorms.  Summing against a Schwartz test proves that each $C_j$
is tempered.  This argument needs no exact homogeneity of $T_0$.

The scaling defect in \eqref{R:scaling-defect} has Fourier support
in $K_z$.  Apply Lemma~\ref{R:scaling-spectrum} to the now tempered
coefficient boundary, with $d=d_W-\delta_j$ and $V(\lambda)=1$.
Its local wedge boundary is supplied by
Lemma~\ref{R:coefficient-distributions}.  We obtain
$\supp\widehat C_j\subset K_z$.
Fourier--Laplace continuation and injectivity identify the original
$C_j$ with the corresponding tempered continuation; in particular
it now has ordinary polynomial bounds at ordered approaches.

It follows that $F_0$ is the merged continuation of the globally
tempered distribution
\begin{equation}\label{R:D0}
 D_0(r,z)=T_0(r,z)+\sum_j C_j(z)P_j(\partial_r)\Delta_0(r).
\end{equation}
It is exactly homogeneous, because its merged continuation $F_0$
is.  We next establish the full target spectrum, rather than just
the merged one.  For a covector strictly outside the full target
cone, choose a strict target imaginary direction $(v_r,v_z)$ with
negative pairing.  Local Gaussian testing near $(r,z)=0$ first
moves $z$ to a small height $it_0v_z$, using the already proved
merged temperateness.  At that face, $D_0=F_0$; move $r$ to
$it_0v_r$ using Lemma~\ref{R:face-bound}.  Intermediate fractions of
the latter move remain in the target tube.  The final contour is an
interior $W_2$ contour and has exponential Gaussian decay.  The side
errors are those already estimated in the proof of
Lemma~\ref{R:coefficient-distributions}.  This supplies the local
off-cone Fourier decay used in Lemma~\ref{R:scaling-spectrum}; exact
homogeneity of $D_0$ therefore gives its full target spectral support.
Its full Fourier--Laplace continuation agrees with $W_2$, since they
have the same $r$-boundary at fixed strict merged damping.

One adjacent interchange is therefore complete: it preserves
temperateness, recovers the full ordered spectrum, and retains
spacelike locality.  Iterate this step in the simultaneous
induction stated above.  At fixed number of $L$ entries, begin with
all of them on the right and move them to the prescribed positions.
Every source has one fewer $L$ and one additional old entry $\Theta$,
so it belongs to the preceding induction stage.  Repeat this for
every finite ordering of the old entries.  Tie gluing, followed by the
ordinary merged spectral boundary with a spacelike relative test,
then proves locality also for adjacent old-field interchanges, with
the prescribed grading.  Adjunction and exact similarity covariance
for each homogeneous list follow from those identities of the analytic
accessible products and uniqueness of their boundaries.

We next complete the Hilbert-valued assertion for the same homogeneous
lists.  The squared norm of a mixed analytic vector product is its
reflected scalar split.  A strict vector approach gives a strict
scalar split approach: the central gap is twice the first vector
damping, and the other gaps are the reflected vector gaps.  The scalar
polynomial bound therefore gives a polynomial norm bound for the vector
product.  The Hilbert-valued moderate-growth theorem supplies its local
boundary.  Its exact similarity covariance for this homogeneous list,
with a unitary dilation factor on the output, globalizes the boundary
by dyadic annuli and gives the vector support by
Lemma~\ref{R:scaling-spectrum}.  Thus this boundary is a tempered
Hilbert-valued distribution with the full ordered vector spectrum.
Taking both strong distributional boundaries in the reflected split,
first for factor tests and then for joint tests by continuity, gives
\[
 W(B^*A)=\langle\Psi(B),\Psi(A)\rangle
\]
for any two homogeneous words $A,B$ with their tests.  The concatenated
cross list is itself a list of homogeneous slots, even when the two
words have different total dimensions.

It remains to pass from homogeneous slots to arbitrary original
fields.  Write each such field in the fixed list as
$A_i=\sum_{\Delta\in S_i}A_{i,\Delta}$, with $S_i$ finite, as in
Assumption~\ref{ass:framework}.  Let $\mathbf A$ denote the complete
ordered list, including its $L$ slots, and let
$\mathbf A_{\boldsymbol\Delta}$ replace each old $A_i$ by
$A_{i,\Delta_i}$.  If there are no old slots, the indexing product
below is a singleton.  Multilinearity of the accessible analytic
products already gives the corresponding finite expansion in the
tube.  For an already constructed homogeneous list $\mathbf B$,
write $W_{\mathbf B}$ and $\Psi_{\mathbf B}$ for its scalar and vector
boundary maps.  For every joint Schwartz test $h$, in particular
every compact joint test, define the maps for $\mathbf A$ by
\begin{equation}\label{R:finite-field-expansion}
 \begin{split}
 W_{\mathbf A}(h)
   &=\sum_{\boldsymbol\Delta\in\prod_i S_i}
                 W_{\mathbf A_{\boldsymbol\Delta}}(h),\\
 \Psi_{\mathbf A}(h)
   &=\sum_{\boldsymbol\Delta\in\prod_i S_i}
                 \Psi_{\mathbf A_{\boldsymbol\Delta}}(h).
 \end{split}
\end{equation}
The same joint kernel $h$ occurs in every term.  This is a finite
expansion of field labels, not an assertion that $h$ has finite tensor
rank.  Each summand is an already constructed tempered distribution,
and every summand has the same cone for the chosen ordering.  Their
finite sum therefore has the required temperateness and support and
has the specified analytic product as its Fourier--Laplace continuation.
Adjunction and locality pass through the same finite expansion.
The exact scaling laws of the summands give similarity covariance with
dilations acting on the original field labels; a nonhomogeneous list
is not assigned one total dimension.  Orders and constants may be
chosen by taking a maximum of the finitely many seminorm orders and
a sum of the corresponding constants, without any uniform estimate
over dimensions or lengths.  Finite distributional derivatives in
any slot preserve temperateness and cone support and have the shifted
homogeneous covariance laws term by term.

The reflected identity also passes through this expansion with all
cross terms.  For arbitrary finite linear combinations $P,Q$ of
finite ordered words with joint tests, expand their original labels
finitely and apply the homogeneous split to
each pair of component words.  Here $P^*Q$ uses reversed adjunction
on each bra word and the tensor product of its conjugated, reversed
test with the ket test.  The result is
\begin{equation}\label{R:reflected-polynomials}
 W(P^*Q)=\langle\Psi(P),\Psi(Q)\rangle.
\end{equation}
In particular, the full joint Wightman positivity inequalities are
the Gram inequalities for these vectors in the original $\cH$.
This uses the cross-tuple splittings, not a claim of termwise
positivity for the individual summands.

The homogeneous induction, vector construction, and finite
identities \eqref{R:finite-field-expansion} and
\eqref{R:reflected-polynomials} complete the proof of
Theorem~\ref{R:mixed-products}.

\subsection{The joint domain and affiliation with the given net}

Let $\mathcal D$ be the span of the reconstructed vacuum products
with joint Schwartz test kernels, including compact joint kernels
and finite sums of factorized smearings.

The passage from positive vacuum distributions to an invariant
insertion domain is the Wightman reconstruction mechanism, in the
test-algebra formulation of Borchers~\cite{Wightman1956,Borchers1962}.
Here the vacuum vectors are already in the given $\cH$; after this
domain construction we must still prove affiliation in the fixed net.

For an old vacuum word represented by such a kernel, expand its field
labels as in
\eqref{R:finite-field-expansion} under the same joint kernel.
Old product multilinearity and the old-only identification in
Proposition~\ref{E:accessible-products} give equality of the old
word vector with this finite sum of homogeneous reconstructed word
vectors.  This exact label expansion makes no finite-rank assertion
about the joint kernel. Since $\mathcal D_{\mathrm{old}}$ is exactly the
algebraic span of these original vacuum-word vectors, it follows that
$\mathcal D_{\mathrm{old}}\subset\mathcal D$. In particular,
$\mathcal D$ is dense in the original $\cH$. This inclusion concerns the
specified vacuum-generated domain, not arbitrary additional vectors of
a separately chosen invariant domain.

Regard a test polynomial as a finite algebraic sum of finite ordered
words with the allowed kernels.  Multiplication concatenates the
words and tensors their kernels, and adjunction reverses the entries
and conjugates and reverses the kernel.  All resulting lengths remain
finite, with no fixed upper bound.  Left insertion defines joint
fields on $\mathcal D$.  To check that it is well-defined, let a whole
polynomial $A$ represent a zero vector.  Equation~\eqref{R:reflected-polynomials} gives
$W(A^*A)=0$.  Positivity and Cauchy--Schwarz applied to $A$ and
$B^*BA$, for any test polynomial $B$, give
\[
 |W(A^*B^*BA)|^2
 \leq W(A^*A)\,
       W\bigl((B^*BA)^*(B^*BA)\bigr)=0.
\]
Every term exists by the all-finite-length construction, and
$W(A^*B^*BA)=\|\Psi(BA)\|^2$.  Thus zero states form a left ideal.
This applies to the whole polynomial, even when its zero vector
results from cancellation between different scaling components.
Adjoint insertion gives
$A^\dagger(\bar f)|_{\mathcal D}\subset A(f)^*$, where the right
side is the Hilbert-space adjoint.  Its dense domain therefore makes
each smeared field closable.
For an old field inserted into a vector of
$\mathcal D_{\mathrm{old}}$, the same old-only identification applied to
the longer word, with the inserted test tensored with the original joint
kernel, proves agreement with the old insertion. For compact old smearings,
Section~\ref{sec:foundations} identified that insertion and its adjoint with
the restrictions of the original affiliated realizations. Thus the old fields and adjoints agree
with their original polynomial restrictions on
$\mathcal D_{\mathrm{old}}$, and
\eqref{R:leg-equation} becomes the operator identity
\begin{equation}\label{R:operator-leg}
 \Box L=\Theta\quad\hbox{on }\mathcal D.
\end{equation}
Translation covariance, similarity covariance, and locality are
operator identities on this common invariant domain.  No additional
Hilbert-space sector has been introduced.

This gives the common operator domain but has not yet supplied a
closed realization in a prescribed local algebra.  For that purpose
we prove exchange with every bounded element of that algebra's
commutant and then enlarge the domain by those commutant actions.
The distinction matters because weak field locality on
$\mathcal D$ alone does not establish affiliation.

We will use the same extension argument for the new scalar, the old
fields on the enlarged domain, and compact joint polynomials.  It is
convenient to prove it once for an adjoint pair.

\Needspace{6\baselineskip}
\begin{lemma}[Extension by commutant actions]\label{R:commutant-extension}
Let $\mathcal D$ be dense in $\cH$, and let $A,A^\sharp$ be operators
on $\mathcal D$.  Fix a localization set $K$ and the original
algebras $\cM(O)$ for open neighborhoods $O$ of $K$.  Suppose that
for every such $O$ and every $C\in\cM(O)'$,
\[
 \langle\phi,CA\psi\rangle
 =\langle A^\sharp\phi,C\psi\rangle,
 \qquad\phi,\psi\in\mathcal D,
\]
and that the corresponding identity also holds with $A,A^\sharp$
interchanged.  Then $A$ has a closed extension affiliated with every
$\cM(O)$, whose adjoint extends $A^\sharp$.  The reverse adjoint
pair has the same property.  If $A^\sharp=A$, the extension is
symmetric.
\end{lemma}

\begin{proof}
Let $\mathcal D_K$ be the span of $C_1\cdots C_m\psi$, with
$\psi\in\mathcal D$ and each $C_j$ in the commutant of an allowed
neighborhood algebra, including $m=0$.  Every finite collection of
these neighborhoods has a common smaller open neighborhood of $K$.
By isotony all its commutant coefficients belong to the commutant
of this smaller algebra.  Products and adjoints of the coefficients
therefore obey the assumed exchange identities.

For a finite sum with product coefficients $C_i$, prescribe
\[
 \widetilde A\sum_iC_i\psi_i=\sum_iC_iA\psi_i,
 \qquad
 \widetilde A^\sharp\sum_iC_i\psi_i=\sum_iC_iA^\sharp\psi_i.
\]
If the input sum is zero, pairing the proposed first image with
$\phi\in\mathcal D$ gives
$\langle A^\sharp\phi,\sum_i C_i\psi_i\rangle=0$.
Density proves well-definedness; the reverse exchange proves it
for the second prescription.  For two sums use exchange with
$D_j^*C_i$ in a common smaller algebra.  This gives
\[
 \widetilde A^\sharp\subset\widetilde A^*,\qquad
 \widetilde A\subset(\widetilde A^\sharp)^*.
\]
Both operators are thus closable.  Their common domain is invariant
under every unitary in every allowed commutant, and both operators
commute with that unitary and its inverse.  Their closed graphs retain
these invariances, proving affiliation simultaneously with all the
neighborhood algebras.  The adjoint inclusions survive closure.  In
the symmetric case the two prescriptions coincide, so their closure
is symmetric as well.
\end{proof}

\Needspace{6\baselineskip}
\begin{proposition}\label{R:affiliation}
For every real test $f$ with compact support in an open region $O$,
$L(f)$ has a closed symmetric extension affiliated with the original
algebra $\cM(O)$.  These realizations can be made
compatible for all neighborhoods of $\supp f$.
\end{proposition}

\begin{proof}
We first prove exchange with every bounded operator of the actual
commutant, not merely with opposite-region fields.  If
$C\in\cM(O)'$, then
\begin{equation}\label{R:commutant-exchange}
 \langle\phi,CL(f)\psi\rangle
       =\langle L(f)\phi,C\psi\rangle,
           \qquad \phi,\psi\in\mathcal D.
\end{equation}
First take $\phi,\psi$ to be finite sums of words with factorized
compactly supported tests, and by linearity treat one word on each
side.  Translate their slots into real mutually
spacelike patches, all spacelike from $\supp f$.  Subdivide $f$ into
local patches and choose its self-adjoint bounded approximants in
$O$.  The simultaneous approximating products converge there in
Hilbert norm.  Exchange holds at the bounded level because each
approximant to $L(f)$ commutes with $C$.  The norm limit proves
\eqref{R:commutant-exchange} on that real open placement set.

Continue separately in the translated polynomial slots on either
side.  A fixed $L(f)$ on the left of a polynomial leaves every suffix
spectral cone of that polynomial intact; on a bra the corresponding
opposite cones hold.  The bounded operator $C$ between the vectors
does not affect these two separate vector analyticity statements.
To see the common domain explicitly, use the positive vector tube
of \eqref{E:ordered-tube} for each polynomial before taking the bra
adjoint.  For a polynomial $\psi$ of length $n$, both $\psi(b)$ and
$L(f)\psi(b)$ are holomorphic on that tube.  In the latter product
let $q_1$ be its total momentum and
$q_2,\ldots,q_{n+1}$ its suffix momenta.  Fixing the first, $L$,
coordinate real removes its imaginary damping, but the strict
polynomial gaps still damp all $q_j$ with $j\geq2$.
The real smearing contributes $\widehat f(q_1-q_2)$, whose rapid
decay controls the remaining momentum since
$|q_1|\leq|q_1-q_2|+|q_2|$.  The tempered vector distribution
therefore has this smeared face, with all finite derivative bounds.
The same applies to $\phi$ and $L(f)\phi$.  After conjugating the
bra variables, the common scalar domain is the product of the
negative bra tube and the positive ket tube.  Boundedness of $C$
suffices for these pairings; neither commutation with translations
nor invariance of $\mathcal D$ under $C$ is used.
Their scalar pairings therefore obey boundary uniqueness, which
extends the identity to arbitrary placements and smearings.
For a compact joint kernel, first use a finite partition into product
patches and then approximate each piece by finite sums of factor
tests in the required finite smooth seminorms.  Theorem~\ref{R:mixed-products}
gives Hilbert-norm convergence both for the polynomial vector and
for the vector with the additional $L(f)$ slot.  Boundedness of $C$
therefore passes the identity to the joint kernel.  This test-topology
limit neither asserts finite tensor rank nor uses invariance of
$\mathcal D$ under $C$.  Compact joint tests are dense in the joint
Schwartz space, so the same Hilbert-norm continuity extends exchange
to all joint Schwartz kernels in $\mathcal D$.

Apply Lemma~\ref{R:commutant-extension} with
$A=A^\sharp=L(f)$ and $K=\supp f$.
The exchange identity holds for every neighborhood, so the lemma
gives one compatible closed symmetric affiliated extension.
Essential self-adjointness is neither asserted nor required.
\end{proof}

For a complex test $f=f_1+if_2$, linearity of the exchange identities
for the real tests gives the adjoint pair
$A=L(f)$, $A^\sharp=L(\bar f)$ on $\mathcal D$.
Lemma~\ref{R:commutant-extension} supplies a compatible closed
affiliated realization with the required adjoint.  This uses neither
self-adjointness nor strong commutation of the real and imaginary
parts.

We likewise need compatible realizations of the original entries on
the reconstructed domain before using polar tails on its vectors.
For an original field $A$ and compact test $f$, use the bounded
approximants of Proposition~\ref{E:bounded-approximation}, whose
vacuum and adjoint-vacuum convergence was established before the
mixed reconstruction.  Fix an open $O$ containing $\supp f$ and choose
these approximants $A_R(f)$ in $\cM(O)$.  Initially take the vectors
to be words with factorized compactly supported tests, as above.
At separated real placements the simultaneous
product limits give convergence of $A_R(f)\psi_R$ and
$A_R(f)^*\phi_R$.  Bounded commutation with $C\in\cM(O)'$ therefore
gives
\begin{equation}\label{R:old-commutant-exchange}
 \langle\phi,C A(f)\psi\rangle
   =\langle A^\dagger(\bar f)\phi,C\psi\rangle,
       \qquad \phi,\psi\in\mathcal D.
\end{equation}
Initially this holds on the separated placement set.  The same
separate-vector-tube argument, now with the first slot $A(f)$ or
$A^\dagger(\bar f)$ smeared over real coordinates, and the same
joint-test continuity
extend it to all the stated vectors.  Only the already proved mixed
vector spectra and the boundedness of $C$ enter this continuation.

Apply Lemma~\ref{R:commutant-extension} to the pair
$A(f),A^\dagger(\bar f)$ on $\mathcal D$, using
\eqref{R:old-commutant-exchange} and its adjoint version in every
neighborhood.  This gives compatible affiliated extensions of the
reconstructed restrictions, with adjoints extending the corresponding
insertions.  On $\mathcal D_{\mathrm{old}}$ these restrictions agree
with the original chosen realizations by the old-domain argument
above.  We do not require the newly reconstructed vectors to lie in
those previously chosen closed realizations, or identify their full
graphs with the extensions just constructed.

Only now use polar truncation for these realizations on polynomial
states.  If $\widehat A$ is the chosen closed realization, then
$\widehat A$ and $\widehat A^*$ agree on $\mathcal D$ with the
corresponding insertions, which preserve $\mathcal D$.  Induction gives
\[
 \mathcal D\subset
 \Dom(\widehat A^*\widehat A)^m
       \cap\Dom(\widehat A\widehat A^*)^m
 \qquad(m=1,2,\ldots),
\]
with the powers there equal to the alternating joint polynomial
words.  Their norms are finite by Theorem~\ref{R:mixed-products},
and spectral calculus gives the polar-tail estimates on every
polynomial vector.  For factorized polynomials, the natural-product
and polar-cutoff argument of Section~\ref{sec:foundations} applies to
these compatible realizations.  For compact joint kernels, take finite factor-test
approximations in the required test seminorms and pass both the vacuum
and adjoint-vacuum vectors by joint Hilbert-distribution continuity.
The closed wedge Tomita graph then passes its reality relation from
the factorized affiliated polynomial realizations to Hermitian joint
polynomial vectors.  These are the wedge reality properties needed
for the charge estimates of Section~\ref{sec:charges-cones}.

Finally, affiliation is only one part of the physical-membership
criterion in Assumption~\ref{ass:framework}.  The other part is also
present: $L=S$ is the explicit time-line extraction from actual
modular transforms of the physical virial field in arbitrarily small
diamonds.  Its separated products are the limits of those physical
bounded operations, and their joint continuation is uniquely fixed
by them.  Thus the field satisfies the stated intrinsic membership
criterion in the given net.  In a gauge formulation all these
operations take place in the chosen physical, gauge-invariant
algebras.

All construction requirements have now been checked: localization
in arbitrarily small original-net neighborhoods; joint field distributions
at every finite length on the common invariant domain
$\mathcal D\supset\mathcal D_{\mathrm{old}}$, with the required adjoints,
covariance, spectrum, and locality; and compatible affiliated smeared
realizations in the original net.
The notation $L$ henceforth denotes the resulting physical Hermitian
scalar of dimension two on $\mathcal D$.

\subsection{Improvement and the local Ward algebra}

Define on $\mathcal D$
\begin{equation}\label{R:improvement}
 t_{\mu\nu}=T_{\mu\nu}
       +\frac13(\partial_\mu\partial_\nu
                        -\eta_{\mu\nu}\Box)L.
\end{equation}
It is a physical symmetric conserved tensor, and
\eqref{R:operator-leg} gives $t^\mu{}_{\mu}=0$.
For an affine Poincar\'e Killing vector $X$, its charge difference
from $T$ is the divergence of the antisymmetric superpotential
\begin{equation}\label{R:superpotential}
 \frac13\partial^\lambda
 \left(X_\lambda\partial_\mu L-X_\mu\partial_\lambda L
                              +(\partial_\lambda X_\mu)L\right).
\end{equation}
Its compact commutator flux vanishes, so translation and Lorentz
charges agree.  If the initial stress tensor required the weight replacement of
Section~\ref{sec:foundations}, also include the local difference
between that selected tensor and the initial one.  That difference
is symmetric and conserved and has zero affine charges by the
weight-replacement argument.  This proves the improvement criterion.  We now check the special-current Ward algebra on the
actual reconstructed domain.

We first construct the local Ward action on the represented polynomial
algebra generated by the original fields and $L$; this is distinct from
the vector space $\mathcal F$ of point-field labels.
Proposition~\ref{R:compatible-families} will extend the action to every
further jointly admitted finite family.  For an open $O$, let
$\mathfrak P_c(O)$ consist of
finite sums of finite ordered products of original fields, $L$, and their
covariant derivatives, smeared by compact smooth joint kernels supported
in $O^n$, and identified when their restrictions to $\mathcal D$ agree.
The joint-domain construction makes concatenation and reversed adjunction
well-defined on this quotient and leaves $\mathcal D$ invariant.  Write
$P^\dagger$ for the reversed field-adjoint polynomial, to distinguish it
from the Hilbert adjoint.  Every finite length and every admitted compact
joint kernel are included; the kernel need not have finite tensor rank.

These compact polynomial insertions have compatible affiliated
realizations in the original net.  For a factorized word, take the
natural-domain product of the single-field realizations constructed
above, and take natural sums for finite polynomials.  Their domains
contain $\mathcal D$, and their adjoints contain the reversed adjoint
polynomials there, so they are closable.  Every commutant unitary and
its inverse preserve their natural graphs; their closures are therefore
affiliated.  This uses no commutation among the unbounded factors.
In particular, $t(f)$ is the finite-sum case of this argument.

For a compact joint kernel, partition into product patches inside $O^n$
and approximate in the smooth test topology by finite sums of tensor
tests.  If $P_j$ are the resulting factorized polynomials, then
Theorem~\ref{R:mixed-products}, with any fixed $\mathcal D$ word
appended, gives
\[
 P_j\psi\longrightarrow P\psi,\qquad
 P_j^\dagger\phi\longrightarrow P^\dagger\phi
 \quad(\phi,\psi\in\mathcal D)
\]
in Hilbert norm.  Affiliation of the factorized closures gives exchange
with commutant unitaries, hence with every $C\in\cM(O)'$ by linear
combinations of unitaries.  Boundedness of $C$ passes it to the limit:
\[
 \langle\phi,CP\psi\rangle
       =\langle P^\dagger\phi,C\psi\rangle,
       \qquad \phi,\psi\in\mathcal D.
\]
The reversed adjoint polynomial obeys the reverse exchange identity.
Lemma~\ref{R:commutant-extension}, with $A=P$ and
$A^\sharp=P^\dagger$, therefore supplies compatible affiliated
extensions for every neighborhood of the compact localization.
This is an extension with the required adjoint, not an assertion
about the minimal closure or a limit of the factorized closures.
Its polar truncations are bounded original-net operations converging on
$\mathcal D$, with adjoint convergence there.  All the commutants here
are those of the original net.

For a real conformal Killing field $X$, set
$j_{X,\mu}=t_{\mu\nu}X^\nu$ and
$\sigma_X=(\partial\cdot X)/4$.  Conservation and tracelessness give
$\partial^\mu j_{X,\mu}=0$.  Define
\begin{equation}\label{R:Ward-definition}
 \delta_X A=i[q_X,A]
\end{equation}
as a compact commutator flux enclosing the insertion.  This is an
operator insertion on $\mathcal D$, without assuming the existence
of a global unbounded charge on every state.  More explicitly, smear
the two fields with a compact kernel in their relative coordinate
and the target test in their merged coordinate.  Caps carry the same
unit-flux normalization and orientation.  In relative coordinates the
commutator current is divergence-free and supported in causal positions.
The difference of two normalized enclosing caps is a compact gradient
there, up to spacelike outer cutoff terms.  Conservation and locality
therefore allow exact deformation or shrinking while keeping the target
enclosed.  The resulting expression is a finite sum of joint smeared
products; combining the two momenta preserves the ordered cones for
the remaining slots.  Polynomial coefficients of $X$ give finite
derivatives of test kernels and do not spoil these properties.

For a joint target compactly localized in $O$, partition into product
patches and choose the relative caps within their localization margins.
A flux about a word is the finite sum of the fluxes about its slots, so
these representatives belong to $\mathfrak P_c(O)$.  To check the quotient
by zero operators, choose one absolute normalized compact cap enclosing
the finite union of target projections for the polynomial representatives.
It may lie outside $O$.  Deformation identifies every slotwise cap with
this cap.  The algebraic commutator rule identifies the sum with
$i[Q_X,P]$ on factor tests, and the same joint Hilbert continuity
extends the identity to the joint kernel.  Since $Q_X$ and $P$ preserve $\mathcal D$,
this vanishes if $P|_{\mathcal D}=0$.  Thus the local derivations are
well-defined on $\mathfrak P_c(O)$ and preserve it; their iterates are
finite joint polynomials.  The preceding exchange argument supplies
their compatible original-net affiliated extensions.

The complete vacuum flux vanishes on compact joint products by
Lemma~\ref{C:low-energy} and its subsequent dyadic charge estimate.
That proof applies here because the new compact polynomial vectors
have the wedge reality properties just proved, while the current
vacuum wave is the same purely timelike, translation-smooth wave
used there.  Its mass-band estimate is uniform; the summable
low-energy and high-energy dyadic bounds remain unchanged.
Consequently
\begin{equation}\label{R:vacuum-Ward}
 \sum_{i=1}^n
 W(A_1,\ldots,\delta_XA_i,\ldots,A_n)=0.
\end{equation}
The derivation preserves adjunction.  Its translation, Lorentz, and
dilation parts on the original fields are the physical actions:
\eqref{R:superpotential} handles the first two, and
Proposition~\ref{C:Q-zero} removes the possible conserved virial
charge in the third.  Identities first proved on an old vacuum leg
extend to $\mathcal D$ by locality, old polynomial density, and
ordered boundary uniqueness.

We need the transformation of the current itself to identify the
Lie algebra.  The two required primary identities are
\begin{align}
 \delta_X L&=X^\lambda\partial_\lambda L+2\sigma_XL,
                         \label{R:primary-L}\\
 \delta_X t_{\mu\nu}
   &=X^\lambda\partial_\lambda t_{\mu\nu}
      +(\partial_\mu X^\lambda)t_{\lambda\nu}
      +(\partial_\nu X^\lambda)t_{\mu\lambda}
      +2\sigma_Xt_{\mu\nu}. \label{R:primary-t}
\end{align}
We explain their return from the previously established pair
representation.  All short analytic products now agree with the
joint tempered ones: they agree on separated vacuum-wave products,
then everywhere by continuation.  Thus the Wick sphere/cap
comparison of Proposition~\ref{P:flux-wards} applies directly to
physical joint pair boundaries.  For a translation and target $T$,
the Lorentzian cap difference is the physical translation Ward
identity.  Its sphere version has the common nonzero normalization
fixed there.  The translation sphere identity for $L$ was already
proved, and differentiating it at a fixed enclosing surface gives
the same identity for the derivative terms in
\eqref{R:improvement}.  It therefore holds with target $t$.

Conjugate these translation identities in the positive pair
representation of Proposition~\ref{P:pair-representation}, whose
primary targets include $L$ and $t$.  Transporting the surrounding
sphere and its holomorphic Killing coefficient gives the sphere
identities for every conformal Killing field.  Return to Lorentzian
commutators with the same averaged cylinders as in
Proposition~\ref{P:flux-wards}.  The time caps converge by the joint
tempered boundary theorem; the sides have a fixed spacelike margin
and vanish with their thickness.  The Wick pullback of $X$ is
holomorphic on these compact surfaces, so the identical argument
works for its degree-zero, degree-one, and degree-two coefficients.
The normalization is the translation normalization in every case.
This proves \eqref{R:primary-L}--\eqref{R:primary-t} on the vacuum.
Their discrepancies are relatively local insertions.  Move such an
insertion to an extreme position on a spacelike open set; old
polynomial density and ordered uniqueness then show that its vacuum
vanishing implies vanishing on $\mathcal D$.  The two displayed
identities are therefore operator insertion identities on that
domain.

\Needspace{6\baselineskip}
\begin{proposition}\label{R:Ward-Lie}
The compact-flux Ward derivations realize the conformal Lie algebra
with the active-action convention
\begin{equation}\label{R:Lie-sign}
 [\delta_X,\delta_Y]=-\delta_{[X,Y]}.
\end{equation}
They obey \eqref{R:vacuum-Ward} and extend the physical affine Ward
derivations on the original field algebra.
\end{proposition}

\begin{proof}
Coordinates multiplying fields are held fixed when taking a Ward
variation.  Using \eqref{R:primary-t}, the conformal Killing equation,
and conservation, direct differentiation gives
\begin{equation}\label{R:current-variation}
 \delta_X j_{Y,\mu}
  =\partial^\lambda(X_\lambda j_{Y,\mu}
                              -X_\mu j_{Y,\lambda})
       -j_{[X,Y],\mu}.
\end{equation}
For example, expanding the divergence gives
$4\sigma_X j_{Y,\mu}+X\cdot\partial j_{Y,\mu}
-(\partial^\lambda X_\mu)j_{Y,\lambda}$; subtracting
$j_{[X,Y]}$ cancels the derivative of $Y$, and
$\partial_\lambda X_\mu+\partial_\mu X_\lambda
=2\sigma_X\eta_{\lambda\mu}$ yields exactly the two index terms
and the remaining $2\sigma_X$ in \eqref{R:primary-t}.

Choose one compact flux $Q_Y(F)$ implementing $\delta_Y$ on an
insertion $A$ and its local $X$ variation.  Use a larger compact
flux for $\delta_X$ on $Q_Y(F)$ and these insertions.  The derivation
rule on the joint domain gives
\[
 [\delta_X,\delta_Y]A=i[\delta_X Q_Y(F),A].
\]
The first term on the right of \eqref{R:current-variation} is an
antisymmetric divergence.  Its commutator flux about the compact
insertion is zero: integration by parts leaves only the outer
boundary, where locality permits the cutoff to be removed.
The second term therefore gives
$-i[Q_{[X,Y]}(F),A]=-\delta_{[X,Y]}A$.
All products and iterated insertions in this calculation are defined
by compact joint kernels on $\mathcal D$.  Equation
\eqref{R:vacuum-Ward} and the affine identification were proved
above.
\end{proof}

\subsection{Centered fields and compatible families}

A Ward variation with polynomial coordinate coefficients is an insertion,
but need not itself be a translation-covariant point-field label.  We now
remove that coordinate part and verify the intrinsic membership criterion
for each finite family needed in a finite Ward calculation.  Write
\[
 \begin{gathered}
 p_c(z)=c,\qquad d(z)=z,\qquad
 m_{b,c}(z)=(b\cdot z)c-(c\cdot z)b,\\
 k_b(z)=2(b\cdot z)z-z^2b,\qquad b,c\in\R^{1,3}.
 \end{gathered}
\]
Fix a homogeneous finite Lorentz multiplet $A$ of dimension $\Delta$
among the original fields, $L$, and their covariant derivatives.
Let $N_A$ be the dimension of its component space.  Define the centered insertion
\[
 B_{b,A}(x)=\delta_{k_b^{[x]}}A(x),\qquad
 k_b^{[x]}(z)=k_b(z-x).
\]
For a target test $f$, this denotes $i$ times the difference of joint products
with relative kernel $f(x)h^\mu(r)k_b^\nu(r)$, $r=y-x$, where $h$ is a
normalized enclosing cap.  It involves no pointwise product or
unsmeared surface restriction.  The cap deformation just proved makes
the definition independent of its admissible normalized choice.

Define its iterates on the same polynomial domain by
\[
 \begin{aligned}
 B_A^{(0)}&=A,\\
 B^{(r)}_{b_r\ldots b_1,A}(x)
   &=\delta_{k_{b_r}^{[x]}}
       B^{(r-1)}_{b_{r-1}\ldots b_1,A}(x),\qquad r\ge1.
 \end{aligned}
\]
Every fixed expression is a finite compact joint polynomial because
\(\delta_X\) preserves the local polynomial algebra.

\Needspace{6\baselineskip}
\begin{proposition}[Finite families of centered fields]
\label{R:centered-families}
Let $A$ range over a fixed finite collection of homogeneous original
fields, $L$, and their covariant derivatives.  For every fixed finite
collection of depths, the corresponding centered fields $B_A^{(r)}$
have all mutual and original-field joint products on $\mathcal D$,
with the required adjoints, locality, covariance, ordered spectra,
and compatible original-net affiliated realizations.  If $A$ has
real dimension $\Delta$ and $N_A$ components, depth $r$ has dimension
$\Delta-r$ and at most $4^rN_A$ components before field identities.
These finite families satisfy the intrinsic membership criterion of
Assumption~\ref{ass:framework}.  Every finite Ward word on the
original fields and $L$ is a finite sum of polynomial-coordinate
multiples of derivatives of such centered fields.
\end{proposition}

\begin{proof}

For a similarity $g(z)=\lambda\Lambda z+a$, $\lambda>0$, write $U(g)$
for the already given Poincar\'e/dilation implementer.  Changing variables
in the smooth current test for the dimension-four $t$ sends
$X$ to $g_*X=\lambda\Lambda X\circ g^{-1}$ and an absolute normalized
cap test $h_0$ to $(h_0)_g=\lambda^{-1}\Lambda h_0\circ g^{-1}$,
in the tensor convention of their contraction.  For a relative cap,
the translation cancels in this formula.  Since
$g_*k_b^{[x]}=\lambda^{-1}k_{\Lambda b}^{[gx]}$, cap independence gives
\[
 U(g)B_{b,A}(x)U(g)^{-1}
   =\lambda^{\Delta-1}\rho_A(\Lambda)
                       B_{\Lambda b,A}(gx),
\]
where $\rho_A$ is the Lorentz component matrix of $A$.  Thus $B_{b,A}$
is a possibly zero translation-covariant homogeneous insertion of real
grade $\Delta-1$, in a finite Lorentz family with one additional vector index.
Only the existing similarity covariance and the proved cap deformation
have been used.

Let $s_{b,c}$ denote the constant component matrix in the affine action
\[
 \delta_{p_c}A=c\cdot\partial A,\qquad
 \delta_dA=(x\cdot\partial+\Delta)A,\qquad
 \delta_{m_{b,c}}A=(m_{b,c}(x)\cdot\partial+s_{b,c})A.
\]
Expanding
$k_b(z-x)=k_b(z)-2(b\cdot x)d(z)-2m_{b,x}(z)+p_{k_b(x)}(z)$
in the centered flux gives
\begin{equation}\label{R:centered-Ward}
 \delta_{k_b}A(x)
 =\bigl(k_b(x)\cdot\partial+2\Delta(b\cdot x)+2s_{b,x}\bigr)A(x)
       +B_{b,A}(x).
\end{equation}
The polynomial-coordinate term is an insertion, in general not a
translation-covariant point-field label; its coefficients multiply
the target test after smearing.  For example,
\eqref{R:primary-L} gives $B_{b,L}=0$.

The local affine Ward actions on $B$ also follow from the proved Lie
relation.  Indeed
$[p_c,k_b]=2(b\cdot c)d+2m_{b,c}$ and $[d,k_b]=k_b$.
At a fixed enclosing cap target differentiation commutes with the
commutator, so \eqref{R:Lie-sign} gives
\[
 \begin{aligned}
 (\partial_c-\delta_{p_c})\delta_{k_b}A
    &=2(b\cdot c)\delta_dA+2\delta_{m_{b,c}}A,\\
 \delta_d\delta_{k_b}A
    &=(x\cdot\partial+\Delta-1)\delta_{k_b}A .
 \end{aligned}
\]
The polynomial term in \eqref{R:centered-Ward} has the same two
defects.  Subtraction gives
$\delta_{p_c}B=c\cdot\partial B$ and
$\delta_dB=(x\cdot\partial+\Delta-1)B$.
The Lorentz bracket $[m_M,k_b]=-k_{Mb}$ for $m_M(z)=Mz$ similarly
gives the tensor-product component action.  Thus these affine
identifications are consequences, not an assumed conformal closure.

For any fixed finite collection of centered insertions, substituting
their relative kernels into a joint product expands it into finitely
many of the already constructed products.  This is a continuous map
of compact smooth or Schwartz test spaces: the relative supports are
compact and the coefficients are polynomial.  It supplies all joint
products at every finite length on $\mathcal D$, including adjoints
and all mixed orderings.  Merging each block retains the ordered suffix
cones.  For real $b$ and $h$, reversing the two products gives
$B_{b,A}^\dagger=B_{b,A^\dagger}$.  Positivity holds in the original
Hilbert space, and locality
follows by shrinking the relative supports on spacelike product patches;
a compact test on the spacelike configuration set is first partitioned
into such patches.  The preceding exchange argument supplies compatible
original-net affiliated realizations and their adjoints for these finite
homogeneous families.

At each depth, choose a shrinking representative of the previous depth
inside an inner neighborhood, and its enclosing cap inside the prescribed
target neighborhood.  The previous depth's locality makes its current
commutator causally supported, so conservation gives the same normalized-cap
deformation and shrinking property at the next depth.  Every nested
expression is still a finite joint polynomial in the original fields
and $L$.  The joint continuity and full-commutant construction above
therefore give all mutual products and compatible affiliation for any
fixed finite collection of depths.  The covariance calculation gives
grade $\Delta-r$ and at most $4^rN_A$ components before field identities.
Equation~\eqref{R:centered-Ward}, the affine actions just
proved, and finite Leibniz rules express every fixed finite Ward word
as a finite sum of polynomial-coordinate multiples of finite derivatives
of these families.  For an original
$A=\sum_{\Delta\in S_A}A_\Delta$, do this separately for the finitely
many components, using the same target joint kernel in every component
term and assigning no single weight to their sum.  Thus each finite Ward
calculation uses homogeneous insertions in a jointly constructed finite
family on $\mathcal D$.  This does not identify them with original labels
in $\mathcal F$, assert global closure under centered descendants, or
construct a conformal action on the weakly closed bounded net.
\end{proof}

The same currents also act on other fields already admitted by the
intrinsic criterion.  This requires a joint reconstruction with $L$;
separate admission of two families is not by itself a statement about
their mutual products.

\Needspace{6\baselineskip}
\begin{proposition}[Extension to compatible physical families]
\label{R:compatible-families}
Let $\mathcal G$ be a finite homogeneous finite-component family
jointly admitted with the original fields under
Assumption~\ref{ass:framework}.  The scalar $L$ has all joint products
with this family and the original fields, with compatible adjoints and
original-net affiliated realizations on a common invariant domain
$\mathcal D_{\mathcal G}\supset\mathcal D$ in the original Hilbert
space.  On the resulting represented polynomial algebra, the same
currents $j_X$ give unbroken local conformal Ward derivations,
extending the physical Poincar\'e and dilation actions on
$\mathcal G$ as well as on the original fields.  Every fixed finite
family of their centered Ward descendants satisfies the same intrinsic
membership criterion.  These constructions agree when one compatible
family is contained in another.
\end{proposition}

\begin{proof}
Regard the original fields and $\mathcal G$ as the spectator fields
in the boundary reconstruction of this section.  Their joint products,
common invariant realization, adjoints, and original-net affiliations
are given by the admission criterion.  Each fixed homogeneous list
has tempered vacuum distributions by the scaling argument of
Section~\ref{sec:foundations}; finite sums treat the original labels.
Their affiliated polar truncations obey
Proposition~\ref{E:bounded-approximation}, since the required moments
are alternating words in these already given joint products.  Thus
Proposition~\ref{E:accessible-products} supplies their analytic products
with the same accessible $L$ and the same reflected splittings.
No current Ward identity on $\mathcal G$ is needed for this step.

The pure $L$ construction is unchanged.  In the simultaneous mixed
induction, replacing $L$ by $\Theta$ lowers the number of $L$ entries
and adds an original homogeneous spectator.  The merged-face estimate,
causal ambiguity, Gaussian coefficient recovery, and scaling argument
therefore apply with $\mathcal G$ among the fixed spectator labels.
They prove Theorem~\ref{R:mixed-products} for every such enlarged
list.  Its positivity and full-commutant exchange then give the
invariant joint domain $\mathcal D_{\mathcal G}$ and the compatible
affiliated realizations exactly as above.  Products involving only
the original fields and $L$ agree with the previous ones: their
analytic products are the same approximation-independent limits,
and their tempered boundaries are unique.  Consequently
$\mathcal D\subset\mathcal D_{\mathcal G}$ and all insertion actions
agree there.  Old-and-$\mathcal G$ products likewise retain their
given vacuum-generated restrictions, without requiring agreement
of chosen closed realizations on an independently larger domain.
The same reasoning proves agreement for nested
compatible families.  If derivatives of the added fields are included
as labels, their dimensions lie in the finite union of progressions
$\Delta+\mathbb Z_{\ge0}$; each fixed grade receives only finitely
many derivative components.  This preserves finite scaling support
and the stipulated locally finite, bounded-below grade convention.

Compact commutator fluxes now define $\delta_X$ on the enlarged
polynomial algebra.  Conservation, locality, and joint continuity
give cap deformation, the derivation rule, and preservation of the
quotient by zero operators just as before.  Its compact polynomial
vectors have wedge reality by their affiliated realizations and the
joint graph limits.  Lemma~\ref{C:low-energy} and its summable dyadic
estimate therefore give vanishing complete vacuum flux and the Ward
identity \eqref{R:vacuum-Ward} on the enlarged domain.

We check the physical affine actions on the added fields, rather
than assume them.  Let $X$ be a translation, Lorentz, or dilation
generator, and let $G_X$ be its given self-adjoint generator, with
the convention that the physical variation is $i[G_X,\cdot]$.
For a compact smearing $F$ of a member of $\mathcal G$ and an old
compact polynomial $C$, the vacuum Ward identity gives
\[
 \langle C\Omega,\delta_XF\,\Omega\rangle
 =-\langle(\delta_XC)\Omega,F\Omega\rangle
 =i\langle G_XC\Omega,F\Omega\rangle
 =\langle C\Omega,iG_XF\Omega\rangle.
\]
The middle identity uses the already proved affine action on old
polynomials.  Smooth compact smearing and the given similarity
covariance place $F\Omega$ and $C\Omega$ in the generator domains;
the vacuum is invariant.  Density of old polynomial vectors thus
identifies the vacuum action with the physical affine variation of
$F$, including its component and weight terms.

The discrepancy between these two variations is relatively local
and closable on $\mathcal D_{\mathcal G}$: it is a finite compact
joint insertion with a densely defined reversed adjoint.  It
annihilates the vacuum.  Choose an old double cone spacelike to its
compact localization.  Locality makes it annihilate every old
polynomial vacuum vector from that cone, a dense set by
Reeh--Schlieder.  The closure of its graph therefore contains
$\cH\times\{0\}$, and being an operator graph it can contain no
nonzero output.  The discrepancy vanishes on the whole joint domain.
This proves the required affine identifications without assuming
new current Ward identities.

The primary identities for $L$ and $t$ likewise persist on the
enlarged domain: their relatively local discrepancies vanish on the
original dense domain, and the same adjoint and closed-graph argument
applies.  The current-variation computation and compact nested flux
proof of Proposition~\ref{R:Ward-Lie} consequently give the conformal
Lie relation on every enlarged polynomial insertion.  Finally the
centered construction of Proposition~\ref{R:centered-families} uses
only these Ward identities, similarity covariance, cap shrinking,
and finite joint products.  Repeating it on
$\mathcal D_{\mathcal G}$ proves its conclusion for every fixed
finite family of added targets and finite descendant depths.
\end{proof}

Thus the improvement \eqref{R:improvement} lies in the original
physical theory, has the original translation and Lorentz charges,
and is traceless.  Its special-conformal currents are physical
conserved currents with unbroken local Ward identities.  The
conclusion concerns flat spacetime and does not impose a
curved-background Weyl identity.
==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//sections/08-reconstruction.tex ====

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==== END Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory//references.bib ====