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\hypersetup{
 pdftitle={V: Admissible currents, retarded response, and finite Enter readout},
 pdfsubject={A local adaptation on a prescribed conical spacetime}
}

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\title{\textbf{V: Admissible currents, retarded response,\\
and finite Enter readout}\\[4pt]
\large A local adaptation on a prescribed conical spacetime}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We develop a local spacetime-Enter adaptation of the fixed-time
scalar-response problem of Manuscript V. The central analytical
finding is an exact separation between two obstructions:
the complete distributional Ward defect controls raw compact-current
representability, whereas its causal propagation controls
representability after a harmonic prescription. Initial-surface
terms are retained explicitly. A compact-Cauchy residual wave
makes the obstruction nonzero on an open dense family of
smearings, while compact curvature currents nevertheless
interpolate the same response on every declared two-dimensional
transverse-traceless plane whose separating region is admitted.

A separate calculation constructs the first retarded coefficient
of the compactly switched minimal scalar vertex. Its external-metric
version is linear in the scalar field, has coefficientwise
scaling degree below four, and has no minimal-extension ambiguity.
The one-oscillator version is a mixed quadratic operator with a
proved essentially self-adjoint real polynomial realization.
A distinct Newton--wave composition has a symmetric cubic Wick
form with a joint ultraviolet, transfer, and initial-edge cutoff
limit. A regular-region witness distinguishes states agreeing on
all polynomial forms of degree at most two.

The operational contribution is an exact, reference-preserving
finite instrument obtained by compressing the represented
responses and cubic forms before destructive clock readout.
For the cubic witness this gives an explicit qubit experiment.
This connects admissibility and form-level distinctions to a
complete finite readout without identifying them with a continuum
field apparatus. Its potential value is methodological: it
prevents classical descent, operator representation, and recorded
response from being conflated.

The cone, external schedule, flat continuation, vacuum, and Newton
prescription are declared inputs. No global conical Green theorem,
intrinsic Newton inverse, derived spacetime, Einstein dynamics,
or inherited characteristic quantum coefficient is claimed.
\end{abstract}

\statementbox{Scope of the adaptation}{%
The positive release is local Grade S in the explicit sense of
Definition~\ref{def:grade}. It is not a global intrinsic
reformulation of every Fourier or Newton construction on the cone.
The analytical results are proved on a declared flat presentation;
their admitted local restrictions and finite operational interfaces
are placed in the prescribed conical laboratory.}

\tableofcontents

%=====================================================================
\section{Introduction and separation of objects}
%=====================================================================

The problem has three logically distinct stages:
\[
\text{classical response}
\longrightarrow
\text{current or represented form}
\longrightarrow
\text{specified instrument}.
\]
A result at one stage does not construct the next.

The analytical mechanisms studied here originate in Cheung's
Manuscript V \cite{CheungV}. We reproduce the required arguments
with explicit domains and attach a finite operational interface.
The claims of this adaptation are not assertions that the
underlying analytical findings are new relative to that source.

The five analytical objects remain separate:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a fixed-time classical response, with the scalar held fixed;
\item its restriction to a specified finite-dimensional TT plane;
\item a first retarded scalar coefficient with an external metric;
\item a first retarded coefficient with one metric oscillator;
\item a Newton--wave cubic scalar form.
\end{enumerate}
The last object is not obtained by identifying it with either
retarded coefficient.

\subsection{Enter as an interface discipline}

In the Enter calculus, destructive readout removes the measured
quantum names from the live quantum context. A classical record
and unmeasured quantum factors may survive
\cite{CheungEnter,CheungPhilosophy,CheungEn09}.
We use the explicitly specified finite Born instrument below;
we do not rederive Cheung's characterization theorem from
abbreviated axiom names.

The distinctions are important:
\begin{itemize}
\item a classical record is not a surviving clock;
\item agreement of outcome probabilities is weaker than agreement
of complete residual instruments;
\item a finite encoding is not automatically a field apparatus;
\item a supplied causal geometry is not reconstructed by assigning
locations to quantum names.
\end{itemize}

The geometric setting follows I-B \cite{CheungIB}. The operational
proofs use complete channels, including untouched references, in
the usual finite-dimensional sense \cite{Watrous}.

\begin{definition}[Local Grade S]
\label{def:grade}
A local Grade S construction specifies:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item a prescribed spacetime and regular laboratory;
\item the analytical equations, continuation prescriptions, and
domains used to define the responses;
\item an explicit finite response encoding and complete clock
instrument;
\item time, rod, and causal dictionaries for the admitted events;
\item a reference-preserving finite transport protocol and a
separate resource ledger.
\end{enumerate}
No clause requires the geometry or equations to be inferred
from records.
\end{definition}

%=====================================================================
\section{Geometry, local interpretation, and analytical continuation}
\label{sec:geometry}
%=====================================================================

Fix \(c>0\), \(0<\mu<2\pi\), and
\(\alpha=1-\mu/(2\pi)\). Prescribe
\begin{equation}
 M=\RR_T\times C_\mu^\circ\times\RR_z,\qquad
 \bar g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2,
 \label{eq:cone}
\end{equation}
where \(r>0\) and \(\phi\in\RR/(2\pi\mathbb Z)\).
The missing apex gives the worldsheet
\[
 \mathcal W=\RR_T\times\{o\}\times\RR_z
\]
in the metric completion. It is not a worldline.
Signature, lapse, future orientation, and deficit are inputs.

\begin{proposition}[The full regular cone is not globally hyperbolic]
\label{prop:nonGH}
The regular product \eqref{eq:cone} has a noncompact causal diamond.
\end{proposition}

\begin{proof}
Fix \(r_0>0\), one angular coordinate, and \(z=0\).
Take events \(p,q\) at that spatial point with times
\(-2r_0/c\) and \(2r_0/c\).
For \(r_n\downarrow0\), the events
\((0,r_n,\phi,0)\) lie in \(J^+(p)\cap J^-(q)\), by radial
timelike travel. They have no convergent subsequence in \(M\),
because the apex is excluded.
\end{proof}

Thus a global wave theorem cannot be transferred merely from local
flatness. The standard global Green theory requires an appropriate
globally hyperbolic domain \cite{BGP}.

Choose a regular developing chart
\[
 \chi:U\longrightarrow B_R\subset\RR^3
\]
and a reference time \(T_0\). Write
\[
 t=c(T-T_0),\qquad x=\chi(r,\phi,z).
\]
The local metric becomes
\[
 \eta=-\dd t^2+\dd x^2,\qquad
 \eta_{\mu\nu}=\diag(-1,1,1,1).
\]

\begin{definition}[Presentation and laboratory contract]
\label{def:presentation}
The analytical presentation is the auxiliary Minkowski space
\[
 \widehat M=\RR_t\times\RR^3_x.
\]
Its Cauchy evolutions, Fourier transforms, vacuum representation,
and Newton inverse are specified parts of the model.

A finite admitted packet consists of finitely many response
settings, compact measurement and curvature-test supports, their
finite-time causal shadows, and a finite operational schedule.
These compact regions must lie inside
\[
 (-A,A)\times B_R
\]
with a positive margin from its boundary. They are interpreted
in the conical laboratory through the developing chart.

Noncompact auxiliary waves and Newton potentials are restricted
to that laboratory; they are not extended by zero across its
boundary and are not claimed to define global conical solutions.
\end{definition}

For a fixed packet, the laboratory can be chosen after the compact
regions have been selected. Regular conical charts of any prescribed
finite size can be installed sufficiently far from the apex.

All analytical spaces below refer to \(\widehat M\).
Their restriction supplies one declared model on the cone, not a
classification of arbitrary conical fields.

\begin{lemma}[Local causal interpretation]
\label{lem:causal}
A developed straight displacement with
\[
 \Delta T>0,\qquad |\Delta x|\le c\Delta T
\]
is future causal. At a static port, \(\dd\tau=\dd T\).
\end{lemma}

\begin{proof}
The tangent inequality is
\(-c^2\dot T^2+|\dot x|^2\le0\).
The proper-time statement follows by setting \(\dot x=0\).
\end{proof}

The Newton part will retain nonlocal initial-data dependence.
The finite schedule locality theorem will not be applied to
that elliptic construction.

%=====================================================================
\section{Operators and the fixed-time response}
\label{sec:response}
%=====================================================================

Fix \(m>0\), and set
\[
 \Box=-\partial_t^2+\Delta,\qquad
 P=\Box-m^2,\qquad
 \Omega_m=(-\Delta+m^2)^{1/2}.
\]
For a covector \(\xi_\nu\), define
\[
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
 \qquad X_\xi=\xi^\mu\partial_\mu.
\]
For a symmetric tensor \(h\), write
\[
 (\mathsf rh)_{\mu\nu}
 =h_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta h,
 \qquad
 (Dh)_\nu=\partial^\mu(\mathsf rh)_{\mu\nu}.
\]
Trace reversal is an involution in four dimensions.

The linearized Einstein and scalar-response operators are
\begin{align}
 \Glin(h)
 &=-\frac12\Box\mathsf rh+\frac12\mathsf rKDh,
 \label{eq:Einstein}\\
 L_h\phi
 &=-h^{\mu\nu}\partial_\mu\partial_\nu\phi
 -(Dh)^\nu\partial_\nu\phi.
 \label{eq:L}
\end{align}

\begin{lemma}[Differential identities]
\label{lem:identities}
The following identities hold:
\[
 DK=\Box,\qquad \Glin K=0,\qquad
 \partial^\mu\Glin(h)_{\mu\nu}=0,
 \qquad L_{K\xi}=-[P,X_\xi].
\]
The operator \(\Glin\) is formally self-adjoint.
\end{lemma}

\begin{proof}
In \(DK\xi\), the derivatives of \(\partial_\mu\xi^\mu\)
cancel. Substitution in \eqref{eq:Einstein} proves
\(\Glin K=0\), and taking a divergence proves the Bianchi
identity. Twice applying the product rule gives
\[
 [P,X_\xi]\phi
 =(\Box\xi^\nu)\partial_\nu\phi
 +2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu\phi,
\]
which is \(-L_{K\xi}\phi\).
Integration by parts in the constant-coefficient expression
\eqref{eq:Einstein} proves formal self-adjointness.
\end{proof}

Let \(G_{m,R},G_{m,A}\) be the scalar Green operators for \(P\).
Their spatial Fourier time kernels are
\[
 G_{m,R}(t)=-\Theta(t)\frac{\sin(t\Omega_m)}{\Omega_m},
 \qquad
 G_{m,A}(t)=\Theta(-t)\frac{\sin(t\Omega_m)}{\Omega_m}.
\]
For componentwise massless waves write \(G_{r,R},G_{r,A}\),
where \(r\) indicates tensor rank, and use
\begin{equation}
 E_r=G_{r,A}-G_{r,R}.
 \label{eq:E}
\end{equation}
Its time kernel is
\(S_w(t)=\sin(t|D|)/|D|\).

These formulas define the inverses on compact distributions by
transposition. Smooth Cauchy solutions follow from Fourier
evolution. Finite propagation follows from the wave energy
identity. On finite time intervals, all these maps are continuous
in the relevant compact-data smooth topologies.

Fix a real smooth free scalar
\[
 P\phi=0
\]
with compact smooth Cauchy data, and fix
\(f\in C_c^\infty((0,b)\times\RR^3;\RR)\).
Define the response by
\[
 PS_0(h)\phi=-L_h\phi,\qquad
 S_0(h)\phi|_0=\partial_tS_0(h)\phi|_0=0,
\]
and
\begin{equation}
 B_f(h,\phi)=\int fS_0(h)\phi\,\dd^4x.
 \label{eq:B}
\end{equation}
The scalar is held fixed in this metric functional.

Put
\begin{equation}
 u=-G_{m,A}f,\qquad V=\Theta(t)u.
 \label{eq:u}
\end{equation}
Then \(Pu=-f\), and self-adjointness of the spatial sine
propagator gives
\begin{equation}
 B_f(h,\phi)=\int_0^b uL_h\phi\,\dd^4x.
 \label{eq:adjoint}
\end{equation}
The relevant source region is compact:
\[
 K_f=J^-_\eta(\supp f)\cap\{0\le t\le b\}.
\]

%=====================================================================
\section{Raw current, surface term, and complete Ward defect}
\label{sec:raw}
%=====================================================================

\begin{theorem}[Distributional current and initial-surface term]
\label{thm:raw}
For every smooth \(h\),
\[
 B_f(h,\phi)=\pair{J_{\rm raw}}{h},
\]
where
\begin{equation}
 J_{\rm raw}^{\mu\nu}
 =\partial^{(\mu}V\,\partial^{\nu)}\phi
 -\frac12\eta^{\mu\nu}
       \partial_\alpha(V\partial^\alpha\phi).
 \label{eq:Jraw}
\end{equation}
This is a compact order-zero tensor distribution. It has the form
\[
 J_{\rm raw}=\Theta(t)J_{\rm bulk}+\delta(t)j_\Sigma,
\]
with
\begin{equation}
 j_\Sigma^{00}=\frac12u_0\dot\phi_0,\qquad
 j_\Sigma^{0i}=-\frac12u_0\partial_i\phi_0,\qquad
 j_\Sigma^{ij}=\frac12\delta^{ij}u_0\dot\phi_0.
 \label{eq:surface}
\end{equation}
\end{theorem}

\begin{proof}
Integrate the metric derivative in
\(\pair{V}{L_h\phi}\) by parts. The terms containing
\(Vh^{\mu\nu}\partial_\mu\partial_\nu\phi\) cancel, leaving
\eqref{eq:Jraw}. A cutoff equal to one near \(K_f\) extends the
calculation from compact to arbitrary smooth \(h\).

Since
\[
 \partial^\mu(\Theta u)
 =\Theta\partial^\mu u-\delta(t)\delta^\mu_0u_0,
\]
the singular contribution is exactly \eqref{eq:surface}.
The remaining density is bounded and piecewise smooth on a
compact set.
\end{proof}

Pair vector distributions against covectors and define
\[
 (K^*J)^\nu=-2\partial_\mu J^{\mu\nu},
 \qquad
 (D^*a)^{\mu\nu}
 =-\partial^{(\mu}a^{\nu)}
 +\frac12\eta^{\mu\nu}\partial_\alpha a^\alpha.
\]
Direct differentiation gives \(K^*D^*a=\Box a\).

\begin{theorem}[Complete defect]
\label{thm:defect}
Let \(H=K^*J_{\rm raw}\). Then
\begin{equation}
 H=\gamma+\delta(t)a+\delta'(t)b_0,
 \label{eq:H}
\end{equation}
where, componentwise,
\begin{align}
 \gamma^\nu&=f\partial^\nu\phi,\nonumber\\
 a^\nu&=\dot u_0\partial^\nu\phi_0
          -u_0\partial_t\partial^\nu\phi|_0,\nonumber\\
 b_0^\nu&=u_0\partial^\nu\phi_0.
 \label{eq:Hab}
\end{align}
For every smooth descriptor,
\begin{equation}
 B_f(K\xi,\phi)=\pair{H}{\xi}.
 \label{eq:gaugedefect}
\end{equation}
\end{theorem}

\begin{proof}
For a scalar distribution \(v\), the product rule gives
\[
 2\partial_\mu\left(
 \partial^{(\mu}v\,\partial^{\nu)}\phi
 -\frac12\eta^{\mu\nu}\partial_\alpha(v\partial^\alpha\phi)
 \right)
 =(Pv)\partial^\nu\phi,
\]
using \(P\phi=0\).
Also
\[
 P(\Theta u)=-f-\delta(t)\dot u_0-\delta'(t)u_0.
\]
Multiplying the last term by \(\partial^\nu\phi(t)\) and using
\(c(t)\delta'=c(0)\delta'-\dot c(0)\delta\) proves
\eqref{eq:H}--\eqref{eq:Hab}.
Equation~\eqref{eq:gaugedefect} is compact integration by parts.
\end{proof}

The initial-data content can also be read directly.

\begin{proposition}[Homogeneous subtraction]
\label{prop:subtraction}
Let
\[
 H_m(A,B)(t)=\cos(t\Omega_m)A+
                 \frac{\sin(t\Omega_m)}{\Omega_m}B.
\]
Then
\[
 S_0(K\xi)\phi
 =X_\xi\phi-H_m\bigl(\operatorname{Data}_0(X_\xi\phi)\bigr).
\]
\end{proposition}

\begin{proof}
Both sides have zero Cauchy data and solve
\(Pq=P(X_\xi\phi)=-L_{K\xi}\phi\).
Cauchy uniqueness applies.
\end{proof}

Thus omitting the initial-surface current or the homogeneous
subtraction changes the fixed-time observable. These are the
mechanisms identified in V@1 \cite{CheungV}; they are not
introduced by the clock interface.

%=====================================================================
\section{Admissibility on the declared solution spaces}
\label{sec:admissibility}
%=====================================================================

A smooth field on \(\widehat M\) is spacelike compact if its support
is contained in \(J_\eta(C)\) for some compact \(C\).
Define
\[
 \SSol=\{h\in C_{\rm sc}^\infty:\Glin(h)=0\},\qquad
 \SSol_{\rm dD}=\{h\in\SSol:Dh=0\},
\]
and
\[
 \EE=\{J\in C_c^\infty(S^2T\widehat M;\RR):
                  \partial_\mu J^{\mu\nu}=0\}.
\]

\begin{definition}[Compact-current representability]
A functional on a named metric space \(\mathcal V\) is
representable if one \(J\in\EE\) satisfies
\[
 B(h)=\int h_{\mu\nu}J^{\mu\nu}\,\dd^4x
 \quad(h\in\mathcal V).
\]
For the local conical interpretation, the support of \(J\) and the
response packet must satisfy Definition~\ref{def:presentation}.
\end{definition}

These currents have the test-tensor type used by Fewster and Hunt
\cite{FH}. We do not use a radical-exhaustion theorem.

\subsection{Green identities and compact smoothing}

For a compact distribution \(C\),
\begin{equation}
 E_rC=0
 \quad\Longleftrightarrow\quad
 G_{r,A}C=G_{r,R}C=:A_C\text{ is compact},\quad
 \Box A_C=C.
 \label{eq:compactprimitive}
\end{equation}
Indeed, the common support is contained in the compact causal
diamond between the compact source and itself. The converse follows
from the Green identities on compact distributions.

\begin{lemma}[Smooth propagation of the present sources]
\label{lem:smooth}
The causal propagations of \(J_{\rm raw}\) and \(H\) are smooth.
If \(E_1H=0\), the causal propagation of
\(J_{\rm raw}-D^*G_{1,A}H\) is smooth as well.
\end{lemma}

\begin{proof}
The sources are finite sums of slice distributions with smooth
compact spatial coefficients and piecewise smooth densities on a
bounded time interval. For a density \(c(s)\), propagation is
\[
 \int S_w(t-s)c(s)\,\dd s.
\]
All output-time derivatives act on the smooth Cauchy propagator;
spatial derivatives commute with it. On compact time intervals,
Cauchy estimates control every derivative.
A slice derivative gives a derivative of
\(S_w(t-s_*)c_*\), also smooth.

If \(E_1H=0\), its advanced potential is compact by
\eqref{eq:compactprimitive}. It is piecewise smooth with smooth
spatial coefficients, so the same argument applies after \(D^*\).
\end{proof}

\begin{theorem}[Direct compact-current smoothing]
\label{thm:smoothing}
Let \(C\) be a compact symmetric contravariant distribution with
\(K^*C=0\). Suppose
\[
 q=E_2\mathsf r(C^{\flat\flat})
\]
is smooth. For a smooth temporal cutoff \(\chi\), zero in the
remote past and one in the remote future, define
\begin{equation}
 J_C=2\bigl(\Glin(\chi q)\bigr)^{\sharp\sharp}.
 \label{eq:smoothing}
\end{equation}
Then \(J_C\in\EE\) and
\[
 \pair{J_C}{h}=\pair{C}{h}\qquad(h\in\SSol).
\]
\end{theorem}

\begin{proof}
Conservation of \(C\) implies
\[
 \Box q=0,\qquad Dq=0,\qquad \Glin(q)=0.
\]
Consequently \(\Glin(\chi q)\) is supported in the transition
slab of \(\chi\). Spacelike compactness makes this support compact.
The Bianchi identity proves conservation.

First take \(Dh=0\). Set
\[
 F=\Box(\chi\mathsf rq),\qquad d=D(\chi q).
\]
Both \(F,d\) are compact, and
\[
 2\Glin(\chi q)=-F+\mathsf rKd.
\]
The second term pairs to zero with \(h\), because
\(\partial^\mu(\mathsf rh)_{\mu\nu}=0\).

Retarded and advanced uniqueness give
\[
 G_{2,R}F=\chi\mathsf rq,\qquad
 G_{2,A}F=(\chi-1)\mathsf rq.
\]
Hence \(E_2F=-E_2C^{\flat\flat}\).
By \eqref{eq:compactprimitive},
\(F+C^{\flat\flat}=\Box A_2\) for compact \(A_2\).
Since \(\Box h=0\), it follows that
\(\pair{-F}{h}=\pair{C}{h}\).

For general \(h\in\SSol\), solve
\(\Box w_0=Dh\) with zero data at \(0\).
Then \(h-Kw_0\in\SSol_{\rm dD}\).
Both compact conserved currents annihilate \(Kw_0\), so the
equality extends to all \(h\).
\end{proof}

The trace reversal in the gauge term is retained in this proof;
it cannot be dropped before the harmonic pairing.

\subsection{Raw and harmonic criteria}

Fix a time \(t_-<0\). On
\[
 \SSol_{\rm ret}^{-}
 =\{h\in\SSol:Dh=0\text{ for }t\le t_-\},
\]
let \(w_R=G_{1,R}Dh\). For \(h\in\SSol\), let \(w_0\) solve
\(\Box w_0=Dh\) with zero data at \(0\). Define
\[
 s_Rh=h-Kw_R,\qquad s_0h=h-Kw_0,
\]
\[
 B_R(h)=B_f(s_Rh,\phi),\qquad
 B_0(h)=B_f(s_0h,\phi).
\]
The fixed \(t_-\) ensures that the finite-time laboratory
interpretation has a common lower time bound.

Put
\[
 \zeta=E_1H,\qquad A_H=G_{1,A}H,\qquad
 J_{\rm sec}=J_{\rm raw}-D^*A_H.
\]
Then
\[
 K^*J_{\rm sec}=0.
\]
If \(\zeta=0\), \(A_H\) and \(J_{\rm sec}\) are compact.

The propagated defect has explicit Cauchy data:
\begin{align}
 \zeta(0)&=b_0-\int_0^b S_w(s)\gamma(s)\,\dd s,\nonumber\\
 \dot\zeta(0)&=a+\int_0^b\cos(s|D|)\gamma(s)\,\dd s.
 \label{eq:zetadata}
\end{align}
These follow by convolving \eqref{eq:H} with \(S_w\).

\begin{theorem}[Exact admissibility criteria]
\label{thm:classification}
For the fixed \(f,\phi\):
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(B_f(\,\cdot\,,\phi)\) is representable on \(\SSol\)
if and only if \(H=0\), equivalently \(\gamma=a=b_0=0\).
\item Each of the following is equivalent to \(E_1H=0\):
representability of \(B_R\) on \(\SSol_{\rm ret}^{-}\);
representability of \(B_0\) on \(\SSol\);
and representability of \(B_f|_{\SSol_{\rm dD}}\).
\item The latter condition is exactly the vanishing of the two
Cauchy data in \eqref{eq:zetadata}.
\end{enumerate}
When the conditions hold, representatives are obtained from
\eqref{eq:smoothing}, using \(C=J_{\rm raw}\) in (i) and
\(C=J_{\rm sec}\) in (ii).
\end{theorem}

\begin{proof}
Every compact conserved current annihilates every smooth pure
gauge. In (i), all compact \(K\xi\) belong to \(\SSol\), so
representability forces \(\pair{H}{\xi}=0\) for every compact
\(\xi\). This is \(H=0\).
The bulk and slice supports, and arbitrary first normal jets of
slice tests, separate \(\gamma,a,b_0\).
The converse follows from Lemma~\ref{lem:smooth} and
Theorem~\ref{thm:smoothing}.

For (ii), residual gauges with \(\Box\xi=0\) belong to all three
tested spaces and are fixed by both prescriptions. They must be
annihilated. Such annihilation is equivalent to \(E_1H=0\):
necessity follows by testing \(\xi=E_1v\) and using the
skew-adjointness of \(E_1\); sufficiency follows from
\(H=\Box A_H\) with compact \(A_H\).

Assume \(E_1H=0\). Compact integration by parts gives
\[
 B_R(h)=\pair{J_{\rm sec}}{h}.
\]
For \(B_0\), use
\[
 \pair{H}{w_0}
 =\pair{\Box A_H}{w_0}
 =\pair{A_H}{Dh}.
\]
On \(Dh=0\), the correction \(D^*A_H\) pairs to zero.
Smoothing \(J_{\rm sec}\) now gives one representative for all
three named functionals.
Finally, homogeneous-wave uniqueness proves (iii).
\end{proof}

A conserved section current need not be compact when the
propagated defect is nonzero. A harmonic prescription is not a
section of the complete gauge quotient: a residual wave satisfies
\(s_RK\xi=s_0K\xi=K\xi\).

\subsection{A compact-data residual witness}

\begin{theorem}[Open dense failure]
\label{thm:generic}
There are compact smooth scalar and vector Cauchy data and a
nonzero residual gauge \(K\xi\) for which the three functionals
\(B_f,B_R,B_0\) agree and are positive for some nonnegative compact
\(f\). For those fixed data, nonzero evaluation is open and dense
in a specified fixed-support Fr\'echet space of real smearings.
\end{theorem}

\begin{proof}
Choose real \(F\in C_c^\infty(\RR^3)\) with
\(\beta=\partial_1F\not\equiv0\), and solve
\[
 \phi|_0=0,\quad\dot\phi|_0=F,\qquad P\phi=0,
\]
\[
 \xi^1|_0=0,\quad\dot\xi^1|_0=\beta,\qquad\Box\xi^1=0,
\]
with other vector components zero.
Then \((K\xi)_{01}|_0=\beta\), and
\[
 w=\xi^1\partial_1\phi=t^2\beta^2+O(t^4).
\]
The scalar \(w\) has zero initial data, so
Proposition~\ref{prop:subtraction} gives \(S_0(K\xi)\phi=w\).
It is positive in a small open future region. Choose a
nonnegative nonzero \(f\) supported there.
Both prescriptions fix this residual gauge.

Fix a compact \(Q\Subset(0,b)\times\RR^3\) whose interior meets
that positive region, and let
\[
 \DD_Q=\{f\in C^\infty(\widehat M;\RR):\supp f\subset Q\}.
\]
The functional \(f\mapsto\int fw\) is continuous and nonzero.
Its kernel is a proper closed hyperplane, with empty interior.
Its complement is therefore open and dense.
All Cauchy data used above are literally compact.
\end{proof}

This proves failure on the named spaces, not a dense-TT
nonrepresentability theorem.

%=====================================================================
\section{Compact representatives on a finite TT plane}
\label{sec:TT}
%=====================================================================

Choose independent real waves \(v_1=e,v_2=r\) satisfying
\[
 \Box v_a=0,\qquad (v_a)_{0\mu}=0,\qquad
 \partial_i(v_a)_{ij}=0,\qquad (v_a)_{ii}=0,
\]
with Schwartz Cauchy data and a common open Fourier gap around
zero momentum. Put \(\mathcal V_2=\Span_\RR\{e,r\}\).

Such waves are obtained by applying the smooth TT projector away
from momentum zero to compact smooth Fourier data and then
performing wave evolution. A nonzero Fourier-gap datum is not
compactly supported in position: compact support would make its
Fourier transform entire, and an open gap would force it to vanish.

For a spatially Schwartz descriptor, \(K\xi\) cannot be a nonzero
spatial-TT tensor. Indeed, trace and divergence give
\(\partial_i\xi_i=0\) and \(\Delta\xi_j=0\), hence
\(\xi_j=0\); the temporal-spatial component and decay then give
\(\xi_0=0\).

The actual response coefficients are
\begin{equation}
 \beta_a=B_f(v_a,\phi)
 =-\int_0^b u(v_a)^{ij}\partial_i\partial_j\phi\,\dd^4x.
 \label{eq:plane-values}
\end{equation}
The surface term \eqref{eq:surface} pairs to zero with a spatial-TT
tensor, but that observation alone does not prove conservation.

Define the electric curvature
\[
 \mathcal E(h)_{ij}
 =\frac12\left(
 \partial_0\partial_i h_{0j}
 +\partial_0\partial_j h_{0i}
 -\partial_i\partial_jh_{00}
 -\partial_0^2h_{ij}\right).
\]
It annihilates \(K\xi\), and
\(\mathcal E(v_a)=-\Delta v_a/2\).

For compact smooth symmetric \(C^{ij}\), define
\begin{equation}
 J_C^{00}=-\frac12\partial_i\partial_jC^{ij},\quad
 J_C^{0j}=\frac12\partial_0\partial_iC^{ij},\quad
 J_C^{ij}=-\frac12\partial_0^2C^{ij}.
 \label{eq:curvaturecurrent}
\end{equation}
Then
\[
 \partial_\mu J_C^{\mu\nu}=0,\qquad
 \pair{J_C}{h}=\int C^{ij}\mathcal E(h)_{ij}\,\dd^4x.
\]

\begin{theorem}[Finite-plane extension with an explicit dual family]
\label{thm:plane}
There is \(\rho\ge0\), compactly supported in the open response
slab, for which
\[
 G_{ab}=\int\rho\,\mathcal E(v_a)_{ij}
                       \mathcal E(v_b)_{ij}\,\dd^4x
\]
is positive definite. Define
\[
 C_a^{ij}=\rho\sum_b(G^{-1})_{ab}\mathcal E(v_b)_{ij},
 \qquad J_a=J_{C_a}.
\]
Then \(\pair{J_a}{v_c}=\delta_{ac}\), and
\begin{equation}
 J_{\rm plane}=\beta_1J_1+\beta_2J_2\in\EE
 \label{eq:Jplane}
\end{equation}
represents \(B_f\) on \(\mathcal V_2\).
\end{theorem}

\begin{proof}
The two curvature fields are independent on the full open slab.
Otherwise a nonzero combination \(v\) would have
\(\Delta v=0\) there. Each spatial slice is Schwartz, hence zero,
and wave uniqueness would contradict independence.

Consequently finitely many component evaluations separate the
two-dimensional coefficient space. Small nonnegative bumps
around those points produce a positive definite Gram matrix:
its limiting quadratic form is a positive sum of squares of
separating evaluations. Fix a sufficiently small choice.

Equation~\eqref{eq:curvaturecurrent} and matrix inversion give
the duality. Evaluating \eqref{eq:Jplane} on
\(\lambda_1v_1+\lambda_2v_2\) proves the claim.
\end{proof}

For the conical adaptation the selected bumps must fit the
laboratory. If the laboratory was fixed earlier, positive Gram
rank there is an additional admission condition. Independence
on a full auxiliary slab is not silently replaced by independence
on an arbitrary small patch.

The finite-plane ambiguity is exactly
\[
 J_{\rm plane}+\bigl(\EE\cap\Ann(\mathcal V_2)\bigr).
\]
On compact distributions,
\[
 C\longmapsto C-\pair{C}{e}J_1-\pair{C}{r}J_2
\]
is an idempotent projection onto \(\Ann(\mathcal V_2)\).
Thus the interpolation problem has codimension two. It says
nothing about uniform bounds or compatibility as the tested
plane grows. This reproduces the finite-plane mechanism of V@1,
not a dense-space extension \cite{CheungV}.

%=====================================================================
\section{The external first retarded coefficient}
\label{sec:external}
%=====================================================================

\subsection{Representation and normalization}

Let
\[
 \mathfrak h=L^2(\RR^3,\dd^3p),\qquad
 \HH_\phi=\FF(\mathfrak h),\qquad
 \DD_\phi=\bigoplus_{n\ge0}^{\rm alg}
                    \mathcal S_{\rm sym}(\RR^{3n}).
\]
Use the auxiliary Minkowski vacuum and field
\[
 \widehat\phi(x)=
 \int N_p\left(
 a(p)e^{-i\omega_pt+ip\cdot x}
 +a^\dagger(p)e^{i\omega_pt-ip\cdot x}\right)\dd^3p,
\]
\[
 \omega_p=(|p|^2+m^2)^{1/2},\qquad
 N_p=(2\pi)^{-3/2}(2\omega_p)^{-1/2}.
\]
For a real compact smooth \(q\),
\[
 \widehat\phi(q)=\Phi(\mathsf Kq),\qquad
 (\mathsf Kq)(p)=N_p\int q(t,x)e^{i\omega_pt-ip\cdot x}\,\dd^4x,
\]
where \(\Phi(v)=a(v)+a^\dagger(v)\), with annihilation antilinear.
The map \(\mathsf K\) takes compact smooth tests continuously to
Schwartz vectors.

Let
\[
 T_{\mu\nu}(\phi)
 =\partial_\mu\phi\partial_\nu\phi
 -\frac12\eta_{\mu\nu}
  \left(\partial_\alpha\phi\partial^\alpha\phi+m^2\phi^2\right).
\]
For real \(g,f\in C_c^\infty(\widehat M)\) and a real smooth
external tensor \(h\), define
\[
 \mathcal V(g;h)=\frac12\int gh^{\mu\nu}:T_{\mu\nu}:\,\dd^4x.
\]
The perturbative convention is interaction \(+\lambda\mathcal V\)
and
\[
 R_{1,1}(\mathcal V,\widehat\phi(f))
 =i\left(T_2(\mathcal V,\widehat\phi(f))
                  -\mathcal V\widehat\phi(f)\right).
\]

For \(H_{\mu\nu}=gh_{\mu\nu}\), put
\begin{equation}
 \mathsf Q_H
 =-\partial_\mu\left((\mathsf rH)^{\mu\nu}\partial_\nu\right)
 -\frac12m^2\tr_\eta H.
 \label{eq:Q}
\end{equation}
It is formally self-adjoint and
\[
 \mathsf Q_H=L_H+\frac12(\tr_\eta H)P.
\]

The two-insertion normalization is fixed by vacuum Wick expansion
and the scalar Feynman distribution
\[
 W_m^F(z)=\Theta(z^0)W_m(z)+\Theta(-z^0)W_m(-z),
\]
where
\[
 W_m(z)=\int N_p^2e^{-i\omega_pz^0+ip\cdot z}\,\dd^3p.
\]
In these conventions,
\begin{equation}
 PW_m^F=i\delta,\qquad
 W_m^F(y-x)-W_m(y-x)=iG_{m,R}(x-y).
 \label{eq:Feynman}
\end{equation}
The insertions are formed with off-shell labels before free-field
representation, as in the retarded-product framework
\cite{DF}. We construct only this insertion pair.

\subsection{Subcritical coefficients and representation}

\begin{lemma}[Minimal extension and form domain]
\label{lem:extension}
Every numerical cross-contraction in this insertion pair has
relative scaling degree at most three in four dimensions.
Its extension preserving that coefficientwise degree is unique.
The compact quadratic vertex preserves \(\DD_\phi\), and the
required two-insertion expressions define continuous forms in the
finite-sector Schwartz topology.
\end{lemma}

\begin{proof}
There is only one scalar field at the second insertion.
The cross-contractions are therefore \(W_m\), \(W_m^F\), or one
first derivative of either distribution. No double cross-line
coefficient occurs.

Rescaling momentum gives
\(W_m(\epsilon z)=\epsilon^{-2}W_{\epsilon m}(z)\).
After spacetime testing, rapid momentum decay and the locally
integrable bound \((|p|^2+\nu^2)^{-1/2}\le |p|^{-1}\)
prove convergence to the massless distribution as \(\nu\downarrow0\).
The same argument on the two time half-spaces applies to \(W_m^F\).
Thus their scaling degrees are two, and a first derivative has
degree at most three.

A difference of extensions is supported at the relative origin.
Every nonzero such distribution has scaling degree at least four,
so it is excluded. This is the subcritical extension mechanism
of \cite{BF}.

For the domain statement, expand the compact quadratic vertex in
creation and annihilation operators. Pair kernels contain the
Fourier transform of a compact smooth coefficient at a sum of
positive energies and decrease rapidly in both momenta.
Exchange kernels have polynomial growth along the momentum
diagonal and rapid decay in differences; they map Schwartz
wavefunctions to Schwartz wavefunctions.
Finite particle shifts preserve the algebraic domain.

Completely normal-ordered finite-sector matrix elements are smooth
in the insertion coordinates. In the contracted terms, smearing
the numerical line against \(f\) first produces a smooth function;
compact coefficient multiplication and integration by parts leave
a compactly smeared linear field. These operations give the stated
finite-sector form bounds.
\end{proof}

\begin{theorem}[External first retarded coefficient]
\label{thm:external}
For the stated normalization,
\begin{equation}
 R_{1,1}(\mathcal V(g;h),\widehat\phi(f))
 =\widehat\phi(q_{g,h,f}),\qquad
 q_{g,h,f}=\mathsf Q_{gh}u_f,\qquad
 u_f=-G_{m,A}f.
 \label{eq:external}
\end{equation}
The test \(q_{g,h,f}\) is real, smooth, and compact.
The unsmeared coefficient is
\[
 \widehat\phi^{(1)}_{g,h}
 =-G_{m,R}\mathsf Q_{gh}\widehat\phi,
 \qquad
 P\widehat\phi^{(1)}_{g,h}
 =-\mathsf Q_{gh}\widehat\phi=-L_{gh}\widehat\phi.
\]
For spatial-TT \(h\),
\[
 q_{g,h,f}=-\partial_i(gh^{ij}\partial_ju_f).
\]
\end{theorem}

\begin{proof}
The completely normal-ordered cubic block is identical in the
time-ordered and ordinary products and cancels in their difference.
Contracting the scalar insertion against either differentiated
stress factor, and against the trace and mass terms, gives the
Euler operator \(\mathsf Q_{gh}\).

Equation~\eqref{eq:Feynman}, followed by the prefactor \(i\),
gives \(-G_{m,R}\mathsf Q_{gh}\widehat\phi\).
Smearing with \(f\), transposing the Green operator, and using
formal self-adjointness yields
\[
 -\int(G_{m,A}f)\mathsf Q_{gh}\widehat\phi
 =\widehat\phi(\mathsf Q_{gh}u_f).
\]
The coefficients of \(\mathsf Q_{gh}\) are compact, so the
resulting test is compact smooth.
The remaining assertions follow by applying \(P\) and using
the free equation.
\end{proof}

The sign in \eqref{eq:external} is positive:
the minus sign is already contained in \(u_f=-G_{m,A}f\).
The first retarded coefficient is not the uncancelled
three-scalar normal block \cite{CheungV}.

\subsection{Switching and an off-shell-label witness}

On a free scalar,
\begin{equation}
 \mathsf Q_{gh}\phi
 =gL_h\phi-(\partial_\mu g)(\mathsf rh)^{\mu\nu}\partial_\nu\phi.
 \label{eq:switch}
\end{equation}
Thus replacing \(L_{gh}\) by \(gL_h\) is generally incorrect.

For a pure gauge,
\[
 \mathsf Q_{gK\xi}\phi
 =-gP(X_\xi\phi)
  -(\partial_\mu g)(\mathsf rK\xi)^{\mu\nu}\partial_\nu\phi.
\]
Consequently,
\[
 \delta_\xi\phi_g^{(1)}
 =gX_\xi\phi+
 G_{m,R}\left[
 (\partial_\mu g)(\mathsf rK\xi)^{\mu\nu}\partial_\nu\phi
 -[P,g]X_\xi\phi\right].
\]
This formula retains the trace-reversed switched tensor.
Constancy of \(g\) near \(\supp f\) need not eliminate terms in
its causal past.

\begin{proposition}[A free-null label can have nonzero response]
\label{prop:externalwitness}
For every nonzero spatial-TT wave \(h\), there are real compact
\(f,g\) and a one-particle Schwartz vector \(v\) such that
\[
 \widehat\phi(f)=0,\qquad
 \pair{\Omega}{\widehat\phi(q_{g,h,f})a^\dagger(v)\Omega}\ne0.
\]
\end{proposition}

\begin{proof}
Choose \(x_*,p_*\) with
\(h^{ij}(x_*)p_{*i}p_{*j}\ne0\).
A small momentum bump, with phase centered at \(x_*\), gives
a positive-frequency wave \(w_v\) for which
\(\Rea(h^{ij}\partial_i\partial_jw_v)\) has one nonzero sign
on a small neighborhood.

Take a nonnegative real \(\rho\) supported there,
\(f=P\rho\), and compact \(g=1\) near \(\supp\rho\).
Then \(\widehat\phi(f)=0\), \(u_f=-\rho\), and
\(q_{g,h,f}=-\mathsf Q_h\rho\).
Integration by parts gives
\[
 \int q_{g,h,f}w_v
 =\int\rho h^{ij}\partial_i\partial_jw_v\ne0.
\]
\end{proof}

There is no contradiction: the retarded construction was defined
on off-shell insertion labels before representation.
It is not a map on the unchanged free test quotient in the second
entry.

%=====================================================================
\section{One metric oscillator and the switched identity}
\label{sec:oscillator}
%=====================================================================

Let \(\HH_g=\FF(\CC)\), with
\[
 Q=\frac{b+b^\dagger}{\sqrt2},\qquad
 \Pi=\frac{b-b^\dagger}{i\sqrt2},\qquad [Q,\Pi]=iI.
\]
This infinite-dimensional oscillator is distinct from the finite
clock introduced later.

Set
\[
 \widehat h(x)=Qe(x)+\Pi r(x),\qquad
 \DD_{\rm mix}=\DD_g\otimes_{\rm alg}\DD_\phi,
\]
where \(\DD_g\) is the finite oscillator-number span.
Applying the preceding contraction calculation separately to
\(e,r\) gives
\begin{equation}
 \mathcal R_{g,f}
 =Q\otimes\Phi(v_e)+\Pi\otimes\Phi(v_r),
 \qquad
 v_a=\mathsf K\mathsf Q_{ga}u_f.
 \label{eq:mixed}
\end{equation}
No metric--metric contraction enters this insertion pair.
The oscillator metric is not a local relativistic metric field.

\begin{theorem}[Switched bilinear and polynomial closure]
\label{thm:ESA}
Suppose an initial time \(t_{\rm in}\) precedes the supports of
\(g\) and \(f\). Quantizing the classical bilinear
\[
 (q,\pi,\phi)\longmapsto
 \int fS_{t_{\rm in}}(g(qe+\pi r))\phi
\]
by replacing \(q,\pi\) with \(Q,\Pi\) gives exactly
\(\mathcal R_{g,f}\) on \(\DD_{\rm mix}\).

Writing \(N=N_g+N_\phi\), there is \(C<\infty\) with
\[
 \norm{\mathcal R_{g,f}\Psi}
 \le C\norm{(N+2)\Psi}.
\]
Both \(\mathcal R_{g,f}\) and, for every real \(\lambda\),
\[
 A_f^g(\lambda)=I\otimes\widehat\phi(f)+\lambda\mathcal R_{g,f}
\]
are essentially self-adjoint on \(\DD_{\rm mix}\).
\end{theorem}

\begin{proof}
On free scalar matrix elements,
\(\mathsf Q_{ga}\phi=L_{ga}\phi\).
The source is zero before \(t_{\rm in}\), so
\[
 -G_{m,R}\mathsf Q_{ga}\phi=S_{t_{\rm in}}(ga)\phi.
\]
Smearing proves the switched identity, including derivatives of \(g\).

Creation-annihilation estimates give
\[
 \norm{Q\chi},\norm{\Pi\chi}
 \le\sqrt2\norm{(N_g+1)^{1/2}\chi},
\]
\[
 \norm{\Phi(v)\psi}
 \le2\norm v\,\norm{(N_\phi+1)^{1/2}\psi}.
\]
The two factors commute and
\[
 \sqrt{(N_g+1)(N_\phi+1)}\le\frac{N+2}{2}.
\]
This proves the bound. Schwartz mode vectors and finite particle
shifts prove invariance of the common core. Reality proves symmetry.

Each polynomial changes total particle number by at most two.
For a vector supported in sectors up to \(M\),
\[
 \norm{A^k\Psi}
 \le C_A^k\prod_{j=0}^{k-1}(M+2j+2)\norm\Psi.
\]
After division by \(k!\), the resulting analytic-vector series
converges for a positive range of its parameter.
The core is dense, so Nelson's theorem \cite{Nelson} proves
essential self-adjointness.
\end{proof}

This conclusion concerns the displayed degree-at-most-two
polynomial, not the mixed cubic interaction vertex or an
all-orders interacting Hamiltonian.

For comparison with an unswitched response starting at \(0\), set
\[
 v_a^0=
 \mathsf K\left(-\mathbf1_{[0,b]}(t)
                  \partial_i(a^{ij}\partial_ju_f)\right).
\]
These particular discontinuous tests still give Schwartz
mass-shell vectors: their spatial Fourier transforms decrease
rapidly, and their time support is bounded. Then
\[
 \mathcal R_{g,f}-\widehat B_f^{\,0}
 =Q\otimes\Phi(v_e-v_e^0)+\Pi\otimes\Phi(v_r-v_r^0).
\]
On total-number sectors at most \(M\), its norm is bounded by
\[
 \sqrt2(M+2)
 \left(\norm{v_e-v_e^0}+\norm{v_r-v_r^0}\right).
\]
It is a switching and initial-data remainder, not an independent
renormalization ambiguity.

%=====================================================================
\section{A separate Newton--wave cubic form}
\label{sec:cubic}
%=====================================================================

This section specifies another observable, not another expression
for \(R_{1,1}\). It uses the global auxiliary Newton prescription
and therefore is not an intrinsic cone-local construction.

\subsection{Option A and its constraints}

For a smooth rapidly decreasing source, prescribe
\[
 \mathcal Nb(x)=-\frac1{4\pi}\int\frac{b(y)}{|x-y|}\,\dd^3y,
 \qquad
 \widehat{\mathcal Nb}(q)=-|q|^{-2}\widehat b(q).
\]
The multiplier is locally integrable in three dimensions.
No Fourier distribution supported at \(q=0\) is added.

Let \(Q_{\mu\nu}\) be a conserved symmetric source:
\(\partial^\mu Q_{\mu\nu}=0\).
At time \(0\), set
\[
 \rho=Q_{00}|_0,\qquad J_i=Q_{0i}|_0,\qquad
 a=-2\mathcal N\rho,\qquad V_i=2\mathcal NJ_i,
\]
\[
 B_{ij}=\partial_iV_j+\partial_jV_i-\delta_{ij}\partial_\ell V_\ell.
\]
Define trace-reversed initial data \(A,B\) by
\[
 A_{00}=a,\quad A_{0i}=A_{ij}=0,\qquad
 B_{ij}\text{ as above},\quad B_{00}=B_{0i}=0,
\]
and
\begin{equation}
 \NW Q
 =\mathsf r\left[
 \cos(t|D|)A+S_w(t)B+
             2\int_0^tS_w(t-s)Q(s)\,\dd s\right].
 \label{eq:OptionA}
\end{equation}

\begin{lemma}[Constraint compatibility]
\label{lem:constraints}
For the stated conserved sources,
\[
 D(\NW Q)=0,\qquad \Glin(\NW Q)=Q.
\]
\end{lemma}

\begin{proof}
Writing \(j=\NW Q\), one has
\(\Box\mathsf rj=-2Q\).
The initial data give \(Dj|_0=0\).
Furthermore
\[
 \Delta a=-2\rho,\qquad
 \partial_iB_{ij}=\Delta V_j=2J_j,
\]
so differentiating \(Dj\) once in time gives zero initial data.
Conservation implies \(\Box Dj=0\).
Wave uniqueness yields \(Dj=0\), and
\eqref{eq:Einstein} gives \(\Glin j=Q\).
\end{proof}

For free scalars with Schwartz Cauchy data, polarize the stress:
\[
 T_{\mu\nu}(\phi_1,\phi_2)
 =\partial_{(\mu}\phi_1\partial_{\nu)}\phi_2
 -\frac12\eta_{\mu\nu}
 \left(\partial_\alpha\phi_1\partial^\alpha\phi_2
                         +m^2\phi_1\phi_2\right).
\]
It is conserved. Define
\begin{equation}
 C_f(\phi_1,\phi_2,\phi_3)
 =\frac13\sum_{\rm cyc}
 B_f\bigl(\NW T(\phi_1,\phi_2),\phi_3\bigr).
 \label{eq:cubicclassical}
\end{equation}
Newton potentials are smooth with polynomially controlled
derivatives; multiplying by derivatives of the Schwartz scalar
gives an admissible finite-time response source. Hence this
classical functional is defined.

Changing the Newton boundary convention or the constraint-compatible
initial data changes the inverse. Those changes are not gauge or
prescription equivalences proved here.

\subsection{Explicit signed kernels}

For \(\sigma=\pm1\), put
\[
 u_\sigma(x;p)=N_pe^{i\sigma\omega_pt-i\sigma p\cdot x},
 \qquad a_+(p)=a^\dagger(p),\quad a_-(p)=a(p).
\]
For signed momenta define covectors
\[
 k_a=(\sigma_a\omega_a,-\sigma_ap_a),\qquad
 e_{12}=\sigma_1\omega_1+\sigma_2\omega_2,\qquad
 q=\sigma_1p_1+\sigma_2p_2,\quad r=|q|,
\]
and
\[
 n_{\mu\nu}
 =-k_{1(\mu}k_{2\nu)}
   +\frac12\eta_{\mu\nu}(k_1\cdot k_2-m^2).
\]
Let \(\rho_{12}=n_{00}\), \(J_{12,i}=n_{0i}\).
Then
\begin{align}
 \rho_{12}
 &=\frac12\left[
 m^2-\sigma_1\sigma_2(\omega_1\omega_2+p_1\cdot p_2)\right],
 \nonumber\\
 J_{12,i}
 &=\frac{\sigma_1\sigma_2}{2}
              (\omega_1p_{2i}+\omega_2p_{1i}),
 \label{eq:numerators}
\end{align}
and \(e_{12}n_{0\nu}+q_in_{i\nu}=0\).

After removing \(N_{p_1}N_{p_2}e^{-iq\cdot x}\), the
trace-reversed metric amplitudes are
\begin{align}
 Z_{00}^{\rm in}(s)&=\frac{2\rho_{12}}{r^2}\cos(sr),
 &Z_{0i}^{\rm in}(s)&=0,\nonumber\\
 Z_{ij}^{\rm in}(s)
 &=\frac{2i}{r^2}S_w(s,r)
 (q_iJ_{12,j}+q_jJ_{12,i}-\delta_{ij}q\cdot J_{12}),
 \nonumber\\
 Z_{\mu\nu}^{\rm src}(s)
 &=2n_{\mu\nu}\int_0^s
       S_w(s-u,r)e^{ie_{12}u}\,\dd u,
 \label{eq:Z}
\end{align}
where \(S_w(s,r)=\sin(sr)/r\), continuously defined at zero.

Define
\[
 d_\nu[Z]=-\partial_sZ_{0\nu}-iq_iZ_{i\nu},
\]
\[
 \mathcal L_3[Z]
 =(\mathsf rZ)^{\mu\nu}k_{3\mu}k_{3\nu}
             -i\,d[Z]^\nu k_{3\nu}.
\]
An explicit differentiation of \eqref{eq:Z} gives
\begin{equation}
 d_\nu[Z^{\rm in}]
 =2S_w(s,r)n_{0\nu}
 =-d_\nu[Z^{\rm src}].
 \label{eq:dcancel}
\end{equation}
The full \(D(j)\) term is therefore retained before it cancels.

Write
\[
 F_f(t,w)=\int e^{-iw\cdot x}f(t,x)\,\dd^3x,\qquad
 w=q+\sigma_3p_3,
\]
and \(S_m(\tau,w)=\sin(\tau\omega_w)/\omega_w\).
The ordered kernel is
\begin{equation}
 U_f^\sigma
 =N_{p_1}N_{p_2}N_{p_3}
 \int_0^b\dd t\int_0^t\dd s\,
 F_f(t,w)S_m(t-s,w)e^{i\sigma_3\omega_3s}
 \mathcal L_3[Z^{\rm in}+Z^{\rm src}](s).
 \label{eq:U}
\end{equation}
Let \(K_f^\sigma\) be the cyclic average over the three choices
of stress pair. The first two arguments of \(U\) are symmetric,
so this average is fully symmetric in signed arguments.

\subsection{Transfer estimate}

\begin{lemma}[Integrable transfer singularities]
\label{lem:kernel}
Let
\[
 R_p=(1+|p_1|^2+|p_2|^2+|p_3|^2)^{1/2}.
\]
There is \(C_f\) such that
\begin{equation}
 |K_f^\sigma|
 \le C_fR_p^5
 \left(1+\sum_{i<j}|\sigma_ip_i+\sigma_jp_j|^{-2}\right).
 \label{eq:kernelbound}
\end{equation}
The same bound holds before the outer response-time integration.
For a chosen stress pair, equal signs give locally bounded
transfer behavior. Opposite signs can retain an inverse-square
transfer term.
\end{lemma}

\begin{proof}
Use coordinates
\(p_2=\sigma_2(q-\sigma_1p_1)\), of unit absolute Jacobian.
For equal signs, Taylor expansion of \(\omega_{-p+\sigma q}\)
in \eqref{eq:numerators} gives
\[
 \rho_{12}=O(|q|^2),\qquad J_{12}=O(|q|).
\]
Both initial amplitudes are locally bounded, and the source
amplitude is smooth.

For opposite signs, at \(q=0\),
\[
 p_2=p_1,\qquad e_{12}=0,\qquad
 \rho_{12}=\omega_1^2,\qquad J_{12}=-\omega_1p_1.
\]
Thus \(Z_{00}^{\rm in}\) has a generally nonzero
\(|q|^{-2}\) numerator. The spatial initial term is at worst
\(|q|^{-1}\), while the source term is smooth.
Away from the other transfers the cyclic sum cannot universally
cancel that pole.

For the global bound, finite time gives
\[
 |S_w(s,r)|\le b,\qquad |S_m(t-s,w)|\le b.
\]
Stress numerators cost at most \(CR_p^2\), spatial initial
numerators at most \(CR_p^3\), and the scalar Hessian contraction
at most \(CR_p^2\).
Equation~\eqref{eq:dcancel} bounds the separate divergence terms.
The massive mode normalizations are bounded, and
\(|F_f(t,w)|\le\norm{f(t,\cdot)}_{L^1}\).
This proves \eqref{eq:kernelbound} term by term.

All time integrals remain finite integrals. No artificial
resonance denominator is introduced.
\end{proof}

The three pair singularities are added, not multiplied.
Their local transverse integral is
\[
 \int_{|q|<a}|q|^{-2}\dd^3q=4\pi a.
\]
They need not be square integrable. A form theorem is therefore
the appropriate conclusion.

\subsection{Full Wick ordering and joint cutoff removal}

For \(r+s=3\), use the kernel
\(k_{r,s}=K_f^{+\cdots+-\cdots-}\) in the normally ordered
\((r,s)\) block.
For \(n\ge s\) and \(m_n=n+r-s\), define
\begin{align}
 \mathcal C_{r,s}(\Psi_{m_n},X_n)
 ={}&\binom3r\frac{\sqrt{n!\,m_n!}}{(n-s)!}
 \int \overline{\Psi_{m_n}(p,z)}\,
       k_{r,s}(p;q)X_n(q,z)\,
       \dd p\,\dd q\,\dd z .
 \label{eq:sector}
\end{align}
Their finite occupied-sector sum defines
\(\mathcal C_f^{(3)}\) on \(\DD_\phi\times\DD_\phi\).

For \(n\ge1\), set
\[
 s_n(\psi)=
 \sup_{p_1,\ldots,p_n}
       \left(\prod_{j=1}^n\langle p_j\rangle^{30}\right)|\psi|,
 \qquad s_0(z)=|z|,
\]
\[
 I_*=\int\langle p\rangle^{-60}\dd^3p,\qquad
 \mathfrak p(\Psi)
 =\sum_{n\ge0}(1+I_*)^n(n+3)^3s_n(\Psi_n).
\]
This is a continuous norm on the algebraic direct-sum Schwartz
domain.

In each ordered summand insert:
\begin{itemize}
\item bounded ultraviolet profiles equal to one for
\(|p|\le\Lambda\) and zero for \(|p|\ge2\Lambda\);
\item a bounded transfer profile zero for \(|q|\le\delta\) and
one for \(|q|\ge2\delta\);
\item an outer response-edge profile zero for \(s\le\eps\) and
one for \(s\ge2\eps\).
\end{itemize}
Take these profiles in \([0,1]\).
The initial data of \eqref{eq:OptionA} are not changed.

\begin{theorem}[Symmetric cubic form and joint limit]
\label{thm:cubicform}
For real compact \(f\), the fully Wick-ordered kernel defines a
symmetric continuous sesquilinear form, and
\begin{align}
 |\mathcal C_f^{(3)}(\Psi,X)|
 &\le C_f\mathfrak p(\Psi)\mathfrak p(X),\nonumber\\
 |\mathcal C_{f,\Lambda,\delta,\eps}^{(3)}(\Psi,X)
       -\mathcal C_f^{(3)}(\Psi,X)|
 &\le C_f(\Lambda^{-1}+\delta+\eps)
                      \mathfrak p(\Psi)\mathfrak p(X).
 \label{eq:jointlimit}
\end{align}
The limit is joint and independent of the bounded profiles with
the stated support properties. No transfer-zero distribution is
added.
\end{theorem}

\begin{proof}
The bound \eqref{eq:kernelbound} is integrable with weight
\(R_p^{-20}\). For each pair transfer use \(q\in\RR^3\)
and six remaining variables \(y\); the integral is bounded by
\[
 C\int(1+|q|^{-2})
               \langle(q,y)\rangle^{-15}\,\dd^3q\,\dd^6y<\infty.
\]
The ultraviolet error has \(R_p\gtrsim\Lambda\); extract
\(\Lambda^{-1}\) and retain an integrable weight.
The transfer exclusion contributes
\(C(\delta^3+\delta)\).
The response-edge strip has length at most \(2\eps\);
the pre-integration estimate of Lemma~\ref{lem:kernel} gives
\(C\eps\).

In \eqref{eq:sector}, the two Schwartz bounds yield
\(\langle z_j\rangle^{-60}\) for each spectator, and
\(\prod_{\rm external}\langle p_j\rangle^{-30}\)
dominates \(CR_p^{-30}\).
Spectators therefore contribute \(I_*^{n-s}\).
The combinatorial factor is at most \(3(n+3)^{3/2}\).
The weights in \(\mathfrak p\) dominate these factors, proving
both estimates after summing sectors.

For real \(f\), complex conjugation reverses all mode signs.
Combined with signed permutation symmetry, this interchanges
the \((r,s)\) and \((s,r)\) blocks and proves symmetry.
Weighted integral convergence selects the locally integrable
kernel itself.
\end{proof}

The full cubic ordering is essential:
\[
 :AB:C=:ABC:+\langle AC\rangle B+\langle BC\rangle A.
\]
Ordering only the stress leaves two linear contraction terms.
Their unregulated restrictions are not part of
Theorem~\ref{thm:cubicform}.

\subsection{A regular-region cubic witness}

\begin{theorem}[Separation from every quadratic polynomial form]
\label{thm:cubicwitness}
There are real compact \(f\) and a normalized symmetric
\(\chi_3\in C_c^\infty(\RR^9)\), supported away from all
three-creation transfers, such that
\[
 c_f:=\mathcal C_f^{(3)}(\chi_3,\Omega)>0.
\]
For every \(\epsilon>0\), the states
\[
 \Psi_\pm=\frac{\Omega\pm\epsilon\chi_3}{\sqrt{1+\epsilon^2}}
\]
have equal expectations for every defined polynomial form of
degree at most two, whereas
\[
 \mathcal C_f^{(3)}(\Psi_+,\Psi_+)
 -\mathcal C_f^{(3)}(\Psi_-,\Psi_-)
 =\frac{4\epsilon c_f}{1+\epsilon^2}>0.
\]
\end{theorem}

\begin{proof}
Choose \(p_*\ne0\), all three signs positive, and
\(p_1=p_2=p_3=p_*\). Every pair transfer is \(2p_*\ne0\).
At time zero,
\[
 \rho_{12}=-|p_*|^2,\qquad
 Z_{00}^{\rm in}=-\frac12,\qquad Z_{ij}^{\rm in}=0,
\]
and the total divergence vanishes.
Therefore
\[
 \mathcal L_3[Z^{\rm in}+Z^{\rm src}](0)
 =-\frac{m^2+2|p_*|^2}{4}.
\]
The ordered response at small positive \(\tau\) is
\[
 -\frac{N_{p_*}^3}{8}
       (m^2+2|p_*|^2)\tau^2+O(\tau^3).
\]
All cyclic terms coincide at this point.

Take a sufficiently small even nonnegative spatial bump \(b_0\)
with \(\widehat b_0(3p_*)>0\), and
\[
 f_h(t,x)=h^{-1}g_0(t/h)b_0(x),
\]
where \(g_0\ge0\) is a nonzero smooth bump in \((1,2)\).
For sufficiently small \(h>0\),
\(K_{f_h}^{+++}(p_*,p_*,p_*)\ne0\).
Fix this \(f\).
On a small symmetric neighborhood avoiding all transfers, take a
nonnegative bump \(\beta\) and define
\[
 \chi_3=
 \frac{\beta K_f^{+++}}{\norm{\beta K_f^{+++}}_2}.
\]
Equation~\eqref{eq:sector} gives
\[
 \mathcal C_f^{(3)}(\chi_3,\Omega)
 =\frac{\sqrt{6}}{\norm{\beta K_f^{+++}}_2}
       \int\beta|K_f^{+++}|^2>0.
\]
A degree-at-most-two polynomial cannot connect vacuum and three
particles. Its expectations in \(\Psi_\pm\) therefore agree.
The displayed cubic difference follows by expansion and symmetry.
\end{proof}

This witnesses the complete symmetrized kernel, not one selected
branch. It does not prove that the cubic form is a closable
operator, an operator on the vacuum, or a self-adjoint generator.

%=====================================================================
\section{Finite compression and the complete Enter interface}
\label{sec:interface}
%=====================================================================

\subsection{A common finite response carrier}

Let
\[
 \HH_{\rm big}=\HH_g\otimes\HH_\phi.
\]
The external field acts as \(I_g\otimes\widehat\phi(q)\);
the mixed coefficient acts as in \eqref{eq:mixed}; and the cubic
form is extended by the identity oscillator form.

Choose an isometry
\[
 V_X:\CC^d\longrightarrow\HH_{\rm big},
 \qquad d=2^p,
\]
whose image is spanned by finitely many orthonormal vectors of
\(\DD_{\rm mix}\). For each setting \(k\), choose one of the
symmetric response forms proved above and define its finite matrix
\begin{equation}
 (A_k)_{ij}
 =\mathcal A_k(V_X\ket{i},V_X\ket{j}).
 \label{eq:compression}
\end{equation}
For operator settings this is the usual compression.
For the cubic setting it is a form compression, requiring no
unproved unregulated operator.

The finite matrix is not an invariant subspace realization of the
full response operator. In particular,
\[
 e^{-isA_k}
\]
is not identified with a compression of a continuum unitary.
The compressed matrix is the operational observable declared here.

Choose a common \(B>0\) with \(B>\max_k\norm{A_k}\), and define
\[
 E_{k,y}=\frac12(I+yA_k/B),\qquad
 M_{k,y}=E_{k,y}^{1/2},\qquad y=\pm1.
\]

\subsection{Destructive clock readout}

Let the clock have \(D=2^q\) basis vectors
\(\ket{\tau_k}\), with \(P_k=\proj{\tau_k}\), and let
\[
 D_k=\bra{\tau_k}_C\otimes I_X.
\]
The output record \(R\) has diagonal classical algebra.

\begin{theorem}[Complete clock and response instrument]
\label{thm:instrument}
The destructive clock channel is
\begin{align}
 \Mclock(Z)
 &=\sum_k\proj{k}_R\otimes D_kZD_k^\dagger\nonumber\\
 &=\sum_k\proj{k}_R\otimes
 \Tr_C\left[(P_k\otimes I_X)Z(P_k\otimes I_X)\right].
 \label{eq:clock}
\end{align}
Its composition with the code-selected response reading is
\begin{equation}
 \mathcal I(Z)=
 \sum_{k,y}\proj{k}_R\otimes\proj{y}_Y\otimes
 M_{k,y}D_kZD_k^\dagger M_{k,y}^\dagger.
 \label{eq:instrument}
\end{equation}
Both maps are completely positive and trace preserving, including
every untouched reference.

For
\(\sigma_k=D_k\rho D_k^\dagger\) and
\(p_k=\Tr\sigma_k>0\),
\[
 \mathbb E[By\mid k]=\frac{\Tr(A_k\sigma_k)}{p_k}.
\]
The measured clock names are consumed.
\end{theorem}

\begin{proof}
The recorded Kraus operators for \eqref{eq:clock} are
\(\ket{k}_R\otimes D_k\), with squared sum \(I_{CX}\).
Rank-one clock projection gives the partial-trace identity.

For \eqref{eq:instrument}, use the Kraus family
\(\ket{k}_R\otimes\ket{y}_Y\otimes M_{k,y}D_k\).
Since \(\sum_yM_{k,y}^\dagger M_{k,y}=I\),
its squared sum is again \(I_{CX}\).
Tensoring these identities with any reference identity proves
the complete assertions.

Finally
\(\sum_yByE_{k,y}=A_k\).
The destructive implementation changes the clock input into a
classical code; its quantum output contains \(X\), not \(C\).
No normalized conditional state is assigned when \(p_k=0\).
\end{proof}

The instrument is implementable in the admitted finite gate model
using one fresh probe qubit and the isometry
\[
 \psi\longmapsto
 \sum_{y=\pm1}\ket{y}\otimes M_{k,y}\psi.
\]
Completeness makes this an isometry, which extends to a finite
unitary. Destructive probe readout gives the printed instrument.
This is an exact abstract gate construction, not a hardware
synthesis certificate.

\begin{proposition}[Clock limitations]
\label{prop:clocklimits}
The clock channel has no universal left inverse.
Its record marginal does not determine the residual.
No matrices on the finite clock satisfy \([T_C,H_C]=iI_C\).
\end{proposition}

\begin{proof}
The two inputs
\[
 \frac{\ket{\tau_0}+\ket{\tau_1}}{\sqrt2},
 \qquad
 \frac{\ket{\tau_0}-\ket{\tau_1}}{\sqrt2}
\]
with the same residual state have identical clock outputs.
A left inverse cannot recover both.
Inputs \(P_0\otimes\rho_X\) and \(P_0\otimes\rho_X'\)
have the same deterministic record and different residuals.
Finally the trace of a commutator is zero, while \(\Tr(iI_C)\ne0\).
\end{proof}

\subsection{An explicit compressed cubic qubit}

Use the witness of Theorem~\ref{thm:cubicwitness}, with oscillator
vacuum, and define
\[
 V_X\ket0=\Omega_g\otimes\Omega_\phi,\qquad
 V_X\ket1=\Omega_g\otimes\chi_3.
\]
The compressed cubic matrix is
\[
 A_0=c_f\sigma_x.
\]
Indeed, both diagonal cubic matrix elements vanish by scalar
parity, and the off-diagonal element is \(c_f>0\).

Use a one-qubit clock with two settings \(f\) and \(2f\):
\[
 A_1=2c_f\sigma_x,\qquad B=4c_f.
\]
Then
\[
 E_{0,y}=\frac12(I+y\sigma_x/4),\qquad
 E_{1,y}=\frac12(I+y\sigma_x/2).
\]
For encoded
\[
 \psi_\pm=\frac{\ket0\pm\epsilon\ket1}{\sqrt{1+\epsilon^2}},
\]
the difference of conditional scored means is
\[
 \Delta_k=\frac{4\epsilon(k+1)c_f}{1+\epsilon^2},
 \qquad k=0,1.
\]
The same two continuum preparations agree on all degree-at-most-two
polynomial forms. Thus this is a complete finite readout of an
explicitly compressed cubic distinction, not a claim that such a
distinction is supplied by the external linear retarded field.

\subsection{Cutoff compatibility of the finite instrument}

\begin{proposition}[Compression preserves the proved joint limit]
\label{prop:compressionlimit}
For a fixed finite isometry \(V_X\), the regulated cubic matrices
converge to \eqref{eq:compression} jointly in operator norm.
Using a common \(B\) strictly larger than the limiting matrix
norms, their complete instruments converge in diamond norm.
\end{proposition}

\begin{proof}
Each matrix entry is bounded by
Theorem~\ref{thm:cubicform} with two fixed domain vectors.
There are finitely many entries, so the convergence is uniform
and implies operator-norm convergence.

For all sufficiently refined cutoffs, the effects stay in a
common compact interval inside \((0,1)\).
Their positive square roots therefore converge in norm by
continuous functional calculus.
For contractions \(M,N\),
\[
 \norm{\Ad_M-\Ad_N}_\diamond
 \le(\norm M+\norm N)\norm{M-N}.
\]
Summing the finitely many recorded Kraus branches proves complete
instrument convergence.
\end{proof}

This is a same-target statement for one fixed finite compression.
It is not convergence of an increasing family to a continuum
field instrument.

%=====================================================================
\section{Time, rods, histories, and conditional evolution}
\label{sec:dictionary}
%=====================================================================

Choose distinct external schedule readings \(t_k\), and calibrate
the code by
\[
 k\longmapsto T_k=T_0+t_k/c.
\]
The initial surface, response window, and switch of setting \(k\)
are part of its externally specified setting data.

\begin{definition}[Three dictionary maps]
The dictionary consists of:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the code calibration above, together with a separate history
map \(\kappa(k)=(T_k,x_k)\);
\item a location map from each live carrier name to its port
\(\ell(a)\in U\), followed by its three rod coordinates
\(\chi(\ell(a))\);
\item a port-event map from the finite dependency network into
\(M\), taking every permitted dependency to a future-causal pair.
\end{enumerate}
\end{definition}

The code is not an exact register of arbitrary real times:
\(T_k\) is supplied interpretation data.

\begin{proposition}[Calibration agreement and nonuniqueness]
\label{prop:dictionary}
The dictionaries agree on the admitted events by evaluation:
\(T(\kappa(k))=T_k\), carrier rods equal their assigned coordinates,
and scheduled ancestry lies in the output metric past.

At one fixed readout event, exact event-time calibration with
distinct \(T_k\) permits at most one supported outcome.
A nontrivial multi-outcome reading therefore uses the separate
history map rather than identifying every code with the time of
that fixed event.

The instrument does not uniquely determine any of these dictionaries.
\end{proposition}

\begin{proof}
The first assertions are the definitions and causal transitivity.
For fixed-event calibration, every supported code must satisfy
\(T_k=T(e)\); distinct readings allow at most one.

For nonuniqueness, keep all matrices and preparations unchanged
and permute the history assignments of two codes.
The clock channel is unchanged while its interpreted history
events differ. Likewise the same finite matrices can be installed
at different causally permitted regular ports.
A continuation depending on the dictionary must be separately
specified for either choice.
\end{proof}

The clock labels the prescribed schedule or history. It does not
supply the external time on which those definitions depend.

For completeness, the usual finite relational-evolution statement
can be retained without enlarging that claim.

\begin{proposition}[Covariance plus stationarity]
\label{prop:stationarity}
Let finite generators \(H_C,H_X\) satisfy
\[
 [H_C\otimes I+I\otimes H_X,\rho_{CX}]=0,
 \qquad
 P_k=e^{-iH_Ct_k}P_0e^{iH_Ct_k}.
\]
Then
\[
 \sigma_k=e^{-iH_Xt_k}\sigma_0e^{iH_Xt_k},
 \qquad p_k=1/D.
\]
The normalized conditional residuals obey the same sampled
unitary law.
\end{proposition}

\begin{proof}
Stationarity implies
\[
 (e^{iH_Ct}\otimes I)\rho(e^{-iH_Ct}\otimes I)
 =(I\otimes e^{-iH_Xt})\rho(I\otimes e^{iH_Xt}).
\]
Insert this identity in
\(\Tr_C[(P_k\otimes I)\rho]\).
All probabilities coincide, and their sum is one.
\end{proof}

The channel identity holds with every reference.
The dynamical identity for \(XE\) additionally requires
stationarity of the full \(CXE\) preparation, not merely its
\(CX\) marginal. This is the finite conditioning distinction
emphasized in En09-A \cite{CheungEn09,PageWootters}.

%=====================================================================
\section{Transport, schedule locality, and resource accounting}
\label{sec:transport}
%=====================================================================

\subsection{Complete schedule locality}

Reserve a finite set of sites and records. At each tick the
complete channel is a tensor product over disjoint blocks.
Include control records and all accessible auxiliaries in that
network. Require every input-output dependency to be mapped by
the port-event dictionary into the prescribed causal relation.

For terminal region \(Y\), recursively define
\[
 N_n(Y)=Y,\qquad
 N_s(Y)=\bigcup\{b:b\text{ is a tick-}s\text{ block meeting }
                                      N_{s+1}(Y)\}.
\]

\begin{theorem}[Schedule-past factorization]
\label{thm:locality}
The terminal channel satisfies
\[
 \Phi^*(\mathcal A(Y))\subseteq\mathcal A(N_0(Y)).
\]
Its output reduction depends only on the initial reduction to
\(N_0(Y)\). Joint output with an untouched reference \(E\) depends
only on the initial reduction to \(N_0(Y)E\).
Every scheduled ancestor maps into the terminal metric past.
\end{theorem}

\begin{proof}
On a block disjoint from the current backward support, the
observable is the identity. A complete channel adjoint preserves
that identity. Other blocks enlarge support only within their
union. Backward induction proves the inclusion.

Equality of all pulled-back expectation values proves reduced
state dependence; appending the reference identity proves the
joint statement. Concatenation of causally embedded edges proves
metric containment.
\end{proof}

This is the complete-channel argument of the schedule comparisons
\cite{CheungEn03,CheungEn05}. It does not concern selectively
normalized postselection maps or unpriced access to remote records.

The fixed-time current and its surface support lie in \(K_f\),
so their admitted geometric evaluations remain in the prescribed
past shadow. This statement is distinct from the global
Newton-data dependence of Section~\ref{sec:cubic}.

\subsection{Exact target-adapted payload transfer}

Choose a straight developed segment from \(x_A\) to \(x_B\)
of length \(L>0\), entirely inside \(U\), and an integer \(n\ge1\).
Install \(n+1\) sites along it with spacing \(L/n\).
Each site carries \(A_j\cong\CC^d\).
Prepare \(A_1,\ldots,A_n\) in an independent fixed blank
\(\omega\), and at tick \(j\) SWAP \(A_{j-1}\) with \(A_j\).
Set the tick duration to \(L/(nc)\).

\begin{theorem}[Reference-preserving null endpoint transport]
\label{thm:transport}
After tick \(j\), every payload-reference input has state
\[
 \rho_{A_jE}\otimes
       \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}.
\]
The logical transfer is exactly \(\id_d\), all other payload
registers finish blank, arrival time is \(T_{\rm d}+L/c\),
and all tick-boundary dependencies are causal.
\end{theorem}

\begin{proof}
Each SWAP exchanges the entire current payload factor, including
its reference correlations, with the next independent blank.
Induction gives the state identity.
Each hop has length \(c\) times its tick duration, so
Lemma~\ref{lem:causal} applies.
\end{proof}

To execute a receiver operation selected by \(k\), the code must
also be delivered. Reserve a \(q\)-bit classical register at each
site and copy the code along the same causal route.
This is copying a diagonal classical variable, not cloning an
unknown quantum state. At arrival, apply the selected
\(M_{k,y}\) locally. The complete received instrument is exactly
\eqref{eq:instrument}, with the payload relocated.

For \(d=2^p\), the protocol uses \(n\) register SWAPs and
\(n+1\) payload registers. If all payload registers are reserved
before clock consumption, and the response probe is allocated
afterward, the quantum peak is at most
\[
 Q_{\rm lab}=p(n+1)+\max\{q,1\}.
\]
The retained classical ledger uses at most \(q(n+1)+1\) bits.
External reference storage is listed separately.

This is the target-adapted protocol of the I-B type
\cite{CheungIB}. It does not assert exact reachability on one
fixed lattice; En08-B treats a different approximation contract
\cite{CheungEn08}. No continuous-time Hamiltonian for a remote
within-tick SWAP is constructed.

For compatible implemented channels, replacement of one factor
at a time gives
\[
 d_\diamond(\widetilde\Phi,\Phi)
 \le\min\left\{1,\epsilon_{\rm enc}
              +\sum_j\epsilon_j+\epsilon_{\rm dec}\right\},
 \qquad
 d_\diamond=\tfrac12\norm{\cdot}_\diamond.
\]
This is a conditional error certificate, not a reported experiment.

%=====================================================================
\section{Manipulations and non-identifications}
%=====================================================================

\begin{proposition}[Permitted changes have separate meanings]
The following operations do not identify quantum factors with
geometric coordinates:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a clock basis change replaces \(P_k\) and generally changes
the instrument, not the coordinate \(T\);
\item a coupling before readout replaces \(\Mclock\) by
\(\Mclock\circ\Ad_U\), not by a metric back-reaction law;
\item code-selected continuation chooses the declared response
setting or finite matrix;
\item fresh clock allocation creates new names and a new reading
event, not a continuum clock or a returned old clock.
\end{enumerate}
\end{proposition}

\begin{proof}
The first two assertions follow by substituting the changed
projectors or precomposition into the Kraus formula.
The latter two change the instruction tree and its data.
None changes the supplied spacetime or establishes a new field
equation.
\end{proof}

Thus the statement ``the fourth dimension is the factor Enter
consumes'' is used, at most, as an operational analogy.
Literally, the geometric time coordinate remains present after
the finite clock factor is consumed.

Three spatial rod coordinates likewise do not imply that a
continuum relativistic field has a tensor factor for each rod or
point. The finite name factorization is an independent apparatus
choice.

The classical lapse constructions of Em02-E and XX-A, and the
separations of XIX-A, retain their own backgrounds, rods, and
inverse maps \cite{CheungLapse}. No comparison map identifies
their lapse with the code \(k\), the pointer
\(\sum_kT_kP_k\), or the schedule time.

Record agreement alone is insufficient: identity and computational
dephasing have identical one-outcome coarse records, but on
\(\proj{+}\) their residuals are \(\proj{+}\) and \(I/2\).
Their trace distance is \(1/2\).
The complete instruments, not merely records, are therefore used
throughout \cite{CheungEn03}.

Finally, two distinct conical deficits admit isometric regular
laboratories of any fixed finite size. Installing the same
analytical packet and finite matrices in those laboratories
gives the same local records, although the global conical
completions differ. The present records do not determine
the deficit.

%=====================================================================
\section{Release and precise boundaries}
\label{sec:release}
%=====================================================================

\begin{theorem}[Local spacetime-Enter release]
\label{thm:release}
Under Definition~\ref{def:presentation}, the construction proves:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete raw current, including its initial-surface
measure and full Ward defect;
\item exact raw and harmonic-prescription admissibility criteria
on the named auxiliary solution spaces;
\item compact-data residual obstruction on an open dense
fixed-support smearing family;
\item explicit compact-current interpolation on each admitted
two-dimensional TT plane;
\item the external and one-oscillator first retarded coefficients,
their switched identity, and the stated polynomial
self-adjointness;
\item a distinct symmetric Newton--wave cubic form, its joint
cutoff limit, and its complete regular-region witness;
\item exact finite compressed response instruments with
destructive clock readout, causal code delivery, and
reference-preserving endpoint transport.
\end{enumerate}
These data constitute local Grade S as defined here.
\end{theorem}

\begin{proof}
Items (i)--(iv) are
Theorems~\ref{thm:raw}, \ref{thm:defect},
\ref{thm:classification}, \ref{thm:generic}, and
\ref{thm:plane}.
Items (v)--(vi) are
Theorems~\ref{thm:external}, \ref{thm:ESA},
\ref{thm:cubicform}, and \ref{thm:cubicwitness}.
Item (vii) follows from
Theorem~\ref{thm:instrument},
Proposition~\ref{prop:compressionlimit}, and
Theorems~\ref{thm:locality}--\ref{thm:transport}.
The geometry, continuation prescriptions, finite encoding,
dictionaries, and resource charges have all been specified.
\end{proof}

\subsection{What is new in the adaptation}

The analytical results above reproduce and reorganize the
mechanisms of V@1 \cite{CheungV}. The operational extension is
the explicit form-compression bridge: a symmetric cubic form,
without an unproved self-adjoint continuum generator, still
defines an exact finite instrument after its finite carrier
has been declared. The joint form estimate then controls that
fixed compressed instrument.

The cubic qubit makes the bridge nonvacuous.
It also demonstrates why agreement of lower-degree data cannot
be promoted to agreement of the cubic scalar response.

This is a scoped contribution, not a priority claim about all
finite compressions or measurement theory.

\subsection{Six missing Grade E implications}

The construction does not provide:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item reconstruction of a manifold, dimension, or global topology
from the records;
\item determination of metric scale, signature, or cone from the
finite channel data;
\item derivation of lapse and proper-time calibration without
external geometry;
\item derivation of stress or Einstein dynamics from a residual
matrix response;
\item a same-target continuum field instrument with a proved
physical localization and refinement limit;
\item construction and admission of the separately inherited
characteristic quantum coefficient.
\end{enumerate}

In particular, the auxiliary Minkowski vacuum is supplied, not
selected by the cone records. The Newton prescription is a
nonlocal analytical input, not a finite-speed communication
operation. Removing these inputs would require new theorems.

\subsection{Standing separations}

\begin{itemize}
\item Finite-plane interpolation is not dense-TT representability.
Dense-TT representability remains open in the programme considered
here.
\item The compact switch remains compact. No adiabatic limit
\(g\to1\) is proved.
\item No higher causal product, \(T_2(\mathcal V,\mathcal V)\),
stress--stress product, or graviton-loop calculation is constructed.
\item The Newton--wave cubic is neither the external linear
\(R_{1,1}\) nor the oscillator mixed quadratic \(R_{1,1}\).
It is not identified with a later Einstein cubic.
\item Equality of curvature or compact-current data does not
imply equality of the cubic scalar response.
\item No full Einstein--scalar linearization or consistently
quantized gravitational observable is supplied.
\end{itemize}

III supplies a separate classical descent programme, with its
own equation spaces and unconstructed sourced successor.
IV supplies distinct represented models, including a scalar
quadratic Model 3. Neither is a premise or an identification
of the cubic form here \cite{CheungIII,CheungIV}.

II's finite contact membership logic does not identify a
counterterm class for the present currents \cite{CheungII}.
Oai282-A's finite Ward comparison and the separate OpenAI catalogue
are documentary separation records only
\cite{CheungOaiWard,OpenAIRecord}.
No OpenAI catalogue theorem is a premise or independently verified
by this manuscript.

Hawk04--Hawk07 comparisons concern their separately declared access
and recovery models \cite{CheungHawk}.
There is no Page curve, island, wormhole, interior, or holographic
identification in the present laboratory.

The inherited analytic records distinguish an internal amplitude
construction from the unresolved full joint characteristic map
\cite{CheungAnalytic}. The historical H2 stop in XIV-H is not
a withdrawal of XIV-I's narrower I2 construction.
Neither that construction nor the supported-sector reference of
XIV-J is replaced by our finite readout.

The documentary standing boundaries for this release remain
\[
 \mathrm{O.1}=\source{NOT_CLOSED},\qquad
 \text{XV-B quantum-admission exit: not opened},\qquad
 \text{XVI-N: not discharged}.
\]
Existence of technical XV-B manuscripts is distinct from opening
that admission exit. Differently scoped subsector reports are not
silently upgraded to closure of the full inherited problem
\cite{CheungStops}.

The phrase ``no quantum coefficient admitted'' refers to that
inherited characteristic target. It does not deny the explicit
free-field and first-retarded constructions proved here.
No solution of the nine standing programme problems is claimed.

Finally, no apparatus Hamiltonian, reset protocol, bath, work
account, or heat ledger follows from the destructive channel.
Consumption of names is not a thermodynamic calculation.

%=====================================================================
\appendix
\section{Domain and debugging audit}
%=====================================================================

\subsection{Local versus global geometry}

All conical applications use the regular developing chart.
The analytical theorems quantify over the declared auxiliary
presentation, not over all global conical fields.

The current and finite-time response sectors have compact
dependency regions and can be included in a finite laboratory.
The Fourier-gap plane and Newton construction retain noncompact
continuation data. Restriction does not erase those data.

An intrinsic global construction would need boundary or extension
conditions at the singularity, global Green operators, a global
state prescription, and a conical replacement for the Newton
choice. None is inferred from local flatness.

\subsection{Three different gauge tests}

For the raw response:
\[
\begin{array}{c|c}
\text{tested descriptors}&\text{condition}\\
\hline
C_c^\infty((0,\infty)\times\RR^3)&\gamma=0\\
C_c^\infty(\widehat M)&H=0\\
\Box\xi=0,\ \xi\text{ spacelike compact}&E_1H=0
\end{array}
\]
The first two follow by distributional test separation.
The third uses the compact-source Green criterion.
These are different statements.

\subsection{Why a clock cannot repair nonrepresentability}

Every compact conserved current vanishes on every smooth
\(K\xi\). The residual witness has a nonzero response there.
Encoding that nonzero number in a finite matrix or record does
not alter the mismatch. It records a value of the specified
non-descending functional; it does not manufacture a quotient
identity.

\subsection{What finite matrix convergence means}

The form topology in \eqref{eq:jointlimit} controls each fixed
finite compression. It does not imply operator-norm convergence
of an unregulated cubic operator on Fock space.

Likewise, exact finite instrument identities do not construct a
field--probe interaction implementing those compressions on an
unknown continuum state. Such a realization would need its own
input class, localization, selective-map comparison, and error
topology.

\subsection{Support and source choices are not interchangeable}

Compact Cauchy data, Schwartz data with a Fourier gap, compact
spacetime smearing, and polynomially decaying Newton initial data
have different roles. No proof exchanges them.

In particular, a compactly supported ungauged Einstein perturbation
need not vanish: compact pure gauges exist. The zero-past
uniqueness used above applies to the homogeneous wave equations,
not to an unrestricted gauge equation by itself.

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

\paragraph{Authorship.}
This source is formatted for anonymous review.
Authorized author and affiliation metadata must be supplied for
a signed submission. Citation of N.~Cheung does not assign
authorship of this adaptation.

\paragraph{Analytical provenance.}
The registered source V@1 was consulted for its current,
admissibility, retarded, oscillator, and cubic constructions.
The arguments needed here are presented explicitly.
Other companion references identify documentary comparisons
rather than unprinted proof premises.

\paragraph{Novelty and impact.}
The claimed contribution relative to the source is the explicit
local-domain audit and finite complete-instrument interface,
including the compressed cubic qubit and fixed-compression
cutoff compatibility. No claim of spacetime emergence or
continuum apparatus realization is attached to it.

\paragraph{Verification.}
No successful pdfLaTeX build, proof-assistant verification,
independent symbolic computation, numerical execution,
physical programme compilation, laboratory realization,
referee certification, or acceptance is asserted.

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

\bibitem{CheungV}
N.~Cheung,
\emph{Admissible currents and first retarded corrections for a
fixed-time scalar response}.
Manuscript V, \source{V.tex}, version 1,
registered unpublished manuscript, Enter Calculus Companion.
Relevant source locators include
\source{thm:rawcurrent}, \source{thm:defect},
\source{thm:rawclassification}, \source{thm:harmonic},
\source{thm:planeextension}, \source{thm:external},
\source{thm:ESA}, \source{thm:cubicform}, and
\source{thm:cubicwitness}.
\url{https://enter.grok.me/browse/V}.

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime}.
\source{I-B-v1.tex}, version 1,
registered unpublished manuscript, Enter Calculus Companion.
Prescribed cone, clock instrument, calibration, and
target-adapted transport source record.
\url{https://enter.grok.me/browse/I-B}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement.''
Enter-calculus article identified in the supplied bibliographic
record by DOI \doi{10.3389/fphy.2026.1954943}.
Its supplied record lists \emph{Frontiers in Physics} 14 (2026),
article 1954943. Publisher publication-status metadata is not
independently certified here. The present instrument is proved
directly and does not require a publication-status premise.

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter}.
Enter Calculus Companion introductory source record.
\url{https://enter.grok.me/enter}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{En09-A: Realization of Relational Time by Destructive Enter
on a Clock Register}.
\source{En09-A.tex}, version 1,
registered unpublished manuscript, Enter Calculus Companion.
Complete extraction, covariance, stationarity, and
reference-extension boundaries.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization}.
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007.
\url{https://arxiv.org/abs/0806.1036}.

\bibitem{FH}
C.~J.~Fewster and D.~S.~Hunt,
``Quantization of linearized gravity in cosmological vacuum
spacetimes,''
\emph{Reviews in Mathematical Physics} 25 (2013), 1330003.
DOI: \doi{10.1142/S0129055X13300033}.
\url{https://arxiv.org/abs/1203.0261}.
The compact conserved observable class supplies context;
no radical-exhaustion theorem is imported.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
``Microlocal analysis and interacting quantum field theories:
Renormalization on physical backgrounds,''
\emph{Communications in Mathematical Physics} 208 (2000),
623--661.
DOI: \doi{10.1007/s002200050004}.
\url{https://arxiv.org/abs/math-ph/9903028}.

\bibitem{DF}
M.~D\"utsch and K.~Fredenhagen,
``Causal perturbation theory in terms of retarded products, and
a proof of the Action Ward Identity,''
\emph{Reviews in Mathematical Physics} 16 (2004), 1291--1348.
DOI: \doi{10.1142/S0129055X04002266}.
\url{https://arxiv.org/abs/hep-th/0403213}.

\bibitem{Nelson}
E.~Nelson,
``Analytic vectors,''
\emph{Annals of Mathematics} 70 (1959), 572--615.
DOI: \doi{10.2307/1970331}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's book and errata interface:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{PageWootters}
D.~N.~Page and W.~K.~Wootters,
``Evolution without evolution: Dynamics described by stationary
observables,''
\emph{Physical Review D} 27 (1983), 2885--2892.
DOI: \doi{10.1103/PhysRevD.27.2885}.

\bibitem{CheungEn03}
N.~Cheung,
\emph{En03-A}, \source{En03-A.tex}, version 1.
Registered unpublished manuscript, Enter Calculus Companion.
Supplied records on schedule locality, observation interfaces,
and restricted destructive deferred readout.
\url{https://enter.grok.me/browse/En03-A}.

\bibitem{CheungEn05}
N.~Cheung,
\emph{En05-A}, \source{En05-A.tex}, version 1.
Registered unpublished manuscript, Enter Calculus Companion.
Supplied locator \source{thm:cone}:
complete-channel dependence on a prescribed schedule past.

\bibitem{CheungEn08}
N.~Cheung,
\emph{En08-B}, \source{En08-B.tex}, version 1.
Registered unpublished manuscript, Enter Calculus Companion.
Resource-controlled finite transport toward a prescribed null
boundary; comparison only.
\url{https://enter.grok.me/browse/En08-B}.

\bibitem{CheungIII}
N.~Cheung,
\emph{III}, \source{III.tex}, version 2.
Registered unpublished manuscript, Enter Calculus Companion.
Classical equation spaces, first-order descent, and the separate
sourced successor S1--S3.
\url{https://enter.grok.me/browse/III}.

\bibitem{CheungIV}
N.~Cheung,
\emph{IV}, \source{IV.tex}, version 1.
Registered unpublished manuscript, Enter Calculus Companion.
Three represented models with separately declared observables,
cutoffs, and measurement assumptions. Documentary comparison.

\bibitem{CheungII}
N.~Cheung,
\emph{II}, \source{II.tex}, version 2.
Registered unpublished manuscript, Enter Calculus Companion.
Finite contact-span certificate; no contact-space identification
or global identity is transferred to this paper.

\bibitem{CheungLapse}
N.~Cheung,
\emph{Em02-E}, version 1;
\emph{XX-A}, version 1; and
\emph{XIX-A}, version 1.
Supplied unpublished source records, Enter Calculus Companion.
Classical relational lapse, original rods, and clock-prescription
separations. Documentary comparisons, not premises.

\bibitem{CheungOaiWard}
N.~Cheung,
\emph{Oai282-A}, \source{Oai282-A.tex}, version 1.
Supplied unpublished source record, Enter Calculus Companion.
Locator \source{thm:ward}: finite tested Ward residual for its
own improvement and contact contract. Not a premise here and
not a verification of an OpenAI catalogue family.

\bibitem{OpenAIRecord}
OpenAI,
\emph{OpenAI mathematics catalogue}.
Separately attributed documentary material reproduced in the
supplied companion record.
No catalogue theorem, family proof, or formalization certificate
is a premise or independently authenticated by this manuscript.

\bibitem{CheungHawk}
N.~Cheung,
\emph{Hawk04-A}, version 1;
\emph{Hawk04-B}, version 2; and the supplied
\emph{Hawk05--Hawk07} records.
Unpublished manuscripts, Enter Calculus Companion.
Operational access and recovery comparisons only.

\bibitem{CheungAnalytic}
N.~Cheung,
\emph{XIV-H}, version 1;
\emph{XIV-I}, version 1;
\emph{XIV-J}, version 1; and technical
\emph{XV-B}, version 2.
Supplied unpublished manuscripts, Enter Calculus Companion.
Internal-amplitude, reconstruction, supported-reference, and
relative-lifting records. No full inherited characteristic
coefficient is imported.

\bibitem{CheungStops}
N.~Cheung,
\emph{XVI-N}, revision r12;
\emph{XVI-O}, revision o70-sync; and
\emph{XVI-Q}, release Q11.
Supplied unpublished source records, Enter Calculus Companion.
Conditional insertion obstruction and separately scoped
subsector reports. This paper updates none of their release
conditions.

\end{thebibliography}

\end{document}