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% Principal corrections:
% 1. The host is the prescribed conical product, not Minkowski space.
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% 4. The residue is supported on a reflected observation relation,
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\hypersetup{
  pdftitle={IX-B-ST: Source-locked triangle calibration on a prescribed conical Enter spacetime},
  pdfsubject={Local Grade S calibration, observation descendants, finite records, and causal transport}
}

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\title{\textbf{IX-B-ST: Source-locked triangle calibration\\
on a prescribed conical Enter spacetime}\\[5pt]
\large Observation descendants, finite calibration records,\\
and reference-preserving transport}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a local Grade S adaptation of the IX-B sharp-insertion
triangle calibration to a causally saturated regular laboratory
on the prescribed I-B conical product. The host metric and its
worldsheet defect are inputs; the laboratory closure avoids the
defect. The covariance blocks, regulator placement, and VII-D
parent finite part are supplied comparison data in the developing
chart, not an intrinsic conical field theory.

Relative to the stated parent estimate, the calibrated numerical
distribution transfers to every fixed observation differential
operator \(D\) and smooth multiplier \(a\), with the quantitative
test-order loss \(64\to64+\ord D\), uniformly on bounded
coefficient families and through the reflected-diagonal
intersection. We retain the eight contraction skeletons, the unique
fully contracted triangle, its assignment factor \(2\), and the
admitted first-stage coefficient with sign \(-2\). The residue is
a positive measure on a reflected observation relation, not on an
observation equality diagonal. Its observation-derivative orbit is
infinite-dimensional, although all compatible descendants can
inherit one common parent conversion parameter.

The additional contribution is a supported transfer and finite
certificate interface: explicit numerical enclosure errors become
fixed-interface complete-channel bounds, and an unknown payload
together with its classical calibration history admits exact
reference-preserving buffered transport entirely in the regular
laboratory. This supplies a controlled benchmark for comparing
future product constructions without confusing numerical
calibration, continuum acquisition, and quantum multiplication.
The analytic calibration results are attributed to IX-B and
VII-D; their ultraviolet analysis is not re-proved here.
Equality of the two regularized parenthesizations constructs
neither sharp second multiplication. The VII-C obstruction remains
under its original hypotheses. No renormalized interacting
product, quantum Ward identity, Einstein dynamics, or spacetime
emergence is established.
\end{abstract}

\statementbox{Exact scope}{%
This sheet is placed after IX-B and before the IX-C interface in
the present adaptation programme. It does not replace IX-B and
does not begin IX-C-ST or a unified IX-ST.

The principal analytic release is conditional on the explicitly
stated inherited parent estimate. The admission carrier is the
already adapted IX-A-ST carrier. Neither is reconstructed by
citation or by a finite table.

The host, initial prescription, covariance, regulator profile,
and calibration are fixed inputs. Changing a regulator or
subtraction-scale label changes comparison data, not the host
metric. The quantum certificate remains \loc{RG-open}.}

\tableofcontents

%=====================================================================
\section{Introduction, provenance, and the Enter distinction}
\label{sec:introduction}
%=====================================================================

IX-B calibrates a particular numerical coefficient of the ordered
word
\[
 B_f,\qquad W_g,\qquad B_{f'},
\]
on the actual-functional admission carrier of IX-A
\cite{CheungIXB,CheungIXA}.
Its numerical finite part is inherited from VII-D, while the
normal-topology incompatibility is retained from VII-C
\cite{CheungVIID,CheungVIIC}.
IX-B does not introduce a conical construction.
The purpose of IX-B-ST is to add a prescribed conical host and a
complete finite operational interface without changing that
analytic target.

The Enter background is an interface discipline. A quantum name
denotes declared access to a tensor factor. Destructive readout
consumes the measured names and returns a classical record,
together with the specified unmeasured residual. It is not a
quantum-return decoder for an unknown input.
Cheung's operational characterization has its own structural
hypotheses; they are not re-proved or enlarged here
\cite{CheungEnter}.
The philosophical distinctions between readout, recovery,
implementation, and geometric identification are useful precisely
because none is inferred merely from another
\cite{CheungPhilosophy}.

In spacetime Enter, a finite clock encoding is calibrated against
an already supplied geometry. Consuming that encoding does not
consume the geometric time axis, change the event of readout, or
derive a metric \cite{CheungIB}.
We use finite Born instruments directly and prove their complete
channel identities.

\begin{definition}[Local Grade S]
\label{def:grade}
A local Grade S package here consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item a prescribed spacetime and a supported regular laboratory;
\item explicit classical or numerical objects with stated domains
and proof dependencies;
\item complete finite channels, including untouched references;
\item one destructive clock and an explicit calibration contract;
\item proved computational support and separately certified
physical causal containment;
\item one nonempty finite transport realization and its resource
ledger.
\end{enumerate}
Grade E, understood here as a claim to derive spacetime or its
metric from records or quantum data, is not constructed.
\end{definition}

\paragraph{Conditional proof versus documentary citation.}
A cited manuscript is a source record, not a blanket axiom.
The particular mathematical inputs used below are restated as
Assumptions~\ref{ass:carrier}, \ref{ass:tensor}, and
\ref{ass:VIID}. The conditional conclusions genuinely depend on
those assumptions. No claim of an assumption-free reconstruction
of the VII-D analysis is made.

\paragraph{Version boundary.}
``Neither sharp second multiplication is constructed'' means
that no such map is defined by this sheet.
It does not deny separately versioned later partial-product
constructions. IX-C, IX-Free, and IX-Free-2 are documentary
successors, not premises, and their releases are not reversed
by this adaptation \cite{CheungSuccessors}.

%=====================================================================
\section{Prescribed cone and supported regular laboratory}
\label{sec:host}
%=====================================================================

Use \(\delta_{\mathrm c}\) for the conical deficit and \(\mu\) for
the numerical subtraction scale. Fix
\[
 0<\delta_{\mathrm c}<2\pi,\qquad
 \alpha=1-\frac{\delta_{\mathrm c}}{2\pi},\qquad c>0.
\]
The host is
\begin{equation}
 M_{\delta_{\mathrm c}}
   =\RR_T\times C_{\delta_{\mathrm c}}^\circ\times\RR_z,
 \qquad
 g_{\delta_{\mathrm c}}
   =-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2,
 \qquad r>0.
 \label{eq:host}
\end{equation}
The angular coordinate has period \(2\pi\).
The stipulated completion has defect
\[
 \WW=\RR_T\times\{o\}\times\RR_z.
\]
This is a worldsheet; only fixed-\(z\) sections are worldlines.
The signature is \((-,+,+,+)\), the lapse in \(T\) is one,
and static proper time is \(T\).
These are I-B inputs, not outputs of the calibration
\cite{CheungIB}.

\begin{proposition}[The regular product is not globally hyperbolic]
\label{prop:nonGH}
The regular spacetime in \eqref{eq:host} is not globally
hyperbolic.
\end{proposition}

\begin{proof}
Fix a radius \(r_0>0\), an angle, and an axial coordinate.
Choose events at that spatial point and times
\(T_*\mp2r_0/c\).
Points at time \(T_*\), the same angle and axial coordinate,
and radii \(r_n\downarrow0\), belong to the causal diamond
between those events by radial causal curves.
They have no convergent subsequence in the regular product.
That diamond is noncompact.
\end{proof}

\begin{definition}[Supported developing laboratory]
\label{def:support}
Fix a product-adapted isometric developing embedding
\[
 \iota:U=(-\delta,H)\times B_R
           \longrightarrow M_{\delta_{\mathrm c}},
 \qquad t=c(T-T_\Sigma),
\]
with developed metric
\[
 \eta=\diag(-1,1,1,1).
\]
Choose \(T_0,r_0>0\) and \(R>r_0+T_0\), and set
\[
 V=(0,T_0)\times B_{r_0},\qquad
 \Sigma=\{0\}\times B_R,\qquad
 K_*=[0,T_0]\times\overline B_{r_0+T_0}\Subset U.
\]
All observation tests are supported in \(V\).
The compact analytic and operational laboratory is contained
in \(\iota(U)\), has closure disjoint from \(\WW\), and contains
a neighbourhood of \(\iota(K_*)\).

For \(f\in\DD(V)\), the full truncated developed past
\[
 D_f=J^-_\eta(\supp f)\cap\{t\ge0\}
\]
is retained. It is contained in \(K_*\).
The response prescription has zero value and zero first
\(t\)-derivative at \(t=0\), with no additional incoming
homogeneous or lateral-boundary data.

For each fixed compact test family, the permitted positive
smoothing widths are small enough that the smoothed field
kernels remain inside \(U\).
\end{definition}

The positive support margin is necessary: a chart containing
only the observation points would not determine the prescribed
zero-Cauchy response. Such laboratories are nonempty by choosing
a regular developed ball and a sufficiently short observation
interval, sufficiently far from the defect.

\begin{proposition}[Use of the IX-A-ST saturation contract]
\label{prop:saturation}
For the supported response already supplied by IX-A-ST,
two auxiliary field extensions equal near \(K_*\) give the
same response on \(V\), with the same smooth derivatives and
observation pairings.
The comparison covariances and the residue are not thereby
constructed as host Green functions.
\end{proposition}

\begin{proof}
The local uniqueness and finite-propagation argument in the
IX-A-ST saturation contract applies because every \(D_f\)
lies inside the region on which the extensions agree
\cite[{\loc{lem:saturation}}]{IXAST}.
Their difference has zero source in the relevant truncated
past and zero initial response data, so it vanishes on the
observation region. Only this supported Cauchy statement is
used; it supplies neither a global inverse on
\(M_{\delta_{\mathrm c}}\) nor a conical quantum state.
The ordinary local Cauchy theory is distinct from global
hyperbolicity \cite{BGP}.
\end{proof}

\begin{proposition}[Fixed host under catalogue replacement]
\label{prop:fixedhost}
Replacing a regulator label \(\rho_a\) by \(\rho_b\), or a scale
label \(\mu_1\) by \(\mu_2\), in the comparison catalogue leaves
\(M_{\delta_{\mathrm c}},g_{\delta_{\mathrm c}},\WW,\Sigma\),
the lapse, and the installed causal relation unchanged.
\end{proposition}

\begin{proof}
A comparison command acts on the catalogue and finite programme
registers. The host data in \eqref{eq:host} are held fixed and
are not outputs of that command. No metric-update map is
included.
\end{proof}

This is the fixed-host distinction also documented in VIII-ST
\cite{STInterfaces}.

%=====================================================================
\section{Chart formulae and the exact dependency ledger}
\label{sec:inputs}
%=====================================================================

Write \(x=(t,\mathbf x)\), \(y=(s,\mathbf y)\), and
\(x'=(t',\mathbf x')\), and set
\[
 X=V\times V\times V.
\]
This is a space of observation triples in one supported chart;
it is not the spacetime host.

All comparison quantities are expressed in one fixed system
of units. In particular \(\mu\varepsilon\) and the scale ratios
inside logarithms are dimensionless.

\subsection{Off-shell comparison prescription}

The inherited local formulae are
\begin{align}
 \Box&=-\partial_t^2+\Delta_{\mathbf x},\nonumber\\
 (Ch)^\mu
   &=\partial_\nu h^{\mu\nu}
          -\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}),
 \qquad L=\Ezero C,\qquad w=Lh,\nonumber\\
 B_f[h,\phi]
   &=-\int f(x)w^\mu[h](x)\partial_\mu\phi(x)\,\dd^4x,
 \qquad
 W_g[h]=\int g(y)h_{00}(y)^2\,\dd^4y.
 \label{eq:generators}
\end{align}
Here \(\Ezero\) denotes the supported comparison prescription,
not a newly introduced host inverse.
Both input fields are smooth and off shell.
The source \(j=\Box h\) is not set to zero.

\begin{assumption}[Already adapted carrier]
\label{ass:carrier}
The pair \((\Padm,\taus)\) is the IX-A-ST actual-functional
carrier and topology, with the following inherited properties
\cite{IXAST,CheungIXA}.
\begin{enumerate}[label=\textup{(A\arabic*)}]
\item Its elements are finite polynomial functionals with compact
symmetric distributional coefficient kernels.
Their admission criterion is
\[
 (\Ddir_{\boldsymbol\eta}F)'_h(\Psi)\in\mathscr S_1
 \quad\text{for every smooth }\Psi
 \text{ and finite smooth direction list }\boldsymbol\eta.
\]
Here
\[
 \mathscr S_1=\mathscr M_1+\mathscr Q_\Sigma,\qquad
 \mathscr Q_\Sigma=\{L^*a:a\in\DD(V;T^*U)\},
\]
and \(\mathscr M_1\) consists of compact linear metric kernels
with no nonzero nonspacelike wavefront covectors.
The trace of a represented linear derivative is the intrinsic
advanced-Cauchy trace defined by that carrier.
\item The topology records strong compact-distribution
coefficients and all those trace coordinates. It is Hausdorff.
Fixed-direction differentiation and conjugation are continuous.
Classical pointwise products stay in \(\Padm\) and are separately
continuous. Joint continuity and hypocontinuity are not assumed.
\item The bulk inclusion is faithful. The functionals \(B_f,W_g\)
and the selected one-contraction coefficient with \(W_g\)
belong to \(\Padm\), with all directional descendants required
by admission.
\item For every nonzero real \(f\), the unchanged compact-kernel
approximants satisfy
\[
 B_{f,\varepsilon}\not\longrightarrow B_f
                  \quad\text{in }\taus.
\]
A normalized continuous boundary separator detects this
nonconvergence.
\end{enumerate}
No quantum multiplication on all of \(\Padm\) is part of this
assumption.
\end{assumption}

This recalls the domain and input interface without reconstructing
the carrier, its topology, or its trace proof.

\subsection{Covariance and mixed kernel}

The supplied comparison covariance has blocks
\begin{equation}
 H_{hh,\alpha\beta\rho\sigma}
    =P_{\alpha\beta\rho\sigma}\Delta_0^+,\qquad
 H_{\phi\phi}=\Delta_m^+,\qquad
 H_{h\phi}=H_{\phi h}=0,
 \label{eq:covariance}
\end{equation}
where
\[
 P_{\alpha\beta\rho\sigma}
  =\eta_{\alpha(\rho}\eta_{\sigma)\beta}
          -\frac12\eta_{\alpha\beta}\eta_{\rho\sigma},
 \qquad m\ge0,
\]
and
\[
 \Delta_m^+(x-y)=
 \int_{\RR^3}\frac{\dd^3q}{(2\pi)^3\,2\omega_m(q)}
 e^{-i\omega_m(q)(t-s)+iq\cdot(\mathbf x-\mathbf y)},
 \qquad \omega_m(q)=\sqrt{|q|^2+m^2}.
\]
No positivity of the metric block is assumed.

\begin{assumption}[Locked tensor and old-domain data]
\label{ass:tensor}
Retain the VII-C tensor identities and old Wick-symbol convention
\cite[{\loc{prop:tensor}}, {\loc{eq:K}}]{CheungVIIC}:
\[
 C_x^\mu H_{hh,\rho\sigma}
    =\delta^\mu_{(\rho}\partial_{\sigma)}^x\Delta_0^+,
 \qquad H_{ww}=0.
\]
The nonzero mixed component is
\begin{equation}
 \begin{split}
 K(x,y)&=H_{wh}^{0}{}_{00}(x,y)\\
 &=\int_{\RR^3}\frac{\dd^3p}{(2\pi)^3}
 e^{ip\cdot(\mathbf x-\mathbf y)}
 \left[
 \left(-\frac{t}{4P}+\frac{i}{8P^2}\right)e^{-iP(t-s)}
       -\frac{i}{8P^2}e^{iP(t+s)}
 \right],
 \qquad P=|p|.
 \end{split}
 \label{eq:K}
\end{equation}
The reverse component is \(\overline{K(x,y)}\).
The covariance selects the \(0\)-component at a contraction
with \(h_{00}\).

On the old bulk polynomial domain, retain
\begin{equation}
 F\star_HG
   =\sum_{j\ge0}\frac{\hbar^j}{j!}
           \pair{F^{(j)}}{H^{\otimes j}G^{(j)}}.
 \label{eq:oldproduct}
\end{equation}
Self-contractions within a Wick symbol are excluded.
The usual old-domain operations used at positive cutoff are
defined. The unchanged mixed-kernel approximants converge in
the fixed-support normal stages of VII-C.
\end{assumption}

In \eqref{eq:K}, the infrared powers \(P^{-1},P^{-2}\) are
locally integrable in three momentum dimensions. The high-frequency
terms are understood distributionally.
The reflected term is retained. In particular,
\(H_{ww}=0\) is not the statement \(K=0\).

These are prescribed chart comparisons. They are neither derived
from the conical metric nor claimed to extend to a global conical
two-point function. Functional Wick products and their domain
restrictions are reviewed in \cite{FR}.

\subsection{Regulator placement}

Let \(\rho\in C_c^\infty(\RR^4)\) be real, nonnegative, even,
Euclidean radial, and normalized by \(\int\rho=1\).
Use the same \(\rho\) on both original field slots of each
insertion:
\[
 \rho_\varepsilon(z)=\varepsilon^{-4}\rho(z/\varepsilon),
 \qquad S_\varepsilon=\rho_\varepsilon*,
 \qquad
 B_{f,\varepsilon}[h,\phi]
       =B_f[S_\varepsilon h,S_\varepsilon\phi].
\]
The middle observable \(W_g\) is not smoothed.

Use the coordinate Fourier pairing
\(\widehat\rho(k)=\int e^{-ik\cdot z}\rho(z)\,\dd^4z\).
For \(\mathbf n\in S^2\), put
\[
 \vartheta_\rho(v)=\widehat\rho(-v,v\mathbf n),\qquad
 M_\varepsilon(\lambda)
   =\vartheta_\rho\!\left(
       \varepsilon\sqrt{\lambda^2+m^2/2}\right).
\]
Radiality makes \(\vartheta_\rho\) independent of \(\mathbf n\).
Let \(K_\varepsilon\) be \eqref{eq:K} with multiplier
\(\vartheta_\rho(\varepsilon P)\), and set
\begin{equation}
 D_\varepsilon(z)=
 \int_{\RR^3}\frac{\dd^3q}{(2\pi)^3}
       \frac{\omega_m(q)}2M_\varepsilon(|q|)^2
       e^{-i\omega_m(q)z^0+iq\cdot\mathbf z}.
 \label{eq:Deps}
\end{equation}
The exact numerical family under this placement is
\begin{equation}
 T_\varepsilon^\rho(x,y,x')
   =2K_\varepsilon(x,y)\overline{K_\varepsilon(x',y)}
                               D_\varepsilon(x-x').
 \label{eq:triangle}
\end{equation}
Its positive-cutoff smoothness and its identification with the
old-domain coefficient are included in the following analytic
input.

\subsection{Parent finite part}

Let
\begin{align}
 Z_U&=\{(\mathbf z,\mathbf n,t,s,t'):
                       \Psi(\mathbf z,\mathbf n,t,s,t')\in X\},
 \nonumber\\
 \Psi(\mathbf z,\mathbf n,t,s,t')
  &=\bigl((t,\mathbf z+t\mathbf n),
          (s,\mathbf z-s\mathbf n),
          (t',\mathbf z+t'\mathbf n)\bigr),\nonumber\\
 \dd\nu&=\dd^3z\,\dd\Omega(\mathbf n)\,\dd t\,\dd s\,\dd t',
 \qquad \int_{S^2}\dd\Omega=4\pi,\nonumber\\
 r&=c_0\Psi_*(\nu|_{Z_U}),\qquad
 c_0=\frac1{64(2\pi)^3},\qquad
 \Rel=\Psi(Z_U).
 \label{eq:residue}
\end{align}

For compact \(A\Subset X\), write
\[
 \DD_A(X)=\{\Phi\in C_c^\infty(X):\supp\Phi\subset A\},
 \qquad
 p_{A,N}(\Phi)=\max_{|\beta|\le N}\sup_X|\partial^\beta\Phi|.
\]
Distributional pairings are complex linear in the test.

\begin{assumption}[VII-D parent finite-part input]
\label{ass:VIID}
The prescribed family \eqref{eq:triangle} is smooth for
positive cutoff and agrees with the fully contracted old
coefficient on product tests.
There is a regulator-independent distribution \(U_m\in\DD'(X)\)
such that
\begin{equation}
 u_\mu^\rho=U_m+(\kappa_\rho+\log\mu)r,
 \label{eq:parent}
\end{equation}
where
\[
 \kappa_\rho=
 \int_0^1\frac{\vartheta_\rho(v)^4-1}{v}\,\dd v
 +\int_1^\infty\frac{\vartheta_\rho(v)^4}{v}\,\dd v.
\]
For every compact \(A\Subset X\),
\begin{equation}
 \begin{split}
 \abs{\pair{
   T_\varepsilon^\rho-\log((\mu\varepsilon)^{-1})r-u_\mu^\rho
 }{\Phi}}
 \le C_{A,\rho,m}\,
       \varepsilon(1+|\log\varepsilon|)\,p_{A,64}(\Phi),
 \quad \Phi\in\DD_A(X).
 \end{split}
 \label{eq:parentestimate}
\end{equation}
The range is \(0<\varepsilon\le1/2\), further restricted when
required by the supported smoothing margin.
The estimate includes compact sets meeting
\(\Rel\cap\Delta_{13}\), where \(\Delta_{13}=\{x=x'\}\).

For admissible regulators \(\rho_a,\rho_b\),
\begin{align}
 u_\mu^{\rho_a}-u_\mu^{\rho_b}&=c_{ab}r,
 &
 c_{ab}&=\int_0^\infty
      \frac{\vartheta_{\rho_a}(v)^4-
            \vartheta_{\rho_b}(v)^4}{v}\,\dd v\in\RR,
 \label{eq:parentconversion}\\
 c_{ab}+c_{bc}&=c_{ac},
 &
 u_{\mu_2}^\rho-u_{\mu_1}^\rho
            &=\log(\mu_2/\mu_1)r.
 \label{eq:parentscale}
\end{align}
The regulator conversion is complete within this fixed parent
class: no extra normal derivative, nonconstant weight, or
independent intersection-supported term occurs.
These are the specific VII-D inputs
\cite[{\loc{thm:finitepart}}, {\loc{thm:conversion}},
      {\loc{sc:placement}}, {\loc{prop:positive}}]{CheungVIID}.
\end{assumption}

No profile integral, ultraviolet sector estimate, reflected
intersection estimate, or leading-coefficient ledger is re-proved
by this assumption.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.24\linewidth}XX@{}}
\toprule
Input & Use & Not supplied by that input\\
\midrule
I-B host & Metric, lapse, calibration, regular routes &
A conical Green function or Einstein dynamics\\
IX-A-ST carrier & Admission, traces, classical continuity &
A sharp quantum second multiplication\\
VII-C tensor data & Edge species and reflected mixed kernel &
Vanishing of \(K\) or universal algebra nonexistence\\
VII-D parent input & Numerical finite part and conversion &
Binary factorization or arbitrary decorated-graph extension\\
Later IX records & Documentary successor gates &
Premises for the present release\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{Skeletons and the selected triangle}
\label{sec:skeletons}
%=====================================================================

A contraction graph is a finite loopless multigraph.
Its vertices are insertion symbols, its edges are
cross-contractions, and no field slot is used twice.
Decorations specify the field species, indices, and derivatives.

\begin{theorem}[Eight unlabelled connected skeletons]
\label{thm:skeletons}
Every connected loopless multigraph with at least one and at
most three edges is isomorphic to exactly one of:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item a single edge;
\item two parallel edges;
\item a two-edge path;
\item three parallel edges;
\item a two-edge path with one edge doubled;
\item a triangle;
\item a three-edge path;
\item a three-edge star.
\end{enumerate}
Arbitrary graphs with at most three edges are disjoint unions
of these with total edge count at most three, together with
isolated vertices.
\end{theorem}

\begin{proof}
One edge forces two vertices.
For two edges, connectedness permits either two vertices with
a double edge or three vertices forming a path.

For three edges, connectedness permits at most four vertices.
With two vertices there is a triple edge.
With three vertices, either all vertex pairs are joined or
exactly two pairs are joined, with multiplicities two and one.
With four vertices the graph is a tree, whose degree sequence
is either \((2,2,1,1)\) or \((3,1,1,1)\).
These are respectively a path and a star.
Distinct listed vertex counts, multiplicities, or degree
sequences distinguish the cases.
Connected-component decomposition proves the last assertion.
\end{proof}

This is the IX-B inventory \cite[{\loc{thm:skeletons}}]{CheungIXB}.
It is combinatorial and independent of the conical host.
It is not an existence theorem for all distributionally decorated
versions of those graphs.

\begin{proposition}[Unique fully contracted selected graph]
\label{prop:unique}
For the ordered word
\(B_{f,\varepsilon},W_g,B_{f',\varepsilon}\), the
field-independent order-\(\hbar^3\) coefficient has exactly one
graph type: a scalar edge from the first to the third vertex
and two metric edges, one from each outer vertex to the
middle vertex.
Its numerical density is \eqref{eq:triangle}.
The factor \(2\) is the middle-slot assignment count.
No \(ww\)-edge occurs.
\end{proposition}

\begin{proof}
Each outer insertion has one scalar and one metric-response
slot. The middle insertion has two metric slots.
Since scalar--metric covariances vanish, full contraction
forces the scalar outer slots to pair with one another.
The remaining two outer metric slots must pair with the two
middle slots, in two assignments.

Pairing the two outer metric slots instead would leave
the middle slots unmatched; that edge would also have
covariance zero by Assumption~\ref{ass:tensor}.
The locked tensor identity selects index \(0\).
The scalar derivative edge gives \(D_\varepsilon\), and
the ordered metric edges give \(K_\varepsilon\) and its
reverse conjugate.
The two defining insertion minus signs multiply to \(+1\).
The assigned placement therefore gives \eqref{eq:triangle}.
\end{proof}

There are three single-edge placements and three choices of a
two-edge placement in this word. This identifies placements,
not their sharp composition domains
\cite[{\loc{prop:unique}}]{CheungIXB}.

%=====================================================================
\section{Residue geometry and the observation-diagonal limitation}
\label{sec:geometry}
%=====================================================================

\begin{lemma}[Positive reflected observation measure]
\label{lem:measure}
The map \(\Psi\) in \eqref{eq:residue} is an embedding proper
over compact subsets of \(X\).
The distribution \(r\) is a positive locally finite measure
of order zero, with support exactly \(\Rel\).
For product tests,
\begin{equation}
 \pair r{f\otimes g\otimes f'}
 =c_0\int g(y)\,\dd^4y\int_{S^2}
              A_f(y,\mathbf n)A_{f'}(y,\mathbf n)
                                 \,\dd\Omega(\mathbf n),
 \label{eq:productresidue}
\end{equation}
where
\[
 A_f(y,\mathbf n)
  =\int_\RR f(t,\mathbf y+(t+s)\mathbf n)\,\dd t
\]
uses zero extension of \(f\) outside \(V\).
\end{lemma}

\begin{proof}
The observation times recover \(t,s,t'\). On the image,
\[
 \mathbf n=\frac{\mathbf x-\mathbf y}{t+s},
 \qquad
 \mathbf z=\mathbf y+s\mathbf n.
\]
Since \(t+s>0\), these are smooth inverse formulae.
They also prove immersion and injectivity.

On the inverse image of a compact subset of \(X\), all times
are bounded and separated from zero, \(\mathbf z\) is bounded,
and \(\mathbf n\) lies in the compact sphere.
The inverse image is closed in the resulting compact
parameter region, proving properness there.

The parameter density is positive, so its pushforward is a
positive locally finite measure. Every neighbourhood of an
image point has inverse image of positive parameter measure;
hence its support is the whole image.
Changing \(\mathbf z\) to
\(\mathbf y=\mathbf z-s\mathbf n\) in the defining integral
gives \eqref{eq:productresidue}.
\end{proof}

\begin{proposition}[Relation, equality diagonals, and density]
\label{prop:relation}
The reflected relation has dimension eight and codimension
four in \(X\). It is not an observation equality diagonal.
Moreover,
\[
 x-x'=(t-t')(1,\mathbf n)\quad\text{on }\Rel,
\]
and
\[
 \Rel\cap\Delta_{12}=\Rel\cap\Delta_{23}=\varnothing,
 \qquad
 \dim(\Rel\cap\Delta_{13})=7
\]
where that intersection is present.

Near a point of \(\Rel\), set
\[
 a=t+s,\quad b=t'+s,\quad
 \mathbf n=\frac{\mathbf x-\mathbf y}{|\mathbf x-\mathbf y|},
 \quad \sigma=|\mathbf x-\mathbf y|-a,
\]
\[
 \mathbf w=\mathbf x'-\mathbf y
                   -\frac ba(\mathbf x-\mathbf y),
 \qquad \mathbf z=\mathbf y+s\mathbf n.
\]
Then
\begin{equation}
 \dd X=(a+\sigma)^2
   \dd^3z\,\dd\Omega\,\dd t\,\dd s\,\dd t'\,
                 \dd\sigma\,\dd^3w,
 \qquad
 r=c_0a^{-2}\delta(\sigma)\delta^{(3)}(\mathbf w).
 \label{eq:density}
\end{equation}
In particular, no \(\delta(t-t')\) occurs.
\end{proposition}

\begin{proof}
The parameter space has dimension
\(3+2+3=8\).
The differences
\[
 \mathbf x-\mathbf y=(t+s)\mathbf n,\qquad
 \mathbf x'-\mathbf y=(t'+s)\mathbf n
\]
prove all diagonal statements. Equality \(x=x'\) is exactly
the one additional relation \(t=t'\), when restricted to
\(\Rel\).

For the density, the spatial change to
\((\mathbf y,\mathbf x-\mathbf y,\mathbf w)\), at fixed times,
has determinant of absolute value one.
Write \(\mathbf x-\mathbf y=(a+\sigma)\mathbf n\), producing
the polar factor \((a+\sigma)^2\).
The replacement of \(\mathbf y\) by \(\mathbf z\) is triangular
with unit determinant. The time block is unchanged.
The inverse coordinate formulae are smooth near the relation.
Restricting the density to \(\sigma=\mathbf w=0\) and comparing
with \(\dd\nu\) gives the coefficient \(c_0a^{-2}\).
\end{proof}

The density and its normalization agree with the supplied
VII-D comparison \cite[{\loc{def:residue}}]{CheungVIID}.
The positivity argument is made in parameter measure, not by
assigning a sign to a formal delta expression.

\begin{theorem}[Local observation-diagonal limitation]
\label{thm:diagonal}
There exist nonnegative \(f,g,f'\in\DD(V;\RR)\) with pairwise
disjoint supports such that
\[
 \mathcal C(f,g,f')
     :=\pair r{f\otimes g\otimes f'}>0.
\]
For those tests,
\begin{equation}
 \pair{T_\varepsilon^\rho}{f\otimes g\otimes f'}
       =\mathcal C(f,g,f')\log(\varepsilon^{-1})+O(1).
 \label{eq:log}
\end{equation}
No family of counterterms supported on
\(\Delta_{12}\cup\Delta_{23}\cup\Delta_{13}\) removes this
divergence on all joint tests.
The same conclusion holds for observation faces avoided by
the chosen compact supports.
\end{theorem}

\begin{proof}
Choose
\[
 0<\lambda<\min\{T_0/4,r_0/5\}
\]
and one unit vector \(\mathbf n_0\).
The relation contains the triple
\[
 y_0=(\lambda,0),\qquad
 x_0=(2\lambda,3\lambda\mathbf n_0),\qquad
 x'_0=(3\lambda,4\lambda\mathbf n_0).
\]
All three points lie in \(V\) and are distinct.
Take nonnegative bumps with mutually disjoint sufficiently
small supports, positive near these points.
Their product is positive on an open parameter neighbourhood
of this triple. Lemma~\ref{lem:measure} gives the positive
residue pairing.

Assumption~\ref{ass:VIID} gives the logarithmic term plus a
finite distributional pairing and a vanishing remainder.
The contribution \(-\log\mu\,\mathcal C\) is constant at fixed
\(\mu\), proving \eqref{eq:log}.

The product-test support avoids every observation equality
diagonal. Every distribution supported on their union pairs
to zero with it, even when the distribution depends on
\(\varepsilon\). Such counterterms cannot cancel the nonzero
logarithm.
\end{proof}

This retains the VII-C separated-test obstruction locally
\cite[{\loc{thm:log}}, {\loc{lem:separated}}]{CheungVIIC}.
It does not exclude the supplied subtraction on \(\Rel\), and
it is not a theorem about the conical worldsheet or a curvature
singularity.

%=====================================================================
\section{Observation derivatives and smooth spectators}
\label{sec:descendants}
%=====================================================================

Let
\[
 D=\sum_{|\beta|\le d}b_\beta\,\partial^\beta
\]
be a fixed differential operator on \(X\), with smooth
coefficients, and let \(a\in C^\infty(X)\).
The ordered operation is
\[
 \mathcal O_{a,D}U=aDU.
\]
Its transpose, relative to the displayed coordinate density, is
\[
 D^{\mathrm t}\Phi
   =\sum_{|\beta|\le d}(-\partial)^\beta(b_\beta\Phi),
 \qquad
 \pair{aDU}{\Phi}=\pair U{D^{\mathrm t}(a\Phi)}.
\]
There is no coefficient conjugation in this transpose.

\begin{lemma}[Explicit test-order bound]
\label{lem:leibniz}
For \(\Phi\in\DD_A(X)\),
\begin{equation}
 p_{A,64}\bigl(D^{\mathrm t}(a\Phi)\bigr)
 \le C_d
     \left(\sum_{|\beta|\le d}\norm{b_\beta}_{C^{64+d}(A)}\right)
       \norm a_{C^{64+d}(A)}
          p_{A,64+d}(\Phi),
 \label{eq:leibniz}
\end{equation}
with a combinatorial constant depending only on dimension and
\(d\). Differential operators and multiplication do not enlarge
the test support.
\end{lemma}

\begin{proof}
Apply every derivative of order at most \(64\) to the finite
sum defining \(D^{\mathrm t}(a\Phi)\), then use the product
rule on \(b_\beta a\Phi\).
The total number of derivatives distributed among the three
factors is at most \(64+|\beta|\le64+d\).
Taking suprema and summing the finitely many multinomial
coefficients gives \eqref{eq:leibniz}.
\end{proof}

\begin{theorem}[Derivative--spectator finite part]
\label{thm:descendants}
Under Assumption~\ref{ass:VIID},
\begin{equation}
 aDT_\varepsilon^\rho
  -\log((\mu\varepsilon)^{-1})aDr
       \longrightarrow aDu_\mu^\rho
           \quad\text{strongly in }\DD'(X).
 \label{eq:descendantlimit}
\end{equation}
More precisely,
\begin{equation}
 \begin{split}
 &\abs{\pair{
 aD\bigl(T_\varepsilon^\rho-
       \log((\mu\varepsilon)^{-1})r-u_\mu^\rho\bigr)
 }{\Phi}}\\
 &\qquad\le
 C_{A,\rho,m}\,C_{A,a,D}\,
 \varepsilon(1+|\log\varepsilon|)\,
                         p_{A,64+d}(\Phi),
 \end{split}
 \label{eq:descendantrate}
\end{equation}
where \(C_{A,a,D}\) can be chosen as the coefficient expression
in \eqref{eq:leibniz}.
The estimate is uniform on families with those coefficient
seminorms bounded. It includes compact sets crossing
\(\Rel\cap\Delta_{13}\).
\end{theorem}

\begin{proof}
Apply \eqref{eq:parentestimate} to
\(D^{\mathrm t}(a\Phi)\), whose support remains in \(A\),
and use Lemma~\ref{lem:leibniz}.
A bounded subset of \(\DD(X)\) has a common compact support
and bounded smooth seminorms of every fixed order. Thus the
right side tends to zero uniformly on each bounded test set,
which is the asserted strong convergence.

The parent estimate already holds at the reflected-diagonal
intersection. Applying a support-preserving transpose does
not remove that intersection or introduce a new exclusion.
No new ultraviolet argument is required.
\end{proof}

This is the local transport of the IX-B derivative theorem
\cite[{\loc{thm:descendants}}]{CheungIXB}, not a re-proof of
VII-D's ultraviolet analysis.

\begin{corollary}[Descendant conversion and scale dependence]
\label{cor:conversion}
For the same operation,
\begin{align}
 aD(u_\mu^{\rho_a}-u_\mu^{\rho_b})&=c_{ab}aDr,\nonumber\\
 aD(u_{\mu_2}^\rho-u_{\mu_1}^\rho)
       &=\log(\mu_2/\mu_1)aDr,\label{eq:descconversion}\\
 aD(T_\varepsilon^{\rho_a}-T_\varepsilon^{\rho_b})
       &\longrightarrow c_{ab}aDr
                      \quad\text{strongly}.\nonumber
\end{align}
The last convergence has the summed order-\(64+d\) remainder
bound of the two parent estimates.
\end{corollary}

\begin{proof}
The first two identities follow by applying the linear
operation \(aD\) to
\eqref{eq:parentconversion}--\eqref{eq:parentscale}.
Subtracting the two instances of
\eqref{eq:descendantrate} proves the last assertion, since
the leading logarithmic terms agree.
\end{proof}

The logarithm in \eqref{eq:descconversion} is a numerical
conversion on supplied chart distributions. It is not a
running of the host curvature.

\begin{corollary}[Smooth spectator rule]
\label{cor:spectators}
Suppose that, after fixing smooth configurations
\(\Psi=(h,\phi)\), a selected numerical coefficient is exactly
\[
 a[\Psi]\,DT_\varepsilon^\rho
\]
with \(a[\Psi]\in C^\infty(X)\).
If \(\Psi\mapsto a[\Psi]\) is continuous in the compact smooth
seminorms through order \(64+d\), then its numerical finite
part is
\[
 a[\Psi]\,Du_\mu^\rho.
\]
Convergence is uniform on configuration sets for which those
multiplier seminorms are bounded.
\end{corollary}

\begin{proof}
Insert \(a=a[\Psi]\) in
Theorem~\ref{thm:descendants} and use its uniform coefficient
bound.
\end{proof}

Finite products of smooth field jets satisfy the multiplier
condition under the corresponding compact smooth bounds.
Smooth Cauchy responses may be used when their required
smooth seminorm bounds are supplied.
This rule does not multiply by an additional singular
spectator distribution, classify arbitrary decorated graphs,
or prove carrier admission of every resulting functional.

\subsection{Three derivative operations}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.23\linewidth}XX@{}}
\toprule
Operation & Established statement & Not inferred\\
\midrule
Observation derivative &
\(D\) acts on the joint numerical distribution.
For example,
\[
 \pair{(-\partial_t)^nT_\varepsilon^\rho}
       {f\otimes g\otimes f'}
 =
 \pair{T_\varepsilon^\rho}
       {(\partial_t^nf)\otimes g\otimes f'}.
\]
&
A quantum product on carrier elements.\\
\addlinespace
Configuration derivative &
\(\Ddir_\eta F\) is differentiation of an actual polynomial
along a fixed smooth field direction; its carrier closure is
inherited. &
Differentiability of an unconstructed sharp triple output.\\
\addlinespace
Functional-slot derivative &
Unsmeared functional derivatives expose field slots for later
contractions. &
Existence of the resulting decorated distributional graphs.\\
\bottomrule
\end{tabularx}
\end{center}

An observation derivative such as \(\partial_t\) uses the time
coordinate of the developing chart. Thus its coordinate vector
is related to \(\partial_T\) by the fixed calibration.
Nevertheless, differentiating a kernel argument is not evolving
the host metric, implementing a clock operation, or constructing
quantum evolution without external scheduling.
This is the precise meaning of the time-axis separation.

%=====================================================================
\section{Parent conversion, derivative orbit, and weighted classes}
\label{sec:orbits}
%=====================================================================

\begin{theorem}[Infinite observation-derivative orbit]
\label{thm:independence}
On the supported observation space \(X\), the distributions
\[
 r_n=(-\partial_t)^nr,\qquad n=0,1,2,\ldots,
\]
are linearly independent over \(\CC\).
No finite-dimensional subspace containing \(r\) is invariant
under \(-\partial_t\).
\end{theorem}

\begin{proof}
Let
\[
 F(x,y,x')=|\mathbf x-\mathbf y|^2-(t+s)^2.
\]
On \(\Rel\),
\[
 F=0,\qquad \partial_tF=-2(t+s)\ne0.
\]
The measure \(r\) is nonzero by
Theorem~\ref{thm:diagonal}.

Suppose a finite relation
\(\sum_{j=0}^Nc_jr_j=0\) has \(c_N\ne0\).
For \(N\ge1\), pair the relation with \(F^N\Phi\).
When \(j<N\), every term in
\(\partial_t^j(F^N\Phi)\) still contains a factor \(F\)
and vanishes on \(\Rel\).
At order \(N\), its restriction is
\[
 N!(\partial_tF)^N\Phi|_\Rel.
\]
The relation therefore gives
\[
 c_NN!(\partial_tF)^Nr=0.
\]
A nonnegative compact test positive near a point of \(\Rel\)
detects this nonzero measure, since
\((\partial_tF)^N\) has constant nonzero sign there.
This is a contradiction. The \(N=0\) case follows from
\(r\ne0\).

An invariant finite-dimensional space containing \(r\) would
contain every \(r_n\), contradicting independence.
\end{proof}

This carries the IX-B orbit statement to the regular chart
\cite[{\loc{thm:independence}}]{CheungIXB}.
It is not a renormalization-group trajectory of the host.

\begin{proposition}[One compatible parent parameter]
\label{prop:oneparameter}
Fix \(\rho,\mu\) and define
\[
 U_c=u_\mu^\rho+cr,\qquad c\in\RR.
\]
If observation compatibility is imposed by the actual operation
\(aD\), the compatible descendant family is
\[
 aDU_c=aDu_\mu^\rho+c\,aDr,
\]
with the same \(c\) for every declared derivative and multiplier.
An infinite-dimensional residue orbit therefore does not imply
one independent normalization constant per derivative.
\end{proposition}

\begin{proof}
This is the linear image under \(aD\) of the one-parameter
affine parent family.
Independently choosing a new constant for a derivative would
change, rather than follow from, the stipulated compatibility
identity.
\end{proof}

\begin{proposition}[Unrestricted tangential weights]
\label{prop:weights}
If arbitrary real smooth weights were admitted, the space
\[
 \{ar:a\in C^\infty(X;\RR)\}
\]
would be infinite-dimensional, even among first--third
exchange-symmetric weights.
This proposition does not declare those weights admissible
counterterms.
\end{proposition}

\begin{proof}
Consider \(a_n=(t+t')^n\), which are exchange symmetric.
If \(P(t+t')r=0\) for a polynomial \(P\), positivity of the
parameter measure implies that \(P(t+t')\) vanishes throughout
every open parameter patch of \(Z_U\).
There is such a patch by the explicit interior witness above,
and \(t+t'\) varies over an open interval on that patch.
A polynomial vanishing on an open interval is identically
zero. Thus the weighted distributions are linearly independent.
\end{proof}

The class \(\RR r\) is the complete parent \emph{regulator
conversion class} in Assumption~\ref{ass:VIID}.
It is not the span of all derivatives and is not a classification
of arbitrary weighted or decorated extensions.

In the normal coordinates of \eqref{eq:density}, \(r\) has
transverse scaling degree four. A nonzero pure normal derivative
of order \(k\) has degree \(4+k\).
This follows from the dilation of a delta distribution in four
normal variables.
It is not the test-order loss \(64+d\), and it does not restrict
arbitrary tangential coefficient functions to a finite-dimensional
space.

%=====================================================================
\section{Regularized association and the missing sharp operation}
\label{sec:association}
%=====================================================================

\begin{theorem}[Equality of regularized parenthesizations]
\label{thm:regularassociation}
For every positive cutoff allowed by the supported prescription,
the field-independent order-\(\hbar^3\) coefficients of
\[
 (B_{f,\varepsilon}\star_HW_g)\star_HB_{f',\varepsilon},
 \qquad
 B_{f,\varepsilon}\star_H(W_g\star_HB_{f',\varepsilon})
\]
are equal and have common value
\[
 \mathcal G_\varepsilon(f,g,f')
   =\pair{T_\varepsilon^\rho}{f\otimes g\otimes f'}.
\]
After the single common subtraction their numerical limit is
\[
 \pair{u_\mu^\rho}{f\otimes g\otimes f'}.
\]
The corresponding statement on arbitrary compact joint tests is
the strong distributional limit in
Assumption~\ref{ass:VIID}.
\end{theorem}

\begin{proof}
Use independent fields at the three vertices.
The cross-contraction operators
\(\mathcal P_{12},\mathcal P_{13},\mathcal P_{23}\) have
field-independent coefficients and commute on the
positive-cutoff old-domain expressions.
Both parenthesizations consequently give the finite polynomial
expansion of
\[
 \exp\!\bigl(\hbar(\mathcal P_{12}
                  +\mathcal P_{13}+\mathcal P_{23})\bigr)
\]
followed by field identification.
The old products in this calculation are defined under
Assumption~\ref{ass:tensor}.
Proposition~\ref{prop:unique} identifies their scalar
order-three graph and multiplicity.
The asserted limit is the parent input.
Product-test density identifies the same joint numerical
distribution; no independent graph is assigned to one
parenthesization.
\end{proof}

The theorem carries the IX-B regularized identity
\cite[{\loc{thm:regularassociation}}]{CheungIXB}.
It concerns a schedule of old-domain evaluations, not a
physical causal past.

\begin{remark}[What numerical subtraction changes]
\label{rem:subtraction}
Subtracting
\(\log((\mu\varepsilon)^{-1})r\) from a positive-cutoff
coefficient changes a coefficient of an already fixed old
product.
It is a numerical comparison prescription, not an unchanged
old Wick product.

Conversely, assigning only a finite value at the sharp
kernel cannot make the unchanged divergent regular sequence
sequentially continuous. The finite part does not remove
VII-C's incompatibility \cite{CheungVIIC,CheungVIID}.
\end{remark}

\begin{proposition}[The admitted partial coefficient]
\label{prop:partial}
The selected one-contraction output for the ordered pair
\((B_f,W_g)\) is
\begin{equation}
 Q_{f,g}[h,\phi]
   =-2\int f(x)g(y)K(x,y)
               \partial_t\phi(x)h_{00}(y)\,\dd^4x\,\dd^4y.
 \label{eq:Qpartial}
\end{equation}
It is an actual compact-distribution bilinear functional and
belongs to \(\Padm\).
Thus
\[
 \mathsf P_L(f,g)=B_fW_g+\hbar Q_{f,g}
\]
is a well-defined specified first-stage output.
This does not define a quantum product on the full carrier.
\end{proposition}

\begin{proof}
Existence and full carrier admission of the selected
contraction output are the input in
Assumption~\ref{ass:carrier}, inherited from IX-A-ST.
Multiplying by fixed smooth compact tests preserves its
compact coefficient support.

For its normalization, differentiation of \(W_g\) supplies
the factor \(2\).
The tensor identity in Assumption~\ref{ass:tensor} selects
\(K(x,y)\partial_t\phi(x)\).
There is one defining minus sign from \(B_f\), giving \(-2\).
Classical closure supplies \(B_fW_g\), so the stated formal
first-stage polynomial is well defined.
\end{proof}

The normalization agrees with IX-B
\cite[{\loc{prop:partial}}, {\loc{eq:Qpartial}}]{CheungIXB}.
The positive assignment factor of the full triangle and the
negative first-stage coefficient are consistent: the full
triangle has two insertion minus signs.

\begin{proposition}[Missing sharp second multiplication]
\label{prop:missing}
The parent assignment \(U_c\), the admitted \(Q_{f,g}\), and
Theorem~\ref{thm:regularassociation} do not define either
\[
 (B_f\star_\Sigma W_g)\star_\Sigma B_{f'},
 \qquad
 B_f\star_\Sigma(W_g\star_\Sigma B_{f'}),
\]
where \(\star_\Sigma\) is only a placeholder for a further
construction.
In particular, the trilinear numerical assignment
\begin{equation}
 \mathcal N_c(f,g,f')
    =\pair{u_\mu^\rho+cr}{f\otimes g\otimes f'}
 \label{eq:Nc}
\end{equation}
is not by itself a binary map on actual intermediate
carrier elements.
\end{proposition}

\begin{proof}
The arguments of \(U_c\) are observation tests.
The arguments of the missing second-stage map would instead
include actual functionals such as \(Q_{f,g}\) and \(B_fW_g\).
The parent distribution gives no value on arbitrary such
intermediate elements or their finite relations.

A binary factorization must prove descent from all test
presentations to actual functionals, admission of all
nonconstant outputs and their directional descendants,
compatible overlap with old products, and the required
coherence identities.
None of these maps is defined by the scalar pairing
\eqref{eq:Nc}. Equality of two regularized scalar limits
therefore cannot supply either missing composition.
\end{proof}

This is a boundary of the present construction, not a theorem
excluding every possible binary factorization.
The next IX-C interface asks for both locked first stages,
actual-functional descent, and coefficientwise associativity
conditions. No result from a later IX sheet is used to pass
that interface here \cite{CheungSuccessors}.

\subsection{Numerical involution}

Let \(\tau(x,y,x')=(x',y,x)\), and define
\[
 \pair{U^\dagger}{\Phi}
     =\overline{\pair U{\overline\Phi\circ\tau}}.
\]
The explicit formulae give
\[
 (T_\varepsilon^\rho)^\dagger=T_\varepsilon^\rho,
 \qquad r^\dagger=r.
\]
Indeed \(D_\varepsilon(-z)=\overline{D_\varepsilon(z)}\),
and exchange swaps the two mixed factors.
The strong finite-part limit then gives
\[
 (u_\mu^\rho)^\dagger=u_\mu^\rho.
\]
For \(\mathcal O=aD\), put
\[
 \mathcal O^\dagger U
      =\bigl(\mathcal O(U^\dagger)\bigr)^\dagger.
\]
Then
\[
 (\mathcal O u_\mu^\rho)^\dagger
                      =\mathcal O^\dagger u_\mu^\rho.
\]
An \(x^0\)-derivative is exchanged with an \(x'^0\)-derivative;
a descendant need not itself be exchange invariant.
These are distributional symmetry identities, not a quantum
algebra involution theorem.

%=====================================================================
\section{Three continuity statements and the retained obstruction}
\label{sec:continuity}
%=====================================================================

The topologies and operations in the following table are
different.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.27\linewidth}X@{}}
\toprule
Setting & Statement\\
\midrule
VII-C normal mixed-kernel stages &
The unchanged kernels converge normally, while the unchanged
scalar coefficient diverges on the separated witness.\\
\addlinespace
IX-A-ST admission topology \(\taus\) &
The unchanged regular approximants do not converge to a
nonzero real sharp insertion. Separate classical continuity
does not give quantum associativity.\\
\addlinespace
Numerically subtracted joint-test distributions &
The changed family
\(T_\varepsilon^\rho-\log((\mu\varepsilon)^{-1})r\)
converges strongly. The same is true after \(aD\) with the
proved loss \(64+d\).\\
\bottomrule
\end{tabularx}
\end{center}

\begin{proposition}[Persistence of the normal-topology obstruction]
\label{prop:obstruction}
There is no sequentially continuous extension of the
unchanged regular scalar coefficient to the sharp pair in
the normal kernel stages of Assumption~\ref{ass:tensor},
agreeing with that coefficient on regular kernels.

Consequently the VII-C algebra-extension requirements cannot
all hold simultaneously: unchanged old-product agreement,
exact sharp admission with the specified normal admission
continuity, bounded-set hypocontinuous quantum multiplication,
and continuous scalar coefficient extraction.
\end{proposition}

\begin{proof}
Choose the separated witness in
Theorem~\ref{thm:diagonal}.
Normal convergence of the two insertion kernels is part of
the inherited VII-C input.
Sequential continuity of a scalar extension would make its
values on that convergent pair converge.
Equation~\eqref{eq:log} contradicts this.

For the algebra consequence, hypocontinuity gives separate
continuity and continuity along convergent pairs of sequences:
split a product difference into two terms and use uniformity
over the bounded convergent family in the other factor.
Apply this first with the fixed middle factor, then with the
other convergent insertion.
Continuous scalar extraction would force convergence of the
same divergent coefficient.
\end{proof}

The proposition retains the original quantifiers of VII-C
\cite[{\loc{ass:extension}}, {\loc{cor:noalgebra}}]{CheungVIIC}.
It is not a universal obstruction to all carrier topologies or
all partial products. No global comparison between every
normal topology and \(\taus\) is asserted.
The distinction between separate continuity, hypocontinuity,
and other distribution topologies matters
\cite{BDH}.

%=====================================================================
\section{Finite numerical certificates and a destructive clock}
\label{sec:finite}
%=====================================================================

\subsection{A quantitative enclosure interface}

Fix one operation \(aD\), one compact test \(\Phi\), and one
catalogue choice \((\rho,\mu)\). Set
\[
 z=\pair{aDu_\mu^\rho}{\Phi},\qquad
 A_\varepsilon=\pair{aDT_\varepsilon^\rho}{\Phi},\qquad
 b=\pair{aDr}{\Phi},
 \qquad L_\varepsilon=\log((\mu\varepsilon)^{-1}).
\]

\begin{theorem}[Finite enclosure from the analytic estimate]
\label{thm:enclosure}
Suppose supplied numerical approximants
\(\widehat A_\varepsilon,\widehat b\) obey
\[
 |\widehat A_\varepsilon-A_\varepsilon|\le e_T,
 \qquad |\widehat b-b|\le e_r.
\]
Then
\begin{equation}
 \left|\widehat A_\varepsilon-L_\varepsilon\widehat b-z\right|
 \le
 E_\varepsilon
 :=e_T+|L_\varepsilon|e_r
   +C_{A,\rho,m}C_{A,a,D}
       \varepsilon(1+|\log\varepsilon|)
                         p_{A,64+d}(\Phi).
 \label{eq:certificateerror}
\end{equation}
For \(\varepsilon=2^{-n}\), the analytic term is bounded by
the same constant times
\(2^{-n}(1+n\log2)p_{A,64+d}(\Phi)\).
It tends to zero.
\end{theorem}

\begin{proof}
Subtract \(z\), insert the two exact pairings, apply the
triangle inequality, and use
Theorem~\ref{thm:descendants}.
The dyadic statement is substitution.
\end{proof}

The theorem is an error-propagation statement, not an executed
quadrature or a general computability theorem for arbitrary
smooth inputs. The constants and numerical enclosures used
by an actual finite certificate must be supplied and justified.

\subsection{Installed finite catalogue}

Choose a finite catalogue with classical index \(i\in\mathcal I\).
An entry records:
\begin{itemize}
\item the test and source-description identifiers;
\item the regulator \(\rho\), scale \(\mu\), and cutoff;
\item the observation operator \(D\) and smooth weight \(a\);
\item the selected signs and slot multiplicities;
\item supplied pairings or certified enclosures and the
remainder bound in \eqref{eq:certificateerror}.
\end{itemize}
Only finitely many choices occur in one experiment.
An arbitrary smooth function or real parameter is not claimed
to have been acquired or exactly encoded by finitely many bits.

A finite report word \(r(k,i)\) stores the installed data selected
by a clock code \(k\).
It may include symbolic identifiers and rational interval
endpoints. Its meaning is a certificate about supplied
comparison data, not an observation of an unknown field or
of the residue itself.

\subsection{Complete destructive clock}

Let \(C\) have dimension \(2^m\), with orthonormal calibration
basis \(\{\ket{\tau_k}\}\). Let \(S\) contain all surviving
factors and let \(E\) be any untouched reference. Put
\[
 P_k=\proj{\tau_k},\qquad
 D_k=\bra{\tau_k}_C\otimes I_S.
\]

\begin{theorem}[One destructive clock instrument]
\label{thm:instrument}
The calibrated clock readout is
\begin{equation}
 \mathcal M_C(X)
   =\sum_k\proj k_R\otimes D_kXD_k^\dagger
   =\sum_k\proj k_R\otimes
       \Tr_C[(P_k\otimes I_S)X(P_k\otimes I_S)].
 \label{eq:clock}
\end{equation}
It is CPTP, including after tensoring with \(\id_E\), and
\[
 \Tr_R\mathcal M_C(X)=\Tr_CX.
\]
It returns subnormalized conditional blocks and consumes the
quantum clock names. The geometric coordinates and metric
are not consumed.
\end{theorem}

\begin{proof}
The recorded Kraus operators are \(\ket k_R\otimes D_k\).
Their squared moduli sum to \(I_{CS}\).
Expanding \(X\) in the calibration basis proves the
partial-trace expression and the record-discard identity.
Tensoring these identities with \(I_E\) proves the complete
statements.
The declared destructive output interface has no quantum
\(C\)-factor, whereas the host is fixed metadata.
\end{proof}

This is the I-B instrument interface, not a field measurement
\cite[{\loc{thm:instrument}}]{CheungIB}.
The complete conditioning distinction is also documented in
En09-A and the spacetime observation sheets
\cite{CheungEn09,STInterfaces}.

\begin{proposition}[Calibration limits]
\label{prop:calibration}
If a fixed readout event \(e\) has distinct labels \(t_k\)
satisfying
\[
 p_k>0\Longrightarrow t_k=t(e),
\]
at most one label has positive probability.
Nontrivial multi-outcome programme or history readout therefore
does not assert that all labels equal the time of one event.

The clock channel has no universal left inverse on its full
input space.
\end{proposition}

\begin{proof}
Distinct labels prove the first assertion.
For the second, the two clock states
\((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), tensored with the
same payload, give identical destructive outputs.
A universal left inverse could not recover both.
\end{proof}

We impose no stationarity or covariance condition for an
unscheduled relational evolution interpretation.
External scheduling remains.

\subsection{A fixed-interface evaluation channel}

Let \(G\) be the catalogue register, padded to \(2^g\) labels.
Its intended inputs are classical in the basis \(\{\ket i\}\).
The canonical full-matrix extension pinches that basis.
Let \(P\) be an arbitrary quantum payload.

For each \((k,i)\), choose an installed real score
\(v_{ki}\in[-1,1]\), and put
\[
 p_y(k,i)=\frac{1+yv_{ki}}2,\qquad y=\pm1,
\]
\[
 X_{ki}
 =(\bra{\tau_k}_C\bra i_G\otimes I_P)
        X
   (\ket{\tau_k}_C\ket i_G\otimes I_P).
\]
Define
\begin{equation}
 \mathcal I_v(X)=
 \sum_{k,i,y}\proj{k,i,y,r(k,i)}_W
                       \otimes p_y(k,i)X_{ki}.
 \label{eq:evaluation}
\end{equation}

\begin{proposition}[Complete table instrument]
\label{prop:table}
The map \eqref{eq:evaluation} is CPTP and has conditional
sign mean \(v_{ki}\).
It preserves the payload in each subnormalized branch, including
its correlations with an untouched reference.
A finite controlled probe preparation, followed by destructive
probe readout, realizes this instrument under specified finite
unitary permissions.
\end{proposition}

\begin{proof}
Its Kraus operators are
\[
 \sqrt{p_y(k,i)}\,
   \ket{k,i,y,r(k,i)}_W
        (\bra{\tau_k}_C\bra i_G)\otimes I_P.
\]
Their squared moduli sum to identity.
The reference statement is the same operator identity
tensored with the reference identity.

Prepare the probe vector
\(\sqrt{p_+(k,i)}\ket+
+\sqrt{p_-(k,i)}\ket-\), controlled by the selected classical
indices, and read the probe.
The difference of its outcome probabilities is \(v_{ki}\).
Every finite normalized probe preparation extends to a finite
unitary with a declared blank input.
\end{proof}

\begin{theorem}[Exact fixed-table channel distance]
\label{thm:fixedtable}
Keep the same pinching, history labels, payload interface, and
report words, and change only \(v_{ki}\) to
\(\widetilde v_{ki}\in[-1,1]\).
Then
\begin{equation}
 \ddia(\mathcal I_v,\mathcal I_{\widetilde v})
   =\frac12\max_{k,i}|v_{ki}-\widetilde v_{ki}|,
 \qquad
 \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond.
 \label{eq:exactdistance}
\end{equation}
\end{theorem}

\begin{proof}
Include an arbitrary reference and apply both maps to an
arbitrary operator \(X\).
The output \((k,i,y)\)-flags are orthogonal.
For a fixed input block, the sum of the absolute differences
of the two sign probabilities is
\(|v_{ki}-\widetilde v_{ki}|\).
Hence
\[
 \norm{((\mathcal I_v-\mathcal I_{\widetilde v})
                         \otimes\id)(X)}_1
 \le
 \max_{ki}|v_{ki}-\widetilde v_{ki}|
                        \sum_{ki}\norm{X_{ki}}_1.
\]
The sum is the trace norm of the pinched input and is at most
\(\norm X_1\). Pinching is an average of unitary conjugations,
so this inequality applies to arbitrary operators and
references.
This proves the diamond upper bound.
A basis input supported on a maximizing \((k,i)\), with any
normalized payload, attains it.
\end{proof}

If a table value is the normalized real part of a supplied
finite-part pairing, choose \(M_{ki}>0\) and set
\[
 v_{ki}=\clip(\operatorname{Re}z_{ki}/M_{ki}),\qquad
 \clip(u)=\max\{-1,\min\{1,u\}\}.
\]
For approximants \(\widehat z_{ki}\) with
\(|z_{ki}-\widehat z_{ki}|\le E_{ki}\), clipping is
one-Lipschitz. Therefore
\begin{equation}
 \ddia(\mathcal I_v,\mathcal I_{\widehat v})
       \le\min\left\{1,\max_{ki}\frac{E_{ki}}{2M_{ki}}\right\}.
 \label{eq:enclosurechannel}
\end{equation}
One may use \eqref{eq:certificateerror} for \(E_{ki}\), adding
any supplied finite rounding allowance.

This is a complete-channel comparison for a fixed installed
interface. It does not measure \(z_{ki}\), approximate an
unknown field, or turn the scalar estimate into a norm estimate
for a quantum field product.
Changing a report to a different orthogonal label is not, merely
because its interpreted number is close, a small diamond-norm
change. Standard channel norms and their reference extensions
are treated in \cite{Watrous}.

%=====================================================================
\section{Residual records do not determine residual sources}
\label{sec:residual}
%=====================================================================

Let \(\{\mathcal A_w\}\) and \(\{\mathcal B_w\}\) be finite
instruments with the same input and residual output spaces.
Their record effects are
\(\mathcal A_w^*(I)\) and \(\mathcal B_w^*(I)\).

\begin{proposition}[Record versus residual criterion]
\label{prop:residualcriterion}
The two instruments have equal record laws on every input
exactly when
\[
 \mathcal A_w^*(I)=\mathcal B_w^*(I)\quad\text{for every }w.
\]
Their recorded residual expectations agree on a declared
algebra \(\mathcal O\) exactly when
\[
 \mathcal A_w^*(B)=\mathcal B_w^*(B)
         \quad(w,\ B\in\mathcal O).
\]
The corresponding map identities remain valid under untouched
reference extension.
\end{proposition}

\begin{proof}
Pair an output record block first with \(I\), then with
\(B\), and pull the effect back to the input.
Equality against every density matrix is equality of the
Hermitian effects; complex linearity gives the full identities.
Reference extension tensors the resulting linear-map equalities
with identity.
\end{proof}

\begin{proposition}[A residual separation with identical records]
\label{prop:residualgap}
Equal calibration records need not determine residual quantum
states or residual source expectations.
There is a qubit example with residual trace-distance gap
\(1/2\).
\end{proposition}

\begin{proof}
Compare the identity channel and computational dephasing, each
with a single fixed reported outcome.
Both record effects are \(I\).
On \(\proj+\), their residuals are \(\proj+\) and \(I/2\).
Their difference is \(X/2\), with eigenvalues \(1/2,-1/2\).
The trace distance is \(1/2\), and the residual observable
\(X\) has expectations one and zero.
Tensor the example with any unchanged calibration table to
obtain identical calibration records as well.
\end{proof}

This is the record/residual distinction in the VII and VIII
observation interfaces \cite{CheungObservation,STInterfaces}.

\begin{proposition}[Finite linear observation records do not fix the source]
\label{prop:classicalsource}
Let finitely many linear observation-test records of smooth
compact metric configurations be prescribed.
There are two configurations with identical such records but
different source functionals of \(j=\Box h\).
In particular, an installed calibration table without a
field-acquisition map cannot determine that source.
\end{proposition}

\begin{proof}
If there are \(N\) records, choose \(N+1\) nonzero scalar
bumps \(u_j\) with mutually disjoint compact supports in \(V\).
Use configurations with only \(h_{11}\) nonzero and in the
span of those bumps. The record map from this span to
\(\CC^N\) has a nonzero kernel; let \(h_{11}=u\ne0\) be in it.

A nonzero compact smooth solution of \(\Box u=0\) cannot
exist with zero data before its time support, by local
uniqueness. Thus \(\Box u\ne0\).
The zero configuration and this configuration have the same
records, but the source functional with scalar test
\(b=\overline{\Box u}\) has difference
\[
 \int_V|\Box u|^2\,\dd^4x>0.
\]
Real configurations and real tests may be chosen when the
original records are real linear, by using the corresponding
real finite-dimensional argument.
\end{proof}

This proposition concerns finite linear records, not an
unrestricted statement about arbitrary nonlinear infinite
observation families.
No source-acquisition map is included in the table channel.

%=====================================================================
\section{Scheduled locality and buffered reference-preserving transport}
\label{sec:transport}
%=====================================================================

All programme selectors, record wires, table memories, encoders,
decoders, and controls actually read by an operation are part of
its declared interface. Selective normalized branches are not
substituted for complete channels.

\subsection{Computational support}

For a finite set of sites \(Y\subseteq\mathcal X\), write
\[
 \HH_{\mathcal X}=\bigotimes_{x\in\mathcal X}\HH_x,\qquad
 \AA(Y)=\BB(\HH_Y)\otimes I_{\mathcal X\setminus Y}.
\]

\begin{theorem}[Scheduled locality]
\label{thm:locality}
For layers
\(\Phi_j=\bigotimes_{B\in\mathcal B_j}\Phi_{j,B}\), where
\(\mathcal B_j\) is a partition and every block map is CPTP,
define
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
  \bigcup_{\substack{B\in\mathcal B_{s+1}\\
               B\cap N_{s+1,t}(Y)\ne\varnothing}} B.
\]
Then
\[
 (\Phi_{s+1}^*\cdots\Phi_t^*)(\AA(Y))
                           \subseteq\AA(N_{s,t}(Y)).
\]
Equal input reductions on that ancestor region give equal
output reductions on \(Y\).
For a joint output with reference \(E\), the required equal
input reductions are on the joint ancestor--reference region.

If every block has diameter at most \(\Delta\), then
\[
 N_{s,t}(Y)\subseteq
  \{x:d(x,Y)\le(t-s)\Delta\}.
\]
\end{theorem}

\begin{proof}
The adjoint of a channel is unital.
Blocks disjoint from an observable's support act on identity
and do not change it.
A block meeting that support can enlarge it only within the
block. Reverse induction proves the support inclusion.
The metric bound follows by adding at most one block diameter
per layer. Testing all output observables proves the reduced
state statements; carry \(\BB(\HH_E)\) unchanged for the
reference version.
\end{proof}

The same argument applies to a complete finite acyclic port
network. It uses the ancestors of the full joint output, not
only a selected marginal record
\cite{CheungLocality,STInterfaces}.

\begin{proposition}[Exact deterministic no-signalling]
\label{prop:nosignal}
For input factor \(A\) and output channel \(\Phi_B\), the
following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item every deterministic local intervention on \(A\) leaves
the \(BE\) output unchanged for all inputs and references;
\item \(\Phi_B=\Xi\Tr_A\) for some CPTP map \(\Xi\);
\item \(\Phi_B^*(\BB(\HH_B))
      \subseteq I_A\otimes\BB(\HH_{A^c})\).
\end{enumerate}
\end{proposition}

\begin{proof}
Factorization gives intervention invariance.
Conversely compare identity on \(A\) with replacement by
\(\tau_A=I_A/\dim\HH_A\).
Invariance gives
\[
 \Phi_B(X)=\Xi(\Tr_AX),\qquad
 \Xi(Z)=\Phi_B(\tau_A\otimes Z)
\]
on states and hence, by linearity, on all operators.
Adjoints give equivalence with the algebra containment.
\end{proof}

\begin{theorem}[Separate physical-containment certificate]
\label{thm:containment}
Suppose every port is assigned a regular host event and every
dependency edge is realized by a future-causal curve in the
supported laboratory.
Then actual intervention influence is contained in scheduled
ancestry, which is contained in the prescribed physical
causal past of the joint output.
\end{theorem}

\begin{proof}
The first inclusion follows from
Theorem~\ref{thm:locality} or its port-network version.
Concatenation of the installed future-causal curves proves
the second. Proposition~\ref{prop:nosignal} gives exact
deterministic no-signalling outside that region.
\end{proof}

A triangle contraction graph is not such a physical port
network. Neither its reflected relation nor the two
parenthesizations defines a physical causal past.
The difference between computational dependency and physical
containment is essential in the finite transport records
\cite{CheungLocality,CheungTransport,STInterfaces}.

\subsection{An explicit nonempty regular route}

\begin{assumption}[Finite operational permissions]
\label{ass:permissions}
The laboratory permits declared finite preparations, finite
local unitaries and instruments, positive-duration classical
processing and storage, and exact carriage of finite registers
along specified regular future-causal paths.
Installed lookup matrices and their memory are charged resources.
No continuum-field derivation of these permissions is claimed.
\end{assumption}

Choose an axial regular spatial segment of developed length
\(L>0\), compactly inside \(B_R\).
Place stations \(A_0,\ldots,A_n\) at equal spacing
\(\ell=L/n\).
Let the temporal laboratory window be long enough for the
following finite schedule, fixed before the run.

At \(A_0\), install:
\begin{itemize}
\item the one \(m\)-qubit clock and a \(g\)-bit catalogue;
\item the table and clean work memory;
\item a \(p\)-qubit unknown payload, \(p\ge1\);
\item a probe for the optional normalized score in
\eqref{eq:evaluation}.
\end{itemize}
Every other station has a blank \(p\)-qubit payload register
and a blank classical history buffer.

Carry out the finite clock--table instrument locally, retain all
outcomes, and wait through a prescribed positive buffer interval.
This waiting interval is an installed schedule, not a singular
host event.
Let the complete retained history have \(b\) bits, including the
clock code, catalogue index, sign outcome, and report.
At each stage:
\begin{enumerate}[label=\textup{(\roman*)}]
\item copy the classical history into the next outgoing buffer,
using \(b\) positive-duration local copying operations;
\item carry that buffer to the next station;
\item exchange the two adjacent payload contents by opposite
carriage paths, not by an instantaneous remote gate.
\end{enumerate}
Assign the carriage a duration \(2\ell\) in developed time.
Its speed is \(1/2\).
The complete route and its closure stay in the regular
laboratory.

\begin{lemma}[Exact payload transport]
\label{lem:swap}
The successive adjacent exchanges transport the original
unknown payload to \(A_n\), return all preceding payload
registers to blank, and preserve all payload--reference
operators exactly.
\end{lemma}

\begin{proof}
One exchange swaps complete tensor factors and their
correlations. Induction over the chain moves the original
factor forward and leaves behind the next original blank.
The resulting tensor-factor permutation gives the identity
channel from the initial to the final payload interface,
including arbitrary references.
\end{proof}

\begin{theorem}[Buffered certificate and payload transport]
\label{thm:transport}
Under Assumption~\ref{ass:permissions}, the finite route
implements
\begin{equation}
 \Gamma_v(X)=
 \sum_{k,i,y}
   \left(\proj{k,i,y,r(k,i)}\right)^{\otimes(n+1)}
     \otimes p_y(k,i)\,
              \mathcal U_{0\to n}(X_{ki})
     \otimes\omega_{\mathrm{blank}},
 \label{eq:transport}
\end{equation}
where \(\mathcal U_{0\to n}\) is exact payload-factor
transport and \(\omega_{\mathrm{blank}}\) contains the
returned blank payload registers.
The identity is complete and reference preserving.
The protocol satisfies the physical-containment certificate
of Theorem~\ref{thm:containment}.

For a fixed clock--catalogue input independent of the payload,
the payload communication has a unit signalling witness.
This is a declared transport operation, not
gravity-generated entanglement.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:table} gives the subnormalized branches.
A classical copying isometry copies their already orthogonal
history labels; it does not copy an unknown quantum state.
Lemma~\ref{lem:swap} gives the exact payload action in every
branch. Keeping every history and summing every branch proves
\eqref{eq:transport} on matrix units and reference extensions.

All local operations lie on regular timelike station
worldlines. Every communication path has speed \(1/2\), and
every control arrives before its declared use.
Thus every complete dependency has its required future-causal
implementation.

Finally use payload inputs \(\proj0\) and \(\proj1\) on one
payload qubit, with all other input data identical.
Their final receiver states have trace distance one, attaining
the maximum signalling contrast.
The host metric is unchanged and supplies no entangling
interaction. Pre-existing reference entanglement is preserved,
not attributed to gravity.
\end{proof}

The route is the regular buffered form of the I-B transport
pattern, with the clock and complete history retained in its
channel specification \cite{CheungIB,CheungTransport}.

\subsection{Resources and complete error accounting}

Let \(s\) be the report length and \(w\) a reserved lookup-work
size. One may take \(b=m+g+1+s\).
A conservative finite ledger is
\[
\begin{array}{l|l}
\toprule
\text{resource}&\text{reserve or count}\\
\midrule
\text{destructive clock}&m\text{ qubits, one clock factor}\\
\text{catalogue}&g\text{ classical bits}\\
\text{score probe}&1\text{ qubit}\\
\text{payload registers}&(n+1)p\text{ qubits}\\
\text{lookup work/report reserve}&w+s\text{ bits}\\
\text{retained history buffers}&(n+1)b\text{ bits}\\
\text{history-copy operations}&nb\\
\text{payload exchanges}&n\\
\text{history transmissions}&n\\
\text{clock/probe readouts}&m+1\\
\text{resets during the run}&0\\
\bottomrule
\end{array}
\]
The canonical full-matrix implementation can additionally
destructively read the \(g\) catalogue names; those readouts
are charged if that extension is used.
A truth table requires finite storage proportional to the
number of entries and their chosen finite report precision.
Exact arbitrary real amplitudes are ideal matrix inputs, not
claimed to have finite hardware descriptions.

Counting each classical storage bit conservatively as a
basis-state qubit carrier gives
\[
 Q_{\mathrm{peak}}
 \le m+g+1+(n+1)p+w+s+(n+1)b.
\]
External reference storage is declared separately and is not
silently omitted from the complete correctness statement.
If local preprocessing takes \(\tau_{\mathrm{pre}}\), buffering
takes \(\tau_{\mathrm{buf}}\), and one history-copy stage takes
\(\tau_{\mathrm{copy}}>0\), the route duration is
\[
 \tau_{\mathrm{pre}}+\tau_{\mathrm{buf}}
                   +n\tau_{\mathrm{copy}}+2L.
\]
The supported time window is chosen to contain this finite
duration.

\begin{proposition}[Certificate-to-channel error bridge]
\label{prop:errorbridge}
Suppose the analytic table enclosures obey
\eqref{eq:certificateerror} and all implemented operational
factors, on the same complete interfaces, have summed
half-diamond allowance \(\eta\).
Then the implementation \(\widetilde\Gamma\) satisfies
\begin{equation}
 \ddia(\widetilde\Gamma,\Gamma_v)
 \le
 \min\left\{1,\,
       \max_{ki}\frac{E_{ki}}{2M_{ki}}+\eta\right\}.
 \label{eq:bridge}
\end{equation}
The bound includes arbitrary untouched references.
It compares installed finite channels, not continuum
sharp-product kernels.
\end{proposition}

\begin{proof}
The table comparison is \eqref{eq:enclosurechannel}.
Exact classical copying and payload transport are common
postprocessing, so they cannot increase this distance.
For the remaining implementation differences, replace one
compatible channel factor at a time. Pre- and postcomposition
by channels contract diamond norm. Summing those differences
gives \(\eta\); the distance between channels is at most one
\cite{Watrous}.
\end{proof}

The analytic enclosure width and the channel implementation
allowance enter \eqref{eq:bridge} only after an explicit
normalization and fixed-interface encoding.
They are not equated by analogy.
The ledger is not an energy budget or a Landauer-cost theorem.

%=====================================================================
\section{Supported transfer, no-Einstein separation, and release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Supported chart transfer]
\label{thm:transfer}
Transport the declared comparison data, tensor components,
densities, and observation operators through
\(\iota:U\to M_{\delta_{\mathrm c}}\).
Every result above holds on the corresponding supported
regular host presentation.

For tests and kernels with supports in the stated laboratory:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the inherited Cauchy response is independent of auxiliary
extensions equal near the saturated support;
\item the numerical pairings and the derivative estimates are
unchanged by that presentation transfer;
\item compact coordinate changes give equivalent finite-order
test bounds, with transformed smooth coefficients.
\end{enumerate}
No global conical Green operator, global conical covariance,
or gluing around the defect is inferred.
\end{theorem}

\begin{proof}
The developing map is a diffeomorphism and an isometry on
the complete declared support.
Distribution transport through it is defined by transposition
on tests and densities.
It preserves the relevant compact supports.
The first assertion is
Proposition~\ref{prop:saturation}.
The second is equality of the same pairings written in the
two presentations.
For the third, the chain rule on a fixed compact set bounds
transformed test derivatives through any fixed order by
finitely many original derivatives through that order, with
constants depending on the fixed coordinate change.
The transported differential operator includes the transformed
coefficients. The transfer constructs no data outside the
regular presentation.
\end{proof}

Changing the foliation, the zero-Cauchy prescription, or the
Euclidean-radial regulator convention is not covered by merely
calling it a coordinate change.
Such a change would require an explicit comparison of the
changed data.

\begin{proposition}[Stress identities do not imply Einstein dynamics]
\label{prop:noeinstein}
Symmetry, conservation, and zero trace of a tensor on a
prescribed metric do not imply Einstein's equation.
\end{proposition}

\begin{proof}
In a regular developed chart choose a nonzero constant null
covector \(k\) and put \(t_{\mu\nu}=k_\mu k_\nu\).
It is symmetric, constant and conserved, nonzero, and
traceless. The local Einstein tensor is zero.
Thus
\(G_{\mu\nu}=8\pi G_{\mathrm N}t_{\mu\nu}\) fails when
\(G_{\mathrm N}\ne0\).
No statement about a distributional stress tensor on \(\WW\)
is needed.
\end{proof}

\begin{theorem}[IX-B-ST local Grade S release]
\label{thm:package}
\label{thm:release}
Retain the host contract and
Assumptions~\ref{ass:carrier}, \ref{ass:tensor}, and
\ref{ass:VIID}. Under the finite operational permissions,
IX-B-ST establishes:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a supported local placement of the source-locked
IX-B comparison on the regular part of the prescribed cone,
without changing the host or supplying a global conical inverse;
\item the eight unlabelled connected skeletons and the unique
fully contracted selected triangle, with factor \(2\);
\item a positive reflected-relation residue, its exact local
density, and a separated-support observation-diagonal
limitation;
\item the derivative--spectator finite part and complete
parent-conversion transfer, with order loss \(64\to64+d\)
and explicit smooth bounds;
\item an infinite observation-derivative orbit, distinct from
the one-parameter parent conversion and from an unrestricted
weighted class;
\item equality of the two regularized numerical
parenthesizations and the admitted first-stage output
\eqref{eq:Qpartial}, with sign \(-2\);
\item finite enclosure propagation, one destructive clock,
complete fixed-table channels, and the exact fixed-table
channel-distance formula;
\item residual-record and source separations, scheduled
locality, exact deterministic no-signalling, and separately
certified causal containment;
\item a nonempty regular buffered protocol transporting an
unknown quantum payload and its classical calibration history,
with arbitrary untouched references and a finite resource
ledger.
\end{enumerate}
These statements form local Grade S.
They neither remove the VII-C obstruction under its original
requirements nor construct either sharp second multiplication.
The quantum certificate remains \loc{RG-open}.
\end{theorem}

\begin{proof}
Sections~\ref{sec:host}--\ref{sec:inputs} fix the supported
input and prove its geometric distinctions.
Theorem~\ref{thm:skeletons} and
Proposition~\ref{prop:unique} give (ii).
Section~\ref{sec:geometry} gives (iii).
Section~\ref{sec:descendants} gives (iv), and
Section~\ref{sec:orbits} gives (v).
Section~\ref{sec:association} gives (vi).
Section~\ref{sec:finite} gives (vii).
Sections~\ref{sec:residual}--\ref{sec:transport} give
(viii)--(ix), including the complete error bridge.
Theorem~\ref{thm:transfer} completes (i).

Proposition~\ref{prop:obstruction} retains the specified
normal-topology incompatibility.
Proposition~\ref{prop:missing} identifies the absent binary
maps. None of the finite channel results defines them.
\end{proof}

\subsection{Potential significance and the exact novelty boundary}

The useful output is a controlled separation of four interfaces:
\[
 \text{classical admission}
 \quad\ne\quad
 \text{numerical finite part}
 \quad\ne\quad
 \text{finite encoded evaluation}
 \quad\ne\quad
 \text{sharp quantum multiplication}.
\]
The local transfer and error bridge provide a benchmark against
which a later deformation, acquisition proposal, or implementation
can be compared without changing the source, covariance, regulator
placement, or host by implication.

The skeleton classification, parent calibration, derivative
transfer, and residue-orbit findings are attributed to IX-B and
VII-D. The added result is their supported conical and finite
operational assembly, including the explicit passage from
certified scalar enclosures to fixed-interface channel error.
No general priority claim is made for distributional
differentiation, finite instruments, or tensor-factor transport.

\subsection{The next product obligation}

A subsequent factorization theorem must construct binary maps on
named actual-functional domains, verify every resulting coefficient
and directional descendant in \(\Padm\), prove descent through all
presentation relations and agreement on old-product overlaps,
and establish the coefficientwise coherence equations for the
same fixed parent \(U_c\).
Naming \(\mathcal N_c\) is not that theorem.

Even a separately constructed partial free product would not
automatically give time-ordered products, retarded products,
a switched interacting differential, or a quantum Ward identity
\cite{FR,CheungSuccessors}.
This sheet makes none of those transfers.

\statementbox{Standing stops and non-claims}{%
Grade E is not constructed. This is not spacetime emergence,
Einstein dynamics, a Page curve, an island, an interior,
holographic identification, gravity-generated entanglement,
or an ultraviolet completion.

The finite part is not factored through either sharp second
multiplication in this release.
No joint continuity or hypocontinuity of classical multiplication
is inferred; no quantum associativity is constructed here.
No retarded product, renormalized interacting product, quantum
Ward/BRST/BV identity, radiative coefficient, or continuum
field-acquisition instrument is supplied.

O.1 stays \loc{NOT_CLOSED} in the stated inherited prescription.
XV-B is not opened as a downstream quantum-admission exit.
XVI-N is not discharged.
Historical H2 and the full I3/J2 characteristic-map boundaries
are not discharged by IX-B-ST.
No inherited interacting or radiative quantum coefficient is
admitted. The selected already-defined free comparison
coefficient remains a classical member of the admission carrier.

No compilation, machine check, numerical execution,
experimental implementation, or independent execution is asserted.}

%=====================================================================
\appendix
\section{Debugging checklist}
\label{app:debug}

\begin{longtable}{p{.30\linewidth}p{.63\linewidth}}
\toprule
Potential incorrect inference & Correction used here\\
\midrule
\endfirsthead
\toprule
Potential incorrect inference & Correction used here\\
\midrule
\endhead

The host is \(\RR^{1+3}\) &
The host is \eqref{eq:host}. Flat formulae are supplied in one
supported developing chart.\\[4pt]

The four-dimensional defect is a worldline &
The completed product defect has coordinates \((T,z)\) and
is a worldsheet.\\[4pt]

An observation patch determines the response &
The full truncated Cauchy past is included, with a positive
support margin.\\[4pt]

The local Cauchy prescription constructs a conical covariance &
The comparison covariance is separate supplied data.
No global host inverse or state is constructed.\\[4pt]

The residue lies on an equality diagonal &
It lies on the reflected relation \(\Rel\). Its intersection
with \(\Delta_{13}\) has one less dimension, and the local
density has no \(\delta(t-t')\).\\[4pt]

\(H_{ww}=0\) implies \(K=0\) &
The mixed reflected covariance \eqref{eq:K} remains nonzero.\\[4pt]

The factor \(2\) is an adjustable normalization &
It counts the two assignments of middle metric slots.
The first-stage sign is separately fixed to \(-2\).\\[4pt]

Eight skeletons define every decorated graph &
The inventory is combinatorial. Distributional decorations
have their own existence conditions.\\[4pt]

Order \(64+d\) is a scaling degree &
It is a test-seminorm order. It is not a transverse or
ultraviolet scaling degree.\\[4pt]

The derivative transfer re-proves VII-D &
The parent theorem, including its intersection estimate,
is an explicit analytic input.\\[4pt]

All derivative operations are interchangeable &
Observation, configuration-direction, and unsmeared
functional-slot derivatives have different extension claims.\\[4pt]

A time-coordinate derivative changes the host &
It differentiates a numerical kernel argument in fixed
coordinates. No metric-update or clock operation follows.\\[4pt]

\(\RR r\) contains every descendant &
The derivative orbit is infinite-dimensional.
Compatibility can still impose one common parent parameter.\\[4pt]

Arbitrary weights are admissible &
The weighted-class proposition is a warning, not an
admissibility declaration.\\[4pt]

Regularized associativity gives sharp factorization &
The theorem evaluates one old-domain scalar coefficient.
The sharp binary maps remain undefined here.\\[4pt]

A sharp-only finite value repairs continuity &
The unchanged regular sequence still diverges in the
VII-C normal-continuity comparison.\\[4pt]

A regulator command evolves curvature &
It replaces a catalogue label on a fixed host.\\[4pt]

A table acquires the residue or a continuum field &
It stores supplied evaluations and certified enclosures.
No acquisition map is present.\\[4pt]

Close real labels give close quantum channels &
The complete-channel bound requires the same output labels
and fixed interface. Orthogonal relabelling is different.\\[4pt]

Equal records determine the residual source &
Full residual-effect identities are stronger than equal
record effects; finite source records need not separate
all configurations.\\[4pt]

A contraction graph is a physical causal past &
Computational support and physical causal embedding are
separate certificates.\\[4pt]

Unit payload signalling tests gravity-generated entanglement &
It witnesses an installed carriage-and-exchange channel on
a prescribed metric.\\[4pt]

An unresolved statement here denies later partial results &
The present input boundary does not import or overturn
IX-C, IX-Free, IX-Free-2, or other later releases.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Routing note, obligations, and documentary separations}
\label{app:routing}

The following is proposed metadata. It is not a claim that a
registry has been modified.

\begin{verbatim}
identifier: IX-B-ST
version: 1
placement: after IX-B, before the IX-C adaptation interface
grade: local Grade S
track: product
bearingProblem: products
nearUv: false
timeLandmarks: 1
includeInEmergenceSources: false
emergenceStatus: not constructed
quantumCertificate: RG-open

primarySourceRecords:
  IX-B@2
  IX-A-ST@1
  I-B@1

explicitMathematicalInputs:
  already adapted carrier
  locked chart tensor and old-domain data
  VII-D parent finite-part estimate and conversion

routingBoundary:
  The source records are not blanket premises.
  Only the explicitly restated mathematical assumptions
  enter the conditional release theorem.

openInThisRelease:
  sharp second multiplication on the adapted carrier
  actual-functional binary factorization of the finite part
  renormalized interacting products
  continuum field acquisition
  emergence of metric, lapse, or cone

standingStops:
  Grade E is not constructed.
  O.1 remains NOT_CLOSED in the stated inherited prescription.
  XV-B downstream admission exit is not opened.
  XVI-N is not discharged.
  Historical H2 and full I3/J2 are not discharged here.
  No inherited quantum coefficient is admitted.
  No compilation or execution is asserted.
\end{verbatim}

The flag \loc{nearUv: false} classifies this local host-and-interface
adaptation. It does not deny that the supplied parent triangle has
an ultraviolet logarithm or that VII-D performs an ultraviolet
analysis.

\paragraph{Routing note.}
IX-B-ST adapts the source-locked IX-B numerical triangle to a
supported regular chart of the IX-A-ST/I-B prescribed cone.
The regular product is not globally hyperbolic, and the compact
laboratory avoids its worldsheet.
Covariance blocks and the parent finite part are supplied chart
data, not conical evolution.
The derivative--spectator transfer has test-order loss
\(64\to64+d\).
The parent conversion class is distinct from the infinite
observation-derivative orbit.
The two regularized parenthesizations have one numerical
coefficient; neither sharp second multiplication is constructed
here.
The clock reads an installed finite encoding, not the geometric
time axis. Transport is reference preserving and is not
gravity-generated entanglement.
The VII-C obstruction is retained, the quantum certificate stays
\loc{RG-open}, and no emergence or Einstein-dynamics claim is made.

\paragraph{Emergence walk.}
IX-B-ST certifies a source-locked numerical triangle, a
derivative--spectator finite part, and one admitted partial
coefficient on a prescribed conical chart, with a destructive
clock and reference-preserving transport.
It does not factor that finite part through a sharp binary
multiplication, evolve the host metric by a regulator token,
or establish spacetime emergence.

\paragraph{Home bullet.}
A triangle-calibration adaptation, IX-B-ST, places the IX-B
numerical triangle and derivative--spectator transfer on a
supported chart of the prescribed cone.

\paragraph{Two obligations retained.}
\begin{description}[leftmargin=1.8em,style=nextline]
\item[\loc{triangle-calibration-not-sharp-product}]
Products, with beyond-EFT only as a boundary connection:
construct an actual-functional binary factorization or a
renormalized interacting product with the same source,
normalization, domains, and compatibility obligations.
IX-B-ST does not supply that theorem.

\item[\loc{triangle-calibration-not-emergence}]
Emergence:
derive the metric, lapse, or cone from the relevant quantum
data or records, rather than prescribing them.
IX-B-ST does not supply that theorem.
\end{description}

\subsection{Documentary comparison edges}

Each edge below is a separation, not a blanket theorem import.
For VII-D and the adapted carrier, the independently displayed
mathematical assumptions remain the exact dependencies of this
paper.

\begin{itemize}
\item IX-B \loc{thm:package}, IX-A-ST \loc{thm:release},
IX-A \loc{thm:IX1}, I-B \loc{thm:transport}, VII-D
\loc{thm:finitepart}, and VII-C \loc{thm:log}:
calibration, admission, transport, and obstruction are
different objects.
\item IX-C, IX-Free, and IX-Free-2:
version-qualified actual-functional factorization results are
successor records, not premises or claims negated by this
sheet \cite{CheungSuccessors}.
\item VIII-ST and VII-J-ST:
fixed-host control and causal-interface comparisons only,
not product theorems \cite{STInterfaces}.
\item Em01-B and Em01-C:
moderate smoothing growth and a separate scalar expansion
do not identify this residue with the full physical
characteristic coefficient \cite{CheungCharacteristic}.
\item XIV-I \loc{thm:I2}, XIV-J, and technical XV-B:
a constructed internal amplitude or an extension/descent
interface is not the full characteristic map or its quantum
admission \cite{CheungAnalytic}.
\item XVI-N and XVI-O:
the massless covariance block here does not discharge the
insertion obstruction, and this numerical conversion does
not close the stated bulk O.1 prescription
\cite{CheungStops}.
\item VII-H and VII-I:
a free susceptibility or finite-width free algebra is not
this reflected triangle or an interacting quantum BRST
algebra \cite{CheungFreeComparisons}.
\item En05-A, En08-A/B, and En09-A:
scheduled locality, prescribed null-boundary approximation,
and finite clock conditioning do not derive a Lorentz cone
or remove external scheduling
\cite{CheungLocality,CheungTransport,CheungEn09}.
\item En10-A/B:
a declared source-weight estimate is not acquisition of
this residue or reconstruction of the host metric
\cite{CheungSourceEstimates}.
\end{itemize}

\subsection{OpenAI catalogue separation}
\label{rem:not-premise}

The catalogue and its companion comparison records are documentary
only \cite{OpenAIRecord,OpenAIComparisons}.
No OpenAI theorem is a premise, and no family is independently
verified here.

Oai282-A/B and Oai069-A do not turn the present numerical
identities into a fieldwise Ward theorem, a general
scale-to-conformal theorem, or an ultraviolet completion.
Oai283-A is at most a comparison for a fixed finite-channel
certificate; it supplies no interacting product.
Oai365-A does not derive a host metric from a calibration tag.
The finite-certificate comparisons Oai212-A, Oai265,
Oai272-A, Oai276-A, Oai279-A, Oai227-A, and Oai119-A are not
premises, Page curves, or entanglement tests for this transport.
In particular, a Family 265 static record is not a radiation
history.

\paragraph{Historical gate terminology.}
H2 denotes the historical internal-amplitude stop in XIV-H.
XIV-I records a narrower internal amplitude construction at I2;
that progress is not withdrawn.
The full characteristic maps at I3/J2 are different obligations,
and IX-B-ST discharges neither.
``XV-B is not opened'' refers to the downstream quantum-admission
exit, not to the nonexistence of a technical manuscript with that
identifier \cite{CheungAnalytic}.

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

\paragraph{Anonymous review.}
The author field is intentionally empty.
Citation of N.~Cheung does not assign authorship of this
adaptation. Authorized authorship, affiliation, and submission
metadata must be supplied for a signed submission.

\paragraph{Attribution.}
The journal article, philosophical essay, internal manuscripts,
and supplied anonymous spacetime adaptations are distinguished
in the bibliography.
No internal publication date, DOI, or independent validation is
invented. The inherited estimates are assumptions with explicit
source attribution, not independently re-certified ultraviolet
proofs.

\paragraph{Analytic scope.}
The numerical construction is source locked, local, and tied to
the displayed regulator placement.
It does not classify arbitrary singular spectators, independently
decorated graphs, conical quantum states, global cone products,
or all renormalization freedoms.

\paragraph{Operational scope.}
Finite matrices and report words encode supplied evaluations and
enclosures. An acquisition instrument for an unknown continuum
field is absent. Transport permissions and their causal routes
are declared finite-model data.

\paragraph{Verification.}
This source uses one conventional pdfLaTeX file with an internal
bibliography and no external figures or bibliography database.
No successful pdfLaTeX execution, machine check, independently
executed symbolic or numerical computation, hardware synthesis,
experiment, independent referee certification, or journal
acceptance is asserted.

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

\bibitem{CheungIXB}
N.~Cheung,
\emph{Landmark IX-B: Derivative-stable triangle calibration and
counterterm constraints for sharp Cauchy insertions}.
\loc{IX-B-v2.tex}, version 2,
Enter Calculus Companion, manuscript record.
\url{https://enter.grok.me/browse/IX-B}.

\bibitem{CheungIXA}
N.~Cheung,
\emph{Landmark IX-A: An off-shell admission domain and
boundary-sensitive topology for sharp Cauchy relational insertions}.
\loc{IX-A.tex}, version 1,
Enter Calculus Companion, manuscript record.
\url{https://enter.grok.me/browse/IX-A}.

\bibitem{IXAST}
\emph{IX-A-ST: Boundary-sensitive admission of sharp Cauchy
insertions on a prescribed conical Enter spacetime}.
\loc{IX-A-ST.tex}, version 1.
Supplied local adaptation; author field intentionally empty.
Carrier, support-saturation, separator, and classical topology
inputs only.
\url{https://enter.grok.me/browse/IX-A-ST}.

\bibitem{CheungVIID}
N.~Cheung,
\emph{Finite part of the regulated Cauchy-reflection triangle:
A strong numerical distributional limit for the VII-C family}.
\loc{VII-D.tex}, version 1,
Enter Calculus Companion, manuscript record.
\url{https://enter.grok.me/browse/VII-D}.

\bibitem{CheungVIIC}
N.~Cheung,
\emph{Landmark VII-C: A mixed-Wick obstruction to a
normal-topology boundary extension for sharp Cauchy harmonic
insertions}.
\loc{VII-C.tex}, version 1,
Enter Calculus Companion, manuscript record.
\url{https://enter.grok.me/browse/VII-C}.

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B-v1: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed Conical Product}.
\loc{I-B-v1.tex}, version 1,
Enter Calculus Companion, manuscript 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,''
\emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
DOI: \doi{10.3389/fphy.2026.1954943}.
Accepted publisher record; the consulted record identifies the
final formatted version as forthcoming.
\url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}.

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout, Quantum
Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2,
Enter Calculus Companion, introductory essay.
Philosophical and documentary comparison, not a theorem premise.
\url{https://enter.grok.me/enter}.

\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{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory}.
Lecture notes, revised version 2, 2013.
\url{https://arxiv.org/abs/1208.1428}.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
``Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counterexample by Semyon Alesker),''
\emph{Studia Mathematica} \textbf{232} (2016), 201--226.
DOI: \doi{10.4064/sm8316-3-2016}.
\url{https://arxiv.org/abs/1409.7662}.

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

\bibitem{CheungEn09}
N.~Cheung,
\emph{En09-A: Relational Enter}.
\loc{En09-A.tex}, version 1,
Enter Calculus Companion, supplied manuscript record.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{CheungObservation}
N.~Cheung,
\emph{VII-E}, \loc{VII-E.tex}, version 1;
\emph{VII-G}, \loc{VII-G.tex}, version 1; and
\emph{VIII-A}, \loc{VIII-A-v3.tex}, version 3.
Enter Calculus Companion, supplied manuscript records.
Record compatibility, residual criteria, and complete
finite-instrument comparisons.
\url{https://enter.grok.me/browse/VII-E};
\url{https://enter.grok.me/browse/VII-G};
\url{https://enter.grok.me/browse/VIII-A}.

\bibitem{STInterfaces}
\emph{VIII-ST}, version 1;
\emph{VII-E-ST}, version 1;
\emph{VII-F-ST}, version 1; and
\emph{VII-J-ST}, version 1.
Enter Calculus Companion, supplied spacetime adaptation records.
Fixed-host, observation, clock, and causal-interface comparisons;
not product premises.
\url{https://enter.grok.me/browse/VIII-ST};
\url{https://enter.grok.me/browse/VII-E-ST};
\url{https://enter.grok.me/browse/VII-F-ST};
\url{https://enter.grok.me/browse/VII-J-ST}.

\bibitem{CheungLocality}
N.~Cheung,
\emph{En03-A}, version 1;
\emph{En05-A}, version 1; and
\emph{VII-J}, version 1.
Enter Calculus Companion, supplied manuscript records.
Scheduled locality and separate physical-containment interfaces.
\url{https://enter.grok.me/browse/En03-A};
\url{https://enter.grok.me/browse/En05-A};
\url{https://enter.grok.me/browse/VII-J}.

\bibitem{CheungTransport}
N.~Cheung,
\emph{En08-A}, version 1; and
\emph{En08-B}, version 1.
Enter Calculus Companion, supplied manuscript records.
Scheduler-versus-cone separation and transport relative to a
prescribed null boundary.
\url{https://enter.grok.me/browse/En08-A};
\url{https://enter.grok.me/browse/En08-B}.

\bibitem{CheungSuccessors}
N.~Cheung,
\emph{IX-C}, version 1;
\emph{IX-Free}, version 1;
\emph{IX-Free-2}, version 1; and
\emph{IX-D}, version 3.
Enter Calculus Companion, supplied manuscript records.
Version-qualified successor constructions and interfaces;
not inputs to IX-B-ST.
\url{https://enter.grok.me/browse/IX-C};
\url{https://enter.grok.me/browse/IX-Free};
\url{https://enter.grok.me/browse/IX-Free-2};
\url{https://enter.grok.me/browse/IX-D}.

\bibitem{CheungCharacteristic}
N.~Cheung,
\emph{Em01-B}, version 3; and
\emph{Em01-C}, version 3.
Enter Calculus Companion, supplied manuscript records.
Moderate smoothing growth and separate scale-profile
constructions; documentary separations only.
\url{https://enter.grok.me/browse/Em01-B};
\url{https://enter.grok.me/browse/Em01-C}.

\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.
Enter Calculus Companion, supplied manuscript records.
Historical amplitude and full-characteristic-map boundaries;
not premises of the calibration.
\url{https://enter.grok.me/browse/XIV-H};
\url{https://enter.grok.me/browse/XIV-I};
\url{https://enter.grok.me/browse/XIV-J};
\url{https://enter.grok.me/browse/XV-B}.

\bibitem{CheungStops}
N.~Cheung,
\emph{XVI-N}, revision r12; and
\emph{XVI-O}, revision o70-sync.
Enter Calculus Companion, supplied manuscript records.
Insertion-obstruction and prescription-qualified O.1 comparisons.
\url{https://enter.grok.me/browse/XVI-N};
\url{https://enter.grok.me/browse/XVI-O}.

\bibitem{CheungFreeComparisons}
N.~Cheung,
\emph{VII-H}, version 1; and
\emph{VII-I}, version 1.
Enter Calculus Companion, supplied manuscript records.
Separate free-field benchmarks and finite-width constructions.
\url{https://enter.grok.me/browse/VII-H};
\url{https://enter.grok.me/browse/VII-I}.

\bibitem{CheungSourceEstimates}
N.~Cheung,
\emph{En10-A}, version 1; and
\emph{En10-B}, version 1.
Enter Calculus Companion, supplied manuscript records.
Source-weight estimation and its separate acquisition
requirements; not this triangle residue.
\url{https://enter.grok.me/browse/En10-A};
\url{https://enter.grok.me/browse/En10-B}.

\bibitem{OpenAIRecord}
OpenAI,
\emph{OpenAI mathematics catalogue}.
Separate documentary source collection, recorded in the
Enter Calculus Companion wrapper \loc{OpenAI.tex}.
The wrapper's \loc{rem:not-premise} is a source-separation
record, not verification of the underlying catalogue.
\url{https://enter.grok.me}.

\bibitem{OpenAIComparisons}
N.~Cheung,
\emph{Oai282-A}, \emph{Oai282-B}, \emph{Oai069-A},
\emph{Oai283-A}, \emph{Oai365-A}, \emph{Oai212-A},
\emph{Oai265-A/B}, \emph{Oai272-A}, \emph{Oai276-A},
\emph{Oai279-A}, \emph{Oai227-A}, and \emph{Oai119-A}.
Enter Calculus Companion, supplied comparison records.
Documentary separations only; no catalogue family or theorem
is imported or independently verified here.
\url{https://enter.grok.me/browse}.

\end{thebibliography}

\end{document}