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\hypersetup{
  pdftitle={XXII-D: Operational approximation of an extracted Lorentz cone},
  pdfauthor={Ngo Cheung},
  pdfsubject={Stage 3 of a limited kinematic emergence programme}
}

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\title{\textbf{XXII-D: Operational approximation\\
of an extracted Lorentz cone}\\[4pt]
\large Stage 3 of a limited kinematic emergence programme}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
This note specifies an operational approximation layer that could be
attached to a successful Lorentz-cone extraction. Extraction alone
does not supply the required transport resources, quantum endpoint
access, composition law, or calibration. For the unchanged carrier
of XXII-C, every component has a finite future; consequently the
entire null boundary cannot be approximated by comparable event
differences from one fixed component. A separate example exhibits
a faithful generated Lorentz representation of a classical signalling
order without reference-preserving quantum transmission through its
designated channel. These results identify substantive missing
interfaces, rather than merely missing estimates. Conditional on
a compatible transport realization, we prove complete-channel
telescoping, an endpoint certificate, an arrival-window bound, and
model-dependent locality and resource lower bounds. Boundary
attainment, finite-stencil coverage, moving arrival events, and noise
accumulation receive explicit failure witnesses. No successful
extraction or transport realization on the frozen process family
is constructed, and no metric calibration or invariant physical
speed is derived.
\end{abstract}

\section{Precise claim and logical status}
\label{sec:purpose}

XXII-A specifies a nonempty operational Lorentz-cone selection target
and a faithful generated affine representation
\cite[\loc{target:selection}, \loc{def:representation}]
{CheungXXIIA}.
XXII-B separates the candidate data, additional hypotheses, and
failure tests
\cite{CheungXXIIB}.
XXII-C fixes a tripartite qubit process-matrix family, its one-use
instrument access, and its component-order extraction
\cite{CheungXXIIC}.

The present note does not assert either implication
\[
 \text{cone extraction}
 \Longrightarrow
 \text{quantum transport realization},
\]
or
\[
 \text{finite null-boundary approximation}
 \Longrightarrow
 \text{cone selection}.
\]
Both would omit substantive hypotheses.

En08-B constructs finite transport protocols for a prescribed
geometric boundary and speed
\cite[\loc{thm:main}, \loc{cor:operational-limit}]{CheungEn08B}.
Its result is an approximation theorem on its stated model.
It is not a derivation of those prescribed inputs.

The narrower objective considered here is:
\begin{quote}
After a successful extraction and a separately justified compatible
transport realization, construct finite protocols approximating a
declared nonempty family of future-null targets, with reference-stable
channel error, endpoint and arrival control, and a model-specific
resource ledger.
\end{quote}
This is a research objective, not an existence theorem of XXII-D.

\paragraph{What is proved.}
The proofs below establish scoped obstructions and conditional
certificates on explicitly stated mathematical interfaces.
They do not construct the missing realization.
The approximation layer is therefore not absent only because
extraction remains open: even a successful extraction would leave
the transport and calibration obligations to be discharged.

\paragraph{Source discipline.}
Cheung's manuscripts are identified by supplied filename, version
where available, and internal label. Documentary comparisons are
not proof-transfer premises. Published literature supplies the
process-matrix, quantum-channel, and quantum-network frameworks
\cite{OCB,Watrous,CDP}.
No predecessor proof, empirical implementation, or numerical artifact
is represented as independently verified here.

\section{The unchanged XXII-C interface}
\label{sec:frozen}

For completeness, the relevant frozen objects are stated without
using predecessor auxiliary files.

Let
\[
 \mathsf J=\{A,B,C\},
 \qquad
 H_{a_I}=H_{a_O}=\C^2,
 \qquad
 \mathcal H=
 \bigotimes_{a\in\mathsf J}(H_{a_I}\otimes H_{a_O}).
\]
Use the transposed unnormalized Choi convention
\[
 J(\mathcal M)
 =
 \sum_{i,j=0}^{1}\kb{i}{j}\otimes\mathcal M(\kb{i}{j}),
 \qquad
 M=J(\mathcal M)^{\mathsf T}.
\]
The set \(\ProcessSet\) consists of positive operators \(W\) on
\(\mathcal H\) satisfying
\[
 \Tr[W(M_A\otimes M_B\otimes M_C)]=1
 \quad
 \text{whenever }
 M_a\geq0,\quad \Tr_{a_O}M_a=I_{a_I}.
 \label{eq:process-validity}
\]
Local instrument insertions give the complete probability law
\[
 P_W(\boldsymbol x|\boldsymbol s)
 =
 \Tr\!\left[
 W\bigotimes_{a\in\mathsf J}M^a_{x_a|s_a}
 \right].
 \label{eq:process-law}
\]
This is the process-matrix language of Oreshkov, Costa, and Brukner
\cite{OCB}, with the exact access restrictions of XXII-C
\cite[\loc{spec:access}]{CheungXXIIC}.

The frozen family is
\[
 \ProcessClass=(\ProcessSet)^{\N},
 \qquad
 \boldsymbol W=(W_e)_{e\in\N}.
 \label{eq:process-family}
\]
Each experiment is tested separately. Its identifier is not time,
a refinement scale, or spatial position.

The access \(\Access\) permits one use of each party's process
interface in a selected experiment. The receiver datum is its
local classical instrument record. Other records are summed;
postselection and undeclared interparty feed-forward are excluded.
Local auxiliary implementations are allowed only insofar as they
implement the declared local instrument on that one-use interface.

Let \(\Delta_e(a,b)\) be the supremum of the total-variation distance
between \(b\)'s unconditional laws when a deterministic insertion
at \(a\) is replaced, with all other insertions fixed.
The direct relation on \(\N\times\mathsf J\) is
\[
 (e,a)\Infl_D(f,b)
 \quad\Longleftrightarrow\quad
 e=f,\ a\ne b,\ \Delta_e(a,b)>0.
\]
There are no cross-experiment edges.
Taking strongly connected components and then reachability gives
\[
 (E_D,\preceq_D).
\]

\begin{lemma}[Finite futures of the frozen component order]
\label{lem:finite-futures}
For every datum of \(\ProcessClass\), the component order is a
partial order. Each experiment contributes at most three components,
and
\[
 \bigl|\{q\in E_D:p\preceq_D q\}\bigr|\leq3
 \quad\text{for every }p\in E_D.
 \label{eq:finite-future}
\]
\end{lemma}

\begin{proof}
Mutual reachability is an equivalence relation.
Reachability between its equivalence classes is well defined:
changing either representative adds paths within a component.
It is reflexive and transitive, and mutual comparison of two
classes means that they are the same class.

Every path remains within its experiment identifier.
The corresponding directed graph has three vertices, so its
component quotient has at most three elements. The future of a
component is contained in that quotient.
\end{proof}

This reproduces the relevant content of XXII-C's order theorem
\cite[\loc{thm:order}]{CheungXXIIC}.
The components are not automatically individual source and receiver
laboratories.

\begin{remark}[The proposed protocol interface is not already frozen]
\label{rem:access}
An arbitrary sequence of repeated interparty uses, an external
unknown quantum message, and a retained quantum receiver port are
not specified by \(\Access\).
A local gate sequence implementing one admitted instrument can
be compiled into that instrument; this does not create additional
process uses or new event coordinates.

Thus ``finite local protocols on the same interface'' cannot mean
an undeclared enlargement of \(\Access\).
A transport theorem must state its compatible enlargement and prove
the required complete-channel validity.
\end{remark}

Nor does \eqref{eq:process-family} authorize unrestricted products
or feedback compositions of its members. Such products can fail
process normalization under unrestricted joint operations
\cite{JiaSakharwade}.
A declared composition must have its own validity proof.

\section{Conditional extraction and calibration}
\label{sec:extraction}

\begin{specification}[Conditional extraction output]
\label{spec:extraction}
Fix \(m\geq3\).
For a datum \(D\), suppose a predeclared rule has supplied
\[
 (V_D,\iota_D,K_D,T_D)
\]
with \(\dim V_D=m\), \(\iota_D:E_D\to V_D\) injective, \(K_D\)
proper, and \(T_D:V_D\to\R^m\) invertible, such that
\begin{align}
 p\preceq_D q
 &\quad\Longleftrightarrow\quad
 \iota_D(q)-\iota_D(p)\in K_D,
 \label{eq:faithful}\\
 V_D
 &=\Span\{\iota_D(q)-\iota_D(p):p,q\in E_D\},
 \label{eq:span}\\
 K_D
 &=\overline{\Cone
 \{\iota_D(q)-\iota_D(p):
                 p\preceq_D q,\ p\ne q\}},
 \label{eq:generated}\\
 T_D(K_D)&=\Lor_m
 =
 \{(t,x)\in\R\times\R^{m-1}:t\geq\|x\|_2\}.
 \label{eq:lorentz}
\end{align}
The construction must satisfy the non-circularity and admissibility
audits of XXII-A and XXII-C.
Neither its existence nor a nonempty admissible domain is asserted
in this specification.
\end{specification}

If a revised carrier is used, the carrier, order, and representation
equations must be restated for that carrier.
It is not an unannounced replacement of \(E_D\).

The homogeneous strictly convex cone classification of Ito and
Louren\c{c}o is one conditional route to
\eqref{eq:lorentz}
\cite[Theorem 7]{ItoLourenco}
\cite[\loc{thm:classification}]{CheungXXIIA}.
This note assumes \eqref{eq:lorentz} when discussing the approximation
interface; it does not derive or apply that classification to process
data.

\subsection{What the cone determines}

In coordinates supplied by \(T_D\), write
\[
 T_Dv=(t(v),x(v)),
 \qquad
 q_D(v)=t(v)^2-\|x(v)\|_2^2.
 \label{eq:quadratic}
\]

\begin{proposition}[Conformal information does not fix calibration]
\label{prop:calibration}
The full cone \(K_D\) determines the Lorentzian quadratic form
\eqref{eq:quadratic} up to a positive scalar.
It does not select a preferred Lorentz coordinate frame, a
normalization of that scalar, or physical length and time units.
Consequently it does not determine a numerical speed in physical
units.
\end{proposition}

\begin{proof}
In Lorentz coordinates, any other quadratic form vanishing on the
same null boundary and positive in the future interior has the form
\[
 \widetilde q(t,x)
 =
 a t^2+2t\,b\cdot x+x^{\mathsf T}Cx,
 \qquad C=C^{\mathsf T}.
\]
Vanishing on \((1,u)\) and \((1,-u)\) for every unit \(u\)
gives \(b=0\) and
\[
 u^{\mathsf T}Cu=-a.
\]
Thus \(C=-aI\), and positivity on \((1,0)\) gives \(a>0\).
This is the uniqueness calculation recorded in XXII-A
\cite[\loc{rem:conformal}]{CheungXXIIA}.

For every \(s>0\), the map \(sT_D\) still maps \(K_D\) onto
\(\Lor_m\), but replaces \(q_D\) by \(s^2q_D\).
Lorentz boosts also preserve \(\Lor_m\).
For example, on one temporal and one spatial coordinate,
\[
 (t,x_1)\longmapsto
 (\cosh\eta\,t+\sinh\eta\,x_1,\
  \sinh\eta\,t+\cosh\eta\,x_1)
\]
does not select the same observer when \(\eta\ne0\).

Finally, if coordinate units are converted by
\[
 t_{\mathrm{phys}}=\tau_0 t,
 \qquad
 x_{\mathrm{phys}}=\ell_0 x,
 \qquad \tau_0,\ell_0>0,
\]
then the null equation becomes
\[
 \|x_{\mathrm{phys}}\|_2
 =
 (\ell_0/\tau_0)t_{\mathrm{phys}}.
\]
The constants \(\ell_0,\tau_0\) are not fixed by cone membership.
\end{proof}

\paragraph{Calibration convention of this note.}
All endpoint estimates below are in one declared coordinate frame
\(T_D\). The normalized null slope is one in those coordinates.
A positive comparison unit \(\ell\) is used to make endpoint errors
dimensionless. Its numerical normalization is declared, not derived.

If clocks, rods, tick durations, or physical units are introduced,
their calibration and correspondence to these coordinates belong
to the transport realization. Classification does not fix them.
A Lorentzian quadratic form on one affine carrier is also not a
constructed metric field on a manifold.

\section{Null targets and a frozen-carrier obstruction}
\label{sec:targets}

Define the formal future-null set
\[
 \Null_D=\partial K_D\setminus\{0\}.
\]
A formal anchored target is
\[
 \zeta=(a,v),
 \qquad a\in\iota_D(E_D),\quad v\in\Null_D.
\]
Its target endpoint is \(a+v\), which need not lie in
\(\iota_D(E_D)\).

The represented null-pair set is instead
\[
 \Null_D^{\mathrm{pair}}
 =
 \{(p,q):p\ne q,\
       \iota_D(q)-\iota_D(p)\in\Null_D\}.
 \label{eq:null-pairs}
\]
It may be empty.
Even a member of \eqref{eq:null-pairs} is a represented comparison,
not an identity-transmission protocol.

For a fixed source \(p\), put
\[
 F_p=
 \{\iota_D(q)-\iota_D(p):
                  p\preceq_D q,\ q\ne p\}.
 \label{eq:source-differences}
\]

\begin{theorem}[No full anchored null-boundary approximation on the
unchanged carrier]
\label{thm:anchored-obstruction}
Under Specification~\ref{spec:extraction}, \(F_p\) has at most two
elements. If
\[
 p\preceq_D q_n,\qquad
 \iota_D(q_n)-\iota_D(p)\longrightarrow v\ne0,
\]
then \(v\in F_p\), and the differences are eventually equal to \(v\).

In particular, for \(m\geq3\), the entire nonzero future-null
boundary cannot be approximated from one fixed component by
comparable differences in the unchanged carrier.
\end{theorem}

\begin{proof}
Equation~\eqref{eq:finite-future} gives the cardinality bound.
A finite subset of a finite-dimensional vector space is closed,
so the limit lies in \(F_p\).
Distinct points of a finite set have a positive minimum separation.
Convergence therefore implies eventual equality to the limiting
point.

The boundary \(\partial\Lor_m\setminus\{0\}\) has infinitely many
vectors and infinitely many rays when \(m\geq3\).
An invertible map preserves that fact, whereas \(F_p\) is finite.
\end{proof}

The theorem does not deny approximation on a revised operational
carrier or with moving source labels whose correspondence is
proved. It says that either change is substantive.
It also does not obstruct a theorem restricted to a named subset
of already represented null pairs.

\paragraph{Non-vacuous target declaration.}
A later approximation theorem must state its target set
\(\Targets\) and prove that it is nonempty.
A claim uniform over \(\Null_D^{\mathrm{pair}}\) could otherwise
be vacuous.

For all-direction approximation, a typical coordinate target set
has
\[
 T_Dv=(r,ru),
 \qquad
 \rho\leq r\leq L,\quad \|u\|_2=1,
 \qquad 0<\rho\leq L<\infty.
 \label{eq:target-annulus}
\]
Its endpoints require an operational realization or extension;
\eqref{eq:generated} alone does not provide them.

\section{Geometry and signalling do not certify quantum transport}
\label{sec:capacity}

\begin{proposition}[An entanglement-breaking transport floor]
\label{prop:eb-floor}
Let \(M\cong\C^d\), \(d\geq2\).
If a receiver channel \(\Lambda:L(M)\to L(M)\) is
measure-and-prepare, then
\[
 \ddiamond(\Lambda,\id_M)\geq1-\frac1d.
 \label{eq:eb-floor}
\]
\end{proposition}

\begin{proof}
Write
\[
 \Lambda(X)=\sum_z\Tr(F_zX)\sigma_z,
 \qquad F_z\geq0,\quad \sum_zF_z=I.
\]
For a maximally entangled input
\[
 \ket{\Omega_d}
 =
 d^{-1/2}\sum_{j=1}^{d}\ket j_M\ket j_R,
\]
the receiver--reference output is separable.
This is the measure-and-prepare characterization of
entanglement-breaking channels \cite{HSR}.

A pure product vector has squared overlap with
\(\ket{\Omega_d}\) at most \(1/d\), by Cauchy--Schwarz.
Convexity gives the same bound for a separable state.
Testing the projector \(\kb{\Omega_d}{\Omega_d}\) therefore gives
trace distance at least \(1-1/d\) from the ideal Bell output.
This input is allowed in the diamond norm, proving
\eqref{eq:eb-floor}.
\end{proof}

The Bell-witness argument is also used in Cheung's finite recovery
and streaming obstructions
\cite{CheungEn02C}.
Its use here does not identify those recovery models with the
process family.

\begin{theorem}[A faithful generated Lorentz representation need not
provide quantum transmission]
\label{thm:geometry-not-transport}
There is a datum in the frozen class whose component order has
a faithful generated representation with cone \(\Lor_3\), but whose
designated \(A\)-to-\(B\) connection is classical and cannot transmit
an unknown qubit with half-diamond error below \(1/2\) using local
preprocessing and receiver decoding without an additional
communication resource.

The representation in this example is an independently chosen
mathematical witness, not an operational extraction satisfying
XXII-A's non-circularity requirement.
\end{theorem}

\begin{proof}
For every experiment, use the definite-order circuit that initializes
a qubit at \(A\), connects \(A\)'s output to \(B\)'s input through
\[
 \mathcal Z(X)
 =
 \sum_{j=0}^{1}\bra jX\ket j\,\kb j j,
 \label{eq:classical-channel}
\]
and discards \(B\)'s output.
Party \(C\) receives an independent initialized qubit and has its
output discarded. Definite-order circuits give valid process
matrices \cite{OCB}.

Preparing \(\ket0\) or \(\ket1\) at \(A\) and measuring at \(B\)
gives \(\Delta_e(A,B)=1\).
There is no reverse edge and no edge incident to \(C\).
The extracted order is therefore the disjoint union of
\[
 A_e\prec B_e,\qquad C_e\text{ isolated},
 \qquad e\in\N.
\]

Let \(u_e\) be a countable dense sequence in the unit circle of
\(\R^2\). Define, only for this mathematical example,
\begin{align*}
 \iota(A_e)&=(0,10e,0),\\
 \iota(B_e)&=(1,10e+(u_e)_1,(u_e)_2),\\
 \iota(C_e)&=(0,10e,4).
\end{align*}
The comparable differences are exactly
\[
 \iota(B_e)-\iota(A_e)=(1,u_e).
\]
The \(A_e,B_e\) pair is future-null.
Within one experiment, the spatial distance from \(C_e\) to
\(B_e\) is at least three, exceeding the temporal difference one.
Across experiments, the spatial separation of events at different
time levels is at least nine, again exceeding their temporal
difference. Distinct events at the same time level are also
incomparable. The embedding is therefore injective and faithful
for \(\Lor_3\).

The closure of the cone generated by \((1,u_e)\) is \(\Lor_3\):
density supplies every unit-circle boundary generator, and the
closed unit disk is the convex hull of that circle.
These differences span \(\R^3\).
Thus all three representation equations hold.

Now introduce a qubit message at \(A\) and permit a quantum receiver
output at \(B\), but add no new communication channel or shared
entanglement resource.
Every influence reaching the receiver through
\eqref{eq:classical-channel} is a classical measurement outcome.
Local source processing and receiver decoding leave the resulting
message-to-receiver map in measure-and-prepare form.
Proposition~\ref{prop:eb-floor} with \(d=2\) gives the error floor
\(1/2\).

The spatial placements and dense directions above were chosen
independently of the operational data. They prove a nonimplication
from representation equations and cone shape to transport capacity.
They do not define an admissible data-derived selector.
\end{proof}

The theorem does not refute a future extraction package that includes
a separately proved quantum transmission condition.
It identifies why that condition cannot be replaced by classical
signalling, component reachability, or Lorentz geometry alone.

\section{The missing transport-realization contract}
\label{sec:realization}

\begin{specification}[Compatible transport realization \(\Realization\)]
\label{spec:realization}
For a datum with the conditional output of
Specification~\ref{spec:extraction}, a transport theorem must supply:
\begin{enumerate}[label=\textup{(R\arabic*)}]
\item \textbf{Quantum ports and resources.}
      A named message space \(M\), with \(d=\dim M\geq2\), source
      injection, receiver quantum output, admissible local memories,
      and initial resources independent of the unknown message.
      The qubit case is the minimal extension of the frozen wire
      dimensions. Larger messages require their own capacity and
      implementation statement.

\item \textbf{Validity and composition.}
      The admissible uses of the \(W_e\), their auxiliary extensions,
      and any classical or quantum wiring between uses.
      Prove positivity and normalization of the resulting complete
      maps, including untouched references.
      Reproducing the frozen classical tests after the declared
      reductions must be proved, not assumed from similar notation.

\item \textbf{Carrier correspondence.}
      Source and receiver ports associated with components, or a
      declared extended event carrier \(\widehat E_D\), together
      with its comparison map.
      A component containing several laboratories does not choose
      one of them as a source by itself.

\item \textbf{Geometric compatibility.}
      Coordinates of the realized ports in the extracted affine
      carrier, and the relation between realized dependency or
      influence comparisons and the extracted order.
      If an extension is used, prove the claimed preservation of
      old comparisons and the claimed compatibility of its generated
      cone. Additional ports do not automatically preserve
      \eqref{eq:generated}.

\item \textbf{Execution and calibration.}
      An actual finite execution model for the protocols, with
      elementary communications, local gates, guards, records,
      tick conventions, and arrival readings.
      Its calibration to the chosen extracted coordinates is stated
      separately from cone selection.

\item \textbf{Complete comparison interfaces.}
      The declared retained residual systems, classical histories,
      failure flags, encoders, decoders, and permitted terminal
      reductions. The receiver decoder may not access the source's
      remaining message-bearing systems through an undeclared
      side channel.

\item \textbf{Evidence and invariance.}
      Exact mathematical premises or actual implementation allowances,
      their quantifiers, and any statistical coverage.
      State covariance under permitted operational relabellings and
      coordinate changes.
\end{enumerate}
None of these clauses is certified for a cone-selecting datum here.
\end{specification}

A finite definite-order channel network is a possible realization
language, but not a consequence of a process matrix's signalling
graph. Combs and quantum networks supply frameworks for such
declared sequential interfaces \cite{CDP}.
The scheduler of a realizing protocol is an execution input; it is
not retrospectively claimed as the order derived from the raw data.

If all these structures are additionally assumed, the resulting
approximation theorem remains conditional on them.
To claim a realization from the frozen data, they must instead be
constructed from those data under named additional premises.

\section{Complete-channel transport and error assembly}
\label{sec:channels}

All channels in this section are completely positive and trace
preserving on their stated finite-dimensional interfaces.
Use
\[
 \ddiamond(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond,
 \qquad
 \dtrace(\rho,\sigma)=\frac12\|\rho-\sigma\|_1.
\]
The diamond norm compares channels completely, including an
untouched reference \cite[Chapter 3]{Watrous}.
It is not assigned to an arbitrary process-matrix probability
functional.

\subsection{Reference marginal versus logical transport}

For every channel \(\Lambda:L(M)\to L(M')\),
\[
 \Tr_{M'}[(\Lambda\otimes\id_R)(\rho_{MR})]=\rho_R.
 \label{eq:reference-marginal}
\]
This follows from trace preservation.
It does not establish preservation of the message--reference
correlations.

After a declared identification \(M'\cong M\), identity transport is
the stronger channel equality
\[
 \Lambda=\id_M,
 \qquad
 (\Lambda\otimes\id_R)(X)=X
 \quad\text{for every }X.
 \label{eq:identity}
\]
A replacement channel satisfies \eqref{eq:reference-marginal} and
generally destroys those correlations.
This is the distinction of XXII-B's Candidate D
\cite{CheungXXIIB} and the reference-stable transport statements
of En08-B \cite[\loc{lem:identity}]{CheungEn08B}.

For a declared residual and record system \(F_n\), a complete ideal
transport target is
\[
 \Theta_n(X)=X\otimes\sigma_n,
 \label{eq:complete-target}
\]
where \(\sigma_n\) is input independent.
It may include a declared arrival-record law.
The logical target is \(\id_M\), obtained by tracing out \(F_n\).

\begin{lemma}[Full comparison implies logical and referenced comparison]
\label{lem:full-logical}
If
\[
 \ddiamond(\Gamma_n,\Theta_n)\leq e_n,
 \qquad
 \Gamma_n:L(M)\to L(M\otimes F_n),
\]
then the logical reduction
\(\Lambda_n=\Tr_{F_n}\Gamma_n\) satisfies
\[
 \ddiamond(\Lambda_n,\id_M)\leq e_n.
\]
For every state \(\rho_{MR}\),
\[
 \dtrace((\Lambda_n\otimes\id_R)(\rho_{MR}),\rho_{MR})
 \leq e_n.
 \label{eq:reference-error}
\]
\end{lemma}

\begin{proof}
Partial trace is a channel, so postcomposition contracts the
diamond norm. Tensoring with the reference and evaluating on a
state of trace norm one gives \eqref{eq:reference-error}.
\end{proof}

A logical-output estimate alone is not the same numerical estimate
for the entire retained record--residual channel.
A full comparison must be supplied if that stronger conclusion is
claimed.

\subsection{Telescoping on actual compatible interfaces}

\begin{lemma}[Complete-channel telescoping]
\label{lem:telescoping}
Let
\[
 \overline\Gamma
 =
 D G_k\cdots G_1 E,
 \qquad
 \Gamma
 =
 \widetilde D\widetilde G_k\cdots\widetilde G_1\widetilde E
\]
be compatible channel compositions on matching step interfaces.
Suppose
\[
 \ddiamond(\widetilde E,E)\leq\alpha,\qquad
 \ddiamond(\widetilde G_j,G_j)\leq e_j,\qquad
 \ddiamond(\widetilde D,D)\leq\beta.
\]
Then
\[
 \ddiamond(\Gamma,\overline\Gamma)
 \leq
 \min\left\{1,\alpha+\sum_{j=1}^{k}e_j+\beta\right\}.
 \label{eq:telescoping}
\]
\end{lemma}

\begin{proof}
Insert intermediate compositions replacing one factor at a time.
Each difference contains one changed factor, preceded and followed
by channels. Since channels have diamond norm one, each term is
bounded by the diamond norm of that factor difference.
The triangle inequality and division by two give the sum.
The half-diamond distance between two channels is at most one.
\end{proof}

The same argument is printed in En08-B
\cite[\loc{lem:telescoping}]{CheungEn08B}.
Here it applies only after the realization contract supplies actual
compatible channels.
Memory, classical control histories, and retained residuals must be
present in the step interfaces; an unmodelled correlated memory is
not an independent fresh error term.

\begin{corollary}[Design and implementation allowances]
\label{cor:assembly}
If also
\[
 \ddiamond(\overline\Gamma_n,\Theta_n)\leq\mu_n,
\]
then
\[
 \ddiamond(\Gamma_n,\Theta_n)
 \leq
 \min\left\{
 1,\mu_n+\alpha_n+\sum_{j=1}^{k_n}e_{n,j}+\beta_n
 \right\}.
 \label{eq:assembled}
\]
If \(k_n\leq H_n\) and \(e_{n,j}\leq\epsilon_n\), the sum is at
most \(H_n\epsilon_n\).
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:telescoping}, then the triangle inequality
with \(\Theta_n\), and cap the result at one.
\end{proof}

The allowance \(\mu_n\) is a design comparison, not a geometric
endpoint error. Geometric and channel errors are kept separate.

\subsection{Arrival records and complete attempts}

Let \(Z_n\) be a retained finite classical arrival register with a
declared calibration \(z\mapsto a_n(z)\in\R\).
A failure code is included in its alphabet.
For a target arrival coordinate \(\tau\), let the good codes satisfy
\[
 |a_n(z)-\tau|\leq\omega_n
\]
and exclude failure codes.

\begin{proposition}[Arrival-window transfer]
\label{prop:arrival}
If the ideal record marginal of \(\Theta_n\) is supported entirely
on the good codes and
\(\ddiamond(\Gamma_n,\Theta_n)\leq e_n\), then, for every input
state, including a referenced input,
\[
 \Pr_{\Gamma_n}\{
 \text{failure or arrival outside the declared window}
 \}
 \leq e_n.
 \label{eq:arrival-bound}
\]
\end{proposition}

\begin{proof}
The bad-code projector is an effect on the complete output.
Its ideal probability is zero. Its actual probability differs
from that value by at most the output trace distance, bounded
by the half-diamond error.
\end{proof}

This is an unconditional complete-attempt bound.
It is not a normalized postselected channel guarantee on accepted
trials. The instrument and availability of the decoded output at
the claimed arrival must themselves be supplied by the realization.

\section{Endpoint topology and a conditional certificate}
\label{sec:endpoint}

Choose a nonempty declared target set \(\Targets\) of anchored
future-null targets. In the selected coordinates write
\[
 T_Da=(s,y),\qquad T_Dv=(\tau,x),\qquad
 \tau=\|x\|_2>0.
\]
For each \(\zeta=(a,v)\in\Targets\) and refinement \(n\), a realized
protocol has declared source coordinates \(a_{n,\zeta}\) and
receiver coordinates \(b_{n,\zeta}\) in the same coordinate space.
Write
\[
 b_{n,\zeta}-a_{n,\zeta}
 =
 (\tau_{n,\zeta},x_{n,\zeta}).
\]
The coordinate assignments are part of R3--R5; they are not inferred
from experiment numbering.

Set
\[
 \|(t,x)\|_* = |t|+\|x\|_2.
\]
This is a comparison norm in a chosen frame, not a newly derived
physical spatial metric.

\begin{definition}[Anchored endpoint error]
\label{def:endpoint}
For a decoded logical channel \(\Lambda_{n,\zeta}\), define
\[
 \Err_n(\zeta)
 =
 \frac{
 \|a_{n,\zeta}-T_Da\|_*
 +\|x_{n,\zeta}-x\|_2
 +|\tau_{n,\zeta}-\tau|
 }{\ell}
 +
 \ddiamond(\Lambda_{n,\zeta},\id_M).
 \label{eq:endpoint-error}
\]
\end{definition}

The source term matters: displacement accuracy alone could compare
a protocol beginning at the wrong operational source.
The identity target requires a declared logical identification;
the cone does not assign a quantum transport channel to a vector.

\begin{theorem}[Conditional endpoint approximation certificate]
\label{thm:certificate}
Suppose a compatible transport realization supplies, for every
\(\zeta\in\Targets\) and \(n\):
\begin{enumerate}[label=\textup{(\roman*)}]
\item a finite protocol and complete decoded channel
      \(\Gamma_{n,\zeta}:L(M)\to L(M\otimes F_n)\);
\item an ideal complete target
      \(\Theta_{n,\zeta}(X)=X\otimes\sigma_{n,\zeta}\);
\item matching ideal and implemented channel factorizations with
      at most \(H_n\) steps, uniform design allowance \(\mu_n\),
      encoder allowance \(\alpha_n\), decoder allowance \(\beta_n\),
      and complete step allowance \(\epsilon_n\);
\item uniform endpoint estimates
      \[
       \|a_{n,\zeta}-T_Da\|_*
       +\|x_{n,\zeta}-x\|_2\leq\delta_n,
       \qquad
       |\tau_{n,\zeta}-\tau|\leq\omega_n.
      \]
\end{enumerate}
Then, for the logical reduction
\(\Lambda_{n,\zeta}=\Tr_{F_n}\Gamma_{n,\zeta}\),
\[
 \sup_{\zeta\in\Targets}\Err_n(\zeta)
 \leq
 \frac{\delta_n+\omega_n}{\ell}
 +
 \min\{1,\mu_n+\alpha_n+H_n\epsilon_n+\beta_n\}.
 \label{eq:certificate}
\]
In particular, uniform anchored endpoint approximation holds if
\[
 \delta_n,\omega_n,\mu_n,\alpha_n,\beta_n\longrightarrow0,
 \qquad
 H_n\epsilon_n\longrightarrow0.
 \label{eq:certificate-convergence}
\]
The same channel allowance bounds all message--reference output
errors. If the ideal arrival records satisfy the window condition,
it also bounds the failure and out-of-window probability.
\end{theorem}

\begin{proof}
Corollary~\ref{cor:assembly} gives the uniform complete-channel
allowance. Lemma~\ref{lem:full-logical} transfers it to logical
channels and referenced inputs.
Substituting the endpoint estimates into
\eqref{eq:endpoint-error} gives \eqref{eq:certificate}.
Equation~\eqref{eq:certificate-convergence} makes its right side
tend to zero.
Proposition~\ref{prop:arrival} gives the last statement.
\end{proof}

This is an implication from constructed or explicitly assumed
protocol data. It does not assert that those data exist.
It also proves no global channel evolution at intermediate times.

\section{Locality and resource bounds require a model}
\label{sec:locality}

A cone contains no elementary gate size, no tick duration, and no
storage cost. An unconditional lower bound on gate count or memory
as a numerical function of distance therefore cannot be inferred
from cone selection alone.

For example, a positive rescaling of the affine embedding rescales
all coordinate lengths while leaving its order and conical
generation unchanged. It does not change a given gate count.
Any numerical distance--resource comparison must fix its scale
and allowed elementary operations.

\subsection{A finite-network lower bound}

\begin{specification}[Bounded-step realizing networks]
\label{spec:bounded-step}
The following lower bounds concern finite acyclic networks of
complete channels, not arbitrary process matrices.
Assume:
\begin{enumerate}
\item an external message \(M\cong\C^d\) is introduced at a source;
      other initial resources are fixed independently of that message;
\item all quantum wires, classical controls, routing choices, and
      retained records occur in the dependency graph;
\item source and receiver interfaces have specified spatial
      coordinates in the chosen extracted frame;
\item every spatially nonzero dependency hop advances by at most
      \(a_n>0\);
\item execution has \(N_n\) rounds and every source--receiver
      dependency path contains at most \(N_n\) such hops;
\item no undeclared communication or decoder access bypasses this
      graph.
\end{enumerate}
The step bound and round convention are additional realization
premises. They are not supplied by \eqref{eq:lorentz}.
\end{specification}

\begin{lemma}[No ancestral path gives a constant receiver channel]
\label{lem:no-path}
Under Specification~\ref{spec:bounded-step}, if there is no dependency
path from the message source to the receiver interface, its logical
channel is a replacement channel.
Its error from identity satisfies
\[
 \ddiamond(\Lambda,\id_M)\geq1-\frac1{d^2}.
 \label{eq:constant-floor}
\]
\end{lemma}

\begin{proof}
Pull a receiver observable backwards through the acyclic network.
A channel whose outputs carry only the identity observable
contributes the input identity, because its adjoint is unital.
Including classical and quantum predecessors gives support only
on the receiver's scheduled ancestors.
If the message input is absent from those ancestors, the receiver
law is independent of every message state.
This is the scheduled-support argument of VII-J and En06-A
\cite{CheungVIIJ}
\cite[\loc{thm:order}]{CheungEn06A}.

Linearity then gives
\[
 \Lambda(X)=\Tr(X)\sigma
\]
for one state \(\sigma\).
On a maximally entangled message--reference input, the output is
\(\sigma\otimes I_R/d\), whose Bell overlap is \(1/d^2\).
The Bell-projector test gives \eqref{eq:constant-floor}.
\end{proof}

\begin{theorem}[Conditional propagation-depth lower bound]
\label{thm:depth}
Under Specification~\ref{spec:bounded-step}, suppose
\[
 \ddiamond(\Lambda_{n,\zeta},\id_M)<1-\frac1{d^2}.
\]
Let \(r=\|x\|_2\) be the target spatial displacement and suppose
\(\|x_{n,\zeta}-x\|_2\leq\delta_n\).
Then
\[
 N_n\geq
 \left\lceil
 \frac{\pos{r-\delta_n}}{a_n}
 \right\rceil.
 \label{eq:depth-lower}
\]
If each round has calibrated duration \(h_n>0\), measured from the
declared source-input cut, then that execution duration is at least
the right side of \eqref{eq:depth-lower} multiplied by \(h_n\).
\end{theorem}

\begin{proof}
Lemma~\ref{lem:no-path} forces a dependency path.
If it contains \(k\) spatially nonzero hops, the triangle inequality
gives
\[
 \|x_{n,\zeta}\|_2\leq k a_n\leq N_n a_n.
\]
The reverse triangle inequality gives
\(\|x_{n,\zeta}\|_2\geq\pos{r-\delta_n}\).
This proves the integer bound.
The duration statement is the declared round-duration convention.
\end{proof}

If \(a_n=\ell/n\), the bound is linear in \(n\) for a fixed nonzero
target once \(\delta_n\to0\). At a fixed scale it is linear in
distance. The choice \(a_n=\ell/n\), however, is an additional
realization prescription.

The ratio \(a_n/h_n\) is a model-dependent propagation ceiling.
No invariant physical speed is derived by naming that ratio.

\subsection{Footprint sites are not automatically peak memory}

\begin{proposition}[A static-site footprint bound]
\label{prop:site-lower}
Suppose a realizing network uses \(q_n\) fixed spatial sites.
All quantum and classical communication edges between distinct
sites have length at most \(a_n\).
If a source--receiver dependency path exists and their displacement
is \(x_{n,\zeta}\), then
\[
 q_n\geq
 1+
 \left\lceil
 \frac{\|x_{n,\zeta}\|_2}{a_n}
 \right\rceil.
 \label{eq:sites}
\]
If, additionally, the resource model keeps a \(d\)-dimensional
register at every used site live simultaneously, its live quantum
dimension is at least \(d^{q_n}\).
\end{proposition}

\begin{proof}
The communication path lies in a connected finite graph on the
used sites. Erasing loops gives a simple spatial graph path with
at most \(q_n-1\) edges.
The triangle inequality gives
\(\|x_{n,\zeta}\|_2\leq(q_n-1)a_n\), proving
\eqref{eq:sites}.
The simultaneous-register clause gives the product dimension.
\end{proof}

Without the simultaneous-register premise, a site-count bound is
not a peak-memory bound. Dynamic allocation, retirement, buffering,
and reuse require their actual execution ledger.
Cheung's one-shot and streaming ledgers illustrate precisely this
distinction
\cite{CheungEn02A,CheungEn02B}.
No memory optimum for those models is imported here.

\begin{definition}[Required resource ledger]
\label{def:ledger}
A finite protocol records at least
\[
 \Ledger_{n,\zeta}
 =
 (U_{n,\zeta},G_{n,\zeta},N_{n,\zeta},
  Q_{n,\zeta},q_{n,\zeta},
  C_{n,\zeta},A_{n,\zeta}),
\]
where:
\begin{itemize}
\item \(U\) is the number and type of process-resource uses;
\item \(G\) counts gates under a stated elementary gate convention;
\item \(N\) is execution depth or calibrated rounds;
\item \(Q\) is peak live quantum memory;
\item \(q\) is the distinct-site or port footprint;
\item \(C\) is classical communication, distinguished from storage;
\item \(A\) is total auxiliary allocation, distinguished from peak use.
\end{itemize}
The error allowances and endpoint estimates are recorded alongside
this vector, with their dependence on \(n,\zeta\).
\end{definition}

No \(O(n)\) gate, footprint, or memory upper bound is established
for the frozen process family in this note.
En08-B's resource scaling belongs to its constructed static-site
transport model
\cite[\loc{thm:main}]{CheungEn08B}.

\section{The six approximation obligations}
\label{sec:obligations}

\begin{specification}[Approximation obligations A1--A6]
\label{spec:obligations}
A later operational approximation theorem must discharge:
\begin{enumerate}[label=\textup{(A\arabic*)}]
\item \textbf{Finite protocols and quantum transmission.}
      Construct protocols for one declared nonempty target set and
      supply the compatible access enlargement of
      Specification~\ref{spec:realization}.
      Prove transmission of the unknown message with an untouched
      reference. Do not substitute preservation of the reference
      marginal for identity transport.
      Theorem~\ref{thm:geometry-not-transport} is a standing
      transmission-capability test.

\item \textbf{Anchored endpoint and complete-channel control.}
      Provide actual source and receiver maps, complete comparison
      channels, residual and record conventions, and estimates of
      the form \eqref{eq:certificate}.
      State whether the claim is logical-output comparison or a
      full retained-interface comparison.
      Use an event-aware topology when arrival events move.

\item \textbf{Resource upper ledger.}
      Give explicit resource functions under
      Definition~\ref{def:ledger}, including every operative
      communication and auxiliary resource.
      Prove the error scaling needed for
      \eqref{eq:certificate-convergence}.
      No predecessor asymptotic count is transferred without its
      realizing-model correspondence.

\item \textbf{Model-dependent locality lower bounds.}
      State the elementary step bounds, execution rules, and cost
      model under which lower bounds are claimed.
      Theorems~\ref{thm:depth} and
      Proposition~\ref{prop:site-lower} apply only under their
      stated premises.
      A spatial ceiling does not prove availability of every null
      direction, nor does a depth bound automatically bound memory.

\item \textbf{Arrival and calibration.}
      Construct the declared output availability, arrival window
      or distribution, and calibrated record interface.
      Account for failed and late attempts without normalized
      postselection.
      Frame, scale, units, and clock correspondence must retain
      the status described in Proposition~\ref{prop:calibration}.

\item \textbf{Carrier compatibility and failure tests.}
      Retain Theorem~\ref{thm:anchored-obstruction} and the tests of
      Section~\ref{sec:tests}.
      If the operational carrier or allowed compositions are enlarged,
      prove their compatibility with the extracted representation.
      State separately any exact null-attainment or continuum
      evolution claim.
\end{enumerate}
\end{specification}

These obligations supplement extraction. They do not repair an
unsatisfied extraction target or a circular geometric input.

\section{Failure tests}
\label{sec:tests}

\subsection{F1: generated boundary is not protocol attainment}

Let
\[
 G=\{(1,x):x\in\Q^{m-1},\ \|x\|_2<1\}.
\]
Every nonzero finite conical combination is strictly inside
\(\Lor_m\): the strict triangle estimate gives
\[
 \left\|\sum_j\lambda_jx_j\right\|_2
 <
 \sum_j\lambda_j
\]
when at least one \(\lambda_j>0\).
Nevertheless,
\[
 \overline{\Cone G}=\Lor_m
\]
by density of rational interior points in the closed unit ball.

This is a generator example, not a faithful quantum event
representation. It shows why a closure-produced null boundary need
not contain an attained generator or protocol.
XXII-B gives the same scope test
\cite[Failure test F3]{CheungXXIIB}.

Even convergence of implemented endpoints to a null target does not
assert equality with that target at any finite refinement.
Exact logical transmission and exact null-coordinate attainment
are also different predicates.

\subsection{F2: a fixed finite stencil does not cover every null direction}

\begin{proposition}[Finite-stencil directional delay]
\label{prop:stencil}
In a chosen Lorentz coordinate frame, suppose every moving step has
nonnegative duration and spatial velocity in a fixed finite set
\[
 S\subseteq\{s\in\R^{m-1}:\|s\|_2\leq1\},
 \qquad 0\in S.
\]
For \(m\geq3\), choose a unit vector \(u\notin S\) and set
\[
 h_S(u)=\max_{s\in S}u\cdot s<1.
\]
If a protocol's endpoint has
\[
 \|x_n-ru\|_2\leq\delta_n,\qquad r>0,
\]
and total step duration \(\tau_n\), then
\[
 r-\delta_n\leq h_S(u)\tau_n.
 \label{eq:stencil-bound}
\]
If \(h_S(u)=0\), sufficiently accurate endpoints are impossible.
If \(h_S(u)>0\) and \(\delta_n\to0\), then
\[
 \liminf_n\tau_n\geq r/h_S(u)>r.
\]
Thus this stencil cannot approximate the null target \((r,ru)\)
with vanishing arrival delay.
\end{proposition}

\begin{proof}
For a unit \(u\), equality \(u\cdot s=1\) with \(\|s\|_2\leq1\)
requires \(s=u\). Hence \(h_S(u)<1\).
Because \(0\in S\), it is nonnegative.

Writing step durations as \(\Delta t_j\geq0\) gives
\[
 u\cdot x_n
 =
 \sum_j\Delta t_j\,u\cdot s_j
 \leq h_S(u)\sum_j\Delta t_j
 =
 h_S(u)\tau_n.
\]
The endpoint estimate gives \(u\cdot x_n\geq r-\delta_n\).
The conclusions follow.
\end{proof}

The same reasoning permits velocities in \(\operatorname{conv}S\).
An increasing stencil may avoid this particular obstruction, but
requires an actual routing and resource theorem.
Isotropy is not proved by the upper bound \(\|s\|_2\leq1\) alone.

The outline's En08-B labels
\loc{prop:exact-null} and \loc{prop:fixed-stencil} are treated as
documentary identifiers. The tests in this section are established
by their printed arguments, not by importing unsupplied predecessor
proofs.

\subsection{F3: moving arrivals obstruct uniform comparison across the event}

\begin{proposition}[A moving arrival jump]
\label{prop:jump}
Let \(\tau_n>\tau\) and \(\tau_n\to\tau\).
On a common message-plus-flag interface define
\[
 \mathcal J_\tau(t)(X)=
 \begin{cases}
 \Tr(X)\sigma_0\otimes\kb00,&t<\tau,\\
 X\otimes\kb11,&t\geq\tau.
 \end{cases}
\]
Then endpoint arrival times converge and the post-arrival transport
channels are identical, but
\[
 \sup_t\ddiamond(\mathcal J_{\tau_n}(t),
                 \mathcal J_\tau(t))=1
 \quad\text{for every }n.
\]
\end{proposition}

\begin{proof}
For \(\tau\leq t<\tau_n\), the flags of the two outputs are
orthogonal. Their trace distance is one for every state input.
The half-diamond distance is therefore one.
The channel distance cannot exceed one.
\end{proof}

Uniform estimates away from the arrival event, or a declared
event-aware endpoint topology, can be appropriate.
Arrival-time convergence alone is not uniform channel convergence
through a discontinuity.

\subsection{F4: small errors can accumulate}

\begin{proposition}[An exact accumulation witness]
\label{prop:noise}
On a qubit let
\[
 \mathcal R_0(X)=\Tr(X)\kb00,
 \qquad
 \mathcal F_e=(1-e)\id+e\mathcal R_0,
 \qquad 0\leq e\leq1.
\]
For an unencoded sequence of \(k\) such steps,
\[
 \ddiamond(\mathcal F_e^{\,k},\id)
 =
 1-(1-e)^k.
 \label{eq:noise}
\]
Consequently, if \(k_n\to\infty\), \(e_n\to0\), and
\(k_ne_n\to a>0\), the endpoint error tends to \(1-e^{-a}>0\).
\end{proposition}

\begin{proof}
Since \(\mathcal R_0^2=\mathcal R_0\),
\[
 \mathcal F_e^{\,k}
 =
 \mathcal R_0+(1-e)^k(\id-\mathcal R_0).
\]
The half-diamond distance between \(\mathcal R_0\) and identity
is one: it is at most one for channels, and the input \(\kb11\)
gives orthogonal outputs.
Homogeneity of the norm gives \eqref{eq:noise}.

Finally,
\[
 k_n\log(1-e_n)
 =
 -k_ne_n+O(k_ne_n^2)\longrightarrow-a.
\]
\end{proof}

This is a lower-bound witness for the displayed noise model without
error correction. It does not prove that every nonzero per-step
allowance prevents convergence.

Conversely, if a telescoping upper bound fails to tend to zero,
that fact alone proves only failure of that certificate.
Error cancellation or a separately constructed correction scheme
requires its own analysis.

\subsection{F5: tolerance is not exact causal containment}

For comparison channels independent of an intervention, uniform
implemented allowances \(e\) imply at most a \(2e\) intervention
difference by the triangle inequality. They do not imply exact
absence of influence
\cite[\loc{prop:tolerance}]{CheungXXIIB}.

A protocol can therefore approximate identity on a target while
having small additional signals outside an asserted exact support
region. Exact support and tolerance support require different
guards and different statements.

The separation
\[
 \Infl\subseteq\Sched
\]
holds for the complete definite-order networks used in
Section~\ref{sec:locality}. Containment in an independently supplied
geometric relation \(\Supplied\) requires a separate embedding
certificate \cite{CheungVIIJ}.
No endpoint norm allowance supplies that certificate.

\subsection{F6: endpoint limits are not a unique continuum evolution}

A family of terminal channels and endpoint comparisons need not define
compatible channels at all intermediate times.
Any evolution-limit claim must provide its own encoders, decoders,
complete interfaces, and statement such as
\[
 \sup_{0\leq t\leq T}
 \ddiamond\bigl(
 \mathcal D_n\mathcal C_n(t)\mathcal E_n,\mathcal T_t
 \bigr)\longrightarrow0
 \quad\text{for every finite }T,
\]
together with its composition and continuity hypotheses.

XIX-C's two-refinement and interpolation witnesses show the missing
information in an inference from one coarse channel or finite samples
\cite[\loc{prop:two-limits}, \loc{prop:interpolation},
      \loc{def:limit}]{CheungXIXC}.
They do not exclude uniqueness after appropriate additional data.
Naming a consistency condition is not its construction or a
uniqueness proof.

\section{Component ledger and release}
\label{sec:release}

\begingroup
\small
\renewcommand{\arraystretch}{1.13}
\begin{longtable}{@{}p{.24\linewidth}p{.68\linewidth}@{}}
\toprule
Component & Role and boundary\\
\midrule
\endfirsthead
\toprule
Component & Role and boundary\\
\midrule
\endhead
\bottomrule
\endfoot

XXII-A &
Faithful-generation target and non-circularity audit.
No positive extraction theorem is inherited
\cite{CheungXXIIA}.\\[4pt]

XXII-B &
Boundary, reference, tolerance, and refinement distinctions.
Its catalogue does not supply a transport realization
\cite{CheungXXIIB}.\\[4pt]

XXII-C &
Frozen tripartite family, one-use access, component order,
and open affine-construction rule.
Its finite-future structure is retained here
\cite{CheungXXIIIC}.\\[4pt]

En08-A &
Scheduler obstruction and warnings concerning generated boundaries,
finite samples, and metric interpretation
\cite{CheungEn08A}.\\[4pt]

En08-B &
Documentary approximation pattern and complete-channel telescoping.
Its metric, speed, lattice, and static-site resources are not
outputs of the present programme
\cite{CheungEn08B}.\\[4pt]

VII-J; En06-A &
Scheduled support versus operational influence for complete
definite-order networks; geometric containment is additional
\cite{CheungVIIJ,CheungEn06A}.\\[4pt]

XIX-C &
Refinement and interpolation non-determination, and the complete
evolution-limit interface
\cite{CheungXIXC}.\\[4pt]

En05-B; En06-B &
Their supplied speed and realization records are distinct models.
No identification with the limit of an extracted cone is made
\cite{CheungEn05B,CheungEn06B}.\\[4pt]

Family 265 /
Oai265-A--B &
Static area-law records do not supply transport or metric extraction.
Finite channel certificates require their actual statistical and
calibration premises
\cite{CheungOai265A,CheungOai265B}.\\[4pt]

VIII-A/B/C;
Em02-E &
Documentary conical-family and classical-lapse comparisons as
recorded in XXII-A. They do not supply the transport contract
\cite{CheungXXIIA}.\\
\end{longtable}
\endgroup

\noindent
In the XXII-C row, the bibliographic key denotes the supplied
\loc{XXII-C.tex}; no external LaTeX reference file is used.

\subsection{Exact status}

The completed release consists of:
\begin{enumerate}
\item the finite-future and anchored-boundary obstruction;
\item the separation between cone geometry and quantum transmission;
\item the corrected calibration statement;
\item the compatible transport-realization contract;
\item complete-channel, endpoint, arrival, and bounded-step
      conditional certificates;
\item the six approximation obligations and explicit failure tests.
\end{enumerate}

The abstract conditional certificates are proved.
They are not instantiated by a successful cone-selecting datum and
a compatible protocol family.

\begin{target}[Extraction followed by compatible approximation]
\label{target:approximation}
Supply:
\begin{enumerate}[label=\textup{(D\arabic*)}]
\item one nonempty admissible data domain and a predeclared successful
      extraction meeting the XXII-A and XXII-C target;
\item one separately justified transport realization
      \(\Realization\), including every enlarged access or composition
      interface;
\item one declared nonempty target family and its source/receiver
      correspondence;
\item finite protocols satisfying A1--A6, with the applicable
      convergence, resource, locality, and arrival statements;
\item the required relabelling invariance and evidence allowances.
\end{enumerate}
If the target covers the whole null boundary from a fixed source,
the unchanged carrier obstruction must be repaired by an explicit,
compatible carrier or realization change.
This research target has no existence proof here.
\end{target}

Both non-vacuity gates matter: a nonempty extraction domain is not
a nonempty domain of compatible transport realizations.
Conversely, constructing a transport network for a prescribed
Lorentz cone does not meet the operational extraction target.

\paragraph{Unchanged boundaries.}
No successful operational extraction, invariant physical speed,
metric field, exact null transport event, or geometric causal
embedding is established.
No Einstein dynamics, holographic reconstruction, Page curve,
gravity-generated entanglement, or quantum evolution without
external time is proved.

Nothing here changes O.1, opens XV-B, discharges XVI-N or Route B,
or alters H2, I3, or J2. No quantum coefficient is admitted.
No successful pdfLaTeX compilation, programme execution, laboratory
implementation, or proof-assistant verification is asserted
\cite{CheungXIXE}.

\paragraph{Final gap statement.}
A successful extraction must first be constructed.
A compatible quantum transport realization, calibration, and
nonempty target family must then be supplied independently of the
mere cone classification.
Only after those constructions can
Theorem~\ref{thm:certificate} certify operational endpoint
approximation.

On the unchanged XXII-C carrier, universal fixed-source null-boundary
approximation is obstructed by finite futures.
On a compatible revised realization, its existence remains an
open construction problem.
En08-B remains a theorem about its prescribed boundary, not about
a boundary derived by the present programme.

\begin{thebibliography}{99}

\bibitem{CheungXXIIA}
N.~Cheung.
\emph{XXII-A: Freeze the precise claim and the gap.
Stage 0 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-A.tex}, n.d.
Relevant labels:
\loc{target:selection}, \loc{def:representation},
\loc{thm:classification}, and \loc{rem:conformal};
conditions (T1)--(T5), (A1)--(A6).

\bibitem{CheungXXIIB}
N.~Cheung.
\emph{XXII-B: Catalogue the additional quantum data and the
hypothesis ledger.
Stage 1 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-B.tex}, n.d.
Relevant items: Candidate D, ledger syntax, failure tests F3--F5,
\loc{prop:tolerance}, and \loc{prop:two-limits}.

\bibitem{CheungXXIIC}
N.~Cheung.
\emph{XXII-C: Freeze one quantum process class and the extraction
obligations.
Stage 2 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-C.tex}, n.d.
Relevant labels:
\loc{spec:access}, \loc{def:class}, \loc{thm:order},
\loc{def:feasible}, \loc{def:extraction},
and \loc{target:selection}.

% Alias used in the component ledger; it refers to the same supplied
% manuscript and is included to keep the standalone source resolved.
\bibitem{CheungXXIIIC}
N.~Cheung.
\emph{XXII-C}, supplied manuscript \loc{XXII-C.tex}, n.d.
Same manuscript as the preceding reference; component-ledger
identification only.

\bibitem{OCB}
O.~Oreshkov, F.~Costa, and \v{C}.~Brukner.
Quantum correlations with no causal order.
\emph{Nature Communications}
\textbf{3} (2012), 1092.
\doi{10.1038/ncomms2076}.
Preprint: \url{https://arxiv.org/abs/1105.4464}.

\bibitem{Watrous}
J.~Watrous.
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Chapters 2, 3, and 6.
Author's text:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{CDP}
G.~Chiribella, G.~M.~D'Ariano, and P.~Perinotti.
Theoretical framework for quantum networks.
\emph{Physical Review A}
\textbf{80} (2009), 022339.
\doi{10.1103/PhysRevA.80.022339}.
Preprint: \url{https://arxiv.org/abs/0904.4483}.

\bibitem{JiaSakharwade}
D.~Jia and N.~Sakharwade.
Tensor products of process matrices with indefinite causal structure.
\emph{Physical Review A}
\textbf{97} (2018), 032110.
\doi{10.1103/PhysRevA.97.032110}.
Preprint, under the title \emph{Process Products Need Qualification}:
\url{https://arxiv.org/abs/1706.05532}.

\bibitem{HSR}
M.~Horodecki, P.~W.~Shor, and M.~B.~Ruskai.
Entanglement breaking channels.
\emph{Reviews in Mathematical Physics}
\textbf{15} (2003), 629--641.
\doi{10.1142/S0129055X03001709}.
Preprint:
\url{https://arxiv.org/abs/quant-ph/0302031}.

\bibitem{ItoLourenco}
M.~Ito and B.~F.~Louren\c{c}o.
The \(p\)-cones in dimension \(n\geq3\) are not homogeneous
when \(p\ne2\).
\emph{Linear Algebra and its Applications}
\textbf{533} (2017), 326--335.
\doi{10.1016/j.laa.2017.07.029}.
Preprint: \url{https://arxiv.org/abs/1701.05801}.
Theorem 7 is a classification reference, not an operational
transport premise.

\bibitem{CheungEn08A}
N.~Cheung.
\emph{En08-A}, version 1, manuscript \loc{En08-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:main}, \loc{thm:mainpositive},
\loc{thm:representation}, \loc{thm:weaker},
and the boundary, tolerance, and finite-sample warnings.
Registry: \url{https://enter.grok.me/}.

\bibitem{CheungEn08B}
N.~Cheung.
\emph{En08-B}, version 1, manuscript \loc{En08-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{lem:identity}, \loc{lem:telescoping},
\loc{thm:main}, \loc{def:comparison},
and \loc{cor:operational-limit}.
The outline additionally names \loc{prop:exact-null} and
\loc{prop:fixed-stencil}; no unsupplied proof of those propositions
is used here.

\bibitem{CheungVIIJ}
N.~Cheung.
\emph{VII-J}, version 1, manuscript \loc{VII-J.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:schedule}, \loc{thm:nosignal},
\loc{thm:physical}, and \loc{thm:release}.
Scheduled ancestry, operational influence, and supplied geometric
causality have separate interfaces.

\bibitem{CheungEn06A}
N.~Cheung.
\emph{En06-A}, version 1, manuscript \loc{En06-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:order}, \loc{lem:onelayer},
\loc{cor:influence}, \loc{def:experiment}, and \loc{def:limit}.
Finite complete-network support and the separate coarse
outcome-law completion.

\bibitem{CheungXIXC}
N.~Cheung.
\emph{XIX-C}, version 1, manuscript \loc{XIX-C.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{def:limit}, \loc{prop:two-limits},
\loc{prop:interpolation}, and \loc{thm:main}.
Refinement non-determination and the complete operational-limit
interface.

\bibitem{CheungEn05B}
N.~Cheung.
\emph{En05-B}, version 1, manuscript \loc{En05-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied label: \loc{thm:main}.
Isotropic matched operational existence for a prescribed
constant \(c>0\); documentary comparison only.

\bibitem{CheungEn06B}
N.~Cheung.
\emph{En06-B}, manuscript \loc{En06-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied scope record: \loc{thm:T}.
Finite-network transport on an increasing-stencil family with
prescribed speed; documentary comparison only.

\bibitem{CheungEn02A}
N.~Cheung.
\emph{En02-A}, version 1, manuscript \loc{En02-A-v1.tex}.
Enter Calculus Companion, n.d.
Relevant supplied label: \loc{thm:Dprime}.
One-shot instrument environment ledger and its separation from
adaptive peak memory.

\bibitem{CheungEn02B}
N.~Cheung.
\emph{En02-B}, version 1, manuscript \loc{En02-B-v1.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:batch}, \loc{thm:B5}, and \loc{prop:observable}.
Batch, streaming, observable, and subsystem-recovery resource
tasks have different scopes.

\bibitem{CheungEn02C}
N.~Cheung.
\emph{En02-C}, version 1, manuscript \loc{En02-C-v1.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:C2}, \loc{thm:C3},
\loc{lem:firstmeasurement}, and \loc{cor:robustgap}.
Bell witnesses and a model-specific streaming recovery obstruction;
not a transport realization for the present class.

\bibitem{CheungOai265A}
N.~Cheung.
\emph{Oai265-A}, manuscript \loc{Oai265-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied label: \loc{thm:separation}.
Static area-law records, recovery, and radiation histories are
distinct tasks; documentary negative control.

\bibitem{CheungOai265B}
N.~Cheung.
\emph{Oai265-B}, version 2, manuscript \loc{Oai265-B-v2.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{ass:statistics}, \loc{thm:main},
\loc{eq:coverage}, and \loc{eq:falseaccept}.
Complete-record finite-confidence channel certificate;
not evidence for the realization contract in this note.

\bibitem{CheungXIXE}
N.~Cheung.
\emph{XIX-E}, version 1, manuscript \loc{XIX-E.tex}.
Enter Calculus Companion, n.d.
Relevant supplied passage:
``What remains outside the release.''
Documentary programme boundary record.

\end{thebibliography}

\end{document}