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\hypersetup{
  pdftitle={X-Enter-ST: Readout closure of an obstructed convolution coefficient on a prescribed conical host},
  pdfauthor={Ngo Cheung},
  pdfsubject={Surjective gauge orbit, complete destructive instruments,
  conditional relational readout, and supported operational transport}
}

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\title{\textbf{X-Enter-ST: Readout closure of an obstructed\\
convolution coefficient on a prescribed conical host}\\[5pt]
\large Complete destructive instruments, conditional relational
inputs,\\
and the separation from the fixed-width ramp prescription}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We place Cheung's X-Enter readout-closure problem on the
unchanged supported I-B/X-A-ST conical host. The analytical
coefficient remains the fixed-width convolution gap inherited
from X-A--Convolution; it is not replaced by the ramp
prescription. Its positive charge \(c_\varepsilon\) generates
the coefficient orbit \(a(\lambda)=\lambda c_\varepsilon\),
whose image is the entire real input domain.

For an arbitrary encoding and an arbitrary declared complete
output map, exact constancy on this orbit forces independence
from the scalar input. The conclusion concerns the composite
readout on the encoded family, not constancy of the instrument
on unrelated states. For the retained qubit amplitude encoding,
we give the selective-map criterion, the explicit total-variation
gap, and the second-order test missed by a vanishing first
derivative. We also state a quantitative approximate-orbit
version and retain the exact bounded-segment
tolerance--visibility trade-off.

The positive operational construction is a finite dictionary
of supplied evaluations and status certificates, equipped with
a complete destructive-clock instrument. Its output contains
the classical code and every unmeasured finite residual, and
its channel identities hold with untouched references.
Scheduled locality and a nonempty regular SWAP route attach
this interface to the prescribed host without identifying
transport with gravity-generated entanglement.

An independently charged second coefficient would permit a
conditional relational readout. We prove its two-qubit
CNOT--Hadamard instrument, including the retained failure
outcome and the necessity of respecting the declared
coarse-record boundary. That coefficient and its
whole-algebra charge matching are not supplied here.

The contribution is a supported, complete-interface audit of
negative closure: changing readout cannot repair a transitive
scalar gauge orbit, and implementing a supplied report does not
acquire the continuum coefficient. The parent obstruction and
readout mechanisms are attributed to X-Enter. No convolution
repair, field-to-qubit preparation, interacting differential,
characteristic quantum coefficient, or spacetime-emergence
theorem is constructed. The release is local Grade S.
\end{abstract}

\statementbox{The principal verdict}{%
The convolution coefficient alone has no informative readout
that is invariant on its full inherited witness orbit.

The finite report instrument and its supported route are
constructed under declared operational permissions.

The second charged reference remains an additional classical
input. The ramp prescription supplies neither that reference
nor a repair of the convolution coefficient.}

\tableofcontents

%=====================================================================
\section{Purpose, attribution, and the Enter discipline}
\label{sec:purpose}
%=====================================================================

\subsection{Source and exact object}

The principal source is Cheung's
\emph{Landmark X-Enter: Readout closure tests for the obstructed
convolution coefficient: a named comparison with the ramp
prescription}, version 1 \cite{XEnter}.
Its surjective-orbit lemma, amplitude tests, conditional
relational protocol, and bounded-segment analysis are the
parent results transferred here.

The classical convolution premises are reproduced explicitly
as Assumption~\ref{ass:convolution}. Their role is limited:
the readout proofs require a nonzero real charge and the
stated coefficient orbit. They do not require a new
derivation of the convolution kernels.
The full X-A--Convolution construction is not reconstructed
from an unspecified mollifier \cite{XConv,XEnter}.

The supplied X-A-ST, X-G-ST, and X-H-ST provide the supported
presentation and finite supplied-report comparisons
\cite{XAST,XGST,XHST}. Their identifiers are not registration
or independent-verification certificates.

The present contribution is the attachment to the retained
host, a complete output and reference audit, explicit supplied
report instruments, and the separation of mathematical
encoding from field acquisition. It is not a priority claim
for the parent orbit obstruction or conditional circuit.

\subsection{Four non-identifications}

Cheung's philosophy of Enter distinguishes a destructive
readout interface from quantum return and from geometric
identification \cite{EnterPhilosophy,EnterI}.
Here that discipline has four binding consequences:
\begin{enumerate}
\item a defined coefficient is not acquisition of unknown
continuum field data;
\item a probability law is not a complete instrument
returning its quantum residual;
\item a reference-preserving transport channel is not
entanglement generated by gravity;
\item the prescribed conical host is not derived from a
readout or a gauge complex.
\end{enumerate}

A measured clock name is absent from the returned quantum
interface. This is an access statement, not a claim that
the apparatus, its environment, or the geometric time axis
has been annihilated \cite{IB,En09}.

We use explicitly specified finite Born instruments.
No Enter characterization theorem is reconstructed from
an abbreviated axiom list, and no such characterization
is a premise of the classical gap.

\subsection{Corrections needed for a coherent transfer}

Three distinctions are essential.

First, the orbit theorem concerns the function
\(\Phi\circ\Enc\) on real scalar inputs. It does not say that
\(\Phi\) is constant on its entire quantum input space.

Second, a finite-dimensional qubit can carry a continuously
parameterized family of nonorthogonal states. This is not
an orthogonal finite-basis encoding of every real number,
nor an exact finite gate-list preparation of arbitrary
unknown real data.

Third, placing a supplied numerical report in a regular
laboratory does not establish that the original convolution
witness, its acquisition, or its complete nonlocal
dependence lies in that laboratory. The required support
certificate is stated separately in
Section~\ref{subsec:support}.

These corrections preserve the negative closure result;
they do not weaken it.

%=====================================================================
\section{Stage A: the inherited host and the frozen coefficient}
\label{sec:host}
%=====================================================================

\subsection{Unchanged geometric presentation}

\begin{assumption}[Retained I-B/X-A-ST instance]
\label{ass:host}
Retain exactly the previously supplied instance:
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item a regular time-oriented conical product
\((\mathcal M^\circ,G)\), its stipulated completion defect
\(\mathcal W\), and its supplied future relation;
\item a supported injective developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ,
\]
with the retained regular compact reserve disjoint from
\(\mathcal W\);
\item the independently declared analytical completion
\[
 \widehat M=\RR^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 M_{T_*}=[0,T_*]\times\RR^3,\qquad
 \Sigma=\{t=0\};
\]
\item the signature and time ledger
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma);
\]
\item a nonempty observation region
\[
 \mathcal O\Subset(0,T_*)\times\RR^3,\qquad
 D_*:=J_\eta^-(\overline{\mathcal O})\cap M_{T_*}
                                             \Subset\mathcal L;
\]
\item a regular convex developed spatial tube and a positive
time reserve for the finite protocols below.
\end{enumerate}
All global field equations and Green operators belong to
\(\widehat M\), not to a purported globally hyperbolic
regular cone.
\end{assumption}

In I-B's coordinates, the prescribed regular metric is
locally represented by
\[
 G=-c_{\mathrm{host}}^2\dd T^2+\dd r^2
          +\alpha^2r^2\dd\varphi^2+\dd z^2,
 \qquad
 \alpha=1-\frac{\mu}{2\pi},\quad 0<\mu<2\pi.
\]
The completion locus is
\[
 \mathcal W=\RR_T\times\{o\}\times\RR_z,
\]
a worldsheet. A fixed-\(z\) section is a worldline;
it is not the whole defect \cite{IB}.

The coordinate formula describes the retained instance.
It does not select a new deficit or update its metric.
There is no second cone, no host-update map, and no global
conical inverse.

For comparison with the inherited ramp, retain a fixed
positive width and plateau:
\[
 0<\varepsilon<\delta,\qquad
 \sigR(t)=\chi(t/\varepsilon),\qquad
 \overline U_R\Subset
      \mathcal L\cap\operatorname{int}\{g_R=1\}
                    \cap\{\varepsilon<t<T_*\},
\]
\[
 \gR=C_c^\infty(U_R;T\widehat M).
\]
The profile \(\chi\) is smooth, zero before zero, one after
one, and flat at the endpoints. Its width is not removed.

\subsection{The convolution input contract}

Use \(\Enc\) for state encoding, reserving \(\iota\) for the
geometric chart. Also distinguish \(c_\varepsilon\) from the
host speed \(c_{\mathrm{host}}\).

\begin{assumption}[Frozen convolution data]
\label{ass:convolution}
Fix the source-specific convolution operator
\(S_\varepsilon\), switch and Cauchy prescription,
compact real observation test \(f_C\), and real-linear
supported gauge algebra \(\gC\) of X-A--Convolution.
Retain the following explicit inputs:
\begin{enumerate}[label=\textup{(C\arabic*)}]
\item for the source's operators,
\[
 \Ezero\,\mathsf C\,T_{g,\varepsilon}(K\xi)
          =S_\varepsilon\xi,\qquad \xi\in\gC;
\]
\item an admitted witness \(\xi_*\in\gC\) has the finite
real coefficient
\begin{equation}
 c_\varepsilon
   =\int_{\widehat M}
      f_C(x)\bigl(\xi_*-S_\varepsilon\xi_*\bigr)^\mu(x)
                    \partial_\mu x^1\,\dd^4x>0,
 \qquad 0<\varepsilon\le\frac18;
 \label{eq:gap}
\end{equation}
\item every real multiple \(\lambda\xi_*\) remains in
\(\gC\), and the evaluated first-order coefficient has
the orbit
\begin{equation}
 a(\lambda)=\lambda c_\varepsilon,\qquad\lambda\in\RR.
 \label{eq:orbit}
\end{equation}
\end{enumerate}
The operators, normalization, width range, and witness in
these statements retain their source meanings
\cite{XConv,XEnter}.
\end{assumption}

The bound \(1/8\) is a source-coordinate bound, not a new
physical length or time calibration.
The symbol \(x^1\) in \eqref{eq:gap} is a rod coordinate
on the analytical completion; it is not asserted to be
a global Cartesian function on the cone.

The positive charge \(c_\varepsilon\) is fixed during an
orbit comparison. The variable being read is \(a\).
An output may depend on frozen calibration data such as
\(c_\varepsilon,\varepsilon\), or an encoding scale while
being independent of \(a\). The orbit theorem forbids
informativeness about \(a\), not the reporting of fixed
metadata.

\begin{remark}[What is assumed and what is proved]
Positivity in \eqref{eq:gap} is an inherited classical
input, as it is in X-Enter. It is not inferred here from
the word ``convolution,'' and no new kernel formula is
invented to establish it.

All readout theorems below are proved from the explicit
contract above. Thus their dependence on the unreconstructed
classical source is visible and minimal.
\end{remark}

Equation~\eqref{eq:orbit} is a statement about a first-order
coefficient. It is not a uniform finite-coupling metric
expansion for unbounded \(\lambda\). An assertion through
a finite perturbative order is always made for each fixed
gauge amplitude.

\subsection{Support attachment versus laboratory acquisition}
\label{subsec:support}

The negative readout theorem depends only on the numerical
contract \eqref{eq:gap}--\eqref{eq:orbit}. Therefore its
application does not require a new cone Green operator.

To interpret \eqref{eq:gap} itself as a retained
laboratory-supported field test, one additionally needs a
source-specific certificate placing the smearing and the
convolution/preparation dependence actually used by that
test inside \(\mathcal L\). The ramp causal reserve does
not automatically provide that certificate for a different
convolution prescription.

\begin{proposition}[Exact scope of supported attachment]
\label{prop:attachment}
The following statements are distinct:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the scalar input contract is used in an experiment
whose finite ports lie in the retained regular laboratory;
\item the continuum coefficient has a certified
laboratory-supported realization;
\item an apparatus acquires that coefficient from unknown
continuum field data.
\end{enumerate}
The construction below establishes (i).
It preserves (ii) whenever its separate support certificate
is supplied, and constructs no map proving (iii).
\end{proposition}

\begin{proof}
A supplied number or status certificate can be installed
as finite task data at a regular port without evaluating
its continuum definition there.
This gives (i).

If the source-specific dependence reserve lies in the
chart, the compact density pairing is transported by
change of variables, with its density Jacobian retained.
That transport preserves its numerical value, giving the
conditional assertion in (ii).
It does not replace \(S_\varepsilon\) by a cone operator.

Neither a change of variables nor a supplied finite table
defines a field-to-apparatus acquisition map. Thus neither
argument supplies (iii).
\end{proof}

This qualification prevents a completion-level witness
from being silently renamed a local field detector.

%=====================================================================
\section{Stage B: the encoding-independent orbit obstruction}
\label{sec:orbit}
%=====================================================================

\begin{definition}[Declared complete output]
\label{def:complete}
For an encoding \(\Enc:\RR\to\mathscr S\), let
\(\mathcal F(a)=\Phi(\Enc(a))\) denote the entire declared
output: every reported classical word and every returned
quantum factor.

It is \emph{orbit invariant} if
\[
 \mathcal F(\lambda c_\varepsilon)=\mathcal F(0)
                                  \quad(\lambda\in\RR).
\]
It is \emph{informative about \(a\)} if \(\mathcal F\)
is nonconstant as a function of \(a\).
\end{definition}

The definition is meaningful for arbitrary sets and maps.
Quantum channels are a special case.

\begin{theorem}[Surjective-orbit negative closure]
\label{thm:orbit}
Let \(\Enc:\RR\to\mathscr S\) be any encoding and let
\(\Phi:\mathscr S\to\mathscr Y\) be any declared output
map. Under Assumption~\ref{ass:convolution},
\[
 \Phi(\Enc(\lambda c_\varepsilon))
          =\Phi(\Enc(0))\quad\text{for all }\lambda
\]
implies
\[
 \Phi(\Enc(a))=\Phi(\Enc(0))\quad\text{for all }a\in\RR.
\]
Consequently no scalar-only readout is both invariant on
the full witness orbit and informative about its input.
\end{theorem}

\begin{proof}
For any \(a\in\RR\), take
\(\lambda=a/c_\varepsilon\).
This is possible because \(c_\varepsilon>0\).
Substitution proves the assertion.
\end{proof}

This is X-Enter's \loc{lem:orbit}, included with its proof
and exact quantifier \cite{XEnter}.
No Born rule, continuity, injective encoding, dimensional
bound, linearity, or amplitude model enters it.

The same proof works for
\(a(\lambda)=a_0+\lambda c_\varepsilon\).
It does not prove that \(\Phi\) is constant on states outside
\(\Enc(\RR)\).

\begin{corollary}[No downstream scalar-only repair]
\label{cor:downstream}
Appending fixed postprocessing, adaptive finite control,
or a transport channel cannot turn an invariant
scalar-only complete output into an informative one.

If an appended output remains informative about \(a\),
the resulting complete output is not invariant on the
full witness orbit.
\end{corollary}

\begin{proof}
A fixed appended operation is another map of the same
encoded input. Apply Theorem~\ref{thm:orbit}.
\end{proof}

A randomized preparation is also covered once its
complete state-valued encoding, including any returned
preparation label, is specified as a function of \(a\).
An extra side channel not determined by \(a\) changes the
input contract and must be declared.

\subsection{A quantitative approximate version}

\begin{proposition}[Approximate orbit constancy]
\label{prop:approxorbit}
Suppose the output is a normalized finite
classical--quantum state \(\Omega(a)\) and
\[
 \sup_{\lambda\in\RR}
       \dtr(\Omega(\lambda c_\varepsilon),\Omega(0))
                                                    \le\eta.
\]
Then
\[
 \dtr(\Omega(a),\Omega(0))\le\eta,\qquad
 \dtr(\Omega(a),\Omega(b))\le2\eta
\]
for all real \(a,b\).

For any finite label family \(a_x\), prior \(\pi_x\),
and estimator based on the declared output,
\[
 \Pr\{\widehat x=x\}\le
                 \min\{1,\max_x\pi_x+\eta\}.
\]
\end{proposition}

\begin{proof}
The first conclusion follows by the same substitution
as in Theorem~\ref{thm:orbit}; the second follows by the
triangle inequality.

Let \(M_x\) be the estimator's POVM effects.
Trace distance bounds an effect expectation, so
\[
 \sum_x\pi_x\Tr(M_x\Omega(a_x))
 \le \sum_x\pi_x\Tr(M_x\Omega(0))+\eta.
\]
The first sum is at most
\(\max_x\pi_x\sum_x\Tr(M_x\Omega(0))=\max_x\pi_x\).
\end{proof}

This estimate concerns the full supplied output family.
If additional accessible systems carry the label, they
must be included in that family before the hypothesis
can be applied.

%=====================================================================
\section{Stage C: the retained finite amplitude trial}
\label{sec:amplitude}
%=====================================================================

\subsection{Finite programs and complete selective maps}

A finite program uses fixed finite-register unitaries,
fresh initialized registers, finite classical control,
and destructive computational-basis readout.
Every live quantum name is eventually consumed or
returned. Failure outcomes are included.

For a complete finite history \(h\), compose its
allocation isometries, gates, and measurement bras into
a linear path map \(K_h\). Group histories by the
reported classical output \(y\):
\begin{equation}
 \Phi_y(\rho)=\sum_{h:\mathrm{out}(h)=y}
                        K_h\rho K_h^\dagger,\qquad
 \sum_hK_h^\dagger K_h=I.
 \label{eq:pathmaps}
\end{equation}
The complete recorded channel is
\begin{equation}
 \Rec_\Phi(\rho)
       =\sum_y\proj y_R\otimes\Phi_y(\rho).
 \label{eq:record}
\end{equation}

\begin{lemma}[Finite completeness]
\label{lem:paths}
Equation~\eqref{eq:pathmaps} defines completely positive
selective maps whose recorded channel is trace preserving.
All identities extend by the identity on every untouched
reference.
\end{lemma}

\begin{proof}
Each path gives a completely positive summand.
Backwards induction on the finite instruction tree gives
the completeness relation: isometries preserve the identity
in the adjoint calculation, and at a readout the squared
measurement bras sum to the identity.
Grouping complete histories changes no sum.
Tensoring every path map with a reference identity proves
the extension.
\end{proof}

The parent class with at most four live qubits is a
subclass of this interface \cite{XEnter}.
The proofs here hold for every fixed finite instrument.
Additional task clocks, route registers, or payloads are
charged separately; their attachment does not preserve
an unqualified four-qubit total budget.

\subsection{The encoding}

Freeze \(\sigma>0\), with the units of \(a\), and set
\begin{equation}
 p(a)=\frac{1+\tanh(a/\sigma)}2,\qquad
 \ket{e(a)}=\sqrt{1-p(a)}\ket0+\sqrt{p(a)}\ket1,\qquad
 \tau(a)=\proj{e(a)}.
 \label{eq:encoding}
\end{equation}
The constant comparison encoding is
\(\ket{e_0(a)}=\ket+\).

Let
\[
 D=\proj1-\proj0,\qquad
 X=\kb01+\kb10,\qquad
 t(a)=\tanh(a/\sigma),\qquad s(a)=\sech(a/\sigma).
\]
Then
\begin{equation}
 \tau(a)=\frac12\bigl(I+t(a)D+s(a)X\bigr).
 \label{eq:bloch}
\end{equation}

The basis \(\ket0,\ket1\) may be the two calibrated
basis states of a finite encoding clock. The values
of \(a\) occur in its amplitudes, not as a continuum
of orthogonal clock codes.

For any fixed instrument define
\begin{equation}
 A_y=\Phi_y(I/2),\qquad
 B_y=\Phi_y(D/2),\qquad
 C_y=\Phi_y(X/2).
 \label{eq:ABC}
\end{equation}
Linearity gives the exact selective expression
\begin{equation}
 \Phi_y(\tau(a))=A_y+B_yt(a)+C_ys(a).
 \label{eq:selective}
\end{equation}
Taking traces gives the classical law with coefficients
\(\alpha_y=\tr A_y,\beta_y=\tr B_y,\gamma_y=\tr C_y\).

\begin{theorem}[Amplitude visibility or erasure]
\label{thm:amplitude}
For the family \eqref{eq:encoding}, the following are
equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete output is invariant on
\(a(\lambda)=\lambda c_\varepsilon\);
\item \(B_y=C_y=0\) for every reported output \(y\);
\item the complete output is independent of \(a\) on
\(\RR\).
\end{enumerate}
For the classical law alone the corresponding criterion
is \(\beta_y=\gamma_y=0\) for every \(y\).
\end{theorem}

\begin{proof}
Differentiation of \eqref{eq:selective} gives
\begin{equation}
 \frac{\dd}{\dd\lambda}\Phi_y(\tau(\lambda c_\varepsilon))
  =\frac{c_\varepsilon}{\sigma}
                      \bigl(B_ys^2-C_yst\bigr).
 \label{eq:derivative}
\end{equation}
Different classical blocks cannot cancel one another.
At \(\lambda=0\), a zero derivative forces \(B_y=0\).
At any nonzero \(\lambda\), \(st\ne0\), forcing
\(C_y=0\).
The converse follows from \eqref{eq:selective}.
The same argument after taking traces proves the
law criterion.
\end{proof}

This is the parent amplitude mechanism
\cite[\loc{thm:amplitude}]{XEnter}.
Erasure here means erasure of dependence on the declared
amplitude family. It does not imply a replacement channel
on arbitrary qubit states. For example, a \(Y\)-basis
measurement has a constant law on this real-amplitude
family but not on the whole Bloch sphere.

\subsection{An explicit visible readout}

Direct destructive readout has
\[
 \Pr(1\mid a)=p(a).
\]
Its witness discrepancy is
\begin{equation}
 \Delta_\varepsilon
   :=\dTV(\Law_{\mathrm{read}}(c_\varepsilon),
                       \Law_{\mathrm{read}}(0))
    =\frac12\tanh(c_\varepsilon/\sigma)>0.
 \label{eq:TV}
\end{equation}
In fact,
\[
 \frac{\dd}{\dd\lambda}
       \Pr(1\mid\lambda c_\varepsilon)
   =\frac{c_\varepsilon}{2\sigma}
                  \sech^2(\lambda c_\varepsilon/\sigma)>0.
\]
Thus this readout detects variation of the encoded
gauge coefficient. It does not make that variation
gauge invariant \cite{XEnter}.

For fixed finite \(a\), an ideal mathematical preparation
is a one-qubit rotation taking \(\ket0\) to \(\ket{e(a)}\).
Installing finitely many such rotations or their
approximations is not a construction of a field-to-qubit
preparation for arbitrary smooth input data.

\begin{proposition}[No deterministic decoder of the witness pair]
\label{prop:decoder}
No fixed quantum instrument and classical decoder
perfectly distinguish \(\tau(0)\) from
\(\tau(c_\varepsilon)\) in one execution.
\end{proposition}

\begin{proof}
Let \(E\) be the total effect of outcomes decoded as
the second input. Perfect decoding would require
\[
 0\le E\le I,\qquad
 \bra{e(0)}E\ket{e(0)}=0,\qquad
 \bra{e(c_\varepsilon)}E\ket{e(c_\varepsilon)}=1.
\]
Positivity implies
\(E\ket{e(0)}=0\) and
\(E\ket{e(c_\varepsilon)}=\ket{e(c_\varepsilon)}\).
Hermiticity would make these vectors orthogonal.
Their components in \eqref{eq:encoding} are strictly
positive, so their inner product is positive.
\end{proof}

The result is not an acquisition theorem, and the
visible binary law is not exact real-number output.

\subsection{The retained-order diagnostic}

\begin{theorem}[Second-order encoding test]
\label{thm:secondorder}
Keep \(\sigma\) and the instrument fixed.
Suppose, for each fixed \(\lambda\),
\[
 a(\kappa,\lambda)
    =\kappa\lambda c_\varepsilon
                +\kappa^2b(\lambda)+\ord(\kappa^3),
 \qquad b(0)=0.
\]
If
\[
 \Rec_\Phi(\tau(a(\kappa,\lambda)))
             -\Rec_\Phi(\tau(0))=\ord(\kappa^3)
\]
in trace norm for every fixed \(\lambda\), then
\(B_y=C_y=0\) for all \(y\).
\end{theorem}

\begin{proof}
The expansions
\[
 t(a)=a/\sigma+\ord(a^3),\qquad
 s(a)=1-a^2/(2\sigma^2)+\ord(a^4)
\]
give, in every selective block,
\[
 \frac{\kappa\lambda c_\varepsilon}{\sigma}B_y
 +\kappa^2\left(
       \frac{b(\lambda)}{\sigma}B_y
       -\frac{\lambda^2c_\varepsilon^2}{2\sigma^2}C_y
          \right)+\ord(\kappa^3).
\]
The first coefficient gives \(B_y=0\); the second
then gives \(C_y=0\).
\end{proof}

Hadamard followed by destructive readout is a useful
countercheck:
\begin{equation}
 \Pr(1\mid a)=\frac{1-\sech(a/\sigma)}2
             =\frac{a^2}{4\sigma^2}+\ord(a^4).
 \label{eq:Hadamard}
\end{equation}
Its first derivative vanishes at zero, but its law is
not invariant.

The theorem is an encoding expansion, as in X-Enter
\cite[\loc{thm:secondorder}]{XEnter}.
It is not a new second-order calculation of the
convolution prescription, and its remainder is not
uniform in unbounded \(\lambda\).

%=====================================================================
\section{Stage D: complete instruments for supplied evaluations}
\label{sec:instrument}
%=====================================================================

\subsection{A finite dictionary and its status semantics}

A dictionary is a finite list of tasks. Each entry contains:
\begin{enumerate}
\item a quantity and prescription tag;
\item the frozen width, test-data identifier, and support
contract relevant to that quantity;
\item a supplied exact finite value or rational enclosure,
when a numerical quantity is available;
\item a status certificate with its quantifier;
\item any requested finite encoding and its error allowance.
\end{enumerate}

Permitted entries include the following.
\[
\begin{array}{p{.28\linewidth}|p{.60\linewidth}}
 \text{Entry type}&\text{Meaning}\\ \hline
 \text{Positive convolution charge}&
 A supplied enclosure \(0<L\le c_\varepsilon\le U\).\\[3pt]
 \loc{ORBIT_CONSTANT}&
 A specified scalar-input output is proved constant,
 for example the erasure channel below.\\[3pt]
 \loc{INFORMATIVE_FORBIDDEN}&
 The requested conjunction of full-orbit invariance
 and informativeness about \(a\) is excluded by
 Theorem~\ref{thm:orbit}.\\[3pt]
 \loc{NOT_CONSTRUCTED}&
 A named additional object, such as the independently
 charged second coefficient, has not been supplied.\\[3pt]
 \text{Amplitude witness}&
 A finite supplied evaluation or enclosure of
 \(\Delta_\varepsilon\) in \eqref{eq:TV}.
\end{array}
\]

The status \loc{ORBIT_CONSTANT} is not assigned merely
because several sampled values agree.
For example, a complete erasure channel
\[
 \rho\longmapsto\tr(\rho)\proj{0}_{R}
\]
has a valid universal certificate.
A finite table of endpoint checks alone does not certify
an all-real orbit identity.

Existence of rational \(L,U\) enclosing a fixed positive
finite \(c_\varepsilon\) follows from
Assumption~\ref{ass:convolution}.
An algorithm producing those enclosures from arbitrary
smooth field descriptions is not assumed.

A report of the fixed charge is not a measurement of the
variable orbit value \(a(\lambda)\). A task label identifying
a supplied value is likewise not an acquired continuum
coefficient.

\subsection{The destructive task clock}

Let the dictionary have \(J\) entries.
Choose \(d_C=2^{n_C}\ge J\), \(n_C\ge1\), and an
orthonormal calibrated basis
\(\{\ket{\tau_j}\}_{j=0}^{d_C-1}\).
Unused labels have an explicit unused-task report.
Let \(A\) contain every surviving finite quantum register.

For the installed classical word \(q_j\), define
\[
 D_j=\bra{\tau_j}_C\otimes I_A
\]
and
\begin{equation}
 \mathcal M(X)
   =\sum_j\proj{j,q_j}_R\otimes D_jXD_j^\dagger.
 \label{eq:clockinstrument}
\end{equation}

\begin{theorem}[Complete supplied-evaluation instrument]
\label{thm:clock}
The map \eqref{eq:clockinstrument} is CPTP and
\begin{equation}
 \Tr_R\mathcal M(X)=\Tr_CX
 \label{eq:drop}
\end{equation}
on every input operator. Both identities hold after
tensoring with \(\id_E\) for every untouched reference.

For a basis-task input,
\[
 (\mathcal M\otimes\id_E)
        (\proj{\tau_j}_C\otimes\rho_{AE})
       =\proj{j,q_j}_R\otimes\rho_{AE}.
\]
For a general correlated input the returned blocks are
the actual subnormalized conditional operators
\((D_j\otimes I_E)X(D_j^\dagger\otimes I_E)\).

The returned quantum interface contains \(A\), not \(C\).
\end{theorem}

\begin{proof}
The recorded Kraus maps are
\[
 K_j=\ket{j,q_j}_R\otimes D_j,\qquad
 \sum_jK_j^\dagger K_j
              =\sum_j\proj{\tau_j}_C\otimes I_A=I.
\]
This proves complete positivity and trace preservation.

Expand
\[
 X=\sum_{j,l}\kb{\tau_j}{\tau_l}_C\otimes X_{jl}.
\]
The \(j\)-th output block is \(X_{jj}\).
Summing these blocks gives \(\Tr_CX\), proving
\eqref{eq:drop}. The same expansion with \(A\otimes E\)
proves every reference identity.
The basis-input statement is its special case.
The quantum codomain specifies consumption of \(C\).
\end{proof}

This is the complete I-B/En09-A clock interface
\cite{IB,En09}, with the present dictionary attached.
No normalized state is assigned to a zero-probability
branch. Conditional reference states may be steered;
their unconditional marginal is unchanged.

\subsection{Combining the task clock and an amplitude program}

If a task selects a finite amplitude program, regard
\(A\) initially as containing its signal input \(Q_a\)
and a passive residual \(A_0\).
For common padded output interfaces, the combined map is
\begin{equation}
 \mathcal N(X)
  =\sum_{j,y}\proj{j,q_j,y}_R\otimes
     \bigl(\Phi_y^{(j)}\otimes\id_{A_0}\bigr)
                               (D_jXD_j^\dagger).
 \label{eq:combined}
\end{equation}
It is CPTP by composition and
Lemma~\ref{lem:paths}. Its complete Kraus maps and
reference extensions are therefore explicit.

This formula does not give free access to a
gauge-dependent task selector. During an orbit comparison
the apparatus, task choice, gates, and external
preparations are frozen. If a label or residual varies
with \(a\), it must be included in the encoding and
complete output. The orbit theorem then applies to the
resulting complete scalar-input map.

Returning an unmeasured \(\tau(a)\) while reporting a
constant bit is not erasure. It gives the varying output
\[
 \proj0_R\otimes\tau(a).
\]
A carried qubit is also not Option B's independent
classical reference coefficient \cite{XEnter}.

\subsection{No acquisition and no universal clock return}

\begin{proposition}[Report observables and acquisition]
\label{prop:noacquisition}
The instrument constructs finite report observables,
not an apparatus for acquiring \(c_\varepsilon\) or
\(a\) from unknown continuum fields.
\end{proposition}

\begin{proof}
For displayed real report values \(\widehat a_j\), the
bounded output observable
\[
 B_R=\sum_j\widehat a_j\proj{j,q_j}_R
\]
has pullback
\[
 \mathcal M^*(B_R\otimes I_A)
       =\sum_j\widehat a_j\proj{\tau_j}_C\otimes I_A.
\]
This is an observable of the installed finite task encoding.
The input and Kraus maps contain no field-to-register
acquisition map.
Different hypothetical continuum inputs with the same
supplied finite input therefore produce the same output.
\end{proof}

\begin{proposition}[No universal quantum return]
\label{prop:noreturn}
For \(d_C\ge2\), \(\mathcal M\) has no channel left
inverse on its full clock--residual input space.
\end{proposition}

\begin{proof}
With any fixed residual \(\omega_A\), the distinct clock
states
\[
 \ket{\chi_\pm}
       =\frac{\ket{\tau_0}\pm\ket{\tau_1}}{\sqrt2}
\]
give the same output
\[
 \frac12\left(\proj{0,q_0}_R+\proj{1,q_1}_R\right)
                                      \otimes\omega_A.
\]
No left inverse can send this one output to both inputs.
\end{proof}

This is the elementary no-return mechanism of En09-A
\cite{En09}. It does not exclude recovery of a separately
declared code in another surviving factor.

The calibration labels are finite history or task labels.
If distinct labels are all required to equal the time of
one fixed readout event on positive branches, at most one
can occur. Clock conditioning alone also supplies no
unitary relational dynamics without the extra covariance
and preparation hypotheses of En09-A.

The geometric coordinates \(t,T\), the worldsheet, and
the prescribed future relation survive the readout.

%=====================================================================
\section{Stage E: conditional relational input}
\label{sec:relational}
%=====================================================================

\subsection{The additional classical hypothesis}

\begin{assumption}[Independently charged second coefficient]
\label{ass:reference}
A separately defined second coefficient \(r\) and a proved
nonzero real charge \(r_\varepsilon\) are supplied, with
\[
 a(\lambda)=a_0+\lambda c_\varepsilon,\qquad
 r(\lambda)=r_0+\lambda r_\varepsilon.
\]
For whole-algebra comparisons, their first-order charges
on \(\gC\) are specified linear functionals
\(q_a,q_r\).
\end{assumption}

This assumption changes the input from one coefficient
to a pair. It is not supplied by the present manuscript.

\begin{proposition}[Relational combination and matching]
\label{prop:matching}
Under Assumption~\ref{ass:reference},
\begin{equation}
 I_{\mathrm{rel}}=r_\varepsilon a-c_\varepsilon r
 \label{eq:Irel}
\end{equation}
is invariant on the witness orbit.

It is invariant under every direction of the specified
whole coefficient algebra precisely when
\begin{equation}
 r_\varepsilon q_a(\xi)-c_\varepsilon q_r(\xi)=0
                         \qquad(\xi\in\gC).
 \label{eq:matching}
\end{equation}
\end{proposition}

\begin{proof}
Substituting the affine witness laws cancels the
\(\lambda\)-coefficient in \eqref{eq:Irel}.
For an arbitrary direction \(\xi\), the charge of the
same combination is the left side of
\eqref{eq:matching}; its vanishing is exactly the
stated first-order invariance.
\end{proof}

The baseline symbols \(a_0,r_0\) do not themselves prove
gauge-invariant physical values.
Choosing a compensating number by hand is not a
construction of an independently defined charged
coefficient.

Conditional classical pre-combination followed by
\eqref{eq:encoding} gives
\[
 \Pr(1\mid I_{\mathrm{rel}})
       =\frac{1+\tanh(I_{\mathrm{rel}}/\sigma_I)}2,
 \qquad \sigma_I>0.
\]
The probability is injective in \(I_{\mathrm{rel}}\).
Hence it preserves exactly an established identity of
that combination, not an identity created by readout.

\subsection{The two-qubit protocol}

Freeze
\[
 \sigma>0,\qquad
 \sigma'=\frac{r_\varepsilon}{c_\varepsilon}\sigma\ne0,
\]
where the second scale may be signed.
Define the abstract input family
\begin{equation}
 \ket{\psi(a,r)}
    =\frac12
       (\ket0+e^{i\alpha}\ket1)
            \otimes(\ket0+e^{i\beta}\ket1),
 \quad
 \alpha=\frac a\sigma,\quad
 \beta=\frac r{\sigma'},\quad
 \theta=\alpha-\beta
                  =\frac{I_{\mathrm{rel}}}{\sigma r_\varepsilon}.
 \label{eq:phaseinput}
\end{equation}

Apply CNOT with the first qubit as control and the second
as target, then Hadamard on the first.
Destructively read both qubits, giving raw bits \((b,e)\).
Report
\[
 (b,e)=(0,1)\mapsto+,\qquad
 (1,1)\mapsto-,\qquad
 (0,0),(1,0)\mapsto F.
\]
The failure output \(F\) is retained.
There is no returned signal qubit.

Let
\[
 \ket{\phi_\pm}=(\ket{00}\pm\ket{11})/\sqrt2,\qquad
 \ket{\chi_\pm}=(\ket{01}\pm\ket{10})/\sqrt2.
\]

\begin{theorem}[Conditional complete relational instrument]
\label{thm:relational}
The protocol has selective maps, on arbitrary two-qubit
input operators,
\begin{align}
 \Phi_F(\rho)
   &=\bigl(\bra{00}\rho\ket{00}
                   +\bra{11}\rho\ket{11}\bigr)\one,
 \nonumber\\
 \Phi_+(\rho)&=\bra{\chi_+}\rho\ket{\chi_+}\,\one,
 \qquad
 \Phi_-(\rho)=\bra{\chi_-}\rho\ket{\chi_-}\,\one.
 \label{eq:relmaps}
\end{align}
They form a complete instrument.
On \eqref{eq:phaseinput},
\begin{equation}
 \Pr(F)=\frac12,\qquad
 \Pr(+)=\frac{1+\cos\theta}{4},\qquad
 \Pr(-)=\frac{1-\cos\theta}{4}.
 \label{eq:relprob}
\end{equation}

For
\[
 G_\gamma=\operatorname{diag}(1,e^{i\gamma})^{\otimes2},
\]
the selective-map identities
\begin{equation}
 \Phi_y\circ\Ad_{G_\gamma}=\Phi_y,
                       \qquad y\in\{F,+,-\},
 \label{eq:relcov}
\end{equation}
hold on all operators and with every untouched reference.

Consequently the declared complete output is witness-orbit
invariant under Assumption~\ref{ass:reference}; under
\eqref{eq:matching} the same conclusion holds for the
whole stated first-order coefficient algebra.
\end{theorem}

\begin{proof}
With \(U=(H\otimes I)\mathrm{CNOT}\), the four destructive
path bras are
\(\bra{00}U=\bra{\phi_+}\),
\(\bra{10}U=\bra{\phi_-}\),
\(\bra{01}U=\bra{\chi_+}\), and
\(\bra{11}U=\bra{\chi_-}\).
Summing the two fine failure maps gives the first line
of \eqref{eq:relmaps}.
The four Bell projectors sum to the identity, proving
completeness.

The two odd-parity amplitudes have squared moduli
\(\abs{e^{i\beta}\pm e^{i\alpha}}^2/8\).
The even-parity total weight is \(1/2\).
This gives \eqref{eq:relprob}.

For \(F\), the equivalent computational-basis Kraus bras
\(\bra{00},\bra{11}\) acquire only scalar phases under
\(G_\gamma\); each phase cancels in its selective map.
Both \(\chi_\pm\) acquire the common phase
\(e^{i\gamma}\).
Thus every map in \eqref{eq:relmaps} is invariant.
Tensoring the linear-map identity with a reference
identity proves its complete version.

Under the witness translations, both \(\alpha,\beta\)
increase by \(\lambda c_\varepsilon/\sigma\).
Equation~\eqref{eq:matching} gives the same collective
phase for every permitted gauge direction.
\end{proof}

These are the parent conditional maps of X-Enter
\cite[\loc{thm:relational}]{XEnter}.
With an arbitrary passive finite residual \(A\), replace
each bra in the proof by its tensor product with \(I_A\).
The output retains the corresponding conditional
\(A\)-blocks. Equation~\eqref{eq:relcov} still holds
including arbitrary correlations with a reference.

\subsection{Why the failure merger and output boundary matter}

The raw even-parity probabilities are
\[
 \Pr(0,0)=\frac{1+\cos(\alpha+\beta)}4,\qquad
 \Pr(1,0)=\frac{1-\cos(\alpha+\beta)}4.
\]
They generally vary under the collective gauge phase.
The invariant boundary is therefore the declared
coarse record \(F,+,-\), with no exported raw failure
log and no returned signal qubit.

The merger is an explicit classical coarse-graining:
two selective maps are added, and no branch is removed.
It is not normalized postselection.
If a further accessible apparatus output retains the
raw distinction, that output must be included, and the
larger boundary need not be invariant.

The protocol is informative about \(\cos\theta\), not
globally faithful to the real number \(I_{\mathrm{rel}}\).
Periodicity and evenness produce aliases. Physical
informativeness requires admissible invariant inputs
with different cosines.

The circuit is a constructed finite channel at its
abstract input interface. Its application to the
convolution model remains conditional: no second
coefficient, acquisition map, or whole-algebra matching
certificate has been supplied.

%=====================================================================
\section{Stage F: bounded segments and excluded prescription changes}
\label{sec:bounded}
%=====================================================================

\subsection{A separately imposed comparison restriction}

Suppose, as additional data, that comparisons are restricted
to
\[
 |\lambda|\le\Lambda,\qquad
 \Lambda>0,\qquad d=\Lambda c_\varepsilon.
\]
This is a bounded comparison family, not the original
real-linear gauge algebra and not necessarily a group
closed under composition.

\begin{proposition}[Support is not an amplitude cutoff]
\label{prop:nocutoff}
Neither compact gauge support nor plateau support
implies a finite \(\Lambda\).
\end{proposition}

\begin{proof}
For any admitted nonzero supported vector field \(\xi\)
and every real \(\lambda\), the vector field
\(\lambda\xi\) has the same support.
Thus support alone bounds no coefficient multiplying it.
\end{proof}

\begin{theorem}[Exact tolerance--visibility trade-off]
\label{thm:tradeoff}
For direct readout of \eqref{eq:encoding},
\begin{equation}
 \sup_{|\lambda|\le\Lambda}
   \dTV(\Law_{\mathrm{read}}(\lambda c_\varepsilon),
                          \Law_{\mathrm{read}}(0))
      =\frac12\tanh(d/\sigma)
      \le\frac{d}{2\sigma}.
 \label{eq:segment}
\end{equation}
Let \(0<\delta<1/2\) be an allowed variation and
\(0<v<1/2\) a desired visibility for a separately
specified benchmark magnitude \(A_{\mathrm{sig}}>0\).
Both
\[
 \tfrac12\tanh(d/\sigma)\le\delta,\qquad
 \tfrac12\tanh(A_{\mathrm{sig}}/\sigma)\ge v
\]
hold precisely when
\begin{equation}
 \frac{d}{\artanh(2\delta)}
        \le\sigma\le
           \frac{A_{\mathrm{sig}}}{\artanh(2v)}.
 \label{eq:scales}
\end{equation}
A permitted scale exists if and only if
\begin{equation}
 \frac{A_{\mathrm{sig}}}{d}
         \ge\frac{\artanh(2v)}{\artanh(2\delta)}.
 \label{eq:ratio}
\end{equation}
For \(d>0\), zero tolerance is impossible at finite
\(\sigma\).
\end{theorem}

\begin{proof}
The binary-law distance is the absolute difference
of its success probabilities.
Monotonicity and oddness of \(\tanh\) give
\eqref{eq:segment}; \(\tanh u\le u\) for \(u\ge0\)
gives its upper bound.
Applying the increasing inverse \(\artanh\) to
the two requirements gives \eqref{eq:scales}.
The interval is nonempty exactly under
\eqref{eq:ratio}.
\end{proof}

This is the bounded-segment alternative of X-Enter
\cite{XEnter}. It is approximate closure under a new
restriction, not an exact loophole in
Theorem~\ref{thm:orbit}.

Calling \(A_{\mathrm{sig}}\) a physical signal requires
independent provenance. The scalar input alone does
not label one displacement as physical and another
as gauge. The required signal-to-gap ratio also
depends on \(v\) and \(\delta\); it is not an
unconditional claim that the signal must exceed
the gap.

\subsection{Complete-output bounds for general finite programs}

Set
\[
 \Delta_\Phi(a,b)
   =\frac12\norm{\Rec_\Phi(\tau(a))
                         -\Rec_\Phi(\tau(b))}_1.
\]
For \(J=[a_0-d,a_0+d]\), define
\[
 L_\Phi(J)
    =\sup_{a\in J}\frac1{2\sigma}
          \sum_y\norm{B_ys(a)^2-C_ys(a)t(a)}_1.
\]

\begin{proposition}[Selective and universal continuity bounds]
\label{prop:bounds}
For \(|\lambda|\le\Lambda\),
\[
 \Delta_\Phi(a_0+\lambda c_\varepsilon,a_0)
                  \le|\lambda|c_\varepsilon L_\Phi(J).
\]
For all \(a,h\),
\[
 \Delta_\Phi(a+h,a)\le
                       \min\{1,|h|/(2\sigma)\}.
\]
Exact selective constancy on any nontrivial real interval
already forces \(B_y=C_y=0\) for all \(y\).
\end{proposition}

\begin{proof}
Integrate the derivative of each block in
\eqref{eq:selective}. The trace norm of a
block-diagonal operator is the sum of its block
trace norms.

For a Hermitian \(Z=Z_+-Z_-\), positivity and trace
preservation give
\[
 \norm{\Rec_\Phi(Z)}_1
 \le \tr\Rec_\Phi(Z_+)+\tr\Rec_\Phi(Z_-)=\norm Z_1.
\]
Differentiating \eqref{eq:bloch} gives
\[
 \frac12\norm{\tau'(a)}_1=\frac{s(a)}{2\sigma}
                                                    \le\frac1{2\sigma}.
\]
Integration and the fact that distances of states are
at most one prove the universal bound.

If the derivative is zero throughout an interval,
division by \(s^2>0\) gives
\(B_y=C_y\sinh(a/\sigma)\).
Taking two different points in the interval gives
\(C_y=0\), then \(B_y=0\).
\end{proof}

The contraction argument is standard finite-channel
analysis, proved here on the needed Hermitian
differences \cite{Watrous}.
It also bounds preparation errors for an installed
approximate value \(\widetilde a\):
\[
 \dtr(\tau(a),\tau(\widetilde a))
               \le |a-\widetilde a|/(2\sigma).
\]
This does not imply that distinct deterministic
orthogonal numerical report words are close as
channels.

\subsection{Excluded changes}

The width remains fixed. Positivity at that width
proves no statement about \(\varepsilon\to0\).
No sharp limit is used to avoid the orbit theorem.

The convolution coordinate response
\[
 w^\mu=\Ezero\,\mathsf C\,T_{g,\varepsilon}h
\]
and the ramp response \(\Co_1\) remain their source
definitions. An Enter operation acts on a finite
encoding or record, not on either classical definition.

Subtracting a coordinate term, changing a switch,
or replacing convolution by the ramp would require a
new classical prescription and covariance proof.
Those changes are outside this release, not
measurement updates.

%=====================================================================
\section{The ramp comparison is not a fifth closure}
\label{sec:ramp}
%=====================================================================

The supplied ramp identity is
\begin{equation}
 \Co_1[K\xi]=\xi,\qquad \xi\in\gR.
 \label{eq:ramp}
\end{equation}
It supplies a compensating coordinate response in the
classical covariance calculation. It is not an
invariant numerical value by itself
\cite{XAST,XEnter}.

The excluded-edge comparison has, in its source
discrepancy convention,
\[
 d_{\mathrm{edge}}
       =-\int_0^\varepsilon
            (\sigR)'(s)\alpha_{\mathrm{edge}}(s)\,\dd s\ne0.
\]
This is a different functional with a different
parameter and prescription from \eqref{eq:gap}.
An accidental equality of numerical values would not
identify the two constructions.

\begin{proposition}[Prescription separation]
\label{prop:rampseparation}
The following conclusions are compatible:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the ramp observable has its stated classical
plateau covariance;
\item the convolution coefficient has the surjective
orbit \eqref{eq:orbit};
\item no scalar-only readout closes that convolution
orbit informatively.
\end{enumerate}
Neither the ramp identity nor the finite instruments
supply Assumption~\ref{ass:reference}.
\end{proposition}

\begin{proof}
Theorem~\ref{thm:orbit} uses only
Assumption~\ref{ass:convolution}, not
\eqref{eq:ramp}. The ramp plateau cancellation
uses its own response and gauge algebra.
There is no common defect orbit asserted for its
already invariant coefficient.

Assumption~\ref{ass:reference} requires a separately
defined coefficient and charges. An identity for a
different coordinate response, or a channel acting
on supplied numbers, defines neither.
\end{proof}

If an enlarged ramp comparison admitted the excluded
edge vector and all its real multiples while preserving
the old response, its tested discrepancy would have
an affine real-surjective orbit. The same elementary
orbit proof would apply to that \emph{new} comparison.
It would not identify its edge integral with
\(c_\varepsilon\), nor change the frozen ramp algebra.

An already invariant real ramp coefficient, if supplied
as input, can be encoded and read with equal outputs
on equal invariant values. That preserves an upstream
identity; it does not prove the ramp theorem, acquire
the coefficient, or repair the convolution.

Retaining the cited X-A-ST through X-H-ST interfaces
adds no operation changing the frozen convolution
contract. Their tree, free-BRST, relative-BV, and
supplied-report statements are not a new covariance
proof for \(w^\mu\) \cite{XAST,STRecords,XGST,XHST}.
This is a prescription-level non-identification, not
a claim to have scanned every possible manuscript or
excluded every future changed model.

%=====================================================================
\section{Stage G: complete locality and regular report transport}
\label{sec:transport}
%=====================================================================

\subsection{The scheduled locality theorem}

At every tick, the complete finite channel factors
over a supplied partition \(\mathcal B_s\) of finite
sites:
\[
 \Phi_s=\bigotimes_{B\in\mathcal B_s}\Phi_{s,B}.
\]
Every record, selector, task table, control, reset,
and resource wire read by a block is included in
its declared interface.

For terminal sites \(Y\), define
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
 \bigcup_{\substack{B\in\mathcal B_{s+1}\\
                   B\cap N_{s+1,t}(Y)\ne\varnothing}}B.
\]

\begin{theorem}[Complete scheduled locality]
\label{thm:locality}
The complete record-retaining network obeys
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\Alg(Y))
                               \subseteq\Alg(N_{s,t}(Y)).
\]
Equal input reductions on \(N_{s,t}(Y)\) give equal
corresponding output reductions on \(Y\).
For joint output with a reference \(E\), equality
is required on the joint
\(N_{s,t}(Y)\)--\(E\) input reduction.

A deterministic intervention outside the complete
backwards region cannot change that output law.
If every dependency edge has a separately certified
regular future-causal host route, every scheduled
ancestor is in the prescribed metric causal past
of the corresponding output.
\end{theorem}

\begin{proof}
A channel adjoint is unital.
A block disjoint from the pulled-back observable
support therefore acts on an identity factor.
A meeting block enlarges support only within
that block. Backwards induction gives the inclusion.

Testing all output observables proves the reduced-state
claim. Tensoring with the reference algebra proves the
joint claim. The same identity-factor argument proves
the intervention statement.
Concatenation of the independently certified routes
proves causal containment.
\end{proof}

This is the explicit prescribed-network mechanism of
En03-A and I-B \cite{En03,IB}.
Ancestry need not be minimal influence and is not
identified with the entire physical causal past.
Postselection is not a deterministic intervention.

The metric causal inclusion requires route certificates;
it is not derived from scheduler consistency alone
\cite{En08A}.

\subsection{A nonempty regular SWAP route}

Choose a straight developed spatial segment of length
\(\ell>0\) in the retained regular convex tube.
Choose its departure time \(t_{\mathrm d}\) after
readout, lookup, and a positive processing buffer,
with the entire route in the time reserve.

At stations
\[
 \mathbf x_j=\mathbf x_0+\frac jn
                         (\mathbf x_n-\mathbf x_0),
 \qquad j=0,\ldots,n,
\]
place equal finite payload registers \(A_j\).
Initialize \(A_1,\ldots,A_n\) in independent blanks
\(\omega\).
At hop \(j\), apply SWAP to \(A_{j-1},A_j\) across
developed duration \(\ell/n\).

The payload may contain the classical report, represented
on its diagonal algebra, and any returned finite quantum
residual. It is not the continuum coefficient itself.

\begin{theorem}[Regular reference-preserving report route]
\label{thm:route}
For every initial payload--reference state
\(\rho_{A_0E}\), after hop \(j\) the complete route state is
\[
 \rho_{A_jE}\otimes
        \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}.
\]
The logical channel is the identity, including every
reference and all correlations between reports and
quantum residuals.
Arrival is at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}}.
\]
The route remains regular and future causal.
\end{theorem}

\begin{proof}
A SWAP exchanges the entire current payload factor,
including its reference correlations, with the next
independent blank. It leaves the old factor blank.
Induction gives the stated tensor identity.

Station separation and developed duration both equal
\(\ell/n\), so cross-port dependencies have straight
null certificates in the developed chart.
Same-station dependencies are timelike.
The signature and time ledger converts the developed
arrival time into the host time.
\end{proof}

This is the target-adapted finite SWAP mechanism of
I-B and En08-B \cite{IB,En08B}.
It is not an assertion of exact null attainment on
an arbitrary previously fixed lattice.
The tick-boundary SWAP is a declared permission,
not a microscopic gravitational Hamiltonian.

Reference preservation does not assert
gravity-generated entanglement. It preserves whatever
correlations were supplied at the input.

\subsection{Resources and complete error}

Let \(p\) count payload qubits, \(b_{\mathrm{rep}}\)
the conservatively encoded report width, \(q_{\mathrm{sig}}\)
the signal subroutine width, and \(w_{\mathrm{lookup}}\)
the workspace. A preallocated upper reserve is
\[
 n_C+q_{\mathrm{sig}}+w_{\mathrm{lookup}}
                         +(n+1)(p+b_{\mathrm{rep}}).
\]
A different peak requires its actual schedule.
Reference storage, supplier holdings, task descriptions,
tables, resets, and returned factors are separately
listed when present.

The simple route uses \(n\) register SWAPs.
Clock calibration, finite preparation, lookup,
readout, report forwarding, and processing buffers
have declared costs and durations.

For compatible ideal and implemented complete channels,
\begin{equation}
 \ddia(\widetilde\Gamma,\Gamma)
 \le\min\left\{1,\,
   \epsilon_{\mathrm{prep}}
  +\epsilon_{\mathrm{clock}}
  +\epsilon_{\mathrm{lookup}}
  +\epsilon_{\mathrm{read}}
  +\sum_{j=1}^n\epsilon_j
  +\epsilon_{\mathrm{decode}}\right\},
 \quad
 \ddia(\Lambda,\Gamma)=\tfrac12\norm{\Lambda-\Gamma}_\diamond.
 \label{eq:errors}
\end{equation}
Indeed, replace one factor at a time and use channel
contraction and the triangle inequality
\cite{Watrous,En08B}.
Every comparison interface retains its records and
quantum residuals.

If two ideal witness laws have discrepancy
\(\Delta_\varepsilon\) and their actual complete
comparisons have errors \(e_0,e_1\), their actual
record discrepancy satisfies
\[
 \dTV(\widetilde P_1,\widetilde P_0)
             \ge\max\{0,\Delta_\varepsilon-e_0-e_1\}.
\]
The inequality follows by classical contraction and
the reverse triangle inequality. It certifies
encoded visibility, not invariance or acquisition.

A channel error does not certify a false numerical
enclosure, construct a missing second coefficient,
or prove exact causality of a faulty implementation.

%=====================================================================
\section{Release, significance, and successor obligations}
\label{sec:release}
%=====================================================================

\begin{definition}[Local Grade S]
\label{def:grade}
For this manuscript, local Grade S consists of the
retained supported host, the explicit scalar input
contract, the proved readout-closure statements,
a nonempty finite supplied-evaluation dictionary,
complete destructive instruments with unmeasured
residuals and references, and scheduled locality
with a nonempty regular route under declared
finite permissions.

It does not include continuum field acquisition or
a derivation of the host geometry.
\end{definition}

\begin{theorem}[X-Enter-ST release]
\label{thm:release}
Under Assumptions~\ref{ass:host} and
\ref{ass:convolution}, at fixed positive width:
\begin{enumerate}[label=\textup{(\roman*)}]
\item full witness-orbit invariance of any scalar-only
encoded complete output implies independence from
that scalar;
\item the retained amplitude family has the selective
visibility-or-erasure criterion, explicit discrepancy
\eqref{eq:TV}, and second-order diagnostic;
\item the finite supplied-evaluation dictionary has a
complete destructive-clock instrument, including all
unmeasured finite residuals and untouched references;
\item the relational combination and CNOT--Hadamard
instrument are proved under the separate reference
hypothesis, but that hypothesis is not supplied;
\item the bounded-segment trade-off is proved under
an additional amplitude restriction not implied by
support;
\item the ramp identity, ramp edge integral, and
convolution gap retain their different prescriptions
and quantifiers;
\item scheduled locality and regular reference-preserving
transport hold with the resource and error contracts
above.
\end{enumerate}
The package is local Grade S.

No field-to-qubit preparation, independently charged
second coefficient, convolution repair, interacting
differential, or inherited characteristic quantum
coefficient is constructed.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:orbit} proves (i).
Theorem~\ref{thm:amplitude},
equation~\eqref{eq:TV}, and
Theorem~\ref{thm:secondorder} prove (ii).
Theorem~\ref{thm:clock} and
equation~\eqref{eq:combined} prove (iii), with its
interpretation limited by
Proposition~\ref{prop:noacquisition}.
Proposition~\ref{prop:matching} and
Theorem~\ref{thm:relational} give exactly the
conditional conclusion in (iv).
Proposition~\ref{prop:nocutoff} and
Theorem~\ref{thm:tradeoff} prove (v).
Proposition~\ref{prop:rampseparation} proves (vi).
Theorems~\ref{thm:locality} and \ref{thm:route},
with \eqref{eq:errors}, prove (vii).
None of these constructions has the input or
codomain of an excluded operation.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}p{.23\linewidth}X@{}}
\toprule
Object & Verdict & Boundary\\
\midrule
Scalar-only exact closure &
Closed, negative &
Surjective inherited coefficient orbit\\
Amplitude readout &
Visible or erased &
One specified encoded family\\
Finite report instrument &
Constructed &
Supplied evaluations, not field acquisition\\
Conditional relational circuit &
Constructed at abstract input &
Second classical coefficient and matching absent\\
Bounded-segment comparison &
Conditional quantitative result &
New amplitude restriction required\\
Ramp plateau covariance &
Retained comparison &
Not an Enter proof or convolution repair\\
Regular route &
Constructed under permissions &
Finite records and residuals, not continuum fields\\
Characteristic quantum coefficient &
Not admitted &
No new product or admission theorem\\
Interacting differential &
Not constructed &
Not an implication of any readout\\
Spacetime emergence &
Not constructed &
Host metric and future relation supplied\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Significance and novelty boundary}

The parent negative closure and readout mechanisms are
Cheung's X-Enter results \cite{XEnter}.
The orbit proof is elementary; its usefulness lies
in exposing an invalid upgrade, not in claiming a
new general measurement principle.

The supported transfer supplies an explicit common
ledger for:
\[
 \text{classical coefficient},\qquad
 \text{encoded state},\qquad
 \text{complete instrument},\qquad
 \text{geometric route}.
\]
The mathematical obstruction survives every declared
readout choice because the scalar orbit is already
transitive. The operational construction is positive
without pretending to remove that obstruction.

Two practical audits are particularly important:
a constant reported bit can conceal a varying
returned quantum residual, and a finite task table
can disclose installed data without acquiring the
continuum quantity used to describe those data.
The manuscript makes both tests explicit.

\subsection{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged host, completion, signature, time,
and fixed-width ledger;
\item the unrepaired convolution charge contract and
its real-surjective coefficient orbit;
\item the exact and approximate orbit arguments;
\item the selective amplitude tests and finite
supplied-report instruments;
\item the conditional second-reference hypothesis and
its exact charge-matching condition;
\item the conditional relational circuit with its
failure and coarse-record boundary;
\item the bounded-comparison trade-off and excluded
prescription changes;
\item scheduled locality, a regular route, and the
complete resource/error contract.
\end{enumerate}

It receives no repaired coefficient, physical
relational observable of the cone, field-acquisition
map, or permission to identify the ramp and convolution
prescriptions.

%=====================================================================
\appendix
\section{Debugging and quantifier audit}
\label{app:audit}
%=====================================================================

\begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}}
\toprule
Potential error & Correction\\
\midrule
\endfirsthead
\toprule
Potential error & Correction\\
\midrule
\endhead

Using one symbol for the chart and state encoding &
The chart is \(\iota\); the encoding is \(\Enc\).\\[4pt]

Treating \(c_\varepsilon\) as the varying orbit coordinate &
It is the fixed positive charge. The input coordinate is
\(a=\lambda c_\varepsilon\).\\[4pt]

Claiming that the instrument is constant on every state &
Only its composite with the scalar encoding is forced
to be constant.\\[4pt]

Using support as a gauge-amplitude bound &
Real scaling preserves support; a bounded comparison
requires new data.\\[4pt]

Treating an unbounded coefficient orbit as a uniform
finite-coupling expansion &
The orbit is coefficient-level. Perturbative assertions
are pointwise in gauge amplitude.\\[4pt]

Claiming a continuum of orthogonal values in one qubit &
The amplitude states are nonorthogonal members of a
two-dimensional family.\\[4pt]

Promoting an installed rotation to field acquisition &
No field-to-register preparation map is constructed.\\[4pt]

Checking only the first derivative at zero &
Hadamard--Enter has a nonzero second-order response.\\[4pt]

Equating constant outcome law with complete invariance &
Every returned quantum factor belongs to the output
comparison.\\[4pt]

Treating amplitude-family erasure as universal quantum erasure &
The unseen \(Y\) direction need not be erased.\\[4pt]

Using finite sample agreement as an orbit certificate &
Universal constancy needs an analytic or complete
matrix certificate, not a finite coincidence.\\[4pt]

Exporting a gauge-dependent task label for free &
It is part of the complete input/output and cannot
evade the orbit theorem.\\[4pt]

Calling the fixed charge report an acquired orbit value &
It is supplied calibration data, not an evaluation of
an unknown \(a(\lambda)\).\\[4pt]

Replacing the failure merger by postselection &
Failure has probability \(1/2\) and remains reported.\\[4pt]

Claiming invariance of every possible raw apparatus log &
Only the declared coarse record and returned quantum
interface have the proved identity.\\[4pt]

Using a carried qubit as the charged classical reference &
An independently defined coefficient and its charge
remain missing.\\[4pt]

Inferring whole-algebra invariance from one orbit &
Equation~\eqref{eq:matching} is additionally required.\\[4pt]

Claiming global real-number decoding from the phase circuit &
It sees \(\cos\theta\) and therefore has aliases.\\[4pt]

Identifying the ramp edge integral with the convolution gap &
Their operators, tests, and admissible algebras differ.\\[4pt]

Using a supplied ramp reserve to localize every convolution
dependence &
A source-specific support certificate is separately
required.\\[4pt]

Using a Green operator on the regular cone &
Global analytical operators remain on the declared
Minkowski completion.\\[4pt]

Consuming geometric time &
Only measured finite clock names are consumed.\\[4pt]

Calling identity transport gravity-generated entanglement &
The route preserves input correlations.\\[4pt]

Calling a small channel error exact causal containment &
The error theorem controls complete outputs, not
exact locality of a faulty implementation.\\[4pt]

Promoting source identifiers to registration or verification &
No such status follows from this source file.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Standing boundaries and documentary comparisons}
\label{app:boundaries}
%=====================================================================

\paragraph{Quantum-admission boundaries.}
No quantum coefficient of the inherited characteristic
graph is admitted.
This does not deny the free carrier or the previously
admitted tree coefficients.

O.1 remains \loc{NOT_CLOSED}.
XV-B is not opened as the downstream quantum-admission
exit; that sentence does not deny the existence of a
technical manuscript with that identifier.
XVI-N is not discharged.
The scoped XVI-O kernel result is not bulk BV restoration
or a repair of the present convolution coefficient
\cite{BoundaryRecords}.

This manuscript discharges neither historical H2 nor the
missing full I3/J2 characteristic maps.
The narrower internal-amplitude advance at I2 is not
withdrawn or identified with the full missing map.
These are inherited scope boundaries, not new
nonexistence theorems.

\paragraph{Finite-interface comparisons.}
Oai283-A, Oai282-A, Oai272-A, Oai276-A, Oai279-A,
and Oai119-A remain separately specified finite
comparisons \cite{FiniteComparisons}.
A constant-error circuit certificate, a finite Ward
residual, a Choi certificate, a one-use information
calculation, a reversible lookup, or a Hellinger
comparison is not the surjective-orbit theorem.
None is an analytic premise here.

Oai365-A is a signature and operational-dictionary
comparison, not permission to replace the completion
signature by the host signature.

The separately attributed OpenAI catalogue is
documentary only \cite{OpenAICatalogue}.
No catalogue proof is verified, imported as a premise,
or renamed a convolution repair.

\paragraph{Physical non-claims.}
The prescribed cone is not derived from the readout.
No Einstein dynamics from records, external-time-free
quantum evolution, Page curve, holographic identification,
gravity-generated entanglement, ultraviolet completion,
or spacetime emergence is constructed.

A readout of an already invariant supplied number is not
a construction of the invariant field observable that
might eventually supply it.

%=====================================================================
\section{Proposed integration record}
\label{app:integration}
%=====================================================================

This is a proposed manuscript source, not an executed
repository transaction. A maintainer may install it as
\loc{content/manuscripts/X-Enter-ST.tex}.
Actual metadata, hashes, source imports, schemas,
registration, and tests must come from the repository.

\begin{verbatim}
identifier: X-Enter-ST
sourceStatus: proposed
grade: local Grade S
primarySourceRecord: X-Enter@1
host: unchanged I-B / X-A-ST instance
analyticalCompletion: Minkowski
hostSignature: minus-plus-plus-plus
completionSignature: plus-minus-minus-minus
widthStatus: fixed-positive-width
classicalGap: inherited convolution coefficient
exactScalarOnlyClosure: proved-negative
constructed:
  - encoding-independent orbit transfer
  - selective amplitude and retained-order tests
  - finite supplied-evaluation dictionary
  - complete destructive-clock report instrument
  - scheduled locality
  - regular reference-preserving report route
conditional:
  - relational circuit application requires second charge
  - whole-algebra closure requires charge matching
  - bounded-segment application requires amplitude bound
notConstructed:
  - field-to-qubit acquisition
  - independently charged second classical coefficient
  - convolution repair
  - interacting differential
  - inherited characteristic quantum coefficient
  - spacetime emergence
registryEditsExecuted: false
repositoryTestsExecuted: false
compilationAsserted: false
\end{verbatim}

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

\paragraph{Attribution.}
The principal orbit obstruction, amplitude tests,
conditional relational circuit, and bounded-segment
mechanism are attributed to N.~Cheung's X-Enter.
The convolution premise, ramp result, prescribed
host, clock interface, and transport pattern retain
their distinct source attributions.

\paragraph{Source precision.}
The X-A--Convolution source is used through the
explicit classical premises reproduced in X-Enter
and in Assumption~\ref{ass:convolution}.
Its full kernel derivation is not independently
reconstructed here.
No unprovided publication date, DOI, definitive
title, or registration status is invented for
an internal manuscript.

\paragraph{Mathematical scope.}
The readout results are self-contained relative
to the displayed host and scalar input contracts.
They do not prove positivity of a newly chosen
convolution kernel or identify two different
classical prescriptions.

\paragraph{Operational scope.}
The complete instruments encode supplied values
and status certificates.
They are not continuum field detectors, physical
state selectors, or gravitational observables.
Clock consumption is an access statement.
The regular route uses declared finite permissions.

\paragraph{Verification.}
No successful \loc{pdflatex} execution,
proof-assistant check, symbolic-program execution,
numerical experiment, hardware synthesis,
repository integration, independent execution,
laboratory realization, referee approval, or
journal acceptance is asserted.
The release consists of the definitions and
written proofs in this source.

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

\bibitem{XEnter}
N.~Cheung,
\emph{Landmark X-Enter: Readout closure tests for the
obstructed convolution coefficient: a named comparison
with the ramp prescription}.
\loc{X-Enter.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-Enter}.
Especially \loc{lem:orbit}, \loc{thm:amplitude},
\loc{thm:secondorder}, \loc{thm:relational},
Options B--D, and Section R.
Principal source for the readout-closure results.

\bibitem{XConv}
N.~Cheung,
\emph{X-A--Convolution}.
Descriptive source identifier \loc{X-A-conv}.
The classical intertwining identity, positive
coefficient, and witness orbit are used as reproduced
in X-Enter, version 1, ``Convolution premises:
the inputs of Options A--D.''
No additional bibliographic metadata or independent
kernel verification is asserted.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Version-specific supplied predecessor record.
Inherited host, developing chart, declared completion,
signature ledger, fixed-width plateau covariance,
and the separate edge audit.

\bibitem{IB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed
Conical Product}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I-B}.
Prescribed conical product and worldsheet,
complete clock instrument, calibration,
non-identifications, and regular transport.

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

\bibitem{EnterI}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Operational characterization of destructive readout
within its full stated structural class.
Contextual attribution; no abbreviated
characterization proof is used here.

\bibitem{En09}
N.~Cheung,
\emph{En09-A: Realization of Relational Time by
Destructive Enter on a Clock Register}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete destructive clock instrument,
untouched references, no universal quantum return,
and the distinction between extraction and dynamics.

\bibitem{En03}
N.~Cheung,
\emph{Observation Boundaries for Destructive Quantum
Readout and External Geometry}.
\loc{En03-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En03-A}.
Complete interfaces and prescribed schedule locality.

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
Version 1, supplied documentary record and excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency and the separate hypotheses
needed for geometric cone representation.

\bibitem{En08B}
N.~Cheung,
\emph{Resource-controlled operational approximation
of null boundary transport}.
\loc{En08-B.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}.
Reference-preserving SWAP transport relative to
supplied spatial and temporal calibration,
resource accounting, and complete-channel errors.

\bibitem{XGST}
N.~Cheung,
\emph{X-G-ST: Free causal BRST compatibility and
a variational upgrade obstruction}.
Supplied \loc{X-G-ST.tex}.
Free auxiliary--ghost algebra, separate classical
obstacles, and supplied-evaluation instrument pattern.
No convolution repair is imported.

\bibitem{XHST}
N.~Cheung,
\emph{X-H-ST: Relative classical BV structure and
complete instruments for supplied residual reports}.
Supplied proposed \loc{X-H-ST.tex}.
Relative off-shell classical construction,
free-shell non-descent audit, and typed
supplied-residual reports.
Not an interacting differential on an unchanged
free carrier and not a convolution repair.

\bibitem{STRecords}
N.~Cheung,
\emph{X-B-ST}, \emph{X-C-ST}, \emph{X-D-ST},
\emph{X-E-ST}, and \emph{X-F-ST}.
Version-specific supplied predecessor records.
Separate scalar completion, fixed-support topology,
tree, and background-identity interfaces.
No common prescription identification or independent
registration is inferred from their identifiers.

\bibitem{BoundaryRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1; technical \emph{XV-B}, version 2;
\emph{XVI-N}, revision r12;
\emph{XVI-O}, version-specific supplied record;
and \emph{XIX-E}, version 1.
Enter Calculus Companion, version-specific supplied
records and excerpts.
Historical H2, the narrower I2 advance,
remaining full I3/J2 maps, prepared-insertion
obstructions, and downstream quantum-admission
boundaries remain separate.

\bibitem{FiniteComparisons}
N.~Cheung,
\emph{Oai283-A}, \emph{Oai282-A}, \emph{Oai272-A},
\emph{Oai276-A}, \emph{Oai279-A}, \emph{Oai119-A},
and \emph{Oai365-A}.
Version-specific supplied documentary descriptions,
Enter Calculus Companion.
Finite-interface and signature comparisons only.
No proof transfer into the present orbit theorem.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's book page:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.
Finite instruments, state distinguishability,
trace-norm contraction, and diamond-norm comparison.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary descriptions in the
Enter Calculus Companion.
Separate catalogue attribution only.
No catalogue proof is independently verified
or used as a premise.

\end{thebibliography}

\end{document}