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\hypersetup{
  pdftitle={XVI-O1-ST-v5: Finite Enter Certificates with a Permanent Printed-Array Integrity Gate},
  pdfsubject={Complete instruments, exact certificates, gauge-family targets, and cross-artifact verification},
  pdfkeywords={Enter calculus, quantum instrument, DRED, counterterms, exact certificates, data integrity}
}

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\title{\textbf{XVI-O1-ST-v5}\\[0.4em]
Finite Enter Certificates with a Permanent\\
Printed-Array Integrity Gate\\[0.4em]
\large Complete readout, exact counterterm feasibility,\\
an affine gauge propagator, and an audited failed regression}
\author{}
\date{Version 5}

\begin{document}
\maketitle

\begin{abstract}
We develop an audit-hardened finite certificate architecture for
a prescribed-host perturbative loop sector, preserving the
distinction between mathematical proofs, printed data, supplied
scientific records, and execution artifacts. The central v5
finding is a cross-artifact integrity witness: the v4 typeset
D01 table has sixth coefficient \(-4/3\), whereas its embedded
and separately supplied regression has \(+4/3\). With the
unchanged gauge-table coefficient \(a^{(0)}_6=4/3\), the regression
relation fails by the exact residual
\(a^{(0)}+f^{\mathrm{reg}}=(8/3)e_6\).
Python index 5 is displayed structure \(B_6\).
Both source values are retained; no physical coefficient is
selected by editing a sign.

The supplied v4 transcript reports eight passing suites and one
failed suite. Its phase failure occurs before that suite's
variation-count and derivative checks, so those later predicates
are not described as executed. Their printed-data proofs remain
available. A proposed v5 regression checks the exact known
discrepancy as a negative control and keeps data-integrity failure
separate from successful regression behavior.

The retained mathematical results include complete destructive
report instruments with untouched references, the sharp complex
qubit-report half-diamond distance
\(\max_j|v_j-\widetilde v_j|/(2B)\), its real binary specialization,
original-minor and complete-kernel feasibility certificates,
quotient descent versus non-feedback, primitive-enlargement
criteria, and an exact affine general-\(\xi\) inverse under the
declared Q4S kinetic normalization. The old unchecked affine
candidate still has residual \((0,0,1)^{\mathsf T}\);
finite-field elimination and even/odd vertex-occurrence controls
remain permanent audit components.

The supplied 31-entry second-readout comparison, mutation
discrepancies \(431/30,1/3,5/12\), 13-of-14 D01 variation, and
restricted evanescent rank report \(7\to8\) are preserved with
their precise evidence boundaries. The potential impact is a
reusable release contract that prevents an expected software
failure, a transcription inconsistency, or a missing scientific
matrix from becoming a positive physics certificate. No novelty
is claimed for elementary channel or rank mathematics.
Q03 remains \code{SUPPLIED_CLOSED}; full gauge rows and
counterterm-derivative exactness remain \code{OPEN}; the
clean-machine rebuild remains \code{MISSING}. The new appendix
is \code{PROPOSED} and unexecuted. The cone remains an input,
bulk symmetry is not restored, XVI-N remains frozen, and O.1 is
\code{NOT_CLOSED} outside independently earned scopes.
\end{abstract}

\scopebox{Release boundary and principal correction}{%
The permanent discrepancy is not erased by changing the caption
or overwriting a coefficient.
The typeset vector, the regression vector, and the gauge vector
are separately identified below.

This manuscript contains a self-contained proof of its finite
mathematical statements and exact printed-data witnesses.
It does not supply the absent loop programs, full gauge-image
arrays, enlarged primitive response, or complete 23-row
gauge-family matrix.

A supplied v4 run is an attributed execution record.
It is not a v5 run.
For the new listing,
\code{appendix_status=PROPOSED},
\code{appendix_executed=false}, and
\code{appendix_verified=false}.}

\tableofcontents

%=====================================================================
\section{Question, Enter background, and evidence}
\label{sec:introduction}
%=====================================================================

\subsection{Enter and the refusal of identification}

Cheung's Enter calculus treats runtime quantum names as access
rights to factors of a global state. Destructive computational-basis
readout consumes the measured names from the returned quantum
interface and supplies a classical record. Its measurement
characterization is restricted to a declared strategy class, not
an unrestricted derivation of continuum field measurements
\cite{CheungEnter,CheungI}.

The introductory essay \emph{The Philosophy of Enter} distinguishes
readout from recovery, a classical record from a quantum-return
decoder, a channel from a thermodynamic machine, and an analogy
from an identification \cite{CheungPhilosophy}.
These distinctions organize interpretation; they are not
additional premises of the mathematical proofs here.
Consumption of a name does not assert annihilation of every
physical carrier used by an implementation.

I-B installs finite clocks, registers, calibration, controls, and
resources on a prescribed conical spacetime
\cite{CheungIB}. En03-A separately declares geometry, locations,
accessible algebras, and control/resource interfaces
\cite{CheungEn03}. We retain this shape but prove the needed finite
instrument identities directly. Standard quantum instruments and
completely bounded trace norms provide the general setting
\cite{DaviesLewis,Watrous}.

A numerical loop coefficient is not automatically a probability
or a measurable field effect. The finite report channels below
are explicitly constructed encodings of supplied numbers, not
field-acquisition channels.

\subsection{Designated artifacts}

The evidence package supplied for v5 contains:
\begin{enumerate}
\item \code{XVI-O1-ST-v4.tex};
\item \code{verify_xvi_o1_st_v4.py};
\item \code{v4_verification_output.txt};
\item \code{DETAILED_SUMMARY_v4.txt};
\item the earlier supplied consolidated and second-readout
manuscripts, together with version-qualified source excerpts.
\end{enumerate}
The manuscripts and artifacts are cited under their supplied
identifiers \cite{SuppliedV3,SuppliedBot,SuppliedV4,V4Artifacts}.

The summary is explanatory metadata. It is not an independent
execution record or a substitute for examining statement order
in the script.
No authorship of this anonymous manuscript is inferred from the
authorship of cited sources.

\begin{definition}[Six evidence categories]
\label{def:evidence}
We distinguish:
\begin{enumerate}
\item \textbf{Specifications}: objects, conventions, scopes, and
acceptance predicates fixed before a calculation.
\item \textbf{Mathematical proofs}: written arguments from
explicit hypotheses.
\item \textbf{Exact synthetic witnesses}: specified finite data
with exactly established properties. This category includes the
printed-array comparison, without identifying it with loop data.
\item \textbf{Supplied scientific records}: attributed calculations
or scientific conclusions in supplied manuscripts.
\item \textbf{Execution artifacts}: supplied program text,
environment descriptions, and transcripts, with provenance
qualifications.
\item \textbf{Listed implementations}: new program text without
an executed run.
\end{enumerate}
These categories are not interchangeable.
A supplied transcript is not an independently reproduced run;
a displayed exact residual is not merely a software assertion.
\end{definition}

\subsection{What v5 changes}

The predecessor's successful mathematical results are retained.
Two audit corrections are added:
\begin{enumerate}
\item the failed phase predicate is localized to a difference
between the typeset table and the regression transcription;
\item predicates after the failing instruction are distinguished
from predicates actually reached in that run.
\end{enumerate}
The new regression contract records the discrepancy exactly.
It does not redefine the false equality to be true.

No elementary rank theorem, instrument theorem, encryption
mechanism, or general propagator technology is claimed as new.
The contribution is their joint use in a scope-explicit
counterterm and data-integrity release architecture.

%=====================================================================
\section{The prescribed host and analytical carrier}
\label{sec:host}
%=====================================================================

Fix \(0<\mu<2\pi\), \(a=1-\mu/(2\pi)\), and
\(c_{\mathrm{host}}>0\). Set
\[
 C_\mu^\circ=(0,\infty)_r\times(\R/2\pi\mathbb Z)_\vartheta,
 \qquad
 M^\circ=\R_T\times C_\mu^\circ\times\R_z,
\]
with prescribed metric
\begin{equation}
 G=-c_{\mathrm{host}}^2\dd T^2
       +\dd r^2+a^2r^2\dd\vartheta^2+\dd z^2.
 \label{eq:host}
\end{equation}
Increasing \(T\) is future directed. The completion defect is
\[
 \mathcal W=\R_T\times\{o\}\times\R_z.
\]
The supported laboratory avoids \(\mathcal W\); the inherited
width \(\eps>0\) remains fixed
\cite{CheungIB,CheungXAST}.

\begin{proposition}[Local development, worldsheet, and non-GH host]
\label{prop:geometry}
The metric has signature \((-,+,+,+)\), and \(\mathcal W\) is a
two-dimensional worldsheet. On an injective angular chart,
\[
 t=c_{\mathrm{host}}(T-T_\Sigma),\quad
 x=r\cos(a\vartheta),\quad y=r\sin(a\vartheta)
\]
gives \(G=-\dd t^2+\dd x^2+\dd y^2+\dd z^2\).
The regular product is not globally hyperbolic.
\end{proposition}

\begin{proof}
The signature and dimension are immediate, and differentiation
gives the local development.
For \(r_0>0\), take in geometrized time
\[
 p=(-2r_0,r_0,\vartheta_0,z_0),\qquad
 q=(2r_0,r_0,\vartheta_0,z_0).
\]
The points \(x_n=(0,r_0/n,\vartheta_0,z_0)\), \(n\ge2\),
belong to \(J^+(p)\cap J^-(q)\) by radial causal curves.
They have no convergent subsequence in the regular product,
since their completion limit has \(r=0\). The diamond is not
compact.
\end{proof}

Declare separately
\begin{equation}
 \widehat M=\R^{1+3},\qquad
 \eta=\diag(1,-1,-1,-1),\qquad
 \iota^*G=-\eta
 \label{eq:completion}
\end{equation}
on the supported chart.
Free fields, Green operators, loop integrals, and local
counterterms use this analytical carrier and its prescription,
not an assumed global Cauchy theorem on \(M^\circ\)
\cite{CheungXAST,CheungXGST}.
A signature reversal does not authorize independent changes of
action-Hessian or Green-function signs.

\begin{proposition}[Conservation and trace do not source the host]
\label{prop:notsource}
A conserved traceless tensor need not satisfy Einstein's equation
for the prescribed host.
\end{proposition}

\begin{proof}
For a nonzero constant \(\rho_0\), let
\[
 \tau=\rho_0(\dd t\otimes\dd t+\dd z\otimes\dd z).
\]
The forms are parallel, so \(\tau\) is conserved; its \(G\)-trace
vanishes. The regular host is locally flat and has zero Einstein
tensor. Thus
\(\operatorname{Ein}(G)=\kappa_{\mathrm{grav}}\tau\)
fails for nonzero \(\kappa_{\mathrm{grav}}\) at zero cosmological
constant. No distributional source equation at the defect is
constructed.
\end{proof}

The conical product is not globally identified with Minkowski
spacetime. The analytical completion is not an inferred geometry.

%=====================================================================
\section{Scope, the Q03 record, and finite package semantics}
\label{sec:scope}
%=====================================================================

\begin{definition}[Scope tuple]
\label{def:scope}
A scope is
\[
 s=(H,\mathcal A,\ell,k,\mathcal E,m,\Psi,\mathcal P,
              \mathcal V,\mathcal R,\mathcal J),
\]
specifying host, action or observable, loop order, power of
\(\kappa\), external sector, mass, gauge fermion, analytical
prescription, counterterm presentation, complete residual space,
and separately imposed consistency tests.
Equality includes versions, conventions, and identification maps.
\end{definition}

\begin{record}[Unchanged Q03 scope]
\label{rec:q03}
The supplied XVI-Q Q11 record, read with XVI-P p23, attributes
closure only to the one-loop, order-\(\kappa^2\), massless bulk
two-point retained sector of \(\Sinv\), with frozen gauge
fermion, QdS loop/external conventions, Q4S vertex indices,
\(\MSbar\), and the p22 evanescent contract.
All 23 named identity rows are reported cancelled by local
counterterms. The consistency statement covers the bulk 11
four-dimensional directions.
The used presentation is 11 p14r directions over \(\Q(\ii)\)
plus 10 p22 evanescent operators, 21 in total
\cite{CheungXVIP,CheungXVIQ}.

The v5 status remains \code{SUPPLIED_CLOSED}.
This record does not update XVI-O's different prescription,
restore bulk symmetry, discharge XVI-N, or certify higher arity
or loop order.
\end{record}

QdS denotes the quasi-dimensional scheme, not de Sitter
spacetime. The symbolic dimension and evanescent algebra must be
handled before the prescribed Laurent extraction.
Consistent DRED does not license unrestricted use of
four-dimensional tensor identities \cite{Stockinger}.

\begin{specification}[Finite pinned package]
\label{spec:package}
A package supplies:
\begin{enumerate}
\item the action, gauge fermion, field/antifield normalizations,
Fourier and Grassmann conventions, and subtraction prescription;
\item the complete inventory and the derivation of every zero;
\item complete row assembly and all computational dependencies;
\item counterterm generators, relations, and reconstruction maps;
\item exact arrays or explicit missing entries;
\item original minors, complete kernels, residuals, negative
controls, error conventions, and budgets;
\item a manifest binding exact versions and bytes.
\end{enumerate}
An unfinished task or resource refusal is not zero.
\end{specification}

A digest identifies bytes relative to a manifest. It does not
prove that the manifest is trustworthy or that the bytes
implement the declared physics.

\subsection{Four correctly typed interface structures}

Following the En03-A shape \cite{CheungEn03}, retain:
\begin{enumerate}
\item external geometry \eqref{eq:host} and the separate carrier
\eqref{eq:completion};
\item an actual location map for physical laboratory registers;
\item accessible finite matrix algebras and classical record
algebras;
\item the complete controls, arithmetic, storage, routing,
calibration, and failure interface.
\end{enumerate}
The 12 inventory graphs and 23 named rows are logical tasks,
not spacetime ports. The 21-direction analytical presentation is
not a quantum observable algebra.

\subsection{Counterterm equations}

Work over \(\K=\Q(\ii)\), unless an enlarged exact field is
declared. Let
\[
 V_{\mathrm{pres}}=\K^{11}\oplus\K^{10},\qquad
 \Rec_{\mathrm{ct}}:V_{\mathrm{pres}}\to\mathcal F_{\mathrm{loc}}^0.
\]
For each named row choose a complete finite coefficient space
\(W_\alpha\), and put \(W=\bigoplus_{\alpha=1}^{23}W_\alpha\).
Twenty-three named rows need not mean 23 scalar equations.

Let \(A:V_{\mathrm{pres}}\to W\), \(d\in W\), and let
\(L_0:V_{\mathrm{pres}}\to J_0\) encode only the consistency
tests actually claimed. Define
\begin{equation}
 Ac+d=0,\qquad L_0c=0,\qquad
 M=\begin{pmatrix}A\\L_0\end{pmatrix},\qquad
 y=\begin{pmatrix}-d\\0\end{pmatrix}.
 \label{eq:system}
\end{equation}

\begin{proposition}[Presentation descent and residual coverage]
\label{prop:descent}
The response descends to reconstructed counterterms iff
\[
 \ker\Rec_{\mathrm{ct}}\subseteq\ker M.
\]
A zero tested coefficient residual implies the asserted
functional identity when the coefficient/test map is injective
on a known subspace containing all possible tested residuals.
Without separation, this implication can fail.
\end{proposition}

\begin{proof}
Equal reconstructions differ by a reconstruction-kernel vector;
their responses agree exactly when \(M\) annihilates it.
The coverage assertion is injectivity. A nonzero element in the
test map's kernel supplies the counterexample without it.
\end{proof}

No full O.1 rank or original minor is deduced from the numbers
23 and 21.

%=====================================================================
\section{Complete destructive report instruments}
\label{sec:instrument}
%=====================================================================

Fix a finite dictionary within declared representation and
resource bounds. Let \(C\cong\C^D\), \(D\ge2\), have calibrated
orthonormal basis \(\{\ket{\tau_j}\}_{j=1}^D\).
Let \(A_{\mathrm{lab}}\) contain all surviving laboratory
registers, and install finite words \(q_j\) carrying scope,
package, evidence, status, and coefficient codes.

Put
\begin{equation}
 D_j=\bra{\tau_j}_C\otimes I_{A_{\mathrm{lab}}},\qquad
 \mathcal M_P(X)=
 \sum_j\proj{j,q_j}_R\otimes D_jXD_j^\dagger.
 \label{eq:instrument}
\end{equation}
The record retains \(j\) even when two words coincide.
Unused labels and failures have explicit words.

\begin{theorem}[Complete destructive report]
\label{thm:instrument}
The map \eqref{eq:instrument} is CPTP, and for every finite
untouched reference \(E\),
\[
 \Tr_R[(\mathcal M_P\otimes\id_E)(X)]=\Tr_CX.
\]
For a state \(\rho\),
\[
 \sigma_j=(D_j\otimes I_E)\rho(D_j^\dagger\otimes I_E),
 \qquad p_j=\Tr\sigma_j.
\]
The positive-probability conditional state is \(\sigma_j/p_j\).
No normalized state is assigned to a zero branch.
The returned quantum interface excludes \(C\).
\end{theorem}

\begin{proof}
The Kraus maps \(K_j=\ket{j,q_j}\otimes D_j\) satisfy
\[
 \sum_jK_j^\dagger K_j
 =\sum_j\proj{\tau_j}\otimes I=I.
\]
Writing
\(X=\sum_{j,k}\ket{\tau_j}\bra{\tau_k}\otimes X_{jk}\),
the output blocks are \(X_{jj}\), whose sum is \(\Tr_CX\).
The same calculation with \(A_{\mathrm{lab}}\otimes E\)
proves the reference identity. Positivity supplies the branch rule.
\end{proof}

This is an I-B/En09-A supplied-report interface
\cite{CheungIB,CheungEn09}. It does not evaluate unknown loop
integrals by measurement or acquire an unknown continuum field.

\begin{proposition}[No universal return and no record-only residual]
\label{prop:noreturn}
The report instrument has no universal left inverse.
Its classical marginal does not determine its surviving quantum
state.
\end{proposition}

\begin{proof}
The distinct coherent clock states
\((\ket{\tau_1}\pm\ket{\tau_2})/\sqrt2\), tensored with the same
laboratory state, have identical outputs.
Also, \(\proj{\tau_1}\otimes\rho_A\) and
\(\proj{\tau_1}\otimes\rho'_A\), with \(\rho_A\ne\rho'_A\),
have the same deterministic record and different residuals.
\end{proof}

\begin{proposition}[Calibration and finite clock limitations]
\label{prop:clock}
If distinct readings \(t_j\) must equal a fixed event time
\(T_e\) on every positive-probability branch of a normalized
input, there is one deterministic matching outcome.
A prepared basis clock separately reads its installed word
deterministically.
Finite matrices cannot satisfy
\([T,H]=\ii\hbar I\) for \(\hbar\ne0\).
\end{proposition}

\begin{proof}
At most one distinct reading matches \(T_e\), and probabilities
sum to one. Basis readout follows from \eqref{eq:instrument}.
The trace of the proposed commutator would give
\(0=\ii\hbar D\).
\end{proof}

A gauge-table comparison at \(\xi=1\) is not this event-time
calibration theorem. Repreparing a known basis state is not a
universal inverse for an unknown coherent clock.

%=====================================================================
\section{Numerical reports and sharp complete error}
\label{sec:error}
%=====================================================================

For real \(r_j\) with \(|r_j|\le B\), define the binary report
\[
 p_r(b\mid j)=\frac{1+br_j/B}{2},\quad b=\pm1,
\]
\[
 \mathcal I_r^B(X)=
 \sum_{j,b}p_r(b\mid j)\proj{j,b}\otimes D_jXD_j^\dagger.
\]
The coefficient is a bounded expectation, not an exact rational
value recovered from one shot.

For complex \(v_j\), \(|v_j|\le B\), define
\[
 \tau_B(v_j)=\frac12\left(
 I+\frac{\operatorname{Re}v_j}{B}X_{\mathrm P}
   +\frac{\operatorname{Im}v_j}{B}Y_{\mathrm P}\right),
\]
\[
 \mathcal Q_v^B(X)=
 \sum_j\proj{j}\otimes\tau_B(v_j)\otimes D_jXD_j^\dagger.
\]
The Bloch bound makes \(\tau_B(v_j)\) a state.
This interface returns a classical task label and a quantum
report qubit, not a wholly classical coefficient word.
Both maps are channels: they measure the clock and append valid
classical or quantum states while retaining the branch block.

\begin{theorem}[Sharp reference-stable numerical-report error]
\label{thm:error}
With \(\ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond\)
and one common admissible \(B>0\),
\begin{align}
 \ddia(\mathcal Q_v^B,\mathcal Q_{\widetilde v}^B)
 &=\frac1{2B}\max_j|v_j-\widetilde v_j|,
 \label{eq:complexerror}\\
 \ddia(\mathcal I_r^B,\mathcal I_{\widetilde r}^B)
 &=\frac1{2B}\max_j|r_j-\widetilde r_j|.
 \label{eq:realerror}
\end{align}
These equalities include arbitrary finite untouched references.
\end{theorem}

\begin{proof}
The difference of the two qubit states has eigenvalues
\(\pm|v_j-\widetilde v_j|/(2B)\), and hence trace norm
\(|v_j-\widetilde v_j|/B\).
For any operator \(X\), including a reference extension, the task
blocks are orthogonal. Thus the output difference has trace norm
\[
 \sum_j\frac{|v_j-\widetilde v_j|}{B}\norm{X_{jj}}_1.
\]
Clock pinching is an average of unitary conjugations, so
\(\sum_j\norm{X_{jj}}_1\le\norm X_1\).
This proves the diamond upper bound.
A basis clock input at a maximizing index attains it.
For binary reports, the two scalar differences are
\(\pm(r_j-\widetilde r_j)/(2B)\); the same calculation applies.
\end{proof}

\begin{remark}[Different report interfaces have different metrics]
Distinct exact symbolic words are orthogonal outputs and have
half-diamond distance one on a distinguishing task.
The formulas above do not describe that symbolic metric.
Splitting a complex coefficient into separate real and imaginary
binary tasks gives the maximum component discrepancy on the
expanded dictionary, not automatically the complex modulus.
\end{remark}

\begin{corollary}[Sparse errors are not diluted]
\label{cor:sparse}
An error confined to one task can attain the full maximum in
\eqref{eq:complexerror} or \eqref{eq:realerror}, regardless of its
rarity under a chosen sampling law.
\end{corollary}

\begin{proof}
Prepare the affected basis task.
\end{proof}

\begin{proposition}[Complete-channel implementation budget]
\label{prop:budget}
For compatible ideal and implemented complete channels,
\[
 \ddia(\widetilde\Phi_n\cdots\widetilde\Phi_1,
       \Phi_n\cdots\Phi_1)
 \le
 \min\left\{1,\sum_{j=1}^n
                  \ddia(\widetilde\Phi_j,\Phi_j)\right\}.
\]
\end{proposition}

\begin{proof}
Telescope the difference, use contractivity under channel
composition and the triangle inequality, and use the maximum
one for half-diamond channel distance \cite{Watrous}.
\end{proof}

Records, failures, residual systems, and references remain in the
compared interfaces. Oai283-A is a documentary comparison for this
resource/error discipline, not a thermodynamic or loop premise
\cite{CheungOai283}.

%=====================================================================
\section{Exact rank certificates and the repaired affine family}
\label{sec:rank}
%=====================================================================

The presentation follows II/II-B's original-minor discipline
without importing their numerical certificate
\cite{CheungII,CheungIIB}.

\begin{theorem}[Rank--minor--left-kernel certificate]
\label{thm:rank}
Let \(M\in\K^{n\times p}\), \(y\in\K^n\), and
\(0\le r\le\min(n,p)\).
Suppose original row and column index sets \(I,J\), both of size
\(r\), and \(L\in\K^{(n-r)\times n}\) satisfy
\[
 \det M_{I,J}\ne0,\qquad LM=0,\qquad \rank L=n-r.
\]
Then
\[
 \rank M=r,\qquad
 Mc=y\text{ solvable}
 \iff Ly=0
 \iff \rank[M\mid y]=\rank M.
\]
A solution \(c_\star\) gives the complete affine family
\(c_\star+\ker M\).
A nonzero component of \(Ly\) gives an outside-span separator.
\end{theorem}

\begin{proof}
The minor gives \(\rank M\ge r\).
Since \(\im M\subseteq\ker L\) and \(\dim\ker L=r\),
the opposite inequality holds and \(\im M=\ker L\).
This proves membership and augmented-rank equality.
The affine assertion follows by subtracting two solutions.
A nonzero row pairing separates the target from the image.
\end{proof}

The empty minor has determinant one.
An annihilating matrix is not necessarily a complete left kernel.

\begin{proposition}[Complete right-kernel certificate]
\label{prop:right}
If \(\rank M=r\), \(MZ=0\), and
\(Z\in\K^{p\times(p-r)}\) has rank \(p-r\), then
\(\im Z=\ker M\).
With a verified particular solution, all solutions are
\(c_\star+Zt\).
\end{proposition}

\begin{proof}
The inclusion and dimensions agree.
\end{proof}

For \eqref{eq:system}, this proves exactly the declared
cancellation and consistency equations.
Functional interpretation additionally requires descent and
coverage. Coefficient uniqueness is presentation-relative,
not physical or cohomological uniqueness.

\begin{theorem}[Permanent affine false-positive witness]
\label{thm:affine}
Let
\[
 M_a=\begin{pmatrix}1&2&1\\2&4&2\\1&1&0\end{pmatrix},
 \qquad y_a=(3,6,1)^{\mathsf T}.
\]
The old fallback \(c_{\mathrm{old}}=(1,1,0)^{\mathsf T}\)
has residual
\begin{equation}
 M_ac_{\mathrm{old}}-y_a=(0,0,1)^{\mathsf T}\ne0.
 \label{eq:oldresidual}
\end{equation}
The complete correct family is
\[
 c=(1,0,2)^{\mathsf T}
       +t(1,-1,1)^{\mathsf T},\qquad t\in\K.
\]
\end{theorem}

\begin{proof}
Direct multiplication gives the residual and the corrected
solution. The original minor on rows \(1,3\), columns \(1,2\),
is \(-1\).
The complete left kernel is generated by \((-2,1,0)\).
The displayed nonzero right-kernel vector spans the
one-dimensional kernel. Apply Theorem~\ref{thm:rank} and
Proposition~\ref{prop:right}.
\end{proof}

Adding a kernel vector to the wrong particular solution preserves
its nonzero residual. A solver exception does not validate a
fallback vector.

%=====================================================================
\section{Second-readout tables and mutation sensitivity}
\label{sec:tables}
%=====================================================================

\subsection{Scientific provenance}

The second-readout predecessor attributes its rules to
XVI-Q's \code{p6q05/gravity.py}, derived from the declared
action rather than importing XVI-P's vertex tables, and pole
extraction to \code{q04/uvpole.py}
\cite{SuppliedBot,CheungXVIQ}.
Its reported conventions include
\[
 S_{EH}=-\frac2{\kappa^2}\int\sqrt{-g}\,g^{mn}
  \bigl(\Gamma^a_{mb}\Gamma^b_{na}
              -\Gamma^a_{mn}\Gamma^b_{ab}\bigr),
 \qquad
 S_{\mathrm{gf}}=\int D_\mu[h]D^\mu[h],
\]
\[
 D^\mu[h]=\partial_\nu h^{\mu\nu}
                   -\tfrac12\partial^\mu h.
\]
The corresponding ghost and antifield couplings, auxiliary
elimination, R09 vertex checks, and fresh same-machine run are
supplied scientific records, not newly executed derivations here.

The record discloses a common author, machine, and extraction
route. A second folder is not a clean-machine or heterogeneous
rebuild.

\subsection{Structure dictionary}

Use
\[
\begin{gathered}
 x=\widehat{a_1\cdot a_1},\quad
 y=\widehat{a_2\cdot a_2},\quad
 z=\widehat{a_1\cdot a_2},\\
 u=\widehat{a_1\cdot p},\quad
 v=\widehat{a_2\cdot p},\quad Q=\widehat{p\cdot p},\\
 x_t=\widetilde{a_1\cdot a_1},\quad
 y_t=\widetilde{a_2\cdot a_2},\quad
 z_t=\widetilde{a_1\cdot a_2}.
\end{gathered}
\]
For \(E_i=a_i\otimes a_i\), the structures are
\[
\begin{array}{c|l@{\qquad}c|l}
1&xyQ^2&8&u^2v^2\\
2&xv^2Q&9&u^2y_tQ\\
3&xy_tQ^2&10&uvz_tQ\\
4&z^2Q^2&11&yx_tQ^2\\
5&zuvQ&12&v^2x_tQ\\
6&zz_tQ^2&13&x_ty_tQ^2\\
7&u^2yQ&14&z_t^2Q^2
\end{array}
\]
and are denoted \(B_1,\ldots,B_{14}\).
Hats and tildes are scheme contractions, not host geometries.

\subsection{Unchanged typeset comparison data}

The v4 typeset table uses
\(v_{01,j}=\widetilde v_{01,j}=\ii f_j^{\mathrm{tex}}\)
and
\(v_{03,j}=\widetilde v_{03,j}=\ii g_j\).
Its mutant values are \(\ii f_j^\mu,\ii g_j^\mu\).

\begin{center}
\small
\begin{tabular}{r r r r r}
\toprule
\(j\)&\(f_j^{\mathrm{tex}}\)&\(f_j^\mu\)&\(g_j\)&\(g_j^\mu\)\\
\midrule
1&\(-27/80\)&\(-871/120\)&\(7/48\)&\(-1/48\)\\
2&\(19/60\)&\(911/120\)&\(-1/8\)&\(-7/24\)\\
3&\(-11/48\)&\(-229/48\)&\(5/48\)&\(1/48\)\\
4&\(-71/120\)&\(56/15\)&\(1/12\)&\(5/12\)\\
5&\(11/10\)&\(-829/30\)&\(-1/12\)&\(7/12\)\\
6&\(-4/3\)&\(17/3\)&\(1/3\)&\(2/3\)\\
7&\(19/60\)&\(911/120\)&\(-1/8\)&\(-7/24\)\\
8&\(-13/15\)&\(-99/10\)&\(1/6\)&\(1/6\)\\
9&\(1/6\)&\(91/12\)&\(-1/24\)&\(-5/24\)\\
10&\(4/3\)&\(-65/3\)&\(-1/3\)&\(1/3\)\\
11&\(-11/48\)&\(-229/48\)&\(5/48\)&\(1/48\)\\
12&\(1/6\)&\(91/12\)&\(-1/24\)&\(-5/24\)\\
13&\(-1/8\)&\(-21/8\)&\(1/16\)&\(1/16\)\\
14&\(-3/4\)&\(5/4\)&\(1/4\)&\(1/4\)\\
\bottomrule
\end{tabular}
\end{center}

For D06, use
\[
 C_1=\widehat{\bar u\cdot u}Q^2,\quad
 C_2=\widehat{\bar u\cdot p}\widehat{u\cdot p}Q,\quad
 C_3=\widetilde{\bar u\cdot u}Q^2,
\]
with
\[
 v_{06}=\widetilde v_{06}=(-3/8,1/4,-1/2),\qquad
 v^\mu_{06}=(-19/24,-7/12,-1/2).
\]
These retain all \(14+14+3=31\) displayed comparison entries.

\begin{proposition}[Retained comparison and mutation arithmetic]
\label{prop:mutationdata}
For the printed frozen/own arrays, the advertised comparison
discrepancy is zero.
The component-mutant comparisons give
\[
\begin{array}{c|c|c}
\text{graph}&\text{changed entries}&
 \delta=\frac12\max_j|v_j-v_j^\mu|\\ \hline
D01&14&431/30\\
D03&11&1/3\\
D06&2&5/12
\end{array}
\]
At common bound \(B=32\), the complex-report distances are
\[
 431/960,\qquad 1/96,\qquad 5/384.
\]
The D01 changed-entry count and maximum discrepancy are the
same if the regression vector \(f^{\mathrm{reg}}\) defined below
is used instead of \(f^{\mathrm{tex}}\).
\end{proposition}

\begin{proof}
The displayed own/frozen entries coincide.
For D01, entry 5 changes by \(431/15\), the maximum.
For D03, the maximum is \(2/3\), at entries 5 and 10.
For D06, it is \(5/6\), at entry 2.
All magnitudes are below 32, so Theorem~\ref{thm:error} applies.
Changing only D01 entry 6 from \(-4/3\) to \(+4/3\)
changes that mutant discrepancy from \(7\) to \(13/3\);
neither is the maximum, and both are nonzero.
\end{proof}

This is arithmetic of printed arrays.
The original loop provenance is supplied, not re-established.
Moreover, the v4 regression does not contain independently
constructed own and frozen arrays for all 31 entries; its
passing mutation checks are not an independent replay of the
31-entry scientific comparison.

\subsection{Global signs and component mutations}

The recorded ghost mutation changes selected transport terms in
\(\mathcal L_ch\), and the graviton mutation changes one
Christoffel contribution used in \(V_3\).
They are not common reversals of entire vertex rules.

\begin{theorem}[Even-occurrence blindness]
\label{thm:parity}
For amplitudes multilinear in vertex occurrences,
\[
 V_a\mapsto-V_a
 \quad\Longrightarrow\quad
 \Gamma_g\mapsto(-1)^{n_{ga}}\Gamma_g.
\]
A \(V_3\times V_3\) bubble is blind to a common cubic-rule
reversal. A two-copy ghost loop is similarly blind.
For \(B_{ga}=n_{ga}\bmod2\), sign vectors in
\(\ker_{\F_2}B\) are invisible to these graphwise comparisons.
\end{theorem}

\begin{proof}
Multiply one factor \(-1\) for each occurrence.
The simultaneous parity condition is \(Bs=0\).
\end{proof}

Zero amplitudes and cancellations create further blind spots.
Odd-occurrence tests, direct differentiated vertices, or
independently normalized tree identities supply additional
controls. They are not claimed newly executed loop tests.

%=====================================================================
\section{Permanent printed-array integrity witness}
\label{sec:integrity}
%=====================================================================

\subsection{The three arrays, without silent editing}

\begin{record}[Cross-artifact phase discrepancy]
\label{rec:phase}
The v4 typeset table and v4 embedded/extracted regression do not
encode the same D01 baseline vector
\cite{SuppliedV4,V4Artifacts}.

Using displayed positions \(1,\ldots,14\), they are
\[
\begin{split}
 f^{\mathrm{tex}}={}&
 (-27/80,\ 19/60,\ -11/48,\ -71/120,\ 11/10,\ -4/3,\\
 &\hspace{1.5em}19/60,\ -13/15,\ 1/6,\ 4/3,\ -11/48,\
      1/6,\ -1/8,\ -3/4)^{\mathsf T},
\end{split}
\]
\[
\begin{split}
 f^{\mathrm{reg}}={}&
 (-27/80,\ 19/60,\ -11/48,\ -71/120,\ 11/10,\ +4/3,\\
 &\hspace{1.5em}19/60,\ -13/15,\ 1/6,\ 4/3,\ -11/48,\
      1/6,\ -1/8,\ -3/4)^{\mathsf T}.
\end{split}
\]
The unchanged gauge-table constant vector is
\[
\begin{split}
 a^{(0)}={}&
 (27/80,\ -19/60,\ 11/48,\ 71/120,\ -11/10,\ 4/3,\\
 &\hspace{1.5em}-19/60,\ 13/15,\ -1/6,\ -4/3,\ 11/48,\
      -1/6,\ 1/8,\ 3/4)^{\mathsf T}.
\end{split}
\]
Python index 5 is displayed row 6, namely
\(B_6=\widehat{a_1\cdot a_2}
\widetilde{a_1\cdot a_2}Q^2\).
\end{record}

\begin{theorem}[Exact phase-integrity residual]
\label{thm:phase}
For the arrays in Record~\ref{rec:phase},
\begin{equation}
 a^{(0)}+f^{\mathrm{tex}}=0,\qquad
 f^{\mathrm{reg}}-f^{\mathrm{tex}}
   =a^{(0)}+f^{\mathrm{reg}}=\frac83e_6.
 \label{eq:phaseresidual}
\end{equation}
Thus the v4 regression assertion \(a^{(0)}=-f^{\mathrm{reg}}\)
is false at precisely Python index 5.
There is no single scalar \(\lambda\) with
\(a^{(0)}=\lambda f^{\mathrm{reg}}\).

If these arrays are encoded as the corresponding complex
reports, the cross-artifact coefficient discrepancy is \(4/3\).
At common bound \(B=32\), its half-diamond distance is \(1/24\).
\end{theorem}

\begin{proof}
All entries cancel except the sixth, where
\(4/3+4/3=8/3\).
The first entry forces \(\lambda=-1\), which fails at the sixth.
Under the convention \(v=u/\ii\), compare
\(\ii f^{\mathrm{reg}}\) with \(-\ii a^{(0)}\);
their difference is \((8\ii/3)e_6\).
Apply Theorem~\ref{thm:error}.
\end{proof}

\begin{remark}[What is corrected, and what is not]
The predecessor's proposed caption change from
\(+\ii u^{(0)}\) to \(-\ii u^{(0)}\) follows algebraically from
\(v=u/\ii\), since \(1/\ii=-\ii\).
It remains a documentary normalization observation.
It cannot make \(a^{(0)}=-f^{\mathrm{reg}}\) true.

The typeset and regression values are both retained.
Their physical origin cannot be decided from this comparison:
it does not identify which underlying loop artifact, if any,
contains the correct coefficient.
In particular, it is inaccurate to describe the witness as a
failure of every entry in the typeset gauge table.
The failure is specifically cross-artifact.
\end{remark}

\subsection{Known-discrepancy tests are not data acceptance}

\begin{proposition}[Two different acceptance predicates]
\label{prop:twopredicates}
Let
\[
 P_{\mathrm{consistent}}:
 a^{(0)}+f^{\mathrm{reg}}=0,\qquad
 P_{\mathrm{witness}}:
 a^{(0)}+f^{\mathrm{reg}}=\tfrac83e_6.
\]
For the retained arrays, \(P_{\mathrm{consistent}}\) is false
and \(P_{\mathrm{witness}}\) is true.
Passing a regression that enforces \(P_{\mathrm{witness}}\)
does not establish \(P_{\mathrm{consistent}}\).
\end{proposition}

\begin{proof}
The right sides are different because \(8/3\ne0\).
Equation~\eqref{eq:phaseresidual} proves the assertions.
\end{proof}

The integrity gate therefore records
\code{DISCREPANCY_RECORDED} and
\code{phase_consistent=false}.
A later reconciliation requires a new versioned source record,
the actual original arrays, a justified convention map, and a
fresh exact comparison. Merely treating the old failure as
expected does not discharge that obligation.

%=====================================================================
\section{The supplied v4 run and its execution coverage}
\label{sec:v4run}
%=====================================================================

\begin{record}[Supplied failed regression transcript]
\label{rec:v4run}
The supplied output identifies Python 3.10.21, SymPy 1.14.0,
and a Linux platform, and reports eight passing suites and one
failure \cite{V4Artifacts}.
The complete short transcript is retained in
Appendix~\ref{app:v4log}.

Its reported source digest is
\begin{center}
\scriptsize
\code{e9f661c8f3eb1dc75b580eb756174f4ff5711f172230ae369b17dbf5a2fbb106}.
\end{center}
This is a digest printed by the supplied record, not a digest
independently recomputed here from authenticated original bytes.
No execution date, signed provenance chain, or separately
observed process exit code is supplied.
\end{record}

\begin{proposition}[What the failed table suite reached]
\label{prop:coverage}
In the supplied v4 script, the mutation-count, maximum-discrepancy,
and common-bound checks precede the phase assertion.
The 13-varying-structure and derivative checks follow it.
Consequently, in the reported failing invocation:
\begin{enumerate}
\item the earlier mutation checks were reached;
\item the phase assertion rejected the data;
\item that invocation did not reach the subsequent variation-count
and derivative predicates in the same test function.
\end{enumerate}
The runner continued to the remaining suites.
\end{proposition}

\begin{proof}
The function executes sequentially and raises
\code{CheckFailure} at
\code{require(all(x == -y for x,y in zip(a0,f)), ...)}.
There is no local handler resuming inside that function.
The outer runner catches the exception and proceeds to the
next listed test.
\end{proof}

Thus the supplied summary's claim that all other table checks
``held'' is too broad if read as execution coverage.
The 13-of-14 count and derivative formulas are nevertheless
proved directly from the printed coefficients in
Section~\ref{sec:gaugetables}; they are not lost.

The reported 1397 evaluated predicates include expected
rejections and internal checks. That count does not establish
that every advertised predicate was reached.

\subsection{Two different failures, two different lessons}

The earlier affine example had a false advertised equation
despite attributed PASS.
The v4 phase suite instead rejected a false equation and reported
aggregate failure.
The first is an acceptance-semantics defect; the second is
evidence that the relevant fail-closed check worked.

Neither statement establishes every predicate of the earlier
software or the correctness of the loop data.

%=====================================================================
\section{Direct graded non-feedback and quotient response}
\label{sec:nonfeedback}
%=====================================================================

The supplied R07--R09 descriptions report six even cubic classes,
four odd classes before a Schouten quotient, and two afterwards
\cite{SuppliedBot,CheungXVIR}.
Counts do not provide representatives or response maps.

Let \(\mathscr P_q\) be local polynomial functionals homogeneous
of field/antifield degree \(q\), with any permitted relations
required to preserve that grading.
Let \(P_2\) be homogeneous projection and
\(e:\mathscr P_2\to\K^{21}\) a declared quadratic coefficient map.

\begin{theorem}[Rank-zero direct graded projection]
\label{thm:graded}
For ten representatives \(u_a\in\mathscr P_3\), the matrix of
\(eP_2\) is
\[
 M_{\mathrm{grade}}=0_{21\times10},\qquad
 \rank M_{\mathrm{grade}}=0,\qquad L_{\mathrm{grade}}=I_{21}.
\]
The original minor is empty with determinant one.
The zero map descends through every relation subspace.

In a nonnegative formal \(\kappa\)-series, an order-\(\kappa^3\)
addition cannot change the order-\(\kappa^2\) coefficient under
operations that do not lower the filtration.
\end{theorem}

\begin{proof}
Homogeneity gives \(P_2u_a=0\).
The full left kernel and empty minor certify rank zero.
A zero map annihilates all relations.
For the formal-order assertion, products of nonnegative powers
cannot create a term below the order of the addition.
\end{proof}

The reported cubic slots \(Kch,\bar cch,c^*cc,hhh\) have three
field slots; spectator vectors representing an epsilon tensor
are not fields. The predecessor attributes a slot-count check
on 821 even coordinates and the odd terms to
\code{o1_1c.py}; that archive is not newly inspected.

This theorem concerns direct graded re-expansion, not Wick
contraction, expansion around a nonzero background, or an
unspecified changed identity response.
Such operations need their own maps.
A formal zero projection does not establish the validity of a
strictly four-dimensional Schouten quotient in DRED.

\begin{theorem}[Quotient descent versus non-feedback]
\label{thm:quotient}
Let \(R_U\subseteq U\), let
\(T:V_{\mathrm{allowed}}\to W_{\mathrm{full}}\) be the allowed
counterterm response, and let \(F:U\to W_{\mathrm{full}}\)
be the complete review response.
For a complete left kernel \(L_T\) of \(T\),
\[
 \pi F\text{ descends to }U/R_U
 \iff L_TF(R_U)=0,
\]
where \(\pi:W_{\mathrm{full}}\to W_{\mathrm{full}}/\im T\).
The descended obstruction vanishes on every supplied class iff
\[
 L_TF=0
 \iff \rank[T\mid F]=\rank T.
\]
\end{theorem}

\begin{proof}
Descent requires annihilation of relations after quotienting.
Theorem~\ref{thm:rank} gives \(\im T=\ker L_T\).
Vanishing on all classes requires containment of \(F(U)\), not
only \(F(R_U)\), in that image.
\end{proof}

Consistency restrictions belong in \(V_{\mathrm{allowed}}\) or
in augmented response rows.
No projection may discard outside-span components first.
Parity mutations test selected software behavior; they do not
replace these hypotheses.

%=====================================================================
\section{The exact affine gauge propagator}
\label{sec:propagator}
%=====================================================================

The gauge family is a new scope.
Take real \(\xi\ne0\), scale the eliminated gauge fixing by
\(\xi^{-1}\), and hold the declared vertices and prescription
fixed. An auxiliary/gauge-fermion realization must reproduce
this normalization, not merely share its name.

On symmetric Q4S tensors define
\[
 \mathsf A=I-\tfrac12J,\qquad
 JE=\eta\,\Tr_\eta E,\qquad \Tr_\eta\eta=4.
\]
For momentum \(k\), put
\[
 \mathsf Rv=k_{(a}v_{b)},\qquad
 \mathsf R^*E=E_{ab}k^b,\qquad
 \mathsf FE=E_{ab}k^b-\tfrac12k_a\Tr_\eta E.
\]
Stars denote algebraic adjoints for the stated pairings, not
positive Hilbert-space adjoints. Then
\[
 \mathsf F=\mathsf R^*\mathsf A,\qquad
 \mathsf F^*=\mathsf A\mathsf R,\qquad
 \mathsf G=2\mathsf F^*\mathsf F.
\]

\begin{lemma}[Projector identities]
\label{lem:projector}
For \(k^2\ne0\),
\[
 \mathsf A^2=I,\qquad
 \mathsf F\mathsf R=\frac{k^2}{2}I,\qquad
 \mathsf H=\frac2{k^2}\mathsf F^*\mathsf R^*,
 \qquad \mathsf H^2=\mathsf H.
\]
Also
\[
 \mathsf R\mathsf R^*=\mathsf S_4(k),
\]
where
\[
 \mathsf S_4(k)_{ab,cd}
 =\tfrac14(k_ak_c\eta_{bd}+k_ak_d\eta_{bc}
                 +k_bk_c\eta_{ad}+k_bk_d\eta_{ac}).
\]
\end{lemma}

\begin{proof}
Since \(J^2=4J\), \(\mathsf A^2=I\).
For a covector \(v\),
\[
 \mathsf R^*\mathsf Rv
 =\tfrac12(k^2v+k(k\cdot v)),\qquad
 \Tr_\eta(\mathsf Rv)=k\cdot v.
\]
This gives \(\mathsf F\mathsf R=k^2I/2\).
Furthermore,
\(\mathsf R^*\mathsf F^*=\mathsf R^*\mathsf A\mathsf R
=\mathsf F\mathsf R\), which proves the projector identity.
Expanding \(\mathsf R\mathsf R^*\) gives the four terms.
\end{proof}

\begin{theorem}[Exact affine inverse under the declared normalization]
\label{thm:propagator}
For
\[
 \mathsf K_\xi=k^2\mathsf A+(\xi^{-1}-1)\mathsf G,
\]
the rational inverse at \(k^2\ne0,\xi\ne0\) is
\begin{equation}
 \mathsf P_\xi=
 \frac{\mathsf A}{k^2}
       +2(\xi-1)\frac{\mathsf S_4(k)}{(k^2)^2}.
 \label{eq:propagator}
\end{equation}
\end{theorem}

\begin{proof}
Write \(\mathsf K_1=k^2\mathsf A\),
\(\mathsf P_1=\mathsf A/k^2\), and \(a_\xi=\xi^{-1}-1\).
Then
\[
 \mathsf K_\xi=(I+a_\xi\mathsf H)\mathsf K_1,\qquad
 (I+a_\xi\mathsf H)^{-1}=I+(\xi-1)\mathsf H.
\]
Multiplying by \(\mathsf P_1\) gives
\[
 \mathsf P_\xi=\frac{\mathsf A}{k^2}
   +\frac{2(\xi-1)}{(k^2)^2}\mathsf R\mathsf R^*.
\]
\end{proof}

This is a general algebraic proof in the declared trace-four
algebra, not an interpolation from finite momenta.
The DRED contraction prescription and overall Feynman factor
remain scientific inputs.
On \(k^2=0\), a prescribed boundary value is required; this is
not pointwise inversion there.
At \(\xi=0\), the eliminated kinetic expression is undefined.
The predecessor's action-Hessian check is attributed to
\code{o1_kincheck.py}, not newly executed.

%=====================================================================
\section{Retained gauge-pole data and exact printed consequences}
\label{sec:gaugetables}
%=====================================================================

Put \(\zeta=\xi-1\). With unchanged vertices, a graph with
\(n_h\) graviton lines has degree at most \(n_h\) in \(\zeta\)
under linear pole extraction.

\begin{record}[Supplied gauge calculation]
\label{rec:gauge}
The predecessor attributes the following to its gauge-family
scripts \cite{SuppliedBot}:
\begin{enumerate}
\item D03 and D11 have no graviton line and are \(\xi\)-independent;
\item D04 has zero UV-pole polynomial in the stated family;
\item D06 and D13 are affine, and D01 is quadratic;
\item \(D06=-2D13[K\to T(\bar c)]\) holds coefficientwise in
\(\zeta\), in the incoming-antighost convention.
\end{enumerate}
These remain supplied scientific records.
The D01 base-point identification across all v4 artifacts is
subject to Section~\ref{sec:integrity}.
\end{record}

Here
\[
 T_{mn}(\bar c)=
 \tfrac12(p_m\bar u_n+p_n\bar u_m)
        -\tfrac12\eta_{mn}(p\cdot\bar u).
\]
Momentum orientation and the separately disclosed D13 slot
orientation must be transported consistently
\cite{CheungXVIQ}.

In own-rule normalization \(u=\ii v\), D06 is
\[
 u_{06}(\xi)=\ii\left[
 (-3/8,1/4,-1/2)^{\mathsf T}
 +\zeta(-7/24,-1/12,-1/4)^{\mathsf T}\right].
\]
For \(K=a_K\otimes a_K\), set
\[
 s=\widehat{a_K\cdot p},\quad
 r=\widehat{a_K\cdot u},\quad
 t=\widetilde{a_K\cdot u},\quad
 A_K=\widehat{a_K\cdot a_K},\quad
 u_p=\widehat{u\cdot p}.
\]
On \((sQt,s^2u_p,rsQ,A_Ku_pQ)\), D13 has coefficients
\[
 u_{13}(\xi)=\ii\left[
 (-1/2,1/4,-3/8,0)^{\mathsf T}
 +\zeta(-1/4,1/12,-7/24,1/12)^{\mathsf T}\right].
\]
The fourth structure first appears at order \(\zeta\).

For D01 write
\[
 u_{01}(\xi)=\sum_{j=1}^{14}
  \bigl(a_j^{(0)}+\zeta a_j^{(1)}
                      +\zeta^2a_j^{(2)}\bigr)B_j.
\]
All printed coefficients are retained:

\begin{center}
\small
\begin{tabular}{r r r r}
\toprule
\(j\)&\(a_j^{(0)}\)&\(a_j^{(1)}\)&\(a_j^{(2)}\)\\
\midrule
1&\(27/80\)&\(1/2\)&\(1/2\)\\
2&\(-19/60\)&\(-2/3\)&\(-1/2\)\\
3&\(11/48\)&\(1\)&\(5/6\)\\
4&\(71/120\)&\(1/2\)&\(1/6\)\\
5&\(-11/10\)&\(-1\)&\(-1/3\)\\
6&\(4/3\)&\(1/2\)&\(0\)\\
7&\(-19/60\)&\(-2/3\)&\(-1/2\)\\
8&\(13/15\)&\(4/3\)&\(2/3\)\\
9&\(-1/6\)&\(-5/4\)&\(-5/6\)\\
10&\(-4/3\)&\(-1/2\)&\(0\)\\
11&\(11/48\)&\(1\)&\(5/6\)\\
12&\(-1/6\)&\(-5/4\)&\(-5/6\)\\
13&\(1/8\)&\(3/2\)&\(5/4\)\\
14&\(3/4\)&\(0\)&\(0\)\\
\bottomrule
\end{tabular}
\end{center}

\begin{proposition}[Variation count and derivative]
\label{prop:derivative}
Exactly 13 of these 14 structures vary with \(\xi\);
\(B_{14}\) is constant.
The printed derivative is
\begin{equation}
 b_{hh}(\xi)=
 \sum_j\bigl(a_j^{(1)}+2(\xi-1)a_j^{(2)}\bigr)B_j.
 \label{eq:hhderivative}
\end{equation}
These facts do not require correcting either version of \(f\).
\end{proposition}

\begin{proof}
Only row 14 has both variable coefficients zero.
Differentiate the finite polynomial.
\end{proof}

The cross-artifact witness concerns the displayed constant
coefficient. It does not alter the formal differentiation of the
unchanged \(a^{(1)},a^{(2)}\) arrays. This observation is not a
new verification of their loop origin.

\begin{remark}[Scaleless zeros need their prescription]
Massless one-point and tadpole zeros are retained under the
supplied prescription. Scaleless dimensional integrals can
vanish by UV/IR cancellation; the integral with integrand
\(1/(l^2)^2\) illustrates the logarithmic issue.
No assertion that every scaleless tadpole has no logarithmic UV
or IR contribution is used.
Insertion problems and infrared rearrangements need their own
separation \cite{SmirnovChetyrkin}.
\end{remark}

D04's supplied zero is not a theorem about every gauge-invariant
scalar correlation function. Such observables require additional
contributions and gauge-dependent cancellations
\cite{Frob2018}.

%=====================================================================
\section{Free-complex exactness and the missing enlarged image}
\label{sec:exactness}
%=====================================================================

Before differentiating, identify the gauge descriptions and
their basis transports. Declare a finite complex
\[
 E^{-1}\xrightarrow{S}E^0\xrightarrow{B_0}E^1,\qquad B_0S=0.
\]
The primitive space has ghost number \(-1\); it is not the
21-direction ghost-number-zero counterterm space.

\begin{theorem}[Free exactness and exact tangents]
\label{thm:exactness}
For a differentiable solution
\(A_\xi c(\xi)+d_\xi=0\) in the fixed identified presentation,
\[
 c'(\xi)\in\im S\iff L_Sc'(\xi)=0.
\]
At fixed \(\xi\), put
\(w_\xi=-d_\xi'-A_\xi'c(\xi)\).
An exact tangent \(z=Su\) satisfying differentiated cancellation
exists iff
\[
 w_\xi\in\im(A_\xi S).
\]
Every such tangent is \(B_0\)-closed.
\end{theorem}

\begin{proof}
Apply Theorem~\ref{thm:rank} to \(S\).
Differentiate cancellation and substitute \(c'=Su\).
Finally \(B_0Su=0\).
\end{proof}

If the identified differential varies, use \(S_\xi\) and state
its transport. Parameter-dependent consistency equations must
also be differentiated.
A pointwise tangent is not an interval theorem without smooth
choices and exceptional-parameter control.

Free BRST, relative classical BV, free-equation descent, and
interacting quantum BRST remain different objects
\cite{CheungXGST,CheungXHST}.
No X-G-ST or X-H-ST normalization is substituted for the P/Q
action normalization.

\subsection{The supplied restricted \(hh\) obstruction}

\begin{record}[Restricted image report]
\label{rec:restricted}
The predecessor reports testing the \(hh\) pole derivative
against the image of
\[
 Y=\int\widetilde KXh
\]
in its 13-generator dimension-two \(X\)-presentation.
For the relaxed map allowing independent \(X,X'\) in
\[
 \Sym(K_{EH}\circ X)+\Sym(G\circ X'),
\]
it reports on a 15-structure \(hh\) space
\[
 \rank H_{\mathrm{rel}}=7,\qquad
 \rank[H_{\mathrm{rel}}\mid b_{hh}]=8
 \quad\text{at }\xi=1,2.
\]
Fixed-normalization failures, including the listed
\(\xi=1,2,\tfrac12\) tests, are also reported.
The four-dimensional projection is reported to have rank five
on six structures and to accept both the derivative and the
undifferentiated \(\xi=1\) pole \cite{SuppliedBot}.
\end{record}

The full generator list, ambient basis, matrices, original
minors, and left-kernel rows are absent from the supplied
artifacts. The relaxed rank report is not extended to additional
parameters and is not newly re-certified.

Define the named source-relative obstruction
\[
 \mathfrak o_{hh}^{\mathrm{src}}(\xi)
 =[b_{hh}(\xi)]\in
 W_{hh}^{\mathrm{src}}/\im H_{\mathrm{rel},\xi}.
\]
If the reported ranks hold for those actual arrays, this class is
nonzero at \(\xi=1,2\), because adjoining the target raises rank.
That excludes only the tested class.
The four-dimensional test is nondiscriminating in the reported
comparison, not invalid for every purpose.

A complete rank-seven image certificate on 15 rows needs an
original \(7\times7\) minor and an \(8\times15\) complete
left-kernel matrix, with a nonzero target pairing.

\subsection{Exact enlargement and separator construction}

\begin{theorem}[Projected primitive-enlargement criterion]
\label{thm:enlargement}
Let \(H:Y_{\mathrm{old}}\to W\) have a complete left kernel
\(L\), let \(E:Y_{\mathrm{new}}\to W\) be a specified additional
primitive response, and let \(b\in W\). Then
\[
 b\in\im[H\mid E]
 \iff Lb\in\im(LE)
 \iff \rank[LE\mid Lb]=\rank(LE).
\]
Let \(Q_E\) be a complete left kernel of \(LE\), and set
\[
 D_E=Q_EL.
\]
Then
\begin{equation}
 \ker D_E=\im[H\mid E].
 \label{eq:combinedseparator}
\end{equation}
If \(D_Eb\ne0\), a row \(\ell\) with \(\ell b\ne0\)
satisfies
\[
 \ell H=0,\qquad \ell E=0,
\]
and is an exact separator for the enlarged class.
For several targets \(B\), a successful enlargement needs at
least \(\rank(LB)\) independent new image directions.
\end{theorem}

\begin{proof}
If \(b=Hu+Ev\), then \(Lb=LEv\).
Conversely, \(Lb=LEv\) implies \(b-Ev\in\ker L=\im H\).
Completeness of \(Q_E\) gives
\[
 D_Eb=0\iff Lb\in\ker Q_E=\im(LE),
\]
proving \eqref{eq:combinedseparator}.
The row identities follow from \(LH=0\) and \(Q_ELE=0\).
The multiple-target lower bound follows by dimension.
\end{proof}

This gives a precise construction of a separator \emph{from
supplied arrays}; it does not supply numerical rows for an absent
physical image.

\begin{proposition}[The rank report alone cannot decide an enlargement]
\label{prop:underdetermined}
Knowing only \(\rank H=7\) and \(\rank[H\mid b]=8\)
does not determine whether an unspecified enlargement \(E\)
removes the obstruction.
\end{proposition}

\begin{proof}
In \(\K^{15}\), let \(H=(e_1,\ldots,e_7)\) and \(b=e_8\).
Both rank statements hold.
Taking \(E_+=e_8\) resolves the target; taking \(E_-=e_9\)
does not. For the second choice \(e_8^{\mathsf T}\) annihilates
both \(H,E_-\) and pairs to one with \(b\).
These are synthetic examples, not loop images.
\end{proof}

\scopebox{Honest v5 ending for the requested enlarged primitive}{%
No concrete enlarged ghost-number-\(-1\) \(Y\), its response
matrix \(E\), or numerical physical separator is supplied.
A separator annihilating an unspecified enlargement cannot be
inferred from the old rank report.

The missing deliverables are recorded as null artifacts.
Theorem~\ref{thm:enlargement} specifies exactly how an eventual
separator or primitive solution will be certified.
Full counterterm-derivative exactness remains \code{OPEN}.}

A legitimate enlargement must name its p22-type or
dimension-dependent primitives, actual differential, relations,
ghost-equation partners, and common normalization.
Terms proportional to \(d-4\) cannot be inserted or discarded by
slogan: their contractions and multiplication by Laurent poles
must follow the prescription \cite{DuganGrinstein}.

The full identity
\(\partial_\xi\mathrm{CT}=\sigma_0Y\)
also requires an actual counterterm family and a joint
\(hh,Kc,\bar cc\), auxiliary, and scalar solve where required.
Divisibility of one component by \(p^2\) is not that joint
identity. Oai282-A supplies a documentary local-Ward certificate
shape, not the missing gravitational response
\cite{CheungOai282}.

%=====================================================================
\section{Finite reduction of the full gauge-family row problem}
\label{sec:familysolve}
%=====================================================================

The affine propagator does not imply an affine counterterm
solution or a constant response matrix.

\begin{theorem}[Constant-response polynomial lift]
\label{thm:constantlift}
Let \(M\) be a constant finite matrix and
\(y(\zeta)=\sum_{j=0}^q y_j\zeta^j\), over \(\K\).
For a complete left kernel \(L\), the following are equivalent:
\begin{enumerate}
\item \(Ly_j=0\) for every \(j\);
\item a polynomial solution \(c(\zeta)=\sum_{j=0}^q c_j\zeta^j\)
exists;
\item the target belongs to \(\im M\) at every point of an
infinite parameter set in the field.
\end{enumerate}
The same original minor and left kernel certify each coefficient
solve. With complete right kernel \(Z\), all polynomial solutions
differ by \(Zt(\zeta)\).
\end{theorem}

\begin{proof}
The first condition solves each \(Mc_j=y_j\).
A polynomial solution gives pointwise membership.
Conversely, \(Ly(\zeta)\) vanishes on infinitely many points, so
every entry is the zero polynomial.
The kernel assertion is coefficientwise.
\end{proof}

If constancy of the complete response is established and the
complete target is quadratic, a certified base coefficient leaves
two additional membership tests. Theorem~\ref{thm:phase} prevents
claiming that every supplied D01 artifact already provides a
consistent base coefficient.

\begin{theorem}[Varying-response block system]
\label{thm:blocklift}
Let
\[
 M(\zeta)=\sum_{a=0}^{m}M_a\zeta^a,\qquad
 y(\zeta)=\sum_{b=0}^{q}y_b\zeta^b.
\]
For a declared degree bound \(D\), existence of
\(c(\zeta)=\sum_{j=0}^Dc_j\zeta^j\) is equivalent to
\[
 \sum_{\substack{a+j=t\\0\le a\le m\\0\le j\le D}}
 M_ac_j=y_t,\qquad
 0\le t\le\max(q,m+D),
\]
where absent target coefficients are zero.
\end{theorem}

\begin{proof}
Expand the product and equate every coefficient, including
coefficients above the target degree.
\end{proof}

Theorem~\ref{thm:rank} certifies this finite block system.
Rational families may be treated over \(\K(\xi)\) or after
clearing declared denominators on a specified domain.
Exceptional ranks and denominator zeros require separate checks:
\(\xi c=1\) is generically solvable and fails at zero.

The complete \(M_\xi,y_\xi\), original minor, and complete
kernels for the changed-gauge 23-row problem remain absent.
Full gauge-family row closure stays \code{OPEN}.
D04 and the ghost equation are cross-checks, not replacement rows.

%=====================================================================
\section{Genuine modular rank and permanent verification rules}
\label{sec:modular}
%=====================================================================

\begin{proposition}[Modular original minors]
\label{prop:modular}
For rational \(M\), with all denominators invertible modulo a
prime \(p\), a nonzero reduced original \(r\times r\) minor
proves \(\rank_\Q M\ge r\).
Finite-field rank alone does not upper-bound rational rank.
Taking \(\rank_\Q M\bmod p\) is not computing
\(\rank_{\F_p}\overline M\).
\end{proposition}

\begin{proof}
A zero rational determinant reduces to zero wherever reduction
is defined. The matrix \((p)\) has rational rank one and
finite-field rank zero, while \(1\bmod p=1\).
\end{proof}

For
\[
 M_q=\begin{pmatrix}1/2&1/3\\2/3&1/2\\1&1\end{pmatrix},
\]
the separate rational original minor is \(1/36\).
Modulo five,
\[
 \overline M_q=\begin{pmatrix}3&2\\4&3\\1&1\end{pmatrix},
\]
and the corresponding determinant is one.
Both ranks are two.
Composite moduli and noninvertible denominators are rejected.

For Gaussian-rational coefficients,
\[
 \operatorname{Real}(M)=
 \begin{pmatrix}M_R&-M_I\\M_I&M_R\end{pmatrix},\qquad
 \rank_\Q\operatorname{Real}(M)=2\rank_{\Q(\ii)}M.
\]
Restriction of scalars proves the identity.
Alternatively, a compatible finite-field representation can be
declared. Field labels are not interchanged.

Finite-field reconstruction and IBP reduction are discovery
methods \cite{ManteuffelSchabinger,Peraro}.
A candidate becomes an exact certificate only after checking the
original arrays, characteristic-zero identities, original minor,
complete kernels, and target residual.
Finite kinematic samples need coverage or certified
degree/denominator bounds for a broader identity claim.

\begin{proposition}[An acceptance string cannot override an equation]
\label{prop:pass}
An aggregate acceptance does not certify an advertised conjunction
if one advertised equation is false.
A known-discrepancy regression must not label data consistency
true merely because it successfully rejects that equation.
\end{proposition}

\begin{proof}
The first statement follows from the logic of conjunction,
with \eqref{eq:oldresidual} as an explicit witness.
The second is Proposition~\ref{prop:twopredicates}.
\end{proof}

The proposed implementation uses required predicates rather than
optimizable \code{assert}, residual verification after solving,
genuine modular elimination, and rejection controls with exact
expected reasons \cite{PythonAssert,SymPyMatrices}.

%=====================================================================
\section{Retained exact synthetic examples}
\label{sec:synthetic}
%=====================================================================

These arrays are not O.1 loop matrices.

\subsection{Rank two and complete affine ambiguity}

Let
\[
 M_1=\begin{pmatrix}
 1&2&3&4\\2&4&6&8\\1&1&1&1\\3&5&7&9\\0&0&0&0
 \end{pmatrix},\quad
 L_1=\begin{pmatrix}
 -2&1&0&0&0\\-2&0&-1&1&0\\0&0&0&0&1
 \end{pmatrix},
\]
\[
 Z_1=\begin{pmatrix}1&2\\-2&-3\\1&0\\0&1\end{pmatrix},
 \quad y_1=(1,2,1,3,0)^{\mathsf T},\quad
 c_1=(1,0,0,0)^{\mathsf T}.
\]
The original minor on rows \(1,3\), columns \(1,2\), is \(-1\).
Columns \(2,4,5\) of \(L_1\) form \(I_3\); the last two rows
of \(Z_1\) form \(I_2\). Direct multiplication gives
\[
 L_1M_1=0,\quad L_1y_1=0,\quad M_1c_1=y_1,\quad M_1Z_1=0.
\]
Hence \(\rank M_1=2\) and every solution is \(c_1+Z_1t\).

For
\[
 M_u=\begin{pmatrix}1&2\\2&4\\0&0\end{pmatrix},\quad
 L_u=\begin{pmatrix}-2&1&0\\0&0&1\end{pmatrix},\quad
 y_u=(1,1,1)^{\mathsf T},
\]
the rank is one and \(L_uy_u=(-1,1)^{\mathsf T}\ne0\).

\subsection{Descent without non-feedback}

Take
\[
 T=\begin{pmatrix}1&0&2\\0&1&1\\1&1&3\\0&0&0\end{pmatrix},
 \quad L_T=\begin{pmatrix}-1&-1&1&0\\0&0&0&1\end{pmatrix},
 \quad R=(2,-1)^{\mathsf T}.
\]
The upper-left minor is one and \(L_T\) is complete.
For
\[
 F=\begin{pmatrix}1&2\\0&1\\1&3\\0&0\end{pmatrix},\quad
 F_d=\begin{pmatrix}1&2\\0&1\\1&3\\1&2\end{pmatrix},
\]
\[
 L_TF=0,\qquad FR=(0,-1,-1,0)^{\mathsf T}\ne0,
\]
\[
 L_TF_d=\begin{pmatrix}0&0\\1&2\end{pmatrix}\ne0,\qquad
 L_TF_dR=0.
\]
Thus \(F_d\)'s obstruction descends but does not vanish.
For
\(F_b=\left(\begin{smallmatrix}1&0\\0&1\\1&1\\1&0\end{smallmatrix}\right)\),
one has \(L_TF_bR=(0,2)^{\mathsf T}\ne0\), so descent fails.

\subsection{Free exactness, tangent failure, and closed nonexact vectors}

Let
\[
 S=\begin{pmatrix}1&0\\0&1\\1&1\end{pmatrix},\quad
 B_0=(-1,-1,1),\quad
 A=\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}.
\]
Then \(B_0S=0\), \(AS=I_2\), and
\[
 c(\xi)=\xi(1,-1,0)^{\mathsf T},\quad
 d(\xi)=(-\xi,\xi)^{\mathsf T}
\]
satisfy cancellation and
\(c'=S(1,-1)^{\mathsf T}\).
The vector \((0,0,1)^{\mathsf T}\) is not exact.

With
\[
 A_*=\begin{pmatrix}1&0&0\\0&0&1\\1&0&1\end{pmatrix},\qquad
 A_*S=\begin{pmatrix}1&0\\1&1\\2&1\end{pmatrix},
\]
the left kernel is \((-1,-1,1)\).
Target \((1,0,1)^{\mathsf T}\) is feasible;
\((1,0,0)^{\mathsf T}\) is not.
Also
\[
 S_0=e_1,\quad B_1=(0,0,1),\quad z=e_2
\]
give \(B_1S_0=B_1z=0\) but \(z\notin\im S_0\).

\subsection{Bounded report attainment}

For
\[
 v=(3/10,-1/2,4/5,1/10),\qquad
 \widetilde v=(3/10,-2/5,7/10,1/10),
\]
the maximum discrepancy is \(1/10\); the half-diamond distance
at scale one is \(1/20\), attained at task 2 or 3.
Using \(2v,2\widetilde v\) at scale two gives the same value.

\subsection{Correlated reference branches}

Let the clock basis be the columns of
\[
 U=\tfrac12\begin{pmatrix}
 1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1
 \end{pmatrix}
\]
and let \(A_{\mathrm{lab}}\cong E\cong\C^2\).
For
\[
 \ket\Psi=
 \frac{\ket{\tau_1}\ket0_A\ket0_E+
       \ket{\tau_2}\ket1_A\ket1_E}{\sqrt2},
\]
the nonzero blocks are
\(\frac12\proj{00}\) and \(\frac12\proj{11}\).
Their sum is \(\Tr_C\proj\Psi\).
The unconditional reference is \(I_E/2\), while positive
conditional references are \(\proj0,\proj1\).
Zero blocks are not normalized.

\subsection{Gaussian-rational rank}

For
\[
 G_c=\begin{pmatrix}1&\ii\\\ii&-1\\0&0\end{pmatrix},\quad
 L_c=\begin{pmatrix}-\ii&1&0\\0&0&1\end{pmatrix},\quad
 z_c=(-\ii,1)^{\mathsf T},
\]
the original minor is one,
\(L_cG_c=0\), \(G_cz_c=0\), and
\(\rank_{\Q(\ii)}G_c=1\).
The realification has rational rank two.
The left-kernel multiplication is algebraic, not an unannounced
conjugate-transpose convention.

%=====================================================================
\section{Scheduled locality, transport, and rebuild resources}
\label{sec:locality}
%=====================================================================

Let each complete tick channel factor over a prescribed
partition \(\mathcal B_t\), including all controllers, records,
manifests, shared solvers, selectors, and resource wires.
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 ancestry]
\label{thm:locality}
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\mathcal A(Y))
 \subseteq\mathcal A(N_{s,t}(Y)).
\]
Equal joint reductions on the ancestor region and an untouched
reference give equal corresponding outputs.
A local CPTP intervention outside the full backward region
cannot change those outputs, with schedule and interfaces fixed.
If each dependency edge has a certified regular future-causal
host route, scheduled ancestry is contained in the prescribed
metric causal relation.
\end{theorem}

\begin{proof}
An excluded block acts on an identity factor, preserved by its
unital adjoint. A meeting block enlarges support only within
itself. Backward induction proves support containment.
Joint observables prove the reference and intervention claims.
Concatenation proves metric containment.
\end{proof}

This is the finite mechanism compared in I-B, En03-A, and
X-Enter-ST \cite{CheungIB,CheungEn03,CheungXEnterST}.
Scheduled ancestry bounds influence, not always equals it.
Cancellation can make an ancestor unobservable.
A shared solve can introduce dependencies absent from a row's
direct Feynman sum.

\begin{proposition}[Regular reference-preserving transport]
\label{prop:route}
Let \(n\ge1\). Place \(n+1\) equal payload registers on a straight
developed segment of length \(\ell>0\) in a convex regular reserve.
Initialize all but the source in independent blanks.
Successive SWAPs realize the identity payload channel, including
every reference. Hop duration \(\ell/n\) gives arrival time
\(\ell\) in geometrized units.
\end{proposition}

\begin{proof}
Each SWAP moves the entire payload and reference correlations
to the next register and leaves the old site blank.
Induction gives the channel identity; spacing and declared
duration give the regular null routes.
\end{proof}

SWAP permissions and speed are inputs, not a derived
gravitational Hamiltonian.
Transporting a counterterm code does not define its analytical
insertion.

For \(J\ge1\) dictionary entries and a \(p\)-qubit payload, a
conservative reserve is
\[
 n_C=\max\{1,\lceil\log_2J\rceil\},\qquad
 Q_{\mathrm{reserve}}=
 n_C+(n+1)p+Q_{\mathrm{work}}+Q_{\mathrm{probe}}+Q_{\mathrm{other}}.
\]
Classical dictionaries, manifests, bit lengths, records, source
storage, and any quantum realization of them are charged
separately. References are identified, not treated as workspace.

Step 1b stays \code{MISSING}.
A clean-machine or heterogeneous rebuild must specify changes
in hardware/software, hash the pinned tree, rebuild
intermediates, enforce the same residual checks, retain failures
and refusals, and disclose common conceptual inputs.
Q11's author rebuild is not reclassified as that rebuild.

%=====================================================================
\section{The integrity-aware finite release gate}
\label{sec:gate}
%=====================================================================

A report names its full scope, evidence, provenance, and status.
A regression status is not a scientific release status.

\begin{verbatim}
VERIFY(P, scope, budget):
  resolve scope and bind exact inputs, versions, and manifest
  check cross-artifact consistency required by the claim
  if a required equality fails:
      retain exact residual and report DISCREPANCY_RECORDED
      do not return CERTIFIED_SCOPED
  reject undeclared prescription or domain changes
  require all ancestors of the requested report
  verify inventory, zero provenance, descent, and coverage
  verify original minor and complete left kernel
  verify candidate residual and all declared consistency rows
  verify any complete right-kernel / affine-family claim
  enforce declared independence and mutation obligations
  retain resource, error, and provenance ledgers
  return CERTIFIED_SCOPED for exactly the resolved scope

  missing input / unfinished computation -> MISSING
  unresolved scientific residual -> OPEN
  failed advertised exact check -> FAILED
  resource limit -> MISSING with RESOURCE_REFUSAL
\end{verbatim}

An irrelevant discrepancy need not block an unrelated theorem.
The D01 artifact mismatch blocks claims requiring that
cross-artifact equality; it does not refute the abstract channel
theorems, the synthetic affine repair, or the documentary Q03
record.

\begin{theorem}[Conditional exact-scope soundness]
\label{thm:gate}
Assume the maps implement the declared analytical problem,
required artifact identifications hold, descent and coverage
hold, and every required exact check succeeds.
Then \code{CERTIFIED_SCOPED} proves cancellation and the
listed consistency identities on exactly that scope.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:rank} gives \(Mc=y\).
Its blocks are \eqref{eq:system}.
The semantic, integrity, descent, and coverage hypotheses identify
them with the declared equations.
No premise identifies changed scopes with those equations.
\end{proof}

The gate is a finite specification with possible refusals, not a
general algorithm deciding arbitrary analytical coverage.
A verifier does not formally certify itself by printing PASS.

%=====================================================================
\section{Machine-readable scope and evidence ledger}
\label{sec:ledger}
%=====================================================================

The ledger retains the prior scientific status strings.
Historical v4 production flags and the supplied later v4
execution record are distinct fields.
The current appendix flags refer only to the new v5 listing.
Null means no supplied artifact, never a zero coefficient.

\begin{lstlisting}[style=recordstyle,caption={Embedded v5 ledger.}]
{
  "schema": "XVI-O1-ST.scope-ledger.v5",
  "scope_key_order": [
    "H", "A", "loop", "kappa_power", "external",
    "mass", "Psi", "prescription", "V", "R", "J"
  ],
  "scopes": {
    "s_Q03": {
      "H": "prescribed_cone_and_supported_lab",
      "A": "S_inv_PQ_normalization",
      "loop": 1,
      "kappa_power": 2,
      "external": "retained_bulk_two_point_QdS",
      "mass": 0,
      "Psi": "frozen_base_gauge_xi_1",
      "prescription": "XVI-P_p23_DRED_QdS_Q4S_MSbar_p22",
      "V": "11_p14r_over_Qi_plus_10_p22",
      "R": "23_named_rows_with_complete_scalar_expansions",
      "J": "Q03_bulk_11_4D_consistency_only"
    },
    "s_readout": {
      "base": "s_Q03",
      "patch": {
        "R": "D01_D03_D06_typeset_31_coefficients",
        "J": "component_mutations_and_attributed_ghost_equation"
      }
    },
    "s_phase": {
      "base": "s_readout",
      "patch": {
        "A": "cross_artifact_D01_array_comparison",
        "R": "14_exact_printed_coordinates",
        "J": "distinguish_f_tex_f_reg_and_a0"
      }
    },
    "s_grade": {
      "base": "s_Q03",
      "patch": {
        "A": "direct_cubic_to_quadratic_projection",
        "R": "21_direct_quadratic_coordinates",
        "J": "homogeneous_cubic_inputs_no_lowering_operation"
      }
    },
    "s_xi": {
      "base": "s_Q03",
      "patch": {
        "Psi": "declared_xi_family_xi_real_nonzero",
        "R": "complete_parameter_dependent_23_row_expansion",
        "J": "identified_gauge_family_consistency_maps"
      }
    },
    "s_kinetic": {
      "base": "s_xi",
      "patch": {
        "A": "declared_Q4S_kinetic_form",
        "loop": 0,
        "kappa_power": 0,
        "V": "symmetric_Q4S_tensors",
        "R": "rational_operator_inverse",
        "J": "trace_4_k_squared_nonzero_xi_nonzero"
      }
    },
    "s_xi_hh_restricted": {
      "base": "s_xi",
      "patch": {
        "A": "derivative_of_reported_hh_pole_block",
        "V": "source_restricted_Ktilde_X_h_primitive_image",
        "R": "source_declared_15_hh_structures",
        "J": "relaxed_rank_report_at_xi_1_and_2"
      }
    }
  },
  "results": [
    {
      "id": "Q03",
      "scope": "s_Q03",
      "status": "SUPPLIED_CLOSED",
      "evidence": "supplied_scientific_record",
      "new_replay": false,
      "original_matrix_artifact": null
    },
    {
      "id": "1a_tables",
      "scope": "s_readout",
      "status": "PROVED_FOR_PRINTED_DATA",
      "source_status": "Earned",
      "new_loop_execution": false,
      "independent_31_entry_engine_replay": false,
      "cross_artifact_dependency": "2a_phase_integrity"
    },
    {
      "id": "1b",
      "scope": "s_Q03",
      "status": "MISSING",
      "clean_machine_rebuild": false,
      "heterogeneous_rebuild": false
    },
    {
      "id": "1c",
      "scope": "s_grade",
      "status": "PROVED_GRADED",
      "matrix_shape": [21, 10],
      "rank": 0,
      "minor_rows": [],
      "minor_columns": [],
      "minor_determinant": "1",
      "left_kernel": "I_21",
      "universal_response_nonfeedback": false
    },
    {
      "id": "2a_propagator",
      "scope": "s_kinetic",
      "status": "PROVED_FOR_DECLARED_KINETIC_FORM",
      "action_hessian_reexecution": false
    },
    {
      "id": "2a_poles",
      "scope": "s_xi",
      "status": "SUPPLIED_RECORD",
      "printed_polynomial_arithmetic": "PROVED",
      "arithmetic_scope": [
        "degree", "13_of_14_variation", "formal_derivative"
      ],
      "D01_varying_structure_count": 13,
      "cross_artifact_phase_consistency": false,
      "new_loop_execution": false
    },
    {
      "id": "2a_phase_integrity",
      "scope": "s_phase",
      "status": "DISCREPANCY_RECORDED",
      "evidence": "exact_synthetic_witness",
      "python_index": 5,
      "displayed_structure": "B_6",
      "f_tex": "-4/3",
      "f_reg": "4/3",
      "a0": "4/3",
      "residual": "(8/3)*e_6",
      "original_matrix_artifact": null,
      "phase_consistent": false,
      "physical_sign_selected": false
    },
    {
      "id": "2a_restricted_hh",
      "scope": "s_xi_hh_restricted",
      "status": "SUPPLIED_RESTRICTED_OBSTRUCTION",
      "reported_rank": 7,
      "reported_augmented_rank": 8,
      "original_minor_artifact": null,
      "complete_left_kernel_artifact": null,
      "v4_recertified": false,
      "v5_recertified": false
    },
    {
      "id": "2a_full_exactness",
      "scope": "s_xi",
      "status": "OPEN",
      "enlarged_primitive_artifact": null,
      "enlargement_response_artifact": null,
      "physical_separator_artifact": null,
      "separator_construction": "Q_E*L_when_arrays_are_supplied"
    },
    {
      "id": "2a_full_rows",
      "scope": "s_xi",
      "status": "OPEN",
      "full_matrix_artifact": null
    },
    {"id": "2b_mass", "status": "PROGRAMME"},
    {"id": "2c_externals", "status": "PROGRAMME"},
    {"id": "3_three_point", "status": "OUTSIDE_Q03_SCOPE"},
    {"id": "4_four_point",
     "status": "PROGRAMME_RESOURCE_LIMIT_REPORTED"},
    {"id": "5_two_loop", "status": "PROGRAMME"}
  ],
  "historical_v4_source_flags": {
    "appendix_status": "PROPOSED",
    "appendix_executed": false,
    "appendix_verified": false
  },
  "supplied_v4_run": {
    "status": "SUPPLIED_FAILED_EXECUTION_RECORD",
    "reported_python": "3.10.21",
    "reported_sympy": "1.14.0",
    "reported_passed_suites": 8,
    "reported_failed_suites": 1,
    "reported_predicate_count": 1397,
    "failed_predicate": "corrected D01 phase",
    "post_phase_variation_checks_reached": false,
    "new_replay_in_v5": false,
    "digest_independently_authenticated": false
  },
  "production": {
    "appendix_status": "PROPOSED",
    "appendix_executed": false,
    "appendix_verified": false,
    "compilation_executed": false,
    "formal_proof_check": false,
    "new_loop_execution": false
  },
  "standing_boundaries": {
    "O1_outside_earned_scopes": "NOT_CLOSED",
    "bulk_symmetry_restored": false,
    "XVI_N_insertion": "FROZEN",
    "Route_B_discharged": false,
    "XV_B_admission_exit_opened": false,
    "inherited_characteristic_quantum_coefficient_admitted": false,
    "host_reconstructed": false
  }
}
\end{lstlisting}

Base/patch scope resolution must finish before comparison.
A future executable release must bind these semantic identifiers
to complete byte manifests.

%=====================================================================
\section{Successor calculations and documentary boundaries}
\label{sec:programme}
%=====================================================================

\subsection{Mass and external data}

At nonzero mass, recompute every relevant graph, including D04,
D05a, and D12 where mass enters the actual rules.
Not every massless zero must become nonzero.
List mass monomials and changed consistency maps before testing
the old domain.

General off-shell momentum is distinct from general external
index/polarization conventions.
Construct the full ambient divergence before testing membership
in the old 21-direction image.
The exact old-domain criterion is
\[
 \rank[M_{\mathrm{old}}\mid y_{\mathrm{new}}]
      =\rank M_{\mathrm{old}}.
\]
A failed membership test has a separator; adding columns is a
disclosed domain revision.

\subsection{Three-point chronology}

The pinned Q06--Q11 ledger supersedes the old Q05 task list:
\(V_4\) and the G01/G02 pole results were supplied; the \(hhc\)
assembly includes its corrected auxiliary/gauge channel
\cite{CheungXVIQ}.
Later R07--R09 hat-aware reduced-system records remain separately
attributed \cite{SuppliedBot,CheungXVIR}.

Q11's conditional
\[
 104=46+24-9+43
\]
and the reduced even/odd counts have different presentations and
qualifications. They are not collapsed into one unconditional
cohomology theorem. No unresolved direction is certified physical.

The scientific successor still needs complete pole/zero audit,
all channels, cubic p22 completion, evanescent operator and
invariant-basis checks, and a complete affine certificate.
Pole exactness is not a finite-amplitude theorem or an
interacting quantum master equation.

\subsection{Four points and modular methods}

The predecessor reports resource failures on a 15\,GB machine,
multi-gigabyte ghost-box expressions, stopped symbolic or
numerical-point paths, absent higher vertices, and ansatz widths
\(27243+4992\) for \(hhhh\) and \(4329+981\) for \(Kchh\)
\cite{SuppliedBot,CheungXVIR}.
These are implementation-specific records, not mathematical
impossibility.

A future certificate needs the actual original matrix, exact
field, rank, original minor, complete left kernel, and target
membership. Choosing reconstruction or IBP is not evaluating it.

\subsection{Two-loop subgraphs and analytical insertion}

A validated coefficient code requires an insertion map
\[
 \operatorname{Ins}_\gamma:
 \{\text{validated coefficients}\}\to
 \{\text{local vertices in contracted graphs}\},
\]
with derivative, tensor, antifield, Grassmann, symmetry,
evanescent, and subtraction conventions.

For the declared pole extraction,
\[
 \overline R\Gamma=
 \Gamma+\sum_{\mathcal S}
 \left(\prod_{\gamma\in\mathcal S}C(\gamma)\right)
                      \Gamma/\mathcal S,\qquad
 C(\gamma)=-K_{\mathrm{pole}}\overline R\gamma,
\]
\[
 R\Gamma=(1-K_{\mathrm{pole}})\overline R\Gamma.
\]
The sum uses allowed disjoint proper UV subgraphs; nested
subtractions are recursive
\cite{Zimmermann,SmirnovChetyrkin}.
Every required proper subgraph needs its counterterm and insertion.

For ordinary gravitational vertex weights,
\[
 \sum_v(n_v-2)=2I+E-2V=E+2L-2.
\]
A one-loop cubic counterterm can therefore contribute to the
\(\kappa^4\) two-loop two-point inventory.
Required Laurent depth and evanescent contributions are retained
before multiplication by subdivergent poles
\cite{DuganGrinstein}.
Separate success on gauge, mass, and external axes does not prove
their simultaneous combination.

%=====================================================================
\section{Source-label audit and non-transfer}
\label{sec:sources}
%=====================================================================

The source audit distinguishes supplied passages, accessible
rendered pages, and unsuccessful fetches.
A rendered page is not a byte-authenticated archive or proof
verification. A failed fetch is not proof of HTTP 404,
nonregistration, or mathematical nonexistence.

For this revision, the public essay, I-B, En03-A, X-H-ST,
Oai283-A, publisher record, and official software references were
consulted. Attempts for X-A-ST, X-G-ST, XVI-Q, and Oai282-A
did not provide fresh accessible text.
Their supplied version-qualified material remains usable as
documentary attribution, not as an invented fresh source audit.

\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.21\textwidth}p{0.71\textwidth}@{}}
\toprule
Source & Component retained and non-transfer boundary\\
\midrule
\endfirsthead
\toprule
Source & Component retained and non-transfer boundary\\
\midrule
\endhead
I-B
&
The accessible version-1 page displays \code{thm:instrument}
and the complete clock/readout distinction.
The host, calibration, and metric cone remain inputs.
\\
En03-A
&
The four-part interface and \code{thm:locality} are visible.
Coefficient spaces are not accessible quantum algebras;
logical task names are not ports.
\\
X-H-ST
&
The accessible page displays \code{thm:BV},
\code{thm:nondescent}, \code{thm:instrument}, and
\code{prop:noacquisition}. Relative classical BV and its
residual reports do not construct interacting quantum BRST.
Its reciprocity comparison is not a loop anomaly.
\\
X-G-ST
&
The supplied \code{thm:freeBRST} and
\code{thm:obstruction} descriptions are retained.
No fresh full-source retrieval or normalization transfer is
claimed.
\\
II; II-B
&
Original-minor, complete-kernel, coverage, and affine-family
discipline only. Their numerical certificate is not imported.
\\
Oai283-A
&
The accessible \code{thm:assembly} compares conditional
complete-channel and resource certificates. Its oracle-synthesis
setting is not a loop or thermodynamic result here.
\\
Oai282-A
&
The supplied \code{thm:ward} certificate shape is documentary.
It does not supply an enlarged gravitational primitive or
counterterm map.
\\
X-Enter-ST
&
The supplied route, locality, and attachment descriptions
motivate transport accounting. They do not define analytical
insertion or continuum acquisition.
\\
X-A-ST
&
The supplied host, non-GH, completion, and signature records are
retained. No fresh retrieval success is asserted in v5.
The host locators are not silently replaced by an
\code{ass:host} label from another sheet.
\\
X-I-ST; X-J-ST
&
Optional deformation/coefficient/selective-amplitude descriptions
in the plan are not needed here. No newly verified locator,
registration update, second charged coefficient, or radiative
vacuum-stability theorem is imported.
\\
XVI-P/Q/R
&
Pinned prescription and scientific records remain distinct from
their missing original artifacts and a new replay.
R07--R09 are supplied review-package descriptions.
\\
OpenAI 282/362/376
&
Supplied catalogue comparisons only, not premises or verified
proofs. Their attribution is not merged with Cheung's sources.
\\
\bottomrule
\end{longtable}
\endgroup

The publisher record consulted identifies Cheung's Enter article
as accepted on 28 September 2026 and describes the final formatted
version as forthcoming. No final publication date is invented
\cite{CheungEnter}.

\subsection{Permanent non-claims}

\begin{enumerate}
\item O.1 remains \code{NOT_CLOSED} outside earned scopes;
\(\code{bulk_symmetry_restored}=\code{false}\).
XVI-O's prescription and verdict are not rewritten
\cite{CheungXVIO,CheungXVIQ}.
\item XVI-N remains frozen; Route B and the universal insertion
completion are not discharged \cite{CheungXVIN}.
\item Free exactness, relative classical BV, free-equation
descent, variational reciprocity, and loop anomaly are not
identified \cite{CheungXGST,CheungXHST}.
\item Three inherited equation-level trees are counted once.
Convolution remains unrepaired; the X-D-ST sharp limit is not
metric admission or a loop regulator \cite{CheungXComparisons}.
\item The historical XIV-H H2 stop is retained without erasing
XIV-I's I2 internal-amplitude advance.
The complete characteristic map remains unresolved at I3/J2 in
the supplied releases \cite{CheungXIVH,CheungXIVI,CheungXIVJ}.
\item Technical XV-B lifting and normal-jet interfaces do not
open its inherited downstream quantum-admission exit
\cite{CheungXVB}.
\item Finite matrix certificates do not construct the missing
joint map, joint section and remainder, transverse estimates,
contact compatibility, output wavefront theorem, or admission.
No differential is applied to an absent quantum coefficient.
\item No Einstein equation from records, derived cone, metric
reconstruction, Page curve, entanglement wedge, gravity-generated
entanglement, external-clock-free quantum evolution, ultraviolet
completion, or spacetime emergence is constructed.
\end{enumerate}

For comparison, the supplied XIV-J null identity
\(q\delta'(q)+\delta(q)=0\) illustrates why presentation degree
alone need not descend through reconstruction
\cite{CheungXIVJ}. It is not used to replace the finite matrix
problem by a distributional product theorem.

These are release boundaries, not universal impossibility
theorems.

%=====================================================================
\section{Consolidated v5 release}
\label{sec:release}
%=====================================================================

\begin{theorem}[Integrity-aware finite release]
\label{thm:release}
For the declared host and finite interfaces:
\begin{enumerate}
\item the complete destructive instrument and its conditional
reference blocks are established;
\item real binary and complex qubit reports have the sharp
complete error formulas;
\item feasibility, inconsistency, and full affine ambiguity
have original-minor and complete-kernel certificates;
\item the printed mutation arrays retain their counts and
discrepancies, while the phase inconsistency has the exact
cross-artifact witness \eqref{eq:phaseresidual};
\item direct graded non-feedback has the stated rank-zero
certificate, with the separate quotient-response criterion;
\item the declared Q4S kinetic form has the exact affine inverse;
\item the unchanged printed gauge polynomial has 13 varying
structures and derivative \eqref{eq:hhderivative};
\item free exactness, primitive enlargement, separator
construction, and polynomial family solving have the stated
finite criteria;
\item the old affine fallback, modular-rank confusion, and
v4 execution-coverage overstatement are explicitly audited;
\item complete scheduled ancestry, transport, and conditional
release soundness retain their precise scopes.
\end{enumerate}

The phase-data consistency predicate remains false for the
retained regression array. A successful known-discrepancy test
would not change that predicate.

Q03 remains \code{SUPPLIED_CLOSED}.
The restricted \(hh\) rank increase remains supplied, not newly
re-certified. Full counterterm-derivative exactness and full
gauge-family rows remain \code{OPEN}; step 1b remains
\code{MISSING}. No physical enlarged primitive or numerical
physical separator is invented.

The supplied v4 failed transcript is retained.
No v5 regression run, loop evaluation, compiler execution,
formal proof check, or independent rebuild is asserted.
\end{theorem}

\begin{proof}
Items 1--3 follow from Theorems~\ref{thm:instrument},
\ref{thm:error}, \ref{thm:rank}, and Proposition~\ref{prop:right}.
Item 4 is Proposition~\ref{prop:mutationdata} and
Theorem~\ref{thm:phase}.
Item 5 is Theorems~\ref{thm:graded} and \ref{thm:quotient}.
Items 6--8 are Theorem~\ref{thm:propagator},
Proposition~\ref{prop:derivative}, and
Theorems~\ref{thm:exactness}--\ref{thm:blocklift}.
Item 9 is Theorem~\ref{thm:affine},
Propositions~\ref{prop:modular}, \ref{prop:pass},
and \ref{prop:coverage}.
Item 10 follows from Theorem~\ref{thm:locality},
Proposition~\ref{prop:route}, and Theorem~\ref{thm:gate}.
The missing scientific artifacts are not premises of any claimed
new closure.
\end{proof}

\subsection*{Conclusion}

Version 5 retains the discrepancy rather than manufacturing
agreement. It also retains the distinction between a test that
correctly rejects data and data that satisfy the scientific
claim.

The finite architecture remains useful precisely because its
positive outputs have limited, inspectable meanings.
Neither an expected failure nor an absent response matrix is
hidden by a successful regression contract.

\scopebox{Final v5 verdict}{%
\[
\begin{gathered}
 \text{finite instruments and certificate criteria: proved},\\
 \text{printed-array witness: }(8/3)e_6,\\
 \text{cross-artifact phase consistency: false},\\
 \text{supplied v4 run: eight suites pass, one fails},\\
 \text{new v5 listing: proposed, not executed},\\
 \text{full gauge rows and exactness: \code{OPEN}},\\
 \text{clean-machine rebuild: \code{MISSING}},\\
 \code{bulk_symmetry_restored}=\code{false},\\
 \text{outside earned scopes: O.1 }\code{NOT_CLOSED}.
\end{gathered}
\]
The cone remains an input.}

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

\paragraph{Authorship.}
Cited Cheung manuscripts retain their supplied N.~Cheung
attribution. This anonymous revision does not assign its audit
wording or authorship to that author.
Authorized authorship and affiliation metadata are needed for a
signed submission.

\paragraph{Mathematical and scientific scope.}
The mathematical release consists of the printed statements and
proofs. Missing scientific constructions remain missing.
The evidence package does not permit selecting the physical sign
of the disputed D01 coefficient or supplying the absent enlarged
primitive/row response.

\paragraph{Execution provenance.}
The v4 code and transcript are supplied artifacts, not a new
invocation performed here. Their reported environment and digest
are attributed, not independently authenticated.
The new v5 appendix is unexecuted.

\paragraph{Compilation and submission.}
This source uses standard pdfLaTeX packages, one internal
bibliography, and no external inputs.
No successful compiler run, pagination inspection, proof-assistant
check, referee decision, submission, or journal acceptance is
asserted. Actual two-pass compilation and log inspection remain
pre-submission tasks.

\paragraph{Frozen sources and availability.}
No frozen XVI-P/Q/R or earlier ST source is edited.
A replayable scientific release must archive the original arrays,
complete scripts, manifests, and logs.
The new program below does not depend on those unavailable
scientific files.

%=====================================================================
\appendix
\section{Supplied v4 transcript}
\label{app:v4log}
%=====================================================================

The following is the supplied short output, reproduced as an
execution artifact rather than as a new run
\cite{V4Artifacts}.

\begin{lstlisting}[style=recordstyle]
XVI-O1-ST-v4: SYNTHETIC/PRINTED-DATA REGRESSION ONLY
Python: 3.10.21 (main, Sep 19 2026, 01:08:03) [GCC 12.2.0]
Platform: Linux-6.12.8+-x86_64-with-glibc2.36
SymPy: 1.14.0
Source SHA256: e9f661c8f3eb1dc75b580eb756174f4ff5711f172230ae369b17dbf5a2fbb106
No inherited loop engine, matrix, or archive is evaluated.
PASS_SYNTHETIC: rank / affine repair
PASS_SYNTHETIC: fields / genuine modular rank
PASS_SYNTHETIC: quotient / primitive enlargement
PASS_SYNTHETIC: free complex / polynomial family
FAIL_SYNTHETIC: printed coefficient tables CheckFailure corrected D01 phase
PASS_SYNTHETIC: bounded report witnesses
PASS_SYNTHETIC: complete instrument / reference
PASS_SYNTHETIC: direct grading / sign parity
PASS_SYNTHETIC: finite propagator regression
Evaluated predicates, including expected rejections: 1397
SYNTHETIC SUITE FAILED: 1 case(s)
\end{lstlisting}

This record supersedes any unqualified present-tense statement
that no v4 execution record has been supplied.
It does not establish successful v4 completion, formal
verification, or a v5 execution.

%=====================================================================
\section{Proposed v5 exact regression with retained discrepancy}
\label{app:verifier}
%=====================================================================

Copy this listing to \code{verify_xvi_o1_st_v5.py}.
It is not executed or extracted by LaTeX.

The two D01 baseline vectors are independent retained literals.
The script never overwrites the v4 regression value.
It verifies the exact nonzero witness, then verifies that the old
phase equality is rejected for the intended reason.
Unexpected discrepancies remain fatal.

The exit contract is:
\begin{itemize}
\item exit 1: an unexpected regression or checker failure;
\item exit 2: the listed regression contracts completed, with the
known data-integrity discrepancy still present;
\item exit 0: reserved for a clean contract with no recorded
discrepancy; it is not the expected outcome for the retained data.
\end{itemize}
Thus a future run cannot be read as an all-data-consistent
success merely because the negative control behaved correctly.
There is no newly observed exit result in this manuscript.

The list retains all v4 test categories and separates the phase
witness into its own suite so that other table predicates are not
hidden behind its expected rejection.
The finite-component propagator regression is not the general
Q4S or loop proof.
Official solving and language behavior are documented in
\cite{SymPyMatrices,PythonAssert}.

\begin{lstlisting}[style=verifier,
caption={Proposed v5 regression; no new execution claimed.}]
#!/usr/bin/env python3
"""
XVI-O1-ST-v5: exact finite regression with a retained discrepancy.

No loop engine, original scientific matrix, or R archive is read.
Exit 2 means regression contracts satisfied WITH known inconsistent
data. It is not release acceptance or phase reconciliation.
"""

import hashlib
import json
import platform
import sys
from pathlib import Path

import sympy as sp
from sympy import Matrix, Rational as Q, eye, zeros, I

CHECKS = 0
DISCREPANCIES = {}


class CheckFailure(RuntimeError):
    pass


def require(condition, message):
    global CHECKS
    CHECKS += 1
    if condition is not True and condition is not sp.S.true:
        raise CheckFailure(message)


def same(X, Y, message):
    require(X.shape == Y.shape, message + ": shape")
    require(all(sp.simplify(z) == 0 for z in X-Y),
            message + ": nonzero difference")


def reject(name, action, expected):
    try:
        action()
    except CheckFailure as exc:
        require(str(exc) == expected,
                name + ": unexpected rejection reason")
        return
    raise CheckFailure("invalid control accepted: " + name)


def gaussian_entries(X):
    for z in X:
        re, im = sp.expand_complex(z).as_real_imag()
        require(re.is_Rational is True
                and im.is_Rational is True,
                "entry is not an exact Gaussian rational")


def verify_rank(M, r, rows, cols, L):
    gaussian_entries(M)
    gaussian_entries(L)
    n, p = M.shape
    rows, cols = list(rows), list(cols)
    require(type(r) is int and 0 <= r <= min(n,p),
            "invalid rank parameter")
    require(len(rows) == r and len(cols) == r, "minor size")
    require(all(type(j) is int for j in rows+cols),
            "minor index type")
    require(len(set(rows)) == r and len(set(cols)) == r,
            "duplicate minor index")
    require(all(0 <= j < n for j in rows), "minor row range")
    require(all(0 <= j < p for j in cols), "minor column range")
    require(L.shape == (n-r,n), "left-kernel shape")
    delta = sp.Integer(1) if r == 0 else \
        sp.simplify(M.extract(rows,cols).det())
    require(delta != 0, "original minor is zero")
    same(L*M, zeros(n-r,p), "left-kernel identity")
    require(L.rank() == n-r, "left kernel is incomplete")
    return delta


def verify_solution(M, y, c):
    for X in (M,y,c):
        gaussian_entries(X)
    require(y.shape == (M.rows,1), "target shape")
    require(c.shape == (M.cols,1), "candidate shape")
    same(M*c,y,"candidate solution")


def verify_right(M, r, Z):
    # Precondition: rank(M)=r has a separate certificate.
    gaussian_entries(M)
    gaussian_entries(Z)
    require(type(r) is int and 0 <= r <= min(M.shape),
            "invalid right-kernel rank parameter")
    require(Z.shape == (M.cols,M.cols-r), "right-kernel shape")
    same(M*Z,zeros(M.rows,Z.cols),"right-kernel identity")
    require(Z.rank() == M.cols-r, "right kernel is incomplete")


def reduce_rational(x, p):
    x = sp.sympify(x)
    require(x.is_Rational is True,
            "modular input is not rational")
    num, den = map(int,x.as_numer_denom())
    require(den % p != 0, "denominator is not invertible")
    return (num % p)*pow(den % p,-1,p) % p


def rank_mod(M, p):
    require(type(p) is int and bool(sp.isprime(p)),
            "modulus is not prime")
    A = [[reduce_rational(M[i,j],p)
          for j in range(M.cols)] for i in range(M.rows)]
    r = 0
    for col in range(M.cols):
        pivot = next((j for j in range(r,M.rows)
                      if A[j][col] != 0),None)
        if pivot is None:
            continue
        A[r],A[pivot] = A[pivot],A[r]
        inv = pow(A[r][col],-1,p)
        A[r] = [(x*inv) % p for x in A[r]]
        for j in range(M.rows):
            if j != r:
                f = A[j][col]
                A[j] = [(A[j][k]-f*A[r][k]) % p
                        for k in range(M.cols)]
        r += 1
        if r == M.rows:
            break
    return r


def unit(d,j):
    v = zeros(d,1)
    v[j,0] = 1
    return v


def channel(Ks,X):
    require(len(Ks) > 0,"empty Kraus list")
    out = zeros(Ks[0].rows,Ks[0].rows)
    for K in Ks:
        out += K*X*K.H
    return out.applyfunc(sp.simplify)


def trace_first(X,d,rest):
    require(X.shape == (d*rest,d*rest),"partial-trace shape")
    out = zeros(rest,rest)
    for j in range(d):
        out += X[j*rest:(j+1)*rest,j*rest:(j+1)*rest]
    return out.applyfunc(sp.simplify)


def rats(text):
    return [Q(s) for s in text.split()]


def printed_data():
    # Both original literals are retained; neither is overwritten.
    f_tex = rats(
        "-27/80 19/60 -11/48 -71/120 11/10 -4/3 "
        "19/60 -13/15 1/6 4/3 -11/48 1/6 -1/8 -3/4")
    f_reg = rats(
        "-27/80 19/60 -11/48 -71/120 11/10 4/3 "
        "19/60 -13/15 1/6 4/3 -11/48 1/6 -1/8 -3/4")
    fm = rats(
        "-871/120 911/120 -229/48 56/15 -829/30 17/3 "
        "911/120 -99/10 91/12 -65/3 -229/48 91/12 -21/8 5/4")
    g = rats(
        "7/48 -1/8 5/48 1/12 -1/12 1/3 -1/8 "
        "1/6 -1/24 -1/3 5/48 -1/24 1/16 1/4")
    gm = rats(
        "-1/48 -7/24 1/48 5/12 7/12 2/3 -7/24 "
        "1/6 -5/24 1/3 1/48 -5/24 1/16 1/4")
    h = rats("-3/8 1/4 -1/2")
    hm = rats("-19/24 -7/12 -1/2")
    a0 = rats(
        "27/80 -19/60 11/48 71/120 -11/10 4/3 "
        "-19/60 13/15 -1/6 -4/3 11/48 -1/6 1/8 3/4")
    a1 = rats(
        "1/2 -2/3 1 1/2 -1 1/2 -2/3 4/3 "
        "-5/4 -1/2 1 -5/4 3/2 0")
    a2 = rats(
        "1/2 -1/2 5/6 1/6 -1/3 0 -1/2 2/3 "
        "-5/6 0 5/6 -5/6 5/4 0")
    return f_tex,f_reg,fm,g,gm,h,hm,a0,a1,a2


def test_rank_affine():
    M = Matrix([[1,2,3,4],[2,4,6,8],[1,1,1,1],
                [3,5,7,9],[0,0,0,0]])
    L = Matrix([[-2,1,0,0,0],[-2,0,-1,1,0],[0,0,0,0,1]])
    Z = Matrix([[1,2],[-2,-3],[1,0],[0,1]])
    y,c = Matrix([1,2,1,3,0]),Matrix([1,0,0,0])
    require(verify_rank(M,2,[0,2],[0,1],L) == -1,
            "rank-two original minor")
    verify_solution(M,y,c)
    verify_right(M,2,Z)
    same(L*y,zeros(3,1),"target membership")
    alpha,beta = sp.symbols("alpha beta")
    same(M*(c+Z*Matrix([alpha,beta])),y,"affine family")
    reject("incomplete left kernel",
           lambda: verify_rank(M,2,[0,2],[0,1],zeros(3,5)),
           "left kernel is incomplete")
    reject("zero chosen minor",
           lambda: verify_rank(M,2,[0,1],[0,1],L),
           "original minor is zero")
    verify_rank(zeros(2,3),0,[],[],eye(2))
    verify_right(zeros(2,3),0,eye(3))
    verify_rank(eye(3),3,[0,1,2],[0,1,2],zeros(0,3))
    verify_right(eye(3),3,zeros(3,0))

    Ma = Matrix([[1,2,1],[2,4,2],[1,1,0]])
    ya,ca = Matrix([3,6,1]),Matrix([1,0,2])
    La,Za = Matrix([[-2,1,0]]),Matrix([1,-1,1])
    verify_rank(Ma,2,[0,2],[0,1],La)
    verify_solution(Ma,ya,ca)
    verify_right(Ma,2,Za)
    old = Matrix([1,1,0])
    same(Ma*old-ya,Matrix([0,0,1]),"old residual")
    reject("old unchecked fallback",
           lambda: verify_solution(Ma,ya,old),
           "candidate solution: nonzero difference")
    sol,params = Ma.gauss_jordan_solve(ya)
    same(Ma*sol,ya,"parametric solver residual")
    require(len(params) == 1,"free parameter count")
    verify_solution(Ma,ya,sol.subs({t:0 for t in params}))

    Mu = Matrix([[1,2],[2,4],[0,0]])
    Lu = Matrix([[-2,1,0],[0,0,1]])
    yu = Matrix([1,1,1])
    verify_rank(Mu,1,[0],[0],Lu)
    same(Lu*yu,Matrix([-1,1]),"inconsistency separator")
    require(Matrix.hstack(Mu,yu).rank() == 2,
            "inconsistent augmented rank")


def test_fields_modular():
    M = Matrix([[Q(1,2),Q(1,3)],[Q(2,3),Q(1,2)],[1,1]])
    red = M.applyfunc(lambda z: reduce_rational(z,5))
    same(red,Matrix([[3,2],[4,3],[1,1]]),"rational reduction")
    require(M.extract([0,1],[0,1]).det() == Q(1,36),
            "separate rational minor")
    require(int(red.extract([0,1],[0,1]).det()) % 5 == 1,
            "finite-field minor")
    require(rank_mod(M,5) == 2 and M.rank() == 2,
            "rational and modular rank")
    require(Matrix([[5]]).rank() == 1,"rank over Q")
    require(rank_mod(Matrix([[5]]),5) == 0,"rank over F5")
    reject("bad denominator",
           lambda: rank_mod(Matrix([[Q(1,5)]]),5),
           "denominator is not invertible")
    reject("composite modulus",
           lambda: rank_mod(eye(2),6),"modulus is not prime")
    reject("floating certificate",
           lambda: gaussian_entries(Matrix([[sp.Float("0.5")]])),
           "entry is not an exact Gaussian rational")

    G = Matrix([[1,I],[I,-1],[0,0]])
    L = Matrix([[-I,1,0],[0,0,1]])
    Z = Matrix([-I,1])
    verify_rank(G,1,[0],[0],L)
    verify_right(G,1,Z)
    verify_solution(G,Matrix([1,I,0]),Matrix([1,0]))
    GR,GI = G.applyfunc(sp.re),G.applyfunc(sp.im)
    realified = Matrix.vstack(Matrix.hstack(GR,-GI),
                               Matrix.hstack(GI,GR))
    require(realified.rank() == 2,"realification rank")


def test_quotient_enlargement():
    T = Matrix([[1,0,2],[0,1,1],[1,1,3],[0,0,0]])
    L = Matrix([[-1,-1,1,0],[0,0,0,1]])
    F = Matrix([[1,2],[0,1],[1,3],[0,0]])
    R = Matrix([2,-1])
    verify_rank(T,2,[0,1],[0,1],L)
    same(L*F,zeros(2,2),"nonfeedback")
    same(F*R,Matrix([0,-1,-1,0]),"relation response")
    same(L*F*R,zeros(2,1),"quotient descent")
    Fd = Matrix([[1,2],[0,1],[1,3],[1,2]])
    same(L*Fd,Matrix([[0,0],[1,2]]),"descended obstruction")
    same(L*Fd*R,zeros(2,1),"descent without vanishing")
    require(Matrix.hstack(T,Fd).rank() == 3,"outside-span rank")
    reject("descent is not nonfeedback",
           lambda: same(L*Fd,zeros(2,2),"nonfeedback"),
           "nonfeedback: nonzero difference")
    Fb = Matrix([[1,0],[0,1],[1,1],[1,0]])
    reject("bad descent",
           lambda: same(L*Fb*R,zeros(2,1),"descent"),
           "descent: nonzero difference")

    H = Matrix([1,0,0])
    LH = Matrix([[0,1,0],[0,0,1]])
    E,b = Matrix([0,1,0]),Matrix([1,2,0])
    verify_rank(H,1,[0],[0],LH)
    same(LH*b,LH*E*Matrix([2]),"projected enlargement")
    same(H*Matrix([1])+E*Matrix([2]),b,"enlarged solution")
    QE = Matrix([[0,1]])
    D = QE*LH
    same(D*H,zeros(1,1),"separator old image")
    same(D*E,zeros(1,1),"separator new image")
    require((D*Matrix([1,2,1]))[0] == 1,
            "enlarged obstruction")

    # Same old ranks, opposite enlargement outcomes.
    H15 = eye(15)[:,:7]
    b15 = unit(15,7)
    require(H15.rank() == 7,"old rank seven")
    require(Matrix.hstack(H15,b15).rank() == 8,
            "augmented rank eight")
    require(Matrix.hstack(H15,unit(15,7),b15).rank() == 8,
            "resolving enlargement")
    require(Matrix.hstack(H15,unit(15,8),b15).rank() == 9,
            "nonresolving enlargement")


def test_free_complex_family():
    S = Matrix([[1,0],[0,1],[1,1]])
    B0 = Matrix([[-1,-1,1]])
    A = Matrix([[1,0,0],[0,1,0]])
    verify_rank(S,2,[0,1],[0,1],B0)
    same(B0*S,zeros(1,2),"free complex")
    xi = sp.symbols("xi",real=True)
    c,d = xi*Matrix([1,-1,0]),Matrix([-xi,xi])
    same(A*c+d,zeros(2,1),"gauge cancellation")
    same(c.diff(xi),S*Matrix([1,-1]),"primitive")
    same(A*S,eye(2),"surjective tangent")
    reject("nonexact vector",
           lambda: same(B0*Matrix([0,0,1]),zeros(1,1),"exactness"),
           "exactness: nonzero difference")
    Ast = Matrix([[1,0,0],[0,0,1],[1,0,1]])
    HH,LH = Ast*S,Matrix([[-1,-1,1]])
    verify_rank(HH,2,[0,1],[0,1],LH)
    same(LH*Ast*c.diff(xi),zeros(1,1),"feasible tangent")
    reject("infeasible tangent",
           lambda: same(LH*Matrix([1,0,0]),zeros(1,1),"tangent"),
           "tangent: nonzero difference")
    S0,B1,z = Matrix([1,0,0]),Matrix([[0,0,1]]),Matrix([0,1,0])
    same(B1*S0,zeros(1,1),"second complex")
    same(B1*z,zeros(1,1),"closed vector")
    require(Matrix.hstack(S0,z).rank() == 2,"closed nonexact")

    M,L = S,B0
    zeta = sp.symbols("zeta")
    coeffs = [Matrix([1,0]),Matrix([0,2]),Matrix([3,-1])]
    cp,yp = zeros(2,1),zeros(3,1)
    for j,a in enumerate(coeffs):
        b = M*a
        same(L*b,zeros(1,1),"coefficient membership")
        cp += zeta**j*a
        yp += zeta**j*b
    same(M*cp,yp,"constant-response polynomial lift")
    same(yp.subs(zeta,0),M*coeffs[0],"base point")


def test_table_arithmetic():
    ft,fr,fm,g,gm,h,hm,a0,a1,a2 = printed_data()
    for name,arr in [("ft",ft),("fr",fr),("fm",fm),("g",g),
                     ("gm",gm),("a0",a0),("a1",a1),("a2",a2)]:
        require(len(arr) == 14,"table length "+name)
        gaussian_entries(Matrix(arr))
    for a,b,count,delta in [
        (ft,fm,14,Q(431,30)),(fr,fm,14,Q(431,30)),
        (g,gm,11,Q(1,3)),(h,hm,2,Q(5,12))
    ]:
        require(len(a) == len(b),"mutation array lengths")
        require(sum(x != y for x,y in zip(a,b)) == count,
                "mutation count")
        require(max(abs(x-y) for x,y in zip(a,b))/2 == delta,
                "mutation discrepancy")
        require(max(abs(x) for x in a+b) <= 32,
                "common report bound")

    # Independent from the known phase-failure suite.
    require(sum(x != 0 or y != 0 for x,y in zip(a1,a2)) == 13,
            "D01 varying structure count")
    require(a1[13] == 0 and a2[13] == 0,
            "constant D01 structure")
    zeta = sp.symbols("zeta")
    for j in range(14):
        p = a0[j]+zeta*a1[j]+zeta**2*a2[j]
        require(sp.diff(p,zeta) == a1[j]+2*zeta*a2[j],
                "printed derivative")


def test_phase_integrity():
    ft,fr,fm,g,gm,h,hm,a0,a1,a2 = printed_data()
    FT,FR,A0 = Matrix(ft),Matrix(fr),Matrix(a0)
    expected = Q(8,3)*unit(14,5)
    same(A0+FT,zeros(14,1),"typeset phase relation")
    same(FR-FT,expected,"cross-artifact transcription witness")
    same(A0+FR,expected,"retained phase residual")
    require([j for j,z in enumerate(A0+FR) if z != 0] == [5],
            "unique phase mismatch index")
    require(ft[5] == -Q(4,3) and fr[5] == Q(4,3)
            and a0[5] == Q(4,3),"retained coefficient literals")
    reject("legacy phase equality",
           lambda: same(A0+FR,zeros(14,1),"legacy D01 phase"),
           "legacy D01 phase: nonzero difference")
    require("2a_phase_integrity" not in DISCREPANCIES,
            "duplicate discrepancy record")
    DISCREPANCIES["2a_phase_integrity"] = {
        "status":"DISCREPANCY_RECORDED",
        "python_index":5,
        "displayed_structure":"B_6",
        "residual":[str(z) for z in A0+FR],
        "phase_consistent":False,
        "physical_sign_selected":False
    }
    require(max(abs(z) for z in A0+FR)/2 == Q(4,3),
            "integrity coefficient discrepancy")
    require(max(abs(z) for z in A0+FR)/64 == Q(1,24),
            "integrity report distance at B32")


def bounded_probabilities(v):
    v = [sp.sympify(x) for x in v]
    require(len(v) > 0,"empty report")
    require(all(x.is_Rational is True for x in v),
            "report not exact rational")
    require(all(-1 <= x <= 1 for x in v),
            "report outside [-1,1]")
    return [[(1+b*x)/2 for b in (-1,1)] for x in v]


def test_report():
    v = rats("3/10 -1/2 4/5 1/10")
    vt = rats("3/10 -2/5 7/10 1/10")
    P,PT = bounded_probabilities(v),bounded_probabilities(vt)
    distances = []
    for j in range(4):
        require(sum(P[j]) == 1 and sum(PT[j]) == 1,
                "report normalization")
        tv = sum(abs(P[j][b]-PT[j][b]) for b in range(2))/2
        require(tv == abs(v[j]-vt[j])/2,"basis TV distance")
        distances.append(tv)
    require(max(distances) == Q(1,20),"attaining task")
    reject("bad report range",
           lambda: bounded_probabilities([2]),
           "report outside [-1,1]")
    XP,YP = Matrix([[0,1],[1,0]]),Matrix([[0,-I],[I,0]])
    B = Q(2)
    z,w = Q(3,5)+I*Q(4,5),-Q(1,5)+I*Q(2,5)
    def tau(a):
        return (eye(2)+sp.re(a)*XP/B+sp.im(a)*YP/B)/2
    D = tau(z)-tau(w)
    m2 = sp.expand_complex((z-w)*sp.conjugate(z-w))
    same(D*D,m2*eye(2)/(4*B**2),"qubit difference square")
    require(sp.trace(D) == 0,"qubit difference trace")


def test_instrument():
    D,a,e = 4,2,2
    U = Q(1,2)*Matrix([[1,1,1,1],[1,-1,1,-1],
                       [1,1,-1,-1],[1,-1,-1,1]])
    same(U.H*U,eye(D),"clock calibration")
    Ds = [sp.kronecker_product(U[:,j].H,eye(a)) for j in range(D)]
    Ks = [sp.kronecker_product(unit(D,j),Ds[j]) for j in range(D)]
    total = zeros(D*a,D*a)
    for K in Ks:
        total += K.H*K
    same(total,eye(D*a),"Kraus completeness")
    psi = (sp.kronecker_product(U[:,0],unit(a,0),unit(e,0))
           + sp.kronecker_product(U[:,1],unit(a,1),unit(e,1))) \
          / sp.sqrt(2)
    rho = psi*psi.H
    out = channel([sp.kronecker_product(K,eye(e)) for K in Ks],rho)
    expected = sp.diag(Q(1,2),0,0,Q(1,2))
    same(trace_first(out,D,a*e),expected,"discard record")
    same(trace_first(rho,D,a*e),expected,"discard clock")
    same(trace_first(expected,a,e),eye(e)/2,"reference marginal")
    for j in range(D):
        Dj = sp.kronecker_product(Ds[j],eye(e))
        sigma = (Dj*rho*Dj.H).applyfunc(sp.simplify)
        target = zeros(a*e,a*e)
        if j in (0,1):
            v = unit(a*e,0 if j == 0 else 3)
            target = v*v.H/2
            same(sigma/(Q(1,2)),v*v.H,"positive conditional state")
        same(sigma,target,"actual reference branch")
        if sp.trace(sigma) == 0:
            same(sigma,zeros(a*e,a*e),"zero branch")
    ap = unit(a,0)
    vp = sp.kronecker_product((U[:,0]+U[:,1])/sp.sqrt(2),ap)
    vm = sp.kronecker_product((U[:,0]-U[:,1])/sp.sqrt(2),ap)
    same(channel(Ks,vp*vp.H),channel(Ks,vm*vm.H),
         "clock coherence loss")


def test_grading_parity():
    verify_rank(zeros(21,10),0,[],[],eye(21))
    h,c,K,cb,cs = sp.symbols("h c K cb cs")
    for monomial in (K*c*h,cb*c*h,cs*c*c,h*h*h):
        require(sp.Poly(monomial,h,c,K,cb,cs).total_degree() == 3,
                "formal cubic slot count")
    even,odd = Matrix([[2,0],[0,2]]),eye(2)
    require(rank_mod(even,2) == 0,"even blindness")
    require(rank_mod(odd,2) == 2,"odd occurrence rank")
    for x in (0,1):
        for y in (0,1):
            s = Matrix([x,y])
            require(all(int(z)%2 == 0 for z in even*s),
                    "bubble sign blindness")
            if x or y:
                require(any(int(z)%2 for z in odd*s),
                        "odd control missed sign")


def test_propagator():
    # Finite-component regression, not a loop/DRED implementation.
    pairs = [(a,b) for a in range(4) for b in range(a,4)]
    eta = sp.diag(1,-1,-1,-1)
    k = Matrix([3,1,1,1])
    kc = eta*k
    k2 = (k.T*eta*k)[0]
    require(k2 == 6,"test momentum")
    basis = []
    for a,b in pairs:
        E = zeros(4,4)
        E[a,b] = 1
        E[b,a] = 1
        basis.append(E)
    def coords(E):
        return Matrix([E[a,b] for a,b in pairs])
    A,F = zeros(10,10),zeros(4,10)
    for j,E in enumerate(basis):
        tr = sp.trace(eta*E)
        A[:,j] = coords(E-eta*tr/2)
        F[:,j] = E*k-kc*tr/2
    R = zeros(10,4)
    for j in range(4):
        v = unit(4,j)
        R[:,j] = coords((kc*v.T+v*kc.T)/2)
    same(A*A,eye(10),"trace reversal")
    same(F*R,k2*eye(4)/2,"F R identity")
    G,S4 = 2*A*R*F,R*F*A
    xi = sp.symbols("xi",nonzero=True)
    Kxi = k2*A+(1/xi-1)*G
    Pxi = A/k2+2*(xi-1)*S4/k2**2
    same(Kxi*Pxi,eye(10),"propagator left inverse")
    same(Pxi*Kxi,eye(10),"propagator right inverse")
    wrong = A/k2+(xi-1)*S4/k2**2
    reject("wrong longitudinal coefficient",
           lambda: same(Kxi*wrong,eye(10),"propagator"),
           "propagator: nonzero difference")


def main():
    print("XVI-O1-ST-v5: FINITE REGRESSION, NOT A LOOP GATE")
    print("Python:",sys.version.replace("\n"," "))
    print("Platform:",platform.platform())
    print("SymPy:",sp.__version__)
    print("Source SHA256:",
          hashlib.sha256(Path(__file__).read_bytes()).hexdigest())
    print("The v4 D01 regression coefficient is retained unchanged.")

    reject("checker false",
           lambda: require(False,"deliberate rejection"),
           "deliberate rejection")
    reject("truthy integer",
           lambda: require(1,"non-boolean predicate"),
           "non-boolean predicate")
    reject("undecided predicate",
           lambda: require(sp.Symbol("unknown"),"undecided predicate"),
           "undecided predicate")

    tests = [
        ("rank / affine repair",test_rank_affine),
        ("fields / genuine modular rank",test_fields_modular),
        ("quotient / primitive enlargement",test_quotient_enlargement),
        ("free complex / polynomial family",test_free_complex_family),
        ("printed table arithmetic",test_table_arithmetic),
        ("permanent phase-integrity witness",test_phase_integrity),
        ("bounded report witnesses",test_report),
        ("complete instrument / reference",test_instrument),
        ("direct grading / sign parity",test_grading_parity),
        ("finite propagator regression",test_propagator),
    ]
    failures = []
    for name,test in tests:
        before = CHECKS
        try:
            test()
            require(CHECKS > before,"vacuous test")
        except Exception as exc:
            failures.append((name,type(exc).__name__,str(exc)))
            print("FAIL_REGRESSION:",name,type(exc).__name__,str(exc))
        else:
            print("PASS_REGRESSION_CONTRACT:",name)

    report = {
        "evaluated_predicates":CHECKS,
        "unexpected_failures":failures,
        "known_discrepancies":DISCREPANCIES,
        "phase_consistent":False,
        "scientific_release_accepted":False,
        "loop_evaluated":False,
        "formal_verification":False
    }
    print(json.dumps(report,indent=2,allow_nan=False))
    if failures:
        print("REGRESSION FAILED")
        return 1
    if DISCREPANCIES:
        print("REGRESSION CONTRACTS SATISFIED WITH KNOWN DISCREPANCY")
        print("DATA INTEGRITY NOT CLEARED; EXIT 2; NO LOOP RELEASE")
        return 2
    print("REGRESSION CONTRACTS SATISFIED WITHOUT KNOWN DISCREPANCY")
    print("Still not a loop release or formal verification.")
    return 0


if __name__ == "__main__":
    sys.exit(main())
\end{lstlisting}

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

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026).
DOI: \code{10.3389/fphy.2026.1954943}.
Publisher accepted-article record, accepted 28 September 2026;
the consulted page describes the final formatted version as
forthcoming.
\url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}.

\bibitem{CheungI}
N.~Cheung,
\emph{Manuscript I}, version 2, \code{I.tex}.
Enter Calculus Companion; supplied version-qualified record.
Finite measurement characterization within its stated class.
\url{https://enter.grok.me/browse/I}.

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

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime---Calibrated Destructive Time
Readout on a Prescribed Conical Product},
\code{I-B-v1.tex}, version 1.
Enter Calculus Companion.
Complete clock instrument, calibration, reference-preserving
transport, and prescribed-geometry boundaries.
\url{https://enter.grok.me/browse/I-B}.

\bibitem{CheungEn03}
N.~Cheung,
\emph{Observation Boundaries for Destructive Quantum Readout and
External Geometry},
\code{En03-A.tex}, version 1.
Enter Calculus Companion.
Four-part interface and \code{thm:locality}.
\url{https://enter.grok.me/browse/En03-A}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{Realization of Relational Time by Destructive Enter on a
Clock Register},
\code{En09-A.tex}, version 1.
Enter Calculus Companion; supplied record.
\url{https://enter.grok.me/browse/En09-A}.

\bibitem{DaviesLewis}
E.~B.~Davies and J.~T.~Lewis,
``An operational approach to quantum probability,''
\emph{Communications in Mathematical Physics}
\textbf{17} (1970), 239--260.
DOI: \code{10.1007/BF01647093}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information},
Cambridge University Press, 2018.
DOI: \code{10.1017/9781316848142}.
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{SuppliedV3}
\emph{XVI-O1-ST-v3.tex}.
User-supplied consolidated predecessor.
Finite proofs, synthetic witnesses, and audit of the earlier
affine and modular-rank verification claims.
No authorship or execution is inferred from its filename.

\bibitem{SuppliedBot}
\emph{XVI-O1-ST(bot).tex}.
User-supplied second-readout and gauge-lift predecessor.
Source for the attributed loop calculations and printed
coefficient tables.
Its underlying loop scripts and complete logs are not supplied
as inspected artifacts here.

\bibitem{SuppliedV4}
\emph{XVI-O1-ST-v4.tex}.
User-supplied audit-hardened predecessor, including its tables,
ledger, and embedded regression.
The typeset D01 table and embedded regression are distinguished
in the present integrity audit.

\bibitem{V4Artifacts}
\emph{verify\_xvi\_o1\_st\_v4.py},
\emph{v4\_verification\_output.txt}, and
\emph{DETAILED\_SUMMARY\_v4.txt}.
User-supplied v4 program, failed execution transcript, and
explanatory summary.
Reported environment: Python 3.10.21 and SymPy 1.14.0.
The transcript is attributed, not independently replayed;
the summary is not an independent execution record.

\bibitem{CheungXAST}
N.~Cheung,
\emph{X-A-ST: Supported Classical Covariance of a Fixed-Ramp
Relational Scalar},
\code{X-A-ST.tex}, version 1.
Enter Calculus Companion; supplied host, completion, non-GH, and
signature records.
\url{https://enter.grok.me/browse/X-A-ST}.

\bibitem{CheungXGST}
N.~Cheung,
\emph{X-G-ST: Free Causal BRST Compatibility and a Variational
Upgrade Obstruction},
\code{X-G-ST.tex}, version 1.
Enter Calculus Companion; supplied source descriptions,
including \code{thm:freeBRST} and \code{thm:obstruction}.
No successful fresh full-source retrieval is asserted.
\url{https://enter.grok.me/browse/X-G-ST}.

\bibitem{CheungXHST}
N.~Cheung,
\emph{X-H-ST: Relative Classical BV Structure and Complete
Instruments for Supplied Residual Reports},
\code{X-H-ST.tex}, version 1.
Enter Calculus Companion.
Relevant locators: \code{thm:BV}, \code{thm:nondescent},
\code{thm:instrument}, \code{prop:noacquisition}.
\url{https://enter.grok.me/browse/X-H-ST}.

\bibitem{CheungXEnterST}
N.~Cheung,
\emph{X-Enter-ST: Readout Closure of an Obstructed Convolution
Coefficient on a Prescribed Conical Host},
\code{X-Enter-ST.tex}, version 1.
Enter Calculus Companion; supplied locality, route, and attachment
descriptions.
\url{https://enter.grok.me/browse/X-Enter-ST}.

\bibitem{CheungXVIP}
N.~Cheung,
\emph{Manuscript XVI-P},
\code{XVI-P.tex}, release p23.
Enter Calculus Companion; supplied record.
DRED/\(\MSbar\) prescription, p14r/p22 domains, inventory,
pole records, and Grassmann-orientation errata.
\url{https://enter.grok.me/browse/XVI-P}.

\bibitem{CheungXVIQ}
N.~Cheung,
\emph{Manuscript XVI-Q},
\code{XVI-Q.tex}, release Q11.
Enter Calculus Companion; supplied excerpts and reading map.
Q03 scope, Q05 rules, Q06--Q11 chronology, and conditional
evanescent classification remain version-qualified.
\url{https://enter.grok.me/browse/XVI-Q}.

\bibitem{CheungXVIR}
N.~Cheung, as attributed in the supplied predecessor,
\emph{XVI-R review package}, R07--R09.
Supplied review-package descriptions, not independently inspected
registered manuscripts.
Reduced-system classification and resource reports retain their
own presentations and evidence status.

\bibitem{CheungII}
N.~Cheung,
\emph{Finite Contact-Span Certificates for a Two-Species Spin-Two
Tree},
Manuscript II, version 2.
Enter Calculus Companion; supplied record.
Original-minor, affine-family, and coverage discipline.
\url{https://enter.grok.me/browse/II}.

\bibitem{CheungIIB}
N.~Cheung,
\emph{II-B: Finite Contact-Span Certificates on a Declared Conical
Enter Spacetime---A Grade S Operational Adaptation},
version 1.
Enter Calculus Companion; supplied record.
Presentation template only; its numerical certificate is not
imported.
\url{https://enter.grok.me/browse/II-B}.

\bibitem{CheungOai282}
N.~Cheung,
\emph{Oai282-A: Finite Local-Ward Certificates and Scale--Conformal
Separation},
version 1.
Enter Calculus Companion; supplied excerpts, including
\code{thm:ward} and the finite-release description.
Not a gravitational response map or verification of an external
catalogue family.
\url{https://enter.grok.me/browse/Oai282-A}.

\bibitem{CheungOai283}
\emph{Oai283-A: Conditional Operational Certification of
Constant-Error Oracle Unitary Synthesis},
version 1.
Enter Calculus Companion, under its supplied N.~Cheung manuscript
attribution; the rendered record notes a conflicting authorship
declaration, not resolved here.
Resource/error and conditional-assembly comparison only.
\url{https://enter.grok.me/browse/Oai283-A}.

\bibitem{Stockinger}
D.~St\"ockinger,
``Regularization by dimensional reduction: consistency,
quantum action principle, and supersymmetry,''
\emph{Journal of High Energy Physics}
\textbf{03} (2005), 076.
DOI: \code{10.1088/1126-6708/2005/03/076}.
\url{https://arxiv.org/abs/hep-ph/0503129}.

\bibitem{Frob2018}
M.~B.~Fr\"ob,
``Gauge-invariant quantum gravitational corrections to correlation
functions,''
\emph{Classical and Quantum Gravity}
\textbf{35} (2018), 055006.
DOI: \code{10.1088/1361-6382/aaa74c}.
\url{https://arxiv.org/abs/1710.00839}.

\bibitem{ManteuffelSchabinger}
A.~von Manteuffel and R.~M.~Schabinger,
``A novel approach to integration by parts reduction,''
\emph{Physics Letters B} \textbf{744} (2015), 101--104.
DOI: \code{10.1016/j.physletb.2015.03.029}.
\url{https://arxiv.org/abs/1406.4513}.

\bibitem{Peraro}
T.~Peraro,
``Scattering amplitudes over finite fields and multivariate
functional reconstruction,''
\emph{Journal of High Energy Physics}
\textbf{12} (2016), 030.
DOI: \code{10.1007/JHEP12(2016)030}.
\url{https://arxiv.org/abs/1608.01902}.

\bibitem{Zimmermann}
W.~Zimmermann,
``Convergence of Bogoliubov's method of renormalization in momentum
space,''
\emph{Communications in Mathematical Physics}
\textbf{15} (1969), 208--234.
DOI: \code{10.1007/BF01645676}.

\bibitem{SmirnovChetyrkin}
V.~A.~Smirnov and K.~G.~Chetyrkin,
``\(R^*\) operation in the minimal subtraction scheme,''
\emph{Theoretical and Mathematical Physics}
\textbf{63} (1985), 462--469.
DOI: \code{10.1007/BF01017902}.

\bibitem{DuganGrinstein}
M.~J.~Dugan and B.~Grinstein,
``On the vanishing of evanescent operators,''
\emph{Physics Letters B} \textbf{256} (1991), 239--244.
DOI: \code{10.1016/0370-2693(91)90680-O}.

\bibitem{CheungXVIN}
N.~Cheung,
\emph{Manuscript XVI-N},
\code{XVI-N.tex}, revision r12.
Enter Calculus Companion; supplied excerpts.
Conditional massless special-family obstruction, unresolved
universal completion, and retained Route B boundary.
\url{https://enter.grok.me/browse/XVI-N}.

\bibitem{CheungXVIO}
N.~Cheung,
\emph{Manuscript XVI-O},
\code{XVI-O.tex}, revision \code{o70-sync}.
Enter Calculus Companion; supplied excerpts.
Distinct kernel prescription, insertion freeze, and standing
O.1 boundary.
\url{https://enter.grok.me/browse/XVI-O}.

\bibitem{CheungXComparisons}
N.~Cheung,
\emph{X-A-conv-ST, X-D-ST, and X-F-ST},
supplied version-qualified records.
Fixed-convolution obstruction, sharp-limit versus metric-admission
distinction, and three equation-level trees counted once.
Grouped documentary comparison only.
\url{https://enter.grok.me/browse}.

\bibitem{CheungXIVH}
N.~Cheung,
\emph{Landmark XIV-H: Internal Kernel Relations on the Operator
Route},
\code{XIV-H.tex}, version 1.
Supplied excerpts, including \code{thm:H2}.
Historical amplitude stop, not a finite-matrix premise.
\url{https://enter.grok.me/browse/XIV-H}.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{Landmark XIV-I},
\code{XIV-I.tex}, version 1.
Supplied excerpts, including \code{thm:I2},
\code{thm:I3}, and \code{thm:release}.
Internal amplitude construction does not supply the full joint
characteristic map.
\url{https://enter.grok.me/browse/XIV-I}.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Landmark XIV-J},
\code{XIV-J.tex}, version 1.
Supplied excerpts, including \code{prop:descent},
\code{lem:null}, \code{thm:reference}, \code{thm:J2},
and \code{cor:noadmission}.
Supported reference and reconstruction descent are not complete
characteristic composition or quantum admission.
\url{https://enter.grok.me/browse/XIV-J}.

\bibitem{CheungXVB}
N.~Cheung,
\emph{Manuscript XV-B},
\code{XV-B.tex}, version 2.
Supplied technical-interface descriptions.
Relative lifting and normal-jet interfaces do not open the
inherited downstream quantum-admission exit.
\url{https://enter.grok.me/browse/XV-B}.

\bibitem{SymPyMatrices}
SymPy Development Team,
\emph{Matrices: linear algebra},
official SymPy documentation.
Matrix extraction, rank, nullspaces, and parametric
\code{gauss_jordan_solve}.
\url{https://docs.sympy.org/latest/modules/matrices/matrices.html}.
Documentation consultation is not execution of the v5 listing.

\bibitem{PythonAssert}
Python Software Foundation,
\emph{The Python Language Reference: Simple statements},
section ``The assert statement.''
\url{https://docs.python.org/3/reference/simple_stmts.html}.
Documents assertion behavior under optimized execution.

\end{thebibliography}

\end{document}