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% Finite contact-span certificates for a two-species spin-two tree
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% Principal revisions:
% - Complete response coverage is defined by three explicit clauses.
% - Action equivalence and response-space equivalence are distinguished.
% - The central certificate uses an actual minor of original columns.
% - The entire affine family of finite fits is identified.
% - Injective evaluation is a sufficient, not necessary, route to a
%   global identity; a direct global proof is another route.
% - Rebasings by allowed contacts are distinguished from changes of
%   the response prescription.
% - Weighting is shown explicitly not to preserve the unweighted
%   integration-by-parts quotient in general.
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  pdftitle={Finite contact-span certificates for a two-species spin-two tree},
  pdfauthor={Ngo Cheung},
  pdfsubject={Exact finite contact-span membership and its scope}
}

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\title{\textbf{Finite contact-span certificates\\
for a two-species spin-two tree}}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We solve an exact finite contact-membership problem for a fixed
two-species Pauli--Fierz tree. On an explicit three-label family,
the single-longitudinal reference response is
\(m_S=(0,16/3,16/3)^T\).
A complete redundant catalogue of \(15120\) local, parity-even,
two-derivative quartic contacts has evaluated rank \(2\), equal
to the rank of its target augmentation.
Two original raw columns are \((0,8,0)^T\) and \((0,0,8)^T\);
their minor is \(64\), and the constant coefficients \(2/3,2/3\)
fit the target simultaneously on all three labels.
The complete left kernel is \(\Span\{(1,0,0)\}\), so no
separator exists on this family against the declared class.
We derive the required Einstein cubic contraction, prove the
analytic controls, and provide an integer encoding of every
contact-matrix entry together with an optional exact checker.
The status is \(\SEP\)-finite: the finite-family problem is
closed, while global membership or separation remains undecided
here. We distinguish action representatives, response classes,
affine fitting freedom, class enlargement, and weighted response
prescriptions. Companion results are cited only for comparison;
none supplies a contact column or a premise of the certificate.
\end{abstract}

\statementbox{Principal certificate: \(\SEP\)-finite}{%
For \(S=(\beta_0,\beta_+,\beta_-)\) and the complete declared
contact catalogue,
\[
m_S=\begin{pmatrix}0\\16/3\\16/3\end{pmatrix},
\qquad
\rank N_S=\rank[N_S\mid m_S]=2,
\]
\[
m_S=\frac23(N_S)_{:\!,\ell_a}
+\frac23(N_S)_{:\!,\ell_b},
\qquad
(N_S)_{:\!,\ell_a}=\begin{pmatrix}0\\8\\0\end{pmatrix},
\quad
(N_S)_{:\!,\ell_b}=\begin{pmatrix}0\\0\\8\end{pmatrix}.
\]
This is one simultaneous finite fit, not a global contact identity.}

\statementbox{Scope}{%
No global \(\SEP\) decision, interacting-gravity construction,
physical no-go theorem, quantum coefficient, field instrument,
or measurement-uniqueness theorem is claimed.
The unweighted certificate is not transferred to a weighted
source class, a changed gauge prescription, or an unidentified
counterterm space.
A record interpretation requires the separately stated
coherent-matching hypothesis.}

\tableofcontents

%=====================================================================
\section{Problem and contribution}
\label{sec:introduction}
%=====================================================================

A nonzero response of one specified tree need not survive the
addition of every allowed local contact.
Survival modulo contacts is a membership question in a space
of response functions.
Evaluating that question on a finite family gives a finite
linear-algebra problem, but its interpretation depends on
admissibility, normalization, and complete coverage of the
allowed response changes.

We solve that finite problem for
\[
S=(\beta_0,\beta_+,\beta_-).
\]
The first label is blind by symmetry.
The other two have reference response \(16/3\), but the complete
contact image on this family contains both responses.

Theorem~\ref{thm:certificate} gives the central result.
Its proof derives four principal quantities:
\[
\frac{16}{3},\qquad 8,\qquad 64,\qquad \frac23.
\]
They are respectively a reference-tree response, a raw contact
contraction, an original matrix minor, and fitting coefficients.

\subsection{Relation to established interaction results}

The free starting point is the massless spin-two quadratic action
in the Pauli--Fierz tradition~\cite{FierzPauli}.
Einstein self-interaction and gauge consistency provide historical
context~\cite{Deser}.
The consistency problem for collections of massless spin-two fields
has been studied using local deformation and BRST methods
\cite{BarnichHenneaux,BoulangerEtAl}.

Those works address hypotheses and conclusions different from the
finite response-space calculation below.
In particular, an action-level consistency or classification theorem
is not used as a substitute for evaluating this contact matrix.
No equivalence between the present global response-membership
question and a full deformation problem is assumed.

Cheung's Enter-calculus result concerns a measurement strategy in
a declared quantum-language class~\cite{CheungEnter}.
It supplies neither the Einstein contraction nor the contact
coverage used here.
Related companion manuscripts are discussed only after the
certificate has been proved.

\subsection{Three mathematical layers}

The development separates:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Locked response functions.}
The action, external domain, and contact class determine
\(m,n_1,\ldots,n_{15120}\).

\item \textbf{Evaluation contract.}
Exact inputs, quotient conventions, analytic controls, and
complete coverage determine the meaning of an evaluated matrix.

\item \textbf{Finite certificate.}
A complete encoding, two original columns, an original minor,
and a fitting vector settle membership on the stated family.
\end{enumerate}
None of these layers constructs an interacting quantum field theory
or an apparatus.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=8mm]
\node[certbox,text width=3.7cm] (data)
  {Locked action and contacts\\\(d,\theta;\ q(d)\)};
\node[certbox,text width=3.7cm,right=of data] (functions)
  {Response functions on \(\XX\)\\
   \(r=qm+\sum_\ell\theta_\ell n_\ell\)};
\node[certbox,text width=3.7cm,right=of functions] (matrix)
  {Evaluation on \(S\)\\\(r_S=q\,m_S+N_S\theta\)};
\draw[certarrow] (data)--(functions);
\draw[certarrow] (functions)--(matrix);
\end{tikzpicture}
\caption{The action, the global response function, and its finite
evaluation are distinct objects. The closed result concerns the
right-hand object on one specified family.}
\label{fig:layers}
\end{figure}

\subsection{Quantifiers}

Let
\[
\CC=\Span_{\RR}\{n_\ell:\ell=1,\ldots,15120\}
\subseteq\RR^\XX.
\]
Global membership means
\begin{equation}
\exists\alpha\in\RR^{15120}\quad
\forall\beta\in\XX:\qquad
m(\beta)=\sum_\ell\alpha_\ell n_\ell(\beta).
\label{eq:global-quantifier}
\end{equation}
The coefficients are constant across external labels.

Our certificate replaces \(\XX\) by the particular family \(S\).
It uses one coefficient vector for all three labels, not a separately
chosen vector at each row.
Nevertheless, it does not establish
\eqref{eq:global-quantifier}.

%=====================================================================
\section{Locked kinematics, action, and contacts}
\label{sec:locked}
%=====================================================================

\subsection{Conventions and external domain}

The signature is
\[
\eta=\diag(-1,1,1,1).
\]
All momenta are incoming.
Displayed vector components have upper indices.
All index operations use \(\eta\).
Fourier waves are \(e^{ik\cdot x}\), so
\[
\partial_\mu\longmapsto ik_\mu.
\]
For symmetric tensors,
\(H:K=H_{\mu\nu}K^{\mu\nu}\).

The longitudinal tensor is
\begin{equation}
G_{\mu\nu}(k,\xi)=k_\mu\xi_\nu+k_\nu\xi_\mu.
\label{eq:gauge-tensor}
\end{equation}
It is generated by the Fourier gauge parameter
\(-i\xi e^{ik\cdot x}\).
No restriction \(k\cdot\xi=0\) is imposed.

\begin{definition}[External domain]
\label{def:domain}
A label \(\beta\in\XX\) specifies:
\begin{enumerate}[label=(\roman*)]
\item four species \(J_1,\ldots,J_4\in\{1,2\}\);
\item four nonzero real null momenta such that
\[
\sum_i k_i=0,\qquad
s=(k_1+k_2)^2,\quad t=(k_1+k_3)^2,\quad
u=(k_1+k_4)^2,\qquad stu\ne0;
\]
\item one replaced leg \(j\) and a real gauge vector \(\xi_j\);
\item for every other leg, a real symmetric tensor \(\eps_i\)
and a timelike reference vector \(t_i\) satisfying
\[
k_i^\mu\eps_{i\mu\nu}=0,\qquad
t_i^\mu\eps_{i\mu\nu}=0,\qquad
\eta^{\mu\nu}\eps_{i\mu\nu}=0.
\]
\end{enumerate}
The replaced tensor is \(G(k_j,\xi_j)\).
Polarization normalization is not imposed.
Simultaneous permutations of all external data are admitted.
\end{definition}

The physical tensors are TT relative to their declared references.
The longitudinal insertion need not be trace free.

For \(k\ne0\), the map \(\xi\mapsto G(k,\xi)\) is injective:
choose \(a\) with \(k_a\ne0\).
If \(G=0\), its \(aa\) component gives \(\xi_a=0\), and its
\(ab\) components then give all \(\xi_b=0\).
In particular,
\[
G(k,\xi+\lambda k)=G(k,\xi)+2\lambda k\otimes k.
\]
There is no quotient \(\xi\sim\xi+\lambda k\) in this domain.

\subsection{Quadratic action and conserved-current inverse}

For one species,
\begin{align}
S_2[h]=\int\dd^4x\,\bigg(
&-\frac12\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu}
+\partial_\mu h^{\mu\nu}\partial^\rho h_{\rho\nu}
\nonumber\\
&-\partial_\mu h^{\mu\nu}\partial_\nu h
+\frac12\partial_\lambda h\,\partial^\lambda h
\bigg),
\qquad h=\eta^{\mu\nu}h_{\mu\nu}.
\label{eq:PF}
\end{align}
For two species,
\[
S_{\PF}=S_2[h^1]+S_2[h^2].
\]
The Hessian is
\begin{align}
(K_0h)_{\mu\nu}
={}&\Box h_{\mu\nu}
-\partial_\mu\partial^\rho h_{\rho\nu}
-\partial_\nu\partial^\rho h_{\rho\mu}
+\partial_\mu\partial_\nu h
\nonumber\\
&+\eta_{\mu\nu}
\left(\partial^\rho\partial^\sigma h_{\rho\sigma}-\Box h\right).
\label{eq:K0}
\end{align}
The fixed propagator coefficient is
\begin{equation}
D(p)=-\frac{P}{p^2},
\qquad
P_{\mu\nu,\rho\sigma}
=\frac12\left(
\eta_{\mu\rho}\eta_{\nu\sigma}
+\eta_{\mu\sigma}\eta_{\nu\rho}
-\eta_{\mu\nu}\eta_{\rho\sigma}
\right).
\label{eq:D}
\end{equation}

\begin{lemma}[Inverse on conserved currents]
\label{lem:inverse}
If \(p^\mu J_{\mu\nu}=0\) and \(p^2\ne0\), then
\[
K_0(p)D(p)J=J.
\]
Also \(K_0G(k,\xi)=0\).
\end{lemma}

\begin{proof}
Write \(J_{\mathrm{tr}}=\eta^{\mu\nu}J_{\mu\nu}\).
For \(h=DJ\),
\[
h_{\mu\nu}
=-\frac{J_{\mu\nu}-\eta_{\mu\nu}J_{\mathrm{tr}}/2}{p^2},
\quad
h=\frac{J_{\mathrm{tr}}}{p^2},
\quad
p^\mu h_{\mu\nu}
=\frac{p_\nu J_{\mathrm{tr}}}{2p^2}.
\]
Substitution in the Fourier transform of \eqref{eq:K0} cancels
the \(p_\mu p_\nu J_{\mathrm{tr}}\) terms and the remaining
trace terms, leaving \(J_{\mu\nu}\).
Direct substitution of \eqref{eq:gauge-tensor} proves the second
identity.
\end{proof}

No inverse identity on arbitrary nonconserved tensors is asserted.

\subsection{Einstein coefficients and differentiation conventions}

Fix the curvature sign and normalization by
\begin{equation}
\frac{2}{\kappa^2}\int\dd^4x\,
\sqrt{-\det(\eta+\kappa h)}\,R[\eta+\kappa h]
=
S_2[h]+\kappa E_3[h]+\kappa^2E_4[h]+O(\kappa^3),
\label{eq:EH}
\end{equation}
modulo total derivatives.
Equivalently, expand the first-derivative density
\begin{equation}
\frac{2}{\kappa^2}\sqrt{-g}\,g^{\mu\nu}
\left(
\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha}
-\Gamma^\alpha_{\mu\nu}\Gamma^\beta_{\alpha\beta}
\right),
\qquad g=\eta+\kappa h.
\label{eq:GammaGamma}
\end{equation}

For a homogeneous functional \(E_n\), put
\[
E_n^{[n]}(u_1,\ldots,u_n)
=\frac1{n!}D^nE_n[0](u_1,\ldots,u_n),
\]
\[
v_n(u_1,\ldots,u_n)
=D^nE_n[0](u_1,\ldots,u_n)
=n!E_n^{[n]}(u_1,\ldots,u_n).
\]
Thus \(v_n\) is exactly the coefficient of \(z_1\cdots z_n\)
in \(E_n[\sum_i z_i u_i]\).
No further factorial multiplies that coefficient.

Plane-wave formulas mean the coefficient after removing the common
momentum-conservation delta.
Integration by parts is interpreted algebraically at conserved
vertex momentum.
No convergent integral of four unlocalized plane waves is required.

\subsection{Species couplings and reference quartic}

Let \(d_{IJK}\) be real and completely symmetric, with
\[
d_{111}=a,\qquad d_{112}=b,\qquad d_{122}=c,\qquad d_{222}=e.
\]
Define
\[
S_3(d)=\sum_{I,J,K=1}^2
d_{IJK}E_3^{[3]}(h^I,h^J,h^K),
\]
\begin{equation}
T_{IJKL}(d)=\frac13\sum_{M=1}^2
\left(
d_{IJM}d_{KLM}
+d_{IKM}d_{JLM}
+d_{ILM}d_{JKM}
\right),
\label{eq:T}
\end{equation}
and
\[
S_{4,\sym}(d)
=\sum_{I,J,K,L=1}^2
T_{IJKL}(d)E_4^{[4]}(h^I,h^J,h^K,h^L).
\]
The action through order \(g^2\) is
\begin{equation}
S(d,\theta)
=S_{\PF}+gS_3(d)
+g^2\left(S_{4,\sym}(d)+\sum_\ell\theta_\ell\OO_\ell\right).
\label{eq:action}
\end{equation}
Every \(\theta_\ell\) is constant across the external domain.

The ordered species sums and normalized polarizations ensure that
a differentiated external vertex has coefficient \(d_{J_1J_2J_3}v_3\)
or \(T_{J_1J_2J_3J_4}v_4\).
Repeated species do not introduce an additional factorial:
all assignments to the distinct bookkeeping variables are already
included.

\subsection{Declared local contact class}

Let \(\VV\) be the real vector space of local quartic density
classes with exactly two derivatives, no extra fields, no inverse
operators, constant coefficients, and no background tensors beyond
\(\eta\) and the orientation tensor.
Densities are parity even, translation invariant, and Lorentz
scalar.
The quotient is by total derivatives and genuine tensor identities,
not by equations of motion or field redefinitions.

Equivalently, the declared tensor-contraction class uses complete
metric contractions and even numbers of Levi-Civita tensors.
The metric-pairing description is the standard orthogonal
invariant-tensor framework~\cite{Brauer}; no independence of those
pairings is assumed.

Let \(\mathfrak P_{10}\) be the perfect matchings of
\(\{1,\ldots,10\}\).
For \(I=(I_1,I_2,I_3,I_4)\in\{1,2\}^4\) and
\(P\in\mathfrak P_{10}\), define
\begin{align}
\OO_{I,P}=\int\dd^4x\,
&(\partial_{\lambda_1}h^{I_1}_{\lambda_3\lambda_4})
(\partial_{\lambda_2}h^{I_2}_{\lambda_5\lambda_6})
h^{I_3}_{\lambda_7\lambda_8}
h^{I_4}_{\lambda_9\lambda_{10}}
\nonumber\\
&\hspace{2em}\times
\prod_{\{u,v\}\in P}\eta^{\lambda_u\lambda_v}.
\label{eq:raw}
\end{align}
There are
\begin{equation}
2^4(10-1)!!=16\cdot945=15120
\label{eq:catalogue-count}
\end{equation}
raw labels.

\begin{lemma}[Raw contact coverage]
\label{lem:coverage}
The classes of the raw functionals \eqref{eq:raw} span \(\VV\).
Every raw functional is an allowed contact.
\end{lemma}

\begin{proof}
If both derivatives act on one factor, integration by parts
expresses the density modulo a divergence as a sum with derivatives
on two distinct factors.
Relabel those factors into positions one and two.
There are then ten tensor slots: two derivative indices and eight
field indices.
Every complete metric contraction pairs those slots and is one of
the perfect matchings.

A pair of Levi-Civita tensors reduces, by the determinant identity,
to a signed sum of metric contractions.
Iterating removes every even number of orientation tensors.
This covers the declared parity-even contraction class.
Species choices give every word in \(\{1,2\}^4\).

Conversely, every raw expression has the declared field degree,
derivative degree, locality, covariance, parity, and constant
coefficients.
\end{proof}

The number \(15120\) counts redundant raw coordinates.
It is not the dimension of inequivalent actions or global responses.

\subsection{The exact contact evaluator}

For lower-index momenta \(p,q\), define
\begin{align}
\Contr_P(p,q;e_1,e_2,e_3,e_4)
={}&\sum_{\lambda_1,\ldots,\lambda_{10}=0}^3
p_{\lambda_1}q_{\lambda_2}
\prod_{r=1}^4(e_r)_{\lambda_{2r+1}\lambda_{2r+2}}
\nonumber\\
&\hspace{2em}\times
\prod_{\{u,v\}\in P}\eta^{\lambda_u\lambda_v}.
\label{eq:Contr}
\end{align}
The differentiated response of a raw contact is
\begin{align}
n_{I,P}(\beta)
={}&-\sum_{\sigma\in S_4}
\left(\prod_{r=1}^4\delta_{I_r,J_{\sigma(r)}}\right)
\nonumber\\
&\quad\times
\Contr_P\left(
k_{\sigma(1)},k_{\sigma(2)};
e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)},e_{\sigma(4)}
\right).
\label{eq:contact-evaluator}
\end{align}
One \(e_i\) is the prescribed \(G\).
The momenta in \(\Contr_P\) have lower indices.

To derive this formula, substitute
\[
h^I=\sum_jz_j\delta_{I,J_j}e_j e^{ik_j\cdot x}
\]
in \eqref{eq:raw} and extract \(z_1z_2z_3z_4\).
All assignments to the factor positions appear.
The two derivatives contribute \(i^2=-1\).

\begin{proposition}[Evaluation descends to the action quotient]
\label{prop:quotient}
The response map
\[
\mathcal R:\VV\longrightarrow\RR^\XX
\]
is well defined.
Its image is precisely
\[
\CC=\Span_{\RR}\{n_{I,P}\}.
\]
\end{proposition}

\begin{proof}
The differentiated coefficient of a total derivative has an overall
factor \(i\sum_j k_{j\mu}\), which vanishes on \(\XX\).
A genuine tensor identity vanishes before and after differentiation.
Thus equivalent density representatives have identical responses.

Raw coverage gives every allowed response as a constant combination
of the \(n_{I,P}\), and every such combination comes from allowed
raw contacts.
\end{proof}

Action equivalence and response equivalence remain different:
\(\mathcal R\) need not be injective.

%=====================================================================
\section{Ward reduction and coupling factorization}
\label{sec:factorization}
%=====================================================================

\subsection{Associator coordinate}

On the internal Euclidean space with basis \(e_1,e_2\), set
\[
e_I\circ e_J=\sum_Kd_{IJK}e_K.
\]
The left multiplication matrices are
\[
L_1=\begin{pmatrix}a&b\\b&c\end{pmatrix},
\qquad
L_2=\begin{pmatrix}b&c\\c&e\end{pmatrix}.
\]
Define
\begin{equation}
q(d)=b^2+c^2-ac-be.
\label{eq:q}
\end{equation}
Then
\[
[L_2,L_1]=\begin{pmatrix}0&q\\-q&0\end{pmatrix}.
\]
For
\[
\AA_{IJKL}
=\pair{(e_I\circ e_J)\circ e_K-e_I\circ(e_J\circ e_K)}{e_L},
\]
the nonzero possible components are
\[
\AA_{1122}=\AA_{2211}=-q,\qquad
\AA_{1221}=\AA_{2112}=q.
\]
Hence
\[
\AA=0\iff q=0,\qquad \norm{\AA}_F=2|q|.
\]

When \(q=0\), the symmetric matrices \(L_1,L_2\) commute.
In a common orthonormal eigenbasis \(f_1,f_2\), the product
\(f_1\circ f_2\) is simultaneously proportional to \(f_1\) and
to \(f_2\), and is therefore zero.
Inner-product invariance then gives
\(f_i\circ f_i=\lambda_i f_i\).
Consequently
\[
d_{IJK}=\sum_{i=1}^2\lambda_i O_{Ii}O_{Ji}O_{Ki}
\]
for an orthogonal matrix \(O\).
The cubic and reference quartic in that basis are
\[
\sum_i\lambda_iE_3[H^i],
\qquad
\sum_i\lambda_i^2E_4[H^i].
\]
This elementary decomposition is used as a control, not as an
imported multi-graviton classification.

\subsection{Polarized Einstein identities}

For a gauge parameter \(\zeta\), write
\[
G_\zeta=\partial\zeta+(\partial\zeta)^T
\]
and
\[
(R_\zeta h)_{\mu\nu}
=\zeta^\rho\partial_\rho h_{\mu\nu}
+h_{\rho\nu}\partial_\mu\zeta^\rho
+h_{\mu\rho}\partial_\nu\zeta^\rho.
\]
Under the coordinate variation scaled by \(\kappa\),
\[
\delta h=G_\zeta+\kappa R_\zeta h.
\]
The Einstein density varies by a total derivative.
The first two coefficient identities are
\[
\delta_0E_3+\delta_1S_2=0,
\qquad
\delta_0E_4+\delta_1E_3=0.
\]
Polarizing them gives
\begin{align}
v_3(G_\zeta,u,v)
&=-\pair{K_0u}{R_\zeta v}
  -\pair{K_0v}{R_\zeta u},
\label{eq:ward3}\\
v_4(G_\zeta,u,v,w)
&=-v_3(R_\zeta u,v,w)
  -v_3(u,R_\zeta v,w)
  -v_3(u,v,R_\zeta w).
\label{eq:ward4}
\end{align}
Here pairings include spacetime integration or, for waves, the
corresponding stripped conserved-momentum coefficient.
The factors agree with the definition of \(v_n\): taking two or
three distinct field derivatives introduces no extra factorial.

Define the cubic current by
\[
\pair{J(v,w)}{z}=v_3(z,v,w).
\]
If \(K_0v=K_0w=0\), equation~\eqref{eq:ward3} with arbitrary
\(\zeta\) implies conservation of \(J(v,w)\).
Thus Lemma~\ref{lem:inverse} applies at the physical end of each
exchange.

The stripped four-point prescription is
\begin{equation}
\mathscr T_4
=V_4-\sum_{\{ij|kl\}\in\{12|34,13|24,14|23\}}
V_3(i,j,\cdot)D(k_i+k_j)V_3(\cdot,k,l).
\label{eq:tree}
\end{equation}
The common momentum delta, \(g^2\), and overall Feynman factor
\(i\) have been removed.

For free \(u,v,w\),
\[
v_3(G_\zeta,u,DJ(v,w))
=-\pair{K_0DJ(v,w)}{R_\zeta u}
=-v_3(R_\zeta u,v,w).
\]
Together with the exchange minus sign in \eqref{eq:tree},
this cancels the corresponding term of \eqref{eq:ward4}.

\subsection{Channel reduction}

Take leg one replaced.
For \(\zeta=-i\xi_1e^{ik_1\cdot x}\), define
\begin{equation}
(\mathscr R_{1i})_{\mu\nu}
=(\xi_1\cdot k_i)\eps_{i\mu\nu}
+k_{1\mu}\xi_1^\rho\eps_{i\rho\nu}
+k_{1\nu}\xi_1^\rho\eps_{i\mu\rho}.
\label{eq:R}
\end{equation}
Its momentum is \(k_1+k_i\).
Put
\[
W_2=v_3(\mathscr R_{12},\eps_3,\eps_4),\quad
W_3=v_3(\mathscr R_{13},\eps_2,\eps_4),\quad
W_4=v_3(\mathscr R_{14},\eps_2,\eps_3).
\]
The composite insertion is generally off shell.

Write
\[
D_s=\sum_Md_{J_1J_2M}d_{J_3J_4M},\quad
D_t=\sum_Md_{J_1J_3M}d_{J_2J_4M},\quad
D_u=\sum_Md_{J_1J_4M}d_{J_2J_3M},
\]
and \(\overline D=(D_s+D_t+D_u)/3\).
If \(X_i\) denotes the unweighted exchange contraction before its
tree minus sign, the preceding identity gives
\[
X_i=-W_i.
\]

\begin{proposition}[Reference longitudinal response]
\label{prop:channel}
The response \(F(d)\) of the reference action is
\begin{equation}
F(d)
=(D_s-\overline D)W_2
+(D_t-\overline D)W_3
+(D_u-\overline D)W_4.
\label{eq:channel}
\end{equation}
\end{proposition}

\begin{proof}
The exchanges contribute
\(-D_sX_2-D_tX_3-D_uX_4
=D_sW_2+D_tW_3+D_uW_4\).
The reference quartic coefficient is \(\overline D\).
Its response is
\(-\overline D(W_2+W_3+W_4)\) by \eqref{eq:ward4}.
\end{proof}

Each longitudinal exchange has become a polynomial contraction.
The inverse pole cancels separately in each channel.

\subsection{Factorization on the whole locked domain}

For species \(1122\),
\[
D_s=ac+be,\qquad D_t=D_u=b^2+c^2,
\]
and therefore
\begin{equation}
F_{1122}(d)=q(d)\frac{-2W_2+W_3+W_4}{3}.
\label{eq:m1122}
\end{equation}
The same formula holds for \(2211\).
Other two-and-two placements permute the three coefficients.

For the remaining species patterns, all three \(D\)'s agree:
\[
\begin{array}{c|c}
\text{pattern, up to permutation}&D_s=D_t=D_u\\
\hline
1111&a^2+b^2\\
2222&c^2+e^2\\
1112&ab+bc\\
1222&bc+ce.
\end{array}
\]
Their reference response is zero.

Define
\[
d_*=(a,b,c,e)=(0,1,0,0),\qquad q(d_*)=1,\qquad m=F(d_*).
\]

\begin{proposition}[Complete response factorization]
\label{prop:factorization}
On \(\XX\),
\begin{equation}
\boxed{
r(d,\theta;\beta)
=q(d)m(\beta)+\sum_\ell\theta_\ell n_\ell(\beta).
}
\label{eq:factorization}
\end{equation}
\end{proposition}

\begin{proof}
The displayed species cases exhaust all four-leg words.
They establish \(F(d)=q(d)F(d_*)\) when leg one is replaced.
Simultaneous relabeling moves any replaced leg to position one.
The additional quartic contacts contribute linearly by definition.
\end{proof}

This proof uses the locked vertices and direct species algebra.
No companion obstruction theorem is involved.

%=====================================================================
\section{Coverage and the evaluation contract}
\label{sec:contract}
%=====================================================================

\subsection{Three-clause completeness}

\begin{definition}[Complete response catalogue]
\label{def:complete-catalogue}
For a response problem on a domain \(\mathcal B\), a list
\(n_1,\ldots,n_L\) is complete when:
\begin{enumerate}[label=(\roman*)]
\item every allowed response change is
\(\sum_\ell c_\ell n_\ell\), with coefficients constant on
\(\mathcal B\);
\item every listed direction is allowed, or the permitted coefficient
subspace \(W\subseteq\RR^L\) is explicitly named;
\item all admitted representative variations are covered, either
by those combinations or by a proof that they are response-null
under the same evaluation prescription.
\end{enumerate}
If a proper \(W\) is declared, the allowed image is
\(N_SW\), not automatically the span of all columns.
\end{definition}

For the present class, \(W=\RR^{15120}\).
Lemma~\ref{lem:coverage} proves constant-coefficient spanning and
that every generator is allowed.
Proposition~\ref{prop:quotient} proves that quotient-representative
changes are response-null.
Thus all three clauses hold.

The catalogue count alone proves none of these clauses.
It records the size of the spanning construction after coverage
has been established.

\subsection{Action classes and response classes}

There are two quotients:
\[
\VV
=\frac{\text{allowed quartic densities}}
       {\text{total derivatives and tensor identities}},
\qquad
[m]\in\RR^\XX/\CC.
\]
The response map connects them but need not be injective.
A response test of a density representative tests its action class
because evaluation descends to \(\VV\).
A test of contact-surviving information instead tests \([m]\),
not one arbitrary reference representative.

In particular, no equation-of-motion quotient, source weighting,
or characteristic-product ambiguity is silently added to \(\VV\).

\subsection{Exact inputs and six controls}

Every row specifies species, momenta, reference vectors,
polarizations, replaced leg, and gauge vector.
The central evaluation is rational; contact coefficients may still
be arbitrary real constants.

The analytic controls are:
\begin{description}[style=nextline,leftmargin=2.2em]
\item[\(C_1\)] One-species Einstein longitudinal cancellation.
\item[\(C_2\)] Associative two-species cancellation with the canonical
reference quartic.
\item[\(C_3\)] Rank-one species factorization with its canonical
longitudinal law.
\item[\(C_4\)] Simultaneous permutation of all external labels.
\item[\(C_5\)] Channelwise equality of direct and Ward-reduced exchanges.
\item[\(C_6\)] Zero target and zero response of every raw contact at the
designated blind seed.
\end{description}

These are proved mathematical identities, not reports of executed
regression tests.
An analytic zero on a control domain is stronger than a finite list
of sampled zeros.
Conversely, an analytic identity does not certify an unrun software
implementation.

\subsection{Finite membership and complete separation}

For a finite ordered family \(S=(\beta_1,\ldots,\beta_s)\), put
\[
N_S=(n_\ell(\beta_i))_{i,\ell},\qquad
m_S=(m(\beta_i))_i.
\]

\begin{proposition}[A complete finite separator excludes global membership]
\label{prop:finite-separation}
If the catalogue is complete and
\[
zN_S=0,\qquad zm_S\ne0,
\]
then \(m\notin\CC\).
Moreover,
\[
z\,r_S(d,\theta)=q(d)\,zm_S
\]
for every allowed constant contact vector.
\end{proposition}

\begin{proof}
A global identity \(m=\sum_\ell\alpha_\ell n_\ell\) would imply
\(m_S=N_S\alpha\), contradicting \(zm_S\ne0\).
Multiply the evaluated factorization by \(z\) for the second claim.
\end{proof}

Without complete coverage, annihilation of the displayed columns
establishes blindness only to the displayed correction subspace.

A finite fit has a different direction of implication:
\(m_S=N_S\alpha\) proves finite membership.
Completeness is needed for the claimed full image, rank, and left
kernel; existence of this particular fit requires only that its
used columns be allowed.

\subsection{Certificate terminology}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.19\linewidth}X@{}}
\toprule
Status & Meaning\\
\midrule
\(\SEP\)-yes &
A complete finite separator proves \(m\notin\CC\).\\
\addlinespace
\(\SEP\)-no &
A constant-coefficient identity proves \(m\in\CC\) on all of \(\XX\).\\
\addlinespace
\(\SEP\)-finite &
An exact finite-family result is complete at its stated evaluation
scope, without a global membership decision.\\
\addlinespace
\(\SEP\)-open &
The intended finite result itself lacks required evaluation,
coverage, or verification.\\
\bottomrule
\end{tabularx}
\end{center}

The result here is \(\SEP\)-finite.
The global question is not answered by that status.

%=====================================================================
\section{The three labels and the reference value}
\label{sec:nonzero}
%=====================================================================

\subsection{Rational family}

Use upper-index momenta
\begin{equation}
\begin{aligned}
k_1&=(1,0,0,1),&
k_2&=(1,0,0,-1),\\
k_3&=(-1,-1,0,0),&
k_4&=(-1,1,0,0),
\end{aligned}
\label{eq:momenta}
\end{equation}
and covariant tensors
\begin{equation}
A=\diag(0,1,-1,0),\qquad B=\diag(0,0,-1,1).
\label{eq:AB}
\end{equation}
The physical tensors on legs \(2,3,4\) are \(A,B,B\).
Their references are all \(t_0=(1,0,0,0)\).
Leg one is replaced.

\begin{center}
\begin{tabular}{@{}c c c c@{}}
\toprule
Label & Species & \(\xi_1\) & Physical tensors\\
\midrule
\(\beta_0\)&\(1122\)&\((0,1,0,0)\)&\(A,B,B\)\\
\(\beta_+\)&\(1122\)&\((1,0,0,0)\)&\(A,B,B\)\\
\(\beta_-\)&\(2211\)&\((1,0,0,0)\)&\(A,B,B\)\\
\bottomrule
\end{tabular}
\end{center}
The row order is
\[
S=(\beta_0,\beta_+,\beta_-).
\]

\begin{lemma}[Admissibility]
\label{lem:admissibility}
All three labels belong to \(\XX\), with
\[
s=-4,\qquad t=2,\qquad u=2.
\]
\end{lemma}

\begin{proof}
The momenta are nonzero, null, and sum to zero.
Their pair sums give the stated invariants.
Both tensors are real, symmetric, temporally zero, and Lorentz
trace free.
The spatial support of \(k_2\) is in direction three, where \(A\)
has a zero row.
The spatial supports of \(k_3,k_4\) are in direction one, where
\(B\) has a zero row.
Thus the momentum and reference transversality conditions hold.
Finally \(t_0^2=-1\), and both gauge vectors are admitted.
\end{proof}

The normalized blind seed uses \(A/\sqrt2,B/\sqrt2,B/\sqrt2\).
It has the same symmetry blindness, but the finite matrix below
uses the rational unnormalized tensors.

\subsection{Restricted Einstein cubic}

\begin{lemma}[Trace-free diagonal restriction]
\label{lem:diagonal}
On the linear subspace
\[
h_{\mu\nu}=\diag(0,H_1,H_2,H_3),\qquad H_1+H_2+H_3=0,
\]
the cubic coefficient is, modulo a divergence,
\begin{equation}
E_3
=-\int\dd^4x\,
\left(
H_1\dot H_2\dot H_3+
H_2\dot H_1\dot H_3+
H_3\dot H_1\dot H_2
\right).
\label{eq:diagonal-cubic}
\end{equation}
It may be polarized using any three fields in this subspace,
including an off-shell insertion.
\end{lemma}

\begin{proof}
Set
\[
q_i=1+\kappa H_i,\quad q_0=1,\quad
\phi_i=\tfrac12\log q_i,\quad\phi_0=0.
\]
The nonzero diagonal-metric Christoffel patterns are
\[
\Gamma^a_{aa}=\partial_a\phi_a,\qquad
\Gamma^a_{ab}=\partial_b\phi_a,\qquad
\Gamma^a_{bb}=-\frac{g_{bb}}{g_{aa}}\partial_a\phi_b
\quad(a\ne b).
\]
Three-distinct-index entries vanish.
Substituting these patterns and collecting each derivative direction
gives
\[
g^{\mu\nu}
\left(
\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha}
-\Gamma^\alpha_{\mu\nu}\Gamma^\beta_{\alpha\beta}
\right)
=
2\sum_{a=0}^3g^{aa}
\sum_{\substack{i<j\\i,j\ne a}}
\partial_a\phi_i\,\partial_a\phi_j.
\]
Since
\(\partial_a\phi_i=\kappa\partial_aH_i/(2q_i)\), the density
\eqref{eq:GammaGamma} becomes
\begin{equation}
\sqrt{q_1q_2q_3}
\sum_{a=0}^3\frac{\eta^{aa}}{q_a}
\sum_{\substack{1\le i<j\le3\\i,j\ne a}}
\frac{\partial_aH_i\,\partial_aH_j}{q_iq_j}.
\label{eq:exact-diagonal}
\end{equation}

The coefficient of \(\kappa\) in the prefactor of the term
indexed by \(a,i,j\) is
\[
\tfrac12(H_1+H_2+H_3)-H_a-H_i-H_j,\qquad H_0=0.
\]
For spatial \(a\), the three indices exhaust \(1,2,3\), so this
coefficient vanishes.
For \(a=0\), it equals the remaining \(H_k\).
The factor \(\eta^{00}=-1\) proves \eqref{eq:diagonal-cubic}.

As a normalization check, the time-derivative quadratic part is
\[
-\sum_{i<j}\dot H_i\dot H_j
=\tfrac12\sum_i\dot H_i^2
\]
on the trace-free subspace, in agreement with \eqref{eq:PF}.
Restriction to a linear subspace commutes with polarization.
No field equation for the inserted fields was used.
\end{proof}

\begin{lemma}[The \(A,B,B\) coefficient]
\label{lem:ABB}
For incoming momenta \(p+q+r=0\),
\begin{equation}
v_3(A_p,B_q,B_r)
=-2(p^0q^0+p^0r^0+q^0r^0)
=(p^0)^2+(q^0)^2+(r^0)^2.
\label{eq:ABB}
\end{equation}
\end{lemma}

\begin{proof}
Polarize \eqref{eq:diagonal-cubic}.
For each choice of undifferentiated labeled wave, the sum of its
spatial positions and the two assignments of differentiated waves
has coefficient \(-2\).
For instance, with \(A\) undifferentiated,
\[
A_{11}(B_{22}B_{33}+B_{33}B_{22})=-2.
\]
A time derivative contributes \(ik_0=-ik^0\).
The two derivative factors and the density sign yield the first
formula.
The second follows from \(p^0+q^0+r^0=0\).
\end{proof}

\begin{theorem}[Exact nonzero reference value]
\label{thm:nonzero}
At both \(\beta_+\) and \(\beta_-\),
\[
W_2=-6,\qquad W_3=W_4=2,
\]
and
\begin{equation}
\boxed{m(\beta_+)=m(\beta_-)=16/3.}
\label{eq:nonzero}
\end{equation}
\end{theorem}

\begin{proof}
Because the physical tensors have zero temporal entries,
\eqref{eq:R} gives
\[
\mathscr R_{12}=-A,\qquad
\mathscr R_{13}=B,\qquad
\mathscr R_{14}=B.
\]
The temporal momenta in \(W_2\) are \(2,-1,-1\).
Those in \(W_3,W_4\) are \(0,1,-1\), up to permutation.
Lemma~\ref{lem:ABB} therefore gives
\[
W_2=-(4+1+1)=-6,\qquad W_3=W_4=2.
\]
At \(d_*\), both species channels have
\((D_s,D_t,D_u)=(0,1,1)\).
Equation~\eqref{eq:m1122} gives
\[
m=\frac{-2(-6)+2+2}{3}=\frac{16}{3}.
\]

The tree signs can also be displayed directly:
\[
v_4(G,A,B,B)=-(W_2+W_3+W_4)=2,
\qquad
(X_2,X_3,X_4)=(6,-2,-2).
\]
Thus
\[
F(d_*)=\frac23(2)
-\left(0\cdot6+1\cdot(-2)+1\cdot(-2)\right)
=\frac{16}{3}.
\]
\end{proof}

The second assembly uses the proved Ward and inverse identities.
It is not an independently executed quartic-vertex calculation.

%=====================================================================
\section{Blindness and analytic controls}
\label{sec:controls}
%=====================================================================

\begin{proposition}[Complete blind identity]
\label{prop:blind}
At \(\beta_0\) and its normalized version,
\[
m=0,\qquad n_\ell=0\quad\text{for every raw label }\ell.
\]
The complete response vanishes there for every \(d,\theta\).
\end{proposition}

\begin{proof}
The spatial reflection
\[
\mathsf S=\diag(1,-1,1,1)
\]
preserves \(\eta\), fixes \(k_1,k_2,t_0,A\), interchanges
\(k_3,k_4\), and preserves \(B\).
It negates the blind gauge vector.
After reflecting, exchange the complete data of legs three and four.
Their species and physical tensors agree.
The original label is recovered with \(G\) negated.

Every raw contact is reflection invariant.
Its differentiated response is symmetric under simultaneous external
permutation.
The Einstein vertices and propagator have the same covariance, and
the complete tree contains all three exchanges.
Each scalar response therefore equals its own negative.
Multilinearity gives zero.
The same argument survives polarization rescaling.
\end{proof}

Parity evenness is used in this proof.
An arbitrary parity-odd class enlargement cannot inherit the blind
row without a new argument.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.06\linewidth}p{.43\linewidth}X@{}}
\toprule
Control & Specialization & Result and justification\\
\midrule
\(C_1\)&
One species, unit Einstein cubic, canonical Einstein quartic &
Quartic \(2\), exchange sum \(6-2-2=2\); response zero by
the polarized Ward identity.\\
\addlinespace
\(C_2\)&
\(a=e=1,\ b=c=0\), species \(1122\), no extra contacts &
All channel factors vanish at this point.
For every associative product the orthogonal Einstein decomposition
gives canonical cancellation.\\
\addlinespace
\(C_3\)&
\(d_{IJK}=\chi w_Iw_Jw_K\), canonical reference &
All channel factors equal
\(\chi^2\|w\|^2\prod_iw_{J_i}\).
The response is this scalar times one-sector zero.
For \(\chi=1,w=(1,1)\), it is \(4-4=0\).\\
\addlinespace
\(C_4\)&
Permute momenta, species, tensors, references, and insertion together &
Coefficient extraction permutes all assignments and exchange
partitions; the complete response is unchanged.\\
\addlinespace
\(C_5\)&
Compare each direct exchange with its Ward reduction &
\(X+W=(6,-2,-2)+(-6,2,2)=0\).
The channel identity holds throughout \(\XX\).\\
\addlinespace
\(C_6\)&
Rational and normalized blind seeds &
All raw columns and the target vanish by
Proposition~\ref{prop:blind}.\\
\bottomrule
\end{tabularx}
\end{center}

The associative and rank-one controls concern the canonical quartic,
not arbitrary added contacts.
The permutation control concerns complete data, not a tensor
permutation with other labels held fixed.
The exchange control uses conservation at the physical end before
applying the inverse.
These qualifications prevent weaker or differently normalized
checks from being substituted for the stated identities.

%=====================================================================
\section{Complete matrix encoding and the finite certificate}
\label{sec:matrix}
%=====================================================================

\subsection{All raw entries}

At \(\beta_+\), the lower-index momenta and longitudinal tensor are
\begin{equation}
K=
\begin{pmatrix}
-1&0&0&1\\
-1&0&0&-1\\
1&-1&0&0\\
1&1&0&0
\end{pmatrix},
\qquad
H_1=
\begin{pmatrix}
2&0&0&-1\\
0&0&0&0\\
0&0&0&0\\
-1&0&0&0
\end{pmatrix}.
\label{eq:KH}
\end{equation}
Put \(H_2=A,H_3=H_4=B\),
\(\eta_a=(-1,1,1,1)_a\), and \(J^+=(1,1,2,2)\).

For every original label \((I,P)\), define
\begin{align}
c_{I,P}
={}&-\sum_{\substack{\sigma\in S_4\\
I_r=J^+_{\sigma(r)}\ (r=1,\ldots,4)}}
\sum_{\lambda_1,\ldots,\lambda_{10}=0}^3
K_{\sigma(1),\lambda_1}K_{\sigma(2),\lambda_2}
\nonumber\\
&\quad\times
\prod_{r=1}^4
(H_{\sigma(r)})_{\lambda_{2r+1}\lambda_{2r+2}}
\prod_{\{u,v\}\in P}
\eta_{\lambda_u}\delta_{\lambda_u,\lambda_v}.
\label{eq:integer-encoding}
\end{align}
The first index of \(K\) labels legs \(1,\ldots,4\);
the second labels components \(0,\ldots,3\).
All inputs and terms are integers.

Let
\[
\overline I=(3-I_1,3-I_2,3-I_3,3-I_4).
\]

\begin{theorem}[Complete finite encoding]
\label{thm:encoding}
The target and every original contact column are
\begin{equation}
m_S=\begin{pmatrix}0\\16/3\\16/3\end{pmatrix},
\qquad
(N_S)_{:\!,(I,P)}
=\begin{pmatrix}0\\c_{I,P}\\c_{\overline I,P}\end{pmatrix}.
\label{eq:full-matrix}
\end{equation}
This encodes all \(45360\) entries of the \(3\times15120\) matrix.
\end{theorem}

\begin{proof}
The first row is zero by Proposition~\ref{prop:blind}.
The second row is exactly
\eqref{eq:contact-evaluator} specialized to \eqref{eq:KH};
the diagonal metric gives the printed equality constraints.
Exchanging the species in \(\beta_+\) gives \(\beta_-\), which
complements the raw species word.
The target follows from blindness and Theorem~\ref{thm:nonzero}.

Every species word and every perfect matching is included.
No reduced or selected contact list replaces the raw catalogue.
\end{proof}

This is a complete exact encoding, not a claim that a catalogue-wide
program was executed.
For a compatible species word, there are four compatible external
assignments.
Sparse component enumeration uses at most
\[
2\cdot2\cdot3\cdot2\cdot2\cdot2=96
\]
products per assignment before the matching constraints.

\subsection{Original pivot columns}

Choose
\[
P_0=\{\{1,2\},\{3,4\},\{5,7\},\{6,9\},\{8,10\}\}
\]
and
\begin{equation}
\ell_a=((1,1,2,2),P_0),\qquad
\ell_b=((2,2,1,1),P_0).
\label{eq:labels}
\end{equation}
They correspond to
\begin{align}
\OO_a
&=\int\dd^4x\,
(\partial_\lambda h^1)
(\partial^\lambda h^1_{\mu\nu})
h^{2\,\mu\rho}h^{2\,\nu}{}_\rho,
\nonumber\\
\OO_b
&=\int\dd^4x\,
(\partial_\lambda h^2)
(\partial^\lambda h^2_{\mu\nu})
h^{1\,\mu\rho}h^{1\,\nu}{}_\rho.
\label{eq:pivot-actions}
\end{align}
Here \(h^I\) without indices is the Lorentz trace.

\begin{proposition}[Exact original pivots]
\label{prop:pivots}
\begin{equation}
\boxed{
(N_S)_{:\!,\ell_a}=\begin{pmatrix}0\\8\\0\end{pmatrix},
\qquad
(N_S)_{:\!,\ell_b}=\begin{pmatrix}0\\0\\8\end{pmatrix}.
}
\label{eq:pivots}
\end{equation}
\end{proposition}

\begin{proof}
At \(\beta_+\), the traced factor of \(\OO_a\) must receive
the longitudinal leg; all physical tensors are trace free.
The other species-one factor receives \(A\).
The two species-two factors receive \(B,B\) in two orders.
The required contractions are
\[
\tr_\eta G=-2,\qquad k_1\cdot k_2=-2,\qquad
A_{\mu\nu}B^{\mu\rho}B^\nu{}_\rho=-1.
\]
Each assignment contributes
\[
-(k_1\cdot k_2)(\tr_\eta G)
 A_{\mu\nu}B^{\mu\rho}B^\nu{}_\rho
=-(-2)(-2)(-1)=4.
\]
Thus \(n_a(\beta_+)=8\).

For \(\OO_b\), its traced species-two factor receives a physical
\(B\), so the response is zero.
Species exchange gives the other nonzero entry.
Blindness gives both first entries.
\end{proof}

The insertion has \(k_1\cdot\xi_1=-1\), hence trace \(-2\).
These pivots use the traceful gauge direction.
Adding \(k\cdot\xi=0\) would change the domain and remove these
particular contractions.

\subsection{Rank factorization with an original minor}

Define
\begin{equation}
\Bpiv=
\begin{pmatrix}0&0\\8&0\\0&8\end{pmatrix},
\qquad
T_{:\!,(I,P)}
=\frac18\begin{pmatrix}c_{I,P}\\c_{\overline I,P}\end{pmatrix}.
\label{eq:BT}
\end{equation}

\begin{theorem}[Complete evaluated contact image]
\label{thm:rank}
\[
N_S=\Bpiv T,
\qquad
\im_{\RR}N_S
=\left\{\begin{pmatrix}0\\x\\y\end{pmatrix}:x,y\in\RR\right\},
\]
and
\[
\rank_{\QQ}N_S=\rank_{\RR}N_S=2.
\]
The original minor on rows two and three and columns
\(\ell_a,\ell_b\) is \(64\).
\end{theorem}

\begin{proof}
The product in \eqref{eq:BT} reconstructs every column in
\eqref{eq:full-matrix}.
Its two left-factor columns are original columns of \(N_S\).
Their last two entries form
\[
\begin{pmatrix}8&0\\0&8\end{pmatrix},
\]
whose determinant is \(64\).
This gives the lower rank bound over both fields.
The zero first row, or the factorization, gives the upper bound.
The same two columns span the entire zero-first-entry plane.
\end{proof}

A nonzero minor of an arbitrary left factor would not suffice.
For instance,
\[
N=(0),\qquad B=(1),\qquad T=(0)
\]
has \(N=BT\) but \(\rank N=0\).
Here the lower bound is valid because the minor is an actual minor
of the original evaluated matrix.

\begin{theorem}[Complete-catalogue finite fit]
\label{thm:certificate}
Let \(\alpha_0\in\QQ^{15120}\) have exactly
\[
(\alpha_0)_{\ell_a}=(\alpha_0)_{\ell_b}=2/3
\]
as its nonzero coordinates.
Then
\begin{equation}
\boxed{
N_S\alpha_0=m_S,\qquad
\rank[N_S\mid m_S]=\rank N_S=2.
}
\label{eq:certificate}
\end{equation}
The complete left kernel is
\begin{equation}
\{z\in\RR^{1\times3}:zN_S=0\}
=\Span_{\RR}\{(1,0,0)\}.
\label{eq:left-kernel}
\end{equation}
No separator exists on \(S\) against the declared contact class.
\end{theorem}

\begin{proof}
The original pivots give
\[
N_S\alpha_0
=\frac23\begin{pmatrix}0\\8\\0\end{pmatrix}
+\frac23\begin{pmatrix}0\\0\\8\end{pmatrix}
=\begin{pmatrix}0\\16/3\\16/3\end{pmatrix}=m_S.
\]
Appending a fitted column does not increase rank.
If \(z=(z_1,z_2,z_3)\) annihilates the matrix, the pivots force
\(8z_2=8z_3=0\).
Every \((z_1,0,0)\) annihilates its zero first row.
The target also has zero first entry, so every left annihilator
annihilates the target.
\end{proof}

\begin{corollary}[The full affine family of finite fits]
\label{cor:affine-fits}
Every real fitting vector is in
\[
\{\alpha:N_S\alpha=m_S\}
=\alpha_0+\ker_{\mathrm{right}}N_S,
\]
an affine space of raw-coordinate dimension \(15118\).
\end{corollary}

\begin{proof}
Subtracting the known fit converts the equation into
\(N_S(\alpha-\alpha_0)=0\).
Rank-nullity gives \(15120-2=15118\).
\end{proof}

This dimension includes raw-coordinate redundancy.
It is not the dimension of inequivalent quartic actions.
The displayed sparse vector is one fit, not a uniquely selected
contact correction.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\node[certbox,text width=4.6cm] (fit) at (0,0)
  {Finite fit\\\(N_S\alpha_0=m_S\)\\
   Closed on the three stated labels};
\node[certbox,text width=4.6cm] (global) at (6.2,0)
  {Global identity\\\(m=\sum_\ell\alpha_\ell n_\ell\) on \(\XX\)\\
   Not proved here};
\draw[certarrow,dashed] (fit)--node[above,font=\small]
  {requires an additional argument} (global);
\node[certbox,text width=8.1cm] at (3.1,-2.0)
  {A direct global identity or a suitable injective-evaluation
   certificate would suffice. A finite fit alone does not.};
\end{tikzpicture}
\caption{The scope of the finite certificate. Dashed transport is
not an established implication without its additional hypothesis.}
\label{fig:certificate-scope}
\end{figure}

%=====================================================================
\section{Meaning of the certificate}
\label{sec:meaning}
%=====================================================================

\subsection{Simultaneous cancellation of the selected tests}

For any fixed \(d\), choose
\[
\theta_{\ell_a}=\theta_{\ell_b}=-\frac23q(d),
\qquad
\theta_\ell=0\quad
(\ell\notin\{\ell_a,\ell_b\}).
\]
Then
\[
r_S(d,\theta)=0.
\]
The coefficients may depend on the fixed coupling \(d\), but not
on which row is tested.
This is cancellation of three longitudinal responses, not a proof
of gauge consistency of the modified action.

\subsection{The remaining global function}

Define
\begin{equation}
H(\beta)
=m(\beta)-\frac23n_{\ell_a}(\beta)-\frac23n_{\ell_b}(\beta).
\label{eq:H}
\end{equation}
Exactly the following is proved:
\[
H|_S=0.
\]
A new label with \(H(\beta)\ne0\) would refute this particular
two-contact identity.
It would not refute every member of the affine family in
Corollary~\ref{cor:affine-fits}.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\node[certbox,text width=5.0cm] (tested) at (0,0)
  {Known\\\(H(\beta_0)=H(\beta_+)=H(\beta_-)=0\)};
\node[certbox,text width=5.0cm] (untested) at (6.6,0)
  {Unknown here\\\(H(\beta)\) at general \(\beta\in\XX\)};
\node[certbox,text width=8.4cm] (family) at (3.3,-2.0)
  {Even a nonzero value of this \(H\) excludes only
   \(\alpha_0\). Global separation must exclude the complete
   allowed contact image.};
\draw[certarrow] (tested.south)--(family.north west);
\draw[certarrow] (untested.south)--(family.north east);
\end{tikzpicture}
\caption{The remaining-function test and the larger affine fitting
problem have different quantifiers.}
\label{fig:remaining-function}
\end{figure}

\subsection{Why one nonzero response is insufficient}

At \(\beta_+\), the target is \(16/3\) while the allowed raw
response \(n_a\) is \(8\).
The scalar contact image at that row is already all of \(\RR\).
The second nonzero row is independently spanned by \(n_b\).
The blind row imposes no condition on either the target or the
contact image.

This explains the finite rank: all columns lie in a
two-dimensional row plane, and two original contacts span that
plane.
The proof does not require printing \(45360\) integers.
Complete reconstruction and an original minor supply exact coverage
and both rank bounds analytically.

%=====================================================================
\section{Certificate patterns that do not transfer}
\label{sec:patterns}
%=====================================================================

The finite alternative of Appendix~\ref{app:algebra} is elementary
linear algebra.
Other manuscripts use analogous alternatives on different
domains.
Similarity of the certificate form does not identify their
actions, equation spaces, correction classes, or response maps.

The comparisons in this section are documentary source records.
None is a premise of Theorem~\ref{thm:certificate},
Proposition~\ref{prop:finite-separation}, or the \(16/3\)
calculation.

\subsection{Class-invariant information and reference rebasing}

\begin{proposition}[Allowed reference rebasing preserves membership]
\label{prop:rebase}
Keep the external domain, response prescription, and complete
contact space fixed.
If
\[
m'=m+\sum_\ell\rho_\ell n_\ell
\]
with constant allowed coefficients, then
\[
[m']=[m]\quad\text{in }\RR^\XX/\CC.
\]
Global membership is equivalent for \(m,m'\).
On every finite family, fitting vectors transform by
\(\alpha\mapsto\alpha+\rho\), separators have the same target
pairing, and augmented ranks are unchanged.
\end{proposition}

\begin{proof}
The difference lies in \(\CC\).
At finite evaluation,
\(m'_S=m_S+N_S\rho\).
This proves the fitting statement.
A left annihilator of \(N_S\) annihilates the difference, and an
elementary augmented-column operation preserves rank.
\end{proof}

Thus changing a reference quartic by an allowed local contact
changes the written representative, but does not automatically
create a different quotient-membership problem.
In particular, such a change is not correctly described as
``never a contact correction.''

A changed gauge rule, switching prescription, admissible domain,
or reference modification outside the declared class changes the
evaluation contract and requires a new comparison.
Identical numerical values are then neither automatic nor sufficient
to identify the problems.

The class-invariant principle resembles the role of preserved data
in the supplied XI-G source record~\cite{CheungXI}.
That manuscript's obstruction is not imported as a contact separator.

\subsection{Weighting changes the quotient}

\begin{proposition}[Weighting need not preserve unweighted equivalence]
\label{prop:weighting}
For a smooth weight \(\chi\) and compactly supported vector density
\(B^\mu\),
\[
\int\chi\,\partial_\mu B^\mu\,\dd^4x
=-\int(\partial_\mu\chi)B^\mu\,\dd^4x.
\]
Consequently, the map ``multiply a density by \(\chi\)'' need not
descend to the unweighted integration-by-parts quotient.
\end{proposition}

\begin{proof}
Integrate the derivative of \(\chi B^\mu\).
For an explicit quartic example, take a compactly supported scalar
\(\varphi\) and
\(B^\mu=\varphi^3\partial^\mu\varphi\).
Then
\[
\int\chi\,\partial_\mu B^\mu\,\dd^4x
=-\frac14\int(\partial_\mu\chi)\partial^\mu(\varphi^4)\,\dd^4x
=\frac14\int(\Box\chi)\varphi^4\,\dd^4x.
\]
The unweighted divergence has zero Euler derivative.
The weighted expression has scalar Euler derivative
\((\Box\chi)\varphi^3\), which need not vanish.
\end{proof}

The same elementary warning underlies the source/density weighting
distinction recorded in XII-B~\cite{CheungXII}.
Its integrability or mixed-variation tests are not separators for
\(N_S\).

The present tree is unweighted.
Its catalogue and total-derivative quotient are formed under
ordinary conserved-momentum evaluation.
They are not automatically the catalogue or quotient of a
switched problem.

\subsection{Completeness under class enlargement}

\begin{proposition}[Finite fitting is monotone under compatible enlargement]
\label{prop:enlargement}
Suppose an enlarged correction class uses the same target, domain,
and evaluation prescription and contains the original contacts.
Then \(m_S\) remains fitted, and no separator exists on this same
family against the enlarged class.
Its rank and complete left kernel require a new coverage argument.
\end{proposition}

\begin{proof}
The existing allowed combination still gives \(m_S\).
Any annihilator of the enlarged image also annihilates that
combination.
The enlarged image may contain additional directions, so its rank
and full kernel do not follow from the old matrix.
\end{proof}

In particular, an enlargement may destroy the blind-row proof.
An unbounded derivative or field enlargement may also destroy
finite-dimensionality of the correction space.
A finite displayed list then cannot be declared complete solely by
analogy with the \(15120\)-label construction.

This is the evaluated-coverage qualification in finite-bank
arguments such as the supplied VII-H record
\cite{CheungVII}.
An annihilator is a complete separator only after the full allowed
ambiguity has been accounted for.

\subsection{Descent, currents, and records are separate questions}

The supplied III record treats a gauge-image membership problem and
descent on a coupled equation-space quotient
\cite{CheungIII}.
A closed-range condition there is not a contact-matrix test here.
Formal compensation is not automatically descent on an unchanged
equation space.

The supplied V record distinguishes a raw current, a compact
representative, and a residual witness~\cite{CheungV}.
Its scalar-response cubic is not the Einstein cubic of
Lemma~\ref{lem:diagonal}.
No current-classification result from V enters the Ward reduction.

Measurement uniqueness, including Cheung's Enter characterization,
does not construct the coherent-matching interface that would turn
a tree coefficient into a record coefficient
\cite{CheungEnter,CheungI}.
Likewise, the represented models in IV have their own generators,
domains, and calibration estimates~\cite{CheungIV}.

To state the limited record implication explicitly, suppose two
proposed output descriptions satisfy
\[
\bra f\rho_\ell(g)\ket i
=ig^2t_\ell+O_{\form}(g^3)
\]
with the same code vectors, phase reference, and probes.
For
\[
Y=-i\ket i\bra f+i\ket f\bra i,
\qquad F_{\mathrm{acc}}=(I+Y)/2,
\]
where \(Y=0\) on the code complement,
\[
p_{\mathrm{acc}}(\rho)
=\frac12+\operatorname{Im}\bra f\rho\ket i.
\]
The difference of order-two acceptance coefficients is therefore
\(\operatorname{Re}(t_2-t_1)\).
This follows by taking the trace of the effect.
It does not supply the hypothesized output states or an instrument
realizing them.

\subsection{Other certificate domains}

The supplied VI records use the same finite alternative, but their
classical and quantum catalogues are separately scoped
\cite{CheungVI}.
Finite packet and conical certificates in VIII-A and VIII-B are not
evaluation samples of this contact span~\cite{CheungVIII}.

The X-E--X-G records distinguish a selected tree, its additional
terms and gauge prescription, and a reciprocity/Hessian question
\cite{CheungX}.
A nonzero selected response is not a defect of every completion.
A reciprocity pairing is not \(N_S\), and free BRST nilpotence
does not assert an interacting Ward identity.

The supplied XII-F witness concerns its unchanged observable
\cite{CheungXII}.
It is not a universal exclusion of all completions.
Overlap conditions in XIII-E become a finite membership problem only
after their own complete finite parameterization is supplied
\cite{CheungXIII}.
Their consistency is not membership in the present contact image.

The supplied XIV-I and XIV-J releases explicitly separate an internal
amplitude, a partial reference distribution, reconstruction descent,
and the missing full characteristic map
\cite{CheungXIVI,CheungXIVJ}.
A bound on normal derivative order does not, by itself, make all
coefficient-distribution freedoms finite dimensional.
A linear separator can test a fully defined parameterized problem;
it cannot construct an absent distributional product.

The VII-H, VII-I, and Em01 records concern, respectively, other
response ambiguities, a free algebra, and isolated contraction
certificates~\cite{CheungVII,CheungEm01}.
None is a column of this \(15120\)-label catalogue.

\statementbox{Non-transfer boundary}{%
No companion source supplies a contact column, separator, or global
identity for this locked tree.
The present calculation does not discharge the characteristic-map
obligations at I3 or J2, admit a quantum coefficient, or open XV-B.
The unresolved H2 verdict in the supplied XIV-H version and the
named I2 construction in XIV-I are different versioned source
statements, not statuses updated by this paper
\cite{CheungXIVH,CheungXIVI}.
No conclusion here closes O.1; its supplied status remains
\SourceLabel{NOT_CLOSED}.}

%=====================================================================
\section{Successor questions and global certification}
\label{sec:successor}
%=====================================================================

\subsection{A sufficient finite-to-global criterion}

\begin{theorem}[Injective evaluation makes a fit global]
\label{thm:injective}
Let
\[
U=\Span_{\RR}\{n_1,\ldots,n_L,m\}\subseteq\RR^X.
\]
If evaluation \(E_{S'}:U\to\RR^{|S'|}\) is injective and
\(m_{S'}=N_{S'}\alpha\), then
\[
m=\sum_\ell\alpha_\ell n_\ell\quad\text{on }X.
\]
\end{theorem}

\begin{proof}
The difference belongs to \(U\) and has zero evaluation.
Injectivity makes it the zero function.
\end{proof}

No such injectivity is established for the present three labels.

Injectivity is a sufficient route, not a necessary condition for
one particular identity to be true.
A direct global algebraic proof can establish that identity even
when the chosen evaluation is not injective.

For example, with \(X=\RR\),
\[
n_1(t)=1,\quad n_2(t)=t,\quad m(t)=1,\quad S'=\{0\},
\]
evaluation is not injective on \(U=\Span\{1,t\}\), but
\(m=n_1\) is a global identity.
The correct warning is that a finite fit \emph{alone} does not
provide such a proof.

\subsection{Three distinct successor questions}

\statementbox{Q1: global membership inside the locked class}{%
Determine either:
\begin{enumerate}[label=(\roman*)]
\item constants \(\alpha\in\RR^{15120}\) such that
\[
m=\sum_\ell\alpha_\ell n_\ell\quad\text{on all of }\XX;
\]
\item an admissible finite \(S'\), complete evaluated coverage, and
\[
zN_{S'}=0,\qquad zm_{S'}\ne0.
\]
\end{enumerate}
The current certificate proves neither alternative.
It proves \(H|_S=0\) for one candidate \(H\).}

\statementbox{Q2: a declared enlargement of the correction class}{%
Repeat Q1 after explicitly naming the enlarged fields, derivative
bound, parity sector, coefficient freedoms, quotient, and external
evaluation.
Completeness and, where needed, finite dimensionality must be proved
anew.
Under a compatible inclusion of the original class, the old finite
fit persists by Proposition~\ref{prop:enlargement}; the global answer
need not persist.}

\statementbox{Q3: a changed reference or response prescription}{%
Recompute and compare after changing the reference quartic, gauge
insertion, source switching, or evaluation domain.
If the reference change is an allowed contact rebasing under the
same prescription, Proposition~\ref{prop:rebase} applies.
Otherwise this is a new response problem, and neither identical
numbers nor the old certificate establish equivalence.}

A future global proof must retain constant coefficients.
A fit with coefficients chosen separately at each kinematic point
does not answer Q1.
A failure of one candidate must be distinguished from exclusion of
the complete affine fitting family.
A weighted or characteristic correction class must not be identified
with \(\VV\) without an explicit map and coverage proof.

\paragraph{Conclusion.}
The exact finite answer is
\[
m_S=\begin{pmatrix}0\\16/3\\16/3\end{pmatrix},
\qquad
\Bpiv=\begin{pmatrix}0&0\\8&0\\0&8\end{pmatrix},
\]
\[
N_S=\Bpiv T,\qquad
\det(\Bpiv)_{\{2,3\},:}=64,\qquad
m_S=\Bpiv\begin{pmatrix}2/3\\2/3\end{pmatrix}.
\]
The complete contact image contains the target, and its entire left
kernel annihilates the target.
The result is \(\SEP\)-finite, with the global question and all
changes of contract separately delimited.

%=====================================================================
\appendix
\clearpage
\section{Exact integer reconstruction checker}
\label{app:encoding}
%=====================================================================

\subsection{Authoritative order and data}

Rows are \((\beta_0,\beta_+,\beta_-)\).
Columns are ordered lexicographically by species word and then by
the recursive matching order below.
That recursion pairs the smallest remaining slot with each possible
partner and recurses.

The blind covariant tensor is
\[
G_x=
\begin{pmatrix}
0&-1&0&0\\
-1&0&0&1\\
0&0&0&0\\
0&1&0&0
\end{pmatrix}.
\]
The time-gauge tensor is \(G_t=H_1\) in \eqref{eq:KH}.
The fit always refers to the original labels
\(\ell_a,\ell_b\), regardless of computational ordering.

The listing uses only Python's standard library.
It reconstructs all raw contact responses, checks the original
pivots and minor, and checks the finite fit.
Its target arithmetic implements the proved diagonal cubic formula,
not an independent Einstein-vertex library.
Inclusion is not a claim of execution.

\begin{lstlisting}
from fractions import Fraction
from itertools import permutations, product
from math import prod

ETA = (-1, 1, 1, 1)
PERMS = tuple(permutations(range(4)))

KUP = (
    ( 1, 0, 0,  1),
    ( 1, 0, 0, -1),
    (-1,-1, 0,  0),
    (-1, 1, 0,  0),
)

A_DENSE = (
    (0,0, 0,0),
    (0,1, 0,0),
    (0,0,-1,0),
    (0,0, 0,0),
)
B_DENSE = (
    (0,0, 0,0),
    (0,0, 0,0),
    (0,0,-1,0),
    (0,0, 0,1),
)

def lower(v):
    return tuple(ETA[a]*v[a] for a in range(4))

def dot(v,w):
    return sum(ETA[a]*v[a]*w[a] for a in range(4))

def addvec(v,w):
    return tuple(v[a]+w[a] for a in range(4))

def gauge(k,xi):
    kl, xl = lower(k), lower(xi)
    return tuple(
        tuple(kl[a]*xl[b]+kl[b]*xl[a]
              for b in range(4))
        for a in range(4)
    )

def trace_eta(H):
    return sum(ETA[a]*H[a][a] for a in range(4))

def transverse(v,H):
    return all(
        sum(v[a]*H[a][b] for a in range(4)) == 0
        for b in range(4)
    )

t0 = (1,0,0,0)
xx = (0,1,0,0)

assert all(any(k) and dot(k,k) == 0 for k in KUP)
assert all(sum(k[a] for k in KUP) == 0
           for a in range(4))

invariants = tuple(
    dot(addvec(KUP[0],KUP[j]),
        addvec(KUP[0],KUP[j]))
    for j in (1,2,3)
)
assert invariants == (-4,2,2)
assert sum(invariants) == 0
assert dot(t0,t0) == -1

for k,H in zip(KUP[1:],(A_DENSE,B_DENSE,B_DENSE)):
    assert all(H[a][b] == H[b][a]
               for a in range(4) for b in range(4))
    assert trace_eta(H) == 0
    assert transverse(k,H)
    assert transverse(t0,H)

GT_DENSE = gauge(KUP[0],t0)
GX_DENSE = gauge(KUP[0],xx)
assert trace_eta(GT_DENSE) == -2

def sparse_vector(v):
    return tuple(((a,),v[a])
                 for a in range(4) if v[a] != 0)

def sparse_tensor(H):
    return tuple(
        ((a,b),H[a][b])
        for a in range(4) for b in range(4)
        if H[a][b] != 0
    )

K = tuple(sparse_vector(lower(k)) for k in KUP)
A = sparse_tensor(A_DENSE)
B = sparse_tensor(B_DENSE)
G_TIME = sparse_tensor(GT_DENSE)
G_X = sparse_tensor(GX_DENSE)

assert K == (
    (((0,),-1),((3,), 1)),
    (((0,),-1),((3,),-1)),
    (((0,), 1),((1,),-1)),
    (((0,), 1),((1,), 1)),
)
assert G_TIME == (
    ((0,0),2),((0,3),-1),((3,0),-1)
)

# Matching slots are zero-based in the code.
def matchings(slots):
    slots = tuple(slots)
    if not slots:
        yield ()
        return
    a = slots[0]
    for j in range(1,len(slots)):
        b = slots[j]
        rest = slots[1:j]+slots[j+1:]
        for tail in matchings(rest):
            yield ((a,b),)+tail

P_LIST = tuple(matchings(range(10)))
I_LIST = tuple(product((1,2),repeat=4))
LABELS = tuple((I,P) for I in I_LIST for P in P_LIST)

assert len(P_LIST) == 945
assert len(set(P_LIST)) == 945
assert len(LABELS) == 15120

def contract_sparse(P,factors):
    total = 0
    for chosen in product(*factors):
        indices = tuple(
            a
            for index_tuple,value in chosen
            for a in index_tuple
        )
        assert len(indices) == 10

        if any(indices[u] != indices[v] for u,v in P):
            continue

        value = prod(value for index_tuple,value in chosen)
        value *= prod(ETA[indices[u]] for u,v in P)
        total += value
    return total

def raw_value(I,P,J,Kdata,Hdata):
    if sorted(I) != sorted(J):
        return 0

    total = 0
    for sigma in PERMS:
        if any(I[r] != J[sigma[r]] for r in range(4)):
            continue
        factors = (
            Kdata[sigma[0]], Kdata[sigma[1]],
            Hdata[sigma[0]], Hdata[sigma[1]],
            Hdata[sigma[2]], Hdata[sigma[3]],
        )
        # Two Fourier derivatives contribute i^2 = -1.
        total -= contract_sparse(P,factors)
    return total

J_PLUS = (1,1,2,2)
J_MINUS = (2,2,1,1)
H_TIME = (G_TIME,A,B,B)
H_BLIND = (G_X,A,B,B)

P0 = ((0,1),(2,3),(4,6),(5,8),(7,9))
LABEL_A = ((1,1,2,2),P0)
LABEL_B = ((2,2,1,1),P0)

assert LABEL_A in LABELS
assert LABEL_B in LABELS

# Target arithmetic from the proved restricted Einstein cubic.
# This is not an independent implementation of Einstein vertices.
def v_abb(p0,q0,r0):
    assert p0+q0+r0 == 0
    return -2*(p0*q0+p0*r0+q0*r0)

W = (
    -v_abb(2,-1,-1),
     v_abb(1,0,-1),
     v_abb(1,0,-1),
)
assert W == (-6,2,2)

V4 = -sum(W)
X = tuple(-w for w in W)
M_VALUE = Fraction(-2*W[0]+W[1]+W[2],3)

assert V4 == 2
assert X == (6,-2,-2)
assert M_VALUE == Fraction(16,3)
assert Fraction(2,3)*V4-X[1]-X[2] == M_VALUE

assert V4-sum(X) == 0
assert 0*V4-sum(0*x for x in X) == 0
assert 2*V4-sum(2*x for x in X) == 0

def reconstruct_and_check():
    plus = {}
    blind = {}

    for label in LABELS:
        I,P = label
        plus[label] = raw_value(I,P,J_PLUS,K,H_TIME)
        blind[label] = raw_value(I,P,J_PLUS,K,H_BLIND)
        assert blind[label] == 0

    columns = {}
    for I,P in LABELS:
        Ibar = tuple(3-i for i in I)
        minus = raw_value(I,P,J_MINUS,K,H_TIME)
        assert minus == plus[(Ibar,P)]

        col = (blind[(I,P)],plus[(I,P)],minus)
        columns[(I,P)] = col

        # Complete factorization N = B_piv T.
        t1 = Fraction(col[1],8)
        t2 = Fraction(col[2],8)
        assert (0,8*t1,8*t2) == col

    ca = columns[LABEL_A]
    cb = columns[LABEL_B]
    assert ca == (0,8,0)
    assert cb == (0,0,8)

    # This minor uses ORIGINAL columns, not an arbitrary factor.
    original_minor = ca[1]*cb[2]-cb[1]*ca[2]
    assert original_minor == 64

    target = (Fraction(0),M_VALUE,M_VALUE)
    fitted = tuple(
        Fraction(2,3)*ca[i]+Fraction(2,3)*cb[i]
        for i in range(3)
    )
    assert fitted == target

    N_rows = tuple(
        tuple(columns[label][i] for label in LABELS)
        for i in range(3)
    )
    assert all(value == 0 for value in N_rows[0])

    # Selected permutation regressions; the general identity
    # is proved analytically in the manuscript.
    for label in (LABEL_A,LABEL_B):
        I,P = label
        expected = plus[label]
        for pi in PERMS:
            Jp = tuple(J_PLUS[i] for i in pi)
            Kp = tuple(K[i] for i in pi)
            Hp = tuple(H_TIME[i] for i in pi)
            assert raw_value(I,P,Jp,Kp,Hp) == expected

    # Rank lower bound: original minor 64.
    # Rank upper bound: zero first row.
    # Augmented rank: the target is the printed column fit.
    return {
        "shape": (3,len(LABELS)),
        "labels": LABELS,
        "N_rows": N_rows,
        "target": target,
        "original_minor": original_minor,
        "rank": 2,
        "augmented_rank": 2,
        "nonzero_alpha": (
            (LABEL_A,Fraction(2,3)),
            (LABEL_B,Fraction(2,3)),
        ),
        "left_kernel_basis": ((1,0,0),),
    }

if __name__ == "__main__":
    certificate = reconstruct_and_check()
    print("shape =",certificate["shape"])
    print("target =",certificate["target"])
    print("original minor =",certificate["original_minor"])
    print("rank pair =",(
        certificate["rank"],
        certificate["augmented_rank"],
    ))
    print("status = SEP-finite")
\end{lstlisting}

\subsection{Exhaustiveness and verification limits}

The matching recursion generates each matching exactly once:
the partner of the smallest slot is unique, and induction handles
the remainder.
The species loop covers all sixteen words.
The permutation loop performs full external differentiation.
The sparse products enumerate every nonzero component assignment,
and the diagonal-metric test is precisely the matching contraction.

Thus the listing reconstructs the complete three-row raw matrix.
The analytical completeness proof does not depend on executing it.

The factorization products and final fit are exact:
\[
\Bpiv\begin{pmatrix}c/8\\d/8\end{pmatrix}
=\begin{pmatrix}0\\c\\d\end{pmatrix},
\qquad
\det\begin{pmatrix}8&0\\0&8\end{pmatrix}=64.
\]
The substantive facts are that every raw column has zero first
entry and that the two pivot columns are original allowed columns.
Those facts were proved in the body.

%=====================================================================
\section{Finite certificate principles}
\label{app:algebra}
%=====================================================================

\subsection{Membership or separation}

\begin{theorem}[Finite alternative]
\label{thm:alternative}
For \(N\in K^{s\times L}\), \(u\in K^s\), and a field \(K\),
exactly one of the following holds:
\begin{enumerate}[label=(\roman*)]
\item \(N\alpha=u\) for some \(\alpha\in K^L\);
\item \(zN=0,\ zu=1\) for some \(z\in K^{1\times s}\).
\end{enumerate}
The second case is equivalent to
\(\rank[N\mid u]=\rank N+1\).
\end{theorem}

\begin{proof}
A fit makes every annihilator of \(N\) annihilate \(u\).
Otherwise append \(u\) to a basis of \(\im N\), extend to a basis
of \(K^s\), and define a linear functional equal to one on \(u\)
and zero on the other chosen basis elements.
Its row is the separator.
Appending one column increases rank exactly when it lies outside
the original image.
\end{proof}

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\node[certbox,text width=4.3cm] (question) at (3.2,0)
  {Exact finite data \(N,u\)\\Complete allowed correction image};
\node[certbox,text width=4.8cm] (fit) at (0,-2.0)
  {Membership\\\(N\alpha=u\)\\No separator for these data};
\node[certbox,text width=4.8cm] (sep) at (6.4,-2.0)
  {Separation\\\(zN=0,\ zu=1\)\\Global nonmembership if coverage holds};
\draw[certarrow] (question.south west)--(fit.north);
\draw[certarrow] (question.south east)--(sep.north);
\end{tikzpicture}
\caption{The finite alternative is exact linear algebra. Its global
interpretation additionally depends on the response contract and
coverage.}
\label{fig:finite-alternative}
\end{figure}

\begin{lemma}[Extension of the coefficient field]
\label{lem:field}
If \(K\subseteq\RR\) and \(N,u\) have entries in \(K\), ranks are
unchanged on extending scalars to \(\RR\).
A fit exists over \(\RR\) if and only if it exists over \(K\).
\end{lemma}

\begin{proof}
Each minor vanishes over \(K\) exactly when it vanishes over \(\RR\).
Apply this to \(N\) and \([N\mid u]\).
Equality of their ranks is the consistency criterion over either
field.
\end{proof}

Thus rational certificates remain valid when real contact coefficients
are permitted.

\subsection{Small-support separators}

\begin{theorem}[Support at most \(r+1\)]
\label{thm:sparse}
If \(\rank N=r\) and \(\rank[N\mid u]=r+1\), a separator exists
supported on at most \(r+1\) rows.
\end{theorem}

\begin{proof}
For \(r=0\), \(N=0\) and one nonzero coordinate of \(u\) supplies
a one-row separator.

For \(r>0\), choose \(r\) original independent columns \(B\) and
\(r\) rows \(R\) such that \(M=B_R\) is invertible.
Because \(u\notin\im B\), some \(j\notin R\) satisfies
\[
\delta=u_j-b_jM^{-1}u_R\ne0;
\]
otherwise \(u=BM^{-1}u_R\).
Set \(z_j=1\), \(z_R=-b_jM^{-1}\), and the other entries zero.
Then \(zB=0\), hence \(zN=0\), while \(zu=\delta\).
Normalize by \(\delta\).
\end{proof}

The same residual occurs in the original augmented minor:
\[
\det\begin{pmatrix}M&u_R\\b_j&u_j\end{pmatrix}
=\det(M)\delta.
\]

\begin{proposition}[Upper and lower rank certificates]
If \(N=BT\) and \(B\) has \(r\) columns, this gives
\(\rank N\le r\).
A nonzero \(r\times r\) minor of \(N\) gives the reverse bound.
It suffices that \(B\) consist of original columns of \(N\)
and contain that minor.
\end{proposition}

\begin{proof}
The image of \(N\) lies in the image of \(B\).
A nonzero original minor gives \(r\) independent original columns
and rows.
\end{proof}

\subsection{Finite evaluation versus global functions}

For any prescribed nonempty finite
\(S=\{t_1,\ldots,t_s\}\subset\RR\), let
\[
n_1(t)=1,\qquad m(t)=1+\prod_{i=1}^s(t-t_i).
\]
Then \(m|_S=n_1|_S\), but \(m\) is not a constant function.
This gives a finite fit without a global identity.
The functions are constructed after the sample is prescribed;
it is not an assertion that one fixed nonmember fits every finite
sample.

\begin{proposition}[Finite injective evaluation exists]
\label{prop:finite-injective}
If \(U\subseteq\RR^X\) has dimension \(d<\infty\), some family
of at most \(d\) labels has injective evaluation on \(U\).
\end{proposition}

\begin{proof}
If the common kernel of the labels chosen so far is nonzero,
choose a nonzero function in it and a label where it does not
vanish.
That evaluation decreases the kernel dimension.
Repeat at most \(d\) times.
\end{proof}

Consequently, if a fixed target lies outside a contact space of
dimension \(c<\infty\), some family of at most \(c+1\) labels
has a separator.
This existence result supplies neither rational labels nor a
constructive search on the locked domain.

\subsection{Kernel orientation, rescaling, and modular arithmetic}

For \(N\in K^{s\times L}\),
\[
\ker_{\mathrm{left}}N=\{z:zN=0\},
\qquad
\ker_{\mathrm{right}}N=\{\alpha:N\alpha=0\}.
\]
Their dimensions are \(s-\rank N\) and \(L-\rank N\).
A right-nullspace routine must be applied to \(N^T\) to obtain
left separators.

If row denominator clearing gives
\[
\widetilde N=DN,\qquad \widetilde u=Du,
\]
then a separator \(\widetilde z\) of the cleared data corresponds
to \(z=\widetilde zD\).
The row rescaling must be retained in the certificate.

For a rational matrix and a prime avoiding entry denominators,
\[
\rank_{\mathbb F_p}(N\bmod p)\le\rank_{\QQ}N.
\]
A nonzero modular minor has a nonzero rational lift.
But a modular augmented-rank jump need not prove rational
nonmembership:
\[
N=[p],\qquad u=[1]
\]
has a modular separator, while over \(\QQ\) it has the fit \(1/p\).
Discovery by modular arithmetic must therefore be followed by an
exact characteristic-zero certificate.

%=====================================================================
\section{Comparison example: selected defects and contact quotients}
\label{app:comparison}
%=====================================================================

This appendix preserves the useful interaction comparison without
using it in the certificate.

\subsection{Fixed vector quartic}

Use signature \((+---)\) and waves \(e^{-ik\cdot x}\).
Consider real antisymmetric \(f_{abc}\) and the fixed action
\[
-\frac14\int
\left(F^a_{\mu\nu}+g f^a{}_{bc}A_\mu^bA_\nu^c\right)^2\dd^4x.
\]
Its quartic is fixed, not an arbitrary contact nuisance.

Put
\[
C_s=\sum_e f_{abe}f_{cde},\quad
C_t=\sum_e f_{ace}f_{bde},\quad
C_u=\sum_e f_{ade}f_{bce},\quad
J_{abcd}=C_s-C_t+C_u.
\]
For transverse physical polarizations,
\[
j_{ij}
=(\eps_i\cdot\eps_j)(k_i-k_j)
+2(\eps_i\cdot k_j)\eps_j
-2(\eps_j\cdot k_i)\eps_i.
\]
Direct differentiation gives
\[
\mathscr T_{abcd}=-C_sK_s-C_tK_t-C_uK_u,
\]
where \(E_{ij}=\eps_i\cdot\eps_j\) and
\[
\begin{aligned}
K_s&=\frac{j_{12}\cdot j_{34}}{s_{12}}
     +E_{13}E_{24}-E_{14}E_{23},\\
K_t&=\frac{j_{13}\cdot j_{24}}{s_{13}}
     +E_{12}E_{34}-E_{14}E_{23},\\
K_u&=\frac{j_{14}\cdot j_{23}}{s_{14}}
     +E_{12}E_{34}-E_{13}E_{24}.
\end{aligned}
\]
The longitudinal identity
\[
j_{1j}\big|_{\eps_1=k_1}
=s_{1j}\eps_j-(k_1\cdot\eps_j)(k_1+k_j)
\]
and conservation of the opposite current remove the exchange
denominators.

Define
\[
D=
E_{34}\eps_2\cdot(k_3-k_4)
+E_{24}\eps_3\cdot(k_4-k_2)
+E_{23}\eps_4\cdot(k_2-k_3).
\]
With \(\eps_1\mapsto k_1/\mu\), the three complete \(K\)'s
are \(D/\mu,-D/\mu,D/\mu\).
For example, the first is obtained from
\[
\eps_2\cdot j_{34}
+(k_1\cdot\eps_3)E_{24}
-(k_1\cdot\eps_4)E_{23}=D.
\]
Thus
\[
\delta\mathscr T_{abcd}=-(D/\mu)J_{abcd}.
\]

At momenta \eqref{eq:momenta} scaled by \(\mu>0\), take
\[
\eps_1=\eps_2=(0,1,0,0),\qquad
\eps_3=\eps_4=(0,0,1,0).
\]
Then \(D=-2\mu\), hence \(\delta\mathscr T=2J\).
For the sparse seed \(f_{123}=a,f_{145}=b\), the species channel
\(2345\) has \(J_{2345}=ab\), giving \(2ab\).

This is a defect calculation for that fixed vector quartic.
It does not establish survival against an arbitrary vector contact
class, and it supplies no spin-two separator.

\subsection{A deliberately augmented spin-two representative}

Return to the locked spin-two signature.
For
\(d_{112}=d_{121}=d_{211}=\lambda\) and other independent
components zero, \(q=\lambda^2\).
Add the allowed quartic
\[
-L_0q\int
(h^1:h^2)\,
\partial_\rho h^1_{\alpha\beta}
\partial^\rho h^{2\,\alpha\beta}\,\dd^4x.
\]
At the scaled momenta \(\mu k_i\), take
\[
H_1=H_2=\frac{A}{\sqrt2},\qquad
H_3=H_4=-\frac{B}{\sqrt2},
\]
and replace \(H_1\) by
\[
G_1=k_{1,\mu}k_{1,\nu}/\mu^2.
\]
This is an admitted pure gauge with
\(\xi_\mu=k_{1,\mu}/(2\mu^2)\).

The differentiated contact polynomial before \(-L_0q\) is
\[
-\left[
s_{13}(H_1:H_3)(H_2:H_4)
+s_{14}(H_1:H_4)(H_2:H_3)
\right].
\]
The contraction values are
\[
G_1:H_3=G_1:H_4=-1/\sqrt2,\qquad
H_2:H_3=H_2:H_4=-1/2,\qquad
s_{13}=s_{14}=2\mu^2.
\]
Its value is \(-\sqrt2\,\mu^2\).
The augmentation contributes
\(\sqrt2L_0\lambda^2\mu^2\).

Let \(b_{\EH}\) denote the finite dimensionless unaugmented
coefficient at this label.
Choosing the fixed constant
\[
L_0=\frac{|b_{\EH}|+1}{\sqrt2}
\]
makes the augmented coefficient
\[
\lambda^2\mu^2(\sqrt2L_0+b_{\EH})
\]
strictly positive for \(\lambda\ne0\).
This is a construction of a visible representative, not a numerical
evaluation of \(b_{\EH}\).

The added functional is itself in the declared contact class.
Therefore, with its response denoted \(n_{\mathrm{extra}}\),
\[
[m-L_0n_{\mathrm{extra}}]=[m]
\quad\text{in }\RR^\XX/\CC.
\]
Increasing a selected representative's response cannot establish that
its quotient class is nonzero.
This is precisely why a selected-defect witness and a complete
contact separator are different objects.

%=====================================================================
\section*{Declarations and source provenance}
\addcontentsline{toc}{section}{Declarations and source provenance}

\paragraph{Nature of the work.}
This is an exact theoretical finite-certificate study.
No empirical dataset is used.
All action, domain, catalogue, and certificate definitions needed
for the finite result are included.

\paragraph{Reproducibility.}
The quantities \(16/3\), \(8\), \(64\), and \(2/3\) are derived
analytically.
The embedded checker is optional and is not a theorem premise.

\paragraph{Verification status.}
No successful \texttt{pdflatex} compilation, checker execution,
independent Einstein-vertex implementation, numerical integration,
or mechanized verification is claimed.
The source contains explanatory figures drawn from its displayed
objects; it does not claim recovery of unavailable companion figures.

\paragraph{Companion provenance.}
Companion references identify unpublished registered manuscripts or
collections of source records by their supplied identifiers.
They are not presented as journal articles with invented titles,
dates, or DOIs.
The comparison locators and summaries do not constitute independent
verification of those manuscripts.
No companion theorem enters the central proof.

\paragraph{Relation to the earlier manuscript.}
The principal finite certificate is unchanged from the supplied
version of manuscript II.
The revision clarifies the quotient, coverage, successor questions,
and non-transfer boundaries rather than claiming a new global
contact result.

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

\bibitem{FierzPauli}
M.~Fierz and W.~Pauli.
\newblock On relativistic wave equations for particles of arbitrary spin
in an electromagnetic field.
\newblock \emph{Proceedings of the Royal Society of London A},
173:211--232, 1939.
\newblock doi:\ \doi{10.1098/rspa.1939.0140}.

\bibitem{Deser}
S.~Deser.
\newblock Self-interaction and gauge invariance.
\newblock \emph{General Relativity and Gravitation}, 1:9--18, 1970.
\newblock doi:\ \doi{10.1007/BF00759198}.
\newblock Author's repost: arXiv:\nolinkurl{gr-qc/0411023}.

\bibitem{BarnichHenneaux}
G.~Barnich and M.~Henneaux.
\newblock Consistent couplings between fields with a gauge freedom
and deformations of the master equation.
\newblock \emph{Physics Letters B}, 311:123--129, 1993.
\newblock doi:\ \doi{10.1016/0370-2693(93)90544-R}.
\newblock arXiv:\nolinkurl{hep-th/9304057}.

\bibitem{BoulangerEtAl}
N.~Boulanger, T.~Damour, L.~Gualtieri, and M.~Henneaux.
\newblock Inconsistency of interacting, multi-graviton theories.
\newblock \emph{Nuclear Physics B}, 597:127--171, 2001.
\newblock doi:\ \doi{10.1016/S0550-3213(00)00718-5}.
\newblock arXiv:\nolinkurl{hep-th/0007220}.

\bibitem{Brauer}
R.~Brauer.
\newblock On algebras which are connected with the semisimple
continuous groups.
\newblock \emph{Annals of Mathematics}, 38(4):857--872, 1937.
\newblock doi:\ \doi{10.2307/1968843}.

\bibitem{CheungEnter}
N.~Cheung.
\newblock The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement.
\newblock \emph{Frontiers in Physics}, volume~14, 2026.
\newblock doi:\ \doi{10.3389/fphy.2026.1954943}.
\newblock Accepted-article record; accepted 28 September 2026.
The publisher listed the formatted version as forthcoming at the
bibliographic check on 10 October 2026.
\newblock
\url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/abstract}.

\bibitem{CompanionRegistry}
N.~Cheung.
\newblock \emph{Enter Calculus Companion}.
\newblock Registered-manuscript index and source-record interface.
\newblock \url{https://enter.grok.me/}.
\newblock Registry consulted 10 October 2026.
Individual companion references below are unpublished source records,
not independently machine-verified publications.

\bibitem{CheungI}
N.~Cheung.
\newblock \emph{I}, \SourceLabel{I.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied locator: \SourceLabel{thm:main}.
\newblock Finite measurement characterization in its declared class.

\bibitem{CheungIII}
N.~Cheung.
\newblock \emph{III}, \SourceLabel{III.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied comparison locators:
\SourceLabel{lem:closed-range} and \SourceLabel{thm:jet-descent}.
\newblock Its gauge-image and coupled-descent problems are not the
contact-matrix problem of this paper.

\bibitem{CheungIV}
N.~Cheung.
\newblock \emph{IV}, \SourceLabel{IV.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied locator: \SourceLabel{lem:Ramsey};
three separately declared represented-measurement models.
\newblock No coherent-matching interface for the present tree is
imported.

\bibitem{CheungV}
N.~Cheung.
\newblock \emph{V}, \SourceLabel{V.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied comparison locators:
\SourceLabel{thm:defect}, \SourceLabel{thm:residualwitness},
and \SourceLabel{lem:gaugeidentity}.
\newblock The raw current and scalar-response cubic have their own
definitions and are not the Einstein cubic used here.

\bibitem{CheungVI}
N.~Cheung.
\newblock \emph{VI}, registered manuscript.
\newblock Undated source record, Enter Calculus Companion.
\newblock Supplied comparison locators:
\SourceLabel{def:catalogue},
\SourceLabel{thm:finite-certificate}, and
\SourceLabel{cor:certificate-meaning}.
\newblock Cited for certificate scope, not as a premise or an
identification of quantum correction classes.

\bibitem{CheungVII}
N.~Cheung.
\newblock \emph{VII-H} and \emph{VII-I}, registered manuscripts.
\newblock Undated source records, Enter Calculus Companion.
\newblock Supplied VII-H locator: \SourceLabel{thm:bank}.
\newblock The connected-vacuum ambiguity and finite-width free algebra
are separate from the present contact catalogue.

\bibitem{CheungVIII}
N.~Cheung.
\newblock \emph{VIII-A} and \emph{VIII-B}, registered manuscripts.
\newblock Undated source records, Enter Calculus Companion.
\newblock Supplied comparison locators:
\SourceLabel{A:prop:fixed-certificate} and \SourceLabel{thm:finite}.
\newblock Their finite matching and packet domains are not \(\XX\).

\bibitem{CheungX}
N.~Cheung.
\newblock \emph{X-E}, \emph{X-F}, and \emph{X-G},
registered manuscripts.
\newblock Undated source records, Enter Calculus Companion.
\newblock Supplied comparison locators:
\SourceLabel{thm:ward}, \SourceLabel{thm:main},
\SourceLabel{thm:obstruction}, and \SourceLabel{prop:Dsigned}.
\newblock Selected tree responses, additional prescriptions, and
reciprocity tests are not identified with \(N_S\).

\bibitem{CheungXI}
N.~Cheung.
\newblock \emph{XI-G}, registered manuscript.
\newblock Undated source record, Enter Calculus Companion.
\newblock Supplied comparison locator: \SourceLabel{thm:criterion}.
\newblock Cited for the class-invariant formulation, not for transfer
of its obstruction.

\bibitem{CheungXII}
N.~Cheung.
\newblock \emph{XII-B} and \emph{XII-F}, registered manuscripts.
\newblock Undated source records, Enter Calculus Companion.
\newblock Supplied comparison locators:
\SourceLabel{lem:weighted}, \SourceLabel{lem:boundary},
\SourceLabel{thm:obstruction}, and \SourceLabel{lem:main-shell}.
\newblock Weighted-source and unchanged-observable statements have
their own hypothesis packages.

\bibitem{CheungXIII}
N.~Cheung.
\newblock \emph{XIII-E}, registered manuscript.
\newblock Undated source record, Enter Calculus Companion.
\newblock Cited for the supplied overlap-equation comparison;
no contact-space identification or unfinished limit is imported.

\bibitem{CheungXIVH}
N.~Cheung.
\newblock \emph{XIV-H}, \SourceLabel{XIV-H.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied locators:
\SourceLabel{prop:K5exact} and \SourceLabel{thm:H2}.
\newblock Its versioned H2 verdict is a documentary statement about
its internal-symbol task.

\bibitem{CheungXIVI}
N.~Cheung.
\newblock \emph{XIV-I}, \SourceLabel{XIV-I.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied locators:
\SourceLabel{thm:I2}, \SourceLabel{thm:I3},
and \SourceLabel{thm:release}.
\newblock The named internal amplitude and missing joint
characteristic map are distinct objects.

\bibitem{CheungXIVJ}
N.~Cheung.
\newblock \emph{XIV-J}, \SourceLabel{XIV-J.tex}, version~1.
\newblock Undated registered manuscript, Enter Calculus Companion.
\newblock Supplied locators:
\SourceLabel{prop:descent}, \SourceLabel{thm:reference},
\SourceLabel{thm:conditionalextension},
\SourceLabel{thm:J2}, and \SourceLabel{cor:noadmission}.
\newblock A partial supported-sector reference and conditional
extension interface do not supply the full joint map.

\bibitem{CheungEm01}
N.~Cheung.
\newblock \emph{Em01-A} and \emph{Em01-G},
registered source records.
\newblock Undated manuscripts, Enter Calculus Companion.
\newblock Supplied comparison locator in Em01-A:
\SourceLabel{prop:no-witness-certificate}.
\newblock An isolated contraction certificate is not a complete
classification or a contact column of this paper.

\end{thebibliography}

\end{document}