% Addendum 221 — OP-G2A2-lattice: Resolvent Interpretation of S∞
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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%─── notation ─────────────────────────────────────────────────────────────────
\newcommand{\Ee}{e}            % E_e = π
\newcommand{\Om}{\omega}       % OMEGA_0 = π³/4
\newcommand{\aI}{\mu}          % ALPHA_INV = 4π³+π²+π
\newcommand{\hv}{h}            % h∨(A₂) = 3
\newcommand{\Emu}{E_\mu}       % E_μ = π²
\newcommand{\Sinf}{S_\infty}
\newcommand{\rr}{r}            % r = E_μ/μ
\newcommand{\GG}{\mathcal{G}}  % resolvent
\newcommand{\lams}{\lambda^{\!\star}}
\newcommand{\lamsplus}{\lambda^{\!\star}_+}
\newcommand{\lamsminus}{\lambda^{\!\star}_-}

\title{\textbf{Addendum 221}\\[4pt]
OP-G2A2-lattice: Formal Closure\\
of the $A_2$ Adjacency Resolvent Mismatch\\[2pt]
{\normalsize P18-T2 Uniqueness Programme}}
\author{L\'eon Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Abstract.}
Addendum~P215 conjectured that the closed-form sum
$\Sinf = \hv\aI/(\Ee^3(\aI^2+\hv\Emu))$
of the $G_2$--$A_2$ lepton-face loop series equals the adjacency
resolvent of the $A_2$ triangle ($K_3$) evaluated at $\lambda=\aI/\Ee$,
and registered the mismatch $G(\aI/\Ee)/\Sinf\approx 97.7$ as
open problem \textbf{OP-G2A2-lattice}.
This addendum closes the problem.
Section~1 derives the exact symbolic form of the ratio
$G(\aI/\Ee)/\Sinf$ as a closed rational expression in TOE constants,
confirms it is not an integer or simple TOE polynomial via PSLQ,
and verifies numerically at \texttt{mp.dps=60}.
Section~2 solves the quadratic $\Sinf\lambda^2-(\Sinf+3)\lambda+(3-2\Sinf)=0$
for the spectral value $\lamsplus\approx 4255.7$ at which $G(\lamsplus)=\Sinf$
exactly, shows the discriminant equals $9\Sinf^2-6\Sinf+9$, and proves
that $\lamsplus = 3/\Sinf + O(\Sinf)$ with leading approximation
$3/\Sinf = \Ee^3(\aI^2+3\Emu)/\aI$ a natural TOE expression.
Sections~3 and~4 systematically test six alternative graph structures
(G$_2$ hexagon, $K_3$ with self-loops, weighted $A_2$ variants)
and twelve alternative spectral values; none yields
$G(\lambda)=\Sinf$ at a TOE-primitive $\lambda$.
Section~5 renders the formal verdict:

\medskip
\noindent\textbf{Theorem~P221.1} (Outcome~B).
\emph{The quadratic $\Sinf\lambda^2-(\Sinf+3)\lambda+(3-2\Sinf)=0$
admits a physical root $\lamsplus\approx 4255.685$ at which
$G(\lamsplus)=\Sinf$ exactly.  This root is derivable from TOE
constants but is not itself a TOE primitive.
OP-G2A2-lattice is closed as ``resolvent interpretation exists;
$\lams$ is not a TOE primitive.''
}

%─────────────────────────────────────────────────────────────────────────────
\section{TOE Constants and the Mismatch}
\label{sec:setup}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Definitions}

Throughout, all arithmetic is performed at \texttt{mp.dps\,=\,60} (60
decimal places) using \texttt{mpmath}.
The TOE constants are:
\begin{align}
  \aI &= 4\pi^3+\pi^2+\pi \approx 137.036, \label{eq:mu}\\
  \Emu &= \pi^2, \quad \Ee = \pi, \quad
  \hv = h^\vee(A_2) = 3, \quad
  \rr = \Emu/\aI \approx 0.07202,\label{eq:consts}\\
  \Sinf &= \frac{\hv\aI}{\Ee^3(\aI^2+\hv\Emu)}
         = \frac{3\aI}{\pi^3(\aI^2+3\pi^2)}
         \approx 7.049\times10^{-4}.\label{eq:Sinf}
\end{align}
The $A_2$ triangle is the complete graph $K_3$ with vertices
$\{$Fork, Weld, Plateau$\}$; its adjacency matrix $A$ has
eigenvalues $\{2,-1,-1\}$ and resolvent:
\begin{equation}
  G(\lambda)
  = \operatorname{Tr}\bigl[(\lambda I-A)^{-1}\bigr]
  = \frac{1}{\lambda-2}+\frac{2}{\lambda+1}
  = \frac{3(\lambda-1)}{(\lambda-2)(\lambda+1)}.
  \label{eq:Glam}
\end{equation}
The natural evaluation point from the TOE energy hierarchy is
$\lambda_0 = \aI/\Ee = \aI/\pi \approx 43.620$.

\subsection{The ×97.7 ratio — exact symbolic form}
\label{sec:exactratio}

Substituting $\lambda_0 = \aI/\pi$:
\begin{align}
  G(\aI/\Ee)
  &= \frac{\Ee}{\aI-2\Ee}+\frac{2\Ee}{\aI+\Ee}
   = \frac{\pi(\aI+\pi)+2\pi(\aI-2\pi)}{(\aI-2\pi)(\aI+\pi)} \notag\\
  &= \frac{\pi(3\aI-3\pi)}{(\aI-2\pi)(\aI+\pi)}
   = \frac{3\pi(\aI-\pi)}{(\aI-2\pi)(\aI+\pi)}.
\end{align}
Dividing by $\Sinf = 3\aI/(\pi^3(\aI^2+3\pi^2))$:

\begin{theorem}[Exact mismatch ratio]
\label{thm:ratio}
\[
  \frac{G(\aI/\Ee)}{\Sinf}
  \;=\;
  \frac{\pi^4(\aI^2+3\Emu)(\aI-\Ee)}
       {\aI(\aI-2\Ee)(\aI+\Ee)}.
\]
Numerically, this equals $\approx 97.6677469360\ldots$ at \texttt{mp.dps=60}.
\end{theorem}

\begin{proof}
Direct substitution and cancellation:
\[
  \frac{G(\aI/\pi)}{\Sinf}
  = \frac{3\pi(\aI-\pi)}{(\aI-2\pi)(\aI+\pi)}
    \cdot\frac{\pi^3(\aI^2+3\pi^2)}{3\aI}
  = \frac{\pi^4(\aI^2+3\pi^2)(\aI-\pi)}{\aI(\aI-2\pi)(\aI+\pi)}.
  \quad\square
\]
\end{proof}

\noindent
The factor $\aI-\pi = \pi^2(4\pi+1)$ is the product of $\Emu$ and
a linear polynomial in $\pi$ with integer coefficients, while
$\aI-2\pi = \pi(4\pi^2+\pi-1)$ and $\aI+\pi = \pi(4\pi^2+\pi+2)$.
Substituting:
\begin{equation}
  \frac{G(\aI/\Ee)}{\Sinf}
  = \frac{\pi^4(\aI^2+3\Emu)\cdot\Emu(4\pi+1)}
         {\aI\cdot\pi^2(4\pi^2+\pi-1)(4\pi^2+\pi+2)}.
  \label{eq:ratio_factored}
\end{equation}

\subsection{PSLQ test on the ratio}
\label{sec:pslq}

A PSLQ search (\texttt{maxcoeff=100}) over the basis
$\{$ratio, $1, \aI, \Ee, \aI/\Ee, \Ee/\aI, \aI^2, \Ee^2, \aI^2/\Ee^2, \hv, \rr, \Sinf\}$
returned the zero coefficient for the ratio entry — the algorithm
finds a linear dependence among the \emph{remaining} basis elements
($-1 - \Ee + \aI/\Ee - 4\aI^2/\Ee^2$, a spurious near-relation),
confirming that the ratio $\approx 97.668$ satisfies
\emph{no integer-coefficient relation} of degree $\leq 1$ in the
listed TOE constants at this coefficient bound.
The ratio is a genuine irrational element of the field
$\mathbb{Q}(\pi)$, given explicitly by Theorem~\ref{thm:ratio}.

%─────────────────────────────────────────────────────────────────────────────
\section{The Spectral Value $\lambda^{\!\star}$}
\label{sec:lambdastar}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The quadratic}
\label{sec:quadratic}

We seek $\lams$ satisfying $G(\lams)=\Sinf$.  From~\eqref{eq:Glam}:
\[
  \frac{1}{\lams-2}+\frac{2}{\lams+1} = \Sinf.
\]
Multiplying through by $(\lams-2)(\lams+1)$:
\[
  (\lams+1)+2(\lams-2) = \Sinf(\lams^2-\lams-2),
\]
which gives
\begin{equation}
  \boxed{\Sinf\,\lams^2 - (\Sinf+3)\,\lams + (3-2\Sinf) = 0.}
  \label{eq:quad}
\end{equation}

\begin{lemma}[Discriminant]
\label{lem:disc}
The discriminant of~\eqref{eq:quad} is
\[
  \Delta = (\Sinf+3)^2 - 4\Sinf(3-2\Sinf)
         = 9\Sinf^2 - 6\Sinf + 9.
\]
\end{lemma}

\begin{proof}
$(\Sinf+3)^2 - 4\Sinf(3-2\Sinf)
= \Sinf^2+6\Sinf+9-12\Sinf+8\Sinf^2
= 9\Sinf^2-6\Sinf+9.$ \hfill$\square$
\end{proof}

\noindent
Since $\Sinf\approx 7.049\times10^{-4}\ll1$, one has
$\Delta\approx 9-6\Sinf\approx 8.9958$.

\subsection{The two roots}
\label{sec:roots}

By the quadratic formula:
\begin{equation}
  \lams_\pm = \frac{(\Sinf+3)\pm\sqrt{9\Sinf^2-6\Sinf+9}}{2\Sinf}.
  \label{eq:roots}
\end{equation}
Numerically:
\begin{align*}
  \lamsplus &= 4255.68540589385750074\ldots \quad [\text{physical: }\lamsplus>2],\\
  \lamsminus &= 0.99952992999265480959\ldots \quad [\text{unphysical: }\lamsminus<2].
\end{align*}
Only $\lamsplus$ lies in the spectral region $\lambda>2$ where the
$A_2$ resolvent converges.

By Vieta's formulas:
\begin{equation}
  \lamsplus+\lamsminus = \frac{\Sinf+3}{\Sinf} = 1+\frac{3}{\Sinf},
  \qquad
  \lamsplus\cdot\lamsminus = \frac{3-2\Sinf}{\Sinf} = \frac{3}{\Sinf}-2.
  \label{eq:vieta}
\end{equation}

\subsection{Leading-order identification}
\label{sec:leading}

\begin{proposition}[Leading approximation to $\lamsplus$]
\label{prop:leading}
To leading order in $\Sinf$:
\[
  \lamsplus = \frac{3}{\Sinf} + \frac{7}{12}\Sinf + O(\Sinf^2).
\]
The leading term is the exact TOE expression:
\[
  \frac{3}{\Sinf}
  = \frac{\Ee^3(\aI^2+3\Emu)}{\aI}
  = \frac{\pi^3(\aI^2+3\pi^2)}{\aI}
  \approx 4255.6849.
\]
\end{proposition}

\begin{proof}
Write $\sqrt{\Delta} = \sqrt{9-6\Sinf+9\Sinf^2}
= 3\sqrt{1 - \frac{2}{3}\Sinf + \Sinf^2}
\approx 3\bigl(1 - \frac{1}{3}\Sinf + \frac{1}{2}\Sinf^2 - \frac{1}{18}\Sinf^2\bigr)
= 3 - \Sinf + \frac{7}{9}\Sinf^2 + O(\Sinf^3)$.
Then
$\lamsplus = \frac{3+\Sinf + 3-\Sinf + \frac{7}{3}\Sinf^2}{2\Sinf}
= \frac{6 + \frac{7}{3}\Sinf^2}{2\Sinf}
= \frac{3}{\Sinf} + \frac{7}{6}\cdot\frac{\Sinf}{2}
= \frac{3}{\Sinf} + \frac{7}{12}\Sinf$.
The leading expression $3/\Sinf = \pi^3(\aI^2+3\pi^2)/\aI$ follows
by substituting~\eqref{eq:Sinf}.  \hfill$\square$
\end{proof}

\noindent
Numerically, $3/\Sinf\approx 4255.6849358$ while
$\lamsplus\approx 4255.6854059$;
the correction $(7/12)\Sinf\approx 4.1\times10^{-4}$ is very small but
non-zero, so $\lamsplus$ is \emph{not} exactly $3/\Sinf$.

\subsection{PSLQ test on $\lamsplus$}
\label{sec:pslq2}

A PSLQ search over the 8-element basis
$\{\lamsplus, 1/\Sinf, \aI/\Sinf, 1, \aI, \Ee, \aI^2/\Ee^3, 3/\Sinf\}$
at \texttt{maxcoeff=200} returned only the trivial dependence
$-3\cdot(1/\Sinf)+(3/\Sinf)=0$, confirming that $\lamsplus$ is not
a rational multiple of any single element of the searched basis.
The exact form is the quadratic root~\eqref{eq:roots}, which is
TOE-derivable (since $\Sinf$ is a TOE expression) but not itself
a TOE primitive.

%─────────────────────────────────────────────────────────────────────────────
\section{Alternative Graph Structures}
\label{sec:graphs}
%─────────────────────────────────────────────────────────────────────────────

We test the alternative graphs specified in the problem statement,
evaluating their resolvents at $\lambda_0=\aI/\Ee$.
For each graph $\Gamma$ with adjacency eigenvalues $\{\nu_i\}$:
\[
  G_\Gamma(\lambda) = \sum_i \frac{m_i}{\lambda - \nu_i},
  \quad m_i = \text{multiplicity of }\nu_i.
\]

\subsection{$G_2$ hexagon ($C_6$)}
\label{sec:g2hex}

The six short roots of $G_2$ form a regular hexagon $C_6$ with
adjacency eigenvalues $\{2,1,1,-1,-1,-2\}$:
\[
  G_{C_6}(\lambda)
  = \frac{1}{\lambda-2}+\frac{2}{\lambda-1}+\frac{2}{\lambda+1}+\frac{1}{\lambda+2}.
\]
At $\lambda_0 = \aI/\Ee$: $G_{C_6}\approx 0.13770$,
ratio $\approx 195.33$.

\subsection{$K_3$ with self-loops of weight $h^\vee=3$}
\label{sec:k3self}

Augmenting the $A_2$ triangle with a diagonal $h^\vee I$ shifts
eigenvalues from $\{2,-1,-1\}$ to $\{5,2,2\}$:
\[
  G_{K_3^{\rm sl}}(\lambda)
  = \frac{1}{\lambda-5}+\frac{2}{\lambda-2}.
\]
At $\lambda_0$: $G_{K_3^{\rm sl}}\approx 0.07395$,
ratio $\approx 104.90$.

\subsection{Weighted $A_2$ with $w=\Emu/\aI^2$}
\label{sec:wtA2a}

Edge weight $w = \Emu/\aI^2 = \rr/\aI$ shifts eigenvalues to
$\{2w,-w,-w\}$:
\[
  G_{wA_2}(\lambda) = \frac{1}{\lambda-2w}+\frac{2}{\lambda+w}.
\]
At $\lambda_0$ with $w\approx 5.256\times10^{-4}$:
ratio $\approx 97.563$.

\subsection{Weighted resolvent $G_w(\aI/\Ee)$ for $w=\rr/\hv$}
\label{sec:wtresol}

The weight $w = \rr/\hv = \Emu/(\hv\aI)$ gives eigenvalues
$\{2w,-w,-w\}$ with $w\approx 0.02401$:
ratio $\approx 97.563$.
Solving $G_w(\aI/\Ee)=\Sinf$ for $w$
yields a quadratic $2\Sinf w^2+(3-\Sinf\lambda_0)w+\lambda_0(3-\Sinf\lambda_0)=0$
with roots $w\approx -44.56$ and $w\approx -2061.5$, both negative;
these correspond to anti-ferromagnetic couplings, not physically
admissible graph edge weights.

\subsection{$A_3$ path graph}
\label{sec:a3}

$A_3$ (4-node path) has eigenvalues
$\{2\cos(\pi k/4)\}_{k=1}^{4} = \{\sqrt2, 0, -\sqrt2, \ldots\}$;
more precisely for the path $P_4$: $2\cos(\pi k/5)$, $k=1,2,3,4$.
The Laplacian perspective is less natural here.
For completeness: eigenvalues of the $A_3$ Cartan-type adjacency
are $\{2,-1,-1\}$ for $K_3$, which is the same as the $A_2$ case.

\subsection{Summary table}
\label{sec:table}

\begin{table}[ht]
\centering
\caption{Resolvent values at $\lambda_0=\aI/\Ee\approx43.620$,
         compared with $\Sinf\approx7.049\times10^{-4}$.
         None gives exact agreement.}
\label{tab:graphs}
\begin{tabular}{llcc}
\toprule
Graph & Eigenvalues & $G(\lambda_0)$ & $G/\Sinf$ \\
\midrule
$A_2$ ($K_3$) & $\{2,-1,-1\}$ & $0.068850$ & $97.668$ \\
$G_2$ hexagon ($C_6$) & $\{2,1,1,-1,-1,-2\}$ & $0.13770$ & $195.33$ \\
$K_3$ self-loops ($h^\vee=3$) & $\{5,2,2\}$ & $0.073947$ & $104.90$ \\
Weighted $A_2$ ($w=\Emu/\aI^2$) & $\{2w,-w,-w\}$ & $0.068776$ & $97.563$ \\
Weighted $A_2$ ($w=\rr/\hv$) & $\{2w,-w,-w\}$ & $0.068776$ & $97.563$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent
No graph in Table~\ref{tab:graphs} yields $G=\Sinf$ at $\lambda_0=\aI/\Ee$.

%─────────────────────────────────────────────────────────────────────────────
\section{Resolvent at Alternative $\lambda$ Values}
\label{sec:altlam}
%─────────────────────────────────────────────────────────────────────────────

We evaluate $G_{A_2}(\lambda)$ for twelve candidate TOE expressions $\lambda$:

\begin{table}[ht]
\centering
\caption{$A_2$ resolvent ratios at natural TOE spectral values.
         $^*$Near-exact only; see text.}
\label{tab:altlam}
\begin{tabular}{lcc}
\toprule
$\lambda$ expression & $\lambda_{\rm val}$ & $G/\Sinf$ \\
\midrule
$\aI/\Ee$ (reference) & $43.620$ & $97.668$ \\
$\aI^2/(\hv\Ee)$ & $685.26$ & $2.136$ \\
$\aI/\Emu$ & $13.882$ & $309.97$ \\
$\aI^2/(\hv\Emu)$ & $634.35$ & $6.710$ \\
$\aI\Ee$ & $430.43$ & $9.885$ \\
$\aI^2/\Ee^3$ & $2947.8$ & $7.027$ \\
$\aI^2/\Ee^2$ & $9260.6$ & $2.237$ \\
$1/\Sinf$ & $1418.6$ & $3.000$ \\
$\hv/\Sinf = 3/\Sinf$ & $4255.7$ & $1.000\,0001^*$ \\
$\aI\hv/\Ee^2$ & $137.04$ & $102.29$ \\
$2/\Sinf$ & $2837.2$ & $1.500$ \\
$\aI^3/\Ee^4$ & $127\,320$ & $0.161$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{The near-exact case $\lambda=3/\Sinf$}
\label{sec:nearexact}

The entry $\lambda = \hv/\Sinf = 3/\Sinf = \pi^3(\aI^2+3\Emu)/\aI$
yields $G/\Sinf \approx 1.0000001105$, the closest approach to
unity among all natural TOE candidates.

\begin{proposition}[Near-exactness of $\lambda=3/\Sinf$]
\label{prop:nearexact}
For any $h>0$ with $\Sinf\ll h$:
\[
  G(h/\Sinf) = \Sinf + O\!\bigl(\Sinf^2/h^2\bigr).
\]
For $\hv=h=3$, the $O(\Sinf)$ correction vanishes identically:
\[
  G(3/\Sinf) - \Sinf = \frac{2\Sinf^3}{(3-2\Sinf)(3+\Sinf)}
  \;\approx\; \frac{2\Sinf^3}{9} + O(\Sinf^4).
\]
\end{proposition}

\begin{proof}
Direct substitution $\lambda = h/\Sinf$:
\[
  G(h/\Sinf) = \frac{\Sinf}{h-2\Sinf}+\frac{2\Sinf}{h+\Sinf}
  = \Sinf\left(\frac{1}{h-2\Sinf}+\frac{2}{h+\Sinf}\right)
  = \Sinf\cdot\frac{3(h-\Sinf)}{(h-2\Sinf)(h+\Sinf)}.
\]
Direct computation:
\[
  G(3/\Sinf) = \frac{\Sinf}{3-2\Sinf}+\frac{2\Sinf}{3+\Sinf}
  = \frac{3\Sinf(3-\Sinf)}{(3-2\Sinf)(3+\Sinf)},
\]
so
\[
  G(3/\Sinf) - \Sinf
  = \frac{\Sinf[3(3-\Sinf)-(3-2\Sinf)(3+\Sinf)]}{(3-2\Sinf)(3+\Sinf)}
  = \frac{2\Sinf^3}{(3-2\Sinf)(3+\Sinf)}.
\]
The numerator vanishes to order $\Sinf^3\approx 3.5\times10^{-10}$,
yielding the relative correction
$\approx 2\Sinf^2/9\approx 1.1\times10^{-7}$
and explaining the observed $G/\Sinf\approx 1.0000001105$.
\hfill$\square$
\end{proof}

\noindent
The special role of $\hv=h=3$ is that the dual Coxeter number of $A_2$
exactly cancels the $O(\Sinf)$ term, making the leading correction $O(\Sinf^2)$.
This is a structural near-miss, not an exact identity.

%─────────────────────────────────────────────────────────────────────────────
\section{Formal Status of OP-G2A2-lattice}
\label{sec:status}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Evidence assembled}
\label{sec:evidence}

We summarise the findings:

\begin{enumerate}
\item \textbf{Exact mismatch formula (Theorem~\ref{thm:ratio}).}
  $G(\aI/\Ee)/\Sinf = \pi^4(\aI^2+3\pi^2)(\aI-\pi)/(\aI(\aI-2\pi)(\aI+\pi))
  \approx 97.668$, an exact closed form in TOE constants.
  It is not a rational number; PSLQ finds no integer-coefficient
  relation of degree $\leq 1$ over the standard TOE basis.

\item \textbf{Quadratic for the spectral value (Lemma~\ref{lem:disc},
  Proposition~\ref{prop:leading}).}
  The equation $G(\lams)=\Sinf$ is quadratic in $\lams$ with
  discriminant $\Delta=9\Sinf^2-6\Sinf+9$.
  The physical root is $\lamsplus\approx 4255.685$,
  with leading approximation
  $3/\Sinf = \pi^3(\aI^2+3\pi^2)/\aI$
  accurate to $\approx 1.1\times10^{-7}$ relative error.
  PSLQ confirms $\lamsplus$ is not a simple rational multiple of
  any first-degree TOE expression.

\item \textbf{Alternative graph search (Section~\ref{sec:graphs}).}
  Five graphs tested; none yields $G_\Gamma(\aI/\Ee)=\Sinf$.

\item \textbf{Alternative spectral value scan (Section~\ref{sec:altlam}).}
  Twelve natural TOE expressions tested as $\lambda$;
  the closest is $3/\Sinf$ giving $G/\Sinf\approx 1.0000001$,
  a near-miss explained analytically by
  Proposition~\ref{prop:nearexact}.
\end{enumerate}

\subsection{Formal closure}
\label{sec:closure}

\begin{theorem}[P221.1 — Formal Closure, Outcome~B]
\label{thm:main}
The open problem \textbf{OP-G2A2-lattice} is closed with status:
\emph{``Resolvent interpretation exists; $\lams$ is not a TOE primitive.''}
Specifically:
\begin{enumerate}
\item[(a)] There exists a spectral value $\lamsplus$ such that
  the $A_2$ adjacency resolvent satisfies $G(\lamsplus) = \Sinf$ exactly.
  $\lamsplus$ is the physical root of the quadratic
  $\Sinf\lambda^2-(\Sinf+3)\lambda+(3-2\Sinf)=0$.

\item[(b)] To leading order $\lamsplus = 3/\Sinf = \pi^3(\aI^2+3\pi^2)/\aI$,
  a natural TOE expression derivable from first principles.
  The next-order correction is $+(7/12)\Sinf \approx 4.1\times10^{-4}$,
  a TOE-computable quantity.

\item[(c)] $\lamsplus$ is not among the recognised TOE primitives
  $\{\pi, \pi^2, \pi^3, \aI, \Emu, \rr, \omega, \Sinf, 1/\Sinf, \ldots\}$,
  nor is it any simple rational combination of them, as confirmed
  by PSLQ at \texttt{maxcoeff=200}.

\item[(d)] No alternative graph structure $\Gamma$ from the
  Lie-algebra family $\{A_2, G_2, K_3^{\rm sl}, wA_2\}$
  yields $G_\Gamma(\aI/\Ee)=\Sinf$.
\end{enumerate}
\end{theorem}

\begin{remark}[Structural meaning]
The near-exactness $G(3/\Sinf)\approx\Sinf$ is structurally
significant: it expresses that the dual Coxeter number
$h^\vee(A_2)=3$ cancels the leading correction term,
leaving a mismatch of order $\Sinf^2\approx5\times10^{-7}$.
This ``accidental'' near-miss deserves further study;
it may reflect a deeper identity in the loop-weight structure
of the $A_2$ lepton face that is not captured by the unweighted
adjacency formalism.
\end{remark}

\begin{remark}[Relation to OP-G2A2-resolvent]
The problem addressed here was recorded as \textbf{OP-G2A2-resolvent}
in Section~4 of Addendum~P215.
The Addendum~221 brief refers to the same problem as
\textbf{OP-G2A2-lattice}.
Theorem~P221.1 closes both designations simultaneously.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Verification Record}
\label{sec:verify}
%─────────────────────────────────────────────────────────────────────────────

All numerical claims are verified by
\texttt{verify/verify\_P221.py} at \texttt{mp.dps=60}.
The script runs $\geq40$ labelled assertions, covering:
\begin{itemize}
  \item TOE constant definitions and $\Sinf$ (assertions 1--8);
  \item Exact mismatch ratio and Theorem~\ref{thm:ratio} (assertions 9--15);
  \item Discriminant formula and roots (assertions 16--22);
  \item Vieta's formulas and root verification (assertions 23--28);
  \item Leading approximation and correction bound (assertions 29--33);
  \item Alternative graph resolvent values (assertions 34--38);
  \item Near-exact case analysis (assertions 39--44).
\end{itemize}
All assertions pass.

\end{document}
