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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{\textbf{Addendum 215: OP-G$_2$A$_2$-lattice\\
Lattice Path Derivation of $T_k$:\\
Closed Walks on the $A_2$ Lepton Face}}
\author{L\'{e}on Fernando Vlegels}
\date{18 May 2026}

%─── notation ─────────────────────────────────────────────────────────────────
\newcommand{\Ee}{e}         % E_e = π (electron sector)
\newcommand{\Om}{\omega}    % OMEGA_0 = π³/4
\newcommand{\aI}{\mu}       % ALPHA_INV = 4π³+π²+π
\newcommand{\hv}{h}         % h-dual(A2) = 3
\newcommand{\Emu}{E_\mu}    % E_mu = π² = e²
\newcommand{\Rtarg}{206.7682830}
\newcommand{\ppm}{\,\text{ppm}}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Context.}
Addenda P213 and P214 established the $G_2$--$A_2$ mass-ratio series:
\begin{equation}
  \label{eq:Rfull}
  R = 207\!\left(1 - \frac{\Om}{\Ee^3\aI}
      + \sum_{k=1}^{\infty}(-1)^{k+1}\,T_k\right),
\end{equation}
where $T_k = (-1)^{k+1}\hv^k\Ee^{2k-5}/\aI^{2k-1}$, with
$\Ee = \pi$, $\Om = \pi^3/4$, $\aI = 4\pi^3+\pi^2+\pi = h^\vee(A_2)^{-1}$
in natural TOE units, and $\hv = h^\vee(A_2) = 3$.
The geometric ratio was proved in P214:
$T_{k+1}/T_k = -r$, $r = \hv\Ee^2/\aI^2 \approx 1.577\times10^{-3}$.

This addendum addresses open problem \textbf{OP-G2A2-lattice}: derive $T_k$
as the $k$-th loop correction from the $A_2$ lepton-face geometry, establishing
that the factors $\hv^k$ arise from closed walks on the $A_2$ triangle.

%─────────────────────────────────────────────────────────────────────────────
\section{Path Counting on the $A_2$ Triangle}
\label{sec:paths}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The lepton face as $K_3$}

The $A_2$ lepton face is a triangle with vertices
$\{v_0 = \text{Fork},\; v_1 = \text{Weld},\; v_2 = \text{Plateau}\}$,
each pair connected by an edge of weight $\Ee = \pi$ (the 4-simplex edge
length from P202).  As a graph the lepton face is the complete graph $K_3$;
its unweighted adjacency matrix is:
\[
  A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}.
\]
The eigenvalues of $A$ are $2$ (multiplicity 1) and $-1$ (multiplicity 2),
giving the spectral trace formula:
\begin{equation}
  \label{eq:trace2k}
  \operatorname{Tr}(A^{2k}) = 2^{2k} + 2\cdot(-1)^{2k} = 4^k + 2.
\end{equation}

\subsection{Unoriented 2-step returns and the Coxeter number}

A \emph{2-step return} at vertex $v$ is a walk $v \to w \to v$
for some neighbour $w$.  Each vertex of $K_3$ has degree 2, so there
are exactly 2 such walks per vertex, giving $\operatorname{Tr}(A^2) = 6$.
Dividing by 2 to pass to unoriented loops:
\[
  \frac{\operatorname{Tr}(A^2)}{2} = 3 = h^\vee(A_2) = \hv. \quad\checkmark
\]

\subsection{Higher-order traces}

From \eqref{eq:trace2k}:
\begin{equation}
  \label{eq:trace_table}
  \frac{\operatorname{Tr}(A^{2k})}{2} = \frac{4^k + 2}{2},
  \qquad k = 1, 2, 3, \ldots
\end{equation}
Comparing with $\hv^k = 3^k$:

\begin{table}[ht]
\centering
\caption{Adjacency-matrix traces vs.\ Coxeter powers.}
\label{tab:traces}
\begin{tabular}{ccccc}
\toprule
$k$ & $4^k+2$ & $\operatorname{Tr}(A^{2k})/2$ & $\hv^k = 3^k$ & Match? \\
\midrule
1 & 6  & 3  & 3  & \checkmark \\
2 & 18 & 9  & 9  & \checkmark \\
3 & 66 & 33 & 27 & $\times$ \\
4 & 258 & 129 & 81 & $\times$ \\
5 & 1026 & 513 & 243 & $\times$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}
The unoriented trace $\operatorname{Tr}(A^{2k})/2$ equals $\hv^k$ only for
$k = 1, 2$.  For $k \geq 3$ the adjacency-matrix count grows as $4^k/2$,
which is exponentially larger than $3^k$.  The $T_k$ formula with \emph{exact}
$\hv^k$ cannot, therefore, be derived from the unweighted path count alone.
This is the central difficulty of OP-G2A2-lattice.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Modified Path Count: The Loop-Weight Factorisation}
\label{sec:loop}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Factorisation of $T_k$}

The resolution to the overcounting above lies in a different
factorisation of $T_k$.  Define the \emph{bare propagator seed}:
\begin{equation}
  \label{eq:seed}
  \sigma \;=\; \frac{\aI}{\Ee^5}
           \;=\; \frac{4\pi^3+\pi^2+\pi}{\pi^5}
           \;=\; \frac{4}{\pi^2} + \frac{1}{\pi^3} + \frac{1}{\pi^4}
           \;\approx\; 0.44780\ldots
\end{equation}
and the \emph{one-loop weight}:
\begin{equation}
  \label{eq:loopweight}
  r \;=\; \frac{\hv\,\Ee^2}{\aI^2} \;=\; \frac{3\pi^2}{\aI^2}
     \;\approx\; 1.5767\times10^{-3}
     \;\equiv\; \hv^\vee(A_2)\cdot\alpha^2\cdot\Ee^2,
\end{equation}
where $\alpha = 1/\aI$ is the fine-structure constant.

\begin{lemma}[Seed factorisation]
  \label{lem:seed}
  For all $k \geq 1$:
  \[
    T_k \;=\; (-1)^{k+1}\cdot r^k \cdot \sigma
         \;=\; (-1)^{k+1}\cdot\left(\frac{\hv\,\Ee^2}{\aI^2}\right)^{\!k}
               \cdot\frac{\aI}{\Ee^5}.
  \]
\end{lemma}

\begin{proof}
  Expand:
  \[
    (-1)^{k+1}\cdot r^k\cdot\sigma
    = (-1)^{k+1}\cdot\frac{\hv^k\,\Ee^{2k}}{\aI^{2k}}
      \cdot\frac{\aI}{\Ee^5}
    = (-1)^{k+1}\cdot\frac{\hv^k\,\Ee^{2k-5}}{\aI^{2k-1}} = T_k. \quad\square
  \]
\end{proof}

\subsection{Interpretation}

Lemma~\ref{lem:seed} expresses $T_k$ as a product of three factors:
\begin{enumerate}
  \item \textbf{Sign} $(-1)^{k+1}$: alternating from P213's geometric ratio.
  \item \textbf{Loop weight} $r^k$: each power of $r$ contributes one factor
        of $\hv = h^\vee(A_2) = 3$ (the $A_2$ Coxeter number), one factor
        of $\Ee^2$ (the squared electron-sector energy), and two inverse
        powers of $\aI$.
  \item \textbf{Seed} $\sigma = \aI/\Ee^5$: the bare amplitude, independent
        of $k$; it encodes the overall scale of the series.
\end{enumerate}

The loop weight $r = h^\vee(A_2)\cdot\alpha^2\cdot\Ee^2$ is a product of:
the $A_2$ Coxeter multiplicity ($h^\vee = 3$), one factor of $\alpha^2 = 1/\aI^2$
(fine-structure loop suppression), and one factor of $\Ee^2 = \pi^2$ (muon
energy insertion from the lepton face, $\Emu = \Ee^2$).

The \emph{exact} factor $\hv^k$ in $T_k$ thus arises not from counting
$k$-step paths on the $A_2$ triangle (which would give $(4^k+2)/2$ as shown
in Table~\ref{tab:traces}), but from $k$ independent applications of the
one-loop weight $r$, each carrying one factor of $\hv^\vee(A_2)$.

\subsection{Where does the $\hv^k$ come from at $k \geq 3$?}

The overcounting problem (Table~\ref{tab:traces}) resolves once we recognise
that the correct loop counting is not over all closed walks in $K_3$, but
over a \emph{projected} count: only walks that return to the \emph{same vertex}
via the single-edge loop $v \to w \to v$ contribute to $T_k$.  Such walks
have weight $\Ee^2$ each (one edge out, one back), and per vertex there is
one unoriented such loop; the $A_2$ face has $h^\vee(A_2) = 3$ vertices,
giving one factor of $\hv$ per loop order.  At loop order $k$ one multiplies
$k$ independent such factors, yielding $\hv^k$ — not the trace of $A^{2k}$
(which includes cross-vertex paths) but the $k$-fold product of the
single-vertex loop count $\hv$.

This single-vertex projection is a hypothesis, not a derived fact from
the $G_2$ holonomy integral.  It is stated as Conjecture~\ref{conj:loop}
below.

%─────────────────────────────────────────────────────────────────────────────
\section{Generating Function and Resolvent}
\label{sec:genfun}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The generating function of $T_k$}

From Lemma~\ref{lem:seed}, the infinite sum over the $T_k$ is:
\begin{equation}
  \label{eq:Sclosed}
  S = \sum_{k=1}^{\infty} T_k
    = \sigma \sum_{k=1}^{\infty}(-r)^k \cdot (-1)
    = \sigma \cdot \frac{r}{1+r}
    = \frac{\aI}{\Ee^5}\cdot\frac{\hv\Ee^2/\aI^2}{1+\hv\Ee^2/\aI^2}
    = \frac{\hv\,\aI}{\Ee^3(\aI^2+\hv\Ee^2)}.
\end{equation}
This is the closed form from P213.  The generating function of the
normalised series $\tilde T_k = T_k/\sigma = (-1)^{k+1}(-r)^k$ is:
\[
  \mathcal{G}(x) = \sum_{k=1}^{\infty}\tilde T_k\,x^k
                 = \sum_{k=1}^{\infty}(-1)^{k+1}(-r)^k x^k
                 = \frac{rx}{1+rx},
\]
evaluated at $x=1$ to give $r/(1+r)$, confirming \eqref{eq:Sclosed}.

\subsection{Comparison with the $A_2$ adjacency resolvent}

A natural candidate for the generating function is the resolvent of
the $A_2$ adjacency matrix $A$.  For $K_3$ with eigenvalues $2,-1,-1$:
\[
  G(\lambda) = \operatorname{Tr}\bigl[(\lambda I - A)^{-1}\bigr]
             = \frac{1}{\lambda-2} + \frac{2}{\lambda+1}
             = \frac{3(\lambda-1)}{(\lambda-2)(\lambda+1)}.
\]
Evaluating at the most natural energy scale $\lambda = \aI/\Ee$
(the fine-structure constant inverted in units of $\Ee$):
\begin{align*}
  G(\aI/\Ee) &= \frac{\Ee}{\aI-2\Ee} + \frac{2\Ee}{\aI+\Ee}
             = \frac{3\Ee(\aI-\Ee)}{\aI^2-\aI\Ee-2\Ee^2}
             \approx 0.02192\ldots
\end{align*}
while the closed-form sum satisfies:
\[
  S = \frac{3\aI}{\Ee^3(\aI^2+3\Ee^2)} \approx 0.000705\ldots
\]
The ratio $G(\aI/\Ee)/S \approx 97.7$, showing that the unweighted
adjacency resolvent does \emph{not} reproduce $S$ at $\lambda = \aI/\Ee$.

\begin{remark}[Resolvent mismatch]
Solving for the value of $\lambda$ at which $G(\lambda) = S$ requires
solving a quadratic with solutions
$\lambda \approx 4255.7$ and $\lambda \approx 1.000$, neither of which
is a natural TOE energy scale.  The adjacency-resolvent interpretation
of $S$ is therefore not established; see open problem \textbf{OP-G2A2-resolvent}
in Section~\ref{sec:status}.
\end{remark}

\subsection{Scalar resolvent and the geometric series}

The series $S$ \emph{is} the resolvent of the scalar operator
$\hat R = -r$ at $z = 1$:
\[
  S = \sigma\cdot\operatorname{Tr}\bigl[(z + r)^{-1}\bigr]_{z=1}\cdot r
    = \frac{\sigma\,r}{1+r},
\]
where the ``trace'' is trivially 1 in the 1-dimensional space spanned
by the normalised series.  The A₂ information enters only via the
eigenvalue $-r = -\hv\Ee^2/\aI^2$ of this scalar operator; the
multiplicities from the full $K_3$ adjacency spectrum are not used.

%─────────────────────────────────────────────────────────────────────────────
\section{Loop-Order Interpretation}
\label{sec:looporder}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Loop suppression structure}

In the TOE framework, each loop order carries a suppression factor
$\alpha^2 = 1/\aI^2$ and an energy insertion from the relevant
Lie-algebra face.  For the $A_2$ lepton face, the muon energy insertion
is $\Emu = \Ee^2 = \pi^2$ (the squared electron-sector energy, from
the edge metric of P202).

The one-loop weight is therefore:
\[
  r = h^\vee(A_2)\cdot\alpha^2\cdot\Emu
    = \underbrace{3}_{\text{Coxeter mult.}}
      \times\underbrace{\alpha^2}_{\text{loop supp.}}
      \times\underbrace{\pi^2}_{\text{energy ins.}}
\]
At loop order $k$, $k$ independent insertions give weight $r^k$,
and the bare amplitude $\sigma = \aI/\Ee^5$ sets the overall scale
(encoding the relationship between the muon propagator and the
lepton-face geometry established in P208).

\subsection{The normalised ratio}

From Lemma~\ref{lem:seed}, one has for all $k \geq 1$:
\begin{equation}
  \frac{T_k\,\Ee^3\,\aI}{\hv} = (-1)^{k+1}(-r)^{k-1},
\end{equation}
which is precisely the $(k-1)$-th power of the one-loop weight,
with the alternating sign from the geometric series.  This confirms
that consecutive $T_k$ terms are related by a single factor of $-r$,
as proved algebraically in P214.

\begin{conjecture}[Loop-order lattice, P215.1]
  \label{conj:loop}
  The correction term $T_k$ is the $k$-th loop contribution from
  closed walks of formal length $2k$ on the $A_2$ lepton face,
  with each walk carrying weight:
  \begin{enumerate}
    \item[\textup{(i)}] one factor of $h^\vee(A_2) = 3$ per loop vertex
          (the Coxeter multiplicity of the $A_2$ triangle);
    \item[\textup{(ii)}] one factor of $\Emu = \Ee^2$ per loop
          (the squared electron-sector energy on each traversed edge);
    \item[\textup{(iii)}] one fine-structure suppression $\alpha^2 = 1/\aI^2$
          per loop (the $G_2$ holonomy weight per loop order);
    \item[\textup{(iv)}] a $k$-independent seed $\sigma = \aI/\Ee^5$
          from the bare lepton-face propagator.
  \end{enumerate}
  Under this accounting, $T_k = (-1)^{k+1}\cdot r^k\cdot\sigma$
  with $r = h^\vee(A_2)\cdot\alpha^2\cdot\Emu$.
  Proof requires an explicit computation of the $k$-th order $G_2$
  holonomy integral over the $A_2$ face (open: \textbf{OP-G2A2-lattice}).
\end{conjecture}

\begin{remark}
  The Conjecture accounts for the exact factors $\hv^k$ by assigning
  one power of $\hv = h^\vee(A_2)$ per loop order, rather than counting
  all $2k$-step closed walks (which gives $(4^k+2)/2$; see Table~\ref{tab:traces}).
  The physical distinction is:
  \begin{itemize}
    \item \textbf{Path count} (Section~\ref{sec:paths}): counts all
          distinct $2k$-step closed walks in $K_3$, including walks
          that traverse different edges.  Gives $(4^k+2)/2$.
    \item \textbf{Loop weight} (Conjecture~\ref{conj:loop}): counts
          only the Coxeter multiplicity per loop order — one factor of
          $\hv$ per independent loop, as in the $A_2$ root-system contribution
          to the $G_2$ spectral integral.  Gives $\hv^k = 3^k$.
  \end{itemize}
  Establishing which is correct requires the $G_2$ holonomy integral of P198
  to be computed at each loop order explicitly.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Numerical Verification}
\label{sec:numerical}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Term-by-term table}

The following table verifies the formula $T_k = (-1)^{k+1}\hv^k\Ee^{2k-5}/\aI^{2k-1}$
and the convergence of $R_N = 207(1 + T_0 + \sum_{k=1}^N T_k)$ toward
$R_\infty$ at \texttt{mp.dps=60}.  The CODATA-2018 target is $T = \Rtarg$;
recall $T_0 = -\Om/(\Ee^3\aI) = -1/(4\aI)$.

\begin{table}[ht]
\centering
\caption{Partial-sum convergence of the $G_2$--$A_2$ series
(\texttt{mp.dps=60}, $h = h^\vee(A_2) = 3$).}
\label{tab:convergence}
\begin{tabular}{clllr}
\toprule
$k$ & $T_k$ & $T_{k+1}/T_k$ & $R_N \approx$ & Gap to $T$ (ppm) \\
\midrule
1 & $+7.0605\times10^{-4}$ & $-$ & $206.768\,515\,4$ & $1.1239$ \\
2 & $-1.1132\times10^{-6}$ & $-1.5767\times10^{-3}$ & $206.768\,284\,9$ & $0.009373$ \\
3 & $+1.7552\times10^{-9}$ & $-1.5767\times10^{-3}$ & $206.768\,285\,3$ & $0.011130$ \\
4 & $-2.7675\times10^{-12}$ & $-1.5767\times10^{-3}$ & $206.768\,285\,3$ & $0.011127$ \\
5 & $+4.3635\times10^{-15}$ & $-1.5767\times10^{-3}$ & $206.768\,285\,3$ & $0.011127$ \\
\bottomrule
\end{tabular}
\end{table}

The ratio $T_{k+1}/T_k = -r = -3\pi^2/\aI^2$ is constant to within
$10^{-63}$ from $k=2$ onward (algebraically exact from P214), confirming
the geometric series structure.  The partial sum $R_N$ converges to
$R_\infty \approx 206.768\,285\,3$ (gap $0.01113\ppm$) by $N = 3$;
all further terms are too small to shift the displayed digits.

\subsection{Adjacency matrix eigenvalue verification}

The eigenvalues of the $A_2$ triangle adjacency matrix $A$ are $2$ (once)
and $-1$ (twice), giving:
\[
  \operatorname{Tr}(A^{2k}) = 2^{2k} + 2\cdot1^{2k} = 4^k + 2,
\]
verified for $k = 1, 2, 3$ in \texttt{verify\_P215.py}:
\begin{align*}
  k=1: &\quad\operatorname{Tr}(A^2)  = 6  = 4^1+2 \checkmark \\
  k=2: &\quad\operatorname{Tr}(A^4)  = 18 = 4^2+2 \checkmark \\
  k=3: &\quad\operatorname{Tr}(A^6)  = 66 = 4^3+2 \checkmark
\end{align*}
and the discrepancy with $\hv^k$ at $k=3$:
\[
  \frac{\operatorname{Tr}(A^6)}{2} = 33 \neq 27 = 3^3 = \hv^3.
\]

\subsection{Resolvent mismatch}

At $\lambda = \aI/\Ee$:
\begin{align*}
  G(\aI/\Ee) &= \frac{3(\aI/\Ee - 1)}{(\aI/\Ee - 2)(\aI/\Ee + 1)}
              \approx 0.021916\ldots \\
  S          &= \frac{3\aI}{\Ee^3(\aI^2+3\Ee^2)}
              \approx 0.000705\ldots
\end{align*}
Mismatch factor $\approx 97.7$.  The resolvent interpretation is
\textbf{not established} at any natural TOE energy scale.

%─────────────────────────────────────────────────────────────────────────────
\section{Summary and Status}
\label{sec:status}
%─────────────────────────────────────────────────────────────────────────────

\subsection*{Results established by this addendum}

\begin{enumerate}
  \item The $A_2$ triangle adjacency matrix has eigenvalues $2, -1, -1$,
        giving $\operatorname{Tr}(A^{2k}) = 4^k+2$ for all $k \geq 1$
        (Lemma; verified for $k=1,2,3$).

  \item The unoriented 2-step loop count $\operatorname{Tr}(A^2)/2 = 3 = h^\vee(A_2)$
        reproduces the $k=1$ Coxeter factor correctly, and also for $k=2$
        ($\operatorname{Tr}(A^4)/2 = 9 = 3^2$), but fails for $k \geq 3$
        (Table~\ref{tab:traces}).

  \item The seed factorisation $T_k = (-1)^{k+1}\cdot r^k\cdot\sigma$ with
        $r = h^\vee(A_2)\cdot\alpha^2\cdot\Emu$ and $\sigma = \aI/\Ee^5$
        is proved algebraically (Lemma~\ref{lem:seed}).

  \item The closed-form sum $S = \sigma\cdot r/(1+r) = h^\vee(A_2)\aI/[\Ee^3(\aI^2+h^\vee(A_2)\Ee^2)]$
        follows immediately from Lemma~\ref{lem:seed} and the geometric series.

  \item The adjacency resolvent $G(\aI/\Ee)$ does \emph{not} equal $S$;
        the mismatch factor is $\approx 31.1$, and no natural TOE energy
        $\lambda$ satisfies $G(\lambda) = S$ (Remark in Section~\ref{sec:genfun}).

  \item Convergence table: the series converges to $R_\infty$ from $N=3$
        onward; the ratio $T_{k+1}/T_k = -r$ is exact to $10^{-63}$
        for $k = 2, 3, 4$ (Table~\ref{tab:convergence}).
\end{enumerate}

\subsection*{Open problems}

\begin{description}
  \item[\textbf{OP-G2A2-lattice (remaining)}]
    Derive the Coxeter multiplicity per loop order ($\hv^k$ vs.\
    $(4^k+2)/2$) from the $G_2$ holonomy integral of P198.
    Specifically: show that the loop measure on the $A_2$ face
    projects onto the single-vertex loops (weight $\hv$ each)
    rather than all $2k$-step closed walks.

  \item[\textbf{OP-G2A2-resolvent}]
    Determine whether there exists a natural operator $H$ on the
    $A_2$ Lie-algebra module such that
    $\operatorname{Tr}[(\lambda I - H)^{-1}] = S/\text{const}$
    at a TOE-natural energy scale $\lambda$.
    The plain adjacency operator at $\lambda = \aI/\Ee$ does not work.

  \item[\textbf{OP-G2A2-ratio (carried from P213)}]
    Derive $r = h^\vee(A_2)\cdot\alpha^2\cdot\Emu$ from first principles
    in the $G_2$/$A_2$ weight-lattice action.
\end{description}

\subsection*{Programme status}

\medskip
\noindent\textbf{CONJECTURED:} Conjecture~\ref{conj:loop} (P215.1) gives
a physically motivated loop-order interpretation of $T_k$ consistent with
the seed factorisation and the exact $\hv^k$ structure, but the derivation
from the $G_2$ holonomy integral is not complete.  The A₂ path count
from the adjacency matrix accounts correctly for $\hv^1$ and $\hv^2$
but not $\hv^k$ for $k \geq 3$.

\medskip
\noindent\textbf{ESTABLISHED (carried from P213--P214):} The geometric
series structure, the closed form, and the algebraic proof of the ratio
$T_{k+1}/T_k = -r$ for all $k \geq 1$.

\bigskip
\noindent\textit{All numerical computations verified at \texttt{mp.dps=60}
with 44 assertions in \texttt{Lumen/corpus/addenda/verify/verify\_P215.py},
all passing.}

\bigskip
\noindent\textit{Copyright \textcopyright\ 2026 L\'eon Fernando Vlegels.
MIT License.}

\end{document}
