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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{\textbf{Addendum 214: OP-G$_2$A$_2$-ratio\\
Deriving the Geometric Ratio $r = h^\vee(A_2)\,E_\mu\,\alpha^2$}}
\author{L\'{e}on Fernando Vlegels}
\date{18 May 2026}

%─── notation ─────────────────────────────────────────────────────────────────
\newcommand{\Ee}{e}          % E_e = π  (electron sector energy)
\newcommand{\Om}{\omega}     % OMEGA_0 = π³/4
\newcommand{\aI}{\mu}        % ALPHA_INV = 4π³+π²+π  (monad)
\newcommand{\hv}{h}          % h∨(A₂) = 3
\newcommand{\Emu}{E_\mu}     % E_μ = π²  (muon sector energy)
\newcommand{\Ee@sym}{E_e}    % E_e = π
\newcommand{\alp}{\alpha}    % fine-structure constant  α = 1/μ
\newcommand{\rr}{r}          % geometric ratio

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Context.}
Addendum~213 (P213) established that the $G_2$--$A_2$ muon mass-ratio
correction series $\{T_k\}_{k\ge1}$ is a geometric series with common ratio
\begin{equation}
  \rr \;=\; \frac{T_{k+1}}{T_k}\bigg|_{\text{sign-stripped}}
         \;=\; \frac{\hv^\vee(A_2)\,\Emu}{\aI^2}
         \;=\; \frac{3\pi^2}{\aI^2}
         \;\approx\; 1.577\times10^{-3},
  \label{eq:r_def}
\end{equation}
verified to $10^{-63}$ precision at \texttt{mp.dps\,=\,60}.  The individual
correction terms are
\[
  T_k \;=\; (-1)^{k+1}\,\frac{\hv^k\,\Ee^{2k-5}}{\aI^{2k-1}},
  \qquad k\ge1,
\]
with $\Ee = \pi$, $\aI = 4\pi^3+\pi^2+\pi$, $\hv = h^\vee(A_2) = 3$, and
$\Emu = E_e^2 = \pi^2$.

\medskip
\noindent\textbf{Open problem OP-G2A2-ratio.}
\emph{Why does $r = \hv\,\Emu/\aI^2$?}
Derive this from $G_2$--$A_2$ geometry and/or loop-correction structure.

\medskip
\noindent The present addendum addresses this question in four sections:
\begin{enumerate}
\item numerics and TOE interpretation of $r$;
\item algebraic derivation of $r$ from the $T_k$ formula
      (Proposition~P214.1 — proven);
\item three geometric derivation attempts
      (Conjecture~P214.2 — conjectured);
\item closed-form structure of $R_\infty$ and the deformed-monad denominator.
\end{enumerate}

%─────────────────────────────────────────────────────────────────────────────
\section{The Ratio in TOE Terms}
\label{sec:numerics}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Numerical value and equivalent forms}

Setting $\alp = 1/\aI$ (the fine-structure constant), the ratio admits
several equivalent expressions:
\begin{align}
  \rr &= \frac{3\pi^2}{\aI^2}
       = \hv^\vee(A_2)\cdot\Emu\cdot\alp^2
       = \hv^\vee(A_2)\cdot E_e^2\cdot\alp^2,
  \label{eq:r_forms}
\end{align}
all verified to full \texttt{mp.dps\,=\,60} precision
(see \verb|verify_P214.py|, assertions A1--A10):
\[
  \rr \;=\; 0.001\,576\,702\,397\,389\,542\,007\,6\ldots
\]

\subsection{TOE interpretation}

The formula $r = \hv\cdot E_e^2 \cdot \alp^2$ decomposes into three
factors of clear TOE provenance:

\begin{itemize}
  \item $\hv = h^\vee(A_2) = 3$ is the dual Coxeter number of $A_2$, the
        Lie algebra governing the lepton face of the $G_2$ spectral frame.

  \item $E_e^2 = \pi^2 = \Emu$ is the muon-sector energy scale.  In the
        TOE identification rule, the electron sector carries energy $E_e = \pi$
        at each site.  The muon sector energy $\Emu$ is the squared
        electron-sector energy: $E_\mu = E_e^2$.  Equivalently, $\Emu$ is the
        energy accumulated by traversing \emph{two} electron-sector edges in
        sequence.

  \item $\alp^2 = 1/\aI^2$ is the square of the fine-structure constant.
        Each loop correction in the $G_2$--$A_2$ spectral expansion carries
        one power of $\alp^2$, the standard two-vertex suppression of quantum
        corrections.
\end{itemize}

Thus $r$ is the product of an algebraic $A_2$-face count, a two-step
energy weight, and a two-loop coupling suppression.  It is not a free
parameter: every factor is fixed by the TOE geometry.

%─────────────────────────────────────────────────────────────────────────────
\section{Algebraic Derivation from the $T_k$ Formula}
\label{sec:algebraic}
%─────────────────────────────────────────────────────────────────────────────

\begin{proposition}[P214.1 — Geometric ratio, algebraic proof]
\label{prop:P214.1}
Let
\[
  T_k \;=\; (-1)^{k+1}\,\hv^k\,\Ee^{2k-5}\,\aI^{-(2k-1)},
  \qquad k \ge 1.
\]
Then for every $k \ge 1$:
\[
  \frac{T_{k+1}}{T_k} \;=\; -\rr,
  \qquad
  \rr \;=\; \frac{\hv\,\Ee^2}{\aI^2} \;=\; \frac{\hv\,\Emu}{\aI^2}.
\]
\end{proposition}

\begin{proof}
Direct computation:
\begin{align*}
  \frac{T_{k+1}}{T_k}
  &= \frac{(-1)^{k+2}\,\hv^{k+1}\,\Ee^{2(k+1)-5}\,\aI^{-(2(k+1)-1)}}
          {(-1)^{k+1}\,\hv^k      \,\Ee^{2k-5}    \,\aI^{-(2k-1)}}\\[6pt]
  &= (-1)\cdot \hv^{(k+1)-k}\cdot \Ee^{[2(k+1)-5]-(2k-5)}
              \cdot \aI^{-[2(k+1)-1]+(2k-1)}\\[6pt]
  &= -\hv^1 \cdot \Ee^{2} \cdot \aI^{-2}
   \;=\; -\frac{\hv\,\Emu}{\aI^2},
\end{align*}
which is independent of $k$.  The three exponent increments are:
\begin{enumerate}
  \item $\hv$-exponent: $(k+1) - k = +1$ in numerator, giving factor $\hv$.
  \item $\Ee$-exponent: $[2(k+1)-5] - (2k-5) = +2$ in numerator, giving factor $\Ee^2 = \Emu$.
  \item $\aI$-exponent: $-[2(k+1)-1] - (-(2k-1)) = -2$ in exponent, giving factor $\aI^{-2}$.
\end{enumerate}
Because all increments are constant, the ratio $T_{k+1}/T_k = -\hv\Emu/\aI^2$ is
$k$-independent, establishing the geometric series structure.
\end{proof}

\begin{remark}
Proposition~P214.1 is a \emph{formal} result: it follows purely from the
exponent pattern in the $T_k$ formula and does not yet explain \emph{why}
the exponents of $\Ee$ and $\aI$ shift by exactly $\pm 2$ at each loop
order.  That deeper question is addressed in Section~\ref{sec:geometric}.
\end{remark}

\begin{corollary}
The full geometric series
$\sum_{k=1}^\infty T_k = T_1\sum_{j=0}^\infty (-\rr)^j$
converges absolutely since $\rr \approx 1.577\times10^{-3} \ll 1$, and
its sum is
\[
  S_\infty \;=\; \frac{T_1}{1+\rr}
  \;=\; \frac{\hv\,\aI}{\Ee^3\,(\aI^2 + \hv\Emu)}.
\]
\end{corollary}

%─────────────────────────────────────────────────────────────────────────────
\section{Geometric Derivation Attempts}
\label{sec:geometric}
%─────────────────────────────────────────────────────────────────────────────

We now ask why the specific increment pattern ($\Ee^{+2}$, $\aI^{-2}$,
$\hv^{+1}$) arises.  Three independent lines of attack are explored.

\subsection{Attempt 1: A₂ root-system lengths}

The $A_2$ root system has two root lengths (both equal in the standard
normalization).  In the $G_2$ Killing form with long roots at length$^2 = 2$,
the $A_2$ subalgebra consists of the \emph{long} roots of $G_2$
(length$^2 = 2$), while the short roots have length$^2 = 2/3$.

\medskip
A natural candidate for $r$ via the Killing form is:
\[
  \rr_{\text{Killing}}
  \;=\; h^\vee(A_2)\cdot|\alpha_{\text{long}}|^2_{G_2}\cdot\alp^2
  \;=\; 3\cdot 1\cdot\alp^2
  \;\approx\; 1.598\times10^{-4}.
\]
\textbf{Assessment:} This is a factor of $\Emu = \pi^2 \approx 9.87$ too small.
The Killing-form normalisation of root lengths does not explain the factor $\Emu$.
The loop-suppression per $A_2$-face edge is $\alp$, not $\alp\cdot|\alpha_{\text{root}}|$
in this normalisation.

\subsection{Attempt 2: Monad-energy scaling}

In the TOE, $\aI = 4\pi^3+\pi^2+\pi$ is the monad: the unique rational
polynomial in $\pi$ whose reciprocal equals the fine-structure constant.
Its square $\aI^2$ naturally suppresses two-loop amplitudes.
The question reduces to: why does each successive $T_k$ carry an extra
factor of $\Emu/\aI^2 = \pi^2/\aI^2$ relative to $T_{k-1}$?

In TOE spectral theory, the $G_2$ frame assigns each site one unit of $E_e$.
A \emph{two-site amplitude} (traversal of one face edge plus return) carries
$E_e^2 = \Emu$.  Suppressed by one power of $\alp^2 = 1/\aI^2$ (the
two-vertex coupling), such an amplitude contributes $\Emu\cdot\alp^2 = \Emu/\aI^2$.
Multiplied by $\hv = h^\vee(A_2) = 3$, this is exactly $\rr$.

\medskip
\textbf{Assessment:} This argument is internally consistent and makes
quantitative contact with $r$, but it \emph{assumes} the amplitude
assignment $E_e$ per site and $\alp^2$ per two-vertex insertion.  These
assignments are the TOE identification rule (not derived here from the
$G_2$ geometry itself).  This attempt is therefore not yet a derivation
from first principles, but rather a restatement of $r$ in spectral-theory
language.

\subsection{Attempt 3: Two-step paths on the A₂ lepton triangle}

The $A_2$ root system realised as a lepton face of $G_2$ has exactly three
positive roots, forming a triangle with three vertices and three edges.
In the TOE, each edge carries energy weight $E_e = \pi$.  A
\emph{two-step path} traverses two consecutive edges, accumulating energy
$E_e^2 = \Emu$.

\medskip
\noindent\textbf{Path count.}
On the $A_2$ triangle with vertices $\{v_1, v_2, v_3\}$:
\begin{itemize}
  \item Oriented directed 2-step paths ($v_i \to v_j \to v_k$,
        $j \neq i$, $k \neq j$): there are $3! = 6 = |W(A_2)|$ such paths.
  \item Unoriented 2-step paths (identifying each path with its reversal):
        there are $3 = h^\vee(A_2)$ such paths.
\end{itemize}
Each unoriented 2-step path contributes energy weight $\Emu$ and coupling
suppression $\alp^2$ per pair of vertices.  Summing over all three
unoriented 2-step paths gives:
\[
  \rr \;=\; h^\vee(A_2)\cdot\Emu\cdot\alp^2 \;=\; 3\pi^2\alp^2
  \;\approx\; 1.577\times10^{-3}.
\]
This matches the ratio exactly (verified to $10^{-60}$ in
\verb|verify_P214.py|, assertions A21--A25).

\medskip
\textbf{Assessment:} The path count $h^\vee(A_2) = 3$ equals the number of
unoriented 2-step paths on the $A_2$ triangle, and the energy weight $\Emu$
equals $E_e^2$ (the energy of one 2-step traversal).  This gives a clean
combinatorial derivation: each order in the loop expansion corresponds to
one traversal of the lepton triangle by 2-step paths, with the number of
independent traversals set by the dual Coxeter number.  The coupling suppression
$\alp^2$ per traversal is the standard two-vertex insertion rule.  This
attempt achieves the strongest geometric contact with $r$.

\begin{conjecture}[P214.2 — Two-step path origin of $r$]
\label{conj:P214.2}
Each order $k$ of the $G_2$--$A_2$ mass-ratio loop expansion corresponds
to $h^\vee(A_2)$ unoriented 2-step path traversals of the lepton face
triangle, each carrying energy weight $E_e^2 = \Emu$ and coupling
suppression $\alp^2$.  The ratio of consecutive correction terms is
therefore:
\[
  \frac{T_{k+1}}{T_k} = -h^\vee(A_2)\cdot\Emu\cdot\alp^2 = -\rr.
\]
The specific proof obligation is: derive the identification of
``one loop order'' with ``one set of $h^\vee(A_2)$ 2-step traversals''
directly from the $G_2$--$A_2$ branching rules and/or the spectral
resolution of the Dirac operator on the lepton face.
\end{conjecture}

%─────────────────────────────────────────────────────────────────────────────
\section{Closed-Form Structure}
\label{sec:closed}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The infinite sum in factored form}

The complete mass-ratio formula is:
\begin{equation}
  R_\infty \;=\; 207\!\left(
    1 - \frac{\Om}{\Ee^3\aI}
      + \frac{\hv\,\aI}{\Ee^3\,(\aI^2 + \hv\Emu)}
  \right),
  \label{eq:Rinf}
\end{equation}
where $207 = 14\cdot15 - h^\vee(A_2) = \dim(G_2)\cdot(\dim(G_2)+1) - 3$
is the prefactor from the $G_2$ spectral frame.  Numerically:
\[
  R_\infty \;=\; 206.768\,285\,300\,712\,17\ldots,
  \qquad
  |R_\infty - T|/T \;\approx\; 0.01113\;\text{ppm}.
\]

\subsection{The deformed-monad denominator}

The denominator $\aI^2 + \hv\Emu$ is the monad squared \emph{deformed} by
the $A_2$ face contribution:
\begin{equation}
  \aI^2 + \hv\Emu \;=\; \aI^2(1 + \rr)
  \;=\; \aI^2 + 3\pi^2.
  \label{eq:deformed_monad}
\end{equation}
This identity follows directly from $\rr = \hv\Emu/\aI^2$
(see \verb|verify_P214.py|, assertion A26).

\subsection{Numerator expansion}

Writing the bracket of~\eqref{eq:Rinf} over the common denominator
$\Ee^3\aI(\aI^2+\hv\Emu)$:
\begin{align*}
  \text{Numerator}
  &= \Ee^3\aI(\aI^2+\hv\Emu)
   - \Om(\aI^2+\hv\Emu)
   + \hv\aI^2\\
  &= (\Ee^3\aI - \Om)(\aI^2+\hv\Emu) + \hv\aI^2.
\end{align*}
Using $\Ee^3 = \pi^3 = 4\Om$ so that $\Ee^3\aI - \Om = \Om(4\aI - 1)$:
\begin{equation}
  \text{Numerator} \;=\; \Om(4\aI-1)(\aI^2+\hv\Emu) + \hv\aI^2.
  \label{eq:numerator}
\end{equation}
Numerically this equals $79\,827\,829.477\,966\ldots$, verified at
\texttt{mp.dps\,=\,60} (assertion~A35 of \verb|verify_P214.py|).

\begin{remark}
The factor $\Om(4\aI-1)$ in~\eqref{eq:numerator} reflects the TOE
structure $\Ee^3 = 4\Om$ (i.e.\ $\pi^3 = 4\cdot\pi^3/4$) together with
the monad relation $\aI = 4\Ee^3 + \Ee^2 + \Ee$.  A putative
simplification $\Om(4\aI-1) = 3\Om(\cdots)$ would require $4\aI - 1 = 3$,
i.e.\ $\aI = 1$, which is false ($\aI \approx 137.04$).  The expression
\eqref{eq:numerator} is the correct minimal form.
\end{remark}

\subsection{Alternative factoring via $(1+r)$}

Using $\aI^2 + \hv\Emu = \aI^2(1+\rr)$ and $T_1 = \hv/(\Ee^3\aI)$:
\[
  S_\infty \;=\; \frac{T_1}{1+\rr}
             \;=\; \frac{T_1\,\aI^2}{\aI^2(1+\rr)}
             \;=\; \frac{\hv\aI}{\Ee^3(\aI^2+\hv\Emu)},
\]
and so
\begin{equation}
  R_\infty \;=\; 207\!\left(1 + T_0 + \frac{T_1}{1+\rr}\right),
  \qquad T_0 = -\frac{\Om}{\Ee^3\aI}.
  \label{eq:Rinf_T1r}
\end{equation}
This makes explicit that the closed form is the leading-term approximation
$207(1+T_0+T_1)$ corrected by the geometric resummation factor $1/(1+\rr)$.

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

\noindent\textbf{Proven (Proposition~P214.1):}
The ratio $r = T_{k+1}/T_k = -\hv\Emu/\aI^2$ follows algebraically from
the exponent structure of the $T_k$ formula.  The three exponent increments
(+1 in $\hv$, +2 in $\Ee$, $-2$ in $\aI$) are constant across all $k$,
establishing $r$ as the unique $k$-independent geometric ratio.  This is
a complete, rigorous proof given the $T_k$ formula.

\medskip
\noindent\textbf{Conjectured (Conjecture~P214.2):}
The geometric origin of the exponent-$\pm 2$ shift in $\Ee$ and $\aI$ —
i.e.\ the derivation of the $T_k$ exponent pattern from the $G_2$--$A_2$
branching rules — remains open.  Attempt~3 (Section~\ref{sec:geometric})
provides a combinatorial model: each loop order corresponds to
$h^\vee(A_2) = 3$ unoriented 2-step traversals of the lepton face triangle
(energy weight $E_e^2 = \Emu$, coupling suppression $\alp^2$).  This model
exactly reproduces $r$ but has not been derived from the $G_2$--$A_2$
spectral decomposition.

\medskip
\noindent\textbf{Status: CONJECTURED.}

\medskip
\noindent\textbf{Specific proof obligation:}
Derive the formula $T_k = (-1)^{k+1}\hv^k\Ee^{2k-5}/\aI^{2k-1}$
from first principles — either from the $G_2$--$A_2$ branching rules,
from the spectral resolution of a Dirac-type operator on the lepton face,
or from a loop-integral over the $A_2$ root system — without assuming the
pattern.  Equivalently, show that each perturbative order on the $A_2$ lepton
face contributes exactly $(-\hv\Emu/\aI^2)$ times the previous order,
using only the Lie-algebraic data of $G_2 \supset A_2$ and the TOE
identification rule $E_e = \pi$, $\aI = 4\pi^3+\pi^2+\pi$.

\end{document}
