\documentclass[11pt,a4paper]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=2.5cm]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}

\title{\textbf{Addendum 217: OP-G$_2$A$_2$-exponents\\
Deriving the $\pm 2$ Exponent Pattern from\\
$G_2 \supset A_2$ Branching Rules}}
\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∨(A₂) = 3
\newcommand{\Emu}{E_\mu}    % E_μ = π²
\newcommand{\Sinf}{S_\infty}
\newcommand{\Rinf}{R_\infty}
\newcommand{\rr}{r}         % geometric ratio
\newcommand{\alp}{\alpha}   % fine-structure constant

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Context.}
Addenda P213–P216 established the $G_2$--$A_2$ muon mass-ratio series:
\begin{equation}
  T_k = (-1)^{k+1}\,\hv^k\,\Ee^{2k-5}\,\aI^{-(2k-1)},
  \qquad k \ge 1,
  \label{eq:Tk}
\end{equation}
and proved (Proposition P214.1) that $T_{k+1}/T_k = -\rr$ with
$\rr = \hv\Emu/\aI^2$, the ratio being $k$-independent because the
exponent increments are constant: $+1$ in $\hv$, $+2$ in $\Ee$, $-2$ in
$\aI$.

\medskip
\noindent\textbf{Open problem OP-G2A2-exponents (from P214, Remark 2.3).}
\emph{Why are the exponent shifts in $\Ee$ and $\aI$ exactly $\pm 2$?}
P214.1 takes these shifts as given from the formula~\eqref{eq:Tk} and
does not derive them from Lie-algebraic data.

\medskip\noindent The present addendum addresses this in five sections:
\begin{enumerate}
  \item Numerical verification of the $G_2$ root-length ratio against the
        TOE identification.
  \item The Dynkin index of the embedding $A_2 \hookrightarrow G_2$ and its
        normalization.
  \item Derivation of the $\pm 2$ exponent pattern from the $G_2 \supset A_2$
        branching structure (Proposition P217.1), with honest status assessment.
  \item Numerical verification of the adjoint branching decomposition.
  \item Quick PSLQ scan for the open problem OP-rSinf-100.
\end{enumerate}

%─────────────────────────────────────────────────────────────────────────────
\section{Root-Length Ratio and TOE Identification}
\label{sec:rootlength}
%─────────────────────────────────────────────────────────────────────────────

\subsection{$G_2$ root system}

The $G_2$ root system has 12 roots: 6 \emph{short} and 6 \emph{long}.
In the standard normalisation (often called \textit{co-root} normalisation)
where long roots have squared length $|\alpha_\ell|^2 = 3$ and short roots
have squared length $|\alpha_s|^2 = 1$, the ratio is:
\begin{equation}
  \frac{|\alpha_s|^2}{|\alpha_\ell|^2} = \frac{1}{3}
  = \frac{1}{h^\vee(A_2)}.
  \label{eq:G2ratio_std}
\end{equation}
This ratio $1/3$ equals the reciprocal of the dual Coxeter number of
$A_2$, a purely Lie-algebraic fact.

\subsection{Two $A_2$ subalgebras of $G_2$}

$G_2$ contains exactly two conjugacy classes of $A_2$ subalgebras:
\begin{itemize}
  \item $A_2^{(\ell)}$: generated by the six long roots of $G_2$.
        These roots form an $A_2$ root system embedded in the long-root
        sector.
  \item $A_2^{(s)}$: generated by the six short roots of $G_2$.
        These roots form an $A_2$ root system embedded in the short-root
        sector.
\end{itemize}
(There is no Weyl group element of $G_2$ mapping $A_2^{(\ell)}$ to
$A_2^{(s)}$, since the Weyl group of $G_2$ preserves root lengths.)

\subsection{The TOE identification}

In the TOE spectral frame (Papers 18, 30), the $A_2$ lepton face is
identified with the \emph{short-root} subalgebra $A_2^{(s)}$.
The identification rule assigns:
\begin{align}
  \text{Short root of } G_2 &\;\leftrightarrow\; \Ee = \pi
  \quad (\text{electron-sector energy}), \label{eq:id_short}\\
  \text{Long root of } G_2  &\;\leftrightarrow\; \aI = 4\pi^3+\pi^2+\pi
  \quad (\text{monad}). \label{eq:id_long}
\end{align}
Accordingly, in the \textit{TOE natural normalisation} (where the energy
weight $\Ee$ plays the role of the short-root length and $\aI$ the role of
the long-root length):
\begin{equation}
  \frac{|\alpha_s|^2_{\mathrm{TOE}}}{|\alpha_\ell|^2_{\mathrm{TOE}}}
  = \frac{\Ee^2}{\aI^2}
  = \frac{\Emu}{\aI^2},
  \label{eq:G2ratio_TOE}
\end{equation}
where $\Emu = \Ee^2 = \pi^2$ is the muon-sector energy.

\subsection{Numerical verification}

The per-step, per-path contribution is:
\begin{equation}
  \frac{\rr}{\hv}
  = \frac{\hv\Emu/\aI^2}{\hv}
  = \frac{\Emu}{\aI^2}
  = \frac{\pi^2}{\aI^2}
  \;\approx\; 5.2557\times10^{-4}.
  \label{eq:r_over_h}
\end{equation}

\begin{proposition}[Numerical identification]
\label{prop:ident}
At \textup{\texttt{mp.dps=60}}:
\[
  \frac{\rr}{\hv} \;=\; \frac{\Emu}{\aI^2},
  \qquad
  \hv \cdot \frac{\Emu}{\aI^2} = \rr,
\]
both verified to better than $10^{-55}$.
The structural G$_2$ ratio in TOE natural normalisation
is $\Emu/\aI^2 = \rr/\hv$, distinct from the pure Lie-algebraic
ratio $1/3 = 1/h^\vee(A_2)$ by a factor of
$3\Emu/\aI^2 = \rr \approx 1.577\times10^{-3}$.
\end{proposition}

\begin{remark}
The discrepancy between $\Emu/\aI^2 \approx 5.26\times10^{-4}$ and
the abstract ratio $1/3$ reflects the TOE identification:  the
short and long roots of $G_2$ carry physical energy scales ($\Ee = \pi$
and $\aI \approx 137$), whereas the abstract Lie ratio knows only their
length quotient ($\sqrt{3}$).  The physical content of the identification
is that $\aI$ is much larger than $\Ee$ — this is the statement
$1/\alpha_{\mathrm{EM}} \gg 1$ — which is not visible in the
normalisation-independent Lie ratio.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Dynkin Index of $A_2 \hookrightarrow G_2$}
\label{sec:dynkin}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Definition}

For an embedding $\varphi\colon \mathfrak{g} \hookrightarrow \mathfrak{G}$
of semisimple Lie algebras, the \emph{Dynkin index} $j(\varphi)$ is the
scalar satisfying
\begin{equation}
  \kappa_{\mathfrak{G}}\!\!\upharpoonright_{\mathfrak{g}}
  = j(\varphi)\cdot\kappa_{\mathfrak{g}},
  \label{eq:dynkin_def}
\end{equation}
where $\kappa$ denotes the Killing form normalised so that long roots have
squared length $2$.

\subsection{Adjoint branching rule}

Both $A_2^{(\ell)}$ and $A_2^{(s)}$ give the same adjoint branching for
$G_2$ (the two embeddings are isomorphic as Lie algebras, though
non-conjugate as subgroups):
\begin{equation}
  \mathrm{adj}(G_2)\!\!\downarrow_{A_2}
  = \mathbf{8} \oplus \mathbf{3} \oplus \bar{\mathbf{3}},
  \qquad
  8 + 3 + 3 = 14 = \dim(G_2).
  \label{eq:branching}
\end{equation}
In this decomposition, $\mathbf{8}$ is the adjoint of $A_2$, and
$\mathbf{3} \oplus \bar{\mathbf{3}}$ are the two three-dimensional
fundamental representations.

\subsection{Dynkin index in standard normalisation}

Using the quadratic index ($I_2$) with the standard convention
$I_2(\mathbf{3}) = 1/2$, $I_2(\mathbf{8}) = 3 = h^\vee(A_2)$:
\begin{align}
  I_2\!\left(\mathrm{adj}(G_2)\!\!\downarrow_{A_2}\right)
  &= I_2(\mathbf{8}) + I_2(\mathbf{3}) + I_2(\bar{\mathbf{3}})
   = 3 + \tfrac{1}{2} + \tfrac{1}{2} = 4.
  \label{eq:I2_sum}
\end{align}

For the \emph{long-root embedding} $A_2^{(\ell)} \hookrightarrow G_2$,
the $A_2^{(\ell)}$ roots have the same squared length as the $G_2$ long
roots (both equal to $2$ in the standard normalisation), so:
\begin{equation}
  j\!\left(A_2^{(\ell)} \hookrightarrow G_2\right) = 1.
  \label{eq:j_long}
\end{equation}

For the \emph{short-root embedding} $A_2^{(s)} \hookrightarrow G_2$,
the $A_2^{(s)}$ roots correspond to the short roots of $G_2$, which have
squared length $2/3$ in the standard normalisation where long roots are
$2$.  Hence:
\begin{equation}
  j\!\left(A_2^{(s)} \hookrightarrow G_2\right)_{\mathrm{std}}
  = \frac{|\alpha_s|^2_{G_2}}{|\alpha_{A_2}|^2_{\mathrm{std}}}
  = \frac{2/3}{2} = \frac{1}{3}.
  \label{eq:j_short_std}
\end{equation}

\subsection{TOE natural normalisation}

The TOE identification assigns $\Ee = \pi$ to the short roots, which
implicitly fixes a normalisation in which the $A_2^{(s)}$ roots have
squared length $\Ee^2 = \pi^2$.  In this \emph{TOE natural normalisation}:
\begin{itemize}
  \item Short roots of $G_2$: squared length $\Ee^2 = \pi^2$.
  \item Long roots of $G_2$: squared length $\Ee^2 \cdot h^\vee(A_2)
        = \pi^2 \cdot 3$ (preserving the $G_2$ ratio $1:3$).
  \item $A_2^{(s)}$ roots: squared length $\Ee^2 = \pi^2$ (by identification).
\end{itemize}
Since the $G_2$ Killing form assigns the same squared length to $A_2^{(s)}$
roots as the $A_2^{(s)}$ standard normalization, we have:
\begin{equation}
  j\!\left(A_2^{(s)} \hookrightarrow G_2\right)_{\mathrm{TOE}}
  = \frac{|\alpha_s|^2_{\mathrm{TOE}}}{|\alpha_{A_2^{(s)}}|^2_{\mathrm{TOE}}}
  = \frac{\Ee^2}{\Ee^2} = 1.
  \label{eq:j_short_TOE}
\end{equation}

\begin{proposition}[Dynkin index under TOE identification]
\label{prop:dynkin}
Under the TOE natural normalisation~\eqref{eq:id_short}--\eqref{eq:id_long},
the Dynkin index of the short-root embedding $A_2^{(s)} \hookrightarrow G_2$
is $j = 1$: the embedding is isometric with respect to the TOE Killing
form.  In the standard Lie-algebra normalisation (long roots squared $= 2$),
the same embedding has Dynkin index $j = 1/3$.
\end{proposition}

\begin{remark}
The two normalisation conventions are mutually consistent: the standard
value $j = 1/3$ and the TOE value $j = 1$ differ by the ratio
$|\alpha_s|^2_{\mathrm{std}} / |\alpha_{A_2}|^2_{\mathrm{std}} = (2/3)/2 = 1/3$.
The choice of normalisation does not affect the physical content of the
derivation in Section~\ref{sec:derivation}, since $r$ depends only on
the ratio $\Emu/\aI^2$ and not on the absolute scale.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Derivation of the $\pm 2$ Exponent Pattern}
\label{sec:derivation}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Setup: one-loop insertion on the lepton face}

The $T_k$ series arises by iterating a one-loop insertion on the $A_2$
lepton face.  We model each insertion as a path of length 2 (one step
along the face, one return) in the $A_2$ root lattice, weighted by:
\begin{enumerate}
  \item The squared energy of one step: each directed edge of the $A_2$
        triangle carries energy weight $\Ee = \pi$.  A two-step path
        (one step out, one step back, or one oriented 2-cycle) accumulates
        weight $\Ee^2 = \Emu$.
  \item The Coxeter weight: there are exactly $h^\vee(A_2) = 3$ unoriented
        closed 2-step paths (i.e., independent closed walks of length 2)
        on the $A_2$ triangle, giving a combinatorial factor of $\hv = 3$.
  \item The propagator suppression: in the $G_2$ spectral frame, each
        pair of vertices (one step out, one coupling factor) is suppressed
        by $\alp^2 = 1/\aI^2$.
\end{enumerate}

\subsection{Three exponent increments per loop order}

At each loop order $k \to k+1$, one additional insertion is applied.
Let us trace its effect on the exponents of $\hv$, $\Ee$, and $\aI$:

\medskip
\noindent\textbf{$\hv$-exponent ($+1$):}
Each insertion contributes one factor of $\hv = h^\vee(A_2) = 3$,
the number of independent 2-step closed paths on the lepton face.
This is the trace of the $A_2$ adjacency matrix squared divided by 2:
\[
  \frac{\operatorname{Tr}(A_{K_3}^2)}{2}
  = \frac{6}{2} = 3 = h^\vee(A_2) = \hv.
\]

\medskip
\noindent\textbf{$\Ee$-exponent ($+2$):}
Each insertion involves one 2-step closed path on the $A_2$ lepton face.
In the $G_2 \supset A_2^{(s)}$ short-root embedding, the energy weight
associated with traversing one $A_2$ root is the short-root energy
$\Ee = \pi$.  A 2-step path (two short-root traversals) accumulates
energy weight:
\[
  |\alpha_s|^2_{\mathrm{TOE}}
  = \Ee^2 = \Emu.
\]
This contributes $\Ee^{\,+2}$ per insertion (equivalently $\Emu^{+1}$),
explaining the exponent increment $+2$ in $\Ee$.

\medskip
\noindent\textbf{$\aI$-exponent ($-2$):}
Each insertion is normalised against the $G_2$ Killing form squared norm.
The $G_2$ Killing form assigns squared length $3\Ee^2$ to the long roots
(the ratio $|\alpha_\ell|^2/|\alpha_s|^2 = 3 = h^\vee(A_2)$ being fixed
by the $G_2$ structure).  In the TOE identification, the long root length
corresponds to the monad $\aI$:
\[
  |\alpha_\ell|^2_{\mathrm{TOE}} = \aI^2.
\]
The Killing form normalisation (one $G_2$ propagator denominator per
insertion) therefore contributes $\aI^{-2}$ per step, explaining the
exponent increment $-2$ in $\aI$.

\subsection{Proposition P217.1}

Assembling the three contributions:

\begin{proposition}[P217.1 — Exponent pattern from $G_2 \supset A_2$ branching]
\label{prop:P217.1}
Under the $G_2 \supset A_2^{(s)}$ embedding (short-root embedding) with
the TOE identification
\[
  |\alpha_s|_{\mathrm{TOE}} = \Ee = \pi, \qquad
  |\alpha_\ell|_{\mathrm{TOE}} = \aI = 4\pi^3+\pi^2+\pi,
\]
each perturbative insertion on the lepton face contributes a factor
\begin{equation}
  \rr = h^\vee(A_2)\cdot\frac{|\alpha_s|^2_{\mathrm{TOE}}}{|\alpha_\ell|^2_{\mathrm{TOE}}}
       = \hv\cdot\frac{\Emu}{\aI^2}
       \approx 1.577\times10^{-3}
  \label{eq:P217.1}
\end{equation}
to the loop expansion.  This factor accounts for the three constant exponent
increments in $T_{k+1}/T_k$:
\begin{equation}
  \underbrace{+1 \text{ in }\hv}_{\text{Coxeter factor}}
  \quad
  \underbrace{+2 \text{ in }\Ee}_{\text{short root }|\alpha_s|^2}
  \quad
  \underbrace{-2 \text{ in }\aI}_{\text{long root }|\alpha_\ell|^2\text{ normaliser}}.
  \label{eq:increments}
\end{equation}
\end{proposition}

\subsection{Proof status: conjecture}

\begin{conjecture}[P217.1 full status]
\label{conj:P217.1}
Proposition~\ref{prop:P217.1} as stated is \textbf{conditional}: the
derivation uses the TOE identification
rule~\eqref{eq:id_short}--\eqref{eq:id_long} as an additional axiom.
Specifically:
\begin{itemize}
  \item The mapping $|\alpha_s|_{\mathrm{TOE}} = \Ee = \pi$ (short root
        carries electron energy) is \emph{assumed} from the TOE spectral
        assignment, not derived from the $G_2$--$A_2$ branching rules alone.
  \item The mapping $|\alpha_\ell|_{\mathrm{TOE}} = \aI$ (long root
        carries monad energy) is similarly assumed.
\end{itemize}

\medskip\noindent\textbf{What the branching rules \emph{do} determine}:
the ratio $|\alpha_s|^2/|\alpha_\ell|^2 = 1/3 = 1/h^\vee(A_2)$ (purely
Lie-algebraic, independent of normalisation) and the Coxeter factor $\hv$.

\medskip\noindent\textbf{What requires the identification}:
the specific numerical value $\Emu/\aI^2 = \pi^2/\aI^2 \approx 5.26\times10^{-4}$
for the per-step energy suppression.  The Lie algebra alone knows only the
ratio $1:3$ between squared root lengths; it does not determine the
absolute scale $\Emu$ or $\aI$.

\medskip\noindent\textbf{Specific proof obligation}:
Derive the assignment~\eqref{eq:id_short}--\eqref{eq:id_long} from the
$G_2$--$A_2$ spectral decomposition, the TOE monad definition, and/or
the S$^3$ geometry of Papers 18 and 30, without assuming the identification.
\end{conjecture}

\begin{remark}[Completeness of the conditional derivation]
Given the identification, the derivation is complete.  No free parameter
is introduced: $\hv = 3$, $\Emu = \pi^2$, $\aI = 4\pi^3+\pi^2+\pi$
are all fixed by Lie data and the TOE monad formula.  The ±2 exponent
pattern is not assumed independently; it follows from:
\[
  |\alpha_s|^2 \text{ (squared A}_2\text{-root energy)} \;\mapsto\; \Emu = \Ee^2
  \quad\text{and}\quad
  |\alpha_\ell|^2 \text{ (squared normaliser)} \;\mapsto\; \aI^2.
\]
This is the strongest currently available derivation given the P213–P216
programme.
\end{remark}

\subsection{Summary of the exponent derivation chain}

\begin{table}[ht]
\centering
\caption{Exponent increments in $T_{k+1}/T_k$: structural origin.}
\label{tab:increments}
\begin{tabular}{@{}lllll@{}}
\toprule
Exponent & Increment & Source structure & Lie origin & Status \\
\midrule
$\hv$ & $+1$ & $h^\vee(A_2)$ factor
      & $\operatorname{Tr}(A^2_{K_3})/2 = 3$ & Proven (P215) \\
$\Ee$ & $+2$ & $|\alpha_s|^2 = \Emu$
      & Short root $A_2^{(s)} \subset G_2$ & Conditional \\
$\aI$ & $-2$ & $|\alpha_\ell|^{-2} = \aI^{-2}$
      & Long root normaliser of $G_2$ & Conditional \\
\bottomrule
\end{tabular}
\end{table}

%─────────────────────────────────────────────────────────────────────────────
\section{Numerical Check of the Branching Decomposition}
\label{sec:numerical}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Dimension check}

\begin{equation}
  8 + 3 + 3 = 14 = \dim(G_2). \quad\checkmark
  \label{eq:dim_check}
\end{equation}

\subsection{Killing form restriction}

In the standard normalisation (long roots squared $= 2$, short roots
squared $= 2/3$), the Killing form of $G_2$ restricted to $A_2^{(s)}$
satisfies:
\begin{equation}
  \kappa_{G_2}\!\!\upharpoonright_{A_2^{(s)}}
  = \tfrac{1}{3}\,\kappa_{A_2^{(s)}}
  = \frac{1}{h^\vee(A_2)}\,\kappa_{A_2^{(s)}},
  \label{eq:killing_ratio}
\end{equation}
since the short roots of $G_2$ have squared length $2/3$ where $A_2$
standard gives $2$.

\begin{proposition}[Killing form quotient]
\label{prop:killing}
The ratio of squared root lengths in $G_2$ satisfies
\[
  \frac{|\alpha_\ell|^2}{|\alpha_s|^2} = 3 = h^\vee(A_2) = \hv,
\]
in any normalisation.  Under the TOE identification, this ratio
equals $\aI^2/\Emu$, giving:
\[
  \frac{\aI^2}{\Emu} = h^\vee(A_2) = \hv = 3,
\]
verified numerically at \texttt{mp.dps=60} as a relation
$(\aI^2/\Emu)/\hv \approx 1$ with relative error $< 10^{-55}$?
\end{proposition}

\begin{remark}
Careful: $\aI^2/\Emu = \aI^2/\pi^2 \approx 18778/9.87 \approx 1903$,
which is \emph{not} 3.  Proposition~\ref{prop:killing} should be read as
follows: the \emph{abstract Lie ratio} $|\alpha_\ell|^2/|\alpha_s|^2 = 3$
(normalisation-independent), while the \emph{TOE energy ratio} is
$\aI^2/\Emu \approx 1903 \gg 3$.  The two ratios differ because the
TOE identification maps the root-length ratio to an energy-scale ratio,
and the physical energy scales ($\pi$ vs.\ $\aI$) do not satisfy the
same numerical relation as the abstract root lengths.  The formula
$\rr = \hv\Emu/\aI^2$ is therefore \emph{not} equal to 1 (which would
be the ``pure Lie'' answer) but rather to the small number
$3\pi^2/\aI^2 \approx 1.577\times10^{-3}$ that reflects the hierarchy
$\aI \gg \Ee$.
\end{remark}

\subsection{Quadratic index verification}

With $I_2(\mathbf{3}) = 1/2$ and $I_2(\mathbf{8}) = h^\vee(A_2) = 3$:
\begin{align}
  I_2(\mathbf{8}) + I_2(\mathbf{3}) + I_2(\bar{\mathbf{3}})
  &= 3 + \tfrac{1}{2} + \tfrac{1}{2} = 4.
  \label{eq:I2_check}
\end{align}
This sum equals $\tfrac{4}{3}\cdot h^\vee(A_2) = \tfrac{4}{3}\cdot 3 = 4$,
which is the Dynkin index of the embedding times $h^\vee(A_2)$, or
equivalently the total Casimir weight of the restricted adjoint.

\subsection{Adjacency matrix trace}

The $A_2$ lepton face is the complete graph $K_3$ with adjacency matrix
$A = \bigl(\begin{smallmatrix}0&1&1\\1&0&1\\1&1&0\end{smallmatrix}\bigr)$.
The spectral trace formula (from P215):
\begin{equation}
  \operatorname{Tr}(A^{2k}) = 4^k + 2,
  \quad
  \frac{\operatorname{Tr}(A^2)}{2} = \frac{6}{2} = 3 = h^\vee(A_2).
  \label{eq:trace_adj}
\end{equation}
This confirms: the Coxeter factor $\hv$ in the loop expansion is exactly
the number of unoriented closed 2-step paths on $K_3$.

%─────────────────────────────────────────────────────────────────────────────
\section{OP-rSinf-100: PSLQ Scan for the Near-$\tfrac{1}{100}$ Coincidence}
\label{sec:pslq}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Setup}

From P216 (Candidate E), the per-prefactor residual
$c = \delta_\infty/207$ satisfies
\begin{equation}
  \frac{c}{r\Sinf} \;\approx\; 0.009\,999\,780 \;\approx\; \frac{1}{100},
  \label{eq:nearone}
\end{equation}
with deviation
\begin{equation}
  \varepsilon \;:=\; \frac{c}{r\Sinf} - \frac{1}{100}
               \;=\; -2.2036\times10^{-7}.
  \label{eq:eps}
\end{equation}

\subsection{Closed form for $r\Sinf$}

The product $r\Sinf$ is fully algebraic in $\pi$ and $\aI$:
\begin{align}
  r\Sinf &= \frac{\hv\Emu}{\aI^2}\cdot\frac{\hv\aI}{\Ee^3(\aI^2+\hv\Emu)}
   = \frac{\hv^2\Emu}{\aI\,\Ee^3(\aI^2+\hv\Emu)} \nonumber\\[4pt]
  &= \frac{9\pi^2}{\aI\cdot\pi^3(\aI^2+3\pi^2)}
   = \frac{9}{\aI\,\pi\,(\aI^2+3\pi^2)}.
   \label{eq:rSinf_closed}
\end{align}
Numerically: $r\Sinf = 1.111\,479\,647\,46\ldots\times10^{-6}$.
Verified at \texttt{mp.dps=60}: equation~\eqref{eq:rSinf_closed} matches
$r\Sinf$ to $< 10^{-55}$.

\subsection{The deviation $\varepsilon$ is not algebraic}

The residual $c = \delta_\infty/207 = (\Rinf - T)/207$ depends on the
CODATA-2018 empirical value $T = 206.7682830$.  Since $T$ is a measured
quantity (not derivable from $\pi$ and $\aI$), the ratio
$c/(r\Sinf)$ is not a pure function of TOE algebraic constants, and
$\varepsilon$ has no closed form in any basis built from
$\{\pi, \aI, \Om, \hv, \rr, \Sinf\}$.

\subsection{PSLQ result}

A PSLQ search at \texttt{mp.dps=60} over the 11-element basis
\[
  \mathcal{B} = \bigl\{
    \varepsilon,\; \rr^2,\; \rr^3,\; \rr^2/\hv,\;
    \Emu/\aI^4,\; \Ee/\aI^3,\; \Om/\aI^3,\;
    \Emu^2/\aI^4,\; r\Sinf/100,\;
    1/\aI^2,\; \Ee^2/\aI^4
  \bigr\}
\]
(with \texttt{maxcoeff}$= 500$, \texttt{tol}$=10^{-20}$)
finds the trivial relation $\Emu/\aI^4 = \Ee^2/\aI^4$ (since $\Emu = \Ee^2$),
and \emph{no non-trivial relation involving $\varepsilon$}.
No ratio $\varepsilon/B_i$ for $B_i \in \mathcal{B}\setminus\{\varepsilon\}$
is close to a simple fraction; the closest is
$\varepsilon/\rr^2 \approx -0.0886$, which is not a recognisable
rational multiple.

\begin{proposition}[No algebraic form for $\varepsilon$]
\label{prop:no_alg}
The deviation $\varepsilon = c/(r\Sinf) - 1/100 \approx -2.20\times10^{-7}$
has no integer-linear representation in the natural $G_2$--$A_2$ basis at
\textup{\texttt{mp.dps=60}} with coefficients $\leq 500$.
The near-$1/100$ coincidence is numerically striking but has no currently
identifiable closed-form origin in the TOE algebra.
\end{proposition}

\begin{remark}[Why this is expected]
$\varepsilon$ is non-zero precisely because $\Rinf \neq T$:
if the exact prediction equalled the CODATA value, then $c = 0$ and the
ratio $c/(r\Sinf)$ would be 0 (not $1/100$).  The factor $\approx 1/100$
arises from the accidental coincidence that the gap
$\delta_\infty \approx 2.30\times10^{-6}$ is approximately $1/100$ of
$207\cdot r\Sinf \approx 2.30\times10^{-4}$.  There is no identified
mechanism in the $G_2$--$A_2$ structure that would produce a factor of
exactly 100 at the next perturbative order.
OP-rSinf-100 remains open.
\end{remark}

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

\subsection*{Results established by this addendum}

\begin{enumerate}
  \item The $G_2$ root-length ratio $|\alpha_s|^2/|\alpha_\ell|^2 = 1/3
        = 1/h^\vee(A_2)$ is verified.  Under the TOE identification,
        the corresponding energy ratio is $\Emu/\aI^2 = \rr/\hv$
        (Proposition~\ref{prop:ident}).

  \item The Dynkin index of $A_2^{(s)} \hookrightarrow G_2$ is $j = 1/3$
        in the standard Lie-algebra normalisation and $j = 1$ in the TOE
        natural normalisation (Proposition~\ref{prop:dynkin}).

  \item Proposition~P217.1: under the TOE identification, the $\pm 2$
        exponent pattern in $T_{k+1}/T_k$ is accounted for by the
        $G_2 \supset A_2^{(s)}$ branching structure — specifically, the
        short-root squared energy $|\alpha_s|^2_{\mathrm{TOE}} = \Emu$
        and the long-root normaliser $|\alpha_\ell|^2_{\mathrm{TOE}} = \aI^2$
        (Section~\ref{sec:derivation}).

  \item The proof is \textbf{conditional}: it requires the TOE identification
        rule as an axiom.  The specific proof obligation for upgrading to a
        full proof is identified (Conjecture~\ref{conj:P217.1}).

  \item $r\Sinf = 9/[\aI\pi(\aI^2+3\pi^2)]$ (closed form,
        verified at \texttt{mp.dps=60}).

  \item OP-rSinf-100: $\varepsilon = c/(r\Sinf) - 1/100 = -2.204\times10^{-7}$
        has no TOE closed form (Proposition~\ref{prop:no_alg}).
\end{enumerate}

\subsection*{Programme status}

\begin{description}
  \item[\textbf{OP-G2A2-exponents (partially resolved)}]
    The $\pm 2$ exponent pattern is derived from $G_2 \supset A_2^{(s)}$
    branching, conditional on the TOE identification rule.
    Status: \textbf{CONJECTURED} (conditional derivation).
    Remaining proof obligation: derive the energy assignment
    $|\alpha_s| = \Ee$, $|\alpha_\ell| = \aI$ from the S$^3$ geometry
    of Papers 18/30.

  \item[\textbf{OP-rSinf-100 (open)}]
    $\varepsilon = -2.204\times10^{-7}$ has no closed form in the current
    $G_2$--$A_2$ basis.

  \item[\textbf{P18-T2 uniqueness programme}]
    This addendum closes the last \emph{structural} gap in the programme:
    all three exponent increments ($+1$ in $\hv$, $+2$ in $\Ee$, $-2$ in
    $\aI$) are now accounted for within the $G_2 \supset A_2$ framework,
    subject to the TOE identification.  The programme may be declared
    \textbf{ESTABLISHED} (conditionally) subject to the proof obligation
    of Conjecture~\ref{conj:P217.1}.
\end{description}

\bigskip
\noindent\textit{All numerical computations verified at \texttt{mp.dps=60}
with $\geq 45$ assertions in \texttt{Lumen/corpus/addenda/verify/verify\_P217.py},
all passing.}

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

\end{document}
