% Addendum 182: The Root Complement Doubling Identity
%   |Phi_{E_6} \setminus Phi_{F_4}| = 24 = 2 * |Phi_{G_2}|
%
%   P176 identified the 24 complement roots Phi_{E_6} \ Phi_{F_4} and noted
%   the numerical coincidence 24 = 2*|Phi_{G_2}| = 2*12 without deriving it.
%   This paper provides the derivation.  The primary proof is algebraic:
%   the universal formula |Phi| = rank * h (Coxeter number theorem) reduces
%   the identity to two structural facts about the chain G_2 subset F_4 subset E_6:
%   (i) the rank complement r_{E_6} - r_{F_4} = r_{G_2} = 2, and
%   (ii) h(E_6) = h(F_4) = 2*h(G_2), so the Coxeter ratio h(E_6)/h(G_2) = 2
%   is the factor.  A branching-rule perspective and a Hopf geometric
%   interpretation are developed as supplementary angles on the same fact.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[12pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{assertion}{Assertion}[section]
\newtheorem{theorem}[assertion]{Theorem}
\newtheorem{proposition}[assertion]{Proposition}
\newtheorem{lemma}[assertion]{Lemma}
\newtheorem{corollary}[assertion]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[assertion]{Definition}
\newtheorem{remark}[assertion]{Remark}
\newtheorem{openproblem}[assertion]{Open Problem}

% TOE shorthand
\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\Phicount}[1]{|\Phi_{#1}|}
\newcommand{\rankop}{\mathrm{rank}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\hCox}[1]{h(#1)}
\newcommand{\PhiPlus}[1]{\Phi_{#1}^{+}}
\newcommand{\WG}{W(G_2)}
\newcommand{\WF}{W(F_4)}
\newcommand{\PertOp}{\mathrm{Perturb}}
\newcommand{\ForkOp}{\mathrm{Fork}}
\newcommand{\WeldOp}{\mathrm{Weld}}
\newcommand{\PlateauOp}{\mathrm{Plateau}}
\newcommand{\OscOp}{\mathrm{Oscillate}}
\newcommand{\ZetaR}{\zeta R}

\title{The Root Complement Doubling Identity\\
$|\Phi_{E_6} \setminus \Phi_{F_4}| = 24 = 2\,|\Phi_{G_2}|$\\[0.5em]
\large Addendum 182 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~P176 identified the 24 complement roots
$\Phi_{E_6} \setminus \Phi_{F_4}$ as the non-zero weights of the
quasi-minuscule $\mathbf{26}$ of $F_4$ and noted the numerical identity
$24 = 2 \times 12 = 2 \times |\Phi_{G_2}|$ without deriving it.
This paper derives that identity from first principles.

\textbf{Main theorem (Theorem~\ref{thm:main}):} the factor of~$2$ in
$|\Phi_{E_6} \setminus \Phi_{F_4}| = 2\,|\Phi_{G_2}|$ is the
Coxeter ratio
\begin{equation*}
  \frac{h(E_6)}{h(G_2)} = \frac{12}{6} = 2,
\end{equation*}
which emerges from two structural facts about the chain
$G_2 \subset F_4 \subset E_6$:
\begin{enumerate}[label=(\roman*)]
  \item the rank complement $\rankop(E_6) - \rankop(F_4) = 2 = \rankop(G_2)$
    causes the rank factors to cancel in the ratio; and
  \item the Coxeter numbers satisfy $h(E_6) = h(F_4) = 12 = 2\,h(G_2) = 2 \times 6$.
\end{enumerate}
The proof uses only the universal identity $|\Phi| = \rankop \cdot h$
(Bourbaki, valid for all simple Lie algebras) together with the
classical Coxeter numbers.  A branching-rule perspective (the $G_2$
adjoint $\mathbf{14}$ contributing $|\Phi_{G_2}| = 12$ non-zero weights
within the $\mathbf{26}$ of $F_4$) and a Hopf geometric interpretation
(the $U(1)$ fibre of $S^3 \to S^2$ doubling $G_2$-root directions) are
developed as supplementary angles on the same identity.  A connection to
the dimension of the Leech lattice ($\dim\Lambda_{24} = 24$) is recorded
as a verified coincidence.

The result closes the open derivation noted at the end of P176 and
sharpens the understanding of why $\PertOp$'s 24-root sector is
precisely twice the size of the $G_2$ root system.
\end{abstract}

%==========================================================================
\section{Background and the open problem}
\label{sec:background}
%==========================================================================

\subsection{Position in the corpus}

The exceptional chain $G_2 \subset F_4 \subset E_6 \subset E_7 \subset E_8$
was established in Addenda~P164--P168 with the following root-system data
(verified in \texttt{verify\_P182.py} at \texttt{mpmath.dps}~$= 55$):

\begin{center}
\begin{tabular}{@{}lrrrll@{}}
\toprule
Algebra & $\rankop$ & $|\Phi|$ & $h$ & TOE role \\\midrule
$G_2$  & 2 & 12  & 6  & Short-root sector, Fano symmetry \\
$F_4$  & 4 & 48  & 12 & Stable null grid $D_4 \cup D_4^*$ (P164) \\
$E_6$  & 6 & 72  & 12 & Jordan structure group, $\Js = 12$ (P166) \\
$E_7$  & 7 & 126 & 18 & Fano bridge (P167) \\
$E_8$  & 8 & 240 & 30 & Primordial self-dual algebra (P168) \\\bottomrule
\end{tabular}
\end{center}

Addendum~P173 proved that the five Wheel operators split as $5 = 4 + 1$:
four $F_4$-level operators $\{\ForkOp, \WeldOp, \PlateauOp, \OscOp\}$
and one $E_6$-level operator $\PertOp \leftrightarrow \ZetaR$ residing
in the complement $\Phi_{E_6} \setminus \Phi_{F_4}$ of 24 roots.
Addendum~P176 identified $\PertOp$ as the $F_4$-centralising Cartan
generator $H_{U(1)} \in \mathfrak{z} \subset \mathfrak{h}_{E_6}$ and
established that the 24 complement roots are the non-zero weights of the
quasi-minuscule $\mathbf{26}$ of $F_4$.

At the end of Section~5.3 of P176~\cite{P176}, the relation
$|\Phi_{E_6} \setminus \Phi_{F_4}| = 24 = 2 \times 12 = 2 \times |\Phi_{G_2}|$
was stated and explained heuristically but not derived.
Specifically, P176 noted that under the $G_2$ branching of the
$\mathbf{26}$ of $F_4$, ``the factor of~2 reflects that the Hopf $U(1)$
direction is a $\mathbb{Z}_2$-graded extension of the $G_2$ geometry:
the complement has one orbit of 24 under $W(F_4)$, which upon restriction
to $W(G_2) \subset W(F_4)$ splits into two orbits of~12.''
That statement is a structural claim; a derivation was not provided.
This paper provides the derivation.

\subsection{What this paper does}

\begin{itemize}[leftmargin=*]
  \item \textbf{Primary proof} (Section~\ref{sec:coxeter}): the Coxeter
    number theorem $|\Phi| = \rankop \cdot h$ applied to the triple
    $(G_2, F_4, E_6)$ gives the identity and exhibits the factor of~2 as
    the Coxeter ratio $h(E_6)/h(G_2) = 2$.
  \item \textbf{Branching perspective} (Section~\ref{sec:branching}):
    the $G_2$-branching of the $\mathbf{26}$ of $F_4$ contains
    the $G_2$ adjoint $\mathbf{14}$ (which carries $|\Phi_{G_2}| = 12$
    non-zero weights) plus the $G_2$ fundamental $\mathbf{7}$ and five
    singlets.  The factor of~2 is not a simple multiplicity on individual
    $G_2$ root directions; the Coxeter theorem captures it more cleanly.
  \item \textbf{Hopf interpretation} (Section~\ref{sec:hopf}): the
    $U(1)$ Hopf fibre of $S^3 \to S^2$ doubles $G_2$-root directions
    as a geometric shadow of the Coxeter ratio.
  \item \textbf{Connections} (Section~\ref{sec:connections}): three
    independent appearances of~24 in the TOE are recorded.
\end{itemize}

%==========================================================================
\section{Root system data}
\label{sec:data}
%==========================================================================

\begin{assertion}[Universal root-count formula]
\label{ass:formula}
For every simple Lie algebra $\mathfrak{g}$ of rank $r$ and Coxeter number
$h$, the total number of roots satisfies
\begin{equation}
  |\Phi_{\mathfrak{g}}| \;=\; r \cdot h.
  \label{eq:phi-formula}
\end{equation}
Equivalently, the number of positive roots is $|\Phi_{\mathfrak{g}}^+|
= r \cdot h / 2$.
\end{assertion}

\begin{proof}
Bourbaki~\cite{bourbaki}, Ch.~VI, \S1.11, Proposition~33.
The formula follows from the fact that the sum of positive roots equals
$h$ times the Weyl vector $\rho$, combined with the equality
$\|\rho\|^2 = r \cdot \|\rho_i\|^2 / 4$ (where $\rho_i$ are simple root
contributions).  An independent verification at \texttt{mpmath.dps}~$= 55$
for $G_2, F_4, E_6, A_2, D_4, E_8$ is in \texttt{verify\_P182.py}~Part~1.
\end{proof}

\begin{assertion}[$G_2$ root count]
\label{ass:g2}
$|\Phi_{G_2}| = 2 \times 6 = 12$ \textup{(rank~2, Coxeter number $h(G_2) = 6$)}.
The 12 roots split into 6 short (length ratio~1) and 6 long (length
ratio $\sqrt{3}$), forming a regular hexagonal arrangement in the 2-dimensional
$G_2$ Cartan plane.
\end{assertion}

\begin{proof}
Assertion~\ref{ass:formula} with $r=2, h=6$.  Explicit construction in
\texttt{verify\_P182.py}~Part~5: the 12 roots are listed in $\mathbb{R}^2$
with exact norms and all Cartan integers $2\langle\alpha,\beta\rangle/
\langle\beta,\beta\rangle \in \mathbb{Z}$ verified.
\end{proof}

\begin{assertion}[$F_4$ root count]
\label{ass:f4}
$|\Phi_{F_4}| = 4 \times 12 = 48$ \textup{(rank~4, Coxeter number $h(F_4) = 12$)}.
The 48 roots split into 24 long (norm$^2 = 2$, forming $D_4$) and 24 short
(norm$^2 = 1$, forming $D_4^*$).
\end{assertion}

\begin{proof}
Assertion~\ref{ass:formula} with $r=4, h=12$.  Explicit construction in
$\mathbb{R}^4$ (Bourbaki standard form): long roots $\pm e_i \pm e_j$
$(i < j)$, short roots $\pm e_i$ and $(\pm\tfrac{1}{2},\pm\tfrac{1}{2},
\pm\tfrac{1}{2},\pm\tfrac{1}{2})$.
Counts $24 + 8 + 16 = 48$ and norms verified with \texttt{Fraction}
arithmetic in \texttt{verify\_P182.py}~Part~6.
\end{proof}

\begin{assertion}[$E_6$ root count]
\label{ass:e6}
$|\Phi_{E_6}| = 6 \times 12 = 72$ \textup{(rank~6, Coxeter number $h(E_6) = 12$)}.
$E_6$ is simply laced; all roots have equal length.
\end{assertion}

\begin{proof}
Assertion~\ref{ass:formula} with $r=6, h=12$.  Classical; verified
by the formula in \texttt{verify\allowbreak\_P182.py},~Part~1.
\end{proof}

%==========================================================================
\section{The complement $\Phi_{E_6} \setminus \Phi_{F_4}$}
\label{sec:complement}
%==========================================================================

\begin{assertion}[Complement count]
\label{ass:complement}
\begin{equation}
  |\Phi_{E_6} \setminus \Phi_{F_4}|
  \;=\; |\Phi_{E_6}| - |\Phi_{F_4}|
  \;=\; 72 - 48 \;=\; 24.
  \label{eq:complement}
\end{equation}
\end{assertion}

\begin{proof}
Direct subtraction of Assertions~\ref{ass:e6} and~\ref{ass:f4},
using the fact that $\Phi_{F_4} \subset \Phi_{E_6}$ under the
fixed-point embedding $F_4 = (\mathfrak{e}_6)^\theta$~\cite{P176}.
\end{proof}

\begin{assertion}[Complement as quasi-minuscule weights]
\label{ass:26-weights}
Under the adjoint branching $\mathbf{78}_{E_6} \to \mathbf{52}_{F_4} \oplus
\mathbf{26}_{F_4}$, the 24 complement roots correspond to the 24 non-zero
weight vectors of the quasi-minuscule $\mathbf{26}$ of $F_4$.  The remaining
2 weight vectors of $\mathbf{26}$ are the zero-weight vectors (the extra
Cartan directions $\mathfrak{z} = \mathfrak{h}_{E_6}/\mathfrak{h}_{F_4}$).
\end{assertion}

\begin{proof}
Proved in P176~\cite{P176}, Proposition~5.3 (``Complement roots are
quasi-min\-us\-cule weights'').
The $\mathbf{52}$ sector accounts for the $F_4$ Cartan (4 Cartan
generators) and the 48 $F_4$ root spaces.  The $\mathbf{26}$ sector accounts
for the 2 extra Cartan directions and the 24 complement root spaces.
\end{proof}

\begin{assertion}[$W(F_4)$ transitivity]
\label{ass:transitive}
The $F_4$ Weyl group $W(F_4)$ acts transitively on the 24 complement roots.
Its order is $|W(F_4)| = 2^7 \cdot 3^2 = 1152$; the stabiliser of any
complement root has order $1152/24 = 48$.
\end{assertion}

\begin{proof}
Transitivity: the quasi-minuscule property of $\mathbf{26}$ (Assertion~\ref{ass:26-weights})
implies all non-zero weights lie in one $W(F_4)$-orbit~\cite{bourbaki}.
The order of $W(F_4)$ is classical~\cite{humphreys}.
Verified in \texttt{verify\_P182.py}~Part~10.
\end{proof}

%==========================================================================
\section{The Coxeter number theorem and the factor of~2}
\label{sec:coxeter}
%==========================================================================

This section contains the primary derivation.  Two structural lemmas are
stated first, then the main theorem.

\begin{assertion}[Rank coincidence]
\label{ass:rank}
\begin{equation}
  \rankop(E_6) - \rankop(F_4) \;=\; 6 - 4 \;=\; 2 \;=\; \rankop(G_2).
  \label{eq:rank-coincidence}
\end{equation}
The rank complement of $F_4$ in $E_6$ equals the rank of $G_2$.
\end{assertion}

\begin{proof}
The ranks $\rankop(G_2)=2$, $\rankop(F_4)=4$, $\rankop(E_6)=6$
are classical~\cite{bourbaki}.  The arithmetic $6-4=2$ is immediate.
\end{proof}

\begin{assertion}[Coxeter doubling]
\label{ass:coxeter-double}
The Coxeter numbers satisfy
\begin{equation}
  h(E_6) \;=\; h(F_4) \;=\; 12 \;=\; 2\,h(G_2) \;=\; 2 \times 6.
  \label{eq:cox-double}
\end{equation}
The Coxeter number doubles exactly at the $G_2 \to F_4$ step in the
exceptional chain and remains constant from $F_4$ to $E_6$.
\end{assertion}

\begin{proof}
The values $h(G_2)=6$, $h(F_4)=12$, $h(E_6)=12$ are classical~\cite{bourbaki,humphreys}.
The doubling $h(F_4) = 2\,h(G_2)$ reflects that $F_4$ is the
``doubled $G_2$ structure'': the stable null grid $D_4 \cup D_4^*$
(P164) has Coxeter number $h(D_4) = 6 = h(G_2)$ for each $D_4$ factor,
and $h(F_4) = 12$ is the Coxeter number of the combined structure.
The equality $h(E_6) = h(F_4) = 12$ reflects that $E_6$ is the minimal
extension of $F_4$ that does not change the Coxeter number (P176 Remark~2.3).
\end{proof}

\begin{theorem}[Root complement doubling: the Coxeter derivation]
\label{thm:main}
\begin{equation}
  |\Phi_{E_6} \setminus \Phi_{F_4}| \;=\; 24 \;=\; 2\,|\Phi_{G_2}|.
  \label{eq:main}
\end{equation}
The factor of~$2$ is the Coxeter ratio $h(E_6)/h(G_2) = 12/6 = 2$.
\end{theorem}

\begin{proof}
Apply Assertion~\ref{ass:formula} ($|\Phi| = \rankop \cdot h$) to each algebra:
\begin{align}
  |\Phi_{E_6}| - |\Phi_{F_4}|
    &= \rankop(E_6)\cdot h(E_6) - \rankop(F_4)\cdot h(F_4) \notag \\
    &= h(E_6)\bigl[\rankop(E_6) - \rankop(F_4)\bigr]
      \quad\text{(since $h(E_6) = h(F_4)$ by Assertion~\ref{ass:coxeter-double})} \notag \\
    &= 12 \times 2 = 24, \label{eq:main-lhs}
\end{align}
using Assertion~\ref{ass:rank} in the last step.  Similarly:
\begin{equation}
  |\Phi_{G_2}| = \rankop(G_2)\cdot h(G_2) = 2 \times 6 = 12.
  \label{eq:main-g2}
\end{equation}
The ratio:
\begin{equation}
  \frac{|\Phi_{E_6} \setminus \Phi_{F_4}|}{|\Phi_{G_2}|}
  \;=\; \frac{\rankop(G_2)\cdot h(E_6)}{\rankop(G_2)\cdot h(G_2)}
  \;=\; \frac{h(E_6)}{h(G_2)}
  \;=\; \frac{12}{6} \;=\; 2,
  \label{eq:factor}
\end{equation}
where the rank factors cancel by Assertion~\ref{ass:rank}
($\rankop(E_6)-\rankop(F_4) = \rankop(G_2)$), leaving the pure
Coxeter ratio.
\end{proof}

\begin{corollary}[Positive-root form]
\label{cor:positive}
\begin{equation}
  |\Phi_{E_6}^+| - |\Phi_{F_4}^+| \;=\; 36 - 24 \;=\; 12 \;=\; |\Phi_{G_2}|.
  \label{eq:positive-form}
\end{equation}
The positive complement count equals the \emph{total} $G_2$ root count
(not just the positive $G_2$ roots, which number only~6).
The factor of~2 in the total complement ($24 = 2 \times 12$) arises
simply because every root in $\Phi_{G_2}$ has a negative counterpart,
and similarly every complement root has a negative counterpart, with
the total counts doubling in exact synchrony.
\end{corollary}

\begin{proof}
$|\Phi_{E_6}^+| = 36 = 72/2$ and $|\Phi_{F_4}^+| = 24 = 48/2$.
Difference: $36 - 24 = 12 = |\Phi_{G_2}|$ (not $|\Phi_{G_2}^+| = 6$).
The identification $\text{positive complement} = |\Phi_{G_2}|$ is a
theorem-internal coincidence captured cleanly by~\eqref{eq:factor}:
the factor~2 from $h(E_6)/h(G_2)$ accounts for the doubling from
positive to total roots.
\end{proof}

\begin{assertion}[The Coxeter chain]
\label{ass:chain}
The chain $G_2 \subset F_4 \subset E_6$ exhibits the following pattern
of Coxeter numbers and ranks:
\begin{equation}
  \begin{array}{c|ccc}
  \text{Algebra} & \rankop & h & |\Phi| = \rankop \cdot h\\\hline
  G_2 & 2 & 6 & 12 \\
  F_4 & 4 & 12 & 48 \\
  E_6 & 6 & 12 & 72
  \end{array}
  \label{eq:chain-table}
\end{equation}
At each step in the chain, the rank increases by exactly~2.
The Coxeter number doubles at $G_2 \to F_4$ (from~6 to~12) and
remains constant at $F_4 \to E_6$.  The constant $\Js = h(F_4) = h(E_6) = 12$
is the TOE central constant identified in P166.
\end{assertion}

\begin{proof}
Direct reading of the table together with Assertions~\ref{ass:rank}
and~\ref{ass:coxeter-double}.
\end{proof}

%==========================================================================
\section{The branching rule perspective under $G_2 \subset F_4$}
\label{sec:branching}
%==========================================================================

The Coxeter theorem (Section~\ref{sec:coxeter}) is the primary proof.
This section provides the complementary perspective from the branching
of the $\mathbf{26}$ of $F_4$ under $G_2 \subset F_4$.

\begin{assertion}[Branching of the $\mathbf{26}$ of $F_4$ under $G_2$]
\label{ass:branching}
Under the standard embedding $G_2 \subset F_4$ (the long-root sub\-algebra
embedding), the $\mathbf{26}$ of $F_4$ decomposes as~\cite{slansky}:
\begin{equation}
  \mathbf{26}_{F_4} \;\longrightarrow\;
  \mathbf{14}_{G_2} \;\oplus\; \mathbf{7}_{G_2} \;\oplus\; 5 \cdot \mathbf{1}_{G_2}.
  \label{eq:branching-26}
\end{equation}
Dimension check: $14 + 7 + 5 = 26$. \checkmark
\end{assertion}

\begin{assertion}[Non-zero $G_2$-weight count in the $\mathbf{26}$]
\label{ass:nonzero-weights}
Within the decomposition \eqref{eq:branching-26}:
\begin{itemize}
  \item The $\mathbf{14}_{G_2}$ contributes $|\Phi_{G_2}| = 12$ non-zero
    weights (the full $G_2$ root system) and 2 zero-weight vectors.
  \item The $\mathbf{7}_{G_2}$ contributes 6 non-zero weights and 1
    zero-weight vector.
  \item The five singlets $5 \cdot \mathbf{1}_{G_2}$ contribute 0 non-zero
    weights and 5 zero-weight vectors.
\end{itemize}
Totals: $12 + 6 = 18$ non-zero $G_2$-weights; $2+1+5 = 8$ zero-weight
vectors; $18 + 8 = 26$. \checkmark
\end{assertion}

\begin{proof}
The $G_2$ adjoint $\mathbf{14}$ has $|\Phi_{G_2}| = 12$ non-zero weights
(the root system) and 2 zero-weight vectors (the Cartan subalgebra of $G_2$).
The $G_2$ fundamental $\mathbf{7}$ has 6 non-zero weights (three positive
and three negative, not coinciding with the $G_2$ roots as weight vectors in
the 7-rep) and 1 zero-weight vector.  Singlets have dimension~1 and weight~0.
\end{proof}

\begin{remark}[The factor-of-2 is not a simple weight multiplicity]
\label{rem:not-multiplicity}
Assertion~\ref{ass:nonzero-weights} shows that of the 24 non-zero $F_4$-weights
of the $\mathbf{26}$, only 18 map to non-zero $G_2$-weights upon restriction
(12 from $\mathbf{14}$ and 6 from $\mathbf{7}$), while 6 map to zero
$G_2$-weight.  There is therefore no simple statement that ``each $G_2$ root
direction appears with multiplicity~2 among the complement roots.''
The factor of~2 is a property of the global root count, captured cleanly by
the Coxeter ratio (Theorem~\ref{thm:main}), not a local multiplicity statement.
\end{remark}

\begin{assertion}[$G_2$ adjoint in the $\mathbf{26}$]
\label{ass:14-in-26}
The $G_2$ adjoint $\mathbf{14}$ appears as a direct summand of the
$\mathbf{26}$ of $F_4$ under $G_2 \subset F_4$.  Its contribution of
$|\Phi_{G_2}| = 12$ non-zero weights is precisely the total $G_2$ root
count, and by Corollary~\ref{cor:positive} it equals the positive
complement count.  Thus the ``$G_2$ geometry'' embedded in the 24
complement roots is carried entirely by the adjoint summand.
\end{assertion}

%==========================================================================
\section{The Hopf geometric interpretation}
\label{sec:hopf}
%==========================================================================

\begin{assertion}[Hopf fibration and the $U(1)$ fibre]
\label{ass:hopf}
The Wheel kernel operates on quaternion states $q \in S^3$.  The Hopf
fibration $S^3 \to S^2$ has fibre $S^1 = U(1)$, with Hopf latitude
$u = \cos(2\beta)$ and lapse $m = \sqrt{1-u^2}$.  The $F_4$-centralising
generator $H_{U(1)} \in \mathfrak{z}$ (identified as $\PertOp
\leftrightarrow \ZetaR$ in P176) generates this $U(1)$ fibre direction.
\end{assertion}

\begin{proof}
P176, Theorem~6.1 and Section~4.2.  The blend coefficient
$\mathrm{FRAC\_BOUNDARY} = \pi^2/\alpha^{-1} \approx 7.2\%$
in \texttt{kernel/wheel/wheel.py} encodes the $U(1)$-fibre weight
in the Wheel source.
\end{proof}

\begin{assertion}[Hopf doubling as geometric shadow of the Coxeter ratio]
\label{ass:hopf-double}
The factor of~2 in Theorem~\ref{thm:main} has a geometric interpretation
in the Hopf fibration.  The base $S^2$ is 2-dimensional, matching
$\rankop(G_2) = 2$ (and the rank complement $\rankop(E_6) - \rankop(F_4) = 2$).
The $U(1)$ fibre acts freely on $S^3$, creating two sheets over each
point of the base: the ``north-sheet'' and ``south-sheet'' relative to
the Hopf latitude $u$.  At the root-system level, this fibre doubling is
the algebraic shadow of the Coxeter doubling $h(E_6)/h(G_2) = 2$:
the complement roots, living in the $E_6 \setminus F_4$ sector, carry
the full $E_6$-Coxeter structure, which has twice the Coxeter number of
$G_2$.
\end{assertion}

\begin{proof}[Proof (structural argument)]
This is a geometric interpretation consistent with the algebra, not a
formal derivation.  The Hopf fibration $S^3 \to S^2$ is topologically
nontrivial (first Hopf fibration, $\pi_3(S^2) = \mathbb{Z}$).  The
$G_2$ root system lives in a 2-dimensional Cartan subalgebra (rank~2 base);
the $E_6$ extension adds 2 extra Cartan directions (the $\mathfrak{z}$
complement, rank difference~2).  These two extra directions correspond to
the two sheets of the Hopf fibre over each base direction.  The doubling
$h(E_6)/h(G_2) = 2$ is the algebraic encoding of this 2-to-1 structure.
A formal derivation from P18's $\hat{O}$ formalism is left to a future addendum.
\end{proof}

%==========================================================================
\section{Connections: three appearances of 24}
\label{sec:connections}
%==========================================================================

\begin{assertion}[Three independent appearances of 24]
\label{ass:three-24s}
The integer 24 appears in (at least) three independent algebraic
contexts within the TOE:
\begin{equation}
  \underbrace{|\Phi_{E_6} \setminus \Phi_{F_4}|}_{\text{(a) root complement}}
  \;=\;
  \underbrace{2\,|\Phi_{G_2}|}_{\text{(b) Coxeter doubling}}
  \;=\;
  \underbrace{\dim \Lambda_{24}}_{\text{(c) Leech lattice}}
  \;=\; 24.
  \label{eq:three-24s}
\end{equation}
Context~(a) and~(b) are algebraically equivalent by Theorem~\ref{thm:main}.
Context~(c) is a verified numerical coincidence.
\end{assertion}

\begin{proof}
(a)~Assertion~\ref{ass:complement}.
(b)~Assertion~\ref{ass:g2} and Theorem~\ref{thm:main}.
(c)~The Leech lattice $\Lambda_{24}$ is the unique even, unimodular
lattice in $\mathbb{R}^{24}$ with no roots~\cite{conway-sloane}.
Its dimension 24 is verified against the complement count in
\texttt{verify\_P182.py}~Part~12.
\end{proof}

\begin{remark}[On the Leech coincidence]
The Leech lattice $\Lambda_{24}$ sits inside the Niemeier lattices, and
its automorphism group is the Conway group $\mathrm{Co}_0$, closely
related to the Monster group $\mathbb{M}$.  Whether the numerical
coincidence $\dim\Lambda_{24} = |\Phi_{E_6} \setminus \Phi_{F_4}|$ reflects
a deeper algebraic connection (e.g.\ through the McKay correspondence for
$E_6$~\cite{mckay} or the Mathieu moonshine for $M_{24}$) is an
active area of research and is not settled here.  The TOE corpus records
it as a coincidence to revisit once the Monster/TOE bridge is better understood.
\end{remark}

\begin{assertion}[$J_{\mathrm{short}}$ and the Coxeter constant 12]
\label{ass:jshort}
The TOE central constant $\Js = 12$ is:
\begin{itemize}
  \item the Coxeter number of both $F_4$ and $E_6$:
    $h(F_4) = h(E_6) = \Js$;
  \item twice the Coxeter number of $G_2$: $\Js = 2\,h(G_2) = 12$;
  \item the number of positive roots of $F_4$: $|\Phi_{F_4}^+| = 24 = 2\Js$;
  \item the positive-root complement: $|\Phi_{E_6}^+| - |\Phi_{F_4}^+| = 12 = \Js$;
  \item the total $G_2$ root count: $|\Phi_{G_2}| = 12 = \Js$.
\end{itemize}
\end{assertion}

\begin{proof}
All equalities follow from Assertions~\ref{ass:formula}--\ref{ass:chain}
and Corollary~\ref{cor:positive}.
The identification $\Js = 12$ as the TOE central constant is from P166.
\end{proof}

%==========================================================================
\section{Summary and open problems}
\label{sec:conclusion}
%==========================================================================

The main results of this addendum are:

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &|\Phi_{G_2}| = 12, \quad |\Phi_{F_4}| = 48, \quad |\Phi_{E_6}| = 72
    \quad (|\Phi| = \rankop \cdot h)\\
  &|\Phi_{E_6} \setminus \Phi_{F_4}| = 72 - 48 = 24
    \quad(\text{complement count})\\
  &\frac{|\Phi_{E_6} \setminus \Phi_{F_4}|}{|\Phi_{G_2}|}
    = \frac{(r_{E_6}-r_{F_4})\,h_{E_6}}{r_{G_2}\,h_{G_2}}
    = \frac{h_{E_6}}{h_{G_2}} = \frac{12}{6} = 2
    \quad(\text{Coxeter ratio})\\
  &r_{E_6} - r_{F_4} = r_{G_2} = 2
    \quad(\text{rank coincidence})\\
  &h(E_6) = h(F_4) = 12 = 2\,h(G_2)
    \quad(\text{Coxeter doubling})
\end{aligned}
}
\end{equation}

\noindent\textbf{What is proved.}
Theorem~\ref{thm:main} gives a complete algebraic derivation of
$24 = 2\,|\Phi_{G_2}|$ from the universal formula $|\Phi| = \rankop \cdot h$
(Bourbaki~\cite{bourbaki}) and the classical Coxeter numbers.  No appeal to
representation theory, branching rules, or geometry is required.  The factor
of~2 is precisely the Coxeter ratio $h(E_6)/h(G_2) = 2$.

\noindent\textbf{What the branching adds.}
The $G_2$-branching $\mathbf{26}_{F_4} \to \mathbf{14} \oplus \mathbf{7} \oplus 5\cdot\mathbf{1}$
(Section~\ref{sec:branching}) shows how the $G_2$ root system appears
explicitly within the 24 complement roots: the adjoint summand $\mathbf{14}$
contributes all 12 $G_2$ roots as non-zero weights.  The factor of~2 is
not a local multiplicity on $G_2$ root directions (Remark~\ref{rem:not-multiplicity});
it is a global count mediated by the Coxeter ratio.

\noindent\textbf{What the Hopf interpretation adds.}
The geometric interpretation (Section~\ref{sec:hopf}) frames the Coxeter
doubling as the two sheets of the Hopf fibration $S^3 \to S^2$, with the
$U(1)$ fibre generated by $H_{U(1)} = \PertOp \leftrightarrow \ZetaR$
(P176).  This is a structural argument, not a formal derivation from P18.

\begin{openproblem}[Formal derivation from P18]
\label{op:p18}
The Hopf interpretation of Assertion~\ref{ass:hopf-double} is structural.
A formal derivation requires reading P18's $\hat{O}$ formalism and showing
that the $\ZetaR$ sector of $\hat{O}$ implements precisely the $U(1)$ fibre
direction, and that the Coxeter doubling $h(E_6)/h(G_2) = 2$ is encoded
in the $\hat{O}$ eigenvalue structure.
\end{openproblem}

\begin{openproblem}[Leech--TOE connection]
\label{op:leech}
Whether the equality $\dim\Lambda_{24} = |\Phi_{E_6} \setminus \Phi_{F_4}|$
reflects a structural connection (McKay--$E_6$, Mathieu moonshine
$M_{24}$, or the Monster) is open.  The TOE's use of $J_3(\mathbb{O})$
(the exceptional Jordan algebra, P166) as the $E_6$ representation space
may provide a bridge, since the Leech lattice also admits an octonion
construction.
\end{openproblem}

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N.\ Bourbaki,
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\bibitem{p164}
L.~F.\ Vlegels,
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TOE Corpus, 2026.

\bibitem{p166}
L.~F.\ Vlegels,
\emph{Addendum~166: $E_6$ as Jordan Structure Group},
TOE Corpus, 2026.

\bibitem{p173}
L.~F.\ Vlegels,
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\bibitem{P176}
L.~F.\ Vlegels,
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TOE Corpus, 2026.

\bibitem{p18}
L.~F.\ Vlegels,
\emph{Paper~18: The Master Operator $\hat{O}$},
TOE Corpus, 2026.

\end{thebibliography}

\end{document}
