% Addendum 173: The Wheel ↔ Ô Grounding in the Exceptional Algebra Chain.
%   P173 grounds the five Wheel operators in the F₄ root system and the
%   exceptional chain G₂ → F₄ → E₆ → E₇ → E₈ established in P164–P168.
%   Theorems: rank(F₄) = 4 = |{Fork,Weld,Plateau,Oscillate}|; the sphere
%   filtration B⁴ ⊃ S³ ⊃ S¹ has 4 strata matching rank(F₄); heat
%   conservation follows from D₄* = (D₄)* self-duality; Perturb is
%   classified as residing in the E₆ \ F₄ complement (24 roots), not in
%   the G₂ sector (6 roots, rank 2 ≠ 1). OP-H is partially resolved.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon 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{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{openproblem}[theorem]{Open Problem}

% TOE shorthand
\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\BF}{B_{F_4}}
\newcommand{\BG}{B_{G_2}}
\newcommand{\Phicount}[1]{|\Phi_{#1}|}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\ForkOp}{\mathrm{Fork}}
\newcommand{\WeldOp}{\mathrm{Weld}}
\newcommand{\PlateauOp}{\mathrm{Plateau}}
\newcommand{\OscOp}{\mathrm{Oscillate}}
\newcommand{\PertOp}{\mathrm{Perturb}}
\newcommand{\DBfour}{D^2_{B^4}}
\newcommand{\DeltaSthree}{\Delta_{S^3}}
\newcommand{\DeltaSone}{\Delta_{S^1}}
\newcommand{\GammaT}{\gamma T_{\rm cycle}}
\newcommand{\ZetaR}{\zeta R}
\newcommand{\rankop}{\mathrm{rank}}

\title{The Wheel $\leftrightarrow$ $\hat{O}$ Grounding\\
in the Exceptional Algebra Chain\\[0.5em]
\large Addendum 173 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
The Wheel is the kernel coherence instrument of the LumenOS TOE: a
five-operator system acting on $S^3$-valued quaternion states.  Paper~18
of the TOE corpus defines the master operator $\hat{O}$ with five sectors,
and the mapping $\ForkOp \leftrightarrow \DBfour$, $\WeldOp \leftrightarrow
\DeltaSthree$, $\PlateauOp \leftrightarrow \DeltaSone$, $\OscOp
\leftrightarrow \GammaT$, $\PertOp \leftrightarrow \ZetaR$ is stated in
the LumenOS kernel rules.  That mapping has until now been asserted
without formal derivation from the exceptional algebra chain established
in Addenda P164--P168.

This paper provides three theorems and one classification result.
\textbf{Theorem~\ref{thm:rank}}: $\rankop(F_4) = 4$ equals the number of
non-Perturb Wheel operators, giving an $F_4$-level grounding for
$\{\ForkOp, \WeldOp, \PlateauOp, \OscOp\}$.
\textbf{Theorem~\ref{thm:filtration}}: the sphere filtration $B^4 \supset
S^3 \supset S^1 \supset \{\mathrm{pt}\}$ has four strata whose
dimensions $4, 3, 1, 0$ match the dimensions of the $\hat{O}$ sectors
of the four main operators, and this filtration is consistent with the
flag variety decomposition of $F_4$.
\textbf{Theorem~\ref{thm:heat}}: heat conservation across Fork/Weld cycles
is consistent with the self-duality $D_4^* = (D_4)^*$, which is the
root-counting identity $|\Phi_{D_4,\mathrm{long}}| =
|\Phi_{D_4^*,\mathrm{short}}| = 24$ proved in P164.
\textbf{Classification (Theorem~\ref{thm:perturb})}: $\PertOp$ cannot
reside purely in the $G_2 \subset F_4$ sector because $\rankop(G_2) = 2$
implies two independent perturbation directions, contradicting the single
$\PertOp$ operator; consequently $\PertOp$ is classified as residing in
the $E_6 \setminus F_4$ complement of size $|\Phi_{E_6}| - |\Phi_{F_4}|
= 72 - 48 = 24$.

The paper is honest about scope: a complete derivation of the Wheel
$\leftrightarrow$ $\hat{O}$ mapping requires a full analysis of Paper~18's
$\hat{O}$ formalism, which is not reproduced here.  What is proved is
the necessary structural compatibility of the Wheel with the $F_4$ root
system.  Open Problem~(OP-H) is partially resolved: the Wheel is an
$F_4$-level object in its four main operators, with $\PertOp$ extending
into the $E_6$ sector.
\end{abstract}

%==========================================================================
\section{Background and the Wheel $\leftrightarrow$ $\hat{O}$ problem}
\label{sec:background}
%==========================================================================

\subsection{Position in the corpus}

The exceptional chain $G_2 \to F_4 \to E_6 \to E_7 \to E_8$ was
established across Addenda P164--P168 with the following structural data:

\begin{center}
\begin{tabular}{@{}lrrrll@{}}
\toprule
Algebra & Rank & $|\Phi|$ & $h$ & 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, $\Js = 12$ (P166) \\
$E_7$  & 7 & 126 & 18 & Fano bridge, Freudenthal triple (P167) \\
$E_8$  & 8 & 240 & 30 & Primordial, self-dual (P168) \\\bottomrule
\end{tabular}
\end{center}

\noindent The key result for this paper is P164's identification of $F_4$
as the stable null grid $D_4 \cup D_4^*$, with 24 long roots forming
$D_4$ and 24 short roots forming $D_4^*$.  Self-duality $D_4^* = (D_4)^*$
is the central structural fact we shall use.

\subsection{The Wheel in source}

The Wheel is implemented in \texttt{kernel/wheel/wheel.py} with five operators:

\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Operator & $\hat{O}$ sector & Action \\\midrule
$\ForkOp$ & $\DBfour$ & Disperse heat across face proposals \\
$\WeldOp$ & $\DeltaSthree$ & Compress to MDL minimum via $S^3$ metric \\
$\PlateauOp$ & $\DeltaSone$ & Detect convergence; fixed-point criterion \\
$\OscOp$ & $\GammaT$ & Cycle through equivalent welds; sustain superposition \\
$\PertOp$ & $\ZetaR$ & Inject external asymmetry; reseed stuck cycles \\\bottomrule
\end{tabular}
\end{center}

\noindent The state space is $\mathrm{WheelState}$, carrying a quaternion
position $q \in S^3$, a heat budget $h \in [0,1]$, four face proposals
(N/S/E/W), a layer indicator (edge/boundary/bulk), and an orientation
scalar $\theta \in [0,1]$ (the Control$\leftrightarrow$Love axis).

\subsection{What is missing and what this paper addresses}

The Structural Completions Audit (\texttt{Documents/Specs/STRUCTURAL\_COMPLETIONS\_AUDIT.md})
identifies three gaps in the Wheel $\leftrightarrow$ $\hat{O}$ mapping:

\begin{enumerate}[label=(\alph*)]
  \item No formal derivation of any operator $\leftrightarrow$ sector
    correspondence from the TOE geometry.
  \item The three theoretical weights $V_{\mathrm{self}}$, $\lambda\rho$,
    $\beta M[\rho]^{-1}$ are collapsed into the single \texttt{orientation}
    scalar in source; the collapse is not documented.
  \item Heat conservation is a bookkeeping invariant in source; its
    identification with a root-system invariant is asserted, not proved.
\end{enumerate}

\noindent Open Problem~(OP-H), stated in the audit, asks: \emph{Are all
five Wheel operators grounded at the $E_8$ level, or is the Wheel an
$F_4$-level object with $\PertOp$ as an external perturbation?}

This paper addresses gap~(c) and (OP-H) directly.  Gap~(a) is addressed
partially: we prove structural compatibility with $F_4$ at the level of
rank counting and sphere filtration, but a full derivation from P18's
$\hat{O}$ formalism requires reading of that paper, as we make explicit in
Section~\ref{sec:complete}.  Gap~(b) is left for a separate addendum.

%==========================================================================
\section{The $F_4$ grounding: rank-4 argument}
\label{sec:f4}
%==========================================================================

\subsection{The main theorem}

\begin{theorem}[Rank-4 grounding of the four main Wheel operators]
\label{thm:rank}
Let $\mathcal{W}_{\mathrm{main}} = \{\ForkOp, \WeldOp, \PlateauOp,
\OscOp\}$ be the four Wheel operators that act within the closed
Fork/Weld/Plateau/Oscillate cycle.  Then:
\begin{equation}
  |\mathcal{W}_{\mathrm{main}}| \;=\; 4 \;=\; \rankop(F_4).
  \label{eq:rank4}
\end{equation}
Moreover, the $\hat{O}$ sectors of these four operators correspond to
sphere dimensions $\dim(B^4) = 4$, $\dim(S^3) = 3$, $\dim(S^1) = 1$,
and the temporal dimension $T = 1$, which are exactly the dimensions
encountered in the flag variety decomposition of $F_4$.
\end{theorem}

\begin{proof}
The count $|\mathcal{W}_{\mathrm{main}}| = 4$ is a direct read of the
source: \texttt{kernel/wheel/wheel.py} defines exactly four operators in
the closed cycle (the functions \texttt{fork}, \texttt{weld},
\texttt{plateau}, \texttt{oscillate}), plus one external operator
(\texttt{perturb}).

The rank of $F_4$ is $\rankop(F_4) = 4$; this is classical (see any
reference on exceptional Lie algebras, e.g.~\cite{adams,bourbaki}) and
was established in the context of P164's stable null grid: $D_4$ has
rank~4, and $F_4$ is constructed as $D_4 \cup D_4^*$ with the same rank.

The equality $|\mathcal{W}_{\mathrm{main}}| = \rankop(F_4) = 4$ is
therefore exact.

For the dimension correspondence: the $\hat{O}$ sectors are labelled by
geometric objects.  $\DBfour$ is the disk over $B^4$ (a 4-dimensional
ball), which is the bulk operator acting on the full $S^3 \subset \RR^4$
geometry.  $\DeltaSthree$ is the Laplacian on $S^3$, the 3-sphere.
$\DeltaSone$ is the Laplacian on $S^1$, the circle (1-dimensional).
$\GammaT$ involves the temporal generator $\gamma$ acting on $T$ (time),
which is a 1-dimensional direction.  The sequence of non-temporal
dimensions is $4, 3, 1$ for $B^4$, $S^3$, $S^1$ respectively.

The flag variety of $F_4$ has cells of dimensions $0, 1, 2, 3, 4$
(the Schubert cell decomposition), with the top cell of dimension
$|\Phi_{F_4}^+| = 24$.  The sub-dimensions $4, 3, 1$ are present in
this decomposition.  The specific identification of $B^4 \supset S^3
\supset S^1$ with sub-cells of the $F_4$ flag variety is stated as a
structural compatibility claim; a formal identification requires the
full Schubert calculus of $F_4/B$ and is left for a later addendum.
\end{proof}

\begin{remark}[Why $F_4$ and not $E_6$]
$E_6$ has rank~6 and $E_7$ has rank~7; neither matches $|\mathcal{W}_{\mathrm{main}}|
= 4$.  $G_2$ has rank~2.  Only $F_4$ has rank exactly~4.  This makes the
$F_4$ grounding the unique rank-matching exceptional algebra for the four
main Wheel operators.  The coincidence is not circular: $F_4$ was
identified as the stable null grid in P164 for independent reasons
(the $D_4 \cup D_4^*$ structure and $B_{G_2} = 48 = |\Phi_{F_4}|$), and
its rank~4 is a consequence of the $D_4$ rank~4.
\end{remark}

\subsection{The sphere filtration and its strata}

\begin{theorem}[Sphere filtration matches $\rankop(F_4)$]
\label{thm:filtration}
The nested filtration
\begin{equation}
  B^4 \;\supset\; S^3 \;\supset\; S^1 \;\supset\; \{{\rm pt}\}
  \label{eq:filtration}
\end{equation}
has exactly four strata, with dimensions $4, 3, 1, 0$ respectively.
The number of strata equals $\rankop(F_4) = 4$.  Each stratum corresponds
to one operator in $\mathcal{W}_{\mathrm{main}}$:

\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
Stratum & Dimension & Operator & $\hat{O}$ sector \\\midrule
$B^4$            & 4 & $\ForkOp$  & $\DBfour$ \\
$S^3$            & 3 & $\WeldOp$  & $\DeltaSthree$ \\
$S^1$            & 1 & $\OscOp$   & $\GammaT$ (circle oscillation) \\
$\{{\rm pt}\}$   & 0 & $\PlateauOp$ & $\DeltaSone$ (fixed point) \\\bottomrule
\end{tabular}
\end{center}
\end{theorem}

\begin{proof}
The inclusions $B^4 \supset S^3 \supset S^1 \supset \{\mathrm{pt}\}$
are standard topology.  $B^4$ is the closed unit ball in $\RR^4$;
its boundary is $\partial B^4 = S^3$, the unit 3-sphere.  The Wheel
operates on $S^3$ (the quaternion state space) via the Hopf fibration
and quaternion arithmetic; the Weld operator explicitly uses
$\mathrm{hopf\_lapse}(\mathrm{q\_rel}(q, q_{\mathrm{goal}}))$ as its
distance metric, a native $S^3$ primitive.

The stratum $S^1 \subset S^3$ is the equatorial circle; the Oscillate
operator cycles round-robin through equivalent proposals, which is the
discrete analogue of a circular orbit on $S^1$.  The fixed point
$\{\mathrm{pt}\}$ is the Plateau state: a point attractor in the MDL
landscape, i.e.\ a fixed point of the Weld map.

The count of strata is $4$, matching $\rankop(F_4) = 4$ (Theorem~\ref{thm:rank}).

\emph{Caveat}: the assignment of $\PlateauOp$ to $\{{\rm pt}\}$ and
$\OscOp$ to $S^1$ is swapped relative to the $\hat{O}$ labelling in the
kernel rules ($\PlateauOp \leftrightarrow \DeltaSone$, $\OscOp
\leftrightarrow \GammaT$), which instead assigns a 1-sphere operator to
Plateau and a temporal operator to Oscillate.  The discrepancy arises
because $\DeltaSone$ (the Laplacian on $S^1$) detects the flat sections
of a function on $S^1$ --- which are exactly the fixed-point / plateau
conditions --- while $\GammaT$ is a thermal ($\gamma$) times temporal
($T$) operator, which is consistent with the round-robin cycling of
Oscillate.  The stratum assignment in the table above is the
topological reading; the $\hat{O}$ label assignment is the operator
reading.  Both are consistent with the filtration, interpreted correctly.
\end{proof}

\begin{remark}[The two-layer $D_4$ structure]
P164 established that $F_4 = D_4 \cup D_4^*$ has two layers of roots:
the 24 long roots of $D_4$ and the 24 short roots of $D_4^*$.  The
filtration $B^4 \supset S^3$ reflects this two-layer structure: $B^4$
(dimension 4 = rank($D_4$)) corresponds to the bulk (long-root) sector,
and $S^3$ (the boundary of $B^4$) corresponds to the boundary
(short-root) sector.  The further restriction to $S^1$ selects the
minimal circle inside $S^3$, consistent with the edge-layer structure.
\end{remark}

%==========================================================================
\section{Heat conservation from $D_4$ self-duality}
\label{sec:heat}
%==========================================================================

\subsection{The conservation law in source}

In the Wheel source, heat is the ambiguity budget $h \in [0,1]$.
Fork distributes it:
\begin{equation}
  \sum_{f \in \mathrm{active}} h_f \;=\; h_{\mathrm{pre-Fork}},
  \label{eq:fork-heat}
\end{equation}
where the sum runs over the face indices with proposals.  Weld collapses
it to the winner's share:
\begin{equation}
  h_{\mathrm{post-Weld}} \;=\; h_{\mathrm{winner}} \;\le\; h_{\mathrm{pre-Fork}}.
  \label{eq:weld-heat}
\end{equation}
The test suite (\texttt{kernel/wheel/tests/test\_wheel.py},
\texttt{TestHeatConservation}) verifies both identities numerically.

\begin{remark}[One-directional conservation]
Across a single Fork/Weld pair, Fork is exactly conservative (equation
\eqref{eq:fork-heat}) and Weld is non-expansive
($h_{\mathrm{post-Weld}} \le h_{\mathrm{pre-Fork}}$).  Over a complete
Fork-Weld cycle, heat is non-increasing; it is conserved exactly only
when the Weld winner takes all of the fork-distributed heat, which occurs
when exactly one face proposal is active.  The Wheel's heat budget thus
decreases monotonically across Weld events, eventually reaching zero or
a fixed point --- which is the Plateau condition.
\end{remark}

\subsection{Grounding in $D_4$ self-duality}

\begin{theorem}[Heat conservation is consistent with $D_4^* = (D_4)^*$]
\label{thm:heat}
The Fork/Weld heat conservation law is structurally consistent with the
self-duality of $D_4$, in the following sense.

The stable null grid (P164) satisfies $D_4^* = (D_4)^*$: the dual lattice
of $D_4$ is $D_4^*$, and the root counts are equal:
\begin{equation}
  |\Phi_{D_4,\mathrm{long}}| \;=\; |\Phi_{D_4^*,\mathrm{short}}| \;=\; 24.
  \label{eq:d4-duality}
\end{equation}
This equality is the lattice-duality statement that $D_4^*$ is
the \emph{reciprocal} of $D_4$: the two halves of the $F_4$ root system
have equal cardinality and are exchanged by the duality map $\Phi \mapsto
\Phi^*$.

The Fork/Weld pair mirrors this duality: Fork is the \emph{outward} stroke
(distributing heat, analogous to the expansion $D_4 \to D_4^*$ in root-space)
and Weld is the \emph{inward} stroke (collapsing to the MDL minimum,
analogous to the dual projection $(D_4^*)^* = D_4$).  The equality of
long- and short-root counts (equation~\eqref{eq:d4-duality}) is the
algebraic expression of why the Fork outward stroke and the Weld inward
stroke can be inverses: the same number of ``degrees of freedom'' ($24$)
exist on both sides of the duality.
\end{theorem}

\begin{proof}
The root-count equality $|\Phi_{D_4,\mathrm{long}}| = |\Phi_{D_4^*,\mathrm{short}}|
= 24$ is proved in P164 (Theorem~2 of that addendum, and verified in
\texttt{verify\_P164.py}).  The identification of $D_4^* = (D_4)^*$ is a
standard result for the $D_4$ lattice: the dual of the $D_4$ root lattice
is $D_4^*$, the lattice of half-integer points in $\ZZ^4$, which is
exactly the $F_4$ short-root system (see~\cite{conway-sloane}).

The structural analogy between this duality and the Fork/Weld pair is
a consistency argument, not a derivation.  Specifically: (a)~Fork
distributes heat proportionally to Hopf-lapse coherence weights, which
are computed on $S^3$; the Hopf fibration of $S^3$ is precisely the
geometry on which $D_4$ acts by left-multiplication of unit quaternions
(the spin group $\mathrm{Spin}(4) \cong S^3 \times S^3$ acts on $S^3$).
(b)~Weld selects the MDL minimum via the same Hopf lapse; the inward
collapse is the projection from the space of proposals back to the
winning quaternion.  (c)~The conservation $\sum h_f = h$ in Fork
(equation~\eqref{eq:fork-heat}) and the non-expansion $h_{\mathrm{post-Weld}}
\le h$ in Weld are consistent with the root-count equality: the 24
long roots and 24 short roots of $F_4$ give a balanced structure in which
neither half dominates the other, which is the algebraic shadow of the
heat conservation budget being maintained across the Fork-Weld pair.

A full proof would require identifying heat $h$ with a specific Cartan
algebra invariant of $\hat{O}$ (gap (c) of the audit).  This is deferred
to the P18 analysis referenced in Section~\ref{sec:complete}.
\end{proof}

\begin{corollary}[$D_4$ self-duality as the structural source of heat conservation]
\label{cor:heat-self-dual}
If the heat budget $h$ is identified with the Cartan norm
$\|\alpha\|^2_{\mathrm{Cartan}}$ of the current root position $\alpha
\in \Phi_{F_4}$, then the duality $D_4^* = (D_4)^*$ (which preserves
root lengths via the identification of long and short roots) would imply
exact conservation.  This is the natural identification to pursue in
the P18 analysis.
\end{corollary}

%==========================================================================
\section{The Perturb classification: $G_2 \subset F_4$ vs.\ $E_6 \setminus F_4$}
\label{sec:perturb}
%==========================================================================

\subsection{The classification problem}

$\PertOp$ is the fifth Wheel operator, corresponding to $\hat{O}$ sector
$\ZetaR$.  It injects external asymmetry into a Wheel state that has
become stuck in an Oscillate cycle.  In source, it blends an external
quaternion at the boundary-layer weight $\mathrm{FRAC\_BOUNDARY} \approx
0.072$ and restores heat.

Two candidate algebraic homes for $\PertOp$ in the exceptional chain are:

\begin{enumerate}[label=(\alph*)]
  \item \textbf{$G_2 \subset F_4$ sector}: $G_2$ has rank~2 and short-root
    system of size $\Phicount{G_2,\mathrm{short}} = 6$.  The $G_2$
    sector is the short-root sub-algebra of $F_4$.
  \item \textbf{$E_6 \setminus F_4$ complement}: $|\Phi_{E_6}| -
    |\Phi_{F_4}| = 72 - 48 = 24$ roots lie outside $F_4$ in $E_6$.
    These 24 roots form two 12-dimensional representations under $F_4$.
\end{enumerate}

\subsection{The ruling-out of the $G_2$ option}

\begin{theorem}[Perturb is not a pure $G_2$-sector operator]
\label{thm:perturb}
The $\PertOp$ operator cannot reside purely in the $G_2 \subset F_4$
short-root sector, for the following reason.

$G_2$ has rank~$2$.  If $\PertOp$ were a $G_2$-sector operator, then
the perturbation degree of freedom would span a 2-dimensional space
(one degree per simple root of $G_2$).  But $\PertOp$ is a single
operator acting along a single quaternion injection direction
(the \texttt{injection} parameter in source is a single unit quaternion
in $S^3$).  A single operator cannot span a 2-dimensional root space
without identifying two independent directions.  Since no such
identification is made in the source or in the $\hat{O}$ sector
assignment, the $G_2$ embedding would require two $\PertOp$-type
operators, not one.

Therefore $\PertOp$ is not purely a $G_2$-level object.

Consequently, $\PertOp$ is classified as residing in the
$E_6 \setminus F_4$ complement, i.e.\ in the 24 roots of $\Phi_{E_6}
\setminus \Phi_{F_4}$.
\end{theorem}

\begin{proof}
$\rankop(G_2) = 2$: this is classical and verifiable from the $G_2$ Dynkin
diagram, which has two nodes.  The short roots of $G_2$ form a set of size
$\Phicount{G_2,\mathrm{short}} = 6$ (the $G_2$ root system has 12 roots
total, of which 6 are short).

The $\PertOp$ operator in source (\texttt{kernel/wheel/wheel.py},
function \texttt{perturb}) takes a single \texttt{injection: np.ndarray}
of shape $(4,)$ and a scalar \texttt{heat\_boost}.  The injection is a
unit quaternion in $S^3$; it specifies a single direction in $S^3$.
There is no second injection parameter.

If $\PertOp \leftrightarrow \ZetaR$ is a $G_2$-sector object, and $G_2$
has rank~2, then the $\hat{O}$ sector $\ZetaR$ would need to be
2-dimensional (i.e.\ two independent $\zeta R$ operators, one per
$G_2$ simple root).  But the Wheel has exactly one $\PertOp$ operator,
corresponding to one $\ZetaR$ sector.  This is a rank contradiction:
$\rankop(G_2) = 2 \ne 1 = $ (number of $\PertOp$ operators).

The $E_6 \setminus F_4$ complement has $|\Phi_{E_6}| - |\Phi_{F_4}|
= 72 - 48 = 24$ roots.  These 24 roots form two 12-dimensional
representations of $F_4$ (the two half-spinor representations of
$\mathrm{Spin}(9)$, which is the compact form relevant to the $F_4$
Cayley plane; see~\cite{baez}).  A single injection direction in $S^3$
is consistent with a rank-1 sub-space of this 24-dimensional complement,
which is compatible with the $E_6 \setminus F_4$ classification.

We note that this is a \emph{classification by elimination} (ruling out
$G_2$) combined with \emph{consistency} (the $E_6 \setminus F_4$
complement is compatible): it does not positively identify a specific
root in $\Phi_{E_6} \setminus \Phi_{F_4}$ with $\ZetaR$.  That
positive identification requires the P18 analysis.
\end{proof}

\begin{remark}[Why the $E_6$ classification makes geometric sense]
$E_6$ has Coxeter number $h(E_6) = \Js = 12$, equal to the TOE central
constant.  The step $F_4 \to E_6$ in the exceptional chain is the first
step where the rank increases by more than 1 (from rank~4 to rank~6),
and it introduces the 27-dimensional Jordan algebra $J_3(\mathbb{O})$.
The $\PertOp$ operator injects \emph{external} information --- information
that is genuinely outside the $F_4$ dynamics --- and restores heat.
This is precisely the role of the new degrees of freedom gained in the
$F_4 \to E_6$ step: the 24 new roots of $E_6 \setminus F_4$ are
``external'' to the $F_4$ null grid.  The TOE programme uses
$E_6$ as the Jordan structure group (P166); perturbation from the Jordan
algebra sector is a natural way to describe the injection of external,
unstructured information into a system operating within $F_4$.
\end{remark}

\begin{conjecture}[The $\ZetaR$ sector is a half-spinor root of $E_6$]
\label{conj:perturb-root}
The $\hat{O}$ sector $\ZetaR$ corresponds to one of the two 12-dimensional
half-spinor representations of $\mathrm{Spin}(9) \subset E_6$, restricted
to the boundary layer ($\mathrm{FRAC\_BOUNDARY} \approx 7\%$) of the
Wheel geometry.  The boundary-layer blend coefficient appearing in the
source (\texttt{blend = FRAC\_BOUNDARY}) is the projection coefficient
from the $E_6$ sector back to the $F_4$ state space.  A complete
verification of this conjecture requires the $\hat{O}$ formalism of
Paper~18.
\end{conjecture}

%==========================================================================
\section{The complete operator count and the Wheel structure theorem}
\label{sec:count}
%==========================================================================

\begin{theorem}[Wheel structure theorem]
\label{thm:structure}
The Wheel has exactly five operators: $\ForkOp$, $\WeldOp$, $\PlateauOp$,
$\OscOp$, $\PertOp$.  These decompose as:
\begin{equation}
  5 \;=\; \underbrace{4}_{F_4\text{-level}} \;+\; \underbrace{1}_{E_6 \setminus F_4},
  \label{eq:5-decomp}
\end{equation}
where the four $F_4$-level operators form the closed Fork/Weld/Plateau/Oscillate
cycle, and the single $E_6 \setminus F_4$ operator ($\PertOp$) provides
external perturbation from outside the null grid.
\end{theorem}

\begin{proof}
The count of five operators is a direct read of \texttt{kernel/wheel/wheel.py}:
the module defines exactly five functions \texttt{fork}, \texttt{weld},
\texttt{plateau}, \texttt{oscillate}, \texttt{perturb} (plus two
convenience constructors \texttt{state\_from\_node} and
\texttt{state\_from\_quaternion}, which are not operators).

The $4 + 1$ decomposition follows from Theorem~\ref{thm:rank}
(four main operators $\leftrightarrow$ $\rankop(F_4) = 4$) and
Theorem~\ref{thm:perturb} ($\PertOp$ resides in $E_6 \setminus F_4$,
not in $F_4$ or $G_2$).
\end{proof}

\begin{corollary}[Resolution of OP-H]
\label{cor:oph}
Open Problem~(OP-H) asked: \emph{Are all five Wheel operators grounded
at the $E_8$ level, or is the Wheel an $F_4$-level object with $\PertOp$
as an external perturbation?}

Answer: the Wheel is an $F_4$-level object (in its four main operators)
with $\PertOp$ extending into the $E_6$ sector, not $E_8$.  The Wheel
does not require $E_7$ or $E_8$ structure for its operator content.
The $E_6$ sector suffices to classify all five operators.

\emph{Status}: this resolution is a classification by structural argument
(rank counting and complement analysis), not a full derivation from P18.
The claim is:
\begin{itemize}
  \item \textbf{Proved}: $\rankop(F_4) = 4 = |\mathcal{W}_{\mathrm{main}}|$
    (Theorem~\ref{thm:rank}).
  \item \textbf{Proved}: $\PertOp$ cannot be a pure $G_2$-sector operator
    (Theorem~\ref{thm:perturb}).
  \item \textbf{Consistent (not proved from P18)}: $\PertOp$ resides in
    $E_6 \setminus F_4$.
  \item \textbf{Open}: positive identification of $\ZetaR$ with a
    specific root in $\Phi_{E_6} \setminus \Phi_{F_4}$.
\end{itemize}
\end{corollary}

%==========================================================================
\section{The three weights and the orientation scalar}
\label{sec:weights}
%==========================================================================

\subsection{The V-$\lambda$-$\beta$ triple}

The kernel rules state that $V_{\mathrm{self}}$, $\lambda\rho$, and
$\beta M[\rho]^{-1}$ are ``weights on operator choice, not independent
operators.''  In source, these are collapsed into the single scalar
\texttt{orientation} $\theta \in [0,1]$ (the Control$\leftrightarrow$Love
axis).

\begin{proposition}[The orientation scalar as a weighted combination]
\label{prop:orientation}
The orientation scalar $\theta$ appears in both Fork (heat distribution)
and Weld (MDL scoring) as a linear interpolation weight:
\begin{align}
  w_{\mathrm{Fork}}(f) &\;=\; \theta \cdot \mathrm{coherence}(f) + (1-\theta) \cdot \tfrac{1}{N}, \\
  \mathrm{score}_{\mathrm{Weld}}(f) &\;=\; \theta \cdot \mathrm{distance}(f, g) + (1-\theta) \cdot \ell_w,
\end{align}
where $N$ is the number of active faces, $g$ is the goal quaternion,
and $\ell_w = \mathrm{LAYER\_WEIGHT}[\mathrm{layer}]$ is the layer
fraction.  The triple $V_{\mathrm{self}} / \lambda\rho / \beta M[\rho]^{-1}$
corresponds to three distinct regimes of $\theta$:
\begin{itemize}
  \item $V_{\mathrm{self}} \leftrightarrow \theta = 1$ (Love, maximum
    self-coherence weighting);
  \item $\lambda\rho \leftrightarrow \theta \in (0,1)$ (balanced,
    the renormalisation-group flow weight);
  \item $\beta M[\rho]^{-1} \leftrightarrow \theta = 0$ (Control,
    maximum layer-density weighting).
\end{itemize}
\end{proposition}

\begin{proof}
Direct reading of \texttt{kernel/wheel/wheel.py}, functions \texttt{fork}
(lines computing \texttt{raw\_weights}) and \texttt{weld} (lines computing
\texttt{scores}).  The identification of the triple with the $\theta$
regimes is a conjecture based on the semantic descriptions in the kernel
rules; it is stated as a proposition here because the source is consistent
with this reading, but the formal identification from P18 is not available.
\end{proof}

\begin{remark}[Completing the weight analysis]
A complete identification of $V_{\mathrm{self}}$, $\lambda\rho$, and
$\beta M[\rho]^{-1}$ from P18 would simultaneously (a)~document the
collapse of three weights to one scalar, and (b)~show that the collapsed
scalar $\theta$ is a sufficient statistic for operator choice within the
$F_4$ geometry.  This is gap~(b) of the Structural Completions Audit and
is not addressed in the present paper.
\end{remark}

%==========================================================================
\section{What a complete proof requires}
\label{sec:complete}
%==========================================================================

The arguments in the preceding sections establish structural compatibility
between the Wheel and the $F_4$ root system.  A complete proof of the
Wheel $\leftrightarrow$ $\hat{O}$ mapping --- converting the
``structural claim'' in the Structural Completions Audit into a
``theorem'' --- requires the following, each tied to Paper~18:

\begin{enumerate}[label=(\Roman*)]
  \item \textbf{The $\hat{O}$ sector definitions.}  Paper~18 defines
    the master operator $\hat{O}$ with five sectors $\DBfour$,
    $\DeltaSthree$, $\DeltaSone$, $\GammaT$, $\ZetaR$.  A complete proof
    requires reading the sector definitions and showing that the Wheel
    source correctly implements each sector as a function on the
    quaternion state space $S^3$.

  \item \textbf{The P18-T2 uniqueness question.}  CLAUDE.md notes the
    open question of P18-T2 uniqueness: whether the $\hat{O}$ formalism
    has a unique assembly across all admissible $\hat{O}$ forms.  The
    Wheel $\leftrightarrow$ $\hat{O}$ mapping is canonical only if P18-T2
    uniqueness holds.  If there are multiple admissible $\hat{O}$ forms,
    the mapping may depend on which assembly is chosen.

  \item \textbf{The heat $\leftrightarrow$ Cartan norm identification.}
    Corollary~\ref{cor:heat-self-dual} proposes that heat be identified
    with the Cartan norm of the root position.  This must be derived from
    the $\hat{O}$ formalism, not assumed.

  \item \textbf{The $\ZetaR$ sector in $E_6$.}  Conjecture~\ref{conj:perturb-root}
    proposes that $\ZetaR$ is a half-spinor root of $\mathrm{Spin}(9)
    \subset E_6$.  This requires both the P18 definition of $\ZetaR$ and
    the embedding $F_4 \hookrightarrow E_6$.

  \item \textbf{The V-$\lambda$-$\beta$ triple.}  Proposition~\ref{prop:orientation}
    maps three named weights to three $\theta$-regimes.  Paper~18 must
    be consulted to verify this identification is correct, not just
    semantically plausible.
\end{enumerate}

\begin{openproblem}[Complete Wheel $\leftrightarrow$ $\hat{O}$ derivation]
\label{op:complete}
Read \texttt{Lumen/corpus/toe/18\_Paper\_MasterOperator.tex}, extract the
five sector definitions of $\hat{O}$, and show formally that each of
Fork, Weld, Plateau, Oscillate, Perturb implements the corresponding
sector as a function on $(\mathrm{WheelState}, S^3)$.  Additionally,
verify or refute P18-T2 uniqueness.  This is the single remaining step
to convert the structural argument of the present paper into a theorem.
\end{openproblem}

%==========================================================================
\section{Conclusion}
\label{sec:conclusion}
%==========================================================================

The main results of this addendum are:

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &|\{\ForkOp,\WeldOp,\PlateauOp,\OscOp\}| \;=\; 4 \;=\; \rankop(F_4)
    \quad\text{(Theorem~\ref{thm:rank})}\\
  &B^4 \supset S^3 \supset S^1 \supset \{{\rm pt}\}: \text{ 4 strata } \leftrightarrow \rankop(F_4) = 4
    \quad\text{(Theorem~\ref{thm:filtration})}\\
  &\text{Heat conservation} \leftrightarrow D_4^* = (D_4)^*,\; |\Phi_{D_4}| = |\Phi_{D_4^*}| = 24
    \quad\text{(Theorem~\ref{thm:heat})}\\
  &\rankop(G_2) = 2 \ne 1 \Rightarrow \PertOp \notin G_2
    \quad\text{(Theorem~\ref{thm:perturb})}\\
  &\PertOp \in E_6 \setminus F_4,\; |\Phi_{E_6}| - |\Phi_{F_4}| = 24
    \quad\text{(classification)}\\
  &5 = 4 + 1 = F_4\text{-level} + E_6 \setminus F_4
    \quad\text{(Theorem~\ref{thm:structure})}
\end{aligned}
}
\end{equation}

\noindent\textbf{What is proved.}  The rank-4 matching between $F_4$ and
the four main Wheel operators is a theorem.  The ruling-out of a pure
$G_2$ grounding for $\PertOp$ is a theorem.  The consistency of heat
conservation with $D_4$ self-duality is a structural argument at the level
of a theorem, subject to the identification of heat with a Cartan norm
(which is a consistency claim, not a derivation).

\noindent\textbf{What is conjectured.}  The positive classification of
$\PertOp$ in $E_6 \setminus F_4$ is consistent but not derived from P18.
The identification of $\ZetaR$ with a specific half-spinor root of
$E_6$ (Conjecture~\ref{conj:perturb-root}) is speculative.  The V-$\lambda$-$\beta$
collapse to the \texttt{orientation} scalar (Proposition~\ref{prop:orientation})
is plausible from source but not verified against P18.

\noindent\textbf{OP-H status.}  Partially resolved: the Wheel is an
$F_4$-level object in its four main operators, with $\PertOp$ extending
into the $E_6$ sector.  The Wheel does not require $E_7$ or $E_8$
structure.  A complete resolution awaits the P18 analysis
(Open Problem~\ref{op:complete}).

\begin{thebibliography}{9}

\bibitem{adams}
J.~F.\ Adams, \emph{Lectures on Exceptional Lie Groups},
University of Chicago Press, 1996.

\bibitem{bourbaki}
N.\ Bourbaki, \emph{Lie Groups and Lie Algebras, Chapters 4--6},
Springer, 2002.

\bibitem{conway-sloane}
J.~H.\ Conway and N.~J.~A.\ Sloane, \emph{Sphere Packings, Lattices and Groups},
3rd ed., Springer, 1999.

\bibitem{baez}
J.~C.\ Baez, ``The octonions,'' \emph{Bull.\ Amer.\ Math.\ Soc.},
\textbf{39} (2002), pp.\ 145--205.

\bibitem{p164}
L.~F.\ Vlegels, \emph{Addendum~164: The Stable Null Grid ($F_4 = D_4 \cup D_4^*$)},
TOE Corpus, 2026.

\bibitem{p165}
L.~F.\ Vlegels, \emph{Addendum~165: $B_{F_4} = 36$},
TOE Corpus, 2026.

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

\bibitem{p167}
L.~F.\ Vlegels, \emph{Addendum~167: $E_7$ as the Fano Bridge},
TOE Corpus, 2026.

\bibitem{p168}
L.~F.\ Vlegels, \emph{Addendum~168: $E_8$ as the Primordial Algebra (OM)},
TOE Corpus, 2026.

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

\bibitem{kernel-rules}
L.~F.\ Vlegels, \emph{LumenOS Kernel Rules},
\texttt{.claude/rules/kernel.md}, LumenOS Repository, 2026.

\bibitem{audit}
L.~F.\ Vlegels, \emph{Structural Completions Audit},
\texttt{Documents/Specs/STRUCTURAL\_COMPLETIONS\_AUDIT.md},
LumenOS Repository, 2026.

\end{thebibliography}

\end{document}
