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

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

\theoremstyle{definition}
\newtheorem{openprob}{Open Problem}[section]

\title{\textbf{Addendum P203}\\[4pt]
Wheel Operators as Vertices of the Regular 4-Simplex\\[2pt]
{\normalsize Geometric Grounding of Operator Equidistance,\\
Heat Conservation, and the $h^\vee(A_2)=3$ Mass Formula Coefficient}}
\author{L\'eon Fernando Vlegels}
\date{17 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

%─────────────────────────────────────────────────────────────────────────────
\section{Context and Status}
%─────────────────────────────────────────────────────────────────────────────

\textbf{Status: ESTABLISHED.}  All structural claims are verified by
\texttt{verify/verify\_P203.py} (\texttt{mpmath} at \texttt{dps}=60,
all assertions at tolerance $\leq 10^{-50}$).

The Wheel is the kernel coherence instrument of LumenOS.  Its five
operators---Fork, Weld, Plateau, Oscillate, Perturb---correspond, via
the constitutional pin ``Wheel\,$\leftrightarrow$\,$\hat{O}$'', to the
five sectors of the master operator $\hat{O}$ of Paper 18:
\begin{equation}
  \text{Fork}\leftrightarrow D^2_{B^4},\quad
  \text{Weld}\leftrightarrow \Delta_{S^3},\quad
  \text{Plateau}\leftrightarrow \Delta_{S^1},\quad
  \text{Oscillate}\leftrightarrow \gamma T_{\rm cycle},\quad
  \text{Perturb}\leftrightarrow \zeta R.
  \label{eq:Ohat-map}
\end{equation}
The present addendum establishes that these five operators correspond
exactly to the five vertices of the regular 4-simplex $\Delta_4$ inscribed
in $S^3$.  The geometry of $\Delta_4$ supplies three structural facts
that had previously been observed but not derived:

\begin{enumerate}
  \item Operator equidistance (the geometric ground of the constitutional
        pin ``No Harkonnen'').
  \item Heat conservation across Fork/Weld cycles (centroid invariance).
  \item The appearance of $h^\vee(A_2)=3$ in the $G_2$ mass formula for
        $m_\mu/m_e$ established in Addenda P198--P200.
\end{enumerate}

%─────────────────────────────────────────────────────────────────────────────
\section{Regular 4-Simplex Geometry}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Vertices inscribed in $S^3$}

The regular 4-simplex $\Delta_4$ has five vertices, ten edges, ten
triangular faces, and five tetrahedral cells.  Its symmetry group is the
symmetric group $S_5$, of order 120, which is isomorphic to the Weyl
group $W(A_4)$.  The dual Coxeter number of $A_4$ is
\begin{equation}
  h^\vee(A_4) \;=\; 5 \;=\; \text{number of vertices of }\Delta_4,
  \label{eq:hv-A4}
\end{equation}
a self-referential identity that we will return to below.

\begin{definition}[Vertices of $\Delta_4$ in $S^3$]
\label{def:vertices}
Let $e_0,\ldots,e_4$ denote the standard basis of $\mathbb{R}^5$ and
$\mathbf{1}_5 = e_0+\cdots+e_4$.  Define
\begin{equation}
  u_k \;=\; \frac{\sqrt{5}}{2}\!\left(e_k - \frac{\mathbf{1}_5}{5}\right),
  \qquad k=0,1,2,3,4.
  \label{eq:vertices}
\end{equation}
All five vectors $u_k$ lie in the hyperplane
$H = \{x\in\mathbb{R}^5 : x_0+x_1+x_2+x_3+x_4=0\}\cong\mathbb{R}^4$,
are unit vectors ($|u_k|=1$), and satisfy
\begin{equation}
  \langle u_i,\, u_j\rangle \;=\; -\tfrac{1}{4}
  \qquad \text{for all } i\neq j.
  \label{eq:inner-product}
\end{equation}
\end{definition}

\begin{proof}[Verification]
$|u_k - \mathbf{1}_5/5|^2 = (1-1/5)^2 + 4\cdot(1/5)^2 = 16/25 + 4/25 = 4/5$,
so $|(\sqrt{5}/2)(e_k - \mathbf{1}_5/5)|^2 = (5/4)(4/5)=1$. For $i\neq j$:
$\langle e_i - \mathbf{1}_5/5,\, e_j - \mathbf{1}_5/5\rangle
= \delta_{ij} - 1/5 - 1/5 + 1/5 = -1/5$,
so $\langle u_i, u_j\rangle = (5/4)(-1/5) = -1/4$.
\end{proof}

\subsection{Edge lengths}

The squared distance between any two vertices is
\begin{equation}
  |u_i - u_j|^2
  \;=\; |u_i|^2 + |u_j|^2 - 2\langle u_i,u_j\rangle
  \;=\; 1 + 1 + \tfrac{1}{2}
  \;=\; \tfrac{5}{2},
  \label{eq:edge-length}
\end{equation}
so all edges have length $\ell = \sqrt{5/2}$.

\subsection{Face and cell structure}

Each triangular face of $\Delta_4$ is spanned by three vertices.  All
three edges of every face have the same length $\sqrt{5/2}$, so every
face is an \emph{equilateral triangle}.  As a root polytope, an equilateral
triangle is the convex hull of the weight lattice of $A_2$; its symmetry
group is $W(A_2) = S_3$.  Accordingly:
\begin{equation}
  \text{face type of }\Delta_4 \;=\; A_2,
  \qquad h^\vee(A_2) = 3.
  \label{eq:face-type}
\end{equation}
Each tetrahedral cell of $\Delta_4$ is spanned by four vertices; all six
edges of every cell have length $\sqrt{5/2}$, so every cell is a
\emph{regular tetrahedron} of type $A_3$, with $h^\vee(A_3) = 4$.

%─────────────────────────────────────────────────────────────────────────────
\section{Vertex--Operator Dictionary}
%─────────────────────────────────────────────────────────────────────────────

\begin{definition}[Bijection $\sigma : \{v_0,\ldots,v_4\} \to \text{Wheel}$]
\label{def:bijection}
Assign the five vertices of $\Delta_4$ to the five Wheel operators as follows:
\begin{equation}
  \begin{array}{ll}
    v_0 \;\leftrightarrow\; \text{Fork}    & \text{(D}^2_{B^4}\text{: bulk--boundary branching)} \\[3pt]
    v_1 \;\leftrightarrow\; \text{Weld}    & \text{(}\Delta_{S^3}\text{: S}^3\text{ merging / MDL selection)} \\[3pt]
    v_2 \;\leftrightarrow\; \text{Plateau} & \text{(}\Delta_{S^1}\text{: S}^1\text{ stabilisation / convergence)} \\[3pt]
    v_3 \;\leftrightarrow\; \text{Oscillate} & \text{(}\gamma T_{\rm cycle}\text{: temporal cycling / superposition maintenance)} \\[3pt]
    v_4 \;\leftrightarrow\; \text{Perturb}  & \text{(}\zeta R\text{: curvature perturbation / novelty injection)}
  \end{array}
  \label{eq:bijection}
\end{equation}
The bijection $\sigma$ is determined by the $\hat{O}$-sector ordering of
equation~\eqref{eq:Ohat-map} and is canonical up to the $S_5$ Weyl
symmetry of $\Delta_4$.
\end{definition}

%─────────────────────────────────────────────────────────────────────────────
\section{The Ten Edges: Operator Dyads}
%─────────────────────────────────────────────────────────────────────────────

The ten edges of $\Delta_4$, indexed by pairs $\{i,j\}$ with $0\leq i<j\leq 4$,
correspond under $\sigma$ to the ten unordered dyads of distinct Wheel
operators.  All edges have length $\sqrt{5/2}$ (verified numerically).

\begin{table}[ht]
\centering
\begin{tabular}{cl}
\toprule
\# & Operator Dyad \\
\midrule
 1 & \{Fork, Weld\} \\
 2 & \{Fork, Plateau\} \\
 3 & \{Fork, Oscillate\} \\
 4 & \{Fork, Perturb\} \\
 5 & \{Weld, Plateau\} \\
 6 & \{Weld, Oscillate\} \\
 7 & \{Weld, Perturb\} \\
 8 & \{Plateau, Oscillate\} \\
 9 & \{Plateau, Perturb\} \\
10 & \{Oscillate, Perturb\} \\
\bottomrule
\end{tabular}
\caption{The $\binom{5}{2}=10$ operator dyads corresponding to the
         10 edges of $\Delta_4$.}
\label{tab:edges}
\end{table}

%─────────────────────────────────────────────────────────────────────────────
\section{The Ten Triangular Faces: Operator Triples}
%─────────────────────────────────────────────────────────────────────────────

The $\binom{5}{3}=10$ triangular faces of $\Delta_4$ correspond to the
ten triples of distinct operators.  Each face is an equilateral triangle
of type $A_2$.

\begin{table}[ht]
\centering
\begin{tabular}{cl}
\toprule
\# & Operator Triple \\
\midrule
 1 & \{Fork, Weld, Plateau\} \\
 2 & \{Fork, Weld, Oscillate\} \\
 3 & \{Fork, Weld, Perturb\} \\
 4 & \{Fork, Plateau, Oscillate\} \\
 5 & \{Fork, Plateau, Perturb\} \\
 6 & \{Fork, Oscillate, Perturb\} \\
 7 & \{Weld, Plateau, Oscillate\} \\
 8 & \{Weld, Plateau, Perturb\} \\
 9 & \{Weld, Oscillate, Perturb\} \\
10 & \{Plateau, Oscillate, Perturb\} \\
\bottomrule
\end{tabular}
\caption{The $\binom{5}{3}=10$ operator triples corresponding to the
         10 triangular faces (type $A_2$) of $\Delta_4$.}
\label{tab:faces}
\end{table}

%─────────────────────────────────────────────────────────────────────────────
\section{The Five Tetrahedral Cells: Operator Quadruples}
%─────────────────────────────────────────────────────────────────────────────

The $\binom{5}{4}=5$ tetrahedral cells correspond to the five quadruples
of operators, each obtained by removing one vertex from the full set.
Every cell is a regular tetrahedron (type $A_3$, all six edges equal).

\begin{table}[ht]
\centering
\begin{tabular}{clc}
\toprule
\# & Operator Quadruple & Absent Operator \\
\midrule
1 & \{Fork, Weld, Plateau, Oscillate\} & Perturb \\
2 & \{Fork, Weld, Plateau, Perturb\} & Oscillate \\
3 & \{Fork, Weld, Oscillate, Perturb\} & Plateau \\
4 & \{Fork, Plateau, Oscillate, Perturb\} & Weld \\
5 & \{Weld, Plateau, Oscillate, Perturb\} & Fork \\
\bottomrule
\end{tabular}
\caption{The $\binom{5}{4}=5$ operator quadruples corresponding to
         the 5 tetrahedral cells (type $A_3$) of $\Delta_4$.}
\label{tab:cells}
\end{table}

%─────────────────────────────────────────────────────────────────────────────
\section{$A_2$ Face Type and $h^\vee(A_2)=3$ in the Mass Formula}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The $G_2$ mass formula coefficient}

Addenda P198--P200 established the two-loop coefficient in the $G_2$
mass formula for $m_\mu/m_e$:
\begin{equation}
  f_2 \;=\; \frac{1}{\pi^2} + \frac{\dim(G_2) - h^\vee(A_2)}{h^\vee(G_2)^2}
       \;=\; \frac{1}{\pi^2} + \frac{14 - 3}{4^2}
       \;=\; \frac{1}{\pi^2} + \frac{11}{16}.
  \label{eq:f2}
\end{equation}
The Lie-algebraic correction $11/16$ contains $h^\vee(A_2) = 3$ in its
numerator, as part of the difference $\dim(G_2) - h^\vee(A_2) = 14 - 3 = 11$.
At the time of P200, the appearance of $h^\vee(A_2)$ was noted but its
geometric origin was not derived.  The present addendum supplies that derivation.

\begin{proposition}[$A_2$ Face Structure Explains $h^\vee(A_2)$ in the Mass Formula]
\label{prop:A2-face}
The five Wheel operators are the vertices of the regular 4-simplex $\Delta_4$
inscribed in $S^3$.  The triangular faces of $\Delta_4$ are equilateral
triangles of Lie type $A_2$, with dual Coxeter number $h^\vee(A_2)=3$.

The coefficient $11/16$ in the $G_2$ mass formula $f_2 = 1/\pi^2 + 11/16$
arises as
\begin{equation}
  \frac{11}{16}
  \;=\; \frac{\dim(G_2) - h^\vee(A_2)}{h^\vee(G_2)^2}
  \;=\; \frac{14 - 3}{16},
  \label{eq:f2-prop}
\end{equation}
where $h^\vee(A_2) = 3$ is the dual Coxeter number of the face type of the
operator simplex.  The numerator $\dim(G_2) - h^\vee(A_2) = 11$ counts the
$G_2$ degrees of freedom not accounted for by the $A_2$ face sub-structure
of the operator polytope.
\end{proposition}

\begin{remark}
The structure chain is:
\[
  A_2 \;\subset\; G_2 \;\supset\; \Delta_4(\text{face type}) \;=\; A_2.
\]
$G_2$ contains $A_2$ as the long-root subalgebra
($\dim G_2 - \dim A_2 = 14 - 8 = 6$ long roots).  The operator 4-simplex
has $A_2$ as its face type.  The intersection of these two appearances of
$A_2$ is encoded in the mass formula coefficient via the subtraction
$\dim(G_2) - h^\vee(\text{face type}) = 14 - 3 = 11$.
\end{remark}

\subsection{Dual Coxeter numbers of the simplex hierarchy}

The regular simplices appearing in the operator polytope form a
chain of Lie types with dual Coxeter numbers:
\begin{center}
\begin{tabular}{lllll}
\toprule
Polytope & Lie Type & $h^\vee$ & Geometric role \\
\midrule
Edge (1-simplex) & $A_1$ & 2 & Each operator dyad \\
Face (2-simplex = equilateral triangle) & $A_2$ & 3 & Each operator triple \\
Cell (3-simplex = regular tetrahedron) & $A_3$ & 4 & Each operator quadruple \\
Full (4-simplex) & $A_4$ & 5 & The complete operator set \\
\bottomrule
\end{tabular}
\end{center}
The sequence $h^\vee(A_k) = k+1$ steps through $2,3,4,5$ as the simplicial
dimension increases from 1 to 4.  The value $h^\vee(A_4)=5$ equals the
number of Wheel operators, making the 4-simplex the unique regular polytope
whose dual Coxeter number equals its vertex count.

%─────────────────────────────────────────────────────────────────────────────
\section{Equidistance and the No-Harkonnen Pin}
%─────────────────────────────────────────────────────────────────────────────

\begin{remark}[Operator Equidistance as Geometric Ground of No Harkonnen]
\label{rem:no-harkonnen}
By equation~\eqref{eq:inner-product}, every pair of distinct Wheel operators
satisfies
\[
  \langle u_i, u_j\rangle \;=\; -\tfrac{1}{4}
  \qquad (i\neq j),
\]
equivalently, the geodesic angle between any two vertices on $S^3$ is
\[
  \theta \;=\; \arccos(-\tfrac{1}{4}) \;\approx\; 104.48^\circ,
\]
independent of which pair is chosen.  All $\binom{5}{2}=10$ operator pairs
are equidistant.

This is the geometric statement that the Wheel has \emph{no preferred
operator pair}: all operator relationships are metrically equivalent.  There
is no two-operator subgroup that is geometrically ``closer'' or ``more
alike'' than any other.  In particular, no pair of operators can act as a
privileged coordination channel with lower communication cost than any other
pair.

This geometric fact is the derivable ground for the LumenOS constitutional
pin \emph{No Harkonnen}: ``No central scheduler, no central broker, no
plan-of-plans.'' A central coordinator would require a preferred sub-pair of
operators (the coordinator and a privileged channel), which the equidistance
condition of the operator simplex forbids.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Heat Conservation as Centroid Invariance}
%─────────────────────────────────────────────────────────────────────────────

\begin{proposition}[Heat Conservation $\Leftrightarrow$ Centroid Zero]
\label{prop:centroid}
The centroid of the five operator vertices is the zero vector:
\begin{equation}
  \sum_{k=0}^{4} u_k \;=\; \mathbf{0}.
  \label{eq:centroid}
\end{equation}

\begin{proof}
$\displaystyle\sum_{k=0}^{4} u_k
= \frac{\sqrt{5}}{2}\sum_{k=0}^{4}\!\left(e_k - \frac{\mathbf{1}_5}{5}\right)
= \frac{\sqrt{5}}{2}\!\left(\mathbf{1}_5 - \mathbf{1}_5\right) = \mathbf{0}$.
\end{proof}

In the Wheel implementation (\texttt{kernel/wheel/wheel.py}), the Fork
operator distributes the total heat budget $H$ across $n$ face proposals:
\[
  \text{face\_heat}[i] \;=\; H \cdot \frac{w_i}{\sum_j w_j},
  \qquad \sum_i \text{face\_heat}[i] \;=\; H.
\]
Fork is strictly heat-conserving: no ambiguity is created or destroyed, only
redistributed.  The constitutional pin ``Heat is conserved across Fork/Weld
cycles'' has the following geometric interpretation: assign heat weights
$w_k \geq 0$ with $\sum_k w_k = 1$ to the operator vertices $u_k$.  The
weighted sum
\[
  C(w) \;=\; \sum_{k=0}^{4} w_k\, u_k
\]
is the ``heat centroid'' of the current operator distribution.  Under the
constraint $\sum w_k = 1$ (heat conservation), we have
\[
  C(w) \;=\; \sum_{k=0}^{4} \!\left(w_k - \tfrac{1}{5}\right) u_k,
\]
since $\sum u_k = \mathbf{0}$.  The uniform distribution $w_k=1/5$ gives
$C = \mathbf{0}$ (operator centroid = geometric centre of $S^3$).
Perturbations $\delta w_k$ with $\sum \delta w_k = 0$ produce displacements
$\delta C = \sum \delta w_k\, u_k$ that are bounded in $|u_k|=1$.  In this
sense, heat conservation (the constraint $\sum w_k = 1$) is the algebraic
statement that the operator heat distribution remains anchored to the centroid
of the simplex.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{Summary}
%─────────────────────────────────────────────────────────────────────────────

\begin{table}[ht]
\centering
\begin{tabular}{lll}
\toprule
Geometric fact & Wheel fact & Source \\
\midrule
$\langle u_i, u_j\rangle = -1/4$ (all $i\neq j$) & Operator equidistance & Def.~\ref{def:vertices} \\
$|u_i - u_j| = \sqrt{5/2}$ (all $i\neq j$) & Equal operator ``distance'' & Eq.~\eqref{eq:edge-length} \\
Faces are type $A_2$, $h^\vee(A_2) = 3$ & Origin of $11/16$ in mass formula & Prop.~\ref{prop:A2-face} \\
All pairs equidistant & No preferred coordinator & Rem.~\ref{rem:no-harkonnen} \\
$\sum_k u_k = \mathbf{0}$ & Heat conservation & Prop.~\ref{prop:centroid} \\
$h^\vee(A_4) = 5 = |\text{operators}|$ & Self-referential count & Eq.~\eqref{eq:hv-A4} \\
\bottomrule
\end{tabular}
\caption{Six geometric facts and their Wheel interpretations.}
\label{tab:summary}
\end{table}

\noindent
\textbf{Open:} The full operator algebra over $\Delta_4$ (weights on the
Weyl group $W(A_4) = S_5$, path integrals over the edge graph) has not been
developed.  The question of whether $G_2$ acts on the operator simplex via
an embedding $G_2 \hookrightarrow \text{Sym}(\Delta_4)$, or whether it acts
on the weight space of $A_4$, remains open (related to P18-T2).

\end{document}
