% P202 — The Regular 4-Simplex on S³: Foundation Geometry
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\title{\textbf{Addendum P202}\\[4pt]
The Regular 4-Simplex on $S^3$: Foundation Geometry\\[2pt]
{\normalsize P18-T2 Programme: Geometric Underpinning of the Wheel Operator Algebra}}
\author{L\'eon Fernando Vlegels}
\date{17 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

%─────────────────────────────────────────────────────────────────────────────
\section{Introduction}
%─────────────────────────────────────────────────────────────────────────────

The P18-T2 programme establishes the mass-ratio formula $m_\mu/m_e$ from first
principles of the TOE, with the algebraic skeleton carried by the Lie algebra
$G_2$ and its distinguished subalgebra $A_2$. Addenda P200 and P201 fixed the
perturbative structure through four loops and initiated a PSLQ search for the
four-loop coefficient $c_4$.

The present addendum steps back to document the foundational geometry that
underlies the operator algebra of the Wheel subsystem: the \emph{regular
$4$-simplex} inscribed in the unit $3$-sphere $S^3 \subset \mathbb{R}^4$. This
polytope — five vertices, ten edges, ten triangular faces, five tetrahedral
cells — concentrates into a single object three Lie-algebraic structures that
recur throughout the programme:

\begin{itemize}
  \item Its vertex inner product is $-1/h^\vee(G_2) = -1/4$.
  \item Its Gram-matrix non-zero eigenvalue is $h^\vee(A_4)/h^\vee(G_2) = 5/4$.
  \item Its face-count total is the Coxeter number $h(E_8) = 30$.
\end{itemize}

The symmetry group is the Weyl group $W(A_4) \cong S_5$, of order $120$. Its
ten edges are in natural bijection with the ten positive roots of $A_4$.

All claims are verified numerically by the companion script
\texttt{verify/verify\_P202.py} at \texttt{mpmath} precision \texttt{dps}$=60$.

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\section{Vertex construction}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Parametrisation in $\mathbb{R}^5$}

Let $e_0, \ldots, e_4 \in \mathbb{R}^5$ be the standard basis vectors, and let
\[
  c \;=\; \frac{1}{5}\sum_{i=0}^{4} e_i \;=\; \tfrac{1}{5}(1,1,1,1,1)
\]
be their centroid. Define
\begin{equation}\label{eq:vertices}
  v_i \;=\; \frac{\sqrt{5}}{2}\,(e_i - c), \qquad i = 0,1,2,3,4.
\end{equation}
Each centred vector $e_i - c$ lies in the hyperplane
$H = \bigl\{x \in \mathbb{R}^5 : \sum_{k=0}^{4} x_k = 0\bigr\}$,
which is a four-dimensional linear subspace. Via any orthonormal basis $\mathcal{B}$ of
$H$, the images of $v_0, \ldots, v_4$ are five unit vectors in $\mathbb{R}^4$
forming a regular $4$-simplex inscribed in $S^3$. All metric properties below
are independent of the choice of $\mathcal{B}$.

\subsection{Unit-norm verification}

A direct computation gives
\[
  |v_i|^2 \;=\; \frac{5}{4}\,|e_i - c|^2
          \;=\; \frac{5}{4}\!\left(\frac{16}{25} + 4\cdot\frac{1}{25}\right)
          \;=\; \frac{5}{4}\cdot\frac{4}{5} \;=\; 1,
\]
so each $v_i$ is a unit vector.

\subsection{Explicit coordinates in $\mathbb{R}^4$}

Taking $v_0 = (1,0,0,0) \in \mathbb{R}^4$ as anchor, the remaining vertices
satisfy $\langle v_0, v_i \rangle = -\tfrac{1}{4}$ for $i=1,\ldots,4$, which
forces the first coordinate of each $v_i$ to be $-1/4$. Writing
$v_i = (-1/4,\, w_i)$ with $w_i \in \mathbb{R}^3$, unit norm requires
$|w_i|^2 = 15/16$. The four vectors $\{v_1,\ldots,v_4\}$ form a regular
tetrahedron in the affine hyperplane $\{x_0 = -1/4\} \subset \mathbb{R}^4$,
with $w_i = \tfrac{\sqrt{15}}{4}\,t_i$ and $\{t_i\}$ any unit-norm tetrahedron
vertices satisfying $\langle t_i, t_j\rangle = -\tfrac{1}{3}$.

%─────────────────────────────────────────────────────────────────────────────
\section{Inner product structure}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Pairwise inner products}

\begin{proposition}\label{prop:inner-product}
For the vertices $v_0,\ldots,v_4$ defined by \eqref{eq:vertices},
\[
  \langle v_i,\, v_j \rangle \;=\; -\,\frac{1}{4}
  \;=\; -\,\frac{1}{h^\vee(G_2)}, \qquad 0 \leq i < j \leq 4,
\]
where $h^\vee(G_2) = 4$ is the dual Coxeter number of the exceptional Lie
algebra $G_2$.
\end{proposition}

\begin{proof}
For $i \neq j$:
\[
  \langle v_i, v_j \rangle
    \;=\; \frac{5}{4}\,\langle e_i - c,\; e_j - c\rangle
    \;=\; \frac{5}{4}\!\left(
      \langle e_i, e_j\rangle - \langle e_i, c\rangle
      - \langle c, e_j\rangle + |c|^2
    \right).
\]
Since $e_i \perp e_j$, $\langle e_i, c\rangle = \langle e_j, c\rangle = 1/5$,
and $|c|^2 = 5 \cdot (1/5)^2 = 1/5$:
\[
  \langle v_i, v_j\rangle
    \;=\; \frac{5}{4}\!\left(0 - \frac{1}{5} - \frac{1}{5} + \frac{1}{5}\right)
    \;=\; \frac{5}{4}\cdot\left(-\frac{1}{5}\right)
    \;=\; -\frac{1}{4}.
\]
The identification $-1/4 = -1/h^\vee(G_2)$ follows from $h^\vee(G_2) = 4$.
\end{proof}

\begin{remark}
The value $-1/4$ is the \emph{unique} pairwise inner product compatible with five
unit vectors in $\mathbb{R}^4$ forming a regular simplex. It is determined by
the constraint that the $5 \times 5$ Gram matrix has rank at most four, which
forces the zero eigenvalue established in Proposition~\ref{prop:gram}.
\end{remark}

\subsection{Centroid}

\begin{proposition}\label{prop:centroid}
The centroid of the five vertices is the origin:
\[
  \sum_{i=0}^{4} v_i \;=\; \mathbf{0}.
\]
\end{proposition}

\begin{proof}
$\displaystyle\sum_{i=0}^{4} v_i
  = \frac{\sqrt{5}}{2}\sum_{i=0}^{4}(e_i - c)
  = \frac{\sqrt{5}}{2}\!\left(\sum_{i=0}^{4} e_i - 5c\right)
  = \frac{\sqrt{5}}{2}\bigl(5c - 5c\bigr)
  = \mathbf{0}.$
\end{proof}

%─────────────────────────────────────────────────────────────────────────────
\section{Gram matrix spectrum}
%─────────────────────────────────────────────────────────────────────────────

\begin{proposition}\label{prop:gram}
Let $G \in \mathrm{Mat}_{5\times 5}(\mathbb{R})$ be the Gram matrix with
$G_{ij} = \langle v_i, v_j\rangle$. Then:
\begin{enumerate}
  \item[(i)] $G = \tfrac{5}{4}\,I_5 - \tfrac{1}{4}\,J_5$, where $I_5$ is the
    identity and $J_5$ is the $5\times 5$ all-ones matrix.
  \item[(ii)] The eigenvalues of $G$ are $0$ with multiplicity $1$ and
    $\tfrac{5}{4}$ with multiplicity $4$.
  \item[(iii)] The non-zero eigenvalue satisfies
    \[
      \frac{5}{4}
      \;=\; \frac{h^\vee(A_4)}{h^\vee(G_2)},
    \]
    where $h^\vee(A_4) = 5$ is the dual Coxeter number of $A_4$ and
    $h^\vee(G_2) = 4$.
\end{enumerate}
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:inner-product}, $G_{ii} = 1$ and $G_{ij} = -1/4$ for
$i\neq j$, giving $G = I_5 - \tfrac{1}{4}(J_5 - I_5) = \tfrac{5}{4}I_5 - \tfrac{1}{4}J_5$,
which is (i).

For the eigenvalues of $J_5$: the all-ones vector $\mathbf{1} = (1,1,1,1,1)$
is an eigenvector with eigenvalue $5$; every vector orthogonal to $\mathbf{1}$
is an eigenvector with eigenvalue $0$ (multiplicity $4$). Therefore $G$ has
eigenvalues
\[
  \frac{5}{4} - \frac{1}{4}\cdot 5 \;=\; 0 \quad(\text{mult.\ }1),
  \qquad
  \frac{5}{4} - \frac{1}{4}\cdot 0 \;=\; \frac{5}{4} \quad(\text{mult.\ }4),
\]
establishing (ii). Part (iii) follows from $h^\vee(A_4) = 5$ and $h^\vee(G_2)=4$.
\end{proof}

\begin{remark}
The rank of $G$ equals $4$, consistent with the fact that five unit vectors
in $\mathbb{R}^4$ spanning a $4$-dimensional space cannot be linearly
independent. The trace identity $\mathrm{tr}(G) = 5 = 0 + 4\cdot(5/4)$ is an
immediate check.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{$A_4$ root system and Weyl group}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Identification}

The symmetry group of the regular $4$-simplex is the group of all permutations
of its five vertices, which is the symmetric group $S_5$. This coincides with
the Weyl group of the root system $A_4$:
\[
  \mathrm{Sym}(\text{4-simplex}) \;\cong\; W(A_4) \;\cong\; S_5,
  \qquad |W(A_4)| = 5! = 120.
\]

The relevant numerical invariants of $A_4$ (the root system of $\mathfrak{sl}_5$)
are collected in Table~\ref{tab:A4}.

\begin{table}[h]
\centering
\caption{Invariants of the $A_4$ root system and their geometric realisations.}
\label{tab:A4}
\begin{tabular}{@{}llll@{}}
\toprule
Invariant & Value & Formula & Geometric meaning \\
\midrule
Rank & $4$ & $n-1$, $n=5$ & Dimension of the simplex \\
Dimension ($\dim \mathfrak{sl}_5$) & $24$ & $n^2-1 = 5^2-1$ & --- \\
Dual Coxeter number $h^\vee(A_4)$ & $5$ & $n = 5$ & Eigenvalue numerator \\
Weyl group order $|W(A_4)|$ & $120$ & $n! = 5!$ & Symmetry count \\
Positive roots $|\Phi^+(A_4)|$ & $10$ & $\binom{5}{2}$ & Edges of 4-simplex \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Edges as positive roots}

The ten positive roots of $A_4$ can be taken as $e_i - e_j$ for $0 \leq i < j \leq 4$
(in the ambient $\mathbb{R}^5$). There is a canonical bijection
\[
  \bigl\{\text{positive roots of } A_4\bigr\}
  \;\longleftrightarrow\;
  \bigl\{\text{edges of the regular 4-simplex}\bigr\},
\]
sending $e_i - e_j \mapsto \{v_i, v_j\}$. Both sets have cardinality
$\binom{5}{2} = 10$.

\begin{remark}
The order $|W(A_4)| = 120$ equals the number of positive roots of $E_8$,
i.e.\ $|\Phi^+(E_8)| = 120$. Whether this numerical coincidence has a deeper
geometric or representation-theoretic explanation in the LumenOS operator
algebra is left as an open question for P203 and P205.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Metric properties}
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\subsection{Edge length in $\mathbb{R}^4$}

\begin{proposition}\label{prop:edge}
The squared Euclidean distance between any two vertices in $\mathbb{R}^4$ is
\[
  |v_i - v_j|^2
    \;=\; |v_i|^2 - 2\langle v_i, v_j\rangle + |v_j|^2
    \;=\; 1 + \frac{1}{2} + 1 \;=\; \frac{5}{2}.
\]
Hence the edge length is $\sqrt{5/2} = \sqrt{10}/2$.
\end{proposition}

\begin{proof}
Substitute $\langle v_i, v_j\rangle = -1/4$ and $|v_i|^2 = |v_j|^2 = 1$:
\[
  |v_i - v_j|^2 = 2 - 2\cdot(-\tfrac{1}{4}) = 2 + \tfrac{1}{2} = \tfrac{5}{2}. \qquad\square
\]
\end{proof}

\begin{remark}
The edge-length formula $|v_i - v_j|^2 = 2 - 2\langle v_i, v_j\rangle$ is a
general identity for unit vectors. For the $4$-simplex the inner product
$-1/h^\vee(G_2) = -1/4$ directly determines the edge length via the dual
Coxeter number of $G_2$.
\end{remark}

\subsection{Geodesic arc length on $S^3$}

The geodesic distance on $S^3$ between $v_i$ and $v_j$ is the great-circle
angle
\begin{equation}\label{eq:theta}
  \theta \;=\; \arccos\!\bigl(\langle v_i, v_j\rangle\bigr)
          \;=\; \arccos\!\bigl(-\tfrac{1}{4}\bigr)
          \;\approx\; 1.8235\;\text{rad}
          \;\approx\; 104.48^\circ.
\end{equation}
Since $\langle v_i, v_j\rangle < 0$, the angle is obtuse: $\theta \in (\pi/2, \pi)$.

%─────────────────────────────────────────────────────────────────────────────
\section{$f$-vector and the $E_8$ connection}
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\subsection{Face counts}

The $f$-vector of the regular $4$-simplex lists the number of faces in each
dimension:
\begin{equation}\label{eq:fvec}
  f = (f_0, f_1, f_2, f_3)
    = \bigl(\tbinom{5}{1},\; \tbinom{5}{2},\; \tbinom{5}{3},\; \tbinom{5}{4}\bigr)
    = (5,\; 10,\; 10,\; 5).
\end{equation}
Concretely: $5$ vertices, $10$ edges, $10$ triangular faces, and $5$ tetrahedral
cells. The total face count is
\[
  f_0 + f_1 + f_2 + f_3 \;=\; 5 + 10 + 10 + 5 \;=\; 30.
\]

\begin{remark}\label{rem:E8}
The Coxeter number of $E_8$ is $h(E_8) = 30$. It is defined by
$h(E_8) = |\Phi(E_8)|/\mathrm{rank}(E_8) = 240/8 = 30$. Therefore
\[
  f_0 + f_1 + f_2 + f_3 \;=\; h(E_8) \;=\; 30.
\]
The $4$-simplex, the symmetry group of $A_4$, and the exceptional algebra $E_8$
are thus linked through this identity.
\end{remark}

\subsection{Symmetry across the $f$-vector}

The $f$-vector $(5,10,10,5)$ is palindromic: $f_k = f_{3-k}$. This reflects the
self-duality of the $4$-simplex (the dual polytope is again a $4$-simplex).

%─────────────────────────────────────────────────────────────────────────────
\section{Status}
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\textbf{Status: ESTABLISHED.}

All claims in this addendum are:
\begin{enumerate}
  \item Proven analytically (Propositions~\ref{prop:inner-product},
    \ref{prop:centroid}, \ref{prop:gram}, \ref{prop:edge}).
  \item Verified numerically to $60$ decimal places by
    \texttt{verify/verify\_P202.py} (\texttt{mpmath}, \texttt{dps}$=60$).
\end{enumerate}

The established results are summarised in Table~\ref{tab:summary}.

\begin{table}[h]
\centering
\caption{Summary of verified geometric identities for the regular 4-simplex on $S^3$.}
\label{tab:summary}
\begin{tabular}{@{}lll@{}}
\toprule
Quantity & Value & Lie interpretation \\
\midrule
Pairwise inner product $\langle v_i, v_j\rangle$ & $-1/4$ & $-1/h^\vee(G_2)$ \\
Gram eigenvalue (non-zero) & $5/4$ & $h^\vee(A_4)/h^\vee(G_2)$ \\
Gram eigenvalue (zero) & $0$ & rank deficiency in $\mathbb{R}^4$ \\
Edge length squared & $5/2$ & $2 + 1/h^\vee(G_2)$ \\
Geodesic angle $\theta$ & $\arccos(-1/4)$ & $\arccos(-1/h^\vee(G_2))$ \\
$f$-vector sum & $30$ & $h(E_8)$ \\
Symmetry group order & $120$ & $|W(A_4)| = 5!$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Programme connections}

This addendum opens the following downstream papers:

\begin{itemize}
  \item \textbf{P203 — Wheel $\leftrightarrow$ 4-simplex identification.}
    The five Wheel operators (Fork, Weld, Plateau, Oscillate, Perturb) map to
    the five vertices of the $4$-simplex; the ten operator transitions correspond
    to the ten edges. The Gram matrix eigenvalue $5/4$ appears in the heat
    conservation identity across Fork/Weld cycles.

  \item \textbf{P205 — PSLQ augmented with 4-simplex constants.}
    The constants $\sqrt{5/2}$, $\arccos(-1/4)$, and $5/4$ are candidate basis
    elements for a PSLQ search augmenting the $c_4$ hunt from P201. The
    edge-length $\sqrt{10}/2$ and geodesic angle $\theta$ are algebraically
    independent of $\{\pi, \alpha^{-1}, \Omega_0\}$, providing genuinely new
    directions in the search space.
\end{itemize}

\end{document}
