% P210 — OP-3 Resolution Attempt: h∨(A₂) and the Lepton Face Correction
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\title{\textbf{Addendum P210}\\[4pt]
OP-3 Resolution Attempt:\\
The Dual Coxeter Number $h^\vee(A_2) = 3$ and\\
the Lepton Face Mass-Ratio Correction\\[2pt]
{\normalsize Why the Muon/Electron Formula Selects the\\
Triangular Topology of the $A_2$ Root System}}
\author{L\'eon Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

%─────────────────────────────────────────────────────────────────────────────
\section{Setup and Restatement}
\label{sec:setup}
%─────────────────────────────────────────────────────────────────────────────

\textbf{Status: OPEN.}  The integer $3$ in $R^\star$ is identified as
$h^\vee(A_2) = 3$, the dual Coxeter number of the root system carried by
the lepton face of the 4-simplex on~$S^3$.  No proof of this identification
has been established; the structural argument of Section~\ref{sec:argument}
reaches conjecture level only.

\subsection{Background from prior addenda}

The prior addenda contributing to this programme are:

\begin{itemize}
  \item \textbf{P202}: The 4-simplex inscribed in $S^3$ has inner product
    $\langle v_i, v_j\rangle = -\tfrac{1}{4}$ for all pairs $i\neq j$, geodesic
    angle $\theta_4 = \arccos(-\tfrac{1}{4})$, and $f$-vector $(5,10,10,5)$.

  \item \textbf{P203}: Of the 10 triangular 2-faces, all carry $A_2$-type
    geometry.  The dual Coxeter number $h^\vee(A_2) = 3$ appears as the
    coupling ratio in the mass formula; equidistance between vertices
    ($\langle v_i, v_j\rangle = \text{const}$) is the geometric realisation of
    the No-Harkonnen pin and ensures heat conservation across Fork/Weld cycles.

  \item \textbf{P208}: The lepton face is the triangular 2-face
    $\{\text{Plateau}\leftrightarrow v_2,\, \text{Weld}\leftrightarrow v_1,\,
    \text{Fork}\leftrightarrow v_0\}$, a triangular 2-face of $A_2$ type.
    The Fork-dominant energy $E_F = 4\pi^3 \gg E_W = \pi^2$.

  \item \textbf{P209}: The formula
    \begin{equation}
      R^\star = 207\!\left(1 + \frac{3 - \omega}{e^3\,\mu}\right)
              = 207\!\left(1 - \frac{\omega - 3}{e^3\,\mu}\right)
              \approx 206.768\,515\ldots
      \label{eq:Rstar}
    \end{equation}
    achieves $1.124\,\text{ppm}$ agreement with CODATA ($206.768\,283$),
    using only the three TOE primitives $\mu$, $e = \pi$, $\omega = \pi^3/4$
    and the integer~$3$.
\end{itemize}

\subsection{TOE primitives}

All computations use \texttt{mpmath} at \texttt{mp.dps}\,=\,60.

\begin{align}
  \mu &= 4\pi^3 + \pi^2 + \pi \approx 137.036\,304
    \quad\text{(monad / fine-structure inverse)},\label{eq:mu}\\
  e   &= \pi \approx 3.141\,593
    \quad\text{(electron sector energy / $E_e$)},\label{eq:e}\\
  \omega &= \frac{\pi^3}{4} \approx 7.751\,569
    \quad\text{(fundamental frequency, $\Omega_0$)}.\label{eq:omega}
\end{align}

Their mutual relation $\mu = 4e^3 + e^2 + e$ makes $\mu$ a polynomial in $e$
with integer coefficients.  The relation $\omega = e^3/4$ is exact, giving
$e^3 = 4\omega$ and $e^3\mu = 4\omega\mu$.

\subsection{Restatement in structural terms}

The question addressed by this addendum is:

\begin{quote}
  Can the integer $3$ in $R^\star$ be identified as $h^\vee(A_2) = 3$,
  where $h^\vee$ is the dual Coxeter number of the $A_2$ root system carried
  by each triangular face of the 4-simplex?
\end{quote}

With this identification the formula reads
\begin{equation}
  \boxed{R^\star = 207\!\left(1 - \frac{\omega - h^\vee(A_2)}{e^3\,\mu}\right)}
  \label{eq:Rstar_structural}
\end{equation}
and the correction $(\omega - h^\vee)/({e^3\mu})$ is the ratio of
\emph{excess fundamental frequency over dual Coxeter number} to
\emph{cubic electron energy times monad}.

%─────────────────────────────────────────────────────────────────────────────
\section{Geometric Analysis of $(\omega - h^\vee(A_2))$}
\label{sec:geom}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Exact value}

Using $\omega = \pi^3/4$ and $h^\vee(A_2) = 3$:
\begin{equation}
  \omega - h^\vee(A_2) = \frac{\pi^3}{4} - 3
  \approx 4.751\,569\,170\,074\,955.
  \label{eq:diff}
\end{equation}
Multiplying by 4:
\begin{equation}
  4(\omega - h^\vee) = \pi^3 - 12 = e^3 - 4h^\vee(A_2)
  \approx 19.006\,277.
  \label{eq:4diff}
\end{equation}
This exposes the correction as proportional to $e^3 - 4h^\vee$, measuring
how far the cube of the electron energy exceeds four times the dual Coxeter
number.

\subsection{Key reformulation}

Using $e^3 = 4\omega$ the denominator simplifies:
\begin{equation}
  \frac{\omega - h^\vee}{e^3\,\mu}
  = \frac{\omega - h^\vee}{4\omega\,\mu}
  = \frac{1}{4\mu}\!\left(1 - \frac{h^\vee}{\omega}\right)
  = \frac{1}{4\mu}\!\left(1 - \frac{12}{\pi^3}\right).
  \label{eq:reform}
\end{equation}

Numerically:
\begin{equation}
  \frac{h^\vee(A_2)}{\omega} = \frac{3}{\pi^3/4} = \frac{12}{\pi^3}
  \approx 0.387\,018.
  \label{eq:hvfrac}
\end{equation}
This ratio is not a standard constant of physics; it is a purely geometric
quantity measuring what fraction of the fundamental frequency is ``used up''
by the dual Coxeter number of the triangular face.

\subsection{Defect $1 - h^\vee/\omega$}

The quantity
\begin{equation}
  \delta \equiv 1 - \frac{h^\vee(A_2)}{\omega} = 1 - \frac{12}{\pi^3}
  \approx 0.612\,982
\end{equation}
is the fractional excess of $\omega$ over $h^\vee$.  The correction becomes
$\delta/(4\mu)$.  No identification of $\delta$ with a Lie-algebraic or
spectral quantity has been found; it is recorded here as an open datum.

\subsection{Euler characteristic of the lepton face}

The lepton triangle (boundary only) has $V = 3$ vertices and $E = 3$ edges,
giving Euler characteristic
\begin{equation}
  \chi_{\partial} = V - E = 3 - 3 = 0.
\end{equation}
Including the interior face: $\chi = V - E + F = 3 - 3 + 1 = 1$ (contractible
disk).  The boundary Euler characteristic $\chi_\partial = 0$ is consistent
with the $A_2$ root system having $h^\vee = 3 = $ the number of vertices
$=$ the number of edges, making the triangle a balanced object in which no
vertex or edge class is privileged.

\subsection{Gram matrix and eigenvalue structure}

The $3\times 3$ Gram matrix $G$ of the lepton face, with
$G_{ii} = 1$ and $G_{ij} = \langle v_i, v_j\rangle = -\tfrac{1}{4}$ for
$i \neq j$, is a circulant.  Its eigenvalues are computed analytically:

\begin{equation}
  \lambda_{\max} = 1 - \left(-\tfrac{1}{4}\right) = \tfrac{5}{4}
  \quad(\text{multiplicity }2),\qquad
  \lambda_{\min} = 1 + 2\!\left(-\tfrac{1}{4}\right) = \tfrac{1}{2}
  \quad(\text{multiplicity }1).
  \label{eq:evals}
\end{equation}

Trace check: $2\cdot\tfrac{5}{4} + \tfrac{1}{2} = 3 = V$.  Determinant:
$(\tfrac{5}{4})^2\cdot\tfrac{1}{2} = \tfrac{25}{32}$.

The minimal eigenvalue $\lambda_{\min} = \tfrac{1}{2}$ belongs to the uniform
eigenvector $\tfrac{1}{\sqrt{3}}(1,1,1)$; the two raised eigenvectors span the
orthogonal complement.  The trace equals the vertex count $V = h^\vee(A_2) = 3$.

\subsection{Heat budget on the lepton face}

The vertex energies from P208 are $E_F = 4\pi^3$, $E_W = \pi^2$,
$E_P = \pi$, with total $E_F + E_W + E_P = \mu$.  The heat budget is:
\begin{equation}
  E_{\text{Fork}} : E_{\text{Weld}} : E_{\text{Plateau}}
  \;=\; 4\pi^3 : \pi^2 : \pi
  \;=\; 4\pi^2 : \pi : 1.
\end{equation}
Fork dominates ($E_F/\mu \approx 90.5\%$), consistent with $E_F = 4\pi^3
= e^3/4 \cdot 16 = 16\omega$ greatly exceeding the other two.  The three
vertex energies are not equal; the equidistance is in the inner product
geometry (all $\langle v_i, v_j\rangle = -1/4$), not in the energy
assignment.  Heat conservation is guaranteed by the Wheel's
Fork$\leftrightarrow$Weld cycle operating on this face.

%─────────────────────────────────────────────────────────────────────────────
\section{Integer Scan: $n = 1,\ldots,10$}
\label{sec:scan}
%─────────────────────────────────────────────────────────────────────────────

The family of candidate formulas indexed by integer $n$ is:
\begin{equation}
  R(n) = 207\!\left(1 + \frac{n - \omega}{e^3\,\mu}\right).
  \label{eq:Rn}
\end{equation}
We compare each to the CODATA value $206.768\,283\,0$ in parts per million.

\begin{longtable}{@{}rccc@{}}
\toprule
$n$ & $R(n)$ & $|R(n)-\text{TARGET}|/\text{TARGET}$ [ppm] & Lie-algebraic meaning \\
\midrule
\endhead
1  & $206.671\,080$ & $470.104$ & --- \\
2  & $206.719\,798$ & $234.490$ & $\mathrm{rank}(A_2)$ \\
3  & $206.768\,515$ & $\mathbf{1.124}$ & $h^\vee(A_2) = h(A_2) = |\Phi^+(A_2)|$ \\
4  & $206.817\,232$ & $236.738$ & --- \\
5  & $206.865\,950$ & $472.352$ & --- \\
6  & $206.914\,668$ & $707.966$ & $|W(A_2)| = |\Phi(A_2)|$ \\
7  & $206.963\,385$ & $943.580$ & --- \\
8  & $207.012\,103$ & $1179.194$ & $\dim(A_2)$ \\
9  & $207.060\,820$ & $1414.808$ & --- \\
10 & $207.109\,538$ & $1650.422$ & --- \\
\bottomrule
\caption{Gap table for $R(n) = 207(1 + (n-\omega)/(e^3\mu))$ versus CODATA.
  The unique minimum occurs at $n = 3 = h^\vee(A_2)$.}
\label{tab:scan}
\end{longtable}

The minimum is sharp and isolated at $n = 3$.  The second-closest candidate,
$n = 2$ ($\mathrm{rank}(A_2)$), is 208 times worse.  Among all Lie-algebraic
invariants of $A_2$ — rank, Coxeter number, dual Coxeter number, number of
positive roots, Weyl group order, total root count, dimension — only
$h^\vee(A_2) = h(A_2) = |\Phi^+(A_2)| = 3$ selects the sub-ppm regime.
(For $A_2$, all three of these invariants coincide at $3$; no test can
distinguish them on this basis alone.)

\paragraph{Remark on the $A_2$ triple coincidence.}
For $A_2 \cong \mathfrak{su}(3)$:
\begin{equation}
  h^\vee(A_2) \;=\; h(A_2) \;=\; |\Phi^+(A_2)| \;=\; 3.
\end{equation}
This triple coincidence holds only for $A_n$ (where $h = h^\vee = n+1$ and
$|\Phi^+| = n(n+1)/2$; equal to $h^\vee$ only for $n = 2$).  The scan
cannot resolve which of the three is ``really'' the relevant invariant.
P203 gives independent evidence for $h^\vee$ via the heat-conservation
argument; we adopt $h^\vee(A_2)$ as the primary label throughout.

%─────────────────────────────────────────────────────────────────────────────
\section{Structural Argument}
\label{sec:argument}
%─────────────────────────────────────────────────────────────────────────────

We state the best available structural argument for the identification
$3 = h^\vee(A_2)$ in $R^\star$.  We label this a conjecture; the proof
obligations are stated explicitly.

\subsection{Context}

The mass ratio formula arises from the spectral weight of the lepton sector
in the 4-simplex.  The leading term $207$ selects the lepton face (the
Fork/Weld/Plateau triangle, three of the five simplex vertices).  The
correction term is a ratio of frequencies and energies derived from the
same geometric objects.

\begin{conjecture}[OP-3: Dual Coxeter coupling]
\label{conj:op3}
The integer $3$ appearing in the mass ratio correction
$(\omega - 3)/(e^3\mu)$ is the dual Coxeter number $h^\vee(A_2) = 3$ of
the root system carried by each triangular 2-face of the 4-simplex on~$S^3$.
\end{conjecture}

\subsection{Supporting evidence}

\textbf{(i) Integer scan uniqueness.}  Of all integers $n = 1,\ldots,10$, only
$n = 3$ gives sub-10-ppm agreement with CODATA; the gap of $1.124\,\text{ppm}$
at $n = 3$ is 200$\times$ better than the next integer.

\textbf{(ii) $A_2$ face type.}  P202--P203 established that every triangular
2-face of the 4-simplex carries the $A_2$ root system.  The correction involves
the lepton face directly; it is structurally consistent that the Lie-algebraic
invariant of that face appears in the formula.

\textbf{(iii) Dual Coxeter number as coupling.}  In gauge theory, one-loop
corrections to propagators carry factors of $h^\vee(G)$ for the gauge group~$G$.
For $G = SU(3)$, i.e.\ $A_2$, $h^\vee = 3$.  The correction $(\omega -
h^\vee)/(e^3\mu)$ is formally analogous to a one-loop correction with coupling
$\omega$ normalised by $h^\vee$: the face carries a ``loop weight''~$h^\vee$
which is subtracted from the bare frequency $\omega$ before the correction is
divided out by the cubic energy-scale $e^3\mu$.

\textbf{(iv) Heat conservation.}  P203 showed that equidistance
($\langle v_i, v_j\rangle = -\tfrac{1}{4}$ for all pairs) is the geometric
form of the No-Harkonnen pin.  The $A_2$ Gram matrix (Section~\ref{sec:geom})
with all three pairs equal realises this exactly.  The dual Coxeter number
$h^\vee = 3$ counts the number of positive roots, which equals the number of
face edges (3), which equals the number of vertices (3): the triangle is
``balanced'' in the sense that no pairing is privileged.  This balance is what
heat conservation requires.

\textbf{(v) Reformulation as defect.}  Using $e^3 = 4\omega$, the correction
becomes
\begin{equation}
  \frac{\omega - h^\vee}{e^3\,\mu}
  = \frac{1}{4\mu}\!\left(1 - \frac{h^\vee}{\omega}\right),
\end{equation}
so $R^\star = 207\left(1 - \frac{1-h^\vee/\omega}{4\mu}\right)$.  The
factor $(1 - h^\vee/\omega) = 1 - 12/\pi^3 \approx 0.613$ is the fractional
excess of the fundamental frequency over the Coxeter-normalised frequency
$h^\vee$.  This decomposition suggests a renormalisation interpretation:
the ``bare'' mass ratio $207/1 = 207$ is corrected by the fractional
defect of $\omega$ relative to its minimal Lie-algebraic anchor $h^\vee$.

\subsection{Gap to proof}

The following steps are required to elevate Conjecture~\ref{conj:op3}
from ``supported by evidence'' to an established result:

\begin{enumerate}
  \item \textbf{Derivation of the 207 prefactor} from the 4-simplex geometry.
    P208 presented a candidate derivation; it has not been fully formalised.
    Without the 207 prefactor being derived, the formula $R^\star$ as a whole
    cannot be proved.

  \item \textbf{Operator integral giving $(\omega - h^\vee)$.}  A computation
    over the lepton face — for instance, a Weyl-character-type integral of the
    $A_2$ root system evaluated at the geodesic angle $\theta_4 =
    \arccos(-\tfrac{1}{4})$ — should reproduce the factor $\omega - h^\vee$.
    No such computation has been attempted.

  \item \textbf{Ruling out the triple coincidence.}  Since $h^\vee(A_2) =
    h(A_2) = |\Phi^+(A_2)| = 3$, the conjecture is ambiguous between three
    Lie-algebraic invariants.  A derivation must identify which one is
    structural.  The heat-conservation argument favours $h^\vee$ (via the
    loop-correction analogy), but this is heuristic.

  \item \textbf{Connection to P18-T2.}  The parent open problem OP-3 asks
    whether $R^\star$ is derivable from the Wheel $\leftrightarrow$
    $\hat{O}$ identification.  Establishing this would require showing that the
    $A_2$ root data of the lepton face follow from the algebraic structure of
    $\hat{O}$ (Paper~18), not merely from geometric coincidence.
\end{enumerate}

Until items (1)--(4) are resolved, Conjecture~\ref{conj:op3} remains OPEN.

%─────────────────────────────────────────────────────────────────────────────
\section{Open Problems Update}
\label{sec:op}
%─────────────────────────────────────────────────────────────────────────────

\begin{openprob}[OP-3: P18-T2 and the mass ratio correction integer]
\label{op:op3}
\textbf{Status: OPEN (conjectured).}

The formula $R^\star = 207(1 - (\omega - h^\vee(A_2))/(e^3\mu))$ achieves
$1.124\,\text{ppm}$ agreement with CODATA.  The identification $3 =
h^\vee(A_2)$ is supported by:
\begin{itemize}
  \item The integer scan (unique minimum at $n = 3$; 200$\times$ better than
    $n = 2$).
  \item The $A_2$ face type of the lepton triangle (P203).
  \item The heat-conservation / loop-correction analogy.
  \item The reformulation $R^\star = 207(1 - (1-h^\vee/\omega)/(4\mu))$
    as a ``renormalised defect'' formula.
\end{itemize}
Not yet proved: derivation of the $207$ prefactor; operator integral
reproducing $\omega - h^\vee$; disambiguation of $h^\vee$ vs $h$ vs
$|\Phi^+|$; connection to the $\hat{O}$ algebraic structure of Paper~18.
\end{openprob}

\begin{openprob}[OP-3b: The $1.124\,\text{ppm}$ residual]
\textbf{Status: OPEN.}

The gap $|R^\star - \text{CODATA}| = 1.124\,\text{ppm}$ is below the
$\sim\!10\,\text{ppm}$ one-loop QED correction scale but above the
$\sim\!0.001\,\text{ppm}$ two-loop scale.  Its origin is unknown.
Candidates: (a) CODATA uncertainty in the input value used
($206.768\,283\,0$, 6 significant figures); (b) a genuine two-loop correction
involving a second face of the 4-simplex; (c) a correction from the
non-lepton vertices~$v_3, v_4$.
\end{openprob}

\begin{openprob}[OP-3c: $A_2$ triple coincidence disambiguation]
\textbf{Status: OPEN.}

For $A_2$: $h^\vee = h = |\Phi^+| = 3$.  The structural argument cannot
distinguish these three invariants using numerical data alone.  An
independent derivation — most likely through the Weyl denominator formula
or the quadratic Casimir — is needed.
\end{openprob}

%─────────────────────────────────────────────────────────────────────────────
\section{Summary}
\label{sec:summary}
%─────────────────────────────────────────────────────────────────────────────

This addendum attempts to resolve Open Problem OP-3 by identifying the
integer~$3$ in the mass ratio correction with the dual Coxeter number
$h^\vee(A_2) = 3$ of the root system carried by the triangular faces of
the 4-simplex on~$S^3$.

\paragraph{What was established.}
\begin{itemize}
  \item The integer scan $n = 1,\ldots,10$ has a unique minimum at $n = 3$
    ($1.124\,\text{ppm}$); the next closest is $n = 2$ at $234\,\text{ppm}$.
  \item The correction $(\omega - h^\vee)/(e^3\mu)$ admits the
    exact reformulation $(1 - h^\vee/\omega)/(4\mu)$ using $e^3 = 4\omega$.
  \item The $3\times 3$ Gram matrix of the lepton face has eigenvalues
    $\tfrac{5}{4}$ (×2) and $\tfrac{1}{2}$ (×1); its trace equals $V =
    h^\vee(A_2) = 3$.
  \item The boundary Euler characteristic $\chi_\partial = V - E = 0$
    reflects the balanced ($A_2$-equidistant) structure of the lepton face.
\end{itemize}

\paragraph{What remains open.}
The identification is a conjecture.  Four proof obligations are listed in
Section~\ref{sec:argument}.  OP-3 is recorded as \textbf{CONJECTURED}: an
integer scan argument and structural analogy support $h^\vee(A_2)$, but no
derivation from first principles has been achieved.

\end{document}
