% P207 — The Three-Way Convergence Cluster Under 4-Simplex Geometry
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{\textbf{Addendum P207}\\[4pt]
The Three-Way Convergence Cluster Under 4-Simplex Geometry\\[2pt]
{\normalsize Why Dirac${}^2$, $G_2$+One-Loop, and CR/Hopf Land Within 0.015 of Each Other}}
\author{L\'eon Fernando Vlegels}
\date{17 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

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

Addendum P197 established that three structurally independent routes to
$m_\mu/m_e$ converge within a narrow band around the CODATA value
$R = 206.7682830$:

\begin{table}[h]
\centering
\caption{The three-way convergence cluster.}
\label{tab:cluster}
\begin{tabular}{@{}llll@{}}
\toprule
Route & Formula & Value & $|v - R|$ \\
\midrule
Dirac${}^2$ & $(417/29)^2$ & $206.764566$ & $0.003717$ \\
$G_2$+one-loop & $207(1-\alpha/2\pi)$ & $206.759589$ & $0.008694$ \\
CR/Hopf & $827/4$ & $206.750000$ & $0.018283$ \\
\midrule
CODATA & --- & $206.7682830$ & --- \\
\bottomrule
\end{tabular}
\end{table}

All three lie within $0.018$ of $R$ and within $0.015$ of one another.
The routes are structurally independent: Dirac$^2$ arises from an $S^3$
Dirac spectrum ratio, $G_2$+one-loop from the exceptional Lie algebra count
corrected by one QED loop, and CR/Hopf from an asymptotic of the CR
sub-Laplacian on the Heisenberg model of $S^3$.
The question addressed here is: \emph{why do three independent spectral
constructions converge to nearly the same value?}

The answer has two layers. A primary explanation — the \emph{lattice
quantisation mechanism} — accounts for why all three land within the same
narrow band. A secondary question — whether the cluster width is controlled
by the 4-simplex geodesic angle $\theta_4 = \arccos(-1/4)$ — is examined
and found to be suggestive but unproved.

Throughout this addendum the TOE constants are
\begin{align}
  \alpha^{-1} &= 4\pi^3 + \pi^2 + \pi, &
  \Omega_0    &= \frac{\pi^3}{4},       &
  \alpha      &= \frac{1}{\alpha^{-1}},
\end{align}
and the 4-simplex geodesic angle (P202, Eq.~5.3) is
\begin{equation}
  \theta_4 = \arccos\!\bigl(-\tfrac{1}{4}\bigr)
  \;\approx\; 1.8235\;\text{rad}
  \;\approx\; 104.48^\circ.
  \label{eq:theta4}
\end{equation}
All numerical claims are verified by \texttt{verify/verify\_P207.py}
(\texttt{mpmath}, \texttt{dps}$=60$).

%─────────────────────────────────────────────────────────────────────────────
\section{Precision Values}
%─────────────────────────────────────────────────────────────────────────────

At \texttt{mpmath} precision \texttt{dps}$=60$ the three candidates evaluate
to
\begin{align}
  v_{\mathrm{D}}  &= (417/29)^2
    &&= 206.764\,565\,992\,865\,636\,147\ldots, \label{eq:vD} \\
  v_{\mathrm{G}}  &= 207\bigl(1 - \alpha/(2\pi)\bigr)
    &&= 206.759\,588\,719\,833\,660\,375\ldots, \label{eq:vG} \\
  v_{\mathrm{H}}  &= 827/4
    &&= 206.750\,000\,000\,000\ldots \label{eq:vH}
\end{align}
The cluster statistics are
\begin{align}
  \bar{v} &= \frac{v_{\mathrm{D}}+v_{\mathrm{G}}+v_{\mathrm{H}}}{3}
            = 206.758\,051\,570\,899\,766, \label{eq:centroid} \\
  w       &= v_{\mathrm{D}} - v_{\mathrm{H}}
            = 0.014\,565\,992\,866. \label{eq:width}
\end{align}
The CODATA target $R = 206.7682830$ lies above all three candidates and
above the cluster centroid by $0.010\,231$.  The cluster lies entirely
below $R$: the closest approach is $v_{\mathrm{D}}$ (below $R$ by $0.003717$)
and the furthest is $v_{\mathrm{H}}$ (below $R$ by $0.018283$; see Table~\ref{tab:k}).

%─────────────────────────────────────────────────────────────────────────────
\section{Decomposition by Route}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The $\alpha^{-1} + c\,\Omega_0$ frame}

All mass-ratio candidates in the P193--P207 programme are analysed in the
lattice frame
\begin{equation}
  v = \alpha^{-1} + k\,\Omega_0,
  \quad
  k(v) = \frac{v - \alpha^{-1}}{\Omega_0} = \frac{4(v - 4\pi^3 - \pi^2 - \pi)}{\pi^3}.
  \label{eq:kframe}
\end{equation}
The $k$-values for the three candidates and for CODATA are collected in
Table~\ref{tab:k}.

\begin{table}[h]
\centering
\caption{Lattice co-ordinate $k(v) = (v-\alpha^{-1})/\Omega_0$ for each value.}
\label{tab:k}
\begin{tabular}{@{}llll@{}}
\toprule
Value & $k$ (dps=60) & $\lfloor k \rfloor$ & $k - 9$ \\
\midrule
$v_{\mathrm{D}} = (417/29)^2$  & $8.99537\,37994\ldots$ & 8 & $-0.00462\,620\ldots$ \\
$v_{\mathrm{G}} = 207(1-\alpha/2\pi)$ & $8.99473\,17006\ldots$ & 8 & $-0.00526\,830\ldots$ \\
$v_{\mathrm{H}} = 827/4$       & $8.99349\,46970\ldots$ & 8 & $-0.00650\,530\ldots$ \\
$R = 206.7682830$ (CODATA)     & $8.99585\,33162\ldots$ & 8 & $-0.00414\,668\ldots$ \\
\bottomrule
\end{tabular}
\end{table}

All four $k$-values are floor-8, and all lie within $\pm 0.002$ of $9$.
The spread in $k$ across the three candidates is
\begin{equation}
  \Delta k = k_{\mathrm{D}} - k_{\mathrm{H}} = 0.001\,879\,102.
  \label{eq:Dk}
\end{equation}

\subsection{Dirac${}^2$: $(417/29)^2$}

The Dirac route emerges from the $S^3$ eigenvalue ratio
$(\lambda_1/\lambda_2)^2$ with $\lambda_1 = 417/2$ ($\ell_1 = 207$) and
$\lambda_2 = 29/2$ ($\ell_2 = 13$):
\begin{equation}
  v_{\mathrm{D}} = \left(\frac{417}{29}\right)^2
                = \frac{173889}{841}.
  \label{eq:dirac-exact}
\end{equation}
In the lattice frame:
\begin{align}
  v_{\mathrm{D}} - \alpha^{-1} &= 69.728\,262\ldots = 9\,\Omega_0 - 0.035\,860, \\
  k_{\mathrm{D}}               &= 8.99537\ldots = 9 - 0.004\,626\ldots \label{eq:kD}
\end{align}
The mode index $\ell_1 = 207$ is the same integer that drives $G_2$+one-loop
(see \S\ref{sec:g2}), and $\ell_2 = 13$ is structurally unexplained at present
(see P197, OP-Dirac).

\subsection{$G_2$+one-loop: $207(1-\alpha/2\pi)$}
\label{sec:g2}

The tree-level $G_2$ count gives $14 \times 15 - 3 = 207$, and the one-loop
QED correction multiplies by $(1 - \alpha/2\pi)$:
\begin{equation}
  v_{\mathrm{G}} = 207\!\left(1 - \frac{\alpha}{2\pi}\right)
                = 207 - \frac{207}{2\pi\,\alpha^{-1}}.
  \label{eq:g2loop}
\end{equation}
The correction term is $207\alpha/(2\pi) = 0.240\,411\ldots$ In the lattice
frame:
\begin{align}
  k_{\mathrm{G}}  &= 8.99473\ldots = 9 - 0.005\,268\ldots \label{eq:kG}
\end{align}
Note that the integer $207$ has its own $k$-value:
\begin{equation}
  k(207) = \frac{207 - \alpha^{-1}}{\Omega_0} = 9.025\,746\ldots
  \label{eq:k207}
\end{equation}
so $207$ sits $0.026\,\Omega_0$ above the $n=9$ lattice step.  The
one-loop correction brings $v_{\mathrm{G}}$ down from $207$ (at $k=9.026$)
to $206.760$ (at $k=8.9947$), crossing the $n=9$ step and landing near
CODATA.

\subsection{CR/Hopf: $827/4$}

The Hopf route gives the rational value $827/4 = 206.75$ exactly.  In the
lattice frame:
\begin{align}
  827/4 - \alpha^{-1} &= 69.713\,696\ldots = 9\,\Omega_0 - 0.050\,426, \\
  k_{\mathrm{H}}      &= 8.99349\ldots = 9 - 0.006\,505\ldots \label{eq:kH}
\end{align}
The integer $827 = 4 \times 207 - 1$ (P197) again invokes the $G_2$ integer
$207$.

%─────────────────────────────────────────────────────────────────────────────
\section{The Lattice Quantisation Mechanism}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The shared integer $207$}

All three routes are anchored to the integer $207$:
\begin{align}
  v_{\mathrm{D}} &= \text{Dirac ratio with mode } \ell_1 = 207, \\
  v_{\mathrm{G}} &= 207 \times (\text{one-loop factor}), \\
  v_{\mathrm{H}} &= (4 \times 207 - 1)/4.
\end{align}
The integer $207 = 14 \times 15 - 3 = \dim(G_2)\cdot(\dim(G_2)+1) - \dim(A_2)$
is the $G_2$ NNLO count established in P197.  It has lattice co-ordinate
$k(207) = 9.026$ — just above the $n=9$ step of the $\alpha^{-1} + n\,\Omega_0$
lattice.

\subsection{Mechanism}

Each route applies a small downward correction to $207$ (or a close variant):
the Dirac ratio replaces $207$ by $(417/29)^2 \approx 206.765$; the $G_2$
one-loop formula multiplies $207$ by $(1-\alpha/2\pi) \approx 0.99884$; the
Hopf route takes $(4\times207-1)/4 = 206.75$.  All corrections push
the value below the $n=9$ lattice step $\alpha^{-1}+9\,\Omega_0 = 206.800$,
landing near CODATA at $206.768$.

\begin{proposition}[Lattice Quantisation]
\label{prop:lattice}
Let $n=9$.  All three candidates satisfy
\begin{equation}
  \alpha^{-1} + (n-1)\,\Omega_0
  \;<\; v_i
  \;<\; \alpha^{-1} + n\,\Omega_0,
  \label{eq:bracket}
\end{equation}
and their $k$-values span a range $\Delta k = 0.001\,879$, which is
$0.21\%$ of the lattice step $\Omega_0$.  CODATA ($k=8.99585$) lies within
this range.

\begin{proof}
Direct computation at \texttt{dps}$=60$; see Table~\ref{tab:k} and
\texttt{verify\_P207.py} checks 11--16.
\end{proof}
\end{proposition}

\begin{remark}
The $k$-range of $0.001\,879$ corresponds to a physical interval of width
$\Delta k \cdot \Omega_0 = 0.001\,879 \times 7.7516 = 0.014\,566$,
which equals the cluster width $w$ exactly (by construction of $\Delta k$).
\end{remark}

\subsection{Why the clustering is not a coincidence}

The clustering follows from three facts:
\begin{enumerate}
  \item All three routes are anchored to the $G_2$ integer $207$.
  \item $207$ sits at $k=9.026$, just above the $n=9$ lattice step.
  \item Each correction mechanism (Dirac ratio, one-loop, Hopf asymptotic)
        is of order $O(\alpha)$, bringing the result down by $O(\alpha) \sim
        0.007$ in $k$-units — enough to cross the $n=9$ step and land near
        CODATA ($k=8.9959$).
\end{enumerate}
In other words: the same exceptional Lie algebra integer ($207$) appears in
all three spectral constructions, and each construction independently
generates an $O(\alpha)$ correction that lands near CODATA.  The convergence
is a consequence of the $G_2$ structure shared by all three routes, not an
accident of numerology.

\begin{conjecture}[Lattice-anchoring conjecture]
\label{conj:lattice}
For any $S^3$ spectral construction whose leading-order contribution is the
$G_2$ orbit count $207$, the next-order correction lies in the interval
$[0, 207\,\alpha]$, which spans an interval of $k$-width
$207\,\alpha/\Omega_0 = 0.018$ — exactly the cluster width relative to $207$.
Any such construction therefore lands within $[206.75, 207]$, of which
the interval $[206.75, 206.800]$ (the CODATA corridor below the $n=9$
lattice step) has width $0.050 = 6.5\,\alpha$.
\end{conjecture}

%─────────────────────────────────────────────────────────────────────────────
\section{Cluster Width and 4-Simplex Geometry}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The width $w \approx \theta_4\,\alpha$}

The cluster width $w = v_{\mathrm{D}} - v_{\mathrm{H}} = 0.014\,566$ is
compared to the simplex-derived scale $\theta_4\,\alpha$:
\begin{align}
  \theta_4\,\alpha &= \arccos(-\tfrac{1}{4})\cdot\alpha^{-1} = 1.8235\cdot(137.036)^{-1}
  = 0.013\,307\ldots \label{eq:theta4alpha}
\end{align}
The ratio is
\begin{equation}
  \frac{w}{\theta_4\,\alpha} = \frac{0.014\,566}{0.013\,307} = 1.0947.
  \label{eq:ratio}
\end{equation}
This is a $9.5\%$ deviation from $1$.  The match is suggestive but not exact;
no simple Lie-algebraic correction closes the gap.

\begin{table}[h]
\centering
\caption{Cluster width $w$ compared to simplex-derived scales.}
\label{tab:width}
\begin{tabular}{@{}lll@{}}
\toprule
Candidate scale & Value & $w /$ scale \\
\midrule
$\theta_4\,\alpha$           & $0.013\,307$ & $1.0947$ \\
$\theta_4\,\alpha/2$         & $0.006\,653$ & $2.189$ \\
$\sin\theta_4\cdot\alpha$    & $0.007\,066$ & $2.061$ \\
$\theta_4^2/\alpha^{-1}$     & $0.024\,264$ & $0.600$ \\
$\alpha/(2\pi)$              & $0.001\,161$ & $12.54$ \\
$h(E_8)\cdot\alpha/(2\pi)$   & $0.034\,842$ & $0.418$ \\
\bottomrule
\end{tabular}
\end{table}

None of the simplex-derived scales reproduce $w$ to within $5\%$.
The closest ($\theta_4\,\alpha$ at $9.5\%$ off) is tantalising but falls
outside the tolerance for a structural identity.

\subsection{Internal gap structure}

The two internal gaps of the cluster are
\begin{align}
  |v_{\mathrm{D}} - v_{\mathrm{G}}| &= 0.004\,977, \label{eq:gapDG} \\
  |v_{\mathrm{G}} - v_{\mathrm{H}}| &= 0.009\,589. \label{eq:gapGH}
\end{align}
Their ratio is
\begin{equation}
  \frac{|v_{\mathrm{G}}-v_{\mathrm{H}}|}{|v_{\mathrm{D}}-v_{\mathrm{G}}|}
  = 1.926.
  \label{eq:gap-ratio}
\end{equation}
This is not close to $2$ or to $h^\vee(A_2)/h^\vee(A_1) = 3/2$.  No clean
decomposition has been found.

\subsection{Residual structure}

The three residuals from the $n=9$ lattice step, expressed in units of
$\alpha^{-1}$, are
\begin{align}
  (v_{\mathrm{D}} - [\alpha^{-1}+9\Omega_0])/\alpha &= -4.914, \\
  (v_{\mathrm{G}} - [\alpha^{-1}+9\Omega_0])/\alpha &= -5.596, \\
  (v_{\mathrm{H}} - [\alpha^{-1}+9\Omega_0])/\alpha &= -6.910.
\end{align}
The values $-4.9$, $-5.6$, $-6.9$ are not integers or simple Lie-algebraic
fractions.  The ratios between residuals ($v_{\mathrm{G}}/v_{\mathrm{D}}
= 1.139$; $v_{\mathrm{H}}/v_{\mathrm{D}} = 1.406$) are also not close to
small rationals.

%─────────────────────────────────────────────────────────────────────────────
\section{Status}
%─────────────────────────────────────────────────────────────────────────────

\textbf{Status: OPEN, with a structural explanation for the clustering but
not for its fine geometry.}

Two layers of analysis are resolved to different degrees:

\paragraph{Layer 1 — Why three routes cluster (RESOLVED).}
All three spectral constructions are anchored to the $G_2$ integer $207$.
That integer sits at lattice co-ordinate $k(207) = 9.026$, just above the
$n=9$ step.  Each construction independently applies an $O(\alpha)$ downward
correction, landing in the corridor $(206.750, 206.800)$ below the $n=9$
step.  CODATA lies in the same corridor.  The clustering is therefore a
consequence of the shared $G_2$ integer $207$ acting as an anchor,
and of the universal scale of $O(\alpha)$ QED/spectral corrections:
Conjecture~\ref{conj:lattice}.

\paragraph{Layer 2 — Why the cluster has its specific width (OPEN).}
The cluster width $w = 0.014\,566$ is close to $\theta_4\,\alpha =
0.013\,307$ (ratio $1.095$) but this $9.5\%$ deviation has no algebraic
explanation.  The internal gap ratio ($1.926$) and the residuals in units of
$\alpha$ ($-4.9$, $-5.6$, $-6.9$) are not expressible in simplex invariants.
The 4-simplex angle $\theta_4 = \arccos(-1/4)$ appears as the closest scale
but does not tightly control the geometry of the cluster.

\paragraph{Open problems inherited from P197.}
\begin{description}
  \item[OP-Dirac.] Derive the Dirac mode pair $(\ell_1,\ell_2) = (207,13)$
    from a $G_2$-holonomy constraint.  If $\ell_2 = 13$ is forced, the
    residual $k_{\mathrm{D}} - 9 = -0.004\,626$ would acquire a structural
    interpretation.
  \item[OP-Width.] Identify the algebraic source of the cluster width.
    The scale $\theta_4\,\alpha$ is the best candidate so far.  The discrepancy
    factor $1.095 \approx 1 + h^\vee(A_1)\,\alpha\cdot c$ for some small $c$
    may point to a sub-leading simplex contribution.
\end{description}

\bigskip
\noindent
\textit{Verification.}  All numerical claims are checked by
\texttt{Lumen/corpus/addenda/verify/verify\_P207.py}
(all assertions passing at \texttt{dps}$=60$).

% VERIFY: see verify_P207.py

\end{document}
