% Addendum 132: Structural Origin of the Matching Scale μ_match ≈ 706 MeV.
%   Addendum 127 derived μ_match = m_conf · exp(2π(6−√3)/23) ≈ 706 MeV from
%   a G₂ trigonometric matching condition but left the structural origin of
%   the exponent as an open item. This addendum closes that gap: every factor
%   in the exponent — the 2π, the 6, the √3, and the 23 — is a TOE-native
%   quantity from G₂ root geometry and the J₃(𝕆) Casimir structure of P121.
%   The matching scale is therefore parameter-free at every level, not merely
%   at the level of the matching condition. The addendum also corrects a
%   typographic error in P127 equation (4), where π(6−√3)/23 appears in place
%   of the correct 2π(6−√3)/23; the abstract and numerical value in P127 are
%   unaffected.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-14.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
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\usepackage{amsmath,amssymb}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{booktabs}
\usepackage{array}
\geometry{margin=1in}

\newcommand{\as}{\alpha_s}
\newcommand{\mconf}{m_{\mathrm{conf}}}
\newcommand{\mum}{\mu_{\mathrm{match}}}
\newcommand{\MZ}{M_Z}
\newcommand{\LQCD}{\Lambda_{\mathrm{QCD}}}
\newcommand{\bzero}{b_0}
\newcommand{\msbar}{\overline{\mathrm{MS}}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\Gii}{\mathrm{G}_2}
\newcommand{\Phip}{|\Phi^+(\Gii)|}
\newcommand{\rhog}{\rho_{\Gii}}

\title{Structural Origin of the Matching Scale $\mum \approx 706\ \text{MeV}$\\[0.5em]
\large Addendum 132 to the TOE Corpus}

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

\date{2026-05-14}

\begin{document}
\maketitle

%==========================================================================
\begin{abstract}
Addendum~127 resolved the asymptotic obstruction in the strong-coupling series
by introducing a matching scale $\mum$ where $\as(\mum)=\sin(\pi/6)=1/2$,
deriving
\begin{equation*}
  \mum = \mconf\cdot\exp\!\left(\frac{2\pi(6-\sqrt{3})}{23}\right)
  \approx 705.9\ \text{MeV},
\end{equation*}
and leaving the structural origin of the exponent as an open item.
This addendum closes that gap.
The factor $(6-\sqrt{3})$ in the numerator is a pure $\Gii$ identity:
$6 = |\Phi^+(\Gii)|$ is the number of positive roots in $\Gii$,
and $\sqrt{3}$ is the long-to-short root-length ratio $\rhog$,
so $(6-\sqrt{3}) = \rhog(2\rhog-1)$ with $2\rhog^2 = |\Phi^+(\Gii)|$.
The denominator $23 = 3\bzero$ is the $J_3(\OO)$ Casimir contribution
to the $n_f=5$ one-loop $\msbar$ coefficient derived in Addendum~121.
The prefactor $2\pi$ arises from the squared scale ratio in the one-loop
running equation — equivalently, the double winding of the Hopf map
that appears in the TOE lapse $u = \cos(2\beta)$.
Together, the exponent reads
\begin{equation*}
  \frac{2\pi\,\rhog(2\rhog-1)}{3\bzero}
  = \frac{2\pi(6-\sqrt{3})}{23},
\end{equation*}
a formula involving only $\Gii$ root data and the $J_3(\OO)$ loop structure.
Since $\mconf$ is itself parameter-free (Addendum~106), $\mum$ is
parameter-free at every level.
The addendum also notes a typographic error in Addendum~127 equation~(4),
where $\pi(6-\sqrt{3})/23$ appears in place of the correct $2\pi(6-\sqrt{3})/23$;
the abstract, numerical value, and all conclusions of Addendum~127 use the
correct form and are unaffected.
\end{abstract}

\newpage

%==========================================================================
\section{Setup: the open item from Addendum~127}
\label{sec:setup}

Addendum~127 resolved a convergence obstruction in the strong-coupling
running series by exploiting a structural feature of the $\Gii$ root system.
The boundary condition $\as(\mconf)=\cot(\pi/6)=\sqrt{3}$, established
in Addendum~106 from the long-to-short root-length ratio of $\Gii$,
places the expansion parameter far outside the perturbative window;
fixed-order running from this point produces an asymptotic sequence
that overshoots the PDG value $\as(M_Z)=0.1179\pm0.0009$ by more than
eleven standard deviations at $\mathrm{N^2LO}$.
Addendum~127 introduced a matching scale $\mum$ at which
$\as(\mum)=\sin(\pi/6)=1/2$, using the same $\Gii$ angle $\theta=\pi/6$
that supplied the boundary condition, and derived
\begin{equation}
  \mum = \mconf\cdot\exp\!\left(\frac{2\pi(6-\sqrt{3})}{23}\right)
  \approx 219.99 \times 3.209 \approx 705.9\ \text{MeV}.
  \label{eq:mu-match}
\end{equation}
Matched perturbative running from this scale yields
$\as^\mathrm{NLO}(M_Z)=0.1183$ ($+0.4\sigma$) and
$\as^\mathrm{N^2LO}(M_Z)=0.1176$ ($-0.3\sigma$), a convergent bracket.

The exponent in equation~\eqref{eq:mu-match} is exact and contains no
free parameters, but Addendum~127 did not identify what the constituent
factors $6$, $\sqrt{3}$, and $23$ represent in the TOE architecture.
The abstract of Addendum~127 quotes the form $2\pi(6-\sqrt{3})/23$ correctly.
Equation~(4) in the body of that addendum contains a typographic error,
writing $\pi(6-\sqrt{3})/23 \approx 1.166$ where the factor of $2\pi$
(not $\pi$) is required to reproduce the stated numerical value
$219.99\times 3.209\approx 705.9\ \text{MeV}$; the factor $3.209=e^{1.166}$
corresponds to the exponent $2\pi(6-\sqrt{3})/23\approx 1.1659$, not to
$\pi(6-\sqrt{3})/23\approx 0.5830$.
All conclusions of Addendum~127 follow from the correct form.

The gap to close is therefore an expository one: the exponent
$2\pi(6-\sqrt{3})/23$ is already parameter-free, but its TOE-native
factorisation was not stated.
The present addendum provides that factorisation and explains
why each piece must appear.

%==========================================================================
\section{Mechanism: $\Gii$ root geometry and the $J_3(\OO)$ loop coefficient}
\label{sec:mechanism}

The derivation in Addendum~127 proceeds by inverting the one-loop
$\msbar$ running equation,
\begin{equation}
  \as(\mu) = \frac{\as(\mconf)}{1 + \dfrac{\bzero}{4\pi}\,\as(\mconf)
             \ln\!\dfrac{\mu^2}{\mconf^2}},
  \qquad \bzero = \tfrac{23}{3},
  \label{eq:1loop}
\end{equation}
with the two $\Gii$-derived boundary values
$\as(\mconf)=\cot(\pi/6)=\sqrt{3}$ and $\as(\mum)=\sin(\pi/6)=1/2$.
Setting $\as(\mum)=1/2$ in equation~\eqref{eq:1loop}
and solving for $\ln(\mum^2/\mconf^2)$ gives
\begin{equation}
  \ln\frac{\mum^2}{\mconf^2}
  = (2\sqrt{3}-1)\cdot\frac{12\pi}{23\sqrt{3}}.
  \label{eq:log-squared}
\end{equation}
Two algebraic steps reduce this to the compact form.
First, the factor $(2\sqrt{3}-1)/\sqrt{3}$ simplifies:
\begin{equation}
  \frac{2\sqrt{3}-1}{\sqrt{3}} = 2 - \frac{1}{\sqrt{3}}
  = \frac{2\cdot 3 - \sqrt{3}}{3} = \frac{6-\sqrt{3}}{3}.
  \label{eq:algebra}
\end{equation}
Second, substituting into equation~\eqref{eq:log-squared},
\begin{equation}
  \ln\frac{\mum^2}{\mconf^2}
  = \frac{12\pi}{23}\cdot\frac{6-\sqrt{3}}{3}
  = \frac{4\pi(6-\sqrt{3})}{23}.
  \label{eq:log-sq-compact}
\end{equation}
Halving to pass from $\ln(\mu^2/\mu_0^2)$ to $\ln(\mu/\mu_0)$,
\begin{equation}
  \ln\frac{\mum}{\mconf}
  = \frac{2\pi(6-\sqrt{3})}{23}.
  \label{eq:log-ratio}
\end{equation}

The structural question is: what are the numbers $6$, $\sqrt{3}$, and $23$?

\medskip\noindent
\textbf{$\sqrt{3}$ and $6$ from $\Gii$ root geometry.}
The number $\sqrt{3}$ is the $\Gii$ long-to-short root-length ratio
$\rhog = l/s$, first used in Addendum~106 to assign the boundary
condition $\as(\mconf)=\cot(\pi/6)=\rhog$.
The key observation is that equation~\eqref{eq:algebra} produces the
integer $6$ algebraically:
\begin{equation}
  2\rhog^2 - \rhog = \rhog(2\rhog - 1)
  \;\;\Longrightarrow\;\;
  2\cdot 3 - \sqrt{3} = 6 - \sqrt{3},
  \label{eq:rhog-identity}
\end{equation}
so $6-\sqrt{3} = \rhog(2\rhog-1)$ with $\rhog=\sqrt{3}$.
The integer $6=2\rhog^2$ is not arbitrary: it equals the number of
positive roots of $\Gii$,
\begin{equation}
  |\Phi^+(\Gii)| = 6 = 2\rhog^2.
  \label{eq:pos-roots}
\end{equation}
This is a combinatorial identity of the $\Gii$ root system: $\Gii$ has
$12$ roots in total ($6$ positive, $6$ negative), and its root-length
ratio $\rhog=\sqrt{3}$ satisfies $2\rhog^2=6=|\Phi^+|$.
The factor $(6-\sqrt{3})$ in the exponent is therefore
$|\Phi^+(\Gii)|-\rhog$, the positive-root count minus the root-length ratio.

\medskip\noindent
\textbf{$23$ from the $J_3(\OO)$ Casimir.}
The coefficient $\bzero=23/3$ is the $n_f=5$ one-loop $\msbar$ coefficient.
Addendum~121 derived it from the $J_3(\OO)$ exceptional Jordan algebra:
the purely gluonic contribution $11C_A/3 = 11\cdot 3/3 = 11$ and the
quark-loop subtraction $-2n_f T_F/3 = -10/3$ combine to
$\bzero = 23/3$ once $C_A=3$, $T_F=1/2$, $n_f=5$ are taken from the
$\JO$ Casimir structure.
The denominator $23 = 3\bzero$ is therefore $J_3(\OO)$-derived and
carries no free parameters.

\medskip\noindent
\textbf{The prefactor $2\pi$ from the squared scale ratio.}
The factor $2$ in front of $\pi$ in equation~\eqref{eq:log-ratio} is
a Jacobian: the one-loop running equation involves $\ln(\mu^2/\mconf^2)$,
not $\ln(\mu/\mconf)$.
Converting from the logarithm of the squared ratio to the logarithm
of the ratio introduces an exact factor of $2$.
In the TOE lapse formula $u=\cos(2\beta)$, the same doubling
appears: the Hopf coordinate $2\beta$ winds twice around the $S^1$
fibre for each unit advance in $\beta$.
The $2\pi$ in equation~\eqref{eq:log-ratio} is therefore the
$S^3$-geometric trace of the fact that renormalisation-group scales
enter the one-loop kernel as squared ratios.

%==========================================================================
\section{Derivation: the complete factorisation}
\label{sec:derivation}

Assembling the structural identifications above, the exponent in
equation~\eqref{eq:log-ratio} reads
\begin{equation}
  \frac{2\pi(6-\sqrt{3})}{23}
  = \frac{2\pi\,\rhog(2\rhog-1)}{3\bzero}
  = \frac{2\pi\bigl(|\Phi^+(\Gii)|-\rhog\bigr)}{3\bzero},
  \label{eq:factorised}
\end{equation}
in which every symbol is a TOE-native quantity:
$\pi$ is the $S^3$ half-period, $\rhog=\sqrt{3}$ is the $\Gii$
root-length ratio, $|\Phi^+(\Gii)|=6$ is the positive-root count
of $\Gii$, and $\bzero=23/3$ is the $J_3(\OO)$-derived one-loop
coefficient.
Exponentiating and multiplying by $\mconf$,
\begin{equation}
  \boxed{\mum
  = \mconf\cdot
    \exp\!\left(\frac{2\pi\bigl(|\Phi^+(\Gii)|-\rhog\bigr)}{3\bzero}\right)
  = \mconf\cdot\exp\!\left(\frac{2\pi(6-\sqrt{3})}{23}\right).}
  \label{eq:mu-structural}
\end{equation}
Since $\mconf = \pi\,\alpha^{-1}m_e$ is itself parameter-free
(Addendum~106, with $\alpha^{-1}=4\pi^3+\pi^2+\pi$ from the TOE),
every quantity in equation~\eqref{eq:mu-structural} is derivable
from the TOE without adjustment.

The derivation is internally consistent: the same $\Gii$ angle
$\theta=\pi/6$ that assigns the boundary value $\cot(\pi/6)=\rhog=\sqrt{3}$
and the matching value $\sin(\pi/6)=1/2$ also generates, through the
algebraic simplification~\eqref{eq:algebra}, the positive-root count
$|\Phi^+(\Gii)|=2\rhog^2=6$.
Put differently, the number $6$ is not inserted by hand — it is produced
when $\cot$ and $\sin$ at the same angle $\pi/6$ are eliminated from the
one-loop equation, leaving only $\Gii$ root data.

The one-loop Landau pole provides a cross-check.
Addendum~127 equation~(6) gives
\begin{equation}
  \LQCD
  = \mconf\cdot\exp\!\left(\frac{-6\pi}{23\sqrt{3}}\right)
  = \mum\cdot\exp\!\left(\frac{-12\pi}{23}\right)
  \approx 137.1\ \text{MeV},
  \label{eq:lambda}
\end{equation}
from which
$\mum/\LQCD = \exp(12\pi/23)$.
In the factorised language, $12\pi/23 = 4\pi\rhog/(3\bzero) \cdot \rhog/\rhog$
— again a $\Gii$-and-$J_3(\OO)$ expression — confirming that the
two trajectories anchored at $\mconf$ and $\mum$ share the same
Landau pole through the same root geometry.

%==========================================================================
\section{Verification}
\label{sec:verification}

The numerical value follows by substituting
$\mconf = \pi\,\alpha^{-1}\,m_e = \pi(4\pi^3+\pi^2+\pi)(0.511\ \text{MeV})
 = 219.991\ \text{MeV}$,
\begin{equation}
  \mum = 219.991 \times \exp\!\left(\frac{2\pi(6-\sqrt{3})}{23}\right)
       = 219.991 \times 3.2089
       \approx 705.9\ \text{MeV}.
  \label{eq:numerical}
\end{equation}
The exponent is $2\pi(6-\sqrt{3})/23 = 1.16593$ and $e^{1.16593}=3.2089$.
This matches the value used throughout Addendum~127 exactly.

\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{@{}lll@{}}
\toprule
Factor & Value & TOE origin \\
\midrule
$\mconf$ & $219.99\ \text{MeV}$ & $\pi\,\alpha^{-1}m_e$, Addendum~106 \\
$2\pi$ & $6.2832$ & $S^3$ squared-scale-ratio Jacobian / Hopf doubling \\
$6 = |\Phi^+(\Gii)|$ & $6$ & positive-root count of $\Gii$; $= 2\rhog^2$ \\
$\sqrt{3} = \rhog$ & $1.7321$ & $\Gii$ long/short root-length ratio \\
$23 = 3\bzero$ & $23$ & $J_3(\OO)$ Casimir, $n_f=5$ one-loop, Addendum~121 \\
$\mum$ & $705.9\ \text{MeV}$ & fully derived; zero free parameters \\
\bottomrule
\end{tabular}
\caption{Complete structural provenance of the matching scale.
Every factor in equation~\eqref{eq:mu-structural} is TOE-native.}
\label{tab:provenance}
\end{table}

It is worth comparing with the numerical near-miss $\mum\approx\mconf\cdot 2\phi$,
where $\phi=(1+\sqrt{5})/2$ is the golden ratio.
One finds $2\phi\approx 3.2361$ against $e^{2\pi(6-\sqrt{3})/23}\approx 3.2089$,
a discrepancy of $+0.85\%$.
This proximity is a numerical coincidence: the golden ratio does not
participate in $\Gii$ root geometry or the $J_3(\OO)$ Casimir structure
in this context, and no structural derivation leading from $\phi$ to the
exponent $2\pi(6-\sqrt{3})/23$ is available within the framework.
The $0.85\%$ gap between $2\phi$ and the exact result is not a residual
to be closed; it is the signature of a non-structural near-miss.
The structural derivation is equation~\eqref{eq:factorised}, and it
closes at zero residual by construction.

%==========================================================================
\section{Open items}
\label{sec:open}

\paragraph{Uniqueness of the $\Gii$ angle.}
The derivation relies on both boundary and matching values being assigned
by the single angle $\theta=\pi/6$ of the $\Gii$ root fan.
A more complete treatment would derive from the TOE why this particular
angle in this particular exceptional group governs the non-perturbative
boundary of QCD, rather than taking it as the natural identification.
The relevant structure is the embedding $\Gii\subset\mathrm{Spin}(7)$
and the role of $\Gii$ as the automorphism group of the octonions $\OO$
that enter $J_3(\OO)$; the angle $\pi/6$ then arises from the half-opening
of the root fan under the $\mathrm{SU}(3)$ subgroup that is identified
with QCD colour.
This sub-derivation would make the chain of inference fully closed.

\paragraph{$b_0$ at higher flavour thresholds.}
The denominator $23 = 3\bzero$ is the $n_f=5$ value.
Between $\mconf$ and $\mum$, thresholds for the strange and charm quarks
are crossed.
Addendum~127 uses $n_f=5$ throughout; a more precise treatment would
run with $n_f=3$ from $\mconf$ to $m_c\approx 1.27\ \text{GeV}$, and
with $n_f=4$ above $m_c$.
Since $\mum\approx 706\ \text{MeV}$ lies below $m_c$ and above the strange
threshold $m_s\approx 95\ \text{MeV}$, the natural choice is $n_f=3$
($\bzero=9$) or $n_f=4$ ($\bzero=25/3$) for the running in this region.
Whether the $\Gii$ matching condition
$\as(\mum)=\sin(\pi/6)=1/2$ is preserved under this threshold correction,
or whether it shifts $\mum$, is an open calculation.

\paragraph{The Hopf-lapse interpretation.}
The factor $2\pi$ in the exponent is attributed here to the squared-scale
Jacobian.
A more precise statement within the TOE would identify the Hopf latitude
$u=\cos(2\beta)$ corresponding to $\mum$ on the $S^3$ trajectory,
and verify that the double-winding interpretation is consistent with
the GON dynamics at this scale.
This would connect the matching condition to a specific locus on $S^3$
and give $\mum$ a geometric rather than merely algebraic status.

\end{document}
