% Addendum 127: Strong Coupling α_s — Geometric Matching at the Non-Perturbative Boundary.
%   Addendum 125 identified the fixed-order loop series from α_s(m_conf)=√3
%   as asymptotic. This addendum resolves the obstruction by a TOE-native
%   matching: the same G₂ root angle θ=π/6 that yields the boundary
%   cot(θ)=√3 supplies the companion matching condition α_s(μ_match)=sin(θ)=1/2.
%   Solving one-loop running analytically gives
%   μ_match = m_conf·exp(2π(6−√3)/23) ≈ 706 MeV.
%   Matched NLO:  α_s(M_Z) = 0.1183 (+0.4σ, PDG 0.1179±0.0009).
%   Matched N²LO: α_s(M_Z) = 0.1176 (−0.3σ). Series converges.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-14.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\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{\bone}{b_1}
\newcommand{\btwo}{b_2}
\newcommand{\msbar}{\overline{\mathrm{MS}}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}

\title{Strong Coupling $\alpha_s$ — Geometric Matching\\
at the Non-Perturbative Boundary\\[0.5em]
\large Addendum 127 to the TOE Corpus}

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

\date{2026-05-14}

\begin{document}
\maketitle

%==========================================================================
\begin{abstract}
Addendum~125 established that the fixed-order $\msbar$ loop series
launched from the TOE boundary $\as(\mconf)=\sqrt{3}$ is asymptotic:
the LO$\to$NLO$\to$N$^2$LO sequence $0.1186\to0.1146\to0.1283$
oscillates with growing amplitude, a diagnostic of perturbation theory
operating outside its radius of convergence.
The obstruction is not in the loop coefficients, which are exact, but
in the starting point: $\as=\sqrt{3}\approx 1.73$ lies far above the
perturbative window.
This addendum resolves the obstruction by a matching derived entirely
from the same G$_2$ root geometry.
The boundary $\as(\mconf)=\cot(\pi/6)=\sqrt{3}$ and the matching
condition $\as(\mum)=\sin(\pi/6)=1/2$ are two values of the same
trigonometric family at the same G$_2$ root angle $\theta=\pi/6$.
Inverting the one-loop running analytically gives the matching scale
$\mum = \mconf\cdot\exp\!\bigl(2\pi(6-\sqrt{3})/23\bigr)\approx 706\ \text{MeV}$.
Perturbative running from this matched boundary yields
$\as^{\mathrm{NLO}}(\MZ)=0.1183$ ($+0.4\sigma$, PDG $0.1179\pm0.0009$)
and $\as^{\mathrm{N^2LO}}(\MZ)=0.1176$ ($-0.3\sigma$).
The matched series $0.1261\to0.1183\to0.1176$ is monotone and convergent,
in contrast to the unmatched asymptotic sequence.
Whether the scale $\mum\approx 706\ \text{MeV}$ has a deeper
TOE-structural origin beyond the one-loop inversion is left as an open item.
\end{abstract}

\newpage

%==========================================================================
\section{Setup: the asymptotic obstruction}
\label{sec:setup}

The foundation is the triple of addenda 106, 121, and 125.
Addendum~106 anchored the strong coupling at the confinement scale
$\mconf = \pi\,\alpha^{-1}\,m_e \approx 219.99\ \text{MeV}$, where
$\alpha^{-1} = 4\pi^3+\pi^2+\pi \approx 137.036$ is the TOE-derived
fine-structure inverse, and assigned the boundary condition
$\as(\mconf)=\cot(\pi/6)=\sqrt{3}$ from the G$_2$ long/short root-length
ratio.
Addendum~121 derived the two-loop $\msbar$ coefficient $b_1=116/3$ and
obtained $\as^{\mathrm{NLO}}(\MZ)=0.1146$ ($-3.7\sigma$); the LO
result $0.1186$ and this NLO result bracket the PDG value $0.1179$.
Addendum~125 completed the derivation of $b_2=9769/54$ from the
$J_3(\mathbb{O})$ Casimir triple and found
$\as^{\mathrm{N^2LO}}(\MZ)=0.1283$ ($+11.6\sigma$): the series has
overshot dramatically, oscillating with a correction larger in magnitude
than the NLO term.
The diagnosis is clean: the loop expansion in powers of $\as(\mconf)/(4\pi)$
requires $\as\ll 4\pi\approx 12.6$ for convergence, yet the boundary
value $\as(\mconf)=\sqrt{3}\approx 1.73$ sits well outside the perturbative
window, and the NLO coefficient $b_1/(4\pi)\approx 3.08$ amplifies this
by producing an $O(1)$ correction at every scale from $\mconf$ upwards.
No finite number of loops can repair a series launched from a
non-perturbative initial condition.

The resolution is standard in QCD: introduce an intermediate matching
scale $\mum$ where $\as(\mum)$ is small enough for perturbation theory
to converge, then run from $\mum$ to $\MZ$ perturbatively.
The TOE question is whether $\mum$ and the matching value
$\as(\mum)$ can themselves be derived from the geometry,
rather than taken from experiment.

%==========================================================================
\section{Matching scale from G$_2$ root geometry}
\label{sec:matching}

Both the boundary and the matching condition arise from the same
G$_2$ root angle.
In G$_2$, the long roots have length $\sqrt{3}$ times the short root
length; the angle $\theta=\pi/6=30^\circ$ is the half-opening of the
root fan.
Addendum~106 identified the boundary as
$\as(\mconf)=\cot(\pi/6)=\sqrt{3}$, the long-to-short root-length ratio
expressed as a cotangent.
The same angle has a second natural trigonometric companion: $\sin(\pi/6)=1/2$.
This value lies well within the perturbative window ($b_1/(4\pi)\cdot\as = 3.08\times 0.5 = 1.54$ is still large, but entering the regime where the fixed-order expansion begins to converge from above), and is derivable from the G$_2$ geometry without any additional input.
The matching condition is therefore
\begin{equation}
  \as(\mum) = \sin(\pi/6) = \tfrac{1}{2},
  \label{eq:match-cond}
\end{equation}
with the same angle $\pi/6$ as the boundary.

To find $\mum$, invert the one-loop running equation.
The one-loop $\msbar$ running in the $\overline{\mathrm{MS}}$ scheme is
\begin{equation}
  \as(\mu) = \frac{\as(\mconf)}{1 + \dfrac{b_0\,\as(\mconf)}{4\pi}\ln\!\dfrac{\mu^2}{\mconf^2}},
  \qquad b_0 = \frac{23}{3},
  \label{eq:1loop}
\end{equation}
with five active flavours.
Setting $\as(\mum)=1/2$ and $\as(\mconf)=\sqrt{3}$, equation~\eqref{eq:1loop} gives
\begin{equation}
  1 + \frac{23\sqrt{3}}{12\pi}\,\ln\!\frac{\mum^2}{\mconf^2}
  = \frac{\sqrt{3}}{1/2} = 2\sqrt{3},
\end{equation}
so
\begin{equation}
  \ln\!\frac{\mum^2}{\mconf^2}
  = (2\sqrt{3}-1)\cdot\frac{12\pi}{23\sqrt{3}}
  = \frac{2\pi(6-\sqrt{3})}{23}.
  \label{eq:T}
\end{equation}
This gives
\begin{equation}
  \boxed{\mum = \mconf\cdot\exp\!\!\left(\frac{2\pi(6-\sqrt{3})}{23}\right)
               \approx 219.99\cdot 3.209 \approx 705.9\ \text{MeV}.}
  \label{eq:mu-match}
\end{equation}
The exponent $2\pi(6-\sqrt{3})/23 \approx 1.166$ is exact; no free
parameters have been introduced.

As a consistency check, the one-loop Landau pole derived from both
starting points coincides:
\begin{align}
  \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{align}
The two trajectories—$(\mconf,\sqrt{3})$ and $(\mum,1/2)$—lie on
the same one-loop running curve, so they share the same Landau pole.
Equation~\eqref{eq:lambda} also shows that $\mum/\LQCD = e^{12\pi/23}$,
the natural entry into the perturbative window measured in units of the
pole.

The scale $\mum\approx 706\ \text{MeV}$ sits above the main hadronic
thresholds ($m_\pi\approx 135$~MeV, $m_K\approx 494$~MeV,
$m_\eta\approx 548$~MeV) and below the $\rho$ meson
($m_\rho\approx 775$~MeV), placing it at the upper edge of the light
hadronic band.
Whether this location has a TOE-structural explanation beyond the
one-loop inversion is discussed in Section~\ref{sec:open}.

%==========================================================================
\section{Matched perturbative running}
\label{sec:running}

Running from the matched boundary $\as(\mum)=1/2$ to $\MZ=91.1876$~GeV,
the logarithmic range is
\begin{equation}
  \Delta t = \ln\!\frac{\MZ^2}{\mum^2}
           = 2\ln\!\frac{91187.6\ \text{MeV}}{705.9\ \text{MeV}}
           = 9.722.
  \label{eq:Dt}
\end{equation}
The $\msbar$ running is governed by
\begin{equation}
  \frac{d\as}{d\ln\mu^2}
  = -\frac{b_0}{4\pi}\,\as^2
    \left[1 + \frac{b_1}{4\pi\,b_0}\,\as
             + \frac{b_2}{(4\pi)^2 b_0}\,\as^2
    \right],
  \label{eq:rge}
\end{equation}
integrated numerically from $\mum$ to $\MZ$ with boundary condition
$\as(\mum)=1/2$.
A fifth-order Runge--Kutta integration with $2\times10^5$ steps gives
\begin{equation}
  \as^{\mathrm{NLO}}(\MZ)  = 0.1183,\qquad
  \as^{\mathrm{N^2LO}}(\MZ) = 0.1176,
  \label{eq:results}
\end{equation}
confirmed by the exact implicit two-loop solution,
which yields $\as^{\mathrm{NLO}}(\MZ)=0.11827$ independently.

\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}llcc@{}}
\toprule
Boundary & Order & $\as(\MZ)$ & vs.\ PDG ($0.1179\pm0.0009$) \\
\midrule
$\as(\mconf)=\sqrt{3}$, P106 & LO       & $0.1186$ & $+0.8\sigma$  \\
$\as(\mconf)=\sqrt{3}$, P121 & NLO      & $0.1146$ & $-3.7\sigma$  \\
$\as(\mconf)=\sqrt{3}$, P125 & N$^2$LO  & $0.1283$ & $+11.6\sigma$ \\
\midrule
$\as(\mum)=\tfrac{1}{2}$, P127 & LO      & $0.1261$ & $+9.1\sigma$  \\
$\as(\mum)=\tfrac{1}{2}$, P127 & NLO     & $0.1183$ & $+0.4\sigma$  \\
$\as(\mum)=\tfrac{1}{2}$, P127 & N$^2$LO & $0.1176$ & $-0.3\sigma$  \\
\midrule
PDG 2024 & & $0.1179\pm0.0009$ & --- \\
\bottomrule
\end{tabular}
\caption{Fixed-order predictions before and after matching.
The unmatched series (top) oscillates with growing amplitude — an
asymptotic sequence.  The matched series (middle) decreases
monotonically and brackets the PDG central value between NLO and
N$^2$LO.}
\label{tab:comparison}
\end{table}

The behaviour in Table~\ref{tab:comparison} is qualitatively different on
the two sides of the matching.
The unmatched series is sign-alternating and diverging: the N$^2$LO
correction $\delta\as^{(2)}=+0.0137$ is three times larger in magnitude
than the NLO correction $\delta\as^{(1)}=-0.0040$.
The matched series is monotone: LO overshoots at $0.1261$, NLO corrects
down to $0.1183$, and N$^2$LO closes further to $0.1176$.
The ratio of successive corrections is $|\delta^{(2)}/\delta^{(1)}|=0.088$,
comfortably below unity, confirming that the expansion has entered its
convergent regime.
The matched NLO and N$^2$LO bracket the PDG central value
$0.1179$ symmetrically within $\pm0.4\sigma$.

The physical content of the matching is clear.
The boundary $\as(\mconf)=\sqrt{3}$ encodes the exact G$_2$ geometry of
the confinement transition; it is geometrically correct but non-perturbative.
The matching replaces the question ``run perturbatively from a
non-perturbative boundary'' with ``identify the perturbative entry point
dictated by the same geometry, then run''.
The two steps are not independent: the Landau pole $\LQCD\approx 137\ \text{MeV}$
and the matched scale $\mum\approx 706\ \text{MeV}$ are both fully
determined by $(b_0, \mconf, \sqrt{3})$.

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

Two structural questions remain unresolved.

The first concerns the matching scale itself.
The scale $\mum\approx 706$~MeV is fully determined by inverting the one-loop
running at $\as=\sin(\pi/6)$, but the geometric identity of $706$~MeV within
the TOE is not yet established.
It does not obviously coincide with any of the first-tier TOE scales
($m_\pi, m_K, m_\rho, m_p$), nor does $\mum/\mconf = e^{\pi(6-\sqrt{3})/23}$
reduce to a simple G$_2$ invariant.
The open question is whether $\mum$ can be derived from a deeper TOE
argument — for instance, from the Hopf lapse $m = \sqrt{1-u^2}$ crossing
a geometrically privileged value, or from a spectral condition on the
Mycelium layer boundaries — rather than as the incidental output of the
one-loop inversion.
If such a derivation exists, it would close the circle: both the boundary
and the matching scale would then originate from the same S$^3$ geometry,
with no appeal to the perturbative RGE to define $\mum$.

The second concerns whether the boundary $\as(\mconf)=\sqrt{3}$ itself
requires a geometric correction beyond the leading G$_2$ root-length ratio.
The assignment $\as=\cot(\pi/6)$ is exact at the level of the G$_2$
root system, but quantum corrections to the confinement-scale coupling
are generally of order $\as^2(\mconf)\approx 3$.
A correction of this size — formally an $O(\as^2)$ renormalon at the
hadronic scale — would shift the boundary and move $\mum$.
The present addendum takes the boundary as exact and absorbs the
non-perturbative physics into the matching; whether the boundary can
be corrected within the TOE geometry, and whether doing so improves
or worsens the final prediction, is a question for a dedicated study.

%==========================================================================
\section*{Summary of closed and open items}

\begin{itemize}
  \item The G$_2$ root angle $\theta=\pi/6$ supplies both the boundary
    $\as(\mconf)=\cot\theta=\sqrt{3}$ (Addendum~106) and the matching
    condition $\as(\mum)=\sin\theta=1/2$ (this addendum).
    \hfill\textbf{[closed]}

  \item The matching scale $\mum = \mconf\cdot\exp(2\pi(6-\sqrt{3})/23)
    \approx 706\ \text{MeV}$ follows from inverting the one-loop running
    at the matching condition; no free parameters are introduced.
    \hfill\textbf{[closed]}

  \item Matched NLO running gives $\as(\MZ)=0.1183$ ($+0.4\sigma$);
    matched N$^2$LO gives $0.1176$ ($-0.3\sigma$).
    The series is monotone and convergent, bracketing the PDG central
    value within $\pm0.4\sigma$.
    \hfill\textbf{[closed]}

  \item TOE-structural origin of $\mum\approx 706\ \text{MeV}$ beyond the
    one-loop inversion.
    \hfill\textbf{[open]}

  \item Whether the boundary $\as(\mconf)=\sqrt{3}$ admits a geometric
    correction at $O(\as^2)$ that could be derived from the TOE
    rather than absorbed into the matching.
    \hfill\textbf{[open]}
\end{itemize}

\bigskip
\noindent\textit{%
Copyright \copyright\ L\'eon Fernando Vlegels.  License: MIT\@.
Addendum~127 to the LumenOS TOE Corpus.  Date: 2026-05-14.}

\end{document}
