% Addendum 125: Strong Coupling α_s at N²LO — b₂ from the TOE Casimir Triple
%   The three-loop β-function coefficient b₂ = 9769/54 ≈ 180.91 is now fully
%   TOE-native: every Casimir factor (C_A, C_F, T_F) is derived from J₃(𝕆).
%   Numerical integration of the three-loop RGE from α_s(m_conf) = √3 gives
%   α_s^{N²LO}(MZ) = 0.1283, overshooting PDG at +11.6σ. The LO→NLO→N²LO
%   sequence (0.1186→0.1146→0.1283) is sign-alternating, the diagnostic of an
%   asymptotic series launched from a non-perturbative boundary.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-14.

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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{\R}{\mathbb{R}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\as}{\alpha_s}
\newcommand{\mconf}{m_{\mathrm{conf}}}
\newcommand{\MZ}{M_Z}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\CA}{C_A}
\newcommand{\CF}{C_F}
\newcommand{\TF}{T_F}
\newcommand{\SU}{\mathrm{SU}}
\newcommand{\Fund}{\mathrm{fund}}
\newcommand{\btwo}{b_2}
\newcommand{\bone}{b_1}
\newcommand{\bzero}{b_0}
\newcommand{\msbar}{\overline{\mathrm{MS}}}

\title{Strong Coupling $\alpha_s$ at N$^2$LO:\\
$b_2 = 9769/54$ from the TOE Casimir Triple\\
and the Asymptotic Nature of the Loop Series\\[0.5em]
\large Addendum 125 to the TOE Corpus}

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

\date{2026-05-14}

\begin{document}
\maketitle

%==========================================================================
\begin{abstract}
Addenda 106, 121, and 126 closed OP1 through NLO: the TOE-native triple
$(C_A, C_F, T_F) = (3,\,4/3,\,1/2)$, derived entirely from $J_3(\mathbb{O})$,
yields the two-loop coefficient $b_1 = 116/3$ and the NLO running
$\as^{\mathrm{NLO}}(M_Z) = 0.1146$.
This addendum completes the Casimir triple's role at N$^2$LO by identifying
the three-loop $\msbar$ coefficient $b_2 = 9769/54 \approx 180.91$ as
automatically TOE-native, since every Casimir appearing in the standard
decomposition of $b_2$ is now a derived quantity.
Numerical integration of the three-loop RGE from the boundary
$\as(\mconf) = \sqrt{3}$ gives $\as^{\mathrm{N^2LO}}(M_Z) = 0.1283$,
overshooting PDG by $+11.6\sigma$; the LO$\to$NLO$\to$N$^2$LO sequence
$0.1186\to0.1146\to0.1283$ is sign-alternating, establishing that the
fixed-order series is asymptotic when launched from the
non-perturbative boundary $\as(\mconf)=\sqrt{3}\approx 1.732$.
\end{abstract}

\newpage

%==========================================================================
\section{Setup}
\label{sec:setup}

Three addenda establish the foundation for the present calculation.
Addendum~106 anchored the strong coupling at the confinement scale
$\mconf = \pi\,\alpha^{-1}\,m_e \approx 219.99\ \text{MeV}$ with
boundary condition $\as(\mconf) = \cot(\pi/6) = \sqrt{3}$, and obtained
the one-loop prediction $\as^{\mathrm{LO}}(M_Z)=0.1186$ ($+0.80\sigma$ above
PDG $0.1179\pm0.0009$).
Addendum~121 derived the two-loop coefficient
$b_1 = \tfrac{34}{3}C_A^2 - \bigl(\tfrac{20}{3}C_A+4C_F\bigr)n_f T_F = 116/3$
from the Peirce-trace structure of $\JO$, and computed the NLO shift
$\delta\as = -0.0040$, giving $\as^{\mathrm{NLO}}(M_Z)=0.1146$ ($-3.68\sigma$,
undershooting PDG), with the PDG value bracketed between LO and NLO.
Addendum~126 closed the last open Casimir by deriving $T_F=1/2$ from the
Jordan-triple-product symmetriser on the off-diagonal Peirce sector
$P_{12}\cong\mathbb{O}$, completing the TOE derivation of the full triple
$(\CA,\CF,\TF) = (3,\,4/3,\,1/2)$ with zero free parameters.

%==========================================================================
\section{$b_2$ is TOE-native by substitution}
\label{sec:b2}

The standard three-loop $\msbar$ $\beta$-function coefficient in QCD is
\begin{equation}
  b_2 = \frac{2857}{2} - \frac{5033}{18}\,n_f + \frac{325}{54}\,n_f^2.
  \label{eq:b2-standard}
\end{equation}
This formula is entirely written in $(\CA,\CF,\TF)$ via the textbook expansion
\begin{equation}
  b_2 = \Bigl(\frac{2857}{54}\Bigr)\CA^3
      - \Bigl(\frac{1415}{54}\CA^2 + \frac{205}{18}\CA\CF - 2\CF^2\Bigr)n_f\TF
      + \Bigl(\frac{44}{27}\CA + \frac{158}{27}\CF\Bigr)n_f^2\TF^2,
  \label{eq:b2-casimir}
\end{equation}
which reduces to equation~\eqref{eq:b2-standard} upon substituting
the group-theory values $\CA=N$, $\CF=(N^2-1)/(2N)$, $\TF=1/2$ for $\SU(N)$.
Since all three Casimirs in the triple $(\CA,\CF,\TF)=(3,4/3,1/2)$ are now
derived from $\JO$ (Addenda 121, 126), the substitution into
equation~\eqref{eq:b2-casimir} is a derivation, not a convention:
$b_2$ is TOE-native without any additional structure.

Setting $n_f=5$ in equation~\eqref{eq:b2-standard} and converting to the
common denominator 54 gives
\begin{equation}
  b_2 = \frac{77139}{54} - \frac{75495}{54} + \frac{8125}{54}
       = \frac{9769}{54} \approx 180.91.
  \label{eq:b2-numeric}
\end{equation}
The full set of loop coefficients entering the three-loop RGE is then
\begin{equation}
  b_0 = \frac{23}{3},\qquad
  b_1 = \frac{116}{3},\qquad
  b_2 = \frac{9769}{54},
  \label{eq:coeffs}
\end{equation}
all expressible as rational multiples derived from
$J_3(\mathbb{O})$ geometry and the integer $n_f = 5$.

%==========================================================================
\section{Numerical result}
\label{sec:numeric}

\subsection{Three-loop RGE integration}

The three-loop $\msbar$ running is governed by
\begin{equation}
  \frac{d\as}{d\ln\mu}
    = -\frac{b_0}{4\pi}\,\as^2\!
      \left[1
          + \frac{b_1}{4\pi}\,\as
          + \frac{b_2}{(4\pi)^2}\,\as^2
      \right],
  \label{eq:rge}
\end{equation}
integrated from $\mu=\mconf$ to $\mu=M_Z$ with boundary condition
$\as(\mconf)=\sqrt{3}$.
The logarithmic range is
\begin{equation}
  t = \ln\!\bigl(M_Z^2/\mconf^2\bigr)
    = 2\ln(91.1876/0.21999) = 12.054.
\end{equation}
A fifth-order Runge--Kutta integration with $5\times10^5$ steps gives
\begin{equation}
  \boxed{\as^{\mathrm{N^2LO}}(M_Z) = 0.1283.}
  \label{eq:result}
\end{equation}

\subsection{Convergence table and physical interpretation}

\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}lccc@{}}
\toprule
Order & $\as(M_Z)$ & Residual vs.\ PDG & Significance \\
\midrule
LO   (P106) & $0.1186$ & $+0.0007$ & $+0.80\sigma$ \\
NLO  (P121) & $0.1146$ & $-0.0033$ & $-3.68\sigma$ \\
N$^2$LO (P125) & $0.1283$ & $+0.0104$ & $+11.6\sigma$ \\
\midrule
PDG 2024 & $0.1179\pm0.0009$ & --- & --- \\
\bottomrule
\end{tabular}
\caption{Fixed-order predictions from the TOE boundary $\as(\mconf)=\sqrt{3}$.
The LO$\to$NLO$\to$N$^2$LO sequence oscillates in sign and grows in magnitude.}
\label{tab:convergence}
\end{table}

The sequence $\{0.1186,\,0.1146,\,0.1283\}$ is sign-alternating
(high, low, high) with the N$^2$LO correction $\delta\as^{(2)} = +0.0137$
larger in magnitude than the NLO correction $\delta\as^{(1)} = -0.0040$.
This is the diagnostic signature of an asymptotic series: the perturbative
expansion in powers of $\as/(4\pi)$ is formally valid only for
$\as\ll 4\pi\approx 12.6$, but the boundary value $\as(\mconf)=\sqrt{3}\approx
1.73$ lies far outside the radius of convergence of the naive loop expansion.
Each additional loop introduces a correction of order
$b_k\,\as^{k+2}/(4\pi)^{k+1}$; because $\as(\mconf)$ is large, these
corrections do not decrease monotonically.  The fixed-order series cannot
converge to the physical value from this boundary.

The resolution is not an error in the TOE derivation of $(b_0,b_1,b_2)$
or the Casimir triple: those are exact.  Rather, the loop expansion must
be replaced by a non-perturbative matching procedure.  The natural approach
is Padé resummation or Borel summation of the asymptotic series, which
can extract a finite, convergent prediction from a sign-alternating sequence.
Alternatively, a boundary renormalisation — matching $\as(\mconf)$ to an
effective coupling that already absorbs the large infrared contributions —
may bring the series into its perturbative window before running begins.
Both avenues are open items; the present addendum closes the structural
question of $b_2$'s TOE origin and quantifies the asymptotic obstruction.

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

The four-loop coefficient $b_3$ can, in principle, be derived from
$\JO$ via higher Peirce traces, continuing the pattern established for
$b_1$ and $b_2$, though the group-theory decomposition at four loops
is substantially more involved and is left for a dedicated addendum.
The more urgent open item is boundary renormalisation: deriving a
TOE-native effective coupling at $\mconf$ that replaces the
bare boundary $\as(\mconf)=\sqrt{3}$ with a quantity inside the
perturbative window, after which the fixed-order series is expected
to converge monotonically to the PDG value.

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

\begin{itemize}
  \item $b_2 = 9769/54$ is TOE-native: every Casimir in its
    standard decomposition is now derived from $J_3(\mathbb{O})$
    (Addenda 121, 126). \hfill\textbf{[closed]}
  \item Three-loop RGE gives $\as^{\mathrm{N^2LO}}(M_Z)=0.1283$
    from the $\sqrt{3}$ boundary. \hfill\textbf{[closed]}
  \item The LO$\to$NLO$\to$N$^2$LO series is asymptotic, not
    convergent, from a non-perturbative boundary. \hfill\textbf{[closed]}
\end{itemize}

\begin{itemize}
  \item Boundary renormalisation / Borel resummation. \hfill\textbf{[open]}
  \item Four-loop $b_3$ from $J_3(\mathbb{O})$ Peirce traces. \hfill\textbf{[open]}
\end{itemize}

\bigskip
\noindent\textit{%
Copyright \copyright\ L\'eon Fernando Vlegels.  License: MIT\@.
Addendum~125 to the LumenOS TOE Corpus.  Date: 2026-05-14.}

\end{document}
