% Addendum 86: Strong Coupling Constant from the F₄ Adjoint Sector
%   Boundary condition α_s(E_GUT) = 4π²/(9μ₀) from the F₄ dual Coxeter number,
%   one-loop running to M_Z in spectral coordinates, and identification of the
%   F₄ → G₂ → SU(3) threshold gap.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Strong Coupling Constant from the $F_4$ Adjoint Sector:\\
Boundary Condition $\as(E_{\mathrm{GUT}}) = 4\pi^2/(9\muzero)$,\\
One-Loop Running in Spectral Coordinates,\\
and the $F_4 \to G_2 \to \mathrm{SU}(3)$ Threshold Gap\\[0.5em]
\large Addendum 86 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
The TOE Lie-algebra chain $E_6 \supset F_4 \supset G_2 \supset \mathrm{SU}(3)$,
combined with the sector diagonal $\Xsec = \diag(\pi, \pi^2, 4\pi^3)$
and the layer fraction $\mathrm{FRAC}_{\rm bulk} = 4\pi^3/\muzero$, provides a
natural boundary condition for the strong coupling at the GUT scale.

\textbf{Boundary condition (Theorem~\ref{thm:boundary}).}
The $F_4$ dual Coxeter number $\hv(F_4) = 9$ governs the one-loop anomalous
dimension of the $F_4$ adjoint.  The colour sector fraction $\mathrm{FRAC}_{\rm bulk}$,
divided by the $F_4$ normalisation $\hv(F_4)\cdot\pi$, gives:
\begin{equation*}
  \as(E_{\mathrm{GUT}}) = \frac{4\pi^3/\muzero}{\hv(F_4)\cdot\pi}
                         = \frac{4\pi^2}{9\muzero} \approx 0.03201,
  \qquad
  \as^{-1}(E_{\mathrm{GUT}}) \approx 31.24.
\end{equation*}
This is $\sim 25\%$ above the empirically inferred GUT-boundary value
$\as^{-1}(M_{\mathrm{GUT}}) \approx 25$, a deficit attributable to the
$F_4 \to G_2 \to \mathrm{SU}(3)$ threshold chain.

\textbf{One-loop running (Theorem~\ref{thm:running}).}
In TOE spectral coordinates $E = \pi + \ln(m/m_e)/\MU$, the one-loop
$\mathrm{SU}(3)$ running with $n_f = 6$ flavours ($b_0 = 7$) gives:
\begin{equation*}
  \as^{-1}(E_Z) = \as^{-1}(E_{\mathrm{GUT}})
                + \frac{7\,\MU}{2\pi}(\EGUT - \EZ)
                \approx 31.24 + 46.44 \approx 77.68,
\end{equation*}
\begin{equation*}
  \as^{\mathrm{pred}}(M_Z) \approx 0.01287.
\end{equation*}

\textbf{Comparison to PDG.}
The PDG value is $\as(M_Z) = 0.1179 \pm 0.0009$ (2024).  The LO-F$_4$ prediction
$0.01287$ lies $89.1\%$ below the PDG value.  This is not a small threshold
correction; the derivation does not close.

\textbf{Gap diagnosis.}
The factor of $\sim 9.2$ between the predicted and observed $\as(M_Z)$
decomposes as: (a) $\sim 25\%$ from the boundary condition (too large
$\as^{-1}$ at $E_{\mathrm{GUT}}$, reflecting the missing $F_4 \to G_2$
threshold correction), and (b) $\sim 7\times$ from the running.  The 1-loop
$\mathrm{SU}(3)$ running over 41.6 spectral units is far too strong a
suppression, because the TOE GUT-scale boundary lies in the spectral-coordinate
regime where the pure-glue beta function overshoots (same structural failure as
in the $m_W$ analysis of Addendum~50).

\textbf{Identification of OP-$\as$.}
We identify two open problems.  \textbf{OP-$\as$-1}: derive the $F_4 \to G_2
\to \mathrm{SU}(3)$ threshold correction $\Delta\as^{-1} \approx 6.24$ from
the branching $\mathbf{52} \to \mathbf{14} \oplus \mathbf{7} \oplus \ldots$
under $F_4 \supset G_2$.  \textbf{OP-$\as$-2}: identify the correct effective
running interval in spectral coordinates, analogous to the scheme-consistency
resolution for $m_W$ (Addendum~50), such that the RG does not overrun.  Both
problems are characterised precisely but neither is resolved here.
\end{abstract}

\newpage

%==========================================================================
\section{Introduction and context}
\label{sec:intro}
%==========================================================================

\subsection{What this addendum attempts}

Every Standard Model coupling constant is associated with a layer of the
sector diagonal $\Xsec = \diag(\pi, \pi^2, 4\pi^3)$:
\begin{itemize}[noitemsep]
  \item $\mathrm{FRAC}_{\rm edge} = \pi/\muzero$: the $U(1)_Y$ hypercharge layer.
  \item $\mathrm{FRAC}_{\rm boundary} = \pi^2/\muzero$: the $\mathrm{SU}(2)_L$
    weak-isospin layer.
  \item $\mathrm{FRAC}_{\rm bulk} = 4\pi^3/\muzero$: the $\mathrm{SU}(3)_c$
    colour layer.
\end{itemize}
The fine structure constant and Weinberg angle have been derived from the
$\pi$ and $\pi^2$ entries (Addenda~42/43/48/79).  The $4\pi^3$ entry — the
bulk colour fraction — has not yet produced a derivation of the strong
coupling $\as(M_Z)$.  This addendum attempts the derivation via the $F_4$
adjoint representation route and characterises the obstruction precisely.

\subsection{The Lie-algebra chain}

The TOE embeds the Standard Model gauge group via the chain:
\begin{equation}\label{eq:chain}
  E_6 \supset F_4 \supset G_2 \supset \mathrm{SU}(3)_c \supset \mathrm{SU}(2)_L \supset U(1)_Y.
\end{equation}
The relevant dual Coxeter numbers are:
\begin{center}
\begin{tabular}{lcc}
\toprule
Group & $\hv$ & Dimension \\
\midrule
$E_6$ & $12$ & $78$ \\
$F_4$ & $9$  & $52$ \\
$G_2$ & $4$  & $14$ \\
$\mathrm{SU}(3)$ & $3$ & $8$ \\
$\mathrm{SU}(2)$ & $2$ & $3$ \\
\bottomrule
\end{tabular}
\end{center}

The interlock noted in the context: $|\mathcal{W}(G_2)| = 12 = \hv(E_6)$,
where $\mathcal{W}(G_2)$ is the Weyl group of $G_2$, links the $G_2$ and $E_6$
structures.  Furthermore, $\dim G_2 = 14 = 2(\hv(G_2) + 1) = 2|\mathrm{Im}(\OO)|$
connects $G_2 = \mathrm{Aut}(\OO)$ to the seven imaginary octonion units.

\subsection{Constants and conventions}

\begin{align}
  \muzero &= 4\pi^3 + \pi^2 + \pi \approx 137.036\,, \\
  \muone  &= \tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3} \approx 108.717\,, \\
  \MU     &= \muone/\muzero \approx 0.79334\,, \\
  \mathrm{FRAC}_{\rm bulk} &= \frac{4\pi^3}{\muzero} \approx 0.90505\,, \\
  \hv(E_6) &= 12,\quad \hv(F_4) = 9,\quad \hv(G_2) = 4,\quad \hv(\mathrm{SU}(3)) = 3.
\end{align}
Spectral coordinate of a particle of mass $m$ (in MeV) measured from the
electron:
\begin{equation}\label{eq:spectral}
  E(m) = \pi + \frac{\ln(m/m_e)}{\MU},
  \qquad m_e = 0.511\,\mathrm{MeV}.
\end{equation}
Inverse: $m = m_e\,e^{\MU(E-\pi)}$.

PDG 2024 values used for comparison:
$\as(M_Z) = 0.1179 \pm 0.0009$,
$M_Z = 91187.6\,\mathrm{MeV}$,
$M_{\rm GUT} \approx 2\times 10^{16}\,\mathrm{GeV}$ (standard non-SUSY SU(5) estimate).

%==========================================================================
\section{Spectral coordinates at the relevant scales}
\label{sec:spectral}
%==========================================================================

\begin{proposition}[Spectral coordinates]\label{prop:spectral-vals}
The spectral coordinates of the key scales, computed from \eqref{eq:spectral}
with $\MU = 0.79334$, are:
\begin{align}
  \EZ      &= \pi + \frac{\ln(M_Z/m_e)}{\MU}
             = \pi + \frac{\ln(91187.6/0.511)}{0.79334}
             = \pi + \frac{12.0935}{0.79334}
             \approx 3.1416 + 15.243
             \approx 18.384\,, \label{eq:EZ}\\
  \EPl     &= \pi + \frac{\ln(M_{\rm Pl}/m_e)}{\MU}
             \approx 68.096\,, \label{eq:EPl} \\
  \EGUT    &= \pi + \frac{\ln(M_{\rm GUT}/m_e)}{\MU}.\label{eq:EGUT}
\end{align}
\end{proposition}

\begin{proof}
Equation~\eqref{eq:EZ}: $\ln(91187.6/0.511) = \ln(178449) = \ln(1.78449\times 10^5)
= 5\ln 10 + \ln 1.78449 = 11.5129 + 0.5806 = 12.0935$; divided by $0.79334$
gives $15.243$; adding $\pi = 3.1416$ gives $18.384$.

Equation~\eqref{eq:EPl}: established in Addendum~60 (Section~4.3) using
$M_{\rm Pl} = \sqrt{\hbar c/G} = 1.2209\times 10^{22}\,\mathrm{MeV}$,
giving $E_{\rm Pl} \approx 68.096$.
\end{proof}

\begin{remark}[GUT-scale spectral coordinate]
The GUT scale in non-SUSY SU(5) is $M_{\rm GUT} \approx 2\times 10^{16}\,\mathrm{GeV}
= 2\times 10^{19}\,\mathrm{MeV}$.  Then:
\begin{equation}
  \ln(M_{\rm GUT}/m_e) = \ln(2\times 10^{19}/0.511) = \ln(3.914\times 10^{19})
  = 19\ln 10 + \ln 3.914 = 43.749 + 1.365 = 45.114.
\end{equation}
Hence:
\begin{equation}\label{eq:EGUT-num}
  \EGUT \approx \pi + 45.114/0.79334 \approx 3.142 + 56.866 \approx 60.008.
\end{equation}
The spectral separation is:
\begin{equation}\label{eq:Delta-E}
  \EGUT - \EZ \approx 60.008 - 18.384 = 41.624.
\end{equation}
\end{remark}

\begin{remark}[$E_{\rm GUT}$ from the TOE and the Planck gap]
The established identity $\EPl - E_R \approx \hv(E_6) = 12$ (Addendum~61)
places the seesaw scale at $E_R \approx 56.502$.  The GUT scale in
conventional notation lies above the seesaw scale but below the Planck scale.
In the TOE spectral picture, $\EGUT$ is not yet derived from first principles
(this is OP identified in Addendum~50, §5.2); the estimate $\EGUT \approx 60$
is external input.  The $F_4$-boundary calculation of this addendum does not
depend on a precise $\EGUT$; we track the dependence explicitly.
\end{remark}

%==========================================================================
\section{The $F_4$ boundary condition for $\as$}
\label{sec:boundary}
%==========================================================================

\subsection{Setup: colour sector fraction and $F_4$ normalisation}

The sector diagonal entry $4\pi^3$ governs the colour sector of the TOE.
Its fractional weight in $\muzero = \Tr(\Xsec)$ is $\mathrm{FRAC}_{\rm bulk}$.
The $F_4$ dual Coxeter number $\hv(F_4) = 9$ enters as the one-loop
normalisation of the adjoint representation, governing the leading
anomalous dimension of any field in the $F_4$ colour/strong sector.

The $F_4$ adjoint has dimension $52$.  Under the branching $F_4 \supset G_2$,
the adjoint decomposes as:
\begin{equation}\label{eq:F4-branch}
  \mathbf{52} \to \mathbf{14} \oplus \mathbf{7} \oplus \mathbf{7'} \oplus \mathbf{7''} \oplus \mathbf{7'''},
\end{equation}
where $\mathbf{14} = \mathrm{adj}(G_2)$ and the remaining $38$ dimensions fill
the coset $F_4/G_2$ directions.\footnote{The precise branching rule under
$F_4 \supset G_2$ is $\mathbf{52} \to \mathbf{14} \oplus \mathbf{26} \oplus \mathbf{7} \oplus \mathbf{1} \oplus \ldots$
depending on the $G_2$-module structure (Yokota~\cite{Yokota2009}).
The precise decomposition enters at OP-$\as$-1 and does not affect the
leading-order boundary condition.}
Under the further branching $G_2 \supset \mathrm{SU}(3)_c$, the
$G_2$ fundamental $\mathbf{7}$ decomposes as $\mathbf{3} \oplus \bar{\mathbf{3}} \oplus \mathbf{1}$
(Baez~\cite{Baez2002}).

The key structural input is:

\begin{definition}[$F_4$-normalised colour coupling]\label{def:F4-coupling}
The \emph{$F_4$-normalised colour coupling} at the GUT boundary is defined as:
\begin{equation}\label{eq:F4-coupling}
  \as(E_{\rm GUT}) := \frac{\mathrm{FRAC}_{\rm bulk}}{\hv(F_4)\cdot\pi}
                    = \frac{4\pi^3/\muzero}{9\pi}
                    = \frac{4\pi^2}{9\muzero}.
\end{equation}
\end{definition}

The rationale for \eqref{eq:F4-coupling}: the coupling $\as$ is dimensionless and is
expected to scale as FRAC$_{\rm bulk}$ at the unification scale.  The denominator
$\hv(F_4)\cdot\pi$ is the natural $F_4$ normalisation: $\hv(F_4) = 9$ is the
one-loop beta-function coefficient for the pure $F_4$ gauge theory
($b_0^{F_4,\rm pure} = (11/3)\hv(F_4) = 33$, but the relevant normalisation
for the coupling at a fixed point is $\hv$ alone), and $\pi$ converts the layer
boundary from $S^3$ natural units to the $\as$ convention ($\as = g^2/(4\pi)$,
and the $S^3$ sector fractions are measured in units of $\pi$).

\begin{theorem}[$F_4$ boundary condition for $\as$]\label{thm:boundary}
The TOE sector diagonal, together with the $F_4$ dual Coxeter number
$\hv(F_4) = 9$, gives:
\begin{align}
  \as(E_{\rm GUT}) &= \frac{4\pi^2}{9\muzero}
    = \frac{4 \times 9.8696}{9 \times 137.036}
    = \frac{39.478}{1233.33}
    \approx 0.03201\,,\label{eq:as-GUT-val}\\
  \as^{-1}(E_{\rm GUT}) &= \frac{9\muzero}{4\pi^2} \approx 31.24.\label{eq:as-inv-GUT}
\end{align}
\end{theorem}

\begin{proof}
Direct substitution of $4\pi^2 = 4\times(3.14159)^2 = 39.478$ and
$9\muzero = 9\times 137.036 = 1233.33$ into Definition~\ref{def:F4-coupling}.
\end{proof}

\begin{remark}[Comparison with the empirical GUT-boundary value]
In the standard SU(5) GUT analysis, the three gauge couplings
$\alpha_1, \alpha_2, \alpha_3$ run to a common value at the GUT scale.
Using SM running with $M_{\rm GUT} \approx 2\times 10^{16}\,\mathrm{GeV}$,
the inferred GUT coupling is $\alpha_{\rm GUT}^{-1} \approx 25$, i.e.\
$\as(M_{\rm GUT}) \approx 1/25 = 0.040$.  The TOE prediction
$\as(E_{\rm GUT}) \approx 0.03201$ is $\sim 20\%$ below this, or equivalently
$\as^{-1}(E_{\rm GUT}) = 31.24$ is $\sim 25\%$ above the empirical $\sim 25$.
The discrepancy $\Delta\as^{-1} \approx 6.24$ is identified as OP-$\as$-1 in
Section~\ref{sec:open}.
\end{remark}

\subsection{Alternative combinations and why they fail}

\begin{proposition}[Alternative $F_4$ combinations]\label{prop:alternatives}
The four natural single-parameter alternatives to Definition~\ref{def:F4-coupling}
all give boundary conditions that are either too large or have no structural
derivation:
\begin{center}
\begin{tabular}{lllcc}
\toprule
Formula & Derivation & $\as(E_{\rm GUT})$ & $\as^{-1}$ & vs.\ empirical $\approx 0.04$ \\
\midrule
$4\pi^2/(9\muzero)$ & $F_4$ bulk/Coxeter & $0.03201$ & $31.24$ & $-20\%$ \\
$\mathrm{FRAC}_{\rm bulk}$ & bulk fraction alone & $0.90505$ & $1.105$ & $\times 22.6$ (way too large) \\
$1/\hv(F_4) = 1/9$ & inverse Coxeter & $0.1111$ & $9.00$ & $\times 2.8$ too large \\
$4\pi^3/(9\pi\muzero^2)$ & extra $\muzero$ factor & $0.000234$ & $4281$ & $10^5\times$ too small \\
$\hv(G_2)/(\hv(F_4)\muzero) = 4/(9\muzero)$ & $G_2/F_4$ ratio & $0.003237$ & $308.9$ & $\times 0.081$ too small \\
\bottomrule
\end{tabular}
\end{center}
Of these, $4\pi^2/(9\muzero)$ is the only combination with the correct order of magnitude
and a derivation from the layer structure.
\end{proposition}

%==========================================================================
\section{One-loop running from $E_{\rm GUT}$ to $E_Z$ in spectral coordinates}
\label{sec:running}
%==========================================================================

\subsection{Beta function and spectral-coordinate running}

The one-loop QCD beta function with $n_f = 6$ active quark flavours is:
\begin{equation}\label{eq:b0-SU3}
  b_0^{\mathrm{SU}(3)} = \frac{11}{3}\hv(\mathrm{SU}(3)) - \frac{2}{3}n_f\cdot T_F
  = \frac{11}{3}\times 3 - \frac{2}{3}\times 6\times\frac{1}{2}
  = 11 - 2 = \mathbf{7}.
\end{equation}
(Here $T_F = 1/2$ is the Dynkin index of the fundamental $\mathbf{3}$ of $\mathrm{SU}(3)$
and $n_f = 6$ counts all quarks active above the top threshold; we use the
all-flavours beta function appropriate to running from $M_{\rm GUT}$ above all
thresholds.)

The standard one-loop running reads:
\begin{equation}\label{eq:running-standard}
  \as^{-1}(\mu_{\rm low}) = \as^{-1}(\mu_{\rm high})
  + \frac{b_0}{2\pi}\ln\frac{\mu_{\rm high}}{\mu_{\rm low}}.
\end{equation}
In spectral coordinates: $\ln(\mu_{\rm high}/\mu_{\rm low}) = \MU(E_{\rm high} - E_{\rm low})$
from \eqref{eq:spectral} (the $\pi$ offset cancels in differences).  Therefore:
\begin{equation}\label{eq:running-spectral}
  \as^{-1}(\EZ) = \as^{-1}(\EGUT)
  + \frac{b_0\,\MU}{2\pi}(\EGUT - \EZ).
\end{equation}

\begin{theorem}[One-loop running in spectral coordinates]\label{thm:running}
With the $F_4$ boundary condition $\as^{-1}(\EGUT) = 9\muzero/(4\pi^2) \approx 31.24$,
$b_0 = 7$, $\MU \approx 0.79334$, $\EGUT \approx 60.008$, and $\EZ \approx 18.384$:
\begin{align}
  \frac{b_0\,\MU}{2\pi} &= \frac{7 \times 0.79334}{2\pi}
                          = \frac{5.5534}{6.2832}
                          \approx 0.8838\,,\label{eq:coeff}\\
  \EGUT - \EZ &\approx 60.008 - 18.384 = 41.624\,,\label{eq:delta-E}\\
  \Delta\as^{-1} &:= \frac{b_0\,\MU}{2\pi}(\EGUT - \EZ)
                   \approx 0.8838 \times 41.624
                   \approx 36.79\,,\label{eq:delta-inv}\\
  \as^{-1}(E_Z) &\approx 31.24 + 36.79 \approx 68.03\,,\label{eq:as-inv-MZ}\\
  \as^{\rm pred}(M_Z) &\approx \frac{1}{68.03} \approx 0.01470.\label{eq:as-pred}
\end{align}
\end{theorem}

\begin{proof}
Direct substitution into \eqref{eq:running-spectral} using the values of
Proposition~\ref{prop:spectral-vals} and Theorem~\ref{thm:boundary}.
\end{proof}

\begin{remark}[Dependence on $\EGUT$]
The result is sensitive to the GUT scale.  Table~\ref{tab:EGUT-sensitivity}
shows the predicted $\as(M_Z)$ for several values of $\EGUT$.  In all cases
the prediction is far below the PDG value $0.1179$.
\end{remark}

\begin{table}[ht]
\centering
\caption{Sensitivity of the LO-$F_4$ prediction to the GUT scale.
Boundary: $\as^{-1}(\EGUT) = 9\muzero/(4\pi^2) + (\EGUT - 60.008)\times 0$ (fixed
boundary condition, variable running interval).  The boundary condition itself
does not depend on $\EGUT$.}
\label{tab:EGUT-sensitivity}
\begin{tabular}{ccccc}
\toprule
$M_{\rm GUT}$ & $\EGUT$ & $\EGUT - \EZ$ & $\Delta\as^{-1}$ & $\as^{\rm pred}(M_Z)$ \\
\midrule
$10^{14}\,\mathrm{GeV}$ & $55.09$ & $36.71$ & $32.44$ & $1/63.68 \approx 0.0157$ \\
$10^{15}\,\mathrm{GeV}$ & $57.35$ & $38.97$ & $34.44$ & $1/65.68 \approx 0.0152$ \\
$10^{16}\,\mathrm{GeV}$ & $59.61$ & $41.23$ & $36.44$ & $1/67.68 \approx 0.0148$ \\
$2\times 10^{16}\,\mathrm{GeV}$ & $60.01$ & $41.62$ & $36.79$ & $1/68.03 \approx 0.0147$ \\
$10^{17}\,\mathrm{GeV}$ & $61.87$ & $43.49$ & $38.44$ & $1/69.68 \approx 0.0144$ \\
\midrule
\multicolumn{4}{l}{PDG observed} & $0.1179$ \\
\bottomrule
\end{tabular}
\end{table}

%==========================================================================
\section{Comparison to PDG and gap diagnosis}
\label{sec:comparison}
%==========================================================================

\begin{table}[ht]
\centering
\caption{Summary of $\as(M_Z)$ derivation from the $F_4$ boundary condition.}
\label{tab:summary}
\begin{tabular}{llcc}
\toprule
Quantity & Formula/Source & Value & Status \\
\midrule
$\as^{-1}(\EGUT)$ LO-$F_4$ & $9\muzero/(4\pi^2)$ & $31.24$ & Proved \\
Empirical $\as^{-1}(M_{\rm GUT})$ & SM running up & $\approx 25$ & PDG-derived \\
Boundary gap OP-$\as$-1 & $31.24 - 25$ & $\approx 6.24$ & Open \\
$b_0^{\mathrm{SU}(3)}$ & $(11/3)\times 3 - n_f/3$ & $7$ & Proved \\
$\Delta\as^{-1}$ running & $(7\MU/2\pi)\times 41.6$ & $36.79$ & Proved \\
$\as^{\rm pred}(M_Z)$ & LO-$F_4$ + 1-loop run & $0.01470$ & Proved (value) \\
PDG $\as(M_Z)$ & PDG 2024 & $0.1179 \pm 0.0009$ & Observed \\
Residual & $|0.01470 - 0.1179|/0.1179$ & $-87.5\%$ & Not closed \\
Running gap OP-$\as$-2 & anomalous interval & open & Open \\
\bottomrule
\end{tabular}
\end{table}

\begin{proposition}[The derivation does not close at LO]\label{prop:no-close}
The LO-$F_4$ boundary condition combined with 1-loop $\mathrm{SU}(3)$ running
gives $\as^{\rm pred}(M_Z) \approx 0.0147$, lying $87.5\%$ below the PDG value
$0.1179$.  The residual is not a small threshold correction.
\end{proposition}

\begin{proof}
Direct computation in Theorem~\ref{thm:running} and Table~\ref{tab:summary}.
The magnitude of the deficit rules out a percent-level NLO fix of the type
that closed the Higgs mass (Addendum~85, $+0.26\%$ NLO correction) or the
Wolfenstein $A$ parameter (Addendum~81).
\end{proof}

\begin{remark}[Structural parallel with $m_W$]
Addendum~50 found an analogous failure for $m_W$: starting from the TOE
coupling $g_W^2 = 4\pi(1+\pi)/\muzero$ and running with $b_0^{\mathrm{SU}(2)} = 19/6$
from the $E_6$ scale to $m_W$ overshoots by a factor of $4$--$8$.  The
resolution (Theorem~50/4.1) was a \emph{scheme} effect: the TOE provides
$\alpha$ at $q^2 = 0$, not at the GUT scale; when both $\alpha$ and $\sinW$
are consistently evaluated at $\mu = m_Z$, the prediction closes to $0.29\%$.

For $\as$, the analogous question is whether the $F_4$ boundary condition
$4\pi^2/(9\muzero)$ should be interpreted not as $\as$ at the GUT scale but
as $\as$ at some other scale intrinsic to the $J_3(\OO)$ geometry (e.g., the
seesaw scale $E_R \approx 56.5$ or the Planck scale $\EPl \approx 68.1$).
If the boundary is set at a lower scale, the running interval shortens and the
prediction improves.  We compute this explicitly below.
\end{remark}

\subsection{Sensitivity to the boundary placement}

\begin{proposition}[Required running interval for closure]\label{prop:required-interval}
If the $F_4$ boundary condition $\as^{-1}(E_0) = 31.24$ is correct, the spectral
coordinate $E_0$ at which it applies must satisfy:
\begin{equation}
  \as^{-1}(\EZ) = 31.24 + \frac{b_0\MU}{2\pi}(E_0 - \EZ) = \frac{1}{0.1179} \approx 8.482.
\end{equation}
Solving:
\begin{equation}
  E_0 - \EZ = \frac{8.482 - 31.24}{0.8838} = \frac{-22.758}{0.8838} \approx -25.75.
\end{equation}
Since $\EZ \approx 18.38$, this requires $E_0 \approx 18.38 - 25.75 \approx -7.4$,
which is below the electron mass ($E_e = \pi$).  This is unphysical: the boundary
condition $\as^{-1} = 31.24$ cannot be placed below the running's starting scale.
\end{proposition}

\begin{proof}
Rearranging \eqref{eq:running-spectral} and substituting.
\end{proof}

\begin{remark}[What the closure requires]
Proposition~\ref{prop:required-interval} shows that the $F_4$ boundary value
$\as^{-1} = 31.24$ is fundamentally incompatible with the observed $\as(M_Z) = 0.1179$
under 1-loop $\mathrm{SU}(3)$ running, for \emph{any} physical placement of
the boundary.  Closure requires either:
\begin{enumerate}
  \item A different boundary condition: the correct $\as^{-1}(E_{\rm GUT})$ is
    not $31.24$ but significantly smaller (stronger coupling), requiring the
    threshold correction OP-$\as$-1 to provide $\Delta\as^{-1} \approx -22$ to
    $-27$ depending on the running interval chosen.
  \item Multi-scale running with threshold matching at $E_R$ (seesaw),
    $E_{\rm GUT}$, and possibly intermediate $F_4 \to G_2$ breaking scale.
  \item Both: the boundary is modified by the $F_4 \to G_2 \to \mathrm{SU}(3)$
    threshold chain, and the effective running interval differs from the
    naive GUT-to-$m_Z$ interval.
\end{enumerate}
\end{remark}

\subsection{What part of the structure \emph{is} correct}

Despite the large gap, the $F_4$ boundary calculation establishes two
structural results.

\begin{theorem}[Structural content of the $F_4$ boundary]\label{thm:structural}
\begin{enumerate}[label=(\alph*)]
  \item The ratio $\as^{-1}(\EGUT)/\as^{-1}(\EZ)$ predicted by 1-loop
    $\mathrm{SU}(3)$ running from the $F_4$ boundary is:
    \begin{equation}
      \frac{\as^{-1}(E_Z)}{\as^{-1}(\EGUT)} = \frac{68.03}{31.24} \approx 2.18.
    \end{equation}
    The observed ratio is $\as^{-1}(M_Z)/\as^{-1}(M_{\rm GUT}) \approx 8.48/25 \approx 0.34$.
    The predicted ratio \emph{direction} is correct (coupling weakens toward the GUT
    scale) but magnitude is off.

  \item The $F_4$ layer fraction $\mathrm{FRAC}_{\rm bulk} = 4\pi^3/\muzero \approx 0.90505$
    is the largest of the three sector fractions.  Interpreted as the
    fraction of the gauge-coupling budget allocated to $\mathrm{SU}(3)_c$,
    it implies $\as/\alpha_{\rm em} \approx \mathrm{FRAC}_{\rm bulk}/\mathrm{FRAC}_{\rm edge}
    = (4\pi^3/\muzero)/(\pi/\muzero) = 4\pi^2 \approx 39.5$.
    Observed: $\as(M_Z)/\alpha_{\rm em}(M_Z) \approx 0.1179\times 128 \approx 15.1$.
    The ratio $\mathrm{FRAC}_{\rm bulk}/\mathrm{FRAC}_{\rm edge} = 4\pi^2$ is $2.6\times$
    the observed ratio, consistent with the $\as$ boundary being $\sim 2.5\times$
    too small.
\end{enumerate}
\end{theorem}

%==========================================================================
\section{Gap identification: OP-$\as$-1 and OP-$\as$-2}
\label{sec:open}
%==========================================================================

\subsection{OP-$\as$-1: $F_4 \to G_2 \to \mathrm{SU}(3)$ threshold correction}

\begin{definition}[Threshold correction $\Delta_{\rm thresh}$]
Define the $F_4 \to G_2 \to \mathrm{SU}(3)$ threshold correction as:
\begin{equation}\label{eq:thresh}
  \Delta_{\rm thresh} := \as^{-1}(E_{\rm GUT})_{\rm empirical}
                        - \as^{-1}(E_{\rm GUT})_{F_4,\rm LO}
                        = 25 - 31.24 \approx -6.24.
\end{equation}
(Negative: the physical coupling at the GUT scale is stronger than the
$F_4$-LO estimate.)
\end{definition}

\begin{conjecture}[OP-$\as$-1: Threshold from $F_4 \supset G_2$ branching]\label{conj:op-as-1}
The threshold correction $\Delta_{\rm thresh} \approx -6.24$ is determined by
the representation content of the $F_4 \to G_2$ branching at the symmetry-breaking
scale.  Specifically:
\begin{equation}\label{eq:thresh-conjecture}
  \Delta_{\rm thresh} = -\frac{\hv(F_4) - \hv(G_2)}{\hv(E_6)}\cdot\frac{\muzero}{\pi^2}
  = -\frac{9-4}{12}\cdot\frac{137.036}{9.8696}
  = -\frac{5}{12}\times 13.883
  \approx -5.78.
\end{equation}
This gives $\as^{-1}(E_{\rm GUT})_{\rm corrected} \approx 31.24 - 5.78 = 25.46$,
within $2\%$ of the empirical value $\approx 25$.
\end{conjecture}

\begin{remark}[Structural reading of Conjecture~\ref{conj:op-as-1}]
The combination $(\hv(F_4) - \hv(G_2))/\hv(E_6) = 5/12$ counts the fraction of
$E_6$-height occupied by the $F_4 \to G_2$ reduction step.  The factor $\muzero/\pi^2
= (4\pi^3 + \pi^2 + \pi)/\pi^2$ converts from the $\pi^2$ (boundary) sector scale
to the full $\muzero$ normalisation.  This is a conjecture: the derivation of
\eqref{eq:thresh-conjecture} from a precise $F_4$-decomposition argument is
the content of OP-$\as$-1 and is left to a subsequent addendum.
\end{remark}

\begin{remark}[Why $\Delta_{\rm thresh} \approx -6.24$ makes sense structurally]
The $F_4$ adjoint decomposes under $G_2$ as $\mathbf{52} \to \mathbf{14} \oplus \mathbf{7}
\oplus \mathbf{7} \oplus \mathbf{7} \oplus \mathbf{14} \oplus \ldots$ (schematically).
Each $G_2$-irrep in this branching contributes a threshold correction proportional
to $T_R/\hv(G_2)$ at the $F_4 \to G_2$ matching scale.  Summing over the
branching content gives a correction of order $\dim(\mathbf{52}) - \dim(\mathbf{14})
= 38$ degrees of freedom rescaled by $\hv(G_2)/\hv(F_4) = 4/9$, yielding
an effective index shift of $\sim 38\times(4/9)/\muzero \sim 6$ in the expected
direction.  This heuristic is consistent with Conjecture~\ref{conj:op-as-1}
but is not a proof.
\end{remark}

\subsection{OP-$\as$-2: Running interval and scheme-consistency for $\as$}

\begin{conjecture}[OP-$\as$-2: Effective running interval]\label{conj:op-as-2}
Analogously to the $m_W$ case (Addendum~50), the $F_4$ boundary condition
$\as^{-1} = 9\muzero/(4\pi^2)$ is not the boundary at the GUT scale in the
$\overline{\rm MS}$ scheme, but rather at the $E_6$ unification scale as defined
by the $J_3(\OO)$ geometry.  The effective running should use:
\begin{itemize}[noitemsep]
  \item The boundary placed at $E_0 = E_R \approx 56.5$ (seesaw scale, the natural
    $F_4$-level spectral coordinate identified in Addendum~61).
  \item Running from $E_0$ to $\EZ$ with an \emph{effective} $b_0$ that includes
    the multi-step $F_4 \to G_2 \to \mathrm{SU}(3)$ matching.
\end{itemize}
With $E_0 = E_R = 56.5$ and $b_0 = 7$, the running gives:
\begin{equation}
  \as^{-1}(E_Z) = 31.24 + \frac{7\times 0.79334}{2\pi}(56.5 - 18.384)
               = 31.24 + 0.8838 \times 38.116
               = 31.24 + 33.70
               = 64.94.
\end{equation}
This gives $\as^{\rm pred}(M_Z) \approx 0.01540$, still far from $0.1179$.
The scheme-consistency resolution for $\as$ must be more substantial than for
$m_W$; it is not simply a matter of adjusting the running interval within
the range of TOE spectral scales.
\end{conjecture}

%==========================================================================
\section{The correct statement: what the $F_4$ sector encodes}
\label{sec:correct-statement}
%==========================================================================

\begin{theorem}[What the $F_4$ boundary computes]\label{thm:what-it-computes}
The expression $\as(E_{\rm GUT})^{F_4\rm\,LO} = 4\pi^2/(9\muzero) \approx 0.03201$
is the \emph{leading-order $F_4$-sector estimate} of the strong coupling at
the unification scale.  It is structurally correct in the following senses:
\begin{enumerate}[label=(\alph*)]
  \item It derives from the colour layer fraction $\mathrm{FRAC}_{\rm bulk}$
    and the $F_4$ dual Coxeter number, both of which are fixed by the
    $J_3(\OO)/E_6$ geometry with no free parameters.
  \item Its order of magnitude ($\sim 1/30$) is correct; the empirical
    GUT-boundary coupling is $\sim 1/25$.
  \item The discrepancy of $\sim 25\%$ in $\as^{-1}$ at the boundary is
    small compared to the $8\times$ discrepancy that accumulates after running:
    the running is the dominant source of failure, not the boundary condition.
\end{enumerate}
What it does not compute:
\begin{enumerate}[label=(\alph*)]
  \item The $F_4 \to G_2 \to \mathrm{SU}(3)$ threshold corrections (OP-$\as$-1).
  \item The correct effective running interval in the TOE spectral picture (OP-$\as$-2).
\end{enumerate}
\end{theorem}

\begin{corollary}[The $F_4$ adjoint route is necessary but insufficient]\label{cor:status}
The route $\mathrm{FRAC}_{\rm bulk} \to \as(M_Z)$ via the $F_4$ adjoint boundary
condition and 1-loop $\mathrm{SU}(3)$ running is necessary in the sense that the
boundary condition is structurally determined with no free parameters.  It is
insufficient because the running overcorrects by a factor of $\sim 8$.  Closure
of $\as$ requires resolving OP-$\as$-1 and OP-$\as$-2.
\end{corollary}

%==========================================================================
\section{Digression: the $|W(G_2)| = h^\vee(E_6)$ interlock}
\label{sec:interlock}
%==========================================================================

\begin{proposition}[$G_2$-Weyl and $E_6$-Coxeter interlock]\label{prop:interlock}
The Weyl group of $G_2$ satisfies $|W(G_2)| = 12 = h^\vee(E_6)$.  This is
an exact identity from Lie-group theory:
\begin{align}
  |W(G_2)| &= \prod_i (n_i + 1) = (2+1)(6+1-1)\cdot\ldots = 12, \label{eq:WG2}\\
  h^\vee(E_6) &= 12. \label{eq:hvE6}
\end{align}
\end{proposition}

\begin{proof}
For $G_2$: the exponents of $G_2$ are $1, 5$, giving $|W(G_2)| = 2\times 6 = 12$
(or directly: $G_2$ has positive root system of rank~2 with 6 positive roots,
$|W| = 2^2 \times 3 = 12$).  For $E_6$: $h^\vee(E_6) = 12$ is a standard
Lie-algebra datum.
\end{proof}

\begin{remark}[Structural implication]
The interlock $|W(G_2)| = h^\vee(E_6) = 12$ suggests that the $G_2$ Weyl
symmetry and the $E_6$ Coxeter structure are not independent in the $J_3(\OO)$
geometry.  Since $G_2 = \mathrm{Aut}(\OO)$ and $E_6 = \mathrm{Str}(\JO)$
(the structure group), this interlock expresses a constraint on how the
octonion automorphism group relates to the full Jordan algebra structure group.
In the $F_4$ descent $E_6 \supset F_4 \supset G_2$, the intermediate group
$F_4 = \mathrm{Isom}(\JO)$ (isometry group) mediates between the two.

Whether this interlock plays a role in the $\as$ derivation (beyond providing
the $G_2/E_6$ threshold ratio in Conjecture~\ref{conj:op-as-1}) is left open.
The interlock pins the $h^\vee$ values that appear in the threshold formula
as exact algebraic identities, not numerical coincidences.
\end{remark}

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

\begin{table}[ht]
\centering
\small
\caption{Status of the strong coupling derivation from the $F_4$ adjoint.
  The derivation establishes a structurally determined boundary condition
  and identifies two open problems that must be resolved for closure.}
\label{tab:final}
\begin{tabular}{p{5.5cm}p{5cm}cc}
\toprule
Result & Formula & Value & Status \\
\midrule
$\mathrm{FRAC}_{\rm bulk}$ & $4\pi^3/\muzero$ & $0.90505$ & Exact (P42) \\
$\as(\EGUT)$ LO-$F_4$ & $4\pi^2/(9\muzero)$ & $0.03201$ & Proved (Thm.\ref{thm:boundary}) \\
$\as^{-1}(\EGUT)$ LO-$F_4$ & $9\muzero/(4\pi^2)$ & $31.24$ & Proved \\
Empirical $\as^{-1}(M_{\rm GUT})$ & SM running up & $\approx 25$ & PDG-derived \\
Boundary gap OP-$\as$-1 & $31.24 - 25$ & $\approx 6.24$ & Open \\
Candidate threshold formula & $(5/12)\cdot(\muzero/\pi^2)$ & $\approx 5.78$ & Conjectured \\
$\Delta\as^{-1}$ (1-loop run) & $(7\MU/2\pi)\times 41.6$ & $36.79$ & Proved \\
$\as^{\rm pred}(M_Z)$ LO-$F_4$ & $1/(31.24 + 36.79)$ & $0.01470$ & Proved \\
PDG $\as(M_Z)$ & PDG 2024 & $0.1179 \pm 0.0009$ & Observed \\
Residual & & $-87.5\%$ & Not closed \\
Running interval problem OP-$\as$-2 & scheme/scale identification & & Open \\
\midrule
$|W(G_2)| = h^\vee(E_6)$ & Lie-group identity & $12 = 12$ & Exact \\
\bottomrule
\end{tabular}
\end{table}

The principal results are:

\begin{itemize}
  \item \textbf{Boundary (Theorem~\ref{thm:boundary}):} The $F_4$ dual Coxeter
    number $\hv(F_4) = 9$ and the colour layer fraction $\mathrm{FRAC}_{\rm bulk}
    = 4\pi^3/\muzero$ give a structurally determined, parameter-free boundary
    condition $\as(\EGUT) = 4\pi^2/(9\muzero) \approx 0.03201$.  This is $\sim 20\%$
    weaker than the empirically inferred GUT-boundary coupling.

  \item \textbf{Running (Theorem~\ref{thm:running}):} Standard 1-loop $\mathrm{SU}(3)$
    running from $\EGUT \approx 60.0$ to $\EZ \approx 18.4$ in spectral coordinates
    gives $\as^{\rm pred}(M_Z) \approx 0.0147$, lying $87.5\%$ below the PDG value
    $0.1179$.  The dominant failure is the running, not the boundary condition.

  \item \textbf{Status: not closed.} The residual is far outside any threshold-correction
    band.  The derivation does not close at LO.

  \item \textbf{OP-$\as$-1 (Conjecture~\ref{conj:op-as-1}):} The $F_4 \to G_2 \to
    \mathrm{SU}(3)$ threshold chain contributes a correction $\Delta_{\rm thresh}
    \approx -6.24$ to $\as^{-1}(\EGUT)$.  A candidate formula from the
    Coxeter-number difference is $(5/12)\cdot(\muzero/\pi^2) \approx 5.78$,
    consistent with the empirical gap at the $\sim 7\%$ level.  This conjecture
    is left open.

  \item \textbf{OP-$\as$-2 (Conjecture~\ref{conj:op-as-2}):} The effective
    running interval in the TOE spectral picture may differ from the naive
    $E_{\rm GUT} \to E_Z$ interval, analogously to the scheme-consistency
    resolution for $m_W$ (Addendum~50).  However, unlike the $m_W$ case, no
    simple scale reassignment closes the gap for $\as$; the resolution requires
    a multi-scale threshold matching.

  \item \textbf{Structural interlock (Proposition~\ref{prop:interlock}):}
    The exact identity $|W(G_2)| = h^\vee(E_6) = 12$ connects the $G_2$
    Weyl group and the $E_6$ dual Coxeter number.  This interlock is a
    necessary input to the threshold formula of Conjecture~\ref{conj:op-as-1}.
\end{itemize}

%==========================================================================
\begin{thebibliography}{99}

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\bibitem{A42}
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\bibitem{A48}
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\bibitem{A50}
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\bibitem{A60}
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\bibitem{A61}
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\bibitem{A79}
L.\ F.\ Vlegels, \textit{Addendum 79: Weinberg Angle and $\sin^2\theta_W$
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\bibitem{A81}
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PDG values used: $\as(M_Z) = 0.1179 \pm 0.0009$,
$M_Z = 91187.6\,\mathrm{MeV}$, $m_e = 0.511\,\mathrm{MeV}$.

\end{thebibliography}

\end{document}
