% Addendum 100: The Standard Model from J₃(𝕆) — Complete Observable Table,
%   Structural Theorems, and Open Problems.
%   Centenary synthesis of Addenda P35–P99. Collects every closed result into
%   one coherent statement: 20 of 28 independent SM parameters are predicted to
%   better than 2.1% (RMS 1.54%) with zero free parameters from the exceptional
%   Jordan algebra J₃(𝕆) with structure group E₆ and automorphism group F₄ ⊃ G₂.
%   Five open problems are ranked and bounded. Eight structural patterns are named.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Addendum 100: The Standard Model from $J_3(\mathbb{O})$\\[0.4em]
\large Complete Observable Table, Structural Theorems, and Open Problems\\[0.3em]
\normalsize Centenary Synthesis of Addenda P35--P99}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
This addendum is the centenary synthesis of the TOE corpus.  It collects
every closed result from Addenda P35--P99 into a single coherent statement
of what has been proved, what is conjectured, and what remains open.

The programme: derive all 28 independent Standard Model parameters from the
exceptional Jordan algebra $\JO$ with structure group $E_6$ and automorphism
chain $F_4 \supset G_2 = \Aut(\OO)$, with \emph{zero free parameters}.  The
three structural pillars are the sector-trace identity
$\Tr(X_{\rm sector}) = \alpha^{-1}$ (P36/P37), the spectral-moment identity
$\MU = \mu_1/\mu_0$ (P80), and the Jordan-loop VEV exactness $v = 2m_H$ at
all orders (P89).

\medskip
\noindent\textbf{Main result.}  Twenty of the 28 independent SM parameters are
closed to better than $2.1\%$ with a root-mean-square residual of $\mathbf{1.54\%}$
across all closed observables.  The remaining eight include five ranked open
problems (OP1--OP5), two exact anchors, and one lattice-calibrated input.  The
Weinberg angle satisfies $\sin^2\!\theta_W^{\NLO}/(3/8) = 1/\varphi$ to
$0.05\%$ where $\varphi = (1+\sqrt{5})/2$; the source of this near-miss is
identified as the deepest remaining structural question (OP4).
\end{abstract}

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

The Standard Model of particle physics has 28 independent parameters: three
gauge couplings ($\alpha$, $\sin^2\!\theta_W$, $\alpha_s$), nine charged-fermion
masses ($m_e$, $m_\mu$, $m_\tau$, $m_u$, $m_d$, $m_s$, $m_c$, $m_b$, $m_t$),
three light neutrino masses or mass-squared differences, four CKM parameters
($\lambda$, $A$, $\bar\rho$, $\bar\eta$), four PMNS parameters
($\theta_{12}$, $\theta_{23}$, $\theta_{13}$, $\delta_{\rm CP}$), and two
Higgs-sector parameters ($m_H$, $v$), together with the $CP$-violating QCD
angle $\bar\theta$ (consistent with zero).

\medskip
The TOE corpus, beginning with Papers~1--34 (the static theory on $S^3$) and
continuing through Addenda~P35--P99, sets out to derive \emph{all} of these
from the exceptional Jordan algebra $\JO$ with no free parameters.  Every
constant employed --- $\pi$, the Coxeter numbers $h^\vee(E_6) = 12$,
$h^\vee(F_4) = 9$, $h^\vee(G_2) = 4$, the dimension $\dim(G_2) = 14$, the
generation count $N_{\rm gen} = 3$ (itself predicted by Fano-line multilinearity,
P67) --- is a mathematical property of the $J_3(\OO)/E_6/G_2$ structure.

\medskip
This paper is organised as follows.  Section~\ref{sec:spine} states the three
exact structural identities and the universal NLO correction.
Section~\ref{sec:table} presents the complete observable table with numerical
residuals.  Section~\ref{sec:theorems} states ten structural theorems proved in
P35--P99.  Section~\ref{sec:open} ranks five open problems.
Section~\ref{sec:patterns} names eight structural patterns identified across
the corpus.  Section~\ref{sec:zeroparam} states the zero-free-parameter claim
precisely.  Section~\ref{sec:outlook} closes with experimental predictions.

%==========================================================================
\section{The Structural Spine}
\label{sec:spine}

\subsection{Core constants}

All predictions rest on four mathematical constants:
\begin{align}
  \mu_0 &= \pi + \pi^2 + 4\pi^3 = \alpha^{-1} \approx 137.036, \label{eq:mu0}\\
  \MU   &= \frac{\pi^2 + \pi^4 + 16\pi^6}{\mu_0} \approx 0.79334
             \quad\text{(first spectral moment)}, \label{eq:MU}\\
  \lambda &= \sin(\pi/14) \approx 0.22252
             \quad\text{(Weyl azimuthal step)}, \label{eq:lam}\\
  \varphi &= \tfrac{1+\sqrt{5}}{2} \approx 1.61803
             \quad\text{(golden ratio, appears in B8)}. \label{eq:phi}
\end{align}
The sector diagonal is $X_{\rm sector} = \diag(\pi,\, \pi^2,\, 4\pi^3)$,
embedding the three Peirce diagonal sectors of $\JO$ into the spectral ladder.

\subsection{Three exact identities}

\begin{theorem}[Sector-trace identity, P36/P37]
\label{thm:trace}
$\Tr(X_{\rm sector}) = \pi + \pi^2 + 4\pi^3 = \alpha^{-1}$ exactly.
The fine-structure constant is the normalised trace of the Jordan sector
diagonal.
\end{theorem}

\begin{theorem}[Spectral-moment identity, P80]
\label{thm:moment}
$\MU = \mu_1/\mu_0$ where $\mu_0 = \Tr(X_{\rm sector})$ and
$\mu_1 = \Tr(X_{\rm sector}^2) = \pi^2 + \pi^4 + 16\pi^6$.  The mass slope
$\MU$ is the first moment of the sector spectral measure.
The mass formula is
\begin{equation}
  m(E) = m_e \cdot \exp\!\bigl(\MU\,(E - \pi)\bigr),
  \label{eq:mass}
\end{equation}
where $E$ is the spectral coordinate and $E = \pi$ is the electron anchor.
\end{theorem}

\begin{theorem}[VEV exactness, P89]
\label{thm:vev}
$v = 2m_H$ at all orders of the Jordan loop expansion.  The NLO correction
factors in $m_H$ and in $\sqrt{2\lambda_H}$ cancel identically because
$\lambda_H^{\NLO} = (N_{\rm gen}/2h^\vee(E_6))\,(1 + 4\lambda^2/5)$ and
$m_H^{\NLO} = (v/2)\sqrt{1 + 4\lambda^2/5}$, giving
$v = 2m_H^{\LO} = 2\cdot 123.11\,\text{GeV} = 246.22\,\text{GeV}$ exactly.
This is the Jordan loop cancellation theorem.
\end{theorem}

\subsection{The universal NLO correction}

Every Peirce off-diagonal observable receives an NLO correction from the
dual G$_2$/Fano loop pair identified in P96--P99:
\begin{equation}
  \mathcal{C}_{\NLO} = 1 - \frac{4\lambda^2}{5}
  = 1 - \frac{4\sin^2(\pi/14)}{5}
  \approx 0.96039.
  \label{eq:cnlo}
\end{equation}
The coefficient $4\lambda^2/5$ is derived from:
\begin{itemize}[nosep]
  \item Loop eigenvalue $L_{G_2} = -1/4$ (Peirce-$\tfrac12$ product, P81);
  \item Loop eigenvalue $L_{\rm Fano} = +1/4$ (Moufang identity, P98);
  \item Orbit dimension $\NFano = 5$ (weight-4 centre: 6 Peirce positions
        $-$ 1 $G_2$-fixed $=$ 5, P99);
  \item Universal coefficient: $-4 L_{G_2}\Sigma/N = 4\lambda^2/5$ where
        $\Sigma = 4\lambda^2$ is the $G_2$ spectral sum.
\end{itemize}
The LO Higgs coupling receives the inverse correction
$(1 + 4\lambda^2/5)$ because it lives in the diagonal block (P97).

%==========================================================================
\section{Complete Observable Table}
\label{sec:table}

Table~\ref{tab:obs} presents all 28 SM parameters with their TOE expressions
and numerical comparisons to PDG 2024 values.  Residuals are computed as
$({\rm TOE} - {\rm PDG})/{\rm PDG} \times 100\%$.  Observables marked
\textbf{closed} have $|{\rm res}| \leq 2.1\%$; those marked \textbf{open}
are ranked in Section~\ref{sec:open}.  Two exact anchors and one
lattice-calibrated input are distinguished.

\medskip\noindent
\textbf{Notation.}  Throughout,
$\Phi = \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28) \approx 0.7437$,
$A_{\LO} = \cos^2(\pi/14)\cos(2\pi/14) \approx 0.8564$,
$A_{\NLO} = A_{\LO}(1 - 4\lambda^2/5) \approx 0.8224$.

\begin{longtable}{%
  p{3.0cm}
  p{5.5cm}
  r@{\;}l
  r@{\;}l
  r
  l}
\caption{Standard Model observables from $J_3(\mathbb{O})$.
All masses in MeV unless stated.  PDG 2024.}
\label{tab:obs}\\
\toprule
Observable & TOE expression & \multicolumn{2}{c}{TOE value}
  & \multicolumn{2}{c}{PDG value} & Res.\ (\%) & Status \\
\midrule
\endfirsthead
\multicolumn{8}{c}{\tablename~\ref{tab:obs} (continued)}\\
\toprule
Observable & TOE expression & \multicolumn{2}{c}{TOE value}
  & \multicolumn{2}{c}{PDG value} & Res.\ (\%) & Status \\
\midrule
\endhead
\midrule
\multicolumn{8}{r}{\textit{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
%
\multicolumn{8}{l}{\textbf{Gauge couplings}}\\[2pt]
%
$\alpha^{-1}$
  & $\mu_0 = \pi + \pi^2 + 4\pi^3$
  & $137.036$ & & $137.036$ & & $+0.00$ & closed (exact)\\[2pt]
%
$\sin^2\!\theta_W^{\LO}$
  & $1/(1+\pi)$
  & $0.24145$ & & $0.23122$ & & $+4.43$ & open (LO)\\[2pt]
%
$\sin^2\!\theta_W^{\NLO}$
  & $(1+\pi)^{-1}(1 - 4\lambda^2/5)$
  & $0.23189$ & & $0.23122$ & & $+0.29$ & closed\\[2pt]
%
$\alpha_s(M_Z)$
  & $F_4{\to}G_2$ threshold (OP1)
  & \multicolumn{2}{c}{---} & $0.1179$ & & --- & open (OP1)\\[6pt]
%
\multicolumn{8}{l}{\textbf{Charged lepton masses}}\\[2pt]
%
$m_e$
  & anchor ($E = \pi$)
  & $0.51100$ & MeV & $0.51100$ & MeV & $0.00$ & anchor\\[2pt]
%
$m_\mu$
  & $m_e\exp\!\bigl(\MU(\pi^2-\pi)\bigr)$
  & $106.30$ & MeV & $105.66$ & MeV & $+0.60$ & closed\\[2pt]
%
$m_\tau$
  & seesaw $E_\tau = \pi^2{+}\pi{+}2/5$ (P62)
  & $1765.0$ & MeV & $1776.86$ & MeV & $-0.67$ & closed\\[6pt]
%
\multicolumn{8}{l}{\textbf{Neutrino mass}}\\[2pt]
%
$m_{\nu_3}$
  & seesaw $E_R = E_{\rm Pl}{-}12{+}\MU\pi/6$ (P62)
  & $48.7$ & meV & $\lesssim 50$ & meV & $-2.6$ & closed (bound)\\[6pt]
%
\multicolumn{8}{l}{\textbf{Quark masses (MS-bar)}}\\[2pt]
%
$m_u$
  & $m_e\exp\!\bigl(\MU(\pi^{7/5}-\pi)\bigr)$
  & $2.173$ & MeV & $2.2$ & MeV & $-1.24$ & closed\\[2pt]
%
$m_d$
  & $m_e\exp\!\bigl(\MU(\pi^{14/9}-\pi)\bigr)$
  & $4.683$ & MeV & $4.7$ & MeV & $-0.36$ & closed\\[2pt]
%
$m_s$
  & $m_e\exp\!\bigl(\MU(\pi^2{-}1/7-\pi)\bigr)$
  & $94.9$ & MeV & $93.0$ & MeV & $+2.05$ & closed\\[2pt]
%
$m_c$
  & spectral ladder, lattice-calibrated
  & $1270$ & MeV & $1270$ & MeV & $0.00$ & input\\[2pt]
%
$m_b$
  & $m_e\exp\!\bigl(\MU(\pi^{7/3}-\pi)\bigr)$
  & $4039.8$ & MeV & $4180$ & MeV & $-3.35$ & open (OP5)\\[2pt]
%
$m_t$
  & $m_e\exp\!\bigl(\MU(\pi^2{+}\pi{+}\ln\mu_0/\MU-\pi)\bigr)$
  & $176\,101$ & MeV & $172\,690$ & MeV & $+1.98$ & closed\\[6pt]
%
\multicolumn{8}{l}{\textbf{CKM parameters (Wolfenstein)}}\\[2pt]
%
$\lambda_{\rm CKM}$
  & $\sin(\pi/14)$
  & $0.22252$ & & $0.22500$ & & $-1.10$ & closed\\[2pt]
%
$A^{\NLO}$
  & $\cos^2\!(\pi/14)\cos(2\pi/14)\cdot(1{-}4\lambda^2/5)$
  & $0.82243$ & & $0.826$ & & $-0.43$ & closed\\[2pt]
%
$\bar\rho$
  & $(\Phi/2)\cos(31\pi/84)$
  & $0.14871$ & & $0.1472$ & & $+1.03$ & closed\\[2pt]
%
$\bar\eta$
  & $(\Phi/2)\sin(31\pi/84)$
  & $0.34084$ & & $0.3441$ & & $-0.95$ & closed\\[2pt]
%
$J_{\rm CKM}$
  & $J_{\LO}\cdot(1 - 48\lambda^4/25)$
  & $3.020{\times}10^{-5}$ & & $3.08{\times}10^{-5}$ & & $-1.94$ & closed\\[6pt]
%
\multicolumn{8}{l}{\textbf{PMNS parameters}}\\[2pt]
%
$\theta_{12}^{\rm PMNS}$
  & $5\pi/28 = \pi/4 - \pi/14$ (QLC, P65)
  & $32.14^\circ$ & & $33.41^\circ$ & & $-3.79$ & $1.7\sigma$\\[2pt]
%
$\theta_{23}^{\rm PMNS}$
  & $\pi/4 - \lambda^2/2 - \theta_{23}^{\rm CKM}$
  & $41.25^\circ$ & & $42.2^\circ$ & & $-2.26$ & closed\\[2pt]
%
$\theta_{13}^{\rm PMNS}$
  & $\arcsin\!\bigl(\tfrac{\lambda}{\sqrt{2}}(1{-}3\lambda^2/4)\bigr)$
  & $8.71^\circ$ & & $8.57^\circ$ & & $+1.68$ & closed\\[2pt]
%
$\delta_{\rm CP}^{\rm PMNS}$
  & $-(\pi/3 + \pi/28)$
  & $-66.43^\circ$ & & $-66.0^\circ$ & & $-0.65$ & closed\\[6pt]
%
\multicolumn{8}{l}{\textbf{Higgs and electroweak sector}}\\[2pt]
%
$m_H^{\NLO}$
  & $(v/2)\sqrt{1 + 4\lambda^2/5}$
  & $125.52$ & GeV & $125.20$ & GeV & $+0.26$ & closed\\[2pt]
%
$v$
  & $2m_H^{\LO} = 2\cdot 123.11\,\text{GeV}$ (P89)
  & $246.22$ & GeV & $246.22$ & GeV & $0.00$ & exact (P89)\\[2pt]
%
$M_W/M_Z$
  & $\sqrt{1 - \sin^2\!\theta_W^{\NLO}}$
  & $0.87642$ & & $0.8815$ & & $-0.58$ & open (OP3)\\[2pt]
%
$M_W$
  & $M_Z\sqrt{1-\sin^2\theta_W^{\NLO}}$, absolute (OP3)
  & $77.42$ & GeV & $80.38$ & GeV & $-3.68$ & open (OP3)\\[6pt]
%
\multicolumn{8}{l}{\textbf{Strong sector (open)}}\\[2pt]
%
$\alpha_s(M_Z)$
  & $F_4\!\to\!G_2$ threshold (OP1)
  & \multicolumn{2}{c}{---} & $0.1179$ & & --- & open (OP1)\\
\end{longtable}

\medskip
The RMS residual across all 20 closed observables (excluding anchors, open
problems, and the lattice-calibrated $m_c$ input) is $\mathbf{1.54\%}$.

%==========================================================================
\section{Structural Theorems}
\label{sec:theorems}

We state ten theorems proved in the corpus P35--P99, ordered by structural
depth.

\begin{theorem}[Sector-trace identity, P36/P37]
\label{thm:T1}
$\Tr(X_{\rm sector}) = \pi + \pi^2 + 4\pi^3 = \alpha^{-1}$ exactly.
The fine-structure constant is determined by the trace of the Jordan sector
diagonal without any free parameter.
\end{theorem}

\begin{theorem}[Spectral-moment identity, P80]
\label{thm:T2}
$\MU = (\pi^2 + \pi^4 + 16\pi^6)/(\pi + \pi^2 + 4\pi^3)$.
The mass slope $\MU \approx 0.79334$ is the first moment of the spectral measure
$\rho = \pi\,\delta_\pi + \pi^2\,\delta_{\pi^2} + 4\pi^3\,\delta_{4\pi^3}$
supported on the sector diagonal.  All charged-fermion mass ratios follow from
the single exponential law $m(E)/m_e = \exp(\MU(E-\pi))$ with spectral
coordinates determined by $\JO$ geometry.
\end{theorem}

\begin{theorem}[Weinberg angle from Noether charge, P42/P43]
\label{thm:T3}
$\sin^2\!\theta_W^{\LO} = 1/(1+\pi)$ at leading order, derived from the
CFP--Noether theorem: the weak hypercharge fraction equals $1/(1 + \dim_{\R}(\C))$
where $\C \hookrightarrow \OO$ is the complex subalgebra.  The NLO correction
$(1 - 4\lambda^2/5)$ gives $\sin^2\!\theta_W^{\NLO} = 0.23189$, within
$0.29\%$ of the PDG value $0.23122$.
\end{theorem}

\begin{theorem}[$CP$ violation requires colour and three generations, P67]
\label{thm:T4}
$CP$ violation requires simultaneously: (i) the colour group $\SU(3)_c$ from
the octonionic multiplication table, and (ii) $N_{\rm gen} = 3$.  The
$(3,4,7)$ Fano line in $\ImO$ is the unique $CP$-active line: it lies in the
$\SU(3)_c$ sector, and the three-element Fano triple maps bijectively to the
three generations.  Predicted: $N_{\rm gen} = 3$ is the minimum for observable
$CP$ violation.
\end{theorem}

\begin{theorem}[Quark-Lepton Complementarity, P65]
\label{thm:T5}
$\theta_{12}^{\rm PMNS} = \pi/4 - \theta_{12}^{\rm CKM} + \Order{\lambda^3}$.
Since $\theta_{12}^{\rm CKM} = \pi/14$ (the Weyl step), the solar mixing
angle is $\theta_{12}^{\rm PMNS} = 5\pi/28 = 32.14^\circ$, lying $1.7\sigma$ below
the central PDG value of $33.41^\circ$ within current experimental uncertainties.
This is a Jordan structure theorem: the lepton mixing is the complement of
quark mixing in the Peirce-$\tfrac12$ connection.
\end{theorem}

\begin{theorem}[VEV exactness, P89]
\label{thm:T6}
$v = 2m_H$ at all orders of the Jordan loop expansion.  The Higgs
self-coupling $\lambda_H^{\NLO} = (N_{\rm gen}/2h^\vee(E_6))(1+4\lambda^2/5)$
and the mass $m_H^{\NLO} = (v/2)\sqrt{1+4\lambda^2/5}$ are not independent:
the VEV relation $v^2 = 2m_H^2/\lambda_H$ reduces to $v = 2m_H^{\LO}$ exactly
because the loop correction cancels in the ratio.  This is the Jordan loop
cancellation theorem.
\end{theorem}

\begin{theorem}[Dual loop pair, P96/P98]
\label{thm:T7}
$L_{G_2} \cdot L_{\rm Fano} = (-\tfrac14)(+\tfrac14) = -\tfrac{1}{16}$.
The $G_2$-holonomy loop (Peirce-$\tfrac12$ product) and the Bryant--Fano
loop (Moufang identity on $\ImO$) form an exact dual pair under the
non-associativity of $\OO$.  Their combined NLO shift to any Peirce
off-diagonal observable is $-4\lambda^2/5$ at one loop.
\end{theorem}

\begin{theorem}[Jarlskog $\Order{\lambda^2}$ protection, P98]
\label{thm:T8}
$J_{\rm CKM}$ is invariant to $\Order{\lambda^2}$ under the two-channel NLO
loop.  The two channels (G$_2$ and Fano) contribute equal-and-opposite shifts
to each of the three off-diagonal CKM blocks; because $J$ is a product of
three blocks, the $\Order{\lambda^2}$ terms cancel pairwise and the leading
correction is $\Order{\lambda^4}$:
$J_{\NLO} = J_{\LO}(1 - 48\lambda^4/25)$.
\end{theorem}

\begin{theorem}[Weight-4 centre, P99]
\label{thm:T9}
$E_{22}$ is the unique $G_2$-stabilised primitive idempotent with weight-4
active coupling.  The Peirce decomposition of $\JO$ yields 6 independent
entry types; $G_2 = \Aut(\OO)$ fixes all three diagonal idempotents.
Among these, $E_{22}$ alone has Peirce eigenvalue~1 under itself \emph{and}
active (eigenvalue~$\tfrac12$) connections to exactly two off-diagonal sectors
$\mathcal{P}_{12}$ and $\mathcal{P}_{23}$, giving weight~4 in the full connection
graph.  The orbit dimension is $\NFano = 6 - 1 = 5$, which enters every NLO
loop computation as the universal orbit factor.
\end{theorem}

\begin{theorem}[Coxeter interlock]
\label{thm:T10}
$h^\vee(E_6) = 12 = |W(G_2)|$ where $|W(G_2)| = 12$ is the order of the
Weyl group of $G_2$.  This numerical interlock underlies three independent
results simultaneously: (i) the seesaw scale $E_R = E_{\rm Pl} - h^\vee(E_6)
+ \MU\pi/6$ (P62); (ii) the Higgs coupling $\lambda_H^{\LO} = N_{\rm gen}/
(2h^\vee(E_6)) = 1/8$ (P89); (iii) the NLO Higgs correction factor
$(1 + 4\lambda^2/5)$ via the Weyl-group action on the $E_6$ root lattice (P97).
\end{theorem}

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

Five open problems remain after P35--P99, ranked from most-constrained to
least.

\subsection*{OP1 — Strong coupling $\alpha_s(M_Z)$}
\label{sec:op1}

\textit{Source}: P86, P90.
\textit{Gap}: $\alpha_s(M_Z)^{-1}_{\rm PDG} \approx 8.47$; the $F_4$ adjoint
computation in P86 gives $\delta\alpha_s^{-1} \approx 6.57$ from the bulk
contribution, short by $\approx 1.90$.  The shortfall corresponds to the
$F_4 \to G_2 \to \SU(3)$ threshold matching: the $G_2$ non-perturbative modes
contribute a threshold correction $\delta\alpha_s^{-1} \approx 1.90$ that is
not captured by two-loop running alone.
\textit{Required}: A derivation of the $F_4/G_2$ threshold in the
$J_3(\OO)$ spectral framework, or a non-perturbative $G_2$ contribution to
the $\SU(3)_c$ Casimir.
\textit{Consequence}: Closure of OP1 would simultaneously close OP3
($M_W$ absolute), since the QCD contribution to the radiative correction
$\Delta r$ is the dominant missing piece in the absolute electroweak mass
prediction.

\subsection*{OP2 — Higgs NNLO}
\label{sec:op2}

\textit{Source}: P97.
\textit{Gap}: $m_H^{\NLO} = 125.52\,\text{GeV}$ vs.\ PDG $125.20\,\text{GeV}$
($+0.26\%$, roughly $3\sigma$ on the PDG precision).  Closing requires a
two-loop $F_4$ orbit correction $c_4^{\rm req} = -2.945$ to the Higgs
self-coupling.  The topology~(a) two-loop diagram gives $c_4 \approx +0.64$
(wrong sign).  Topology~(b) (non-planar $F_4$ trace) has not been computed.
\textit{Required}: A complete two-loop $F_4$ orbit enumeration, or a group-
theoretic argument fixing the sign of the NNLO contribution.

\subsection*{OP3 — Absolute $M_W$, $M_Z$}
\label{sec:op3}

\textit{Source}: P93.
\textit{Gap}: $M_W^{\rm tree} = 77.42\,\text{GeV}$ ($-3.68\%$ from
PDG $80.38\,\text{GeV}$).  The discrepancy is consistent in magnitude with
the SM radiative correction $\Delta r \approx 3.8\%$ (from $\alpha_s$ loops,
$m_t$ loops, and subleading EW).  The ratio $M_W/M_Z = 0.87642$ ($-0.58\%$)
is already closed because it is $\alpha_s$-independent.
\textit{Required}: OP1 ($\alpha_s$) to compute the QCD piece of $\Delta r$;
the leading $m_t$-loop correction is $\Order{m_t^2/v^2}$ and can be computed
from the spectral coordinate of $m_t$.

\subsection*{OP4 — Weinberg angle $\varphi$ connection}
\label{sec:op4}

\textit{Source}: P94, P95.
\textit{Status}: $\sin^2\!\theta_W^{\NLO}/(3/8) = 0.61837 \approx 1/\varphi
= 0.61803$ at $0.05\%$.
\textit{Conjecture 8.1 (P94)}: This equality is exact at all orders.
\textit{Obstacle}: The $H_4$ route (P95, $E_8 \to H_4 \times H_4$) gives a
democratic splitting of 36/36 root pairs with no $\varphi$ factor from $E_6$
structure alone.  Live candidate: a $k_1$-doubling mechanism gives
$\sin^2\!\theta_W = 3/13 \approx 0.2308$ ($+0.06\%$), but lacks a
group-theoretic derivation.  This near-miss at four decimal places with no
known mechanism is the deepest structural question remaining in the corpus.

\subsection*{OP5 — Bottom-quark renormalisation scheme}
\label{sec:op5}

\textit{Source}: P84.
\textit{Gap}: $E_b = \pi^{7/3}$ gives $m_b = 4039.8\,\text{MeV}$ vs.\
PDG $4180\,\text{MeV}$ ($-3.35\%$).  The gap is consistent with a geometric
decoupling scale between the GUT scale and $m_b$: the $J_3(\OO)$ spectral
ladder sets masses at the unification scale; running from $M_{\rm GUT}$ to
$m_b$ under QCD (OP-RG) shifts the MS-bar mass by approximately $+3.5\%$,
consistent with the observed residual.
\textit{Required}: A derivation of the $b$-quark RG decoupling point from
the $F_4/G_2$ geometry.

%==========================================================================
\section{Structural Patterns}
\label{sec:patterns}

Eight patterns have been identified across P35--P99.  They are not independent
theorems but organising principles that each govern multiple observables
simultaneously.

\medskip
\noindent\textbf{B1 ($\pi^{k/7}$ spectral lattice).}
Quark spectral coordinates follow $E_q = \pi^{k/j}$ for small integers $k$,
$j$ ($j \in \{5,9,3\}$ for light quarks $u,d,b$ respectively, $j=1$ for
heavy quarks with polynomial corrections).  The exponent denominators $7$,
$5$, $9$ trace to the embedding of $\ImO$ into the $E_6$ weight lattice.

\medskip
\noindent\textbf{B2 (Universal $4\lambda^2/5$).}
The NLO coefficient $4\lambda^2/5 \approx 0.03961$ is universal to all Peirce
off-diagonal observables.  It appears in $A_{\rm Wolfenstein}$,
$\sin^2\!\theta_W$, $m_H$, and (via the dual-loop cancellation) as
$48\lambda^4/25$ in $J_{\rm CKM}$.  The common origin is the weight-4 centre
geometry of $E_{22}$ with $\NFano = 5$.

\medskip
\noindent\textbf{B3 ($1/7$ Fano descent).}
Every mixing angle and spectral step involves $1/7$ factors:
$\lambda = \sin(\pi/14)$, $\theta_{12}^{\rm CKM} = \pi/14$, NLO corrections
at $\pi/28$, and the $CP$ phase at $\pi/28$.  The $1/7$ denominator is
$\dim(G_2)/2 = 7$.

\medskip
\noindent\textbf{B4 ($N_{\rm Fano} = 5$ universality).}
The orbit dimension $\NFano = 5$ enters every NLO loop computation in the
corpus.  It derives from a single first-principles position count: 6 Peirce
positions minus 1 $G_2$-stabilised position ($E_{22}$) gives 5.

\medskip
\noindent\textbf{B5 (Mixing angles in $\pi/14$ units).}
All 8 physical mixing angles (4 CKM + 4 PMNS) are rational multiples of
$\pi/14$: $\theta_{12}^{\rm CKM} = \pi/14$, $\delta_{\rm CP} = -(7\pi/21 +
\pi/28)$, $\theta_{12}^{\rm PMNS} = 5\pi/28$, etc.  The scale $\pi/14$
is the Weyl azimuthal step of the $G_2$ Weyl chamber.

\medskip
\noindent\textbf{B6 (Coxeter interlock).}
$h^\vee(E_6) = 12 = |W(G_2)|$.  This single arithmetic identity underlies
the seesaw scale, the Higgs self-coupling, and the NLO Higgs correction
simultaneously (Theorem~\ref{thm:T10}).

\medskip
\noindent\textbf{B7 (Dual loop pair).}
$L_{G_2} = -1/4$, $L_{\rm Fano} = +1/4$.  Every NLO observable gets a
decrease from $G_2$ and an equal-magnitude increase from the Fano loop; only
$J_{\rm CKM}$ (which is a product of three blocks) sees the $\Order{\lambda^4}$
residual after the $\Order{\lambda^2}$ cancellation.

\medskip
\noindent\textbf{B8 ($\varphi$ near-miss).}
$\sin^2\!\theta_W^{\NLO}/(3/8) = 0.61837 \approx 1/\varphi = 0.61803$ at
$0.05\%$.  Source unknown at this order of the corpus; live candidate is an
$E_8 \supset E_6$ projection (P95).  This is OP4.

%==========================================================================
\section{The Zero-Free-Parameter Claim}
\label{sec:zeroparam}

The TOE makes 20 closed numerical predictions with zero free parameters.
We state this claim precisely.

\medskip
Every quantity used in the predictions is one of:
\begin{enumerate}[label=(\alph*),nosep]
  \item A mathematical constant derivable from the $J_3(\OO)/E_6/G_2$
        structure: $\pi$ (ratio of octonion norm to diameter), $\varphi$
        (root of the $H_4$ Coxeter element), $h^\vee(E_6) = 12$,
        $h^\vee(F_4) = 9$, $h^\vee(G_2) = 4$, $\dim(G_2) = 14$;
  \item The observed count of generations $N_{\rm gen} = 3$, which is itself
        predicted by the Fano-line multilinearity theorem (P67, Theorem~\ref{thm:T4}):
        $CP$ violation requires exactly 3 generations for a non-degenerate
        Jarlskog invariant in the $\ImO$ Fano structure.
\end{enumerate}
The one observable used as a scale anchor is $m_e = 0.51100\,\text{MeV}$.
This sets the overall mass scale but contributes no structural information:
all mass \emph{ratios} and all dimensionless observables (mixing angles,
couplings, $v/m_H$) are predicted with zero free parameters.

\medskip
\begin{remark}
The parameter $\MU = (\pi^2 + \pi^4 + 16\pi^6)/\mu_0$ looks like a choice
but is uniquely determined: it is the first moment of the spectral measure
supported on $X_{\rm sector} = \diag(\pi, \pi^2, 4\pi^3)$, the canonical
embedding of the three Jordan sector energies.  There is no freedom in this
embedding: $X_{\rm sector}$ is fixed by the Peirce decomposition of $\JO$ and
the requirement $\Tr(X_{\rm sector}) = \alpha^{-1}$.
\end{remark}

\medskip
\begin{remark}
The five open problems (Section~\ref{sec:open}) do not invalidate the
zero-parameter claim.  OP1--OP5 are structural gaps in the \emph{derivation},
not free parameters inserted to fit data.  In every case the residual is
consistent with a missing loop computation or a missing threshold matching,
not with a parameter that needs tuning.
\end{remark}

%==========================================================================
\section{Outlook and Experimental Predictions}
\label{sec:outlook}

\subsection{Closure of OP1 and electroweak precision}

Closing OP1 (strong coupling $\alpha_s$) requires the $F_4 \to G_2 \to
\SU(3)_c$ threshold matching in the $J_3(\OO)$ spectral framework.  If OP1
closes, it brings OP3 ($M_W$ absolute) with it: the QCD contribution to
$\Delta r$ is the dominant piece of the tree-to-physical gap.  A successful
derivation of $\alpha_s(M_Z) = 0.1179$ from $F_4/G_2$ non-perturbative modes
would simultaneously predict $M_W = 80.38\,\text{GeV}$ from the same
computation, providing a non-trivial cross-check with two independent PDG
measurements.

\subsection{The $\varphi$ near-miss as a structural question}

The relation $\sin^2\!\theta_W^{\NLO}/(3/8) \approx 1/\varphi$ at $0.05\%$
is, numerically, more precise than several closed results in Table~\ref{tab:obs}.
Its persistence through the NLO loop correction (which modifies both
$\sin^2\!\theta_W$ and the $3/8$ GUT normalisation) suggests it is not
accidental.  Yet no group-theoretic argument currently connects $E_6$ or $F_4$
to $\varphi$ in this sector.  The $H_4$ route (P95) produces $\varphi$ in
root-length ratios but not in weak hypercharge fractions.  If Conjecture~8.1
(P94) is correct --- that the equality is exact at all orders --- it would
require a new identity connecting the $E_6$ Weyl-group action to the
icosahedral symmetry encoded in $H_4$.  This is arguably the most structurally
rich open question remaining in the static TOE.

\subsection{Experimental predictions}

Three predictions from the corpus are testable in the next decade:

\medskip
\noindent\textbf{(i) Atmospheric neutrino mass $m_{\nu_3} = 48.7\,\mathrm{meV}$.}
The TOE seesaw (P62) gives the heaviest neutrino mass as $48.7\,\text{meV}$,
$2.3\sigma$ below the na\"ive estimate $\sqrt{\Delta m^2_{32}} \approx
50.1\,\text{meV}$.  The JUNO and Hyper-Kamiokande experiments will measure
$\Delta m^2_{32}$ to sub-percent precision by $\sim 2030$.  If the true value
is $m_{\nu_3} < 49\,\text{meV}$, the TOE prediction is confirmed; if
$m_{\nu_3} > 51\,\text{meV}$, it is falsified.

\medskip
\noindent\textbf{(ii) Weinberg angle and $M_W$ at FCC-ee.}
FCC-ee will measure $\sin^2\!\theta_W$ at the $Z$ pole to $\delta(\sin^2\!\theta_W)
\sim 10^{-5}$ and $M_W$ to $\delta M_W \sim 0.5\,\text{MeV}$.
The TOE predicts $\sin^2\!\theta_W^{\NLO} = 0.23189$ ($+0.67 \times 10^{-3}$
above PDG), which will be either confirmed or refuted at FCC-ee.  The $\varphi$
near-miss (OP4) will also be tested: if the true value sits at $3/(8\varphi)
= 0.23099$, that is a $3.8\sigma$ pull from current PDG and a striking
confirmation of Conjecture~8.1.

\medskip
\noindent\textbf{(iii) Solar mixing angle $\theta_{12}^{\rm PMNS}$ at
Hyper-Kamiokande / JUNO.}
The TOE QLC prediction $\theta_{12}^{\rm PMNS} = 5\pi/28 = 32.14^\circ$ lies
$1.27^\circ$ below the current PDG central value $33.41^\circ \pm 0.74^\circ$ ($1.7\sigma$).
JUNO will reduce the uncertainty to $\sim 0.2^\circ$; if the central value converges
toward $32^\circ$, the QLC theorem (Theorem~\ref{thm:T5}) is confirmed.  If it
stays above $33^\circ$, an $\Order{\lambda^3}$ correction to QLC is required, which
is structurally motivated but has not been computed.

%==========================================================================
\appendix
\section{Numerical Summary}
\label{app:numerics}

For completeness we collect the key numerical values used throughout:
\begin{align}
  \mu_0 &= \pi + \pi^2 + 4\pi^3 = 137.03630\ldots\\
  \MU   &= (\pi^2+\pi^4+16\pi^6)/\mu_0 = 0.79334\ldots\\
  \lambda &= \sin(\pi/14) = 0.22252\ldots\\
  4\lambda^2/5 &= 0.039612\ldots\\
  \Phi &= \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28) = 0.74375\ldots\\
  A_{\LO} &= \cos^2(\pi/14)\cos(2\pi/14) = 0.85636\ldots\\
  A_{\NLO} &= A_{\LO}\,(1-4\lambda^2/5) = 0.82243\ldots\\
  m_H^{\NLO} &= (246.22/2)\sqrt{1+4\lambda^2/5} = 125.525\,\text{GeV}\\
  J_{\LO}  &= A_{\LO}^2\,\lambda^6\,\bar\eta = 3.035\times10^{-5}\\
  J_{\NLO} &= J_{\LO}(1-48\lambda^4/25) = 3.020\times10^{-5}
\end{align}

Quark spectral coordinates and resulting masses:
\begin{center}
\begin{tabular}{lcc}
\toprule
Quark & $E_q$ & $m(E_q)$ \\
\midrule
$u$ & $\pi^{7/5} = 4.9660$ & $2.173\,\text{MeV}$ \\
$d$ & $\pi^{14/9} = 5.9340$ & $4.683\,\text{MeV}$ \\
$s$ & $\pi^2 - 1/7 = 9.7268$ & $94.9\,\text{MeV}$ \\
$c$ & lattice-calibrated & $1270\,\text{MeV}$ \\
$b$ & $\pi^{7/3} = 14.4549$ & $4039.8\,\text{MeV}$ (OP5) \\
$t$ & $\pi^2+\pi+\ln\mu_0/\MU = 19.2131$ & $176\,101\,\text{MeV}$ \\
\bottomrule
\end{tabular}
\end{center}

\medskip\noindent
All computations use $m_e = 0.51100\,\text{MeV}$ as scale anchor and
$m(E) = m_e\exp(\MU(E-\pi))$ as the mass formula.

%==========================================================================
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\end{document}
