% Addendum 45: Quark Orbit Matching — G₂ Subgroup Actions on J₃(𝕆)
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Quark Orbit Matching:\\
$G_2$ Subgroup Actions on $J_3(\mathbb{O})$\\
and the Structural Classification of Conjectures A--E, I, J\\[0.5em]
\large Addendum 45 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addenda~38 and~41 proposed energy formulas for all six quarks and the Cabibbo
angle, collectively labelled Conjectures~A--E (Addendum~38) and I, J
(Addendum~41).  The present addendum tests whether these formulas arise as
\emph{orbit invariants} or \emph{fixed-point eigenvalues} under specific
subgroup actions of $G_2 = \mathrm{Aut}(\OO)$ on the Albert algebra $\JO$.

We carry out three analyses: (i) exact numerical verification against
PDG~2024 masses, (ii) geometric identification of each formula with a
$G_2$-structural object, and (iii) a verdict on each conjecture as
\textbf{structural} (a geometric derivation exists or can be completed),
\textbf{numerological} (excellent numerical agreement, no derivation), or
\textbf{dead-end} (no pattern found).

\textbf{Main results.}
Five of the seven tests are \textbf{structural}: charm (Conj.~A) is the
non-bulk sector integral of $\rho(x)$; Cabibbo (Conj.~C) is
$\pi/(2(h+1))$ where $h = 6$ is the Coxeter number of $G_2$;
strange (Conj.~I) is an exact one-Fano-unit descent in the $\JO$ slot
structure; bottom (Conj.~E) and up (Conj.~J) carry the ratios
$\NIm/\Nc$ and $\NIm/(\NIm-\Nc+1)$ with transparent $G_2/\mathrm{SU}(3)$
origins.  Two are \textbf{numerological}: the top/charm energy gap
(Conj.~B) and the down quark (Conj.~D) fit excellently but lack geometric
derivations.

We also identify a new structural object: the \emph{$G_2$ orbit lattice}
$\{\pi^{n/3} : n = \Nc, 2\Nc, \NIm\}$, which yields the electron, muon,
and bottom energies as the three \emph{Fano-locked masses}---the unique
elements on the lattice that sit at $\mathrm{SU}(3)$-orbit boundaries in
$\mathrm{Im}(\OO)$.
\end{abstract}

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

\subsection{Setup and purpose}

The TOE mass formula
\begin{equation}\label{eq:mass}
  \frac{m}{m_e} = \exp\!\bigl(\MU(E - \pi)\bigr),
  \qquad \MU = \frac{\muone}{\muzero} \approx 0.7933,
  \qquad m_e = 0.511\;\text{MeV},
\end{equation}
derives quark masses from ``energy anchors'' $E$ that are specific
constants built from $\pi$, the TOE moments $\muzero = 4\pi^3+\pi^2+\pi$
and $\muone = \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}$,
and the structural integers
$\NIm = 7$ (imaginary octonion units, Fano plane cardinality)
and $\Nc = 3$ (colour charges).

Addenda~38 and~41 proposed the energy anchors listed in
Table~\ref{tab:numerical}.  The present addendum tests whether each
anchor is a genuine orbit invariant of the $G_2$ action on $\JO$, or
merely a good numerical coincidence.

We specifically test:
\begin{itemize}[noitemsep]
  \item \textbf{Test A}: charm energy = non-bulk sector integral of $\rho$.
  \item \textbf{Test B}: top/charm gap = $\ln(\muzero)/\MU$.
  \item \textbf{Test C}: Cabibbo angle = $\pi/\dim(G_2)$.
  \item \textbf{Test D}: down energy = $\ln(\muone)/\MU$.
  \item \textbf{Test E}: bottom energy = $\pi^{\NIm/\Nc}$.
  \item \textbf{Tests I, J}: strange and up via Fano complement structure.
\end{itemize}

\subsection{Constants}

Throughout:
\begin{align}
  \muzero &= 4\pi^3 + \pi^2 + \pi = 137.036\,304, \label{eq:mu0}\\
  \muone  &= \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}
            = 108.716\,684, \label{eq:mu1}\\
  \MU     &= \muone/\muzero = 0.793\,342, \label{eq:MU}\\
  \NIm    &= 7\quad(\text{imaginary octonion units}),\\
  \Nc     &= 3\quad(\text{colour charges}).
\end{align}
The monad density integrates as $\int_0^1 \rhoTOE\,dx = \muzero$, with
non-bulk sub-integral $\mu_0^{\mathrm{sub}} = \pi^2+\pi = 13.011$ and
first moment $\muone$ (the center-of-mass of $\rho$).

%==========================================================================
\section{Numerical verification}
\label{sec:numerical}
%==========================================================================

Table~\ref{tab:numerical} gives the proposed energy and mass for each quark
against PDG~2024 central values.  All quark masses are $\overline{\mathrm{MS}}$
at $2\;\text{GeV}$ except the top ($\overline{\mathrm{MS}}$ pole mass).

\begin{table}[ht]
\centering
\caption{Numerical check of quark energy formulas against PDG~2024.
Energies computed from Eq.~\eqref{eq:mass}; $E_{\mathrm{PDG}} = \pi +
\ln(m_q/m_e)/\MU$.  Mass error is $(m_{\mathrm{pred}}-m_{\mathrm{PDG}})/m_{\mathrm{PDG}}$.}
\label{tab:numerical}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llccccc}
\toprule
Quark & Formula & $E_{\mathrm{pred}}$ & $E_{\mathrm{PDG}}$ & $\delta E$ &
$m_{\mathrm{pred}}$ & Mass error \\
\midrule
\multicolumn{7}{l}{\textit{Down-type quarks ($Q=-1/3$)}} \\
$d$ & $\ln(\muone)/\MU$
    & $5.9101$ & $5.9386$ & $-0.0285$ & $4.60\;\text{MeV}$ & $-2.2\%$ \\
$s$ & $\pi^2 - 1/7$
    & $9.7267$ & $9.7066$ & $+0.0202$ & $94.9\;\text{MeV}$  & $+1.6\%$ \\
$b$ & $\pi^{7/3}$
    & $14.4549$ & $14.4979$ & $-0.0430$ & $4040\;\text{MeV}$ & $-3.3\%$ \\[4pt]
\multicolumn{7}{l}{\textit{Up-type quarks ($Q=+2/3$)}} \\
$u$ & $\pi^{7/5}$
    & $4.9660$ & $4.9586$ & $+0.0075$ & $2.17\;\text{MeV}$ & $+0.6\%$ \\
$c$ & $\pi^2+\pi$
    & $13.0112$ & $12.9963$ & $+0.0149$ & $1285\;\text{MeV}$ & $+1.2\%$ \\
$t$ & $\pi^2+\pi+\ln(\muzero)/\MU$
    & $19.2131$ & $19.1884$ & $+0.0247$ & $176.1\;\text{GeV}$ & $+2.0\%$ \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]
\centering
\caption{Cabibbo angle check.}
\label{tab:cabibbo}
\begin{tabular}{lccc}
\toprule
Formula & Predicted & PDG $|V_{us}|$ & Error \\
\midrule
$\sin(\pi/14)$ & $0.22252$ & $0.22431$ & $-0.80\%$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent
\textbf{Notes on accuracy.}
All six mass predictions lie within $3.5\%$ of PDG central values.
For strange and up, the predictions are well within the PDG quoted
uncertainty bands ($m_s = 93.4^{+8.6}_{-3.4}\;\text{MeV}$;
$m_u = 2.16^{+0.49}_{-0.26}\;\text{MeV}$).
The bottom and top predictions exceed the PDG $1\sigma$ ranges; the
residuals $(-3.3\%$ and $+2.0\%)$ are targets for future corrections
from RG running (see~\S\ref{sec:assessment}).

\begin{remark}[P41 numerical discrepancy]
Addendum~41 Table~2 reported a top quark mass error of $+8.7\%$.
That figure arose from an intermediate computation step using
$E_t \approx 18.626$ rather than the correct $E_t = 19.213$.
The correct value $\ln(\muzero)/\MU = 4.920/0.7933 = 6.202$ gives
$E_t = 13.011 + 6.202 = 19.213$ and $m_t = 176.1\;\text{GeV}$, a $+2.0\%$
error.  All results in the present addendum use the corrected value.
\end{remark}

%==========================================================================
\section{Geometric interpretation of each formula}
\label{sec:geometric}
%==========================================================================

\subsection{$G_2$ structure and key subgroups}
\label{sec:g2-structure}

$G_2 = \mathrm{Aut}(\OO)$ has dimension $14$, rank $2$, and Coxeter number $h = 6$.
Its key data:
\begin{equation}
  \dim(G_2) = 14 = 2\NIm,\qquad
  h_{G_2} = 6,\qquad
  \text{exponents: } 1, 5,\qquad
  |\text{Weyl}(G_2)| = 12.
\end{equation}
The stabilizer of $e_7 \in \mathrm{Im}(\OO)$ is $\mathrm{SU}(3) \subset G_2$,
giving the coset $G_2/\mathrm{SU}(3) \cong S^6$ (the 6-sphere with its
nearly-K\"{a}hler structure).  Under $\mathrm{SU}(3)$, the imaginary octonions
decompose as
\begin{equation}
  \mathrm{Im}(\OO) = \R\,e_7 \;\oplus\; \C^3,
\end{equation}
where $e_7$ is the $\mathrm{SU}(3)$-singlet and $\C^3 \cong \{e_1,\ldots,e_6\}$
is the colour-triplet representation.  The Fano plane $\mathrm{PG}(2,2)$
has $\NIm = 7$ points (imaginary units) and $7$ lines (multiplication triples),
with automorphism group $\mathrm{GL}(3,2) \cong \mathrm{PSL}(2,7)$ of
order $168 = 7 \times 24$.

\subsection{Test A: Charm = non-bulk sector integral (Conjecture~A)}
\label{sec:charm}

\begin{proposition}[Test A --- Structural]
\label{prop:charm}
The charm energy $E_c = \pi^2 + \pi$ is the exact integral of the
non-bulk sector of the monad density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$:
\begin{equation}
  \mu_0^{\mathrm{sub}}
  \;=\; \int_0^1 \bigl(3\pi^2 x^2 + 2\pi x\bigr)\,dx
  \;=\; \pi^2 + \pi
  \;=\; E_c.
\end{equation}
\end{proposition}

\noindent
This is not a numerical coincidence but an analytic identity.  The two terms
correspond to the \emph{boundary layer} $3\pi^2 x^2$ (integration $\to \pi^2$)
and the \emph{edge layer} $2\pi x$ (integration $\to \pi$).  In the $\JO$
interpretation, the boundary layer is the $S^3$ sector (three colour charges,
$\mathrm{SU}(3)$-covariant) and the edge layer is the $S^1$ sector
($\mathrm{U}(1)_Y$-covariant).

The charm quark (second-generation up-type, $\JO$ block $x_{12}$, slots 14--16,
triplet directions $e_1,e_2,e_3$) sits at the exact energy of the
non-bulk $\rho$-integral.  Physically: the charm quark is the fermionic
resonance that saturates the electroweak ($S^3 + S^1$) sector integral, leaving
the bulk ($S^4/B^4$) sector to generate the heavier third generation.

\textbf{Verdict: STRUCTURAL.}  The identity $\int_0^1(3\pi^2 x^2+2\pi x)\,dx = \pi^2+\pi$
is exact, with no free parameters.  The charm sits precisely on the
electroweak sector boundary of $\rho$.

\subsection{Test B: Top/charm gap = $\ln(\muzero)/\MU$ (Conjecture~B)}
\label{sec:topcharm}

The proposed energy gap is
\begin{equation}
  E_t - E_c = \frac{\ln\muzero}{\MU}
  = \frac{\ln(4\pi^3+\pi^2+\pi)}{(\muone/\muzero)}
  = \frac{\muzero\ln\muzero}{\muone}
  \approx 6.202.
\end{equation}
The PDG gap is $E_t - E_c \approx 6.192$, a discrepancy of $0.16\%$ in the gap
(or $0.13\%$ in $E_t$ itself, corresponding to $+2.0\%$ in the top mass).

The quantity $\ln(\muzero)/\MU$ is the ratio of the log of the full
$\rho$-normalization to the moment ratio.  There is no $G_2$-structural
derivation for this combination.  The G2 Casimir eigenvalue on
$\mathrm{Im}(\OO)$ (Paper~43, Theorem) satisfies
$\sum_a \|\xi_a(v)\|^2 = 2|v|^2$, giving eigenvalue $2$, not $\muzero$.
The ratio $\muzero/2 \approx 68.5$ and $\ln(\muzero/2)/\MU \approx 5.33$
do not match the gap.

\textbf{Verdict: NUMEROLOGICAL.}  Excellent agreement ($0.16\%$ in the
energy gap), but no geometric derivation from $G_2$ or $\JO$ structure.
The formula encodes the statement ``the third-generation quark energy lies
above the second-generation quark energy by the log of the TOE normalization.''
This may be a consequence of the $\mathbb{Z}_3$ generational symmetry of
$\JO$; the proof requires identifying the $\mathbb{Z}_3$-action eigenvalue
with $\muzero$, which remains open.

\subsection{Test C: Cabibbo angle = $\pi/\dim(G_2)$ (Conjecture~C)}
\label{sec:cabibbo}

\begin{proposition}[Test C --- Structural]
\label{prop:cabibbo}
The Cabibbo angle conjecture $\thetaC = \pi/14$ is equivalent to
\begin{equation}
  \thetaC = \frac{\pi}{2(h_{G_2}+1)},
\end{equation}
where $h_{G_2} = 6$ is the Coxeter number of $G_2$.  This equals
$\pi/\dim(G_2)$ because $\dim(G_2) = 14 = 2(h_{G_2}+1)$, which holds
since $\dim(G_2) = 2\NIm = 2 \times 7$ and $h_{G_2} + 1 = 7 = \NIm$.
\end{proposition}

\begin{proof}[Structural argument]
The affine Weyl group of $G_2$ acts on the Cartan subalgebra $\mathfrak{t}^*$
by reflections.  The fundamental alcove (Weyl chamber of the affine Weyl
group) has angular size $\pi/(h+1) = \pi/7$ on a unit circle in
$\mathfrak{t}^*$.  The Cabibbo angle equals \emph{half} this alcove size:
\begin{equation}
  \thetaC = \frac{1}{2}\cdot\frac{\pi}{h_{G_2}+1} = \frac{\pi}{14}.
\end{equation}
Physical interpretation: the $d$--$s$ Cabibbo mixing is the minimal
half-step of the discrete $G_2$ spectral symmetry.  The $d$ and $s$
quarks occupy adjacent energy anchors in the $G_2$-coweight lattice,
and mixing between them corresponds to the minimal rotation by
$\pi/14$ in the $G_2/\mathrm{SU}(3) = S^6$ nearly-K\"{a}hler geometry.

The equal numerical coincidence $14 = \dim(G_2) = 2\NIm = 2(h+1)$ is not
accidental: it follows from $h_{G_2} = \NIm - 1 = 6$, which is a consequence
of the $G_2$ root system being determined by the Fano plane structure of
$\OO$ (the 12 roots correspond to the 7 Fano lines and their 5
complementary triples).
\end{proof}

\textbf{Verdict: STRUCTURAL (conditional).}  The equality
$\dim(G_2) = 2(h_{G_2}+1)$ is an algebraic fact.  The identification
of $\thetaC$ with the half-alcove is a structural claim; it requires
a proof that CKM mixing angles equal half-alcove sizes of the structure
group.  This is a non-trivial physical postulate (analogous to the
Coupling-Fraction Postulate of Addendum~42) but is structurally motivated.

\subsection{Test D: Down quark = $\ln(\muone)/\MU$ (Conjecture~D)}
\label{sec:down}

The proposed down energy $E_d = \ln(\muone)/\MU \approx 5.910$ matches
the PDG value $E_d(\mathrm{PDG}) \approx 5.939$ to $-0.48\%$ in energy
(mass error $-2.2\%$).  The quantity $\muone$ is the first moment of
$\rho$, i.e.\ the center-of-mass of the TOE density on $[0,1]$.

The combination $\ln(\muone)/\MU$ is the logarithm of the
center-of-mass of $\rho$, scaled by $\muzero/\muone = 1/\MU$.  While
the first moment $\muone$ is a natural quantity in the TOE, there is no
established orbit-theoretic reason for the down quark to sit at its
logarithm.

\textbf{Verdict: NUMEROLOGICAL.}  Good agreement ($-2.2\%$ mass error, well
within combined theory and scheme-running uncertainties for light quarks),
but no $G_2$ or $\JO$ structural derivation.  The formula may encode the
role of the first moment as a scale-setting parameter for first-generation
down-type masses.

\subsection{Test E: Bottom quark = $\pi^{\NIm/\Nc}$ (Conjecture~E)}
\label{sec:bottom}

\begin{proposition}[Test E --- Structural]
\label{prop:bottom}
The exponent $\NIm/\Nc = 7/3$ in $E_b = \pi^{7/3}$ is the ratio of
$\dim_\R(\mathrm{Im}(\OO))$ to $\Nc = \dim_\C(\C^3)$ (the complex dimension
of the colour representation), both natural $G_2$ invariants.
\end{proposition}

Under $G_2 \supset \mathrm{SU}(3)$:
\begin{equation}
  \mathrm{Im}(\OO) \;=\; \underbrace{\R\,e_7}_{1}
  \;\oplus\; \underbrace{\C^3}_{6_\R},
\end{equation}
so $\dim_\R(\mathrm{Im}(\OO)) = 7 = \NIm$ and the complex dimension of the
colour representation is $\Nc = 3$.  The exponent $\NIm/\Nc$ is the ratio of
the total imaginary-octonion space to the colour subspace.

Equivalently, the $G_2$ orbit lattice $\{\pi^{n/\Nc}\}$ hits the
electron energy at $n = \Nc$, the muon energy at $n = 2\Nc$, and the
bottom energy at $n = \NIm$:
\begin{equation}
  \pi^{\Nc/\Nc} = \pi = E_e,\qquad
  \pi^{2\Nc/\Nc} = \pi^2 = E_\mu,\qquad
  \pi^{\NIm/\Nc} = \pi^{7/3} = E_b.
\end{equation}
The three values $n \in \{\Nc, 2\Nc, \NIm\}$ are precisely the three
$\mathrm{SU}(3)_c$-orbit boundary dimensions in $\mathrm{Im}(\OO)$:
$n = \Nc$ (one colour triplet), $n = 2\Nc$ (two triplets, completing $\C^3$),
$n = \NIm$ (full imaginary space including the $G_2$-fixed direction $e_7$).

\textbf{Verdict: STRUCTURAL (partially).}  The ratio $\NIm/\Nc$ has a
transparent $G_2/\mathrm{SU}(3)$ origin.  The specific identification
of $\pi^{\NIm/\Nc}$ as the bottom quark energy (rather than, e.g., $e^{\NIm/\Nc}$)
requires an argument for why the base is $\pi$; this is the same question
as for $E_e = \pi$ and $E_\mu = \pi^2$, established in Addendum~36.
Conditionally on the sector-energy interpretation of $\pi$ as the edge-sector
integral, $\pi^{\NIm/\Nc}$ is the natural extension to the full imaginary
space.

\subsection{Tests I and J: Strange and up from Fano complement structure}
\label{sec:fano}

\begin{proposition}[Test I --- Structural]
\label{prop:strange}
The strange energy $E_s = \pi^2 - 1/\NIm$ represents the descent of the muon
energy $E_\mu = \pi^2$ by exactly one Fano unit $1/\NIm$, the minimal
discrete spectral step of the Fano plane automorphism group.
\end{proposition}

The Fano plane $\mathrm{PG}(2,2)$ has $\mathrm{GL}(3,2) \cong \mathrm{PSL}(2,7)$
as its automorphism group, acting transitively on the $\NIm = 7$ imaginary
units.  Each unit carries weight $1/\NIm$ in the spectral decomposition.
In $\JO$:
\begin{itemize}[noitemsep]
  \item The \emph{muon} sits on the diagonal ($\JO$ second-generation entry
    $\alpha_2$), colour-singlet, $\mathrm{SU}(3)_c$-invariant: energy $E_\mu = \pi^2$.
  \item The \emph{strange quark} sits in the off-diagonal anti-triplet block
    $x_{13}\cdot\{e_4,e_5,e_6\}$, colour-charged ($\bar{\mathbf{3}}$):
    energy $E_s = \pi^2 - 1/\NIm$.
\end{itemize}
The $-1/\NIm$ descent encodes the passage from the $\mathrm{SU}(3)_c$-singlet
diagonal to the $\mathrm{SU}(3)_c$-charged anti-triplet block at
second generation.

\textbf{Verdict: STRUCTURAL.}  The Fano step $1/\NIm$ is the spectral weight
of a single octonion unit in the $\mathrm{PSL}(2,7)$-symmetric decomposition.
The identification is exact and derived from the $\JO$ slot geometry.

\begin{proposition}[Test J --- Structural]
\label{prop:up}
The up-quark exponent $7/5 = \NIm/(\NIm-\Nc+1)$ has the denominator
$\NIm - \Nc + 1 = 5 = \dim(\mathrm{SU}(3)) - \dim(\mathrm{SU}(2))$,
the co-dimension of the non-abelian weak sector inside $\mathrm{SU}(3)$.
\end{proposition}

The quantity $\NIm - \Nc + 1 = 7 - 3 + 1 = 5$ counts the imaginary octonion
directions that are neither in the colour triplet $\{e_1,e_2,e_3\}$ nor the
$G_2$-fixed unit $e_7$, but also includes the anti-triplet completion.
Equivalently:
\begin{equation}
  \NIm - \Nc + 1 = \dim(\mathrm{SU}(3)) - \dim(\mathrm{SU}(2)) = 8 - 3 = 5.
\end{equation}
The up quark occupies the $\mathbf{3}$ slots ($e_1,e_2,e_3$ directions in
block $x_{12}$), which are the Fano-complement of the anti-triplet $\{e_4,e_5,e_6\}$.
The exponent $\NIm/(\NIm-\Nc+1)$ reflects all imaginary units ($\NIm$) modulo
the non-abelian weak-sector co-dimension ($\dim\,\mathrm{SU}(3) - \dim\,\mathrm{SU}(2)$).

\textbf{Verdict: STRUCTURAL (partially).}  The denominator $5 = \dim\,\mathrm{SU}(3) -
\dim\,\mathrm{SU}(2)$ has a clean Lie-group origin.  The interpretation of the
exponent as a Fano complement ratio is structurally motivated by the
up-type/$\mathbf{3}$-slot geometry.

%==========================================================================
\section{Assessment: structural vs.\ numerological}
\label{sec:assessment}
%==========================================================================

\subsection{Verdict table}

\begin{table}[ht]
\centering
\caption{Verdict for each conjecture.  ``Structural'' = geometric derivation
exists or can be completed from established TOE objects.
``Numerological'' = good numerical fit, no geometric derivation.}
\label{tab:verdicts}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{llcccl}
\toprule
Conj. & Particle & $E$-formula & $E_{\mathrm{pred}}$ & Mass err & Verdict \\
\midrule
A & charm    & $\pi^2+\pi$                 & $13.011$ & $+1.2\%$ & \textbf{Structural} \\
B & top gap  & $\ln(\muzero)/\MU$          & $+6.202$ & $+2.0\%$ & \textbf{Numerological} \\
C & Cabibbo  & $\pi/14 = \pi/(2(h+1))$    & $0.2225$ & $-0.80\%$& \textbf{Structural}${}^*$ \\
D & down     & $\ln(\muone)/\MU$           & $5.910$  & $-2.2\%$ & \textbf{Numerological} \\
E & bottom   & $\pi^{7/3}$                 & $14.455$ & $-3.3\%$ & \textbf{Structural}${}^{**}$ \\
I & strange  & $\pi^2 - 1/7$               & $9.727$  & $+1.6\%$ & \textbf{Structural} \\
J & up       & $\pi^{7/5}$                 & $4.966$  & $+0.6\%$ & \textbf{Structural}${}^{**}$ \\
\bottomrule
\end{tabular}

\smallskip
\noindent\small
${}^*$ Conditional on the half-alcove CKM postulate (\S\ref{sec:cabibbo}).\\
${}^{**}$ Conditional on the sector-energy interpretation of $\pi$ from Addendum~36.
\end{table}

\subsection{The two numerological residuals}

\textbf{Conjecture B (top gap).}
The formula $E_t - E_c = \ln(\muzero)/\MU$ is numerically excellent (gap error
$0.16\%$, mass error $+2.0\%$), but the appearance of $\ln(\muzero)/\MU$ lacks
a $G_2/\JO$ derivation.  The most natural geometric candidate is the
$\mathbb{Z}_3$-action eigenvalue on the third-generation diagonal entry
$\alpha_3$ of $\JO$; if that eigenvalue equals $\muzero$, the gap follows.
This would require the trace of the $\mathbb{Z}_3$-action on $\JO$ to be
related to $\muzero = \alpha^{-1}$, which is a natural conjecture but
unproven.

\textbf{Conjecture D (down).}
The down quark mass is poorly constrained by QCD ($m_d = 4.70^{+0.50}_{-0.30}$
MeV, $\sim 10\%$ uncertainty).  The formula $\ln(\muone)/\MU$ uses $\muone$,
the first moment of $\rho$.  A potential orbit-theoretic reading: the down
quark is the lightest colour-charged fermion, and its energy is set by the
``typical'' scale of the TOE density (its center-of-mass $\muone$) rather
than a discrete sector integral.  No proof exists.

\subsection{Residual errors and running corrections}

The bottom quark mass error ($-3.3\%$, or $-140$ MeV) and the top quark
error ($+2.0\%$, or $+3.4$ GeV) are the largest residuals.  Both lie outside
the PDG $1\sigma$ experimental uncertainties.  However:
\begin{enumerate}[noitemsep]
  \item The TOE energy $E$ is defined at a \emph{geometric scale}, not
    the $\overline{\mathrm{MS}}$ scheme at $2\;\text{GeV}$.
  \item QCD running from the geometric decoupling scale to
    $2\;\text{GeV}$ (for $b$) or the pole mass (for $t$) introduces
    corrections of order $\alpha_s \sim 0.1$--$0.3$, which correspond
    to $\sim 1$--$5\%$ mass shifts.
  \item For the bottom quark, the $\overline{\mathrm{MS}}$ mass at $m_b$
    vs.\ at $2\;\text{GeV}$ differs by $\sim 2\%$, consistent with the
    $-3.3\%$ residual.
\end{enumerate}
A one-loop RG correction to $E_b$ and $E_t$ is the most natural next step,
analogous to the Weinberg angle running analysed in Addendum~42.

%==========================================================================
\section{New conjecture: the $G_2$ orbit lattice}
\label{sec:lattice}
%==========================================================================

\subsection{The Fano-locked mass lattice}

The numerical computations reveal a striking pattern in the $\pi$-power
lattice $\{\pi^{n/\Nc} : n \in \mathbb{Z}_{>0}\}$:

\begin{equation}
  \pi^{3/3} = \pi = E_e,\qquad
  \pi^{6/3} = \pi^2 = E_\mu,\qquad
  \pi^{7/3} = E_b.
\end{equation}

The three occupied values are at $n = \Nc = 3$, $n = 2\Nc = 6$, and
$n = \NIm = 7$.  We identify these as the \emph{Fano-locked masses}:

\begin{conjecture}[Fano-locked mass lattice]
\label{conj:fano-lattice}
The $G_2$ orbit lattice $\mathcal{L} = \{\pi^{n/\Nc} : n \in \mathbb{Z}_{>0}\}$
has exactly three physically occupied nodes at $n \in \{\Nc, 2\Nc, \NIm\}$,
corresponding to the electron, muon, and bottom quark energies respectively:
\begin{align}
  n = \Nc   &:\quad E = \pi^1 = E_e  &&(\text{one colour triplet = }S^1\text{ sector}),\\
  n = 2\Nc  &:\quad E = \pi^2 = E_\mu &&(\text{two triplets = }S^3\text{ sector}),\\
  n = \NIm  &:\quad E = \pi^{7/3} = E_b &&(\text{full }G_2\text{ orbit on }\mathrm{Im}(\OO)).
\end{align}
These are the three $\mathrm{SU}(3)_c$-orbit boundary dimensions in
$\mathrm{Im}(\OO)$: one complex colour triplet ($n=3$), the full colour
triplet in $\C^3$ ($n=6$), and the complete imaginary space ($n=7$).
All other nodes of $\mathcal{L}$ are unoccupied.
\end{conjecture}

\subsection{Structural interpretation}

The lattice $\mathcal{L}$ encodes how the $\mathrm{SU}(3)_c$ action builds up
the imaginary octonion space:
\begin{equation}
  \{e_1,e_2,e_3\} \hookrightarrow \{e_1,\ldots,e_6\} \hookrightarrow
  \{e_1,\ldots,e_6,e_7\},
\end{equation}
with dimensions $3 \to 6 \to 7$ corresponding to $n = 3 \to 6 \to 7$.
At each stage a new structural boundary is crossed: one colour triplet
($S^1$ edge sector, electron), the full colour space ($S^3$ boundary
sector, muon), and the $G_2$-singular direction $e_7$ (bottom quark,
first particle requiring the full $G_2$ symmetry).

The remaining quarks and leptons lie \emph{off} the lattice, at
energies derived from Fano complements ($\pi^{7/5}$ for up, $\pi^2 - 1/7$
for strange) or from sector integrals and moment logarithms.  The Fano-locked
lattice thus identifies the three \emph{maximally symmetric} mass values:
those where the energy is a pure $G_2$-orbit boundary invariant.

\subsection{The tau lepton anomaly}

Notably, the tau lepton energy $E_\tau \approx 13.419$ (Addendum~36) does
\emph{not} sit on $\mathcal{L}$.  The nearest lattice point is
$\pi^{7/3} = 14.455$ (bottom).  This is consistent with the tau's origin
from the self-lensing fold-map at the $B^4/S^3$ boundary (Addendum~36),
a mechanism entirely distinct from the Fano-locked lattice.  The tau
is the \emph{only} charged lepton off the lattice, matching the physical
fact that it is the only lepton heavy enough to decay into hadrons.

%==========================================================================
\section{Summary and open problems}
\label{sec:summary}
%==========================================================================

\subsection{Summary of verdicts}

Of the seven tests carried out:
\begin{itemize}[noitemsep]
  \item \textbf{5 structural}: Conjectures~A (charm), C (Cabibbo), E (bottom),
    I (strange), J (up).  Each has a geometric identification with a specific
    $G_2$, $\JO$, or Fano-plane object.
  \item \textbf{2 numerological}: Conjectures~B (top gap) and D (down quark).
    Both fit PDG within $2.2\%$ in mass; neither has a geometric derivation.
  \item \textbf{0 dead-ends}: all conjectures are numerically consistent.
\end{itemize}

The key structural identifications are:
\begin{enumerate}[noitemsep]
  \item $E_c = \pi^2+\pi$ is the exact non-bulk sector integral of $\rho(x)$
    (Proposition~\ref{prop:charm}).
  \item $\thetaC = \pi/14 = \pi/(2(h_{G_2}+1))$ is the half-alcove of $G_2$
    (Proposition~\ref{prop:cabibbo}).
  \item $E_s = E_\mu - 1/\NIm$ is a one-Fano-unit descent in $\JO$
    (Proposition~\ref{prop:strange}).
  \item The exponents $7/3$ and $7/5$ are $G_2$-subgroup dimension ratios
    (Propositions~\ref{prop:bottom},~\ref{prop:up}).
  \item The $G_2$ orbit lattice $\pi^{n/3}$ at $n = 3, 6, 7$ gives the
    electron, muon, and bottom as Fano-locked masses
    (Conjecture~\ref{conj:fano-lattice}).
\end{enumerate}

\subsection{Open problems}

\begin{description}[noitemsep]
  \item[OP-B] Derive the top/charm gap $\ln(\muzero)/\MU$ from the
    $\mathbb{Z}_3$-action on $\JO$.  The $\mathbb{Z}_3$ clock decomposition
    (\S\ref{sec:Z3-decomp}) places the top in $V_1$ and the charm in
    $V_{\omega^2}$; the conjecture reduces to showing the Jordan
    triple product intertwiner norm $\|\Phi: V_{\omega^2}\to V_1\|^2 = \muzero$
    (OP-B', \S\ref{sec:BD-open}).
  \item[OP-D] Derive the down quark energy $\ln(\muone)/\MU$ from the
    moment structure of $\rho$.  By the $G_0$--$G_1$ identity
    (\S\ref{sec:Gk}), OP-D reduces to OP-B plus showing $-\ln\MU/\MU$
    is a $\JO$ invariant (OP-D', \S\ref{sec:BD-open}).
  \item[OP-E2] Complete the bottom quark structural argument by deriving
    the base $\pi$ (rather than $e$ or another constant) as the natural
    exponential base for the $G_2$ orbit lattice.  This is the same
    derivation needed for $E_e = \pi$ and follows from Addendum~36
    conditionally.
  \item[OP-C2] Prove the half-alcove CKM postulate: mixing angles equal
    half the affine Weyl alcove of the structure group.  Extend from
    Cabibbo to the full CKM matrix.
  \item[OP-RG] Compute one-loop RG corrections to $E_b$ and $E_t$ from
    the geometric decoupling scale to the experimental scheme, to
    account for the $-3.3\%$ and $+2.0\%$ residuals.
\end{description}

%==========================================================================

%==========================================================================
\section{Deeper structural investigation of Conjectures~B and~D}
\label{sec:BD-investigation}
%==========================================================================

This section extends the analysis of Conjectures~B and~D beyond the initial
assessment of Section~\ref{sec:assessment}.  We apply three new tools:
(i)~reframing in terms of mass ratios rather than energy gaps,
(ii)~the $\mathbb{Z}_3$ clock decomposition of $\JO$, and
(iii)~the unified moment-logarithm structure $G_k = \ln(\mu_k)/\MU$.

\subsection{Conjecture~B reframed: $m_t/m_c = \alpha^{-1}$}
\label{sec:B-reframed}

The top/charm energy gap $E_t - E_c = \ln(\muzero)/\MU$ is equivalent,
via the TOE mass formula \eqref{eq:mass}, to the mass-ratio statement
\begin{equation}\label{eq:B-mass}
  \frac{m_t}{m_c} = \exp\!\bigl(\MU(E_t - E_c)\bigr)
  = \exp\!\bigl(\ln\muzero\bigr) = \muzero = \alpha^{-1}.
\end{equation}
This is an \emph{exact algebraic identity} given the formula for $E_t$:
if $E_t$ is defined by Conjecture~B, then $m_t/m_c = \muzero$ holds
with no approximation.  The conjecture is therefore precisely:

\begin{conjecture}[Conjecture~B reframed]\label{conj:B-mass}
The top-to-charm quark mass ratio equals the fine-structure constant inverse:
\begin{equation}
  m_t / m_c = \muzero = \alpha^{-1} = 4\pi^3 + \pi^2 + \pi.
\end{equation}
\end{conjecture}

\noindent
\textbf{Numerical status.}  With PDG~2024 values:
\[
  (m_t/m_c)_{\mathrm{PDG}} = 172760/1275 = 135.50 \quad(\text{pole}/\overline{\mathrm{MS}}),
  \qquad
  \muzero = 137.036.
\]
The discrepancy is $+1.1\%$ in the mass ratio, corresponding to the $+1.9\%$
error in $m_t$ already tabulated in Section~\ref{sec:numerical}.  Using the
$\overline{\mathrm{MS}}$ top mass $m_t^{\overline{\mathrm{MS}}} \approx 162.5\;\text{GeV}$
instead of the pole mass gives a ratio of $127.45$ and a discrepancy of $+7.5\%$,
which is worse.  The pole-mass comparison is therefore more natural.

\subsection{$\mathbb{Z}_3$ clock decomposition of $\JO$}
\label{sec:Z3-decomp}

The generational $\mathbb{Z}_3$ symmetry of $\JO$ is best described by the
\emph{clock action}: the generator $T$ acts on a $\JO$ matrix
$X = (\alpha_k, x_{ij})$ by the phase rule
\begin{equation}\label{eq:T-clock}
  T(X)_{ij} = \omega^{i-j}\, X_{ij}, \qquad \omega = e^{2\pi i/3}.
\end{equation}
Under this action, the complexification $\JO \otimes_\R \C$ decomposes as

\begin{proposition}[$\mathbb{Z}_3$ clock decomposition]\label{prop:Z3-decomp}
The clock generator $T$ acting on $\JO \otimes_\R \C$ ($27_\C$-dimensional)
has the eigenspace decomposition
\begin{equation}\label{eq:Z3-split}
  \JO \otimes \C \;=\; V_1 \;\oplus\; V_\omega \;\oplus\; V_{\omega^2},
\end{equation}
with dimensions
\begin{align}
  \dim V_1       &= 3 \quad (\alpha_1, \alpha_2, \alpha_3),\label{eq:V1}\\
  \dim V_\omega  &= 8 \quad (x_{13}\text{ off-diagonal block}),\label{eq:Vomega}\\
  \dim V_{\omega^2} &= 16 \quad (x_{12} \oplus x_{23}\text{ off-diagonal blocks}).\label{eq:Vomega2}
\end{align}
\end{proposition}

\begin{proof}
For the diagonal entries $\alpha_k$ (real scalars): $T(\alpha_k)_{kk} = \omega^{k-k}\alpha_k = \alpha_k$,
so all diagonals have eigenvalue $1$.  For the off-diagonal block $x_{12}$:
$T(x_{12})_{12} = \omega^{1-2} x_{12} = \omega^{-1}x_{12} = \omega^2 x_{12}$
(using $\omega^{-1} = \omega^2$).  Similarly $x_{23} \to \omega^{2-3}x_{23} = \omega^2 x_{23}$.
For $x_{13}$: $T(x_{13})_{13} = \omega^{1-3}x_{13} = \omega^{-2}x_{13} = \omega x_{13}$.
Since each octonion entry contributes $8_\C$ dimensions,
the total dimensions are as stated.
\end{proof}

\noindent
\textbf{Quark slot assignments.}  From the $\JO$ slot geometry
(Addenda~38, 41):
\begin{itemize}[noitemsep]
  \item The \emph{top quark} sits in $\alpha_3 \in V_1$ (diagonal, eigenvalue $1$).
  \item The \emph{charm quark} sits in $x_{12} \in V_{\omega^2}$ (off-diagonal,
    eigenvalue $\omega^2$).
  \item The \emph{up quark} sits in $x_{12}$ triplet directions, also in $V_{\omega^2}$.
  \item The \emph{bottom quark} sits in $x_{23} \in V_{\omega^2}$.
  \item The \emph{strange quark} sits in $x_{13} \in V_\omega$.
  \item The \emph{down quark} sits in $x_{12}$ anti-triplet directions, in $V_{\omega^2}$.
\end{itemize}
The \emph{diagonal} entries $\alpha_1, \alpha_2, \alpha_3$ form the $V_1$ subspace
and carry the charged-lepton sector; quarks occupy $V_\omega \cup V_{\omega^2}$.
In particular, the top and charm quarks sit in \emph{different} $T$-eigenspaces
($V_1$ and $V_{\omega^2}$ respectively), while the down quark also sits in $V_{\omega^2}$.

\subsection{The $V_1$/$V_{\omega^2}$ sector ratio and Conjecture~B}
\label{sec:B-sector-ratio}

The $\mathbb{Z}_3$ decomposition \eqref{eq:Z3-split} gives the sector dimensions
\begin{equation}
  \dim V_{\omega^2} / \dim V_1 = 16/3, \qquad
  \dim V_\omega / \dim V_1 = 8/3.
\end{equation}
These are \emph{not} equal to $\muzero \approx 137$.  The clock decomposition
therefore does \emph{not} produce $\muzero$ as a dimension ratio.

A weaker structural hook does exist, however.  The charm quark ($V_{\omega^2}$,
eigenvalue $\omega^2$) and the top quark ($V_1$, eigenvalue $1$) are separated
by a $T$-eigenvalue ratio of $\omega^{-2} = \omega$.  For the mass ratio to equal
$\muzero$, one would need:
\begin{equation}\label{eq:B-hook}
  \frac{m_t}{m_c} = \exp\!\bigl(\mathrm{energy\ distance\ from\ }V_{\omega^2}\text{ to }V_1\bigr)
  = \muzero.
\end{equation}
This ``energy distance'' must be $\ln\muzero / \MU$ in $E$-space,
or equivalently $\ln\muzero$ in $\MU$-units.  Structurally, this
would follow from a theorem of the form:

\begin{remark}[Open — OP-B extended]\label{conj:B-hook}
Let $T$ be the $\mathbb{Z}_3$ clock generator on $\JO$, and let
$\Phi: V_{\omega^2} \to V_1$ be the natural $T$-intertwiner
(the restriction of the $\JO$ bilinear to off-diagonal$\times$diagonal sectors).
Then $\|\Phi\|^2 = \muzero$ in the $\JO$ operator norm induced by
the trace form $\mathrm{Tr}(X^2)$.
\end{remark}

\noindent
This would make $m_t/m_c = \muzero$ a consequence of the
$\mathbb{Z}_3$-intertwiner norm in $\JO$.  The proof would require:
(a)~defining $\Phi$ explicitly as a bilinear involving the $\JO$ Jordan product,
(b)~computing $\|\Phi\|^2$ using the $\JO$ trace form, and
(c)~showing this equals $\muzero = \int_0^1 \rho(x)\,dx$.
Step~(c) requires connecting the algebraic norm to the TOE monad density,
which is the same open problem as for all $\rho$-dependent formulas.

\textbf{Verdict (updated).}  Conjecture~B remains \textbf{numerological} but
now has a sharper structural target: the $\mathbb{Z}_3$-intertwiner
norm $\|\Phi: V_{\omega^2} \to V_1\|^2$ in $\JO$.  If this norm equals
$\muzero$, the conjecture becomes structural.

\subsection{The unified $G_k$ moment spectrum}
\label{sec:Gk}

Define the \emph{moment-logarithm spectrum}
\begin{equation}\label{eq:Gk}
  G_k \;=\; \frac{\ln\mu_k}{\MU}, \qquad
  \mu_k = \int_0^1 x^k\,\rho(x)\,dx, \quad k = 0, 1, 2, \ldots
\end{equation}
so that $G_0 = \ln(\muzero)/\MU$ (the Conj.~B gap) and
$G_1 = \ln(\muone)/\MU$ (the Conj.~D energy).
Numerical values of the first six terms:

\begin{table}[ht]
\centering
\caption{Moment-logarithm spectrum $G_k = \ln\mu_k/\MU$ and raw moments $\mu_k$.}
\label{tab:Gk}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{cccc}
\toprule
$k$ & $\mu_k = \int_0^1 x^k\rho\,dx$ & $G_k$ & Physical identification \\
\midrule
0 & $137.036 = \muzero$ & $6.2019$ & $E_t - E_c$ (Conj.~B) \\
1 & $108.717 = \muone$  & $5.9101$ & $E_d$ (Conj.~D) \\
2 & $90.176$            & $5.6744$ & --- \\
3 & $77.063$            & $5.4764$ & --- \\
4 & $67.290$            & $5.3054$ & --- \\
5 & $59.721$            & $5.1550$ & --- \\
\bottomrule
\end{tabular}
\end{table}

\noindent
The key structural identity relating consecutive terms is:
\begin{equation}\label{eq:G-shift}
  G_{k+1} - G_k \;=\; \frac{\ln(\mu_{k+1}/\mu_k)}{\MU}
  \;=\; \frac{\ln(\muone/\muzero)}{\MU} + O(\delta)
  \;=\; \frac{\ln\MU}{\MU} + O(\delta),
\end{equation}
where the dominant step between $G_0$ and $G_1$ is exact:
\begin{equation}\label{eq:G01-diff}
  G_0 - G_1 = \frac{\ln(\muzero/\muone)}{\MU}
  = -\frac{\ln\MU}{\MU}
  \approx 0.2918.
\end{equation}
This identity is algebraic: it holds with no approximation.

\textbf{Structural consequence.}
If \emph{either} Conjecture~B or Conjecture~D can be given a
structural derivation, the other follows from \eqref{eq:G01-diff}
together with the independently derivable relation
\begin{equation}\label{eq:G01-derivable}
  G_0 - G_1 = -\frac{\ln\MU}{\MU},
\end{equation}
\emph{provided} $-\ln\MU/\MU$ has its own structural derivation.
The quantity $-\ln\MU/\MU$ is the \emph{Bregman divergence} of $\muone$
from $\muzero$ rescaled by $\MU^{-1}$:
\begin{equation}\label{eq:bregman}
  -\frac{\ln\MU}{\MU}
  = -\frac{\ln(\muone/\muzero)}{\muone/\muzero}
  = \frac{\muzero}{\muone}\,\ln\!\left(\frac{\muzero}{\muone}\right).
\end{equation}
This is the function $f(t) = -t^{-1}\ln t$ evaluated at $t = \MU = \muone/\muzero$.
Since $f(t)$ is the derivative of the entropy function at $t$, there is a mild
information-theoretic hook: $-\ln\MU/\MU$ is the ``self-information''
of the event $\{x = \MU\}$ under the distribution $\rho(x)/\muzero$,
scaled by $\muzero/\muone$.  However, this hook remains interpretive
rather than geometric.

\subsection{Conjecture~D: moment-theoretic interpretation}
\label{sec:D-moment}

Conjecture~D can be stated equivalently as a condition on $m_d$:
\begin{equation}\label{eq:D-mass}
  \frac{m_d}{m_e}
  = \exp\!\bigl(\MU E_d - \MU\pi\bigr)
  = \exp\!\bigl(\ln\muone - \MU\pi\bigr)
  = \frac{\muone}{\exp(\MU\pi)}.
\end{equation}
This is the \emph{first moment of $\rho$, deflated by the exponential
of $\MU\pi$}.  The down quark mass is therefore:
\begin{equation}\label{eq:D-mass2}
  m_d = m_e \cdot \frac{\muone}{\exp(\MU\pi)}.
\end{equation}

\textbf{Structural interpretation attempt.}
The quantity $\exp(\MU\pi)$ is the ``zero-energy'' mass scale in the TOE:
$\exp(\MU \cdot \pi) = m_{E=\pi}/m_e$ (the mass at the \emph{edge-sector
energy} $E = \pi$, one step below the electron mass in the sector-energy
hierarchy).  Rewriting:
\begin{equation}
  m_d = m_e \cdot \muone \cdot \exp(-\MU\pi)
  = \frac{\muone}{m_{E=\pi}/m_e} \cdot m_e
  = \frac{\muone \cdot m_e^2}{m_{E=\pi}}.
\end{equation}
This says: the down quark mass is the first moment of $\rho$,
measured in units of ``one electron mass below the electron scale.''

At inverse temperature $\beta = \MU$ (the natural TOE temperature), the
Boltzmann weight of the energy $E_d = G_1$ is:
\begin{equation}
  e^{-\MU G_1} = e^{-\ln\muone} = \frac{1}{\muone}.
\end{equation}
So the Boltzmann weight of the down quark energy at $\beta = \MU$
is the \emph{reciprocal of the first moment of $\rho$}.
Similarly, $e^{-\MU G_0} = 1/\muzero$: the Boltzmann weight of the top-charm
gap at $\beta = \MU$ is the fine-structure constant.
Both facts hold by construction (they are algebraic identities),
but they reveal that the natural inverse temperature of the TOE mass
formula ($\beta = \MU = \muone/\muzero$) \emph{selects} the moment-reciprocals
as Boltzmann weights for the two conjectured energies.

\textbf{Verdict (updated).}  Conjecture~D remains \textbf{numerological}
(mass error $-2.2\%$, well within QCD scheme uncertainty for light quarks).
The formula $E_d = G_1$ is moment-theoretically natural and is coupled to
Conjecture~B by the exact identity \eqref{eq:G01-diff}.  If either formula
is given a structural derivation, the other will follow once
$-\ln\MU/\MU$ is derived as a $\JO$ invariant.

\subsection{Updated open problems}
\label{sec:BD-open}

The investigations above sharpen the original open problems OP-B and OP-D
of Section~\ref{sec:summary} to:

\begin{description}
  \item[OP-B'] (\emph{$\mathbb{Z}_3$-intertwiner norm}).
    In $\JO$, let $\Phi: V_{\omega^2} \to V_1$ be the Jordan product
    $\Phi(y) = \{\alpha_3, y, \alpha_3\}$ (the Jordan triple product
    restricted to $V_{\omega^2} \ni y$, $V_1 \ni \alpha_3$).
    Does $\|\Phi\|^2_{\mathrm{Tr}} = \muzero$?
    If yes, Conjecture~B is structural.

  \item[OP-D'] (\emph{Reduction to OP-B'}).
    Given OP-B', does $-\ln\MU/\MU$ arise as a natural $\JO$ invariant
    (e.g., the difference of log-norms of $V_{\omega^2}$ and $V_\omega$
    under the $\JO$ trace form)?  If yes, Conjecture~D is structural.

  \item[OP-B''] (\emph{$m_t/m_c = \alpha^{-1}$ via CKM structure}).
    An independent route: the top/charm ratio appears in the CKM element $V_{cb}$.
    Does the $G_2$ half-alcove structure (which gives $\theta_C$ from
    Proposition~\ref{prop:cabibbo}) also constrain $|V_{cb}|$ in a way
    that forces $m_t/m_c = \alpha^{-1}$ via the GIM mechanism?
    This would be a structural derivation via flavor physics rather than
    $\JO$ geometry.
\end{description}

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\end{document}
