% Addendum 38: Quark Sector and Neutrino Structure from J_3(O) Geometry
% L. F. Vlegels, Independent Researcher

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\title{Quark Sector and Neutrino Structure from $J_3(\mathbb{O})$ Geometry\\[0.5em]
\large Addendum 38 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
We extend the TOE corpus mass derivations to the quark sector and neutrino structure,
using the geometry of $J_3(\OO)$ (the Albert algebra) established in Addendum~37.
The mass formula $m/m_e = \exp(\mathrm{MU} \cdot (E - \pi))$ with
$\mathrm{MU} = \mu_1/\mu_0 = 0.7933$ and quark energies derived from PDG masses
reveals three structural relationships.
\textbf{Conjecture~A}: the charm quark energy $E_c = 12.996$ coincides with the
non-bulk sector sum $\mu_0^{\mathrm{sub}} = \pi^2 + \pi = 13.011$ (difference $-0.015$),
predicting $m_c = 1285$~MeV against PDG $1270$~MeV ($+1.2\%$).
\textbf{Conjecture~B}: the top/charm mass ratio $m_t/m_c = 135.98 \approx \mu_0 = \alpha^{-1} = 137.036$
(error $-0.77\%$), arising because the energy gap $E_t - E_c = \ln(\mu_0)/\mathrm{MU}$.
\textbf{Conjecture~C}: the Cabibbo angle $\theta_C = \pi/\dim(G_2) = \pi/14$, giving
$\sin(\pi/14) = 0.22252$ against PDG $|V_{us}| = 0.22431$ (error $-0.80\%$).
For the neutrino sector, the $\mathbb{Z}_3$ symmetry of $J_3(\OO)$ gives tribimaximal
mixing as the exact fixed point, in agreement with $\sin^2\theta_{12}$ and
$\sin^2\theta_{23}$ to $\sim 8\%$.
A Phase~1.5 analysis proves that the muon energy $E_\mu = \pi^2$ lies entirely below
the self-lensing range $[13.177, 13.617]$ and cannot be derived from an $S^3/S^1$
sub-fold balance; deriving $m_\mu$ requires a new fold formalism extending Paper~04.
\end{abstract}

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

\subsection{Corpus context}

The TOE is grounded in the monad density
\begin{equation}\label{eq:rho}
\rhoTOE, \qquad x \in [0,1],
\end{equation}
whose integral gives $\muzero = 4\pi^3 + \pi^2 + \pi = \alpha^{-1} = 137.036$,
and whose first moment gives $\muone = 16\pi^3/5 + 3\pi^2/4 + 2\pi/3 = 108.717$.
The moment ratio $\mathrm{MU} = \muone/\muzero = 0.7933$ enters every mass formula.

The charged lepton mass sector is now fully derived:
the electron energy $E_e = \pi$ is the edge layer integral, the muon energy $E_\mu = \pi^2$
is the boundary layer integral, and the tau mass was derived in Addendum~36 via the self-lensing
fold-map at the $B^4/S^3$ boundary, with $\lambda_\tau = \mu_1/(\mu_0 + \mu_1) = 0.4424$
and $E_\tau = 13.419$.
Addendum~37 derived the Standard Model gauge group $SU(3)_c \times SU(2)_L \times U(1)_Y$
from the division algebra tower $\R \subset \C \subset \HH \subset \OO$ and established
$J_3(\OO)$ as the home of three generations and color degrees of freedom.

The present paper addresses four open items left by Addendum~37:
\begin{itemize}
\item the muon sub-fold conjecture (Phase~1.5, \S\ref{sec:muon-subfold});
\item neutrino mixing from $\mathbb{Z}_3$ and absolute mass scale (Phase~3, \S\ref{sec:neutrinos});
\item quark mass relationships from sector geometry (Phase~4, \S\ref{sec:charm}--\S\ref{sec:cabibbo}).
\end{itemize}

\subsection{Constants and notation}

We use throughout:
\begin{align}
\muzero &= 4\pi^3 + \pi^2 + \pi = 137.036, \label{eq:mu0}\\
\muone  &= \tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3} = 108.717, \label{eq:mu1}\\
\mathrm{MU} &= \muone/\muzero = 0.7933, \label{eq:MU}\\
\mu_0^{\mathrm{sub}} &= \pi^2 + \pi = 13.011, \label{eq:mu0sub}\\
\mu_1^{\mathrm{sub}} &= \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3} = 9.497. \label{eq:mu1sub}
\end{align}
The quantity $\mu_0^{\mathrm{sub}}$ is the integral of the non-bulk sector
$3\pi^2 x^2 + 2\pi x$ over $[0,1]$ (boundary + edge layers only).
The quantity $\mu_1^{\mathrm{sub}}$ is its first moment.

The quark energy for a quark of mass $m$ is defined by the same mass formula as
for the charged leptons:
\begin{equation}\label{eq:energy-def}
E_q = \pi + \frac{\ln(m_q/m_e)}{\mathrm{MU}},
\qquad m_e = 0.51099895\;\text{MeV}.
\end{equation}
All quark masses are PDG~2024 values (MS-bar at $2\;\text{GeV}$, except the
top quark at the pole mass $m_t$).

\subsection{Summary of results}

\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Result} & \textbf{Status} & \textbf{Ref} & \textbf{Error} \\
\midrule
Muon sub-fold failure & Proven dead-end & Prop.~\ref{prop:muon-subfold} & structural gap \\
Muon from cruciform $C_2$ & Structural ID & Prop.~\ref{prop:cruciform} & $E_\mu=\pi^2$ exact \\
Neutrino oscillation from $\mathbb{Z}_3$ & Consistent & Prop.~\ref{prop:neutrino-z3} & $\sim 8\%$ \\
Absolute neutrino mass (Type~1 gap) & Open & Prop.~\ref{prop:neutrino-mass} & --- \\
Down at $\ln\mu_1/\mathrm{MU}$ & Conjecture~D & Conj.~\ref{conj:down} & $+0.34\%$ \\
Charm at $\mu_0^{\mathrm{sub}}$ & Conjecture~A & Conj.~\ref{conj:charm} & $+1.2\%$ \\
$m_t/m_c = \alpha^{-1}$ & Conjecture~B & Conj.~\ref{conj:topcharm} & $-0.77\%$ \\
Cabibbo $= \pi/14$ & Conjecture~C & Conj.~\ref{conj:cabibbo} & $-0.80\%$ \\
\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\section{Phase 1.5: Muon Sub-Fold Analysis}\label{sec:muon-subfold}
%==========================================================================

\subsection{Setup}

Addendum~37 Remark~\ref{rem:muon-structural} (carried over from Addendum~36) proposed
that the muon mass might follow from a fold-map balance on the $S^3/S^1$ sub-boundary,
analogous to how the tau mass follows from the $B^4/S^3$ fold.  We prove here that this
approach fails.

The self-lensing energy function (Paper~04 \S7, Addendum~36 \S8) is
\begin{equation}\label{eq:eself}
E_{\mathrm{self}}(\lambda) = (1-\lambda)\,E_{\mathrm{low}} + \lambda\,E_{\mathrm{high}},
\qquad \lambda \in [0,1],
\end{equation}
where the endpoints bracket the self-lensing range.  For the full $B^4/S^3$ fold:
$E_{\mathrm{low}} = 13.177$, $E_{\mathrm{high}} = 13.617$, so
$E_{\mathrm{self}} \in [13.177, 13.617]$ for all $\lambda \in [0,1]$.

The tau derivation uses the fixed-point weight
\begin{equation}\label{eq:lambda-tau}
\lambda_\tau = \frac{\muone}{\muzero + \muone} = \frac{108.717}{137.036 + 108.717} = 0.4424,
\end{equation}
yielding $E_{\mathrm{self}}(\lambda_\tau) = 13.419$, which reproduces
$m_\tau/m_e = \exp(\mathrm{MU} \cdot (13.419 - \pi))$ to $< 0.04\%$.

\subsection{Sub-fold balance candidate}

Define the $S^3/S^1$ sub-fold weight by analogy, replacing $(\muzero, \muone)$
with $(\mu_0^{\mathrm{sub}}, \mu_1^{\mathrm{sub}})$:
\begin{equation}\label{eq:lambda-sub}
\lambda_{\mathrm{sub}} = \frac{\mu_1^{\mathrm{sub}}}{\mu_0^{\mathrm{sub}} + \mu_1^{\mathrm{sub}}}
= \frac{9.497}{13.011 + 9.497} = \frac{9.497}{22.508} = 0.422.
\end{equation}

\begin{proposition}[Sub-fold failure]\label{prop:muon-subfold}
The muon energy $E_\mu = \pi^2 \approx 9.870$ cannot be derived from the sub-fold
balance condition $E_{\mathrm{self}}(\lambda_{\mathrm{sub}})$ applied to the $S^3/S^1$ boundary.
\end{proposition}

\begin{proof}
The self-lensing function $E_{\mathrm{self}}$ maps $[0,1]$ into the range
$[13.177, 13.617]$ (Paper~04 \S7).  This range is entirely above $E_\mu = \pi^2 \approx 9.870$.
Evaluating at the sub-fold weight $\lambda_{\mathrm{sub}} = 0.422$ gives
\begin{equation}
E_{\mathrm{self}}(0.422) = (1 - 0.422) \times 13.177 + 0.422 \times 13.617 = 13.36,
\end{equation}
which is in the tau-mass region, not the muon-mass region.  Concretely:
\begin{equation}
\frac{m_{\mathrm{pred}}}{m_e} = \exp(\mathrm{MU} \cdot (13.36 - \pi))
= \exp(0.7933 \times 10.22) = \exp(8.106) \approx 3300,
\end{equation}
whereas $m_\mu/m_e = 206.77$.  The predicted mass is $\sim 1690$~MeV, of order the
tau-mass scale ($m_\tau/m_e = 3477$), not the muon scale.

No choice of $\lambda \in [0,1]$ can produce $E_{\mathrm{self}}(\lambda) = 9.870$
because $9.870 < 13.177 = \min E_{\mathrm{self}}$.  The self-lensing mechanism is
inoperative at the muon energy.
\end{proof}

\begin{remark}[Why the approach fails structurally]
The muon energy $\pi^2$ lies in the \emph{boundary layer sector} of $\rho(x)$, at
$\int_0^1 3\pi^2 x^2\,dx = \pi^2$.  Self-lensing in Paper~04 operates at the
$B^4/S^3$ boundary by balancing bulk and boundary energies, which are both above
$4\pi$.  The boundary-to-edge sub-fold operates at $S^3/S^1$, where one sector
energy is $\pi^2$ and the other is $\pi$ — a ratio of $\pi$, not of order 1.
No self-lensing in this sub-fold can produce an energy as low as $\pi^2$ from
above; the mechanism would need to work \emph{from below}, requiring a new
formalism.
\end{remark}

\begin{remark}[Open problem: muon from $S^3/S^1$ fold]\label{rem:muon-open}
The muon mass problem requires extending the Paper~04 fold-map to the $S^3/S^1$
sub-boundary.  The fixed-point condition
\begin{equation}
\lambda_\mu = \frac{E_\mu}{E_e + E_\mu} = \frac{\pi^2}{\pi + \pi^2} = \frac{\pi}{1+\pi} \approx 0.759
\end{equation}
(Addendum~37 Remark~\ref{rem:muon-structural}) is the natural candidate for such
a fixed point, but the function $E_{\mathrm{self}}^{(S^3)}(\lambda)$ for the
boundary-to-edge fold does not exist in the current corpus.  Its derivation,
starting from the sub-density $3\pi^2 x^2 + 2\pi x$ on $S^3/S^1$, is
designated \textbf{Phase~1.5} of the TOE completion roadmap.
The numerical target is:
\begin{equation}
\frac{m_\mu}{m_e} = 206.768 \quad \Longleftrightarrow \quad
E_\mu = \pi + \frac{\ln(206.768)}{\mathrm{MU}}
= \pi + 6.721 \approx \pi^2 = 9.870,
\end{equation}
which lies \emph{below} the existing $B^4/S^3$ self-lensing range $[13.177, 13.617]$.
The new $S^3/S^1$ fold mechanism must therefore produce energies in a lower range,
making it structurally distinct from the existing $B^4/S^3$ fold.
\end{remark}

%==========================================================================
\section{Cruciform Projection: Structural Identification of the Muon}\label{sec:cruciform}
%==========================================================================

\subsection{The two Clifford parallels of $S^3$}

The Hopf fibration $S^1 \hookrightarrow S^3 \xrightarrow{\pi} S^2$ determines a canonical
pair of orthogonal great circles on $S^3$, known as the \textbf{Hopf link} or Clifford parallels.
In Hopf coordinates $(\eta, \theta_1, \theta_2)$ with $\eta \in [0, \pi/2]$, these are:
\begin{align}
C_1 &= \{\eta = 0\}: \quad \text{the Hopf fiber above the north pole of } S^2,\label{eq:c1}\\
C_2 &= \{\eta = \pi/2\}: \quad \text{the orthogonal Hopf pre-image above the south pole.}\label{eq:c2}
\end{align}
$C_1$ and $C_2$ are linked with linking number $1$ (the Hopf link).
The \textbf{cruciform projection} of $S^3$ is the restriction of the fold map
$F_\Theta = \Pi_\downarrow \circ R_\Theta \circ \Pi_\uparrow$ (Paper~04 Definition~7.1)
to this pair of circles.  Since $R_\Theta$ is rotation along $C_1$, the cruciform
projection naturally distinguishes $C_1$ (rotation axis) from $C_2$ (orthogonal orbit).

\subsection{Energy assignment from $\rho(x)$ layer structure}

The volume element on $S^3$ in Hopf coordinates factors as
\begin{equation}
d\mathrm{Vol}_{S^3} = \sin\eta\,\cos\eta\,d\eta\,d\theta_1\,d\theta_2.
\end{equation}
At the two circles, the Jacobian of the projection to $x \in [0,1]$ degenerates to
the single-layer density:
\begin{equation}
J_{C_1}(x) = 2\pi x \quad (\text{edge layer}), \qquad
J_{C_2}(x) = 3\pi^2 x^2 \quad (\text{boundary layer}),
\end{equation}
matching the edge and boundary terms of $\rhoTOE$ respectively.
The Clifford torus ($\eta = \pi/4$, equidistant from $C_1$ and $C_2$) has Jacobian
$J_{\mathrm{CT}}(x) = (3\pi^2 x^2 + 2\pi x)/2$.

\begin{proposition}[Cruciform energy identification]\label{prop:cruciform}
The energies of the two Clifford parallels of $S^3$, computed from the monad density
$\rho(x)$, are
\begin{equation}
E_{C_1} = \int_0^1 2\pi x\,dx = \pi = E_e, \qquad
E_{C_2} = \int_0^1 3\pi^2 x^2\,dx = \pi^2 \approx E_\mu.
\end{equation}
The electron occupies $C_1$ (the Hopf fiber) and the muon occupies $C_2$ (the
orthogonal Clifford parallel).  The fold map $F_\Theta$ of Paper~04 rotates along
$C_1$; $C_2$ is structurally orthogonal to this rotation axis.
\end{proposition}

\begin{proof}
Direct integration: $E_{C_1} = \int_0^1 2\pi x\,dx = \pi x^2\big|_0^1 = \pi$.
$E_{C_2} = \int_0^1 3\pi^2 x^2\,dx = \pi^2 x^3\big|_0^1 = \pi^2$.
Both are exact, with no free parameters.  The layer decomposition
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ assigns the terms $3\pi^2 x^2$
and $2\pi x$ to the boundary and edge layers of $(B^4, S^3)$ respectively;
these are precisely the Jacobians of $C_2$ and $C_1$.
\end{proof}

\begin{remark}[Resolution of Phase 1.5]
Proposition~\ref{prop:cruciform} provides the structural reason why $E_\mu \approx \pi^2$:
it is the boundary-layer integral, the energy of the orthogonal Clifford parallel $C_2$.
No fold-map balance condition or perturbative correction is needed — the muon energy
is identified by the cruciform geometry of $S^3$ itself.
The mass formula $m_\mu/m_e = \exp(\mathrm{MU}\cdot(\pi^2 - \pi)) = 208.0$ overshoots
CODATA $206.77$ by $0.60\%$.  This residual arises because $\mathrm{MU} = \mu_1/\mu_0$
includes the bulk contribution $16\pi^3/5$ in $\mu_1$, while the muon lives on the
purely boundary/edge sub-manifold $C_2 \subset S^3$.  The overshoot is intrinsic to
applying the bulk-inclusive propagation factor to a boundary-only state; closing it
would require either (a) deriving an effective $\mathrm{MU}^{(S^3)}$ for propagation
restricted to $S^3$, or (b) identifying a Hopf-holonomy correction
$\delta E = E_\mu^{\mathrm{exact}} - \pi^2 = -0.007584$.
\end{remark}

\begin{remark}[The Clifford torus has no SM counterpart]
The midpoint circle at $\eta = \pi/4$ (Clifford torus) has energy
$E_{\mathrm{CT}} = (\pi + \pi^2)/2 = 6.506$, corresponding to
$m_{\mathrm{CT}}/m_e = \exp(0.7933 \times 3.365) = 14.4$, or $\approx 7.4$~MeV.
No SM particle corresponds to this energy; the Clifford torus is not a
particle state but the equipotential surface separating the electron and muon sectors.
\end{remark}

%==========================================================================
\section{Neutrino Sector}\label{sec:neutrinos}
%==========================================================================

\subsection{Oscillation from $\mathbb{Z}_3$ symmetry}

Paper~28 Proposition~6.2 gives the exact two-flavor oscillation formula:
\begin{equation}\label{eq:p28-oscillation}
P(\nu_e \to \nu_e;\, L) = \frac{1 + v(q_{\mathrm{rel}}(L))}{2},
\end{equation}
where $v$ is the Bloch-vector magnitude of the relative state $q_{\mathrm{rel}}(L)$
evolving on $S^3$ as a function of propagation length $L$.

The three-generation case requires three families, provided by the $\mathbb{Z}_3$
permutation symmetry of the $J_3(\OO)$ diagonal (Addendum~37 Proposition~\ref{prop:j3})
via $\pi_1(L(3,1)) = \mathbb{Z}_3$ (Paper~06).

\begin{proposition}[Tribimaximal mixing as $\mathbb{Z}_3$ fixed point]\label{prop:neutrino-z3}
The $\mathbb{Z}_3$-symmetric fixed point of the neutrino mixing matrix, derived from the
cyclic permutation symmetry $\alpha_1 \leftrightarrow \alpha_2 \leftrightarrow \alpha_3$
of the $J_3(\OO)$ diagonal, is the tribimaximal (TBM) mixing matrix with
\begin{equation}\label{eq:tbm}
\sin^2\theta_{12} = \frac{1}{3}, \qquad
\sin^2\theta_{23} = \frac{1}{2}, \qquad
\theta_{13} = 0.
\end{equation}
\end{proposition}

\begin{proof}
The $\mathbb{Z}_3$ cyclic permutation $T: \alpha_i \mapsto \alpha_{i+1 \bmod 3}$ generates
a symmetry group $\langle T \rangle \cong \mathbb{Z}_3$ acting on the flavor basis of
$J_3(\OO)$.  The unique PMNS matrix invariant under this cyclic permutation (up to
rephasing) is the tribimaximal matrix:
\begin{equation}
U_{\mathrm{TBM}} = \begin{pmatrix}
\sqrt{2/3} & \sqrt{1/3} & 0 \\
-\sqrt{1/6} & \sqrt{1/3} & -\sqrt{1/2} \\
-\sqrt{1/6} & \sqrt{1/3} & \sqrt{1/2}
\end{pmatrix}.
\end{equation}
Reading off the mixing angles: $|U_{e2}|^2 = 1/3$ gives $\sin^2\theta_{12} = 1/3$;
$|U_{\mu 3}|^2 = 1/2$ gives $\sin^2\theta_{23} = 1/2$;
$|U_{e3}|^2 = 0$ gives $\theta_{13} = 0$.
The derivation is equivalent to the standard group-theoretic construction of TBM
from $S_4 \supset \mathbb{Z}_3 \times \mathbb{Z}_2$; the $\mathbb{Z}_3$ here is that of
$L(3,1)$.  See \cite{Harrison2002} for the TBM matrix; the $\mathbb{Z}_3$ identification
with $J_3(\OO)$ is as in Addendum~37 Proposition~\ref{prop:j3}.
\end{proof}

\begin{remark}[Comparison with PDG 2024]
The PDG values (NuFIT~5.3, normal ordering) are:
\begin{center}
\begin{tabular}{lllll}
\toprule
\textbf{Angle} & \textbf{PDG} & \textbf{TBM} & \textbf{Difference} & \textbf{Status} \\
\midrule
$\sin^2\theta_{12}$ & $0.307$ & $1/3 = 0.333$ & $-8\%$ & TBM overestimates \\
$\sin^2\theta_{23}$ & $0.545$ & $1/2 = 0.500$ & $+9\%$ & TBM underestimates \\
$\sin^2\theta_{13}$ & $0.0218$ & $0$ & off by $\sim 0.022$ & TBM predicts zero \\
\bottomrule
\end{tabular}
\end{center}
The $\sim 8\%$ deviations from TBM are of the order expected from off-diagonal
$J_3(\OO)$ phases not yet fixed by the $\mathbb{Z}_3$ constraint alone.
In particular, $\theta_{13} \neq 0$ (the ``reactor angle'') requires a nonzero
CP-violating phase $\delta$ that presumably arises from the octonion phase structure
of $x_{ij} \in \OO$ in the off-diagonal entries of $J_3(\OO)$.  This constitutes
an open problem requiring a full representation-theoretic treatment of the $G_2$
action on the off-diagonal elements.
\end{remark}

\subsection{Absolute neutrino mass scale}

\begin{proposition}[Type~1 gap: absolute mass scale]\label{prop:neutrino-mass}
The absolute neutrino mass scale is not constrained by the density $\rho(x)$ alone.
\end{proposition}

\begin{proof}
Applying the mass formula~\eqref{eq:energy-def} to a representative neutrino mass
$m_\nu \sim 0.05$~eV (consistent with oscillation data):
\begin{equation}
E_\nu = \pi + \frac{\ln(0.05\;\text{eV} / 0.511\;\text{MeV})}{\mathrm{MU}}
= \pi + \frac{\ln(9.78 \times 10^{-8})}{0.7933}
= \pi + \frac{-16.14}{0.7933}
= 3.1416 - 20.35 = -17.2.
\end{equation}
The result $E_\nu \approx -17$ is far outside the domain $[0, \infty)$ of the
self-lensing function $E_{\mathrm{self}}$ and below the floor $E_e = \pi \approx 3.14$
set by the edge layer.  The formula predicts no quanta in the density $\rho(x)$
at such an energy — the neutrino mass is not encoded in the monad density.

This is a \textbf{Type~1 gap}: the formula breaks down because the relevant physics
(the Higgs mechanism and/or the seesaw mechanism) is not yet incorporated in $\rho(x)$.
The Yukawa coupling of neutrinos to the Higgs field would require a separate sector
of the operator $\hat{O}$ (Paper~18) not treated in the current corpus.
\end{proof}

\begin{remark}[Neutrino mass differences vs.\ absolute scale]
Although the absolute scale is a Type~1 gap, the oscillation mass \emph{differences}
$\Delta m_{21}^2$ and $\Delta m_{31}^2$ may yet be derivable from the $J_3(\OO)$
off-diagonal octonion phases $x_{ij}$.  The off-diagonal entries carry $8$ real
degrees of freedom each; the three off-diagonal entries together carry $24$ DOF
that are constrained only by $G_2 \times \mathbb{Z}_3$ symmetry.  Identifying the
mass-squared differences with geometric invariants of this action is an open problem,
designated \textbf{Phase~3.5} of the completion roadmap.
\end{remark}

%==========================================================================
\section{Charm Quark at the Non-Bulk Sector Sum}\label{sec:charm}
%==========================================================================

\subsection{Quark energies from PDG masses}

Using the mass formula~\eqref{eq:energy-def} with PDG~2024 values:
\begin{center}
\begin{tabular}{lllr}
\toprule
\textbf{Quark} & \textbf{PDG mass} & \textbf{Energy $E_q$} & \\
\midrule
$u$ & $2.16$~MeV  & $E_u = \pi + \ln(2.16/0.511)/0.7933 = 4.959$ & \\
$d$ & $4.67$~MeV  & $E_d = \pi + \ln(4.67/0.511)/0.7933 = 5.931$ & \\
$s$ & $93.4$~MeV  & $E_s = \pi + \ln(93.4/0.511)/0.7933 = 9.701$ & \\
$c$ & $1270$~MeV  & $E_c = \pi + \ln(1270/0.511)/0.7933 = 12.996$ & \\
$b$ & $4180$~MeV  & $E_b = \pi + \ln(4180/0.511)/0.7933 = 14.498$ & \\
$t$ & $172{,}690$~MeV & $E_t = \pi + \ln(172690/0.511)/0.7933 = 19.187$ & \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Down quark from first-moment logarithm}

A numerical scan of TOE energies against all six quark energies reveals one
additional strong candidate beyond the three principal conjectures.

\begin{conjecture}[Down quark at $\ln\mu_1/\mathrm{MU}$]\label{conj:down}
The down quark energy equals the logarithm of the first-moment integral $\mu_1$,
scaled by $\mathrm{MU}$:
\begin{equation}\label{eq:down-conjecture}
E_d = \frac{\ln \mu_1}{\mathrm{MU}}.
\end{equation}
Numerically: $\ln(108.717)/0.7933 = 4.689/0.7933 = 5.910$, against
$E_d^{\mathrm{PDG}} = 5.931$; error $+0.34\%$.
\end{conjecture}

\begin{remark}[Physical interpretation and relation to Conjecture~\ref{conj:topcharm}]
Conjecture~\ref{conj:topcharm} (Conjecture~B) reads $E_t - E_c = \ln\mu_0/\mathrm{MU}$,
i.e., the top/charm gap is the logarithm of the full monad integral.
Conjecture~\ref{conj:down} reads $E_d = \ln\mu_1/\mathrm{MU}$,
i.e., the down energy is the logarithm of the first-moment integral.
These two results together suggest a logarithmic ladder
$\ln\mu_0$ and $\ln\mu_1$ as intrinsic energy scales for the quark sector,
distinct from the direct integral scales $\mu_0^{\mathrm{sub}}$ used for the charm.
The ratio $\ln\mu_0/\ln\mu_1 = 4.919/4.689 = 1.049$ is close to but not exactly $1$;
the two logarithms are not simply related.  A theoretical grounding would require
identifying why the $d$ quark couples to the first-moment of $\rho(x)$
rather than its integral.
\end{remark}

\subsection{The charm--boundary coincidence}

\begin{conjecture}[Charm at non-bulk sector sum]\label{conj:charm}
The charm quark energy equals the non-bulk sector integral $\mu_0^{\mathrm{sub}} = \pi^2 + \pi$:
\begin{equation}\label{eq:charm-conjecture}
E_c = \mu_0^{\mathrm{sub}} = \pi^2 + \pi = 13.011,
\end{equation}
up to a deviation of $\delta E_c = E_c^{\mathrm{PDG}} - \mu_0^{\mathrm{sub}} = 12.996 - 13.011 = -0.015$.

The predicted charm mass is
\begin{equation}\label{eq:charm-mass}
m_c^{\mathrm{pred}} = m_e \cdot \exp\!\bigl(\mathrm{MU}\cdot(\mu_0^{\mathrm{sub}} - \pi)\bigr)
= m_e \cdot \exp\!\bigl(\mathrm{MU} \cdot \pi^2\bigr).
\end{equation}
Numerically: $\mathrm{MU} \cdot \pi^2 = 0.7933 \times 9.8696 = 7.828$, so
$m_c^{\mathrm{pred}} = 0.511 \times e^{7.828} = 0.511 \times 2513 = 1285$~MeV.
The PDG value is $1270$~MeV; the error is $+1.2\%$.
\end{conjecture}

\begin{remark}[Physical interpretation]
The non-bulk integral $\mu_0^{\mathrm{sub}} = \pi^2 + \pi$ is the total monad density
in the boundary and edge sectors, i.e., the total measure of the $S^3$ boundary plus
the $S^1$ fiber.  The charm quark sitting at this energy means it is the
\emph{quark-sector analog of the electron}: the lowest-mass quark at the interface
between the boundary and edge layers.  In the lepton sector, the electron energy
$E_e = \pi$ is the edge integral alone; the charm energy
$E_c \approx \pi^2 + \pi$ is the combined non-bulk integral.

This is structurally consistent with the $J_3(\OO)$ assignment in Addendum~37
Conjecture~\ref{conj:j3}: the charm quark belongs to the up-type doublet and is
the lowest-mass quark that can be produced in electroweak interactions,
corresponding to the boundary/edge sub-system.
\end{remark}

\begin{remark}[Closure condition]
Conjecture~\ref{conj:charm} requires showing that the charm sector energy is fixed by
a balance condition on the $S^3 \cup S^1$ sub-manifold, analogous to how
$E_e = \int_0^1 2\pi x\,dx = \pi$ is fixed by the edge layer.  The identity
$\mu_0^{\mathrm{sub}} = \int_0^1 (3\pi^2 x^2 + 2\pi x)\,dx$ is exact;
the physical argument is that the charm is the lightest quark for which color
confinement operates in the boundary/edge regime rather than the bulk.
A proof requires the complete quark-sector treatment of $J_3(\OO)$ under
$SU(3)_c$ (Addendum~37 Conjecture~\ref{conj:color}).
\end{remark}

%==========================================================================
\section{Top/Charm Mass Ratio and $\alpha^{-1}$}\label{sec:topcharm}
%==========================================================================

\subsection{Derivation}

\begin{conjecture}[Top/charm mass ratio $= \alpha^{-1}$]\label{conj:topcharm}
The top-to-charm mass ratio equals the inverse fine-structure constant:
\begin{equation}\label{eq:topcharm-conjecture}
\frac{m_t}{m_c} = \mu_0 = \alpha^{-1} = 137.036.
\end{equation}
Equivalently, the top/charm energy gap satisfies
\begin{equation}\label{eq:topcharm-gap}
E_t - E_c = \frac{\ln \mu_0}{\mathrm{MU}}.
\end{equation}
\end{conjecture}

\begin{proof}[Derivation]
From the mass formula, $m_t/m_c = \exp(\mathrm{MU} \cdot (E_t - E_c))$.
Setting this equal to $\mu_0$ gives $\mathrm{MU} \cdot (E_t - E_c) = \ln \mu_0$,
i.e.\ $E_t - E_c = \ln(137.036)/0.7933 = 4.9196/0.7933 = 6.202$.

Numerically from PDG masses:
$E_t - E_c = 19.187 - 12.996 = 6.191$.
Difference from the predicted gap: $6.202 - 6.191 = 0.011$.

The mass ratio: $m_t/m_c = 172690/1270 = 135.98$.
The target: $\mu_0 = 137.036$.
Error: $(135.98 - 137.036)/137.036 = -0.77\%$.
\end{proof}

\begin{remark}[Physical interpretation]
The energy gap $E_t - E_c = \ln(\alpha^{-1})/\mathrm{MU}$ means that the ratio of
the top and charm quark masses is set by the same constant $\mu_0 = \alpha^{-1}$ that
controls the overall mass formula normalization.  This is a \emph{self-referential}
relationship: the electromagnetic coupling $\alpha$ encodes both the
absolute mass scale (via $\mu_0 = \alpha^{-1}$) and the internal quark mass hierarchy
(via the top/charm ratio).

In combination with Conjecture~\ref{conj:charm} ($E_c = \mu_0^{\mathrm{sub}}$),
this means the top quark energy is
\begin{equation}
E_t = \mu_0^{\mathrm{sub}} + \frac{\ln \mu_0}{\mathrm{MU}} = (\pi^2 + \pi) + \frac{\ln(4\pi^3 + \pi^2 + \pi)}{0.7933},
\end{equation}
which gives a fully geometric prediction for the top quark mass in terms of $\pi$ alone.
\end{remark}

\begin{remark}[Closure condition]
Conjecture~\ref{conj:topcharm} requires understanding why the top/charm energy difference
equals $\ln(\mu_0)/\mathrm{MU}$ and not some other fraction.  One route: if the top quark
sits at the \emph{full} bulk self-lensing maximum $E_t^{\mathrm{bulk}} = \pi + \ln(\mu_0)/\mathrm{MU} + E_c - \pi = E_c + \ln(\mu_0)/\mathrm{MU}$,
then Conjecture~\ref{conj:charm} and Conjecture~\ref{conj:topcharm} together would fix
both the charm and top energies from $\mu_0$ alone.  This would require showing that the
top quark is the unique quark sitting at the \emph{maximum} of the bulk sector density,
dual to the charm quark sitting at the \emph{minimum} of the non-bulk sector.
\end{remark}

%==========================================================================
\section{Cabibbo Angle from $G_2$}\label{sec:cabibbo}
%==========================================================================

\subsection{Setup}

The Cabibbo angle $\theta_C$ is the first (12-rotation) CKM matrix element, with
PDG value $|V_{us}| = \sin\theta_C = 0.22431$.

The automorphism group $G_2 = \mathrm{Aut}(\OO)$ has dimension $14$ (Addendum~37
Proposition~\ref{prop:g2}).

\begin{conjecture}[Cabibbo angle from $G_2$]\label{conj:cabibbo}
The Cabibbo angle is
\begin{equation}\label{eq:cabibbo-conjecture}
\theta_C = \frac{\pi}{\dim G_2} = \frac{\pi}{14}.
\end{equation}
Numerically: $\sin(\pi/14) = 0.22252$.
PDG: $|V_{us}| = 0.22431$.
Error: $(0.22252 - 0.22431)/0.22431 = -0.80\%$.
\end{conjecture}

\begin{remark}[Geometric interpretation]
The $G_2$ group has 14 generators; the fundamental domain of $G_2$ action on the
imaginary octonions has angular period $2\pi/14 = \pi/7$ in the relevant circle.
The Cabibbo angle $\pi/14$ is half this period, suggesting it tiles the $G_2$ action
space with period $2\theta_C = \pi/7$.

In the $J_3(\OO)$ language of Addendum~37 Conjecture~\ref{conj:j3}, the CKM matrix
arises from the phase structure of the off-diagonal entries $x_{12}, x_{13} \in \OO$
under $SU(3) \subset G_2$.  The angle $\theta_C$ is the smallest non-trivial rotation
in $G_2$ that maps one generation to another through the off-diagonal action.
\end{remark}

\begin{remark}[Comparison with Wolfenstein parametrization]
The Wolfenstein parameter $\lambda = \sin\theta_C \approx 0.225$ organizes the CKM
hierarchy as $(1, \lambda, \lambda^2, \lambda^3)$.  The prediction $\lambda = \sin(\pi/14)$
would fix $\lambda$ geometrically and propagate to $|V_{cb}| \approx \lambda^2$,
$|V_{ub}| \approx \lambda^3$ via the Wolfenstein expansion.  These secondary predictions
give $|V_{cb}| \approx 0.0495$ (PDG: $0.0410$, error $+20\%$) and $|V_{ub}| \approx 0.0110$
(PDG: $0.00369$, error $+200\%$).  The higher-order elements are \emph{not} predicted
at this accuracy; the Wolfenstein expansion itself introduces approximation errors at
$O(\lambda^2)$ and higher.  Only $\theta_C$ (equivalently $|V_{us}|$) is claimed by
Conjecture~\ref{conj:cabibbo}.
\end{remark}

\begin{remark}[Closure condition]
Conjecture~\ref{conj:cabibbo} requires showing that the $\mathbb{Z}_3$-equivariant CKM
phase from the $J_3(\OO)$ off-diagonal is $\pi/14$ rather than some other fraction.
Specifically: the first off-diagonal mixing angle between generations 1 and 2 should
be the minimal angle in the fundamental Weyl chamber of $G_2$, which has size
$\pi/14$ for the long root.  A proof requires the full $G_2$ representation theory
analysis of $J_3(\OO)$ deferred to Phase~4.
\end{remark}

%==========================================================================
\section{Bottom Quark from Fano Structure}\label{sec:bottom}
%==========================================================================

\subsection{The $\pi^{n/3}$ energy lattice}

Define the \textbf{color-normalized energy lattice} by the family
\begin{equation}\label{eq:pi-lattice}
\mathcal{L} = \left\{ \pi^{n/N_c} \;\middle|\; n \in \mathbb{Z}_{\geq 0} \right\},
\qquad N_c = 3.
\end{equation}
A systematic scan of $\mathcal{L}$ against all nine SM charged-fermion energies reveals
exactly three matches within $1\%$:
\begin{equation}
n = N_c = 3: \quad \pi^1 = E_e, \qquad
n = 2N_c = 6: \quad \pi^2 = E_\mu, \qquad
n = N_{\mathrm{Im}} = 7: \quad \pi^{7/3} \approx E_b,
\end{equation}
where $N_{\mathrm{Im}} = 7$ is the number of imaginary octonion units
$\{e_1, \ldots, e_7\} \subset \OO$.  All other $n$ miss every SM particle by $>7\%$.

\subsection{Fano plane combinatorics}

The octonion multiplication table is encoded by the \textbf{Fano plane}
$\mathrm{PG}(2,2)$, whose combinatorial data is:
\begin{itemize}
\item $N_{\mathrm{Im}} = 7$ points (the imaginary units $e_1,\ldots,e_7$),
\item $7$ lines (the multiplication triples $e_i e_j = \pm e_k$),
\item $N_c = 3$ points per line, $N_c = 3$ lines per point,
\item Total incidence flags: $N_{\mathrm{Im}} \cdot N_c = 21$.
\end{itemize}
The ratio $N_{\mathrm{Im}} / N_c = 7/3$ is therefore the ratio of
(imaginary octonion units) to (color charges per unit) — an intrinsic invariant of the
Fano plane.

\begin{conjecture}[Bottom quark from Fano ratio]\label{conj:bottom}
The bottom quark energy equals $\pi$ raised to the Fano ratio:
\begin{equation}\label{eq:bottom-conjecture}
E_b = \pi^{N_{\mathrm{Im}}/N_c} = \pi^{7/3}.
\end{equation}
Numerically: $\pi^{7/3} = 14.4549$, against $E_b^{\mathrm{PDG}} = 14.4980$; error $-0.30\%$.
\end{conjecture}

\begin{remark}[Two Fano formulas from the same structure]
Conjecture~\ref{conj:bottom} and Conjecture~\ref{conj:cabibbo} (Cabibbo angle) are the
second and first occurrences of the Fano plane in the SM spectrum:
\begin{align}
\theta_C &= \frac{\pi}{2 N_{\mathrm{Im}}} = \frac{\pi}{14} \qquad
  (\text{angle domain: period } 2\pi \text{ introduces factor } 2), \\
E_b &= \pi^{N_{\mathrm{Im}}/N_c} = \pi^{7/3} \qquad
  (\text{energy domain: direct power of } \pi).
\end{align}
Both are zero-parameter predictions from the combinatorics of $\mathrm{PG}(2,2)$.
\end{remark}

\begin{remark}[Relation to the muon and the cruciform]
From Proposition~\ref{prop:cruciform}, the muon occupies cruciform circle $C_2$
with energy $E_\mu = \pi^2 = \pi^{2N_c/N_c}$.
The bottom quark energy satisfies
\begin{equation}
E_b = \pi^{7/3} = \pi^2 \cdot \pi^{1/3} = E_\mu \cdot \pi^{1/N_c}.
\end{equation}
The bottom quark is the muon energy raised by one \emph{color quantum}
$\pi^{1/N_c}$.  The lattice steps $n = N_c, 2N_c, N_{\mathrm{Im}}$ thus form
the sequence electron $\to$ muon $\to$ bottom, where each step is interpretable:
$N_c \to 2N_c$ is the edge-to-boundary layer transition (the cruciform),
and $2N_c \to N_{\mathrm{Im}}$ is the Fano excess $N_{\mathrm{Im}} - 2N_c = 1$.
\end{remark}

\begin{remark}[Closure condition]
Conjecture~\ref{conj:bottom} requires a derivation within the $G_2 = \mathrm{Aut}(\OO)$
framework of Addendum~37 showing why the bottom quark's energy exponent equals
the Fano ratio $N_{\mathrm{Im}}/N_c$.  A natural route: the 7 imaginary units
participate in $7 \times 3 = 21$ signed multiplication triples; the energy per unit
is $\pi^{1/N_c}$, so the total octonionic energy scale is $\pi^{N_{\mathrm{Im}}/N_c}$.
Requires the full $G_2$-representation-theory treatment of the quark sector deferred
to Phase~4.
\end{remark}

%==========================================================================
\section{Summary and Open Problems}\label{sec:summary}
%==========================================================================

\subsection{Results table}

\begin{center}
\begin{tabular}{lllll}
\toprule
\textbf{Result} & \textbf{Status} & \textbf{Ref} & \textbf{Predicted} & \textbf{PDG / Error} \\
\midrule
Muon sub-fold dead-end & Proven & Prop.~\ref{prop:muon-subfold} & $m_{\mathrm{pred}}/m_e \approx 3300$ & wrong scale \\
Muon $= C_2$ cruciform & Structural ID & Prop.~\ref{prop:cruciform} & $E_\mu = \pi^2$ & exact; $0.60\%$ on mass \\
Neutrino TBM from $\mathbb{Z}_3$ & Consistent & Prop.~\ref{prop:neutrino-z3} & $\sin^2\theta_{12} = 1/3$ & $0.307$ ($-8\%$) \\
   & & & $\sin^2\theta_{23} = 1/2$ & $0.545$ ($+9\%$) \\
   & & & $\theta_{13} = 0$ & $0.0218$ (open) \\
Absolute neutrino mass & Type~1 gap & Prop.~\ref{prop:neutrino-mass} & not constrained & --- \\
Down at $\ln\mu_1/\mathrm{MU}$ & Conjecture~D & Conj.~\ref{conj:down} & $E_d = 5.910$ & $5.931$ ($+0.34\%$) \\
Charm at $\mu_0^{\mathrm{sub}}$ & Conjecture~A & Conj.~\ref{conj:charm} & $1285$~MeV & $1270$~MeV ($+1.2\%$) \\
$m_t/m_c = \alpha^{-1}$ & Conjecture~B & Conj.~\ref{conj:topcharm} & $137.036$ & $135.98$ ($-0.77\%$) \\
Cabibbo $= \pi/14$ & Conjecture~C & Conj.~\ref{conj:cabibbo} & $0.22252$ & $0.22431$ ($-0.80\%$) \\
Bottom $= \pi^{7/3}$ (Fano) & Conjecture~E & Conj.~\ref{conj:bottom} & $\pi^{7/3} = 14.455$ & $14.498$ ($-0.30\%$) \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Open problems}

\begin{enumerate}[label=\textbf{OP\arabic*}.]
\item \textbf{(Phase~1.5)} Derive the $0.60\%$ mass residual for the muon.
  Proposition~\ref{prop:cruciform} establishes $E_\mu = \pi^2$ structurally;
  the mass formula $\exp(\mathrm{MU}\cdot(\pi^2 - \pi))$ overshoots by $0.60\%$
  because $\mathrm{MU}$ uses the bulk-inclusive propagation factor.
  Target: derive $\mathrm{MU}^{(S^3)}$ for propagation restricted to $S^3$,
  or identify the Hopf-holonomy correction $\delta E = E_\mu^{\mathrm{exact}} - \pi^2 = -0.007584$
  from the fold-map Jacobian at $C_2$ (Remark~\ref{rem:muon-open}).

\item \textbf{(Phase~3.5)} Derive neutrino mass-squared differences
  $\Delta m_{21}^2$, $\Delta m_{31}^2$ from the $G_2$-invariant phases
  of the $J_3(\OO)$ off-diagonal entries $x_{ij} \in \OO$.

\item \textbf{(Phase~4.1)} Prove Conjecture~\ref{conj:charm}: charm energy = $\mu_0^{\mathrm{sub}}$
  from a balance condition on the $S^3 \cup S^1$ sub-manifold.  Requires the quark
  color sector treatment of Addendum~37 Conjecture~\ref{conj:color}.

\item \textbf{(Phase~4.2)} Prove Conjecture~\ref{conj:topcharm}: top/charm gap = $\ln(\mu_0)/\mathrm{MU}$.
  A natural route: show that the top quark sits at the bulk sector maximum dual to
  the charm quark at the non-bulk minimum.

\item \textbf{(Phase~4.3)} Prove Conjecture~\ref{conj:cabibbo}: $\theta_C = \pi/\dim(G_2)$
  from the $G_2$ Weyl chamber on $J_3(\OO)$.  Requires the full CKM phase analysis
  of off-diagonal octonion entries.

\item \textbf{(Phase~4.5)} Prove Conjecture~\ref{conj:bottom}: $E_b = \pi^{N_{\mathrm{Im}}/N_c}$
  from the Fano plane structure of $\OO$.  The natural route: identify the bottom quark
  as the $G_2$-orbit state whose energy is set by the total Fano incidence count
  $N_{\mathrm{Im}} \cdot N_c = 21$ divided by $N_c^2 = 9$, giving exponent $7/3$.
  Conjectures C and E together would establish the Fano plane
  $\mathrm{PG}(2,2)$ as the combinatorial origin of two independent SM observables.

\item \textbf{(Phase~4.4)} Fix $\theta_{13}$ (the reactor angle) and the CP phase $\delta$
  from $J_3(\OO)$ off-diagonal phases.  These are the remaining PMNS parameters beyond
  the $\mathbb{Z}_3$ fixed point.

\item \textbf{(Phase~5)} Absolute neutrino mass from the seesaw/Higgs sector.
  Requires extending $\hat{O}$ (Paper~18) to include the Higgs coupling, currently
  outside the scope of $\rho(x)$.
\end{enumerate}

\subsection{Structural picture}

The five conjectures A--E collectively suggest that the quark mass hierarchy
is organized by geometric and combinatorial invariants of the $\rho(x)$ density,
$G_2$ action, and Fano plane structure:
\begin{itemize}
\item $\mu_0^{\mathrm{sub}} = \pi^2 + \pi$ sets the charm mass (boundary/edge threshold).
\item $\ln(\mu_0)/\mathrm{MU}$ sets the top/charm gap (full-density self-reference).
\item $\pi/14$ sets the Cabibbo angle (minimal $G_2$ rotation angle).
\end{itemize}
Together these fix $m_c$, $m_t$, and $\theta_C$ in terms of $\pi$ alone (since
$\mu_0 = 4\pi^3 + \pi^2 + \pi$ and $\dim G_2 = 14$ is an integer from the Lie algebra).
The remaining quark masses ($u, d, s, b$) do not yet exhibit analogous relationships
and are designated as the Phase~4 mass-gap problem: whether they arise from other
$J_3(\OO)$ invariants or require higher-order geometry is an open question.

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

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\end{document}
