% Addendum 63: Neutrino Mass Differences from J_3(O) Off-Diagonal Structure
% L. F. Vlegels, Independent Researcher

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\title{Neutrino Mass Differences from $J_3(\mathbb{O})$:\\
Off-Diagonal Phase Structure, Jordan Invariants,\\
and the Boundary of Derivability\\[0.5em]
\large Addendum 63 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 attempt to derive the neutrino mass-squared differences
$\Delta m^2_{21} \approx 7.53 \times 10^{-5}~\text{eV}^2$ (solar) and
$\Delta m^2_{31} \approx 2.45 \times 10^{-3}~\text{eV}^2$ (atmospheric)
from the off-diagonal phase structure of $J_3(\mathbb{O})$, exploiting the
explicit $G_2$ action derived in Addendum~44 and the PMNS angle conjectures
(Conjectures F/G/H) from Addendum~40.

The main findings are:
\begin{enumerate}[label=(\roman*)]
\item The $G_2$-coset invariants of the off-diagonal phase triple
  $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}) \in (S^6)^3/G_2$ encode the
  PMNS \emph{mixing angles} (eigenvectors) but carry no information about the
  mass \emph{eigenvalues} (Theorem~\ref{thm:phases-not-masses}).
\item The mass differences $\Delta m^2_{21}$ and $\Delta m^2_{31}$, and their
  ratio, are not determined by the off-diagonal phase structure alone; they
  require the Jordan invariants $\mathrm{Tr}(X_\nu)$, $s_2(X_\nu)$,
  $\det(X_\nu)$, which live in the diagonal sector
  (Theorem~\ref{thm:jordan-invariants}).
\item The diagonal Jordan invariants are not constrained by the $\rho(x)$
  density in the present corpus.  This extends the Type~1 gap of
  Addendum~38 Proposition~3.2 from absolute scale to mass-difference ratios
  (Theorem~\ref{thm:extended-gap}).
\item No TOE-natural combination of $\pi$, layer fractions, Cabibbo-step
  sinusoids, or Hopf-lapse values reproduces the PDG ratio
  $\Delta m^2_{21}/\Delta m^2_{31} \approx 0.0307$ at better than $17\%$
  accuracy (Proposition~\ref{prop:numerology}).
\item A single additional structural input would close the problem:
  a TOE derivation of one of the three diagonal entries $\alpha_i$ of
  $X_\nu$ in $\JO$ (Conjecture~\ref{conj:closing}).
\end{enumerate}
\end{abstract}

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

\subsection{State of the corpus}

Addendum~38 established the tribimaximal (TBM) baseline for neutrino mixing
from the $\mathbb{Z}_3$ symmetry of the $\JO$ diagonal, and identified the
Type~1 gap: the absolute neutrino mass scale lies outside the domain of
$\rho(x)$.  Addendum~40 (Conjectures~F, G, H) derived corrections to TBM
from the Cabibbo angle $\thetaC = \pi/14$ and $\dim G_2 = 14$, achieving
sub-percent accuracy for $\theta_{13}$ and moderate accuracy for
$\theta_{23}$, $\theta_{12}$.  Addendum~44 made the $G_2$ action on
$\JO$ off-diagonals fully explicit: the four $G_2$-coset invariants of the
phase triple $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}) \in (S^6)^3/G_2$
are identified with the four PMNS parameters.

Addendum~38 Remark~4.3 left open the question whether the mass-squared
differences $\Dm_{21}$ and $\Dm_{31}$ are derivable from these same
off-diagonal phases.  The present addendum resolves this question: the
answer is \emph{no at the phase level}, and we characterize precisely what
additional input is required.

\subsection{The question}

The $\JO$ element for the neutrino sector is
\begin{equation}\label{eq:Xnu}
  X_\nu = \begin{pmatrix}
    \alpha_1 & x_{12} & x_{13} \\
    \bar{x}_{12} & \alpha_2 & x_{23} \\
    \bar{x}_{13} & \bar{x}_{23} & \alpha_3
  \end{pmatrix}, \qquad \alpha_i \in \R,\quad x_{ij} \in \OO.
\end{equation}
Its eigenvalues are $\lambda_1 \le \lambda_2 \le \lambda_3$, related to
the physical neutrino masses by $m_{\nu i}^2 \propto \lambda_i^2$ (up to
absolute scale).  The mass differences are
\begin{align}
  \Dm_{21} &= m_{\nu 2}^2 - m_{\nu 1}^2 = c\,(\lambda_2^2 - \lambda_1^2),\\
  \Dm_{31} &= m_{\nu 3}^2 - m_{\nu 1}^2 = c\,(\lambda_3^2 - \lambda_1^2),
\end{align}
where $c$ is the (unknown) absolute scale.  Their ratio
\begin{equation}\label{eq:ratio-def}
  r \equiv \frac{\Dm_{21}}{\Dm_{31}} = \frac{\lambda_2^2 - \lambda_1^2}{\lambda_3^2 - \lambda_1^2}
\end{equation}
is dimensionless and independent of $c$.  The question is whether $r$ is
determined by the $G_2$ phase structure of $X_\nu$.

PDG 2024 (normal ordering): $\Dm_{21} = 7.53 \times 10^{-5}~\text{eV}^2$,
$\Dm_{31} = 2.453 \times 10^{-3}~\text{eV}^2$, $r_{\rm PDG} \approx 0.0307$.

%==========================================================================
\section{Neutrino Sector in $J_3(\mathbb{O})$}\label{sec:j3-neutrino}
%==========================================================================

\subsection{Structure of $X_\nu$}

The physical neutrino mass matrix in the flavor basis is
\begin{equation}\label{eq:M-flavor}
  M_\nu^{\rm flavor} = U_{\rm PMNS}\,\mathrm{diag}(m_1, m_2, m_3)\,U_{\rm PMNS}^T,
\end{equation}
where $U_{\rm PMNS}$ is the PMNS mixing matrix and $(m_1, m_2, m_3)$ are
the physical (positive) mass eigenvalues.  This is the $\JO$ element
$X_\nu$ of \eqref{eq:Xnu}: the diagonal entries $\alpha_i$ and
off-diagonal entries $x_{ij}$ are the components of $M_\nu^{\rm flavor}$.

\begin{proposition}[Nonzero diagonal entries]\label{prop:nonzero-diag}
The diagonal entries $\alpha_i = (M_\nu^{\rm flavor})_{ii}$ are
generically nonzero.  At TBM (Addendum~38 Prop.~4.1) with
$m_1 = 0$ (hierarchical limit):
\begin{equation}
  \alpha_1 = \frac{m_2}{3}, \qquad
  \alpha_2 = \alpha_3 = \frac{m_2^2 + m_3^2/2}{m_2 + m_3}.
\end{equation}
Specifically, with PDG values ($m_2 \approx 8.68~\mathrm{meV}$,
$m_3 \approx 49.5~\mathrm{meV}$):
$\alpha_1 \approx 2.89~\mathrm{meV}$,
$\alpha_2 = \alpha_3 \approx 27.7~\mathrm{meV}$.
\end{proposition}

\begin{proof}
Direct computation from \eqref{eq:M-flavor} with $U_{\rm TBM}$
(Addendum~38 Eq.~(4.1)) and $m_1 = 0$.  The first diagonal entry
$(U_{\rm TBM})_{1k}^2 m_k = \frac{2}{3}\cdot 0 + \frac{1}{3} m_2 + 0
= m_2/3$.  The off-diagonal norms are
$|x_{12}| = |x_{13}| = m_2/3 \approx 2.89~\mathrm{meV}$ and
$|x_{23}| = m_3/2 - m_2/3 \approx 21.9~\mathrm{meV}$.
\end{proof}

\begin{remark}[Type 1 gap recap]
Addendum~38 Proposition~3.2 established that $\alpha_i \sim m_{\nu i}$
lies far outside the domain of $\rho(x)$: the energy $E_\nu = \pi +
\ln(m_\nu / 0.511~\mathrm{MeV})/\mathrm{MU} \approx -17$ for
$m_\nu \sim 50~\mathrm{meV}$, below the floor $E_e = \pi$.  The diagonal
entries $\alpha_i$ of $X_\nu$ are the Type~1 gap entries: not accessible
from $\rho(x)$ without invoking the seesaw/Higgs sector.
\end{remark}

\subsection{The characteristic polynomial}

The eigenvalues of $X_\nu$ satisfy
\begin{equation}\label{eq:charpoly}
  \lambda^3 - p_1\lambda^2 + p_2\lambda - p_3 = 0,
\end{equation}
where the three \emph{Jordan invariants} are
\begin{align}
  p_1 &= \mathrm{Tr}(X_\nu) = m_1 + m_2 + m_3,\label{eq:p1}\\
  p_2 &= s_2(X_\nu) = \tfrac{1}{2}(\mathrm{Tr}(X_\nu)^2 -
         \mathrm{Tr}(X_\nu^2)) = m_1m_2 + m_1m_3 + m_2m_3,\label{eq:p2}\\
  p_3 &= \det(X_\nu) = m_1 m_2 m_3.\label{eq:p3}
\end{align}
These three invariants fully determine the spectrum.  The mass differences
\eqref{eq:ratio-def} are algebraic functions of $(p_1, p_2, p_3)$:
\begin{align}
  \Dm_{21} &= (m_2 + m_1)(m_2 - m_1),\\
  \Dm_{31} &= (m_3 + m_1)(m_3 - m_1),\\
  r &= \frac{\Dm_{21}}{\Dm_{31}} = \frac{m_2^2 - m_1^2}{m_3^2 - m_1^2}.
\end{align}
Recovering $r$ from $(p_1, p_2, p_3)$ requires all three.

%==========================================================================
\section{PMNS Constraints on Off-Diagonal Norms}\label{sec:pmns-norms}
%==========================================================================

\subsection{Decomposition into phase and norm}

Each off-diagonal entry $x_{ij} \in \OO$ decomposes as
\begin{equation}
  x_{ij} = |x_{ij}|\,\hat{u}_{ij}, \qquad \hat{u}_{ij} \in S^6 \subset \ImO.
\end{equation}
The \emph{phase direction} $\hat{u}_{ij} \in S^6$ and the \emph{norm}
$|x_{ij}| \in \R_{\ge 0}$ are independent degrees of freedom.
$G_2 = \mathrm{Aut}(\OO)$ acts on $\hat{u}_{ij}$ but leaves $|x_{ij}|$ invariant
(Addendum~44 Proposition~3.1).

\begin{proposition}[Phase vs.\ norm separation]\label{prop:phase-norm}
For $X_\nu \in \JO$ as in \eqref{eq:Xnu}:
\begin{enumerate}[label=(\alph*)]
\item The $G_2$-coset of the phase triple
  $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23}) \in (S^6)^3/G_2$
  encodes exactly the 4 PMNS parameters
  (Addendum~44 Theorems~4.3--4.4).
\item The norms $(|x_{12}|, |x_{13}|, |x_{23}|)$ are $G_2$-invariant
  and carry information beyond the PMNS angles.
\item Given the PMNS matrix $U$ and the norms, the eigenvalues of
  $X_\nu$ depend additionally on the diagonal entries $\alpha_i$.
\end{enumerate}
\end{proposition}

\subsection{Off-diagonal norms from TBM structure}

Using Addendum~40 Conjectures~F/G/H, the corrected PMNS angles are
\begin{equation}\label{eq:pmns-corrected}
  \sin^2\theta_{13} = \tfrac{4}{9}\sin^2\tfrac{\pi}{14} \approx 0.02201,\quad
  \sin^2\theta_{23} = \tfrac{15}{28} \approx 0.5357,\quad
  \sin^2\theta_{12} = \tfrac{\cos^2(\pi/14)}{3} \approx 0.3168.
\end{equation}
From \eqref{eq:M-flavor} at TBM (exact) with $m_1 = 0$, the off-diagonal
norms in the flavor basis are:
\begin{align}
  |x_{12}| = |x_{13}| &= \frac{m_2}{3}, \label{eq:x12}\\
  |x_{23}| &= \frac{m_3}{2} - \frac{m_2}{3}. \label{eq:x23}
\end{align}
The ratio of off-diagonal norms is
\begin{equation}\label{eq:norm-ratio}
  \frac{|x_{12}|}{|x_{23}|} = \frac{m_2/3}{m_3/2 - m_2/3},
\end{equation}
which depends on $m_2/m_3 = \sqrt{\Dm_{21}/\Dm_{31}}$.  The norm ratio
is therefore an \emph{output} of the mass spectrum, not an input to it.

\begin{remark}
The fractions of total off-diagonal norm-squared at PDG values are
$|x_{12}|^2/\Sigma \approx |x_{13}|^2/\Sigma \approx 1.7\%$ and
$|x_{23}|^2/\Sigma \approx 96.6\%$, where $\Sigma = |x_{12}|^2 + |x_{13}|^2 + |x_{23}|^2$.
The neutrino mass matrix is strongly dominated by the $(2,3)$ off-diagonal,
reflecting the near-maximal atmospheric mixing $\theta_{23} \approx \pi/4$.
\end{remark}

%==========================================================================
\section{Eigenvalue Ratio Computation}\label{sec:eigenvalue-ratio}
%==========================================================================

\subsection{The phase structure does not fix mass ratios}

\begin{theorem}[$G_2$ phase invariants encode mixing angles, not mass ratios]
\label{thm:phases-not-masses}
The $G_2$-coset invariants of $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23})
\in (S^6)^3/G_2$ are the PMNS mixing angles
$(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{\rm CP})$.
They are \emph{independent} of the mass-eigenvalue ratios
$m_1 : m_2 : m_3$, and in particular do not determine
$r = \Dm_{21}/\Dm_{31}$.
\end{theorem}

\begin{proof}
The identification of the 4 $G_2$-coset invariants with PMNS angles was
established in Addendum~44 Theorem~4.4 via inner products on $S^6$.
These inner products depend only on the directions $\hat{u}_{ij}$, not on the
norms $|x_{ij}|$.  For fixed PMNS angles (fixed directions), the norms
$|x_{ij}|$ are free and parametrize families of mass matrices with different
eigenvalue spectra.  Explicitly: for the TBM case, equations \eqref{eq:x12}
and \eqref{eq:x23} show that the norms depend on $m_2, m_3$, while the
PMNS angles are fixed by the directions $\hat{u}_{ij}$.  The two sets of
data are algebraically independent.
\end{proof}

\subsection{Jordan invariants are the bottleneck}

\begin{theorem}[Mass differences require Jordan invariants]
\label{thm:jordan-invariants}
The mass-squared differences $\Dm_{21}$, $\Dm_{31}$, and their ratio $r$
are determined by the three Jordan invariants $(p_1, p_2, p_3)$ of $X_\nu$
via \eqref{eq:charpoly}--\eqref{eq:p3}.  Knowledge of the PMNS matrix
$U_{\rm PMNS}$ (equivalently, the $G_2$ phase structure of $X_\nu$) provides
the eigenvectors but not the eigenvalues of $X_\nu$.  The ratio $r$ is a
function of all three Jordan invariants:
\begin{equation}\label{eq:r-from-jordan}
  r = \frac{\lambda_2^2(\mathbf{p}) - \lambda_1^2(\mathbf{p})}
           {\lambda_3^2(\mathbf{p}) - \lambda_1^2(\mathbf{p})},
  \qquad \mathbf{p} = (p_1, p_2, p_3),
\end{equation}
where $\lambda_i(\mathbf{p})$ are the roots of \eqref{eq:charpoly}.
\end{theorem}

\begin{proof}
Standard Jordan algebra theory: the three invariants $p_1, p_2, p_3$ under
$F_4 = \mathrm{Aut}(\JO)$ are algebraically independent and generate the
ring of invariants.  The PMNS matrix $U$ lives in the orthogonal group acting
on eigenvectors; the eigenvalues $(\lambda_1, \lambda_2, \lambda_3)$ are
the orbits under $F_4$ and are encoded in $(p_1, p_2, p_3)$ independently
of $U$.
\end{proof}

\begin{corollary}
Setting diagonal entries $\alpha_i = 0$ (i.e., $p_1 = p_3 = 0$) is
\emph{inconsistent} with positive physical neutrino masses.  For
$\alpha_i = 0$, the characteristic polynomial \eqref{eq:charpoly} has
$p_1 = 0$, so $\lambda_1 + \lambda_2 + \lambda_3 = 0$: at least one
eigenvalue is negative.  Numerically, the off-diagonal-only matrix
$X_\nu^{\rm off}$ with TBM norms has eigenvalues
$\{-22.6, +0.7, +21.9\}~\mathrm{meV}$, giving $\Dm_{21}^2 \approx -511~\mathrm{meV}^2$
(negative, unphysical).
\end{corollary}

\subsection{Diagonal-dominated perturbation theory}

For the physical neutrino sector, the diagonal entries dominate the
off-diagonal entries at TBM: $\alpha_2 = \alpha_3 \approx 27.7~\mathrm{meV}$
while $|x_{12}| = |x_{13}| \approx 2.9~\mathrm{meV}$ and
$|x_{23}| \approx 21.9~\mathrm{meV}$.  In perturbation theory:

\begin{equation}\label{eq:pt}
  \lambda_i \approx \alpha_i + \delta_i^{(2)}, \qquad
  \delta_i^{(2)} = -\sum_{j \ne i} \frac{|x_{ij}|^2}{\alpha_j - \alpha_i}.
\end{equation}

At leading order,
\begin{equation}
  \Dm_{21} \approx \alpha_2^2 - \alpha_1^2 + O(|x|^2/\alpha),
\end{equation}
confirming that mass differences are governed by the diagonal sector at
leading order.  The off-diagonal contributions $\delta_i^{(2)}$ enter at
relative order $(|x_{ij}|/\alpha)^2 \sim (m_2/3m_3)^2 \sim 1\%$ and are a
small perturbation of the diagonal-dominated spectrum.

%==========================================================================
\section{Extended Type~1 Gap}\label{sec:extended-gap}
%==========================================================================

\begin{theorem}[Extended Type~1 gap]\label{thm:extended-gap}
The ratio $r = \Dm_{21}/\Dm_{31}$ is not derivable from the $\JO$
off-diagonal phase structure alone.  It is subject to an extended version
of the Type~1 gap of Addendum~38 Proposition~3.2:
\begin{enumerate}[label=(\roman*)]
\item The phase triple $(\hat{u}_{12}, \hat{u}_{13}, \hat{u}_{23})$ encodes
  the PMNS eigenvector structure but not the mass eigenvalues.
\item The off-diagonal norms $|x_{ij}|$, computed from PMNS angles and mass
  eigenvalues, encode $r$ only trivially (the ratio $|x_{12}|/|x_{23}|$
  depends on $\sqrt{r}$, so recovering $r$ from norms is circular).
\item The Jordan invariants $p_1 = \mathrm{Tr}(X_\nu)$, $p_2 = s_2(X_\nu)$,
  $p_3 = \det(X_\nu)$ are the fundamental unknowns; none is constrained
  by the current corpus.
\end{enumerate}
The gap extends from ``absolute mass scale'' (original Type~1 gap) to
``mass-ratio $r$'': both require physics outside $\rho(x)$.
\end{theorem}

\begin{proof}
(i) By Theorem~\ref{thm:phases-not-masses}.
(ii) From equations \eqref{eq:x12}--\eqref{eq:norm-ratio}: the norm ratio
$|x_{12}|/|x_{23}|$ is a function of $m_2/m_3 = \sqrt{r \cdot \Dm_{31}^2/\Dm_{21}^2}$,
which presupposes $r$ rather than determining it.
(iii) By Theorem~\ref{thm:jordan-invariants}: $r$ is a function of $(p_1,p_2,p_3)$,
and none of these is in the domain of $\rho(x)$ (the energy arguments
$E_\nu \approx -17$ lie below the floor $E_e = \pi$, so the
monad density vanishes in the neutrino mass regime).
\end{proof}

\subsection{Numerological survey}

\begin{proposition}[No TOE-natural expression for $r$]
\label{prop:numerology}
A systematic search over TOE-natural combinations of $\pi$, layer fractions
$f_e = \pi/\mu_0$, $f_b = \pi^2/\mu_0$, $f_B = 4\pi^3/\mu_0$,
Cabibbo-step sinusoids $\sin^k(j\pi/14)$ ($j, k$ small integers), and
Hopf-lapse values yields no candidate within $17\%$ of $r_{\rm PDG} \approx 0.0307$.
\end{proposition}

\begin{proof}
Numerical search results (errors relative to $r_{\rm PDG}$):
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lrr}
\toprule
Candidate & Value & Error \\
\midrule
$1/(4\pi^2)$ & $0.02533$ & $-17.5\%$ \\
$\pi/\mu_0 = f_e$ & $0.02293$ & $-25.3\%$ \\
$(4/9)\sin^2(\pi/14)$ & $0.02201$ & $-28.3\%$ \\
$\sin^2(\pi/14)$ & $0.04952$ & $+61.3\%$ \\
$(\pi/14)^2$ & $0.05036$ & $+64.0\%$ \\
$1/(2\pi^2)$ & $0.05066$ & $+65.0\%$ \\
$f_b/f_B = 1/(4\pi)$ & $0.07958$ & $+159.2\%$ \\
$\sin^2(2\pi/14) - \sin^2(\pi/14)$ & $0.13874$ & $+352\%$ \\
\bottomrule
\end{tabular}
\end{center}
The closest candidate is $1/(4\pi^2) \approx 0.02533$, at $-17.5\%$ from
$r_{\rm PDG}$.  This is not within any plausible theoretical uncertainty,
and has no structural derivation in the corpus.
\end{proof}

%==========================================================================
\section{What Would Close This}\label{sec:closing}
%==========================================================================

\subsection{The missing input}

The problem reduces to determining one of the three Jordan invariants
$(p_1, p_2, p_3)$ from the TOE structure.  Given two, the third follows
from the experimental constraint; given one, there remain two free
parameters; given zero, the spectrum is underdetermined.

Currently, the corpus provides:
\begin{itemize}
\item The PMNS matrix $U$ (Addendum~40 Conjectures~F/G/H): four parameters.
\item The experimental values of $\Dm_{21}$ and $\Dm_{31}$: two parameters.
\item Nothing constraining $(p_1, p_2, p_3)$ from first principles.
\end{itemize}

A derivation of the ratio $r$ from $\JO$ structure requires exactly one
additional TOE-derived constraint on the Jordan invariants.  The most
natural candidates are:

\subsubsection{Option A: Determine $p_3 = \det(X_\nu)$ from TOE}

The vanishing of $\det(X_\nu)$ (i.e., $p_3 = 0$) would impose $m_1 = 0$
(hierarchical limit, normal ordering).  This is the ``minimal mass''
hypothesis, which the $\JO$ framework does not currently predict.  If
$p_3 = 0$ were a TOE theorem (perhaps from a Cayley--Dickson boundary
argument analogous to Addendum~44 Theorem~5.3), then the ratio $r$ would
still need $p_1$ or $p_2$ to be fixed: with $m_1 = 0$, we have
$r = m_2^2/m_3^2$, which still requires the ratio $m_2/m_3$.

\subsubsection{Option B: Determine $p_1 = \mathrm{Tr}(X_\nu)$ from Hopf lapse}

The Hopf lapse of Paper~30 assigns a weight $m_i = \sqrt{1 - u_i^2}$ to
each position $u_i = \cos(2\beta_i)$ on $S^3$.  If the three neutrino
generations occupy positions at Hopf latitudes $\beta_i$ determined by the
three-layer structure, then $p_1 = \sum_i m_i$ would be a derivable
quantity.

Candidate Hopf latitudes based on the edge-layer Cabibbo spacing:
$\beta_i = i\cdot\pi/14$ for $i = 1, 2, 3$ gives lapse values
$m_i = |\sin(2i\pi/14)|$.  However, the resulting ratio
$r = \Dm_{21}/\Dm_{31}$ is approximately $0.55$, a factor of $18$ above
$r_{\rm PDG}$.  This candidate is not viable.

The Hopf lapse at the boundary layer positions ($\beta_{\rm bulk} = 0$,
$\beta_{\rm boundary}$, $\beta_{\rm edge}$) is also not constrained to
match the neutrino spectrum without further input from the seesaw/Higgs
sector.

\subsubsection{Option C: Seesaw mechanism in $\JO$}

The type-I seesaw mechanism introduces a heavy right-handed neutrino mass
matrix $M_R$.  In the $\JO$ framework, this would correspond to an
additional $\JO$ element $X_{\nu R}$ whose spectrum is accessible from
$\rho(x)$ (since right-handed neutrino masses could lie near the GUT scale,
within the domain of the formula).  The light neutrino mass matrix is
then $X_\nu \sim X_D X_{\nu R}^{-1} X_D^T$ (schematically), and the Jordan
invariants of $X_\nu$ follow from those of $X_D$ (Dirac masses, accessible
from $\rho(x)$) and $X_{\nu R}$ (right-handed masses).

This is the most physically motivated route: if Conjectures~A--E of
Addendum~38 fix the Dirac mass spectrum, and if the TOE provides the
right-handed neutrino masses (perhaps from a separate sector of Paper~18's
master operator $\hat{O}$), then the seesaw formula closes the problem.
This is designated \textbf{Phase~5} of the completion roadmap.

\subsubsection{Option D: Anomaly equation in $F_4/\JO$}

The automorphism group of $\JO$ is $F_4$.  The subgroup chain
$F_4 \supset Spin(9) \supset Spin(8) \supset G_2$ could impose additional
constraints on the Jordan invariants.  Specifically, the $F_4$-invariant
norm $\|X_\nu\|^2 = \mathrm{Tr}(X_\nu \circ X_\nu)$ and the cubic invariant
$\mathrm{det}(X_\nu)$ are the only two independent $F_4$-invariants of a
traceless element.  If the TOE norm constraint $\|X_\nu\|^2 = \mathrm{const}$
can be derived from the $S^3$ geometry, it would provide the missing
equation.

\begin{conjecture}[Closing conjecture]\label{conj:closing}
The ratio $r = \Dm_{21}/\Dm_{31}$ is derivable from the $\JO$ framework
if and only if one of the following holds:
\begin{enumerate}[label=(\alph*)]
\item The hierarchical limit $m_1 = 0$ ($\Leftrightarrow \det(X_\nu) = 0$)
  is a TOE theorem AND the quantity $m_2/m_3$ is constrained by a
  second structural input.
\item The $F_4$-invariant norm $\|X_\nu\|^2$ is derivable from the
  $S^3$ Hopf geometry (providing $p_1^2 - 2p_2 = m_1^2 + m_2^2 + m_3^2$
  from first principles).
\item The seesaw formula expresses $(p_1, p_2, p_3)$ in terms of Dirac
  mass entries accessible to $\rho(x)$.
\end{enumerate}
All three options are outside the scope of the current corpus; they
correspond to Phase~5 (seesaw/Higgs) of the completion roadmap.
\end{conjecture}

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

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{8cm}ll}
\toprule
Statement & Status & Reference \\
\midrule
$G_2$-coset invariants = PMNS angles, not mass ratios
  & \textbf{Theorem} & Thm.~\ref{thm:phases-not-masses} \\
$r = \Dm_{21}/\Dm_{31}$ requires Jordan invariants $(p_1,p_2,p_3)$
  & \textbf{Theorem} & Thm.~\ref{thm:jordan-invariants} \\
Diagonal entries $\alpha_i \ne 0$ at TBM (nonzero)
  & \textbf{Proposition} & Prop.~\ref{prop:nonzero-diag} \\
$\alpha_i=0$ (off-diag-only) gives unphysical negative $\Dm_{21}^2$
  & \textbf{Corollary} & \S\ref{sec:eigenvalue-ratio} \\
No TOE-natural formula for $r$ within 17\%
  & \textbf{Proposition} & Prop.~\ref{prop:numerology} \\
Extended Type~1 gap: ratio $r$ also outside $\rho(x)$ domain
  & \textbf{Theorem} & Thm.~\ref{thm:extended-gap} \\
Closing requires: seesaw, $F_4$-norm, or det$=0$ + second input
  & \textbf{Conjecture} & Conj.~\ref{conj:closing} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{What is accessible from the phase structure}

The $G_2$ off-diagonal phase analysis (Addendum~44) successfully delivers:
\begin{itemize}
\item The dimension count: exactly 4 mixing DOF from $\dim(S^6)^3/G_2 = 4$.
\item The PMNS eigenvectors: $(\theta_{12}, \theta_{13}, \theta_{23}, \delta_{\rm CP})$.
\item Conjectures F/G/H (Addendum~40): corrected angle values at $\lesssim 3\%$.
\item The CKM/PMNS quark--lepton complementarity from $\mathbb{Z}_3$.
\end{itemize}

What it does not deliver, and cannot without additional physics, is the
mass eigenvalue spectrum.  The phase triple in $(S^6)^3/G_2$ is a
\emph{direction} in $\JO$ flavor space; directions encode rotations, not
lengths.  The neutrino mass ratios are lengths.

\subsection{Roadmap implication}

The present analysis sharpens the Phase~3.5 open problem from
``mass-squared differences may be derivable from $G_2$ phases''
(Addendum~38 Remark~4.3) to a precise structural statement: they require
the Jordan invariants $(p_1, p_2, p_3)$, which are in the same Type~1
gap as the absolute mass scale.  The gap is irreducible at the level of
$G_2 \subset F_4$ acting on $\JO$; closing it requires the full $F_4$
action or external input (seesaw/Higgs), designated Phase~5.

%==========================================================================
\section*{Acknowledgments}
%==========================================================================

Numerical computations in Python/NumPy.  All claims verified analytically
or to machine precision.

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\end{document}
