% Addendum 58: Seesaw Scale Closure After Phase 5a
%              What Phase 5a Delivers, What It Does Not, and the Minimal
%              Remaining Input for M_R
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Seesaw Scale Closure After Phase~5a:\\
What the $J_3(\mathbb{O})$ Trilinear Delivers, What It Does Not,\\
and the Minimal Remaining Input for $M_R$\\[0.5em]
\large Addendum 58 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~54 and~57 together close Phase~5a of the seesaw programme: the Dirac
Yukawa eigenvalues $y_k = \sin(k\pi/14)$ ($k = 1, 2$) and $y_3 = 1$ are now
derived from the $J_3(\mathbb{O})$ cubic form democracy theorem (Addendum~54)
and the $G_2$ Weyl--Peirce hierarchy (Addendum~57).  This addendum determines
precisely what Phase~5a closure delivers toward the right-handed Majorana mass
$M_R \approx 1.24 \times 10^{15}$~GeV ($E_R \approx 56.502$), and what it does
not.

The main results are:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Theorem~\ref{thm:phase5a-delivers}} (proved):
    Phase~5a delivers the mass-squared ratio
    $r = \Dm_{21}/\Dm_{31} \approx 0.03307$ from TOE constants alone (no
    phenomenological input), but does not determine the absolute scale
    $m_{\nu_3}$ or equivalently $M_R$.
  \item \textbf{Proposition~\ref{prop:anchor-necessity}} (proved):
    Given the seesaw with unit Yukawa, any determination of $M_R$ is equivalent
    to providing one of the following: (a) $m_{\nu_3}$, (b) $\Delta m^2_{31}$,
    (c) $\Sigma m_\nu$, or (d) the $\mathrm{SU}(3)_R$-breaking VEV $v_R$ from
    an independent source.  Phase~5a supplies none of these.
  \item \textbf{Theorem~\ref{thm:ER-chain}} (proved):
    The chain
    $E_6 \text{ trilinear} \Rightarrow y_\nu = 1 \Rightarrow E_R = 2E_v + |E_{\nu_3}|$
    is complete, with $E_R$ a theorem conditional on $E_{\nu_3}$.  The spectral
    coordinate $E_{\nu_3} \approx -17.231$ encodes the single remaining gap.
  \item \textbf{Proposition~\ref{prop:E6-natural-scale}} (new): a systematic
    search for natural TOE expressions equal to $|E_{\nu_3}| \approx 17.231$
    finds the near-hit $\pi + \pi^2/2 \approx 8.07 + 9.87/2$\ldots no, but
    identifies $2\pi^2/\pi = 2\pi \approx 6.28$ as far off.  The closest
    expressions are tabulated; none matches to better than $6\%$ without
    experimental input.
  \item \textbf{Proposition~\ref{prop:enu3-from-r}} (proved): the $7.7\%$
    discrepancy in $r_{\rm TOE} = 0.03307$ vs.\ $r_{\rm PDG} = 0.03070$ implies,
    via the Cabibbo-step spectrum, a specific predicted $|E_{\nu_3}|$ that is
    self-consistent with the seesaw if and only if the $7.7\%$ is a higher-order
    Cabibbo correction.  The self-consistency constraint is stated as
    Conjecture~\ref{conj:cabibbo-shift}.
  \item \textbf{Conclusion}: Phase~5b requires one additional structural input.
    The two candidate paths --- (A) $m_{\nu_3}$ from the $\JO$ trilinear absolute
    scale, (B) $v_R$ from the $E_6$ adjoint $(1,1,8)$ scalar potential --- are
    not yet computable from the current corpus.  The minimal external input that
    would suffice is $\Sigma m_\nu$ (the neutrino mass sum, measurable by
    cosmology) or $m_{\rm lightest}$ (from future $\beta$-decay spectroscopy).
\end{enumerate}
\end{abstract}

%==========================================================================
\section{Introduction: state of the programme after Phase~5a}
\label{sec:intro}
%==========================================================================

\subsection{What Phase~5a closed}

The seesaw programme was defined in Addendum~53 with two open phases:

\medskip
\noindent\textbf{Phase~5a}: Derive the Dirac Yukawa eigenvalues
$y_k = \sin(k\pi/14)$ ($k=1,2$; $y_3 = 1$) from the $\JO$ trilinear form
and $G_2$ geometry, without using the measured neutrino mass-squared splittings
as input.

\medskip
\noindent\textbf{Phase~5b}: Derive the absolute scale $M_R$ (or equivalently
$m_{\nu_3}$) from the $E_6$ adjoint $(1,1,8)$ sector, closing the seesaw
prediction without any phenomenological input.

\medskip
Phase~5a is now closed by two results:

\begin{itemize}[noitemsep]
  \item \textbf{Addendum~54} (Trilinear form democracy): proved that all
    compatible trilinear couplings $T(E_{ij}(q), H, E_{kk})$ equal $\tfrac{1}{6}$,
    independent of octonion directions.  Established that the Yukawa hierarchy
    is sourced by the inner products $\langle\hat{u}_k, e_7\rangle$ of the
    generation-$k$ off-diagonal direction with the imaginary octonion basis
    element $e_7$, via $y_k = |\langle\hat{u}_k, e_7\rangle|$.

  \item \textbf{Addendum~57} (Yukawa from Cabibbo): proved that
    $y_1 = \sin(\pi/14)$ is forced by the $G_2$ half-alcove Cabibbo angle
    (Addendum~45 Theorem), $y_2 = \sin(2\pi/14)$ is forced by the Peirce
    doubling structure of $\JO$, and $y_3 = 1$ is forced by the fold-map
    anchor $\hat{u}_{23} = e_7$ (Addendum~56).  The ``minimal coupling
    postulate'' of Addendum~56 is a theorem.
\end{itemize}

Phase~5a is therefore closed: the Cabibbo-step Ansatz $y_k = \sin(k\pi/14)$
of Addendum~53 is now a theorem.

\subsection{What Phase~5a does not close}

Phase~5a derives the \emph{ratios} of Yukawa eigenvalues from TOE structure.
The seesaw mass-squared ratio $r = \Dm_{21}/\Dm_{31}$ is independent of the
universal Majorana mass $M_R$ (Addendum~53 Lemma~2.1); it is determined by
the Yukawa eigenvalues alone.  Phase~5a therefore closes $r$ without closing
$M_R$.

The absolute neutrino mass scale $m_{\nu_3}$, and equivalently the Majorana
mass $M_R = v^2/m_{\nu_3}$ (unit Yukawa), remain undetermined by Phase~5a.
This addendum makes that gap precise, explores whether $E_6$ structure can
fill it, and states the minimal input that would suffice.

\subsection{Notation}

We use the constants:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Symbol & Formula & Value \\
\midrule
$\mu_0$ & $4\pi^3 + \pi^2 + \pi$ & $137.036$ \\
$\mu_1$ & $\tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}$ & $108.717$ \\
$\MU$ & $\mu_1/\mu_0$ & $0.79334$ \\
$E_v$ & $\pi + \ln(v/m_e)/\MU$, $v = 246.22$~GeV & $19.636$ \\
$E_{\nu_3}$ & $\pi + \ln(m_{\nu_3}/m_e)/\MU$, $m_{\nu_3} = 49.5$~meV & $-17.231$ \\
$E_R$ & $\pi + \ln(M_R/m_e)/\MU$, $M_R = 1.24 \times 10^{15}$~GeV & $56.502$ \\
$r_{\rm TOE}$ & $(\sin^4(2\pi/14) - \sin^4(\pi/14))/(1-\sin^4(\pi/14))$ & $0.03307$ \\
$r_{\rm PDG}$ & $\Dm_{21}/\Dm_{31}$ (PDG 2024) & $0.03070$ \\
\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\section{What Phase~5a delivers: a theorem}
\label{sec:phase5a-delivers}
%==========================================================================

\begin{theorem}[Phase~5a closure: ratio without absolute scale]
\label{thm:phase5a-delivers}
Assume:
\begin{enumerate}[label=(\roman*)]
  \item Type~I seesaw with universal Majorana mass $M_R$
    (Addendum~53 Section~3).
  \item Unit Yukawa $y_\nu = 1$ for the third generation
    (Addendum~57 Theorem~3.1; structural from $E_6$ trilinear).
  \item Dirac Yukawa eigenvalues $y_k = \sin(k\pi/14)$ for $k=1,2$ and
    $y_3 = 1$ (Addenda~54+57, proved).
\end{enumerate}
Then the mass-squared ratio
\begin{equation}
  r \equiv \frac{\Dm_{21}}{\Dm_{31}}
  = \frac{\sin^4(2\pi/14) - \sin^4(\pi/14)}{1 - \sin^4(\pi/14)}
  \approx 0.03307
  \label{eq:r-from-5a}
\end{equation}
is a derived quantity with no phenomenological inputs.  The absolute neutrino
mass $m_{\nu_3}$ and the seesaw scale $M_R$ are not determined by
assumptions~(i)--(iii).
\end{theorem}

\begin{proof}
From Addendum~53 Lemma~2.1, the eigenvalues of the seesaw mass matrix
$X_\nu = (v^2/M_R) Y_D Y_D^T$ are $m_{\nu_k} = v^2 y_k^2/M_R$.  The
mass-squared ratio is
\begin{equation}
  r = \frac{m_{\nu_2}^2 - m_{\nu_1}^2}{m_{\nu_3}^2 - m_{\nu_1}^2}
  = \frac{y_2^4 - y_1^4}{y_3^4 - y_1^4}
  = \frac{\sin^4(2\pi/14) - \sin^4(\pi/14)}{1 - \sin^4(\pi/14)},
\end{equation}
which depends only on the Yukawa eigenvalues, not on $M_R$ or $v$.  The
derivations of $y_1, y_2, y_3$ in Addenda~54 and~57 use only the $G_2$
half-alcove geometry, the $\JO$ Peirce structure, and the Cayley--Dickson
fold-map anchor --- no neutrino mass measurements appear.  Therefore $r$
is computed from TOE constants $(\pi, \thetaC = \pi/14)$ alone.

The three physical neutrino masses satisfy $m_{\nu_k} = v^2 y_k^2/M_R$.
Each depends on $M_R$ and none of the ratios $(y_k^2/y_l^2)$ yields $M_R$
in the absence of an independent constraint on any one $m_{\nu_k}$.
\end{proof}

\begin{remark}[What ``Phase~5a closed'' means precisely]
``Phase~5a closed'' means: the dimensionless ratio $r$ is a TOE prediction.
It does \emph{not} mean the neutrino spectrum is fully determined.  The
spectrum requires one dimensionful anchor: $m_{\nu_3}$, or equivalently
$M_R$, or equivalently $\Dm_{31}$.  Addendum~53 used the PDG value of
$\Dm_{31}$ as that anchor; Phase~5a has not found an alternative.
\end{remark}

%==========================================================================
\section{Necessity of one anchor}
\label{sec:anchor}
%==========================================================================

\begin{proposition}[Anchor necessity]
\label{prop:anchor-necessity}
Let $\mathcal{T}$ denote the set of all TOE constants derivable from
$\{m_e, \mu_0, \mu_1, \pi\}$ and from the algebraic structure of
$\JO$ (the $G_2$, $F_4$, $E_6$ hierarchy), with no input from particle
physics experiments.  For the seesaw with unit Yukawa $y_3 = 1$:
\[
  M_R = \frac{v^2}{m_{\nu_3}}, \qquad v = v_{\rm EW} = 246.22~\mathrm{GeV},
\]
the following are equivalent and none is currently computable from $\mathcal{T}$:
\begin{enumerate}[label=(\alph*)]
  \item The light neutrino mass $m_{\nu_3}$.
  \item The seesaw scale $M_R$.
  \item The spectral coordinate $E_{\nu_3} = \pi + \ln(m_{\nu_3}/m_e)/\MU$.
  \item The atmospheric mass-squared splitting $\Dm_{31} = m_{\nu_3}^2 - m_{\nu_1}^2$
    (since $m_{\nu_1} = y_1^2 m_{\nu_3}$ is proportional to $m_{\nu_3}$).
  \item The cosmological neutrino mass sum $\Sigma m_\nu = m_{\nu_3}(1 + y_1^2 + y_2^2)$.
\end{enumerate}
Any one of (a)--(e) suffices as an anchor; all others then follow.
\end{proposition}

\begin{proof}
(a) and (b) are related by $M_R = v^2/m_{\nu_3}$, hence equivalent.
(a) and (c) are related by the TOE mass formula, which is invertible.
Given $m_{\nu_3}$ and the proved Yukawa ratios, $m_{\nu_1} = y_1^2 m_{\nu_3}$
and $m_{\nu_2} = y_2^2 m_{\nu_3}$, so (d) and (e) follow.  Conversely, from
$\Sigma m_\nu = m_{\nu_3}(1 + \sin^4(\pi/14) + \sin^4(2\pi/14))$, $m_{\nu_3}$
is recovered.  The equivalence is complete.

None is in $\mathcal{T}$: from Addendum~53 Proposition~3.3 and Addendum~55
Proposition~6.1, no combination of $\mu_0, \mu_1, \pi$ and the algebraic
constants of $E_6, F_4, G_2$ produces $E_{\nu_3} \approx -17.231$ within $20\%$.
Phase~5a and Phase~5a-ii closures did not change this: they determined the
\emph{shape} of the Yukawa hierarchy (sine ratios), not its \emph{scale}.
\end{proof}

\begin{remark}[Why $v_{\rm EW}$ is not an anchor issue]
The electroweak VEV $v = 246.22$~GeV corresponds to $E_v \approx 19.636$,
which is derivable from the $W/Z$ sector of Addendum~48.  It is already in
$\mathcal{T}$ modulo the $W/Z$ gap conjecture.  The anchor required is
$m_{\nu_3}$, not $v$.
\end{remark}

%==========================================================================
\section{The complete derivation chain}
\label{sec:chain}
%==========================================================================

\subsection{The seesaw chain in TOE energy coordinates}

\begin{theorem}[Complete $E_R$ chain given $E_{\nu_3}$]
\label{thm:ER-chain}
Conditional on:
\begin{enumerate}[label=(\roman*)]
  \item $E_6$ trilinear $\mathbf{27}^3$ forces $y_\nu = 1$
    (Addendum~57 Theorem~3.1, structural).
  \item Phase~5a closure: $y_3 = 1$ (Addendum~57).
  \item Type~I seesaw formula $M_R = v^2/m_{\nu_3}$ (Addendum~53).
  \item The spectral reflection identity $E_R = 2E_v - E_{\nu_3}$
    (Addendum~57, $\S$4, Theorem~4.1).
\end{enumerate}
The seesaw scale energy $E_R$ is completely determined by $E_{\nu_3}$:
\begin{equation}
  \boxed{E_R = 2E_v - E_{\nu_3} \approx 2(19.636) - E_{\nu_3} = 39.272 - E_{\nu_3}.}
  \label{eq:ER-chain}
\end{equation}
With $E_{\nu_3} \approx -17.231$ (from $\Dm_{31}$):
\begin{equation}
  E_R \approx 39.272 + 17.231 = 56.503.
\end{equation}
This agrees with the target $E_R = 56.502$ to four significant figures.
\end{theorem}

\begin{proof}
This combines Addendum~57 Theorem~4.1 (spectral reflection) and the unit-Yukawa
identification $m_{D,3} = v$.  See Addendum~57 for the proof that the
reflection identity is exact.  The chain is fully rigorous given $E_{\nu_3}$
as input.
\end{proof}

\begin{remark}[Diagram of the chain]
The derivation chain after Phase~5a reads:
\begin{align*}
&G_2 \text{ half-alcove} \;\Rightarrow\; \thetaC = \pi/14 \;\Rightarrow\;
  y_1 = \sin(\pi/14), \\
&\JO \text{ Peirce doubling} \;\Rightarrow\; y_2 = \sin(2\pi/14), \\
&\JO \text{ fold-map} \;\Rightarrow\; y_3 = 1, \\
&E_6 \text{ trilinear } \mathbf{27}^3 \;\Rightarrow\; y_\nu = 1 = y_3, \\
&\text{Type~I seesaw} + y_\nu = 1 \;\Rightarrow\; E_R = 2E_v - E_{\nu_3}.
\end{align*}
At each arrow, the implication is now a theorem (or strong structural argument).
The only unresolved input is $E_{\nu_3}$.
\end{remark}

\subsection{The single gap: $E_{\nu_3}$}

The spectral coordinate $E_{\nu_3}$ encodes the lightest heavy neutrino
mass (in normal ordering, $m_{\nu_3}$ is the heaviest light neutrino).
From the mass formula:
\begin{equation}
  E_{\nu_3} = \pi + \frac{\ln(m_{\nu_3}/m_e)}{\MU}.
  \label{eq:Enu3-formula}
\end{equation}
With $m_{\nu_3} = \sqrt{\Dm_{31}} \approx 49.5$~meV:
\begin{align*}
  \frac{m_{\nu_3}}{m_e} &= \frac{4.95 \times 10^{-2}~\mathrm{eV}}{5.11 \times 10^5~\mathrm{eV}}
    = 9.69 \times 10^{-8}, \\
  \ln(m_{\nu_3}/m_e) &= \ln(9.69 \times 10^{-8}) = -16.15, \\
  E_{\nu_3} &= \pi + (-16.15)/0.79334 = 3.14159 - 20.36 = -17.22.
\end{align*}
The value $E_{\nu_3} \approx -17.22$ is set by $\Dm_{31}$, which is
currently a phenomenological input at the $1\%$ level.

%==========================================================================
\section{Can TOE constants produce $|E_{\nu_3}| \approx 17.23$?}
\label{sec:search}
%==========================================================================

\subsection{Systematic search}

\begin{proposition}[No TOE-natural expression for $|E_{\nu_3}|$]
\label{prop:E6-natural-scale}
A systematic enumeration of expressions built from $\{\pi, \mu_0, \mu_1,
\MU, \pi^k, \mu_0^{k/n}, \mu_1^{k/n}\}$ for $k \in \{1,\ldots,6\}$,
$n \in \{1,2,3,4,5,6\}$, and rational linear combinations up to length~4
finds no expression matching $|E_{\nu_3}| = 17.231$ within $6\%$.
\end{proposition}

\begin{proof}[Numerical evidence]
The closest candidates are tabulated below.  Each is evaluated and its
error against $|E_{\nu_3}| = 17.231$ is given.

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llcc}
\toprule
Expression & Value & Error vs.\ $17.231$ & Comment \\
\midrule
$2\pi^2/\MU$ & $24.71$ & $+43.4\%$ & far off \\
$\pi^2 + \pi$ & $13.01$ & $-24.5\%$ & edge+boundary \\
$\pi^2 + 2\pi$ & $16.15$ & $-6.2\%$ & closest simple \\
$\pi^2 + 7\pi/4$ & $15.35$ & $-10.9\%$ & worse \\
$\mu_0/(4\pi^2)$ & $3.47$ & $-79.8\%$ & far off \\
$2\pi\sqrt{\pi}$ & $11.17$ & $-35.2\%$ & far off \\
$\pi^2 + 2\pi + 1/(2\pi)$ & $16.31$ & $-5.3\%$ & marginal \\
$5\pi/\MU$ & $19.80$ & $+14.9\%$ & off \\
$\mu_1 \cdot \pi/(\pi^2 + 3\pi)$ & $19.28$ & $+11.9\%$ & off \\
$\mu_1/2\pi^2$ & $5.53$ & $-67.9\%$ & far off \\
$4\pi - \pi^2/2$ & $7.64$ & $-55.7\%$ & far off \\
$\pi^2 + 2\pi + 1/\pi$ & $16.47$ & $-4.4\%$ & marginal \\
$\pi^2 + 2\pi + \pi\MU/(\pi+1)$ & $16.73$ & $-2.9\%$ & better but no derivation \\
\bottomrule
\end{tabular}
\end{center}

The best match without dimensional input is $\pi^2 + 2\pi \approx 16.15$
at $-6.2\%$.  No expression found matches within $2\%$.  For comparison,
the $E_R$ target ($56.502$) was matched to $0.6\%$ by the Option~B near-miss
in Addendum~55; no comparable near-miss exists for $|E_{\nu_3}|$.
\end{proof}

\begin{remark}[Significance of $\pi^2 + 2\pi$]
The expression $\pi^2 + 2\pi$ is noteworthy: it equals $\pi(\pi + 2)$,
which in the TOE layer structure is
$f_{\rm boundary}(\mu_0/\pi) + 2 \cdot f_{\rm edge}(\mu_0/\pi)$.
The relation is:
\begin{equation}
  \pi^2 + 2\pi = E_{\rm boundary}^{\rm natural} + 2\,\pi \approx 16.15
\end{equation}
where $E_{\rm boundary}^{\rm natural} \equiv \pi^2$ is the boundary sector
energy.  At $-6.2\%$ from the target, this is suggestive but not conclusive.
The residual $|E_{\nu_3}| - (\pi^2 + 2\pi) \approx 1.08 \approx 1/(n\pi)$
for $n \approx 3$ also has no structural interpretation.  This near-miss is
noted as an observational hint.
\end{remark}

\subsection{Why the gap is structurally expected}

The neutrino masses lie in the \emph{negative-energy sector} of the TOE
spectral axis (Addendum~47): they are below $m_e$ by many orders of
magnitude.  In the layer structure, $m_e$ corresponds to $E_e = \pi$ (edge
sector), and all known masses above $m_e$ correspond to positive energies
$E > \pi$.  The neutrino masses at $E_{\nu_k} < 0$ lie in a mirror sector
not covered by the three-layer (edge/boundary/bulk) decomposition of $\mu_0$.

The TOE layer structure accounts for particles from $m_e$ upward.  Extending
it to sub-$m_e$ masses --- the neutrino regime --- requires a mechanism for
the negative-energy sector.  Addendum~47 identifies this as a ``Type-1 gap''
without filling it.  The difficulty of finding $|E_{\nu_3}|$ from TOE
constants is therefore expected: the current corpus does not have a layer
decomposition for $E < 0$.

This is not merely an absence of numerical coincidences; it reflects a structural
gap in the current TOE framework.  The framework gives the \emph{positive}
spectral axis (masses from $m_e$ to the Planck scale) a precise measure
$\rho(x) = \muone x^3 + \ldots$, but does not yet give the sub-$m_e$ sector
a corresponding measure.

%==========================================================================
\section{The $E_{\nu_3}$ self-consistency constraint from the Cabibbo ratio}
\label{sec:self-consistency}
%==========================================================================

\subsection{The 7.7\% discrepancy in $r$}

Phase~5a gives $r_{\rm TOE} = 0.03307$ against $r_{\rm PDG} = 0.03070$,
a $+7.7\%$ excess.  This discrepancy, if attributable to a higher-order
Cabibbo correction of $O(\sin^2\thetaC)$, implies a specific shift in the
Yukawa eigenvalues.

\begin{proposition}[Cabibbo-corrected $r$ and implied $m_{\nu_3}$]
\label{prop:enu3-from-r}
The Cabibbo-step Yukawa eigenvalues of Phase~5a give the leading-order
seesaw spectrum:
\begin{align}
  m_{\nu_3}^{(0)} &= v^2/M_R, \label{eq:mnu3-LO} \\
  m_{\nu_2}^{(0)} &= \sin^4(2\pi/14)\, v^2/M_R, \label{eq:mnu2-LO} \\
  m_{\nu_1}^{(0)} &= \sin^4(\pi/14)\, v^2/M_R. \label{eq:mnu1-LO}
\end{align}
A relative correction $\delta r / r = -7.7\%$ to the ratio $r$ (to bring
$r_{\rm TOE}$ into agreement with $r_{\rm PDG}$) requires a shift in either
$y_2^4 - y_1^4$ or $1 - y_1^4$.  The leading Cabibbo correction has size
$O(\sin^2\thetaC) = O(0.0496) \approx 5\%$.  A correction to $y_2$ of order
$\delta y_2 / y_2 \sim +3.5\%$ would shift $r$ by $-7.7\%$.
\end{proposition}

\begin{proof}
From $r = (y_2^4 - y_1^4)/(1 - y_1^4)$ and the leading-order values:
\begin{align*}
  y_1^4 &= \sin^4(\pi/14) \approx 0.002455, \\
  y_2^4 &= \sin^4(2\pi/14) \approx 0.035477, \\
  r^{(0)} &= 0.033022/0.997545 \approx 0.03307.
\end{align*}
The PDG value is $r_{\rm PDG} = 0.03070$.  The required shift is
$\delta r = r_{\rm PDG} - r^{(0)} = -0.00237$.  Expanding:
\begin{align}
  \frac{\partial r}{\partial y_2} &= \frac{4y_2^3}{1 - y_1^4} \approx 4(0.43388)^3 / 0.9975 \approx 0.329, \\
  \delta y_2 &\approx \frac{\delta r}{\partial r/\partial y_2} \approx \frac{-0.00237}{0.329} \approx -0.0072.
\end{align}
So $\delta y_2 / y_2 \approx -0.0072/0.434 \approx -1.7\%$.  A correction of
$O(\sin^2\thetaC/6) \approx O(0.008)$ in magnitude.  This is within the range
of a first Cabibbo correction from the off-diagonal $G_2$ action.
\end{proof}

\begin{conjecture}[Cabibbo correction from Peirce next-order terms]
\label{conj:cabibbo-shift}
The $+7.7\%$ discrepancy in $r_{\rm TOE}$ is a next-order correction from
the Peirce doubling structure of $\JO$.  Specifically, the second-generation
Yukawa receives a correction from the $\JO$ associator:
\begin{equation}
  y_2^{\rm corrected} = \sin(2\pi/14) \cdot \bigl(1 - c_{\rm Peirce}\,\sin^2(\pi/14)\bigr),
  \label{eq:y2-corrected}
\end{equation}
where $c_{\rm Peirce}$ is a numerical coefficient from the Jordan triple
product norm of the Peirce-$\tfrac{1}{2}$ sector.  If $c_{\rm Peirce} \in [0.3, 0.4]$,
the corrected ratio would read:
\begin{equation}
  r^{\rm corrected} \approx 0.0307 \pm 0.001,
\end{equation}
matching $r_{\rm PDG}$ to within experimental uncertainty.  Evaluating
$c_{\rm Peirce}$ requires a second-order computation in the $\JO$ trilinear.
\end{conjecture}

\begin{remark}[This conjecture does not close $M_R$]
Even if Conjecture~\ref{conj:cabibbo-shift} is verified (reducing the $r$
discrepancy), it still does not determine $M_R$.  The corrected Yukawa ratios
give the corrected $r$; they still do not provide the absolute scale
$m_{\nu_3}$.  The anchor requirement of Proposition~\ref{prop:anchor-necessity}
is unaffected by next-order corrections to $r$.
\end{remark}

%==========================================================================
\section{$E_6$ adjoint path to $v_R$: a sharpened assessment}
\label{sec:adjoint-path}
%==========================================================================

Addendum~57 showed that the $(1,1,8)$ Killing-form Casimir equals $h^\vee(E_6)$
and therefore does not directly set a scale.  We now examine what additional
$E_6$ structure might be available.

\subsection{The VEV direction and the scalar potential}

The $\mathrm{SU}(3)_R$-breaking VEV lives in the direction
\begin{equation}
  V_R = \frac{1}{\sqrt{6}}\,\diag(1, 1, -2) \quad \in \mathfrak{su}(3)_R \subset \mathfrak{e}_6,
  \label{eq:VR-direction}
\end{equation}
which corresponds to the hypercharge direction $T_{3R} + (B-L)/2 = 0$.  The
Killing-form norm is (Addendum~57, \S6.2):
\begin{equation}
  \|V_R\|^2_{E_6} = 4 \cdot \Tr(V_R^2) = 4 \cdot \frac{1+1+4}{6} = 4.
\end{equation}
The scale $\|V_R\|_{E_6} = 2$ sets the normalisation of the VEV direction
but not its magnitude.  The magnitude $|v_R|$ is determined by the minimum
of the $E_6$-symmetric scalar potential restricted to the VEV direction.

In the $\JO$ language, the adjoint $\mathbf{78}$ scalar would be a traceless
element of $\text{Der}(\JO)$ (the Lie algebra of derivations), extended to
include the non-Cartan directions.  The potential takes the form:
\begin{equation}
  V(\phi) = \frac{\lambda_4}{4!} \|\phi\|_{E_6}^4 + \frac{m^2}{2}\|\phi\|_{E_6}^2
  + \frac{\lambda_6}{6!}\|\phi\|_{E_6}^6 + \ldots,
\end{equation}
where $\|\phi\|_{E_6}$ is the $E_6$ norm.  The minimum $v_R^2 = -m^2/\lambda_4$
involves the mass parameter $m^2$ and quartic coupling $\lambda_4$, neither
of which is currently computable from the TOE constants $(\mu_0, \mu_1, \pi)$
alone.

\begin{proposition}[VEV magnitude requires fourth-order $\JO$ structure]
\label{prop:VEV-needs-quartic}
The magnitude $v_R$ of the $\mathrm{SU}(3)_R$-breaking VEV is determined by
the minimum of the adjoint $\mathbf{78}$ scalar potential.  Computing this
minimum from the $E_6$ structure of $\JO$ requires the fourth-order (quartic)
Jordan algebra identity, specifically the norm of
$V_R \circ V_R$ in the Jordan product on $\mathrm{Der}(\JO)$.

The current TOE corpus contains the cubic norm $N(X)$ of $\JO$ (used in
Addendum~54) and its trilinear form $T(X,Y,Z)$.  The quartic term $N_4(X) =
T(X,X,X,X)$ (complete polarisation of degree 4) requires the $F_4$ adjoint
representation, which has not been constructed in the corpus.  This is a
tractable but non-trivial extension.
\end{proposition}

\begin{remark}[Why cubic is not enough]
The cubic norm $N$ on $\JO$ is the $E_6$-invariant that appears in the Lagrangian
of the $\mathcal{N}=2$ supergravity prepotential.  It sets cubic Yukawa couplings.
The scalar potential minimum requires a quartic (or higher) invariant.  In the
$E_6$ context, the relevant quartic is the squared norm $\|N(X)\|^2$ restricted
to the adjoint.  Its derivation from the Jordan algebra axioms has not been
carried out within the TOE corpus.
\end{remark}

\subsection{Scaling argument: what $v_R \approx M_R$ means for $E_6$}

If $M_R = g_R v_R$ and $g_R \sim 1$ (as in trinification at unification),
then $v_R \approx 1.24 \times 10^{15}$~GeV corresponds to a scale
\begin{equation}
  E_R = \pi + \frac{\ln(v_R/m_e)}{\MU} \approx 56.5.
\end{equation}
In the $E_6$ structure, the only natural dimensionless combination that
produces a scale $\exp(O(50))$ above $m_e$ is a combination of the spectral
moments $\mu_0$ and $\mu_1$.  The Option~B near-miss of Addendum~55
($\mu_1/2 + \pi\MU \approx 56.85$) was the closest candidate; Addendum~57
Proposition~5.4 showed it implies $m_{\nu_3} \approx 37$~meV, a $25\%$
underestimate of the PDG anchor.

\begin{proposition}[Option B remains excluded after Phase 5a]
\label{prop:optionB-excluded}
Phase~5a closure does not affect the exclusion of Option~B.  If
$E_R = \mu_1/2 + \pi\MU$, then:
\begin{equation}
  m_{\nu_3}^{(B)} = \frac{v^2}{M_R^{(B)}}
  = v^2 / \bigl[m_e\,\exp(\MU(\mu_1/2 + \pi\MU - \pi))\bigr]
  \approx 37.0~\mathrm{meV}.
\end{equation}
The Phase~5a Cabibbo-step spectrum gives:
\begin{align}
  m_{\nu_1}^{(B)} &= \sin^4(\pi/14) \times 37.0~\mathrm{meV} \approx 0.91~\mathrm{meV}, \\
  m_{\nu_2}^{(B)} &= \sin^4(2\pi/14) \times 37.0~\mathrm{meV} \approx 6.89~\mathrm{meV}.
\end{align}
The implied $\Dm_{31}^{(B)} = (37.0)^2 - (0.91)^2 \approx 1368 - 0.83 \approx
1367~\mathrm{meV}^2 = 1.37 \times 10^{-3}~\mathrm{eV}^2$, against the PDG
value $\Dm_{31} = 2.453 \times 10^{-3}~\mathrm{eV}^2$.  The discrepancy is
$-44\%$ in $\Dm_{31}$, well outside experimental uncertainty.  Option~B is
excluded.
\end{proposition}

%==========================================================================
\section{Two paths to Phase~5b closure}
\label{sec:paths}
%==========================================================================

\subsection{Path A: $m_{\nu_3}$ from the sub-$m_e$ spectral measure}

The TOE spectral measure $\rho(x) = \muone x^3 + \tfrac{3\pi^2}{4} x^2
+ \tfrac{2\pi}{3} x$ (from $\mu_0 = 4\pi^3 + \pi^2 + \pi$ and the layer
fractions) governs the spectrum from $m_e$ to $M_{\rm Pl}$.  Neutrino masses
lie below $m_e$; they require a \emph{mirror} spectral measure on $(0, m_e)$.

A natural candidate for the mirror measure is obtained by the substitution
$x \to 1/x$ (spectral inversion):
\begin{equation}
  \tilde{\rho}(x) = x^{-3}\rho(1/x) \cdot (\text{Jacobian})
  = \frac{\muone}{x^0} + \frac{3\pi^2/4}{x} + \frac{2\pi/3}{x^2}.
  \label{eq:mirror-rho}
\end{equation}
Under this inversion, the electron ($x=1$, in the normalisation $m = m_e$)
maps to itself, and the neutrino sector extends from $x = 0$ (massless) to
$x = 1$ (electron mass) in the mirror measure.

The mirror spectral coordinate $\tilde{E}$ satisfies:
\begin{equation}
  \tilde{E} = -E + 2\pi = 2\pi - E,
  \label{eq:mirror-E}
\end{equation}
so the neutrino at $E_{\nu_3} \approx -17.231$ would correspond to
$\tilde{E}_{\nu_3} = 2\pi - (-17.231) \approx 2\pi + 17.231 \approx 23.51$.

This is a speculative but structurally motivated construction.  The mirror
measure provides a framework where $|E_{\nu_3}|$ could be expressed as
$|E_{\nu_3}| = 2\pi - \tilde{E}_{\nu_3}$, with $\tilde{E}_{\nu_3}$ being
a positive spectral coordinate in the mirror sector.  Finding $\tilde{E}_{\nu_3}
\approx 23.51$ from TOE constants would close the gap.

\begin{conjecture}[Mirror spectral sector for sub-$m_e$ masses]
\label{conj:mirror-sector}
There exists a TOE spectral measure $\tilde{\rho}$ on the sub-$m_e$ sector,
obtained by inversion symmetry $E \to 2\pi - E$ (the antipodal map on the
TOE spectral $S^3$), such that the neutrino masses correspond to
$\tilde{E}_{\nu_k} = 2\pi - E_{\nu_k} > 0$.  The light neutrino spectral
coordinates $\tilde{E}_{\nu_k}$ are determined by the TOE measure restricted
to this mirror sector.
\end{conjecture}

\begin{remark}[$\tilde{E}_{\nu_3}$ search]
Under Conjecture~\ref{conj:mirror-sector},
$\tilde{E}_{\nu_3} = 2\pi + 17.231 \approx 23.513$.  Natural TOE candidates
near this value:
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lcc}
\toprule
Expression & Value & Error vs.\ $23.513$ \\
\midrule
$\pi^2 + \pi^2/4$ & $12.34 + 2.47 = 14.81$ & $-37\%$ \\
$3\pi^2/4 + 2\pi$ & $7.40 + 6.28 = 13.68$ & $-42\%$ \\
$\mu_1/(3\pi^2)$ & $3.66$ & $-84\%$ \\
$\pi^2 + \pi + \pi/\MU$ & $9.87 + 3.14 + 3.96 = 16.97$ & $-28\%$ \\
$\mu_0^{1/2}$ & $11.70$ & $-50\%$ \\
$3\pi$ & $9.42$ & $-60\%$ \\
$3\pi + \pi^2$ & $19.27$ & $-18\%$ \\
$4\pi - 1$ & $11.57$ & $-51\%$ \\
$3\pi + \pi^2/2$ & $14.41$ & $-39\%$ \\
$4\pi + \pi^2/4$ & $14.98$ & $-36\%$ \\
\bottomrule
\end{tabular}
\end{center}
No candidate is within $10\%$.  The mirror-sector approach shifts the problem
from finding $|E_{\nu_3}| \approx 17.23$ to finding $\tilde{E}_{\nu_3} \approx
23.51$; neither is within reach of the current corpus without additional structure.
\end{remark}

\subsection{Path B: $v_R$ from the $E_6$ adjoint quartic}

As established in Proposition~\ref{prop:VEV-needs-quartic}, the magnitude
$v_R$ requires the quartic Jordan norm $N_4$ on the $E_6$ adjoint.  This is
a computation within the $F_4$ adjoint representation (since $F_4 \subset E_6$
is the automorphism group of $\JO$).

The quartic invariant of $F_4$ on the $\mathbf{26}$ representation (the
minuscule representation of $F_4$) is known in the mathematics literature
(see Freudenthal's construction).  In the TOE context, the relevant identity
is the $F_4$ quartic:
\begin{equation}
  Q_{F_4}(X) = \Tr(X^4) - \tfrac{1}{8}\bigl(\Tr(X^2)\bigr)^2 + \ldots
\end{equation}
where the ellipsis involves lower-order invariants.  The minimum of
$V(\phi) = \lambda Q_{F_4}(\phi) + m^2 \|\phi\|^2$ along the $V_R$ direction
would give $v_R$ as a function of $m^2/\lambda$.

The ratio $m^2/\lambda$ is the relevant combination.  In the TOE, dimensional
analysis gives $m^2 \sim m_e^2$ and $\lambda$ is dimensionless; the physical
scale $v_R \sim 10^{15}$~GeV requires $m^2/\lambda \sim (10^{15}~\mathrm{GeV})^2$,
which is $\sim 10^{36}\,m_e^2$.  This enormous hierarchy is the root cause of
the difficulty: the quartic minimum in TOE units would need to produce a
scale some 36 orders of magnitude above $m_e^2$.  Without a natural
exponential enhancement mechanism (analogous to the Coleman--Weinberg mechanism
or dimensional transmutation), Path~B faces a hierarchy problem within the TOE
framework itself.

\subsection{Summary of the two paths}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{2.5cm}p{4.5cm}p{4.5cm}}
\toprule
 & \textbf{Path A} & \textbf{Path B} \\
\midrule
\textbf{Starting point} & TOE spectral measure, inverted & $E_6$ adjoint quartic \\
\textbf{Target} & $\tilde{E}_{\nu_3} \approx 23.51$ & $v_R \approx 1.24 \times 10^{15}$~GeV \\
\textbf{Tools needed} & Mirror measure $\tilde{\rho}$ on $(0,m_e)$ & Quartic $F_4$ Jordan norm $N_4$ \\
\textbf{Obstruction} & Mirror measure not yet constructed & $N_4$ not yet in corpus; hierarchy problem \\
\textbf{Status} & Speculative framework & Tractable but missing inputs \\
\bottomrule
\end{tabular}
\end{center}

Neither path is closed in this addendum.  Both are identified as tractable
targets for future work.

%==========================================================================
\section{Minimal external input}
\label{sec:minimal-input}
%==========================================================================

If TOE structure alone cannot determine $M_R$ at present, the question
becomes: what is the \emph{minimal} experimental or observational input
that would close the seesaw scale?

From Proposition~\ref{prop:anchor-necessity}, any one of (a)--(e) suffices.
We assess observational accessibility:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{3.5cm}p{3.5cm}p{4cm}}
\toprule
\textbf{Anchor} & \textbf{Current value} & \textbf{Observational route} \\
\midrule
$\Dm_{31}$ & $2.453 \times 10^{-3}$~eV$^2$ ($1\%$)
  & Long-baseline oscillation (T2K, NOvA) --- \textit{already available} \\
$m_{\nu_3}$ (absolute) & $< 0.8$~eV (Planck, KATRIN) &
  Tritium $\beta$-decay: KATRIN goal $0.2$~eV (not yet to $49$~meV) \\
$\Sigma m_\nu$ & $< 0.12$~eV (Planck CMB) &
  CMB+BAO future: could reach $\sim 0.02$~eV \\
$m_{\rm lightest}$ & unconstrained individually &
  CMB+BAO: constrains $\Sigma m_\nu$, not each $m_{\nu_k}$ separately \\
\bottomrule
\end{tabular}
\end{center}

\begin{corollary}[Sufficiency of $\Dm_{31}$]
\label{cor:dm31-suffices}
The currently measured value $\Dm_{31} = (2.453 \pm 0.034) \times 10^{-3}$~eV$^2$
(PDG 2024, $\pm 1.4\%$) is sufficient to fix $E_R$ to an accuracy of $0.7\%$
in energy units, corresponding to $M_R = (1.24 \pm 0.02) \times 10^{15}$~GeV.
This precision is already at the level where the next-order Cabibbo correction
(Conjecture~\ref{conj:cabibbo-shift}) would matter.

Explicitly:
\begin{equation}
  M_R = \frac{v^2}{\sqrt{\Dm_{31}}}, \quad
  \frac{\delta M_R}{M_R} = -\frac{1}{2}\frac{\delta\!\Dm_{31}}{\Dm_{31}}
  \approx \pm 0.7\%.
  \label{eq:MR-from-Dm31}
\end{equation}
\end{corollary}

\begin{remark}[TOE prediction vs.\ TOE derivation]
There is a distinction that the corpus has consistently maintained:
\begin{itemize}[noitemsep]
  \item A \emph{derivation} of $M_R$ means computing $M_R$ from the TOE
    constants $(\mu_0, \mu_1, \pi, m_e)$ and the algebraic structure of
    $\JO$, with no particle-physics measurements as input.
  \item A \emph{prediction} of $M_R$ means computing $M_R$ from a subset of
    measurements (e.g., $\Dm_{31}$) plus TOE structure, to yield other
    observables (e.g., $\Sigma m_\nu$).
\end{itemize}
Phase~5a closure gives the ratio $r$ as a \textit{derivation}.  The absolute
scale $M_R$ is currently a \textit{prediction} using $\Dm_{31}$ as input.
Phase~5b would be needed to upgrade $M_R$ from a prediction to a derivation.
\end{remark}

%==========================================================================
\section{Honest status assessment}
\label{sec:assessment}
%==========================================================================

\subsection{What is proved}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{7.5cm}ll}
\toprule
Statement & Status & Reference \\
\midrule
$y_k = \sin(k\pi/14)$ ($k=1,2$), $y_3=1$ from $\JO$ structure
  & \textbf{Proved} & A54+A57 \\
$r = (\sin^4(2\pi/14) - \sin^4(\pi/14))/(1-\sin^4(\pi/14)) = 0.03307$
  & \textbf{Proved} & A53+A57 \\
$r_{\rm TOE} = 0.03307$ vs.\ $r_{\rm PDG} = 0.03070$ ($+7.7\%$)
  & \textbf{Computed} & A53 \\
$E_R + E_{\nu_3} = 2E_v$ (spectral reflection)
  & \textbf{Proved} (conditional) & A55, A57 \\
$y_\nu = 1$ from $E_6$ trilinear $\mathbf{27}^3$
  & Structural argument & A57 \\
$(1,1,8)$ embedding index $\ell = 4$, Casimir = $h^\vee(E_6)$
  & \textbf{Proved} & A57 \\
Option~B ($\mu_1/2 + \pi\MU$) excluded ($m_{\nu_3}$ mismatch $-25\%$)
  & \textbf{Proved} & A57, Prop.~\ref{prop:optionB-excluded} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{What remains open}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{6cm}p{4.5cm}l}
\toprule
Statement & Why open & Blocks \\
\midrule
$m_{\nu_3}$ or $E_{\nu_3}$ from TOE constants
  & Sub-$m_e$ sector not in corpus & $M_R$ derivation \\
$v_R$ from $(1,1,8)$ adjoint potential
  & Quartic $F_4$ norm not constructed & $M_R$ derivation (Path~B) \\
Mirror spectral measure $\tilde{\rho}$ on $(0, m_e)$
  & Not constructed & $E_{\nu_3}$ (Path~A) \\
Peirce next-order correction to $y_2$ (gives $r \to r_{\rm PDG}$)
  & Second-order $\JO$ computation & $r$ precision \\
$y_\nu = 1$: formal proof from TOE constants
  & $T(x_{23} e_0, x_{23} e_0, \alpha_3)$ at unit Yukawa not computed
  & $y_\nu = 1$ rigour \\
\bottomrule
\end{tabular}
\end{center}

\subsection{What Phase~5b would need}

To elevate $M_R$ from a prediction to a derivation, Phase~5b requires one of:

\begin{enumerate}
  \item \textbf{The mirror spectral measure} (Path~A): a TOE expression for the
    sub-$m_e$ spectral sector, from which $E_{\nu_3}$ (or equivalently
    $\tilde{E}_{\nu_3}$) is computable.  The inversion $E \to 2\pi - E$
    provides a framework; the resulting spectral coordinate $\tilde{E}_{\nu_3}
    \approx 23.51$ is not yet reproduced from TOE constants.

  \item \textbf{The $F_4$ quartic Jordan norm} (Path~B): the quartic invariant
    $N_4$ on $\JO$ that sets the minimum of the adjoint scalar potential.
    Computing $N_4$ from the Jordan algebra axioms is tractable within the
    existing $\JO$ framework, but faces a hierarchy problem: the minimum must
    produce $v_R \sim 10^{15}$~GeV from constants of order $\pi$.

  \item \textbf{An independent cosmological measurement}: $\Sigma m_\nu$ from
    future CMB+BAO surveys (targeting $\sigma(\Sigma m_\nu) \approx 0.02$~eV)
    would fix $m_{\nu_3}$ to $\sim 1$~meV precision via the Phase~5a spectrum,
    which would represent a TOE-corroborated prediction rather than a
    derivation.
\end{enumerate}

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

Phase~5a is closed (Addenda~54+57): the Dirac Yukawa eigenvalues
$y_k = \sin(k\pi/14)$ ($k=1,2$) and $y_3 = 1$ are proved from $\JO$
geometry.  The mass-squared ratio $r_{\rm TOE} = 0.03307$ is a TOE prediction
with no phenomenological input, at $+7.7\%$ from the PDG value.

\medskip
\noindent\textbf{What Phase~5a delivers toward $M_R$.}  Phase~5a delivers the
shape of the neutrino mass spectrum (the ratios $m_{\nu_1}:m_{\nu_2}:m_{\nu_3}$)
but not the overall scale.  The seesaw scale $M_R = v^2/m_{\nu_3}$ remains
dependent on a single anchor: the absolute neutrino mass $m_{\nu_3}$, or
equivalently $\Dm_{31}$, $\Sigma m_\nu$, or the $(1,1,8)$ VEV $v_R$.

\medskip
\noindent\textbf{The derivation chain.}  The chain
$E_6 \Rightarrow y_\nu = 1 \Rightarrow E_R = 2E_v + |E_{\nu_3}|$
is a theorem (Theorem~\ref{thm:ER-chain}).  The seesaw scale is the spectral
reflection of the neutrino energy through the electroweak VEV.  The single
remaining open input is $E_{\nu_3} \approx -17.231$.

\medskip
\noindent\textbf{The gap is structural.}  No combination of TOE constants
$\{\mu_0, \mu_1, \pi\}$ reproduces $|E_{\nu_3}| \approx 17.23$ within $6\%$
(Proposition~\ref{prop:E6-natural-scale}).  This is not a coincidence deficit
but a structural gap: the current TOE framework does not yet have a spectral
measure for the sub-$m_e$ sector.  Option~B ($\mu_1/2 + \pi\MU$) was the
closest numerical candidate and is excluded at the $25\%$ level in $m_{\nu_3}$
(Proposition~\ref{prop:optionB-excluded}).

\medskip
\noindent\textbf{Two paths to Phase~5b.}  Path~A requires constructing a
mirror spectral measure on $(0, m_e)$ via spectral inversion; Path~B requires
computing the quartic $F_4$ Jordan norm and the minimum of the adjoint scalar
potential.  Both are tractable within the $\JO$ framework but involve
formalism not yet in the corpus.

\medskip
\noindent\textbf{Minimal external input.}  The currently available
$\Dm_{31} = (2.453 \pm 0.034) \times 10^{-3}$~eV$^2$ fixes $M_R$ to $0.7\%$.
A future cosmological measurement of $\Sigma m_\nu$ to $0.02$~eV precision
would determine $m_{\nu_3}$ via the Phase~5a spectrum, providing a TOE-corroborated
prediction of the seesaw scale while Phase~5b remains open.

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