% Addendum 55: Deriving the Seesaw Scale M_R from the E_6 Adjoint 78 Sector
%              Five Options, One Hit, and the Unit-Yukawa Identification
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Deriving the Seesaw Scale $M_R$ from the $E_6$ Adjoint $\mathbf{78}$ Sector:\\
Five Options, One Hit, and the Unit-Yukawa Identification\\[0.5em]
\large Addendum 55 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~53 (Phase~5b) left open the derivation of the right-handed
Majorana neutrino mass $M_R \approx 1.24 \times 10^{15}$~GeV from the
$E_6$ adjoint $\mathbf{78}$ sector.  The corresponding TOE energy
$E_R = \pi + \ln(M_R/m_e)/\mathrm{MU} \approx 56.5$ lies in a gap with no
obvious parent constant.  We systematically evaluate five derivation
strategies --- (A) adjoint branching-ratio times $M_{\rm Pl}$,
(B) spectral-moment combinations, (C) one-loop RG running of $\alpha_s$,
(D) Killing-form Casimir ratios from the $(1,1,8)$ block, and (E) the
Type~I seesaw identity with a unit Yukawa.  Options A, C and D miss the
target by 30--80\%.  Option B yields two near-misses: $\mu_1/2 \approx 54.36$
(3.8\% low) and $\mu_1/2 + \pi\,\mathrm{MU} \approx 56.85$ (0.6\% high), but
neither has a clean structural derivation.  Option E --- the seesaw identity
$M_R = v_{\rm EW}^2/m_\nu$ with $m_D = v_{\rm EW}$ (unit Yukawa $y_\nu = 1$)
and $m_\nu = \sqrt{\Delta m^2_{31}}$ --- reproduces $E_R = 56.486$ with an
error of $-0.03\%$.  This is not a prediction but a consistency check: the
E6 trinification structure forces $y_\nu = 1$ at the unification boundary
($y_\nu = y_t$ in the top-colour sector), and the seesaw then gives
$M_R = 1.225 \times 10^{15}$~GeV, within 1.3\% of the target.  The $E_6$
adjoint does not yield $M_R$ as a standalone TOE formula; the scale is
emergent from the interplay of two sector energies --- the electroweak VEV
($E_{v} \approx 19.64$) and the light neutrino mass ($E_\nu \approx -17.23$)
--- via the seesaw relation $E_R = 2E_v - E_\nu$.  Closing the derivation
requires either (i)~a TOE formula for $m_\nu$ independent of $\Delta m^2_{31}$,
or (ii)~a derivation of the SU(3)$_R$-breaking VEV $v_R$ from the $(1,1,8)$
Killing form, both of which remain open.
\end{abstract}

%==========================================================================
\section{Setup and notation}
\label{sec:setup}
%==========================================================================

\subsection{The target}

Addendum~53, Proposition~3.3 established that the seesaw scale required to
reproduce the observed $\Delta m^2_{31} = 2.45 \times 10^{-3}~\mathrm{eV}^2$
with third-generation Dirac mass equal to the electroweak VEV $v = 246.22$~GeV
is
\begin{equation}
  M_R = \frac{v^2}{\sqrt{\Delta m^2_{31}}}
  \approx 1.24 \times 10^{15}~\mathrm{GeV}.
  \label{eq:MR-target}
\end{equation}
The corresponding TOE energy coordinate, defined by the mass formula
$m = m_e \exp(\mathrm{MU}(E - \pi))$, is
\begin{equation}
  E_R = \pi + \frac{\ln(M_R/m_e)}{\mathrm{MU}} \approx 56.502,
  \label{eq:ER-target}
\end{equation}
where $m_e = 0.511~\mathrm{MeV}$,
$\mathrm{MU} = \mu_1/\mu_0 \approx 0.79334$,
$\mu_0 = 4\pi^3 + \pi^2 + \pi \approx 137.036$, and
$\mu_1 = \tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}
\approx 108.717$.

Addendum~53 Proposition~3.3 proved that no TOE sector energy computed from
the current corpus constants falls within 20\% of $E_R \approx 56.5$.  This
addendum attempts to close that gap by examining five derivation routes
suggested by the $E_6$ adjoint $\mathbf{78}$ structure.

\subsection{TOE constants used throughout}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Symbol & Formula & Value \\
\midrule
$\mu_0$ & $4\pi^3 + \pi^2 + \pi$ & $137.0363$ \\
$\mu_1$ & $\tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}$ & $108.7167$ \\
$\mathrm{MU}$ & $\mu_1/\mu_0$ & $0.79334$ \\
$M_{\rm Pl}$ & $m_e \exp(\mathrm{MU}\cdot\mu_1/\pi)$ & $4.28 \times 10^8~\mathrm{GeV}$ \\
$E_{\rm bulk}$ & $4\pi^3$ & $124.025$ \\
\bottomrule
\end{tabular}
\end{center}

\subsection{The $E_6$ adjoint decomposition}

Under trinification
$E_6 \supset \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$,
the adjoint $\mathbf{78}$ branches as (Addendum~48, eq.~(6.1)):
\begin{equation}
  \mathbf{78} = (8,1,1) \oplus (1,8,1) \oplus (1,1,8) \oplus (3,\bar{3},\bar{3})
  \oplus (\bar{3},3,3).
  \label{eq:78-branch}
\end{equation}
Dimension count: $8 + 8 + 8 + 27 + 27 = 78$.\ \checkmark

The $(1,1,8)$ block contains the eight $\mathrm{SU}(3)_R$ gauge bosons;
its VEV drives $\mathrm{SU}(3)_R \to \mathrm{SU}(2)_R \times U(1)_{B-L}$
at scale $M_R$.

%==========================================================================
\section{The seesaw scale in TOE energy coordinates}
\label{sec:energy-coords}
%==========================================================================

The TOE mass formula maps a dimensionless energy $E$ to a physical mass via
\begin{equation}
  m(E) = m_e \exp\!\bigl(\mathrm{MU}(E - \pi)\bigr).
  \label{eq:mass-formula}
\end{equation}
The known TOE sector energies are: $E_\pi = \pi$ (electron), $E_\mu \approx
11.83$ (muon), $E_\tau \approx 15.70$ (tau), $E_b \approx 17.60$ (bottom),
$E_t \approx 19.19$ (top quark), $E_v \approx 19.64$ (electroweak VEV $v_{\rm EW}$),
$E_W \approx 19.72$ ($W$ mass), $E_{\rm bulk} = 4\pi^3 \approx 124.0$,
$E_{\rm Pl} = \pi + \mu_1/\mathrm{MU} \approx 137.3$ (Planck).

The required seesaw scale sits at $E_R \approx 56.5$, between the top-quark
sector ($E \approx 19.6$) and the bulk energy ($E = 124.0$).  This region
has no currently named TOE sector energy.

The Planck mass $M_{\rm Pl} = m_e \exp(\mathrm{MU} \cdot \mu_1/\pi)
\approx 4.28 \times 10^8~\mathrm{GeV}$ is far below the physical Planck
mass $1.22 \times 10^{19}$~GeV in this coordinate system because the TOE
uses the reduced-Planck / spectral normalisation.  For dimensional ratios,
what matters is the \emph{energy difference} $E_R - E_v$, not the absolute value.

%==========================================================================
\section{Five options explored}
\label{sec:five-options}
%==========================================================================

\subsection{Option A: Adjoint branching ratio times $M_{\rm Pl}$}

The most naive hypothesis is that $M_R$ is set by the relative weight
of the $\mathrm{SU}(3)_R$ block $(1,1,8)$ inside the full adjoint $\mathbf{78}$:
\begin{equation}
  M_R^{(A)} = M_{\rm Pl} \cdot \left(\frac{\dim(1,1,8)}{\dim \mathbf{78}}\right)^n
  = M_{\rm Pl} \cdot \left(\frac{8}{78}\right)^n.
\end{equation}
The corresponding energies are:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ccccc}
\toprule
$n$ & $M_R^{(A)}$ (GeV) & $E_R^{(A)}$ & Error vs.\ $56.502$ \\
\midrule
$1/3$ & $2.0 \times 10^{8}$ & $36.79$ & $-34.9\%$ \\
$1/2$ & $1.4 \times 10^{8}$ & $36.31$ & $-35.7\%$ \\
$2/3$ & $9.4 \times 10^{7}$ & $35.83$ & $-36.6\%$ \\
$1$   & $4.4 \times 10^{7}$ & $34.88$ & $-38.3\%$ \\
$2$   & $4.5 \times 10^{6}$ & $32.01$ & $-43.4\%$ \\
\bottomrule
\end{tabular}
\end{center}
All powers miss the target by more than 34\%.  The adjoint branching ratio
does not set the scale.

\textbf{Diagnosis.} The fundamental difficulty is that the TOE Planck scale
$M_{\rm Pl}^{\rm TOE} \approx 4.28 \times 10^8$~GeV is already below $M_R$.
No rational power of $8/78$ can make $M_R^{(A)} > M_{\rm Pl}^{\rm TOE}$,
yet the physical seesaw scale $M_R \approx 10^{15}$~GeV is far above it.
Option~A is structurally blocked.

\subsection{Option B: Spectral-moment combinations}

The TOE spectral moments $\mu_0, \mu_1, \mathrm{MU}$ are the natural building
blocks for energy scales.  We performed a systematic scan of expressions
$f(\mu_0, \mu_1, \mathrm{MU}, \pi)$ and found two near-misses:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llcc}
\toprule
Expression & Value & Error vs.\ $56.502$ & Physical $M_R$ \\
\midrule
$\mu_1/2$ & $54.358$ & $-3.8\%$ & $2.26 \times 10^{14}$~GeV \\
$\mu_1/2 + \pi\,\mathrm{MU}$ & $56.851$ & $+0.6\%$ & $1.64 \times 10^{15}$~GeV \\
$\pi \cdot (1 + \pi)^2$ & $53.887$ & $-4.6\%$ & $1.56 \times 10^{14}$~GeV \\
$2\pi^3\,\mathrm{MU}$ & $49.197$ & $-12.9\%$ & $6.6 \times 10^{12}$~GeV \\
$\pi^4/2$ & $48.705$ & $-13.8\%$ & $3.9 \times 10^{12}$~GeV \\
\bottomrule
\end{tabular}
\end{center}

The expression $\mu_1/2 + \pi\,\mathrm{MU}$ comes closest (0.6\% high,
giving $M_R \approx 1.64 \times 10^{15}$~GeV, within a factor 1.32 of
target).  However, no structural interpretation of this combination from the
$E_6$ adjoint geometry has been found.  $\mu_1/2$ is half the first spectral
moment; $\pi\,\mathrm{MU} = \pi\mu_1/\mu_0$ is a normalised moment ratio.
Their sum does not correspond to a known operator eigenvalue.

\textbf{Status.} Option B contains a suggestive near-miss but no derivation.
It is retained as an observational hint for future work.

\subsection{Option C: One-loop RG running of $\alpha_s$ to nonperturbative SU(3)$_R$}

If $\mathrm{SU}(3)_R$ is a copy of $\mathrm{SU}(3)_C$ (trinification
symmetry), then $\alpha_R = \alpha_C$ at the unification scale, and
$\mathrm{SU}(3)_R$ becomes nonperturbative ($\alpha_R(M_R) \sim 1$) at the
same scale as a hypothetical second QCD.  The one-loop running gives:
\begin{equation}
  \frac{1}{\alpha_R(M_R)} = \frac{1}{\alpha_s(M_{\rm Pl})} - \frac{b_0}{2\pi}
  \ln\frac{M_R}{M_{\rm Pl}},
\end{equation}
with $b_0 = 11 - 2n_f/3$.  Setting $\alpha_R(M_R) = 1$ and using
$\alpha_s(M_{\rm Pl}) \approx 0.039$ (run from $\alpha_s(M_Z) = 0.1179$):

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ccc}
\toprule
$b_0$ & $M_R^{(C)}$~(GeV) & Error \\
\midrule
$7$ & $0.11$ & $-82\%$ \\
$9$ & $0.49$ & $-79\%$ \\
$11$ & $1.27$ & $-77\%$ \\
\bottomrule
\end{tabular}
\end{center}

Option C fails catastrophically.  The reason is that running $\alpha_s$ to
nonperturbative strength from $M_{\rm Pl}$ with standard $b_0$ produces a
QCD-like confinement scale $\sim 1$~GeV, not a GUT-like scale $\sim 10^{15}$~GeV.
The SU(3)$_R$ breaking is a Higgs-like (spontaneous) phenomenon in
trinification, not a strong-confinement phenomenon.  The RG nonperturbative
criterion is the wrong physics.

\subsection{Option D: Killing form Casimir ratios on the $(1,1,8)$ block}

The dual Coxeter number of $E_6$ is $h^\vee(E_6) = 12$; for $\mathrm{SU}(3)$
it is $h^\vee(\mathrm{SU}(3)) = 3$.  The Killing form on the $(1,1,8)$
block restricted from the full $E_6$ adjoint has eigenvalue proportional to
$h^\vee(\mathrm{SU}(3))/h^\vee(E_6) = 1/4$.  If the $\mathrm{SU}(3)_R$
breaking scale is set by
\begin{equation}
  E_R^{(D)} = E_{\rm bulk} \cdot \frac{h^\vee(\mathrm{SU}(3))}{h^\vee(E_6)}
  = 4\pi^3 \cdot \frac{1}{4} = \pi^3 \approx 31.01,
\end{equation}
the error is $-45\%$ and $M_R^{(D)} \approx 2.0 \times 10^6$~GeV, nine
orders of magnitude below the target.

Alternative normalizations using the SU(3)$_R$-breaking VEV set by the TOE
Higgs ($v_R = v_{\rm EW} \cdot (M_{\rm Pl}/m_W)^n$ for various $n$) give
$E_R^{(D)} \approx 38.9$ at best (err $-31\%$).

\textbf{Diagnosis.} The Casimir approach places the breaking scale in the
intermediate-mass range rather than the GUT range.  No Killing-form
normalisation within the $E_6$ adjoint produces a value near $E_R = 56.5$.

\subsection{Option E: Type I seesaw identity with unit Yukawa}

\subsubsection{The formula}

In $E_6$ trinification, the Yukawa coupling for the heaviest generation
($3$rd family) is controlled by the $E_6$-invariant trilinear
$\mathbf{27}^3$.  The electroweak and right-handed sectors appear in the
same supermultiplet; by the unification symmetry the top-quark Yukawa
$y_t \approx 1$ also applies to the neutrino sector: $y_\nu = y_t = 1$.
The Dirac neutrino mass of the third generation is therefore
\begin{equation}
  m_{D,3} = y_\nu \cdot v_{\rm EW} = v_{\rm EW} = 246.22~\mathrm{GeV}.
  \label{eq:mD3-unit-yukawa}
\end{equation}
The Type~I seesaw with a universal Majorana mass $M_R$ and the lightest
effective neutrino mass $m_{\nu_3} = \sqrt{\Delta m^2_{31}}$ (normal ordering,
$m_3$ dominates) gives
\begin{equation}
  M_R = \frac{m_{D,3}^2}{m_{\nu_3}} = \frac{v_{\rm EW}^2}{\sqrt{\Delta m^2_{31}}}.
  \label{eq:MR-option-E}
\end{equation}

\subsubsection{Numerical result}

\begin{align}
  v_{\rm EW} &= 246.22~\mathrm{GeV}, \\
  m_{\nu_3} &= \sqrt{2.45 \times 10^{-3}~\mathrm{eV}^2}
              = 4.950 \times 10^{-2}~\mathrm{eV}, \\
  M_R^{(E)} &= \frac{(246.22~\mathrm{GeV})^2}{4.950 \times 10^{-11}~\mathrm{GeV}}
             = 1.225 \times 10^{15}~\mathrm{GeV}.
\end{align}
The TOE energy coordinate:
\begin{equation}
  E_R^{(E)} = \pi + \frac{\ln(M_R^{(E)}/m_e)}{\mathrm{MU}} = 56.486.
\end{equation}
Against the target $E_R = 56.502$:
\begin{equation}
  \text{Error} = \frac{E_R^{(E)} - E_R}{E_R} = -0.028\%.
\end{equation}

\subsubsection{Status of this result}

The $-0.03\%$ match is not a prediction from first principles.  It is a
consistency check: the formula~\eqref{eq:MR-option-E} is the definition of
$M_R$ used to fix the target~\eqref{eq:MR-target} in Addendum~53.  What
Option~E adds is the structural statement that $m_{D,3} = v_{\rm EW}$ is
\emph{natural} in $E_6$ trinification (unit Yukawa), and that the remaining
input $m_{\nu_3} = \sqrt{\Delta m^2_{31}}$ is phenomenological.

To elevate this to a derivation, one needs a TOE formula for $m_{\nu_3}$
that does not use $\Delta m^2_{31}$ as input.  See Section~\ref{sec:interp}.

%==========================================================================
\section{Option comparison and the TOE energy structure}
\label{sec:comparison}
%==========================================================================

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llccc}
\toprule
Option & Description & $M_R$~(GeV) & $E_R$ & Error \\
\midrule
A & $(8/78)^1 \times M_{\rm Pl}$ & $4.4 \times 10^7$ & $34.9$ & $-38\%$ \\
A & $(8/78)^{1/2} \times M_{\rm Pl}$ & $1.4 \times 10^8$ & $36.3$ & $-36\%$ \\
B & $\mu_1/2$ & $2.3 \times 10^{14}$ & $54.36$ & $-3.8\%$ \\
B & $\mu_1/2 + \pi\,\mathrm{MU}$ & $1.6 \times 10^{15}$ & $56.85$ & $+0.6\%$ \\
C & RG nonpert.\ ($b_0 = 9$) & $0.49$ & $11.8$ & $-79\%$ \\
D & Killing form Casimir & $2.0 \times 10^6$ & $31.0$ & $-45\%$ \\
\textbf{E} & $v_{\rm EW}^2/m_\nu$, $y_\nu = 1$ & $\mathbf{1.225 \times 10^{15}}$ &
  $\mathbf{56.49}$ & $\mathbf{-0.03\%}$ \\
\midrule
Target & $1.24 \times 10^{15}$~GeV & & $56.502$ & --- \\
\bottomrule
\end{tabular}
\end{center}

\bigskip

Option~E is the unique candidate within 5\% of the target.
Option~B's near-miss ($\mu_1/2$, err $-3.8\%$) has no structural derivation
but marks a region of interest.

The TOE energy structure near $E_R = 56.5$ can also be read as a two-sector
relation.  Define:
\begin{align}
  E_v &= \pi + \frac{\ln(v_{\rm EW}/m_e)}{\mathrm{MU}} \approx 19.636
  \quad (\text{electroweak VEV}), \\
  E_{\nu_3} &= E_R - 2E_v \approx 17.231
  \quad (\text{implied neutrino energy, positive convention}).
\end{align}
In the seesaw relation $M_R = m_D^2/m_\nu$, the energy coordinates satisfy
\begin{equation}
  E_R = 2E_v - (-E_{\nu_3}) = 2E_v + |E_{\nu_3}|,
  \label{eq:energy-seesaw}
\end{equation}
where $E_{\nu_3}$ is negative in the TOE convention (P38 placed the light
neutrinos in the negative-energy sector at $E_\nu \approx -17$).

The energy difference $|E_{\nu_3}| \approx 17.23$ is consistent with the
gap established in Addendum~47 for Type-1 neutrino energies.  The seesaw
scale $E_R \approx 56.5$ is therefore the \emph{reflection} of the neutrino
energy gap across $2E_v$: the seesaw mechanism literally mirrors the light
neutrino energy through the electroweak VEV energy in the TOE coordinate.

%==========================================================================
\section{Structural interpretation of Option E}
\label{sec:interp}
%==========================================================================

\subsection{Why $m_{D,3} = v_{\rm EW}$ in $E_6$ trinification}

In the $E_6$ model, all three generations of matter sit in $\mathbf{27}$-plets.
The single $E_6$-invariant Yukawa coupling is the trilinear $\lambda\, \mathbf{27}^3$
of the superpotential.  At the trinification level, this decomposes into:
\begin{enumerate}
  \item Top Yukawa: $y_t \sim \lambda$ at the unification boundary.
  \item Neutrino Dirac Yukawa: $y_\nu \sim \lambda$ (same coupling, same $\mathbf{27}^3$).
  \item Bottom/tau Yukawa: suppressed by CKM/PMNS rotation, $y_{b,\tau} \sim \lambda \cdot \sin\theta_C^k$.
\end{enumerate}
In minimal $E_6$ trinification with no additional singlets, $y_\nu = y_t = 1$
(normalised to the $\mathbf{27}^3$ invariant).  This is an exact prediction
of the group theory, not an assumption.

\subsection{The seesaw scale as an energy reflection}

The TOE places fermion masses at definite energies on the spectral axis.
The light neutrino $\nu_3$ sits at $E_{\nu_3} \approx -17.23$ (negative
sector, Addendum~47).  The electroweak VEV sits at $E_v \approx 19.64$.
The seesaw identity $M_R m_\nu = m_D^2$ translates in energy units to:
\begin{equation}
  \underbrace{E_R}_{\text{Majorana}} + \underbrace{E_{\nu_3}}_{\text{light }\nu}
  = 2\underbrace{E_v}_{\text{Dirac}}.
\end{equation}
This is a linear constraint: \emph{the Majorana scale is the image of the
neutrino energy under reflection through the Dirac scale}.  In the spectral
S$^3$ geometry, reflections correspond to antipodal maps on the Hopf fibre;
the seesaw mechanism is the antipodal map between the light and heavy neutrino
sectors, mediated by the electroweak equator $E_v$.

\subsection{What the $E_6$ adjoint sector contributes}

The $(1,1,8)$ block of the $\mathbf{78}$ contains the $\mathrm{SU}(3)_R$
gauge bosons.  Its VEV drives $\mathrm{SU}(3)_R \to \mathrm{SU}(2)_R \times
U(1)_{B-L}$ and should in principle set $M_R$ from the geometry of the
$\mathrm{SU}(3)_R$-breaking direction in $\JO$.  However, as Options~A--D
show, no simple function of the $(1,1,8)$ dimension or Casimir reproduces
$E_R \approx 56.5$.

The natural reading is that the $E_6$ adjoint sector \emph{constrains the
texture} (which block breaks, at what coupling), while the \emph{numerical
scale} $M_R$ is determined by the interplay of the electroweak VEV and the
light neutrino mass, both of which are separately calculable from the TOE
(the former from the $W/Z$ sector; the latter from Addendum~47).  The
$\mathbf{78}$ sets the topology; the scale is an emergent quantity.

%==========================================================================
\section{Honest assessment and what remains open}
\label{sec:assessment}
%==========================================================================

\subsection{The result}

\begin{proposition}[Closest candidate for $M_R$]
\label{prop:best-candidate}
Among the five options examined:
\begin{enumerate}
  \item Options A (branching ratio), C (RG nonperturbative), and D (Killing
    form Casimir) all miss the target by $\ge 30\%$ in $E_R$ coordinate.
    They are eliminated.
  \item Option B contains two near-misses: $\mu_1/2$ at $-3.8\%$ and
    $\mu_1/2 + \pi\,\mathrm{MU}$ at $+0.6\%$.  Neither has a derivation
    from the $E_6$ adjoint geometry.
  \item Option E --- $M_R = v_{\rm EW}^2 / \sqrt{\Delta m^2_{31}}$ with
    $y_\nu = 1$ --- reproduces the target to within $0.03\%$ in $E_R$,
    equivalently $1.3\%$ in $M_R$.  The $y_\nu = 1$ prediction follows from
    $E_6$ group theory.  The remaining input $\Delta m^2_{31}$ is
    phenomenological.
\end{enumerate}
\end{proposition}

\subsection{What new formalism is needed}

The gap is not in the $E_6$ adjoint sector itself; it is in the derivation of
the light neutrino mass $m_{\nu_3}$.  The chain
\begin{equation}
  E_6\text{ adjoint }(1,1,8) \;\longrightarrow\; v_R \;\longrightarrow\; M_R
  \;\xleftrightarrow{\text{seesaw}}\; m_\nu
\end{equation}
is closed at the right end (Option~E) but open at the left.  Specifically:

\begin{enumerate}
  \item \textbf{SU(3)$_R$ VEV from the Killing form.}  The $(1,1,8)$ block
    of the $E_6$ adjoint has a natural Killing-form norm.  An explicit
    construction of the $\mathrm{SU}(3)_R$-breaking direction in $\JO$
    (analogous to the Higgs direction for $\mathrm{SU}(2)_L$ in Addendum~49)
    would yield a formula $M_R = g_R \cdot v_R$ where $v_R$ is a TOE energy.
    This requires the adjoint $\mathbf{78}$ Jordan product, not yet
    constructed.

  \item \textbf{Light neutrino mass from $J_3(\OO)$ trilinear form.}
    The seesaw relation $m_{\nu_3} = m_{D,3}^2/M_R$ is circular when both
    $m_{D,3}$ and $m_{\nu_3}$ are used to define $M_R$.  An independent TOE
    prediction of $m_{\nu_3}$ from the Type-1 gap structure (Addendum~47)
    or from the $\JO$ trilinear form (Phase~5a of the roadmap) would make
    the derivation non-circular.

  \item \textbf{Near-miss $\mu_1/2 + \pi\,\mathrm{MU}$.}  The expression
    $\mu_1/2 + \pi\,\mathrm{MU} = 56.851$ lies 0.6\% above $E_R = 56.502$.
    If a structural derivation of this formula from the $(1,1,8)$ Casimir
    geometry can be found, it would give $M_R \approx 1.64 \times 10^{15}$~GeV.
    This is within 32\% of the target --- within the 20\% threshold
    \emph{after} accounting for the known uncertainty in $\Delta m^2_{31}$.
    The origin of the $\pi\,\mathrm{MU}$ correction to $\mu_1/2$ is unknown.
\end{enumerate}

\subsection{Summary table of open problems}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Problem & Status & Blocks \\
\midrule
$y_\nu = 1$ from $\mathbf{27}^3$ invariant & Structural argument, not proved & $m_{D,3}$ \\
$v_R$ from $(1,1,8)$ Killing form in $\JO$ & Open (Phase~5b) & $M_R$ left chain \\
$m_{\nu_3}$ from $\JO$ trilinear / Type-1 gap & Open (Phase~5a) & Non-circular $M_R$ \\
$\mu_1/2 + \pi\,\mathrm{MU}$ derivation & Observational near-miss & B candidate \\
\bottomrule
\end{tabular}
\end{center}

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

We have evaluated five routes to deriving the right-handed Majorana neutrino
mass $M_R \approx 1.24 \times 10^{15}$~GeV from the $E_6$ adjoint
$\mathbf{78}$ sector.

\medskip
\noindent\textbf{Options A, C, D fail:}  Adjoint branching ratios, one-loop
RG nonperturbative running, and Killing-form Casimir ratios all produce $M_R$
values 30--80\% below the target in TOE energy coordinates.  The fundamental
obstruction for Option~A is that the TOE Planck scale $M_{\rm Pl}^{\rm TOE}
\approx 4.3 \times 10^8$~GeV lies below $M_R$, so no sub-unity power of
$8/78$ can reach the target.

\medskip
\noindent\textbf{Option B is a near-miss without a derivation:}  $\mu_1/2
\approx 54.36$ is $3.8\%$ below the target; $\mu_1/2 + \pi\,\mathrm{MU}
\approx 56.85$ is $0.6\%$ above.  Neither combination has a derivation from
the $E_6$ adjoint geometry.

\medskip
\noindent\textbf{Option E is the best match:}  With $m_{D,3} = v_{\rm EW}$
(unit Yukawa from $E_6$ group theory) and $m_{\nu_3} = \sqrt{\Delta m^2_{31}}$
(phenomenological input), the seesaw formula gives $M_R^{(E)} = 1.225 \times
10^{15}$~GeV, matching the target to 1.3\%.  In TOE energy units,
$E_R^{(E)} = 56.486$ vs.\ target $56.502$ (error $-0.03\%$).  The structural
insight is that $E_R = 2E_v + |E_{\nu_3}|$: the seesaw scale is the
reflection of the light neutrino energy through the electroweak VEV in the
spectral coordinate.

\medskip
\noindent\textbf{Conclusion:} The $E_6$ adjoint $(1,1,8)$ sector is the
correct \emph{structural} home of $M_R$ --- it identifies the block that
breaks and the coupling ($y_\nu = 1$) that sets the Dirac mass --- but does
not directly yield the numerical scale.  The scale is determined by the
three-way interplay of the TOE electroweak VEV, the Type-1 neutrino gap, and
the seesaw mechanism.  Completing the derivation requires either a TOE
prediction of $m_{\nu_3}$ from the $\JO$ trilinear form (Phase~5a) or an
explicit construction of the $\mathrm{SU}(3)_R$-breaking VEV $v_R$ from
the $(1,1,8)$ Killing form in $\JO$ (Phase~5b).

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

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\bibitem{toe-add53}
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\end{thebibliography}

\end{document}
