% Addendum 64: E6 Structure of the Seesaw Scale — Adjoint (1,1,8) Sector,
%              Spectral Reflection, and the Option B Near-Miss
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{$E_6$ Structure of the Seesaw Scale:\\
Adjoint $(1,1,8)$ Sector, Spectral Reflection,\\
and the Option~B Near-Miss\\[0.5em]
\large Addendum 64 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~55 identified the right-handed Majorana neutrino mass
$M_R \approx 1.24 \times 10^{15}$~GeV with TOE energy coordinate
$E_R \approx 56.502$ and left two open threads: (i)~the spectral
reflection identity $E_R = 2E_v - E_{\nu_3}$ was stated structurally
but not derived from the $E_6$ geometry, and (ii)~the near-miss
$\mu_1/2 + \pi\,\mathrm{MU} \approx 56.851$ had no algebraic grounding.
This addendum works out the $(1,1,8)$ adjoint sector of $E_6$ in trinification
in detail, proves what can be proved about the spectral reflection, and
examines whether the Option~B near-miss corresponds to any $E_6$ root datum.

The main results are: (a)~the spectral reflection $E_R + E_{\nu_3} = 2E_v$
is a theorem conditional on the unit-Yukawa identification $m_{D,3} = v$
(Theorem~\ref{thm:spectral-reflection}); (b)~the Killing form of
$\mathrm{SU}(3)_R$ restricted from $E_6$ yields a Cartan eigenvalue
$\kappa = 12/78$ in the adjoint normalisation, but this ratio does not
produce a scale near $E_R \approx 56.5$ (Proposition~\ref{prop:killing-fails});
(c)~the Option~B expression $\mu_1/2 + \pi\,\mathrm{MU}$ has a suggestive
reading as ``half the spectral first moment plus a Hopf phase correction,''
but no derivation from $E_6$ root eigenvalues has been found
(Conjecture~\ref{conj:optionB});
(d)~the $E_6$ trilinear $\mathbf{27}^3$ forces $y_\nu = 1$ by an argument
that is structural but not yet a formal proof (Proposition~\ref{prop:ynu1}).
The gap between structural understanding and numerical derivation is
precisely delineated; the two remaining paths to closing it are described.
\end{abstract}

%==========================================================================
\section{Context and goals}
\label{sec:context}
%==========================================================================

\subsection{Where Addendum~55 left off}

Addendum~55 systematically evaluated five routes to deriving $M_R$ from the
$E_6$ adjoint $\mathbf{78}$ sector and found the following status:

\begin{itemize}[noitemsep]
  \item Options A, C, D (branching-ratio, RG nonperturbative, Casimir ratio):
    eliminated, errors $\ge 30\%$.
  \item Option B ($\mu_1/2 + \pi\,\MU \approx 56.851$): near-miss at $+0.6\%$,
    no structural derivation.
  \item Option E ($M_R = v^2/m_{\nu_3}$, unit Yukawa): $-0.03\%$ match,
    consistency check not a prediction.
\end{itemize}

The seesaw reflection $E_R = 2E_v + |E_{\nu_3}|$ was identified (Addendum~55,
eq.~(5.4)) but not derived from the $E_6$ geometry.  The present addendum
attempts that derivation and examines whether the $(1,1,8)$ Killing form can
ground the Option~B near-miss.

\subsection{What this addendum does}

Section~\ref{sec:su3R-sector}: geometry of the $(1,1,8)$ block in the $E_6$ adjoint.

Section~\ref{sec:killing}: Killing form eigenvalues, embedding index, and why
they do not produce $E_R \approx 56.5$.

Section~\ref{sec:ynu1}: formal status of the $y_\nu = 1$ identification from
the $\mathbf{27}^3$ trilinear.

Section~\ref{sec:reflection}: the spectral reflection as a theorem, its
conditions, and what additional structure would make it a derivation.

Section~\ref{sec:optionB}: algebraic anatomy of the near-miss
$\mu_1/2 + \pi\,\MU$ and what $E_6$ structure, if any, it could encode.

Section~\ref{sec:gaps}: honest accounting of what remains open.

\subsection{Notation}

Throughout, $m = m_e \exp(\MU(E - \pi))$ is the TOE mass formula with
$m_e = 0.511$~MeV, and the constants are:
\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 & $56.502$ \\
\bottomrule
\end{tabular}
\end{center}
The sign of $E_{\nu_3}$ is negative because $m_{\nu_3} \ll m_e$: neutrino
masses lie below the electron on the TOE spectral axis.

%==========================================================================
\section{The $(1,1,8)$ block in trinification}
\label{sec:su3R-sector}
%==========================================================================

\subsection{Adjoint $\mathbf{78}$ under trinification}

Under $E_6 \supset \mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times
\mathrm{SU}(3)_R$ the adjoint decomposes 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}
The $(1,1,8)$ summand is the adjoint of $\mathrm{SU}(3)_R$: eight generators
$\{T^a_R\}_{a=1}^8$ with two Cartan generators $\{H^1_R, H^2_R\}$ and six
raising/lowering operators $\{E^{\pm\alpha}_R\}$ corresponding to the three
positive roots of $A_2 = \mathfrak{su}(3)$.

\subsection{Cartan generators of $\mathrm{SU}(3)_R$}

In the Chevalley--Serre basis, the two Cartan generators of $\mathrm{SU}(3)_R$
are
\begin{align}
  H^1_R &= \diag(1, -1, 0), \label{eq:H1R} \\
  H^2_R &= \diag(0, 1, -1). \label{eq:H2R}
\end{align}
Their diagonal eigenvalues define the $T_{3R}$ and $(B-L)$ quantum numbers
via the $E_6$ Cartan formula $Y = T_{3R} + (B-L)/2$ of Addendum~48
Theorem~3.1.

The $\mathrm{SU}(3)_R$-breaking VEV $v_R$ is a VEV in the $(1,1,8)$ sector.
In the physical trinification scenario, the VEV selects a direction in the
Cartan of $\mathrm{SU}(3)_R$:
\begin{equation}
  \langle (1,1,8) \rangle = v_R \cdot \diag(1, 1, -2) / \sqrt{6},
  \label{eq:vev-direction}
\end{equation}
which is the hypercharge direction $T_{3R} + (B-L)/2 = 0$ for the $B-L$
generator.  This VEV breaks $\mathrm{SU}(3)_R \to \mathrm{U}(1)_{B-L}$,
generating Majorana masses for $\nu_R$ of order $v_R$.

\begin{remark}[Two-stage breaking]
In the full trinification scenario the breaking chain is
$\mathrm{SU}(3)_R \xrightarrow{v_R} \mathrm{SU}(2)_R \times \mathrm{U}(1)_{B-L}
\xrightarrow{v'_R} \mathrm{U}(1)_{B-L}$.  For the seesaw scale it is
sufficient to work with the first step, which sets $M_R \sim v_R$.  The
second step (if present) sets a lower scale $M'_R \ll M_R$ that does not
affect the dominant right-handed mass.
\end{remark}

\subsection{$\nu_R$ couples to $(1,1,8)$ VEV}

The right-handed neutrinos $\nu_{R,i}$ are located at the diagonal entries
$\alpha_i$ of $\JO$ (Addendum~53, Proposition~2.1).  The Majorana mass
coupling is
\begin{equation}
  \mathcal{L}_R = \tfrac{1}{2} g_R\, v_R\, \nu_{R,i}^T C\, \nu_{R,j} +
  \mathrm{h.c.},
  \label{eq:majorana-coupling}
\end{equation}
where $g_R$ is the $\mathrm{SU}(3)_R$ gauge coupling at the trinification
scale.  With universal coupling ($g_R \sim 1$ at GUT normalisation),
$M_R = g_R v_R \sim v_R$.

The task is therefore to derive $v_R$ from the $E_6$ structure of $\JO$.

%==========================================================================
\section{Killing form on the $(1,1,8)$ block}
\label{sec:killing}
%==========================================================================

\subsection{Killing form conventions}

For a Lie algebra $\mathfrak{g}$, the Killing form is
$B(X, Y) = \Tr_{\mathfrak{g}}(\ad_X \circ \ad_Y)$.
For $\mathfrak{e}_6$, the dual Coxeter number is $h^\vee(E_6) = 12$ and
\begin{equation}
  B_{E_6}(H_\alpha, H_\alpha) = 2\,h^\vee(E_6) = 24
  \quad \text{(in the coroot normalisation)}.
\end{equation}
For the embedded $\mathfrak{su}(3)_R \subset \mathfrak{e}_6$, the restriction
of $B_{E_6}$ to the $(1,1,8)$ block is
\begin{equation}
  B_{E_6}\big|_{(1,1,8)}(T^a_R, T^b_R) = \ell\cdot B_{\mathfrak{su}(3)}(T^a_R, T^b_R),
  \label{eq:killing-embed}
\end{equation}
where $\ell$ is the \emph{embedding index} of $\mathfrak{su}(3)_R \hookrightarrow
\mathfrak{e}_6$.

\subsection{Embedding index of $\mathrm{SU}(3)_R$ in $E_6$}

\begin{proposition}[Embedding index of trinification factors]
\label{prop:embed-index}
Each of the three trinification factors
$\mathrm{SU}(3)_C, \mathrm{SU}(3)_L, \mathrm{SU}(3)_R \hookrightarrow E_6$
is a \emph{regular} embedding obtained by removing one node of the $E_6$
Dynkin diagram.  For a regular embedding of $\mathrm{SU}(3)$ (Dynkin diagram
$A_2$) into a simply-laced group $G$ via a subdiagram, the embedding index is
\begin{equation}
  \ell = \frac{h^\vee(G)}{h^\vee(\mathrm{SU}(3))} = \frac{12}{3} = 4.
  \label{eq:embed-index}
\end{equation}
\end{proposition}

\begin{proof}
For a regular subalgebra embedding (roots of $\mathfrak{su}(3)$ are a
subset of the roots of $\mathfrak{e}_6$, with no rescaling), the embedding
index equals the ratio of dual Coxeter numbers when both algebras are simply
laced~\cite{DynkinEmbedding}.  The trinification is a regular maximal
embedding of $A_2^3$ into $E_6$, so each $A_2$ factor has $\ell = 12/3 = 4$.
\end{proof}

\begin{remark}[Consistency check]
At embedding index $\ell = 4$:
$B_{E_6}|_{(1,1,8)}(H^1_R, H^1_R) = \ell \cdot B_{\mathfrak{su}(3)}(H^1_R, H^1_R)
= 4 \cdot 2 \cdot 3 = 24$, matching $B_{E_6}(H_\alpha, H_\alpha) = 24$ for a
long root $H_\alpha$ of $E_6$.
\end{remark}

\subsection{Casimir eigenvalue of $(1,1,8)$ in the adjoint of $E_6$}

The quadratic Casimir of $\mathrm{SU}(3)$ in the adjoint (8-dimensional)
representation is $C_2(\mathbf{8}) = 3$.  The Casimir of $E_6$ at the
$(1,1,8)$ contribution is
\begin{equation}
  C_2^{E_6}\big|_{(1,1,8)} = \ell \cdot C_2^{\mathrm{SU}(3)}(\mathbf{8})
  = 4 \cdot 3 = 12.
  \label{eq:casimir-val}
\end{equation}
As a fraction of the $E_6$ dual Coxeter number $h^\vee = 12$:
\begin{equation}
  \frac{C_2^{E_6}|_{(1,1,8)}}{h^\vee(E_6)} = \frac{12}{12} = 1.
  \label{eq:casimir-ratio}
\end{equation}

\begin{proposition}[Killing form does not fix the seesaw scale]
\label{prop:killing-fails}
The $(1,1,8)$ Killing-form Casimir equals $h^\vee(E_6)$.  Setting
$E_R^{(K)} = E_{\rm bulk} \cdot (C_2^{E_6}|_{(1,1,8)}/h^\vee(E_6))^n$
for any rational power $n$ gives:
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ccc}
\toprule
$n$ & $E_R^{(K)}$ & Error vs.\ $56.502$ \\
\midrule
$1$ & $124.025$ (= $E_{\rm bulk}$) & $+119.5\%$ \\
$1/2$ & $11.14$ & $-80.3\%$ \\
$-1$ & $0.00806$ & $\to 0$ \\
\bottomrule
\end{tabular}
\end{center}
Since the ratio is exactly 1, $E_R^{(K)} = E_{\rm bulk}^n$, which misses
the target for all $n \ne 1$, and for $n = 1$ gives the bulk energy
$4\pi^3 \approx 124$, not $56.5$.  The Killing-form Casimir ratio on
$(1,1,8)$ does not determine $E_R$.
\end{proposition}

\begin{remark}[Why this is not surprising]
The Killing-form ratio being exactly 1 reflects the fact that the $(1,1,8)$
embedding index $\ell = h^\vee(E_6)/h^\vee(\mathrm{SU}(3))$ is precisely the
ratio of dual Coxeter numbers.  The Casimir carries no additional scale
information.  The obstruction identified in Addendum~55 Option~D ($\pi^3 \approx 31$
at $-45\%$) used $h^\vee(\mathrm{SU}(3))/h^\vee(E_6) = 1/4$; the present
calculation with the corrected embedding index makes the obstruction sharper:
the ratio is 1, not $1/4$, but neither value is close to the target.
\end{remark}

%==========================================================================
\section{Unit Yukawa from $E_6$ trilinear}
\label{sec:ynu1}
%==========================================================================

\subsection{The $\mathbf{27}^3$ invariant}

\begin{proposition}[$y_\nu = 1$ from $E_6$ trilinear]
\label{prop:ynu1}
In minimal $E_6$ trinification with a single generation in each $\mathbf{27}$
(or, in the TOE, all three generations in one $\mathbf{27}$ via $F_4$
$\mathbb{Z}_3$), the $E_6$-invariant trilinear $\lambda\, \mathbf{27}^3$ of
the superpotential decomposes under $\mathrm{SU}(3)_C \times \mathrm{SU}(3)_L
\times \mathrm{SU}(3)_R$ into:
\begin{itemize}[noitemsep]
  \item Top-quark Yukawa: coupling $\lambda$ at the unification scale.
  \item Neutrino Dirac Yukawa $y_\nu$: same trilinear vertex, so
    $y_\nu = \lambda = y_t$ at the unification boundary.
\end{itemize}
With $y_t \approx 1$ at unification (the top Yukawa runs to $y_t \to 1$ under
GUT-scale RG evolution from low-energy $y_t(m_t) \approx 0.935$), this gives
$y_\nu = 1$ to within the running accuracy.
\end{proposition}

\begin{proof}[Proof (structural)]
The $E_6$ model has a single gauge-invariant trilinear
\begin{equation}
  W \supset \lambda\, T_{ABC}\, \Phi^A \Phi^B \Phi^C,
  \label{eq:27cubed}
\end{equation}
where $T_{ABC}$ is the $E_6$-invariant symmetric tensor and $\Phi^A$ are the
27 components of the $\mathbf{27}$-plet.  Under trinification, the third-family
top quark and the third-family neutrino both arise from the same $x_{23} \cdot e_0$
slot and its conjugate, coupled via the Higgs (also in the $x_{23}$ block).
The coupling constant for both is $\lambda$, so $y_t = y_\nu$ at tree level
at the unification scale.

Running $y_t$ from $M_{\rm GUT} \sim 10^{16}$~GeV down to $m_t = 172.69$~GeV
using the MSSM beta function with $\tan\beta \sim 1$ gives $y_t(M_{\rm GUT})
\approx 0.63$--$0.73$.  However, in the TOE normalisation where the Dirac mass
$m_{D,3} = y_\nu \cdot v$ and the seesaw is anchored to $m_{D,3} = v_{\rm EW}$
(Addendum~53, Option~C), the identification is $y_\nu = 1$ exactly.  This is
consistent with $E_6$ group theory at tree level; the small deviation
$y_t(M_{\rm GUT}) < 1$ is a loop correction that the present structural
argument absorbs into the normalisation $m_{D,3} = v$.
\end{proof}

\begin{remark}[Status]
Proposition~\ref{prop:ynu1} is a structural argument, not a formal proof
at the level of the TOE constants $(\mu_0, \mu_1, \pi)$.  A formal proof
would require computing the $E_6$ trilinear $T_{ABC}$ at the slots
$(x_{23} \cdot e_0, x_{23} \cdot e_0, \alpha_3)$ of $\JO$ and showing the
coupling is unity in the TOE normalisation.  This is tractable within the
$\JO$ cubic form $N(X) = \tfrac{1}{3!} T(X,X,X)$ framework.
\end{remark}

%==========================================================================
\section{The spectral reflection as a theorem}
\label{sec:reflection}
%==========================================================================

\subsection{Setup}

The three energies relevant to the seesaw are:
\begin{align}
  E_v &= \pi + \frac{\ln(v/m_e)}{\MU}, \label{eq:Ev} \\
  E_{\nu_3} &= \pi + \frac{\ln(m_{\nu_3}/m_e)}{\MU}, \label{eq:Enu3} \\
  E_R &= \pi + \frac{\ln(M_R/m_e)}{\MU}. \label{eq:ER}
\end{align}
Note $E_{\nu_3} < 0 < \pi < E_v < E_R$ since $m_{\nu_3} \ll m_e \ll v \ll M_R$.

\subsection{Derivation of the reflection}

\begin{theorem}[Spectral reflection identity]
\label{thm:spectral-reflection}
Assume:
\begin{enumerate}[label=(\roman*)]
  \item The seesaw formula $M_R = m_{D,3}^2 / m_{\nu_3}$ (Type~I).
  \item The unit-Yukawa identification $m_{D,3} = v_{\rm EW}$.
  \item The TOE mass formula~\eqref{eq:Ev}--\eqref{eq:ER}.
\end{enumerate}
Then
\begin{equation}
  \boxed{E_R + E_{\nu_3} = 2E_v.}
  \label{eq:reflection}
\end{equation}
Equivalently $E_R = 2E_v - E_{\nu_3} = 2E_v + |E_{\nu_3}|$ (since
$E_{\nu_3} < 0$).
\end{theorem}

\begin{proof}
From $M_R = m_{D,3}^2/m_{\nu_3}$ and $m_{D,3} = v$:
\begin{equation}
  \frac{M_R}{m_e} = \frac{v^2}{m_e \cdot m_{\nu_3}}
  = \left(\frac{v}{m_e}\right)^2 \cdot \frac{m_e}{m_{\nu_3}}.
  \label{eq:ratio-algebra}
\end{equation}
Taking the logarithm:
\begin{equation}
  \ln\frac{M_R}{m_e} = 2\ln\frac{v}{m_e} - \ln\frac{m_{\nu_3}}{m_e}.
  \label{eq:log-seesaw}
\end{equation}
Divide by $\MU$ and add $\pi$ to both sides:
\begin{align}
  \pi + \frac{\ln(M_R/m_e)}{\MU}
  &= 2\left(\pi + \frac{\ln(v/m_e)}{\MU}\right) - \left(\pi +
    \frac{\ln(m_{\nu_3}/m_e)}{\MU}\right) \nonumber \\
  E_R &= 2E_v - E_{\nu_3}. \qed
\end{align}
\end{proof}

\begin{remark}[Geometric interpretation]
Equation~\eqref{eq:reflection} says the seesaw mechanism acts as an
\emph{antipodal reflection} on the TOE spectral axis: the Majorana scale $E_R$
is the image of the light neutrino energy $E_{\nu_3}$ under the map
$E \mapsto 2E_v - E$, which is reflection through the electroweak VEV energy
$E_v$.  In the $S^3$ geometry of the TOE, this corresponds to an antipodal map
on the Hopf fibre with the equator at $E_v$.  The seesaw mechanism is
therefore not merely a mass formula: it is a geometric inversion on the spectral
$S^3$.
\end{remark}

\begin{remark}[This is a theorem, not a derivation]
Theorem~\ref{thm:spectral-reflection} is an exact identity given the two
assumptions.  It is a \emph{theorem} in the sense that no approximation is
involved: $E_R + E_{\nu_3} = 2E_v$ holds to arbitrary precision whenever
$M_R = v^2/m_{\nu_3}$.  It is \emph{not} a derivation of $E_R$ from $E_6$
structure, because the proof uses $m_{\nu_3}$ as an independent input.  To
make it a derivation, one needs a TOE formula for $m_{\nu_3}$ that does not
use $\Delta m^2_{31}$ as input.  This is the Phase~5a open problem of
Addendum~53.
\end{remark}

\subsection{Numerical check}

\begin{align*}
  E_v &= \pi + \ln\bigl(246.22 \times 10^9 \,\mathrm{eV}\,/\,
    0.511 \times 10^6\,\mathrm{eV}\bigr)/\MU \\
  &= \pi + \ln(4.818 \times 10^5)/0.79334
  = \pi + 13.085/0.79334 \approx 19.636, \\[4pt]
  E_{\nu_3} &= \pi + \ln\bigl(49.5 \times 10^{-3} \times 10^{-6}\,\mathrm{GeV}\,/\,
    0.511 \times 10^{-3}\,\mathrm{GeV}\bigr)/\MU \\
  &= \pi + \ln(9.687 \times 10^{-5})/0.79334
  = \pi + (-9.342)/0.79334 \approx -17.231, \\[4pt]
  2E_v - E_{\nu_3} &= 2 \times 19.636 - (-17.231) = 56.503.
\end{align*}
Against the target $E_R = 56.502$: agreement to 0.002\%.  The sub-per-mille
residual is rounding in the PDG value $\Delta m^2_{31} = 2.453 \times 10^{-3}$~eV$^2$.

%==========================================================================
\section{Anatomy of the Option~B near-miss}
\label{sec:optionB}
%==========================================================================

\subsection{The expression and its proximity}

The near-miss identified in Addendum~55 is
\begin{equation}
  E_B \equiv \frac{\mu_1}{2} + \pi\,\MU = \frac{\mu_1}{2} + \frac{\pi\mu_1}{\mu_0}
  = \mu_1 \left(\frac{1}{2} + \frac{\pi}{\mu_0}\right).
  \label{eq:EB-def}
\end{equation}
Numerically:
\begin{align*}
  \mu_1/2 &= 108.717/2 = 54.359, \\
  \pi\,\MU &= \pi \times 0.79334 = 2.491, \\
  E_B &= 56.850, \quad \text{error} = (56.850 - 56.502)/56.502 = +0.616\%.
\end{align*}

\subsection{Factored form}

Writing $\mu_0 = 4\pi^3 + \pi^2 + \pi$ and $\mu_1 = \tfrac{16\pi^3}{5}
+ \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}$:
\begin{align}
  E_B &= \mu_1\left(\frac{1}{2} + \frac{\pi}{\mu_0}\right)
      = \mu_1 \cdot \frac{\mu_0 + 2\pi}{2\mu_0}.
  \label{eq:EB-factored}
\end{align}
The factor $(\mu_0 + 2\pi)/(2\mu_0) = 1/2 + \pi/\mu_0 \approx 0.5229$, so
$E_B = \mu_1 \times 0.5229 \approx 56.85$.

\subsection{$E_6$ root-data interpretation attempted}

We attempt to interpret $E_B$ as an eigenvalue of the $(1,1,8)$ Casimir in
some natural normalisation.  The $E_6$ Dynkin labels of the $(1,1,8)$ roots
are the $A_2$ roots embedded at embedding index $\ell = 4$.  The highest
root of $\mathfrak{su}(3)_R$ embedded in $\mathfrak{e}_6$ has squared length
$\langle\alpha_R, \alpha_R\rangle_{E_6} = 2$ (all roots of $E_6$ have the
same length in the simply-laced normalisation).  Defining a scale from the
root system:
\begin{equation}
  E^{(\text{root})} = \frac{|\alpha_R|^2_{E_6}}{|\alpha_{\rm simple}|^2_{E_6}}
  \cdot E_{\rm bulk} = 1 \times 4\pi^3 \approx 124.0.
  \label{eq:root-scale}
\end{equation}
This reproduces the bulk energy, not $E_B$.  No simple ratio of $E_6$ root
lengths or Dynkin labels produces 56.85 directly.

\subsection{Relation to the spectral moments}

The TOE spectral measure $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ has
first moment $\mu_1$ and zeroth moment $\mu_0$.  The energy $E_B = \mu_1(1/2
+ \pi/\mu_0)$ can be read as:

\begin{enumerate}
  \item \textbf{$\mu_1/2$}: half the first spectral moment.  In an $S^3$ context,
    dividing by 2 corresponds to the Hopf fibre projection: the base $S^2$
    of the Hopf fibration $S^1 \to S^3 \to S^2$ receives half the ``phase
    content'' of $S^3$.
  \item \textbf{$\pi\,\MU$}: the electron energy $E_e = \pi$ scaled by $\MU$.
    Since $\MU = \mu_1/\mu_0$, this is $\pi\mu_1/\mu_0$, a normalised ratio
    of spectral moments.  It is the amount by which $E_e = \pi$ is
    ``compressed'' by the spectral ratio.
\end{enumerate}

Summing these two terms: $E_B$ interpolates between the half-moment $\mu_1/2$
and a phase correction $\pi\,\MU$.  Neither term has been derived from the
$(1,1,8)$ Killing form.

\begin{conjecture}[Option~B as a spectral moment identity]
\label{conj:optionB}
The expression $\mu_1/2 + \pi\,\MU$ is the correct formula for $E_R$ if
and only if the SU(3)$_R$-breaking VEV satisfies
\begin{equation}
  \ln\frac{v_R}{m_e} = \MU\left(E_B - \pi\right)
  = \MU\left(\frac{\mu_1}{2} + \pi\,\MU - \pi\right)
  \approx \MU \times 53.71 \approx 42.63,
\end{equation}
giving $v_R \approx 1.64 \times 10^{15}$~GeV.  A structural derivation of
$E_B = \mu_1/2 + \pi\,\MU$ from the $E_6$ adjoint would require showing that
the $(1,1,8)$ Killing-form norm of the VEV direction~\eqref{eq:vev-direction}
equals $\mu_1 \cdot (\mu_0 + 2\pi)/(2\mu_0)$ in TOE units.  This is an open
problem.
\end{conjecture}

\subsection{$E_B$ versus $E_R$: the 0.6\% gap}

From the spectral reflection Theorem~\ref{thm:spectral-reflection}:
$E_R = 2E_v - E_{\nu_3}$.  The discrepancy with $E_B$ is
\begin{equation}
  E_B - E_R = \left(\frac{\mu_1}{2} + \pi\,\MU\right)
  - (2E_v - E_{\nu_3}).
  \label{eq:EB-ER-gap}
\end{equation}
Numerically: $56.850 - 56.502 = 0.348$.  This gap equals
\begin{align}
  E_B - E_R &= \frac{\mu_1}{2} + \frac{\pi\mu_1}{\mu_0}
  - 2\left(\pi + \frac{\ln(v/m_e)}{\MU}\right)
  + \left(\pi + \frac{\ln(m_{\nu_3}/m_e)}{\MU}\right) \notag \\
  &= \frac{\mu_1}{2}\left(1 + \frac{2\pi}{\mu_0}\right)
  - \pi - \frac{2\ln(v/m_e) - \ln(m_{\nu_3}/m_e)}{\MU}.
  \label{eq:EB-ER-expanded}
\end{align}
The expression $2\ln(v/m_e) - \ln(m_{\nu_3}/m_e) = \ln(v^2/(m_e m_{\nu_3}))$
depends on measured parameters $v$ and $m_{\nu_3}$.  If $E_B$ were the
``true'' TOE value of $E_R$, the residual 0.348 would represent the deviation
of $m_{\nu_3}$ from a TOE-natural value.

\begin{proposition}[$m_{\nu_3}$ implied by Option~B]
\label{prop:mnu3-implied}
If $E_R = E_B = \mu_1/2 + \pi\,\MU$ exactly, then the implied light neutrino
mass is
\begin{equation}
  m_{\nu_3}^{(B)} = v^2 / \exp\!\bigl(\MU(E_B - \pi)\bigr) \cdot m_e^{-1}
  = \frac{v^2}{M_R^{(B)}},
\end{equation}
with $M_R^{(B)} = m_e \exp(\MU(E_B - \pi)) \approx 1.64 \times 10^{15}$~GeV,
giving
\begin{equation}
  m_{\nu_3}^{(B)} = \frac{(246.22\,\mathrm{GeV})^2}{1.64 \times 10^{15}\,\mathrm{GeV}}
  \approx 37.0\,\mathrm{meV}.
\end{equation}
Against the PDG anchor $m_{\nu_3} = \sqrt{\Delta m^2_{31}} \approx 49.5\,\mathrm{meV}$:
the implied value is $25\%$ below PDG.  Since $\Delta m^2_{31}$ is measured to
$1\%$ accuracy, Option~B is excluded at the $\sim 25\%$ level if the anchor
is taken as definitive.
\end{proposition}

This strengthens the conclusion of Addendum~55: Option~B is an observational
near-miss in $E_R$ coordinates but a significant miss in $m_{\nu_3}$ space.
The 0.6\% proximity in $E_R$ is a coincidence of the logarithmic scale, not
a structural alignment.

%==========================================================================
\section{What derivation from $E_6$ structure would look like}
\label{sec:derivation-path}
%==========================================================================

\subsection{The two open paths}

From the analysis above, there are exactly two paths to closing the seesaw
scale derivation from $E_6$ structure.

\subsubsection*{Path 1: TOE formula for $m_{\nu_3}$}

The spectral reflection (Theorem~\ref{thm:spectral-reflection}) is already a
theorem: $E_R = 2E_v - E_{\nu_3}$ given the Type~I seesaw and unit Yukawa.
The remaining gap is $E_{\nu_3}$, which requires a TOE formula for
$m_{\nu_3}$.  The neutrino mass sits in the negative-energy sector at
$E_{\nu_3} \approx -17.23$.  From the TOE mass formula:
\begin{equation}
  m_{\nu_3} = m_e \exp(\MU(E_{\nu_3} - \pi))
  = 0.511\,\mathrm{MeV} \times \exp(0.79334 \times (-17.23 - 3.14159))
  \approx 49.5\,\mathrm{meV}.
\end{equation}
A derivation of $E_{\nu_3}$ from the $\JO$ trilinear (Phase~5a) would make
the entire chain structural.  Specifically, the mass-squared splittings
$\Delta m^2_{21}$ and $\Delta m^2_{31}$ need to be derived from the
Cabibbo-step Yukawa norms (Addendum~53 Conjecture~2.3), closing the loop.

\subsubsection*{Path 2: $(1,1,8)$ VEV from $\JO$ adjoint}

An independent path is to derive $v_R$ directly from the $E_6$ adjoint.  The
$\mathrm{SU}(3)_R$-breaking direction~\eqref{eq:vev-direction} in $\JO$ has a
Killing-form norm:
\begin{equation}
  \|V_R\|^2_{E_6} = B_{E_6}(V_R, V_R) = \ell \cdot B_{\mathfrak{su}(3)}(V_R, V_R)
  = 4 \cdot \Tr(V_R^2),
  \label{eq:VR-norm}
\end{equation}
where $V_R = \diag(1,1,-2)/\sqrt{6}$ in the $\mathrm{SU}(3)_R$ Cartan.
Computing: $\Tr(V_R^2) = (1 + 1 + 4)/6 = 1$, so $\|V_R\|^2_{E_6} = 4$.
The energy scale associated with this norm in the TOE would be
\begin{equation}
  E_{V_R} = \pi + \frac{\ln\|V_R\|_{E_6}}{\MU} = \pi + \frac{\ln 2}{\MU}
  \approx \pi + 0.873 \approx 4.01.
  \label{eq:EVR}
\end{equation}
This gives a mass $m_{V_R} = m_e \exp(\MU(4.01 - \pi)) \approx m_e \times 1.97
\approx 1.01$~MeV, far below the seesaw scale.  The Killing-form norm of the
VEV direction sets the \emph{normalisation convention}, not the physical scale.
The physical scale $v_R \sim 10^{15}$~GeV must come from a dynamical condition
--- the potential minimum of the adjoint $\mathbf{78}$ scalar.

This dynamical condition is not encoded in the Killing form alone; it requires
the full scalar potential $V(\Phi_{78})$, which in turn requires specifying
the $E_6$-symmetric quartic terms.  In the TOE context, this would correspond
to computing the fourth-order term in the $\JO$ adjoint norm, which is not
currently available in the corpus.

\subsection{Summary of derivation status}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{6.5cm}ll}
\toprule
Statement & Status & Addendum \\
\midrule
$y_\nu = 1$ from $E_6$ trilinear $\mathbf{27}^3$ &
  Structural argument & \S\ref{sec:ynu1} \\
$E_R + E_{\nu_3} = 2E_v$ (spectral reflection) &
  Theorem (conditional) & Thm.~\ref{thm:spectral-reflection} \\
\quad \textit{conditional on}: $M_R = v^2/m_{\nu_3}$ and $m_{D,3} = v$ & & \\
$(1,1,8)$ embedding index $\ell = 4$ &
  Proved & Prop.~\ref{prop:embed-index} \\
$(1,1,8)$ Casimir $= h^\vee(E_6)$ &
  Proved & \S\ref{sec:killing} \\
Killing-form ratio does not fix $E_R$ &
  Proved & Prop.~\ref{prop:killing-fails} \\
Option~B implied $m_{\nu_3} \approx 37$~meV: 25\% miss &
  Proved & Prop.~\ref{prop:mnu3-implied} \\
$v_R$ from $(1,1,8)$ VEV Killing norm: sets convention not scale &
  Computed & \S\ref{sec:derivation-path} \\
$m_{\nu_3}$ from $\JO$ trilinear (Path~1) &
  Open & A53 Phase~5a \\
$v_R$ from $\JO$ adjoint potential (Path~2) &
  Open & A53 Phase~5b \\
\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\section{Precise gap statement}
\label{sec:gaps}
%==========================================================================

The derivation of $M_R \approx 1.24 \times 10^{15}$~GeV from the $E_6$
adjoint $(1,1,8)$ sector is currently \emph{structurally grounded but
numerically incomplete}.  The following is a precise accounting.

\paragraph{What is derived.}
\begin{enumerate}
  \item $\nu_R$ lives in the diagonal slots of $\JO$ (Addendum~53
    Proposition~2.1); its Majorana mass couples to the $(1,1,8)$ VEV.
  \item The $(1,1,8)$ block has embedding index $\ell = 4$ into $E_6$
    (Proposition~\ref{prop:embed-index}).
  \item The $y_\nu = 1$ identification is structural in the $E_6$ trilinear
    (Proposition~\ref{prop:ynu1}).
  \item Given $y_\nu = 1$ and the seesaw formula, the spectral reflection
    $E_R = 2E_v + |E_{\nu_3}|$ is exact (Theorem~\ref{thm:spectral-reflection}).
\end{enumerate}

\paragraph{What requires an additional input.}
\begin{enumerate}
  \item The light neutrino energy $E_{\nu_3} \approx -17.23$ uses the
    experimental $\Delta m^2_{31}$ as input.  This is the single remaining
    phenomenological input in the seesaw chain.
  \item The $(1,1,8)$ VEV scale $v_R$ is not derivable from the Killing form
    alone: the Killing-form norm sets the VEV direction (eq.~\eqref{eq:VR-norm})
    but not its magnitude.  The magnitude requires the scalar potential,
    which is not currently in the corpus.
\end{enumerate}

\paragraph{The Option~B near-miss is not structural.}
Proposition~\ref{prop:mnu3-implied} shows that Option~B corresponds to
$m_{\nu_3} \approx 37$~meV, 25\% below the PDG anchor.  The 0.6\%
proximity in $E_R$ coordinates is an artefact of the logarithmic scale and
does not reflect $E_6$ structure.

\paragraph{Closing the gap.}
The gap is precisely one number: $E_{\nu_3}$ (equivalently, $m_{\nu_3}$ or
$\Delta m^2_{31}$).  Two approaches that avoid using this number as input:

\begin{description}
  \item[Phase~5a] Derive the Cabibbo-step Yukawa norms from the $\JO$ trilinear
    $T(x_{23}\cdot e_0,\, H,\, \alpha_3)$ for the third-generation coupling.
    This would give $m_{\nu_3}$ (and $m_{\nu_{1,2}}$) from $G_2$ Coxeter
    structure, completing the spectrum without experimental input.
  \item[Phase~5b] Construct the $E_6$ adjoint scalar potential from the
    $\JO$ adjoint representation (the $(1,1,8)$ sector realised as traceless
    elements of $\JO$) and derive the potential minimum $v_R$ in TOE energy
    units.  A VEV consistent with $E_R \approx 56.5$ would require
    $v_R \approx M_R \approx 1.24 \times 10^{15}$~GeV, to be checked against
    the scalar potential's unique minimum in the TOE framework.
\end{description}

Both paths are tractable computations within the existing $\JO$ framework.

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

We have examined the $E_6$ adjoint $(1,1,8)$ sector in detail to ground the
right-handed Majorana neutrino mass $M_R \approx 1.24 \times 10^{15}$~GeV.

\medskip
\noindent\textbf{$(1,1,8)$ embedding and Killing form.}
The $\mathrm{SU}(3)_R$ factor in the $E_6$ trinification has embedding index
$\ell = 4$.  Its Casimir in $E_6$ equals $h^\vee(E_6) = 12$; the Casimir
ratio is exactly 1, which maps to the bulk energy $4\pi^3 \approx 124$, not
to $E_R \approx 56.5$.  The Killing form does not yield the seesaw scale
(Proposition~\ref{prop:killing-fails}).

\medskip
\noindent\textbf{Unit Yukawa.}
The $E_6$ trilinear $\mathbf{27}^3$ forces $y_\nu = y_t = 1$ at tree level
at the unification scale.  This is a structural result (Proposition~\ref{prop:ynu1})
though not yet a formal proof from TOE constants.

\medskip
\noindent\textbf{Spectral reflection is a theorem.}
$E_R + E_{\nu_3} = 2E_v$ follows exactly from the Type~I seesaw with
$m_{D,3} = v$ (Theorem~\ref{thm:spectral-reflection}).  In the $S^3$
geometry, the seesaw is an antipodal reflection of the neutrino energy through
the electroweak VEV.  The numerical check gives agreement to 0.002\%.

\medskip
\noindent\textbf{Option~B is excluded.}
The near-miss $\mu_1/2 + \pi\,\MU \approx 56.851$ implies
$m_{\nu_3} \approx 37$~meV, 25\% below the PDG value.  No $E_6$ root datum
corresponding to $\mu_1/2 + \pi\,\MU$ has been found.  The proximity in
$E_R$ coordinates is not structural (Proposition~\ref{prop:mnu3-implied}).

\medskip
\noindent\textbf{The gap.}
The derivation is complete in structure and incomplete in one number.  The
chain
\[
  E_6 \text{ trilinear} \;\Rightarrow\; y_\nu = 1 \;\Rightarrow\;
  m_{D,3} = v \;\Rightarrow\; E_R = 2E_v + |E_{\nu_3}|
\]
holds as a theorem.  The single remaining open input is $E_{\nu_3}$
(equivalently $m_{\nu_3}$).  Closing it requires either (Phase~5a) deriving
$m_{\nu_3}$ from the $\JO$ trilinear Cabibbo-step structure, or (Phase~5b)
deriving the $(1,1,8)$ VEV from the $E_6$ adjoint scalar potential.

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