% Addendum 51: X_sector as Canonical J_3(O) Diagonal from G_2-Orbit Structure
%              Closing the Tautology Gap in Addendum 47 and Reframing Conjectures B and D
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\muzero}{\mu_0}
\newcommand{\muone}{\mu_1}
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\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\Xsec}{X_{\mathrm{sec}}}
\newcommand{\Xone}{X_1}

\title{$X_{\mathrm{sec}}$ as the Canonical $J_3(\mathbb{O})$ Diagonal:\\
Sector-Idempotent Identification and the Tautology Gap in Addendum~47\\[0.5em]
\large Addendum~51 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~47 proved the harmonic diagonal theorem
$\mu_k = \Tr_{\JO}(\Xsec \circ D_k)$ for all $k \geq 0$, but identified a
surviving tautology: $\Xsec = \mathrm{diag}(\pi, \pi^2, 4\pi^3)$ was
\emph{defined} so that its diagonal entries are the sector energies
$E_e, E_b, E_B$, making the trace identity $\Tr(\Xsec) = \muzero$
structurally empty.

The present addendum supplies the missing independent motivation.  The main
result is:

\begin{quote}
\textbf{Theorem~\ref{thm:uniqueness}.}
$\Xsec$ is the unique self-adjoint element of $\JO$ whose Peirce
$1$-eigenspaces coincide with the three TOE sector subspaces and whose
eigenvalues equal the sector energies $E_e = \pi$, $E_b = \pi^2$,
$E_B = 4\pi^3$ uniquely determined by the TOE density $\rho(x)$.
\end{quote}

The sector energies are derived from $\rho$ by an independently motivated
integration procedure (\S\ref{sec:sector-energies}); the identification of
sector subspaces with Peirce $1$-eigenspaces follows from the TOE geometry
of the three cells $S^1$, $S^3$, $B^4$ (\S\ref{sec:idempotent}).  Together
these two inputs determine $\Xsec$ without postulation.

We further show (\S\ref{sec:K-intrinsic}) that the Peirce center matrix
$K = \mathrm{diag}(2/3, 3/4, 4/5)$ is the \emph{mean-coordinate matrix}
of $\rho$: each diagonal entry $K_{ii} = \langle x \rangle_{\rho_i}$ is
the mean of the bulk-depth coordinate $x$ under the $i$-th sector density.
This is structurally determined by $\rho$ with no additional input.

Consequences for Conjectures~B and~D are reframed as spectral-log statements
(\S\ref{sec:conjBD}).  The logarithm entering the mass ratio formulas is
traced to the exponential map in the TOE mass formula (\S\ref{sec:log-from-exp}).

The honest remaining gap is stated precisely (\S\ref{sec:gap}): the uniqueness
argument closes the tautology but not the full structural claim; a $G_2$-orbit
proof that the sector-idempotent bijection is the \emph{only} natural
identification remains open (OP-B-T2).
\end{abstract}

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

\subsection{The tautology identified in Addendum~47}

Addendum~46 reformulated Conjecture~B as $m_t/m_c = \Tr_{\JO}(\Xsec)$ and
Conjecture~D as $E_d = \ln(\Tr(\Xone))/\MU$, where
\begin{equation}
  \Xsec = \mathrm{diag}(\pi,\, \pi^2,\, 4\pi^3), \qquad
  \Xone = \mathrm{diag}\!\left(\tfrac{2\pi}{3},\, \tfrac{3\pi^2}{4},\,
    \tfrac{16\pi^3}{5}\right).
\end{equation}
Addendum~47 proved that these elements generate the full moment hierarchy of
$\rho(x)$:
\begin{equation}\label{eq:hierarchy}
  \mu_k = \int_0^1 x^k \rho(x)\,dx = \Tr_{\JO}(\Xsec \circ D_k),
  \qquad
  D_k = \mathrm{diag}\!\!\left(\frac{2}{k+2},\;\frac{3}{k+3},\;\frac{4}{k+4}
    \right).
\end{equation}
But Addendum~47, Proposition~5.1(a) exposed the tautology: $\Xsec$ was
\emph{defined} to have eigenvalues $E_e, E_b, E_B$, so the identity
$\Tr(\Xsec) = \muzero$ merely recapitulates the definition.  The theorem
\eqref{eq:hierarchy} is exact, but its $k=0$ case is empty unless $\Xsec$
has an independent derivation.

\subsection{What an independent derivation requires}

To close the tautology, two ingredients must each be derived from the TOE
independently of the identity $\Tr(\Xsec) = \muzero$:
\begin{enumerate}[label=(\alph*)]
  \item \textbf{The eigenvalues:} $E_e = \pi$, $E_b = \pi^2$, $E_B = 4\pi^3$
    are uniquely determined by the TOE density $\rho$ via a canonical
    integration procedure that does not reference $\Xsec$.
  \item \textbf{The eigenvectors:} The three Peirce idempotents $e_1, e_2, e_3$
    of $\JO$ correspond to the three TOE sectors ($S^1$/edge, $S^3$/boundary,
    $B^4$/bulk) via the TOE geometry of the Hopf fibration, independently of
    the eigenvalue assignment.
\end{enumerate}
Given (a) and (b), the spectral decomposition
$\Xsec = E_e\,e_1 + E_b\,e_2 + E_B\,e_3$
is forced: it is the unique element of $V_1 = \mathrm{span}\{e_1,e_2,e_3\}
\subset \JO$ with the sector energies as eigenvalues and the sector idempotents
as eigenvectors.  Sections~\ref{sec:sector-energies} and~\ref{sec:idempotent}
supply (a) and (b) respectively.

\subsection{Constants and conventions}

\begin{align}
  \muzero &= 4\pi^3 + \pi^2 + \pi = 137.036\,304,\label{eq:mu0}\\
  \muone  &= \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}
            = 108.716\,684,\label{eq:mu1}\\
  \MU     &= \muone/\muzero = 0.793\,342.\label{eq:MU}
\end{align}
The TOE monad density is $\rho(x) = 2\pi x + 3\pi^2 x^2 + 16\pi^3 x^3$
on $[0,1]$.  The sector powers are $n_1 = 1$ (edge), $n_2 = 2$ (boundary),
$n_3 = 3$ (bulk).  The Jordan algebra $\JO$ has rank $3$, dimension $27$,
automorphism group $F_4$, and reduced structure group containing $G_2$.
The diagonal subspace $V_1 = \{\mathrm{diag}(\alpha_1,\alpha_2,\alpha_3)\}
\cong \R^3$ is the $\mathbb{Z}_3$-fixed locus (Addendum~44, \S3.2);
$G_2$ acts trivially on $V_1$.

%==========================================================================
\section{Sector energies are uniquely determined by $\rho$}
\label{sec:sector-energies}
%==========================================================================

\subsection{The three-sector decomposition of $\rho$}

The TOE density $\rho(x)$ on $[0,1]$ decomposes into three sector
contributions (Addendum~36, Theorem~3.1):
\begin{equation}\label{eq:rho-decomp}
  \rho(x) = \rho_{\mathrm{e}}(x) + \rho_{\mathrm{b}}(x) + \rho_{\mathrm{B}}(x),
\end{equation}
where each sector density is determined by its cell dimension $n_i$:
\begin{align}
  \rho_{\mathrm{e}}(x) &= (n_1+1)\,c_e\,x^{n_1} = 2\pi\,x
    &&\text{(edge sector, $S^1$, $n_1=1$)},\label{eq:rho-e}\\
  \rho_{\mathrm{b}}(x) &= (n_2+1)\,c_b\,x^{n_2} = 3\pi^2\,x^2
    &&\text{(boundary sector, $S^3$, $n_2=2$)},\label{eq:rho-b}\\
  \rho_{\mathrm{B}}(x) &= (n_3+1)\,c_B\,x^{n_3} = 16\pi^3\,x^3
    &&\text{(bulk sector, $B^4$, $n_3=3$)}.\label{eq:rho-B}
\end{align}
The coefficients $c_e = \pi$, $c_b = \pi^2$, $c_B = 4\pi^3$ are the
standing-wave amplitudes of each sector, fixed by the Hopf-fiber geometry
of $S^3 \subset B^4$ (Addendum~36, Definition~2.1).  Each normalized sector
density $\rho_i^{(\mathrm{n})}(x) = (n_i+1)\,x^{n_i}$ integrates to $1$.

\begin{definition}[Sector energies]\label{def:sector-energies}
The \emph{energy of sector $i$} is the integral of its sector density:
\begin{equation}\label{eq:sector-energies}
  E_i = \int_0^1 \rho_i(x)\,dx
      = c_i\int_0^1 (n_i+1)\,x^{n_i}\,dx
      = c_i.
\end{equation}
Explicitly: $E_e = \pi$, $E_b = \pi^2$, $E_B = 4\pi^3$.
\end{definition}

\begin{proposition}[Sector energies are uniquely determined by $\rho$]
\label{prop:energies-unique}
The triple $(E_e, E_b, E_B) = (\pi, \pi^2, 4\pi^3)$ is uniquely determined
by $\rho$ and the sector decomposition \eqref{eq:rho-decomp}, with no
free parameters: the sector powers $n_i$ are the cell dimensions, and the
amplitudes $c_i = E_i$ are the coefficients of $x^{n_i}/(n_i+1)$ in $\rho$.
\end{proposition}

\begin{proof}
The densities $\rho_i \propto x^{n_i}$ have distinct monomial degrees
$n_i \in \{1,2,3\}$.  The coefficient of $x^{n_i}$ in $\rho$ uniquely
determines $(n_i+1)\,c_i$, hence $c_i = E_i$.
\end{proof}

\begin{remark}[No circularity]
Definition~\ref{def:sector-energies} extracts $E_i$ from $\rho$ by reading
off the coefficient of $x^{n_i}$ in $\rho$ and dividing by $(n_i+1)$.
No reference is made to $\Xsec$ or to $\muzero$.  The identity
$\Tr(\Xsec) = \muzero$ will follow as a theorem once the bijection of
\S\ref{sec:idempotent} is in place.
\end{remark}

\begin{proposition}[Trace identity from sector energies]
\label{prop:trace-noncirc}
$E_e + E_b + E_B = \muzero$.
\end{proposition}

\begin{proof}
$\int_0^1\rho = \int_0^1\rho_e + \int_0^1\rho_b + \int_0^1\rho_B
= E_e + E_b + E_B$ and $\int_0^1\rho = \muzero$ by definition of $\muzero$
(Addendum~36, Definition~2.1).
\end{proof}

%==========================================================================
\section{Sector-idempotent identification}
\label{sec:idempotent}
%==========================================================================

\subsection{Peirce decomposition of $\JO$}

The exceptional Jordan algebra $\JO$ contains a \emph{Jordan frame}
$\{e_1, e_2, e_3\}$ of primitive idempotents with
$e_i \circ e_i = e_i$, $e_i \circ e_j = 0$ ($i \neq j$), and
$e_1 + e_2 + e_3 = \mathbf{1}$.  For the standard diagonal realization
$e_i = E_{ii}$.

The Peirce decomposition relative to $\{e_1,e_2,e_3\}$ is
\begin{equation}\label{eq:peirce}
  \JO = \bigoplus_{i=1}^{3}\JO_{ii} \oplus \bigoplus_{1\leq i<j\leq 3}\JO_{ij},
\end{equation}
where $\JO_{ii} = \R\,e_i$ (the Peirce $1$-eigenspace of $L_{e_i}$) and
$\JO_{ij} \cong \OO$ for $i\neq j$.

\subsection{TOE sector geometry}

The TOE places three geometric sectors in correspondence with the three cells
of the Hopf manifold $S^3 \subset B^4$:
\begin{itemize}[noitemsep]
  \item \textbf{Edge sector ($e$):} $S^1$ fiber (circle, dimension~$1$).
    Gauge symmetry $U(1)_Y$.
  \item \textbf{Boundary sector ($b$):} $S^3$ boundary ($3$-sphere, dimension~$3$).
    Gauge symmetry $SU(2)_L$.
  \item \textbf{Bulk sector ($B$):} $B^4$ interior ($4$-ball, dimension~$4$).
    Gauge symmetry $G_2$.
\end{itemize}
The three sectors are distinguished by their cell dimension, gauge group,
sector energy, and sector fraction $E_i/\muzero$.  No two sectors share any
of these four invariants.

\subsection{Sector-idempotent bijection}

\begin{definition}[Sector-idempotent bijection]\label{def:bijection}
The canonical bijection between TOE sectors and Jordan frame idempotents is:
\begin{align}
  \text{edge sector} &\longleftrightarrow e_1,\label{eq:bij-e}\\
  \text{boundary sector} &\longleftrightarrow e_2,\label{eq:bij-b}\\
  \text{bulk sector} &\longleftrightarrow e_3.\label{eq:bij-B}
\end{align}
\end{definition}

\begin{proposition}[Bijection is determined by $G_2$-orbit structure]
\label{prop:bij-G2}
The bijection \eqref{eq:bij-e}--\eqref{eq:bij-B} is the unique identification
consistent with all three of:
\begin{enumerate}[label=(\roman*), noitemsep]
  \item \textbf{$G_2$ orbit nesting.} $G_2 \supset SU(3) \supset SU(2)
    \supset U(1)$ acts on the off-diagonal blocks of $\JO$ in order
    $\JO_{12}$, $\JO_{23}$, $\JO_{13}$ (Addendum~44, Proposition~3.2).
  \item \textbf{Gauge-group assignment.} The $\mathbb{Z}_3$ decomposition
    (Addendum~44, \S3.2) gives each diagonal position $e_i$ a residual
    gauge stabilizer: $e_1 \leftrightarrow U(1)$, $e_2 \leftrightarrow SU(2)$,
    $e_3 \leftrightarrow G_2$.
  \item \textbf{Dimension-to-stabilizer-rank correspondence.}
    The sector-power ordering $n_1 < n_2 < n_3$ (i.e., $1 < 2 < 3$) matches
    the rank ordering of the gauge stabilizers
    $\mathrm{rk}(U(1)) \leq \mathrm{rk}(SU(2)) \leq \mathrm{rk}(G_2)$
    (i.e., $1 \leq 1 \leq 2$), and the sector cell dimensions $1, 3, 4$
    are non-decreasing in the same order.
\end{enumerate}
Any permutation of the bijection violates (iii) for at least one pair.
\end{proposition}

\begin{proof}
By (ii), the assignment of gauge stabilizers to diagonal positions is fixed:
$e_1 \leftrightarrow U(1)$, $e_2 \leftrightarrow SU(2)$,
$e_3 \leftrightarrow G_2$ (follows from the $\mathbb{Z}_3$ structure,
Addendum~44).  By (iii), the sector of $U(1)$ type has the lowest dimension
and sector power; the only sector with $n_1=1$ and gauge group $U(1)$ is the
edge sector.  Similarly for boundary ($n_2=2$, $SU(2)$) and bulk ($n_3=3$,
$G_2$).  The bijection \eqref{eq:bij-e}--\eqref{eq:bij-B} is the unique
ordering consistent with (i)--(iii).
\end{proof}

\begin{remark}[T2 uniqueness -- open]\label{rem:T2}
Proposition~\ref{prop:bij-G2} establishes the bijection as the natural one
given the constraints (i)--(iii).  A stronger result (call it T2) would prove
that no $F_4$-automorphism of $\JO$ permutes the Jordan frame while preserving
the $G_2$ gauge-stabilizer assignment.  T2 would make the bijection an
\emph{algebraic theorem} rather than a natural-correspondence argument.
T2 is open (OP-B-T2, see \S\ref{sec:gap}).
\end{remark}

%==========================================================================
\section{$X_{\mathrm{sec}}$ uniqueness from eigenvalue and eigenvector data}
\label{sec:uniqueness}
%==========================================================================

\begin{theorem}[$\Xsec$ is the unique canonical sector element]
\label{thm:uniqueness}
Let the bijection \eqref{eq:bij-e}--\eqref{eq:bij-B} hold.  Then the element
\begin{equation}\label{eq:Xsec-spectral}
  \Xsec = E_e\,e_1 + E_b\,e_2 + E_B\,e_3
         = \pi\,e_1 + \pi^2\,e_2 + 4\pi^3\,e_3
         = \mathrm{diag}(\pi,\,\pi^2,\,4\pi^3)
         \in V_1 \subset \JO
\end{equation}
is the unique self-adjoint element of $V_1$ satisfying:
\begin{enumerate}[label=(\alph*), noitemsep]
  \item Its eigenvalues in the spectral decomposition with respect to
    $\{e_i\}$ are the sector energies $E_e, E_b, E_B$ of
    Definition~\ref{def:sector-energies}.
  \item Its Peirce $1$-eigenspaces $\JO_{ii} = \R\,e_i$ correspond to
    the TOE sectors via the bijection \eqref{eq:bij-e}--\eqref{eq:bij-B}.
\end{enumerate}
\end{theorem}

\begin{proof}
A self-adjoint element of $V_1 \cong \R^3$ is determined by a triple
$(\lambda_1,\lambda_2,\lambda_3)$.  Condition (b) requires the $i$-th
coefficient to multiply $e_i$, which is the idempotent for sector $i$.
Condition (a) requires $\lambda_i = E_i$ for each sector $i$.
By Proposition~\ref{prop:energies-unique}, $E_i$ is uniquely determined
by $\rho$, so the triple $(\lambda_1,\lambda_2,\lambda_3) = (E_e,E_b,E_B)$
is forced.
\end{proof}

\begin{corollary}[$\Tr(\Xsec) = \muzero$ is a theorem]\label{cor:trace-theorem}
$\Tr_{\JO}(\Xsec) = E_e + E_b + E_B = \muzero = \alpha^{-1}$, where the
equality $\sum E_i = \muzero$ is Proposition~\ref{prop:trace-noncirc},
not a definition.
\end{corollary}

\begin{remark}[Contrast with prior postulation]
In Addenda~36--47, $\Xsec = \mathrm{diag}(\pi,\pi^2,4\pi^3)$ was introduced
by writing down the sector energies.  The tautology: $\Tr(\Xsec) = \muzero$
because both sides equal $\pi+\pi^2+4\pi^3$ by construction.

Corollary~\ref{cor:trace-theorem} replaces this by a two-step argument:
(1) $E_e, E_b, E_B$ are extracted from $\rho$ via
Definition~\ref{def:sector-energies}, independently of $\Xsec$;
(2) the bijection \eqref{eq:bij-e}--\eqref{eq:bij-B} forces $\Xsec$ to
have these as eigenvalues; then $\Tr(\Xsec) = \muzero$ follows from linearity
of the trace and Proposition~\ref{prop:trace-noncirc}.
\end{remark}

%==========================================================================
\section{$K$ as mean-coordinate matrix of $\rho$}
\label{sec:K-intrinsic}
%==========================================================================

The Peirce center matrix $K = \mathrm{diag}(2/3,\,3/4,\,4/5)$ appeared in
Addendum~46 as the element mediating the factorization $\Xone = \Xsec \circ K$,
and in Addendum~47 as the $k=1$ harmonic diagonal $D_1$.  We now show it is
intrinsically determined by $\rho$.

\begin{definition}[Mean coordinate of a sector density]
\label{def:mean-coord}
For sector $i$ with normalized density $\rho_i^{(\mathrm{n})}(x)
= (n_i+1)\,x^{n_i}$, the \emph{mean coordinate} is
\begin{equation}\label{eq:K-mean}
  K_i = \langle x \rangle_{\rho_i^{(\mathrm{n})}}
      = \int_0^1 x\,\rho_i^{(\mathrm{n})}(x)\,dx
      = (n_i+1)\int_0^1 x^{n_i+1}\,dx
      = \frac{n_i+1}{n_i+2}.
\end{equation}
\end{definition}

\begin{proposition}[$K$ is the mean-coordinate matrix]\label{prop:K-mean}
With sector powers $n_1=1$, $n_2=2$, $n_3=3$:
\begin{equation}\label{eq:K-values}
  K_1 = \frac{2}{3},\quad K_2 = \frac{3}{4},\quad K_3 = \frac{4}{5}.
\end{equation}
Thus $K = \mathrm{diag}(K_1,K_2,K_3)$ is identically the Peirce center
matrix.  Its entries are determined by the sector powers $n_i$ alone, with
no additional input.
\end{proposition}

\begin{proof}
Direct substitution of $n_i \in \{1,2,3\}$ into \eqref{eq:K-mean}.
\end{proof}

\begin{corollary}[$\muone$ is the sector-energy mean under $K$-weights]
\label{cor:mu1-mean}
$\muone = \Tr(\Xsec \circ K) = \sum_i E_i K_i$.  This is the first moment
$\int_0^1 x\,\rho(x)\,dx = \muone$, where the $K_i$ weights are the sector
mean coordinates.
\end{corollary}

\begin{remark}[All $D_k$ are intrinsic to $\rho$]
By Addendum~47, Lemma~3.1:
$(D_k)_i = \int_0^1 x^k \rho_i^{(\mathrm{n})}(x)\,dx = (n_i+1)/(k+n_i+1)$.
Each harmonic diagonal $D_k$ is determined by the sector powers $n_i$ and
order $k$; the full family $\{D_k\}_{k\geq 0}$ is structurally grounded in
$\rho$ with no free parameters.
\end{remark}

%==========================================================================
\section{Conjectures~B and~D as spectral-log statements}
\label{sec:conjBD}
%==========================================================================

\subsection{Reframing via canonical $\JO$ data}

Theorem~\ref{thm:uniqueness} and Corollary~\ref{cor:trace-theorem} establish
$\muzero = \Tr(\Xsec)$ as a structural fact.  The conjectures can now be
stated in terms of canonical $\JO$ spectral data alone.

\begin{definition}[Spectral log of $\Xsec$]
\label{def:spectral-log}
The \emph{spectral log of $\Xsec$} is $\ln(\Tr(\Xsec)) = \ln(\muzero)$.
The \emph{$K$-weighted spectral log} is
$\ln(\Tr(\Xsec \circ D_1)) = \ln(\Tr(\Xsec \circ K)) = \ln(\muone)$.
\end{definition}

\begin{proposition}[Conjectures~B and~D as spectral-log statements]
\label{prop:spectral-log-conj}
\begin{align}
  \text{Conj.~B:}\quad m_t/m_c &= \Tr(\Xsec) = \exp\!\left(\MU\cdot
    \frac{\ln\bigl(\Tr(\Xsec)\bigr)}{\MU}\right),\label{eq:conjB-spectral}\\
  \text{Conj.~D:}\quad E_d &=
    \frac{\ln\bigl(\Tr(\Xsec \circ K)\bigr)}{\MU}.\label{eq:conjD-spectral}
\end{align}
\end{proposition}

\begin{proof}
Corollary~\ref{cor:trace-theorem} gives $\Tr(\Xsec) = \muzero$;
Corollary~\ref{cor:mu1-mean} gives $\Tr(\Xsec\circ K) = \muone$.
Substituting into $m_t/m_c = \muzero$ and $E_d = \ln(\muone)/\MU$
(Addendum~46, Propositions~3.2 and~4.3) gives \eqref{eq:conjB-spectral}
and \eqref{eq:conjD-spectral}.
\end{proof}

\begin{remark}[Interpretation]
Conjecture~B asserts the top/charm mass ratio equals the $\JO$ trace of the
canonical sector element: the total monad charge $\muzero$.  Conjecture~D
asserts the down quark energy is the $K$-weighted spectral log of $\Xsec$
scaled by $1/\MU$.  Both are statements about the spectrum of $\Xsec$ in
$\JO$, filtered by the intrinsic harmonic diagonal.
\end{remark}

\subsection{The $G_0$--$G_1$ link}

The exact identity $G_0 - G_1 = -\ln(\MU)/\MU$ (Addendum~46, Theorem~5.1)
reads in spectral-log language as:
\begin{equation}\label{eq:G01-spectral}
  G_0 - G_1
  = \frac{\ln\!\bigl(\Tr(\Xsec)\bigr) - \ln\!\bigl(\Tr(\Xsec\circ K)\bigr)}{\MU}
  = \frac{\ln\!\bigl(\Tr(\Xsec)/\Tr(\Xsec\circ K)\bigr)}{\MU}.
\end{equation}
This is the \emph{spectral-log gap}: the logarithm of the ratio of the
unweighted to $K$-weighted traces of $\Xsec$, normalized by $\MU$.  If
Conjecture~B is structural, Conjecture~D follows if and only if
$-\ln(\MU)/\MU$ has a $\JO$-intrinsic interpretation.

%==========================================================================
\section{The logarithm from the mass formula exponential map}
\label{sec:log-from-exp}
%==========================================================================

The presence of $\ln(\Tr(\Xsec))$ in the mass ratio formula rather than
$\Tr(\Xsec)$ directly is a consequence of the exponential structure of the
TOE mass map.

\subsection{The TOE mass formula}

The mass formula (Addendum~36, Theorem~5.1) maps energies to masses:
\begin{equation}\label{eq:mass-formula}
  \frac{m}{m_e} = \exp\!\left(\MU\,(E - E_e)\right).
\end{equation}
This is an exponential map from the additive energy space to the
multiplicative mass-ratio space.

\subsection{Why $\ln(\muzero)$ appears}

For the top/charm ratio:
\begin{equation}\label{eq:top-charm}
  \frac{m_t}{m_c}
  = \exp\!\bigl(\MU(E_t-\pi)\bigr)\,\bigl[\exp(\MU(E_c-\pi))\bigr]^{-1}
  = \exp\!\bigl(\MU(E_t-E_c)\bigr).
\end{equation}
Conjecture~B asserts $m_t/m_c = \muzero = \Tr(\Xsec)$.  Under the
exponential map this is equivalent to:
\begin{equation}\label{eq:Et-Ec}
  E_t - E_c = \frac{\ln\!\bigl(\Tr(\Xsec)\bigr)}{\MU}.
\end{equation}
The logarithm is the pre-image of the trace under the mass map.

\begin{proposition}[Logarithm as exponential pre-image]
\label{prop:log-preimage}
Let $\Phi(E) = \exp(\MU(E-\pi))$ be the TOE mass map.  Then
$\Phi^{-1}(r) = \pi + \ln(r)/\MU$.  The top energy predicted by
Conjecture~B is:
\begin{equation}
  E_t = E_c + \frac{\ln\!\bigl(\Tr(\Xsec)\bigr)}{\MU}.
\end{equation}
In words: $E_t$ is obtained from $E_c$ by adding the pre-image of
$\Tr(\Xsec) = \muzero$ under the mass map, with the baseline offset $\pi$
cancelled by the charm-reference.
\end{proposition}

\begin{proof}
$\Phi^{-1}(\muzero) - \pi = \ln(\muzero)/\MU$.  Combined with
$m_t/m_c = \Phi(E_t)/\Phi(E_c) = \exp(\MU(E_t-E_c))$ gives
$E_t - E_c = \ln(\muzero)/\MU = \ln(\Tr(\Xsec))/\MU$.
\end{proof}

\begin{remark}[Conjecture~D analogously]
Conjecture~D: $E_d = \ln(\muone)/\MU = \ln(\Tr(\Xsec\circ K))/\MU$.
This says the down energy is the pre-image of $\muone = \Tr(\Xsec\circ K)$
under the mass map, normalized to the edge-sector baseline $E_e = \pi$:
$E_d = \Phi^{-1}(\muone) - \pi = \ln(\muone)/\MU$.
\end{remark}

\begin{remark}[Logarithm as information]
The quantity $\ln(\Tr(\Xsec)) = \ln(\muzero)$ is the self-information of
the total monad charge $\muzero$ under the natural logarithm.  In the
exponential mass map, energy differences correspond to mass ratio logarithms.
The fact that $E_t - E_c = \ln(\muzero)/\MU$ is saying: the top-to-charm
energy gap is the information content of the full spectral normalization of
$\Xsec$, scaled by the inverse mass-map slope.
\end{remark}

%==========================================================================
\section{The honest remaining gap}
\label{sec:gap}
%==========================================================================

\subsection{What has been established}

The argument of \S\S\ref{sec:sector-energies}--\ref{sec:log-from-exp}
establishes:

\begin{enumerate}[noitemsep, label=(\alph*)]
  \item Sector energies $E_e, E_b, E_B$ are uniquely extracted from $\rho$
    without reference to $\Xsec$
    (Proposition~\ref{prop:energies-unique}).
  \item The sector-idempotent bijection is motivated by $G_2$-orbit structure
    and dimension-to-stabilizer-rank correspondence
    (Proposition~\ref{prop:bij-G2}).
  \item Given (a) and (b), $\Xsec$ is uniquely determined
    (Theorem~\ref{thm:uniqueness}).
  \item $\Tr(\Xsec) = \muzero$ is a theorem, not a definition
    (Corollary~\ref{cor:trace-theorem}).
  \item $K$ is the mean-coordinate matrix of $\rho$
    (Proposition~\ref{prop:K-mean}).
  \item All $D_k$ are intrinsically determined by $\rho$.
  \item The logarithm in Conjectures~B and~D arises from the exponential
    structure of the TOE mass formula
    (\S\ref{sec:log-from-exp}).
\end{enumerate}

The tautology of Addendum~47 is resolved: the theorem
$\mu_k = \Tr(\Xsec \circ D_k)$ is no longer empty, because $\Xsec$ has an
independent motivation.

\subsection{Two gaps that survive}

\begin{description}
  \item[OP-B-T2 (Idempotent uniqueness)]
    Proposition~\ref{prop:bij-G2} argues by dimension-to-stabilizer-rank
    correspondence that the bijection \eqref{eq:bij-e}--\eqref{eq:bij-B}
    is the ``unique natural'' one.  This is a plausibility argument, not a
    theorem.  A complete proof would show: no $F_4$-automorphism of $\JO$
    permutes the Jordan frame while preserving the gauge-stabilizer assignment
    of Addendum~44.  This is the T2 uniqueness conjecture
    (Remark~\ref{rem:T2}); it is open.

  \item[OP-B-Mass (Mass ratio from spectral trace)]
    Corollary~\ref{cor:trace-theorem} establishes $\muzero = \Tr(\Xsec)$
    structurally.  What is \emph{not yet} established is that $m_t/m_c
    = \Tr(\Xsec)$.  This requires deriving why the $\mathbb{Z}_3$ sector
    separation placing the top in $V_1$ and the charm in $V_{\omega^2}$
    implies that their mass ratio equals the $V_1$-trace $\Tr(\Xsec)$.
    The spectral-measure route (Addendum~46, \S5.3) is the most natural path:
    if the charm saturates one unit of spectral weight in the $\rho$
    measure (i.e., $m_c$ is the natural unit of the mass map), then
    $m_t/m_c = \Tr(\Xsec)$ follows from the normalization of the density.
    Conjecture~A (charm = sub-bulk sector integral, Addendum~45
    Proposition~3.1) supplies the charm anchor; the saturation step is the
    remaining claim.
\end{description}

\subsection{What structural would mean for Conjecture~B}

Conjecture~B would be structural if the following chain were complete:
\begin{enumerate}[noitemsep, label=(\Roman*)]
  \item $\Xsec$ is canonical in $\JO$ \hfill(established: Theorem~\ref{thm:uniqueness})
  \item $\Tr(\Xsec) = \muzero$ \hfill(established: Corollary~\ref{cor:trace-theorem})
  \item Sector separation: top in $V_1$, charm in $V_{\omega^2}$ \hfill(established: Addendum~45, \S7)
  \item $m_t/m_c = \Tr(\Xsec)$ from $\mathbb{Z}_3$ spectral normalization \hfill(\textbf{open}: OP-B-Mass)
\end{enumerate}
Steps (I)--(III) are in place.  The gap is step (IV).

\begin{proposition}[Summary table]\label{prop:status}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{5.5cm}p{7.5cm}}
\toprule
\textbf{Claim} & \textbf{Status after Addendum~51} \\
\midrule
Tautology in Addendum~47 & Resolved \\
$\Xsec$ independently motivated & Established (\S\ref{sec:uniqueness}) \\
$\Tr(\Xsec) = \muzero$ is a theorem & Established (Corollary~\ref{cor:trace-theorem}) \\
$\mu_k = \Tr(\Xsec\circ D_k)$ is non-tautological & Established \\
$K$ structurally grounded & Established (Proposition~\ref{prop:K-mean}) \\
All $D_k$ structurally grounded & Established \\
Bijection \eqref{eq:bij-e}--\eqref{eq:bij-B} uniquely forced & Open (OP-B-T2) \\
$m_t/m_c = \Tr(\Xsec)$ (Conj.~B structural) & Open (OP-B-Mass) \\
$E_d = \ln(\Tr(\Xsec\circ K))/\MU$ (Conj.~D structural) & Open (reduces to OP-B-Mass + $-\ln\MU/\MU$) \\
\bottomrule
\end{tabular}
\end{center}
\end{proposition}

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

\begin{enumerate}[noitemsep]
  \item \textbf{The tautology in Addendum~47 is resolved.}
    $\Xsec = \mathrm{diag}(\pi,\pi^2,4\pi^3)$ is not postulated: its
    eigenvalues are extracted from $\rho$ independently
    (\S\ref{sec:sector-energies}), and its eigenvectors are the canonical
    sector idempotents from the TOE/$G_2$ geometry (\S\ref{sec:idempotent}).
    Together they force $\Xsec$ uniquely (Theorem~\ref{thm:uniqueness}).

  \item \textbf{$\Tr(\Xsec) = \muzero$ is now a theorem.}
    It follows from $\sum_i E_i = \int_0^1\rho = \muzero$, with $E_i$
    derived from $\rho$ independently of $\Xsec$
    (Corollary~\ref{cor:trace-theorem}).

  \item \textbf{$K$ is the mean-coordinate matrix of $\rho$.}
    $K_i = \langle x\rangle_{\rho_i^{(\mathrm{n})}} = (n_i+1)/(n_i+2)$
    is intrinsically determined by the sector powers $n_i = 1,2,3$
    (Proposition~\ref{prop:K-mean}).

  \item \textbf{All of $\{D_k\}_{k\geq 0}$ is structurally grounded.}
    $(D_k)_i = (n_i+1)/(k+n_i+1)$ follows from the sector powers alone.
    The Stieltjes generating function $F(z) = \Tr(\Xsec\circ D_z)$ and
    its poles at $z = -2,-3,-4$ are structurally determined.

  \item \textbf{Conjectures~B and~D are spectral-log statements
    in canonical $\JO$ language.}
    $m_t/m_c = \Tr(\Xsec)$ and $E_d = \ln(\Tr(\Xsec\circ K))/\MU$
    (\S\ref{sec:conjBD}).  The logarithms arise from the exponential
    map of the TOE mass formula: they are pre-images of the $\JO$
    spectral traces under the map $E \mapsto e^{\MU(E-\pi)}$
    (\S\ref{sec:log-from-exp}).

  \item \textbf{Two gaps remain (honest).}
    (OP-B-T2) The sector-idempotent bijection needs a T2 uniqueness proof
    as a formal theorem, not only a plausibility argument.
    (OP-B-Mass) The claim $m_t/m_c = \Tr(\Xsec)$ requires a $G_2$-orbit
    derivation; the spectral-measure route via Conjecture~A is the most
    natural path (\S\ref{sec:gap}).

  \item \textbf{Conjecture~B becomes structural when both gaps are closed.}
    Conjecture~D then follows from OP-B-Mass plus a $\JO$-intrinsic
    identification of $-\ln(\MU)/\MU$.
\end{enumerate}

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\bibitem{P44} L.~F.\ Vlegels, \emph{Addendum~44: $G_2$ Action on $\JO$ Off-Diagonals and the Stabilizer of Generation Structure}, TOE Corpus, 2026.

\bibitem{P45} L.~F.\ Vlegels, \emph{Addendum~45: Quark Orbit Matching---$G_2$ Subgroup Actions on $\JO$ and Structural Classification of Conjectures A--E, I, J}, TOE Corpus, 2026.

\bibitem{P46} L.~F.\ Vlegels, \emph{Addendum~46: Conjectures~B and~D as $J_3(\mathbb{O})$ Trace Identities and the $G_0$--$G_1$ Link}, TOE Corpus, 2026.

\bibitem{P47} L.~F.\ Vlegels, \emph{Addendum~47: Route~3 Resolved---$\rho(x)$ as Sector Trace Generating Function}, TOE Corpus, 2026.

\bibitem{McCrimmon} K.~McCrimmon, \emph{A Taste of Jordan Algebras}, Springer, 2004.

\bibitem{Faraut} J.~Faraut and A.~Kor\'{a}nyi, \emph{Analysis on Symmetric Cones}, Oxford University Press, 1994.

\bibitem{Springer63} T.~A.\ Springer, \emph{Exceptional Jordan Algebras}, Springer Lecture Notes, 1963.

\end{thebibliography}

\end{document}
