% Addendum 46: Conjecture B and D as J₃(𝕆) Trace Identities
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
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\newcommand{\muzero}{\mu_0}
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\newcommand{\rhoTOE}{\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x}
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\title{Conjectures~B and~D as $J_3(\mathbb{O})$ Trace Identities:\\
$m_t/m_c = \alpha^{-1} = \mathrm{Tr}(X_{\mathrm{sec}})$\\
and the $G_0$--$G_1$ Link\\[0.5em]
\large Addendum 46 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~45 classified Conjecture~B ($m_t/m_c = \alpha^{-1}$) and
Conjecture~D ($E_d = \ln(\muone)/\MU$) as \emph{numerological}: excellent
numerical agreement with PDG~2024, but no geometric derivation.  It left
OP-B$'$ open: does the $\mathbb{Z}_3$-intertwiner norm
$\|\Phi: V_{\omega^2} \to V_1\|^2$ equal $\muzero$?

The present addendum resolves the intertwiner computation (the answer
is \emph{no}: the natural Jordan triple product intertwiner is zero on
the relevant sector), and replaces it with a sharper structural target:
both conjectures are equivalent to \textbf{trace identities} in $\JO$.

Specifically, define the \emph{sector element}
$\Xsec = \mathrm{diag}(\pi,\pi^2,4\pi^3) \in V_1 \subset \JO$
and the \emph{first-moment element}
$\Xone = \mathrm{diag}(\tfrac{2\pi}{3}, \tfrac{3\pi^2}{4}, \tfrac{16\pi^3}{5}) \in V_1$.
Then:
\begin{itemize}[noitemsep]
  \item $\Tr(\Xsec) = \muzero = \alpha^{-1}$ \quad(exact algebraic identity),
  \item $\Tr(\Xone) = \muone$ \quad(exact algebraic identity),
  \item Conjecture~B $\Longleftrightarrow$ $m_t/m_c = \Tr_{\JO}(\Xsec)$,
  \item Conjecture~D $\Longleftrightarrow$ $E_d = \ln(\Tr_{\JO}(\Xone))/\MU$.
\end{itemize}
The two conjectures are linked by the exact identity
$G_0 - G_1 = -\ln(\MU)/\MU$, where $G_k = \ln(\mu_k)/\MU$,
which follows from $\Xone = \Xsec \circ K$ where
$K = \mathrm{diag}(\tfrac{2}{3}, \tfrac{3}{4}, \tfrac{4}{5})$
is the \emph{Peirce center matrix} of $\JO$.
If either conjecture is given a structural derivation, the other follows
once $-\ln(\MU)/\MU = \ln(\Tr(\Xsec)/\Tr(\Xone))/\MU$ is identified as
a $\JO$ invariant.

This reformulation upgrades both conjectures from ``numerological'' to
``structurally targeted'': each now has a precise algebraic object in $\JO$
whose derivation would close it.
\end{abstract}

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

\subsection{Context}

Addendum~45 established that five of the seven quark-mass conjectures
(A, C, E, I, J) have structural derivations from $G_2 = \mathrm{Aut}(\OO)$
acting on the exceptional Jordan algebra $\JO$.  Two conjectures remain:

\begin{description}[noitemsep]
  \item[Conjecture~B] Top/charm mass ratio $m_t/m_c = \muzero = \alpha^{-1}$.
  \item[Conjecture~D] Down quark energy $E_d = \ln(\muone)/\MU$.
\end{description}

Addendum~45 left these as ``numerological'' but identified a structural
target for Conjecture~B (OP-B$'$): the Jordan triple product intertwiner
$\Phi(y) = \{\alpha_3, y, \alpha_3\}$ mapping $V_{\omega^2} \to V_1$
under the $\mathbb{Z}_3$ clock decomposition of $\JO$.

The present addendum:
\begin{enumerate}[noitemsep]
  \item Computes the intertwiner $\Phi$ explicitly and shows it vanishes
    on the relevant sector (\S\ref{sec:intertwiner}).
  \item Identifies the correct structural object: the $\JO$ trace of the
    sector element $\Xsec$ (\S\ref{sec:trace-B}).
  \item Reformulates Conjecture~D analogously via the first-moment
    element $\Xone$ (\S\ref{sec:trace-D}).
  \item Derives the exact $G_0$--$G_1$ link and identifies the Peirce
    center matrix $K$ as its source (\S\ref{sec:G0G1}).
  \item States what ``structural'' would mean---the additional input
    needed to close each conjecture (\S\ref{sec:closing}).
\end{enumerate}

\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}\\
  \omega  &= e^{2\pi i/3} \quad(\text{primitive cube root of unity}).
\end{align}
The monad density $\rhoTOE$ on $[0,1]$
satisfies
\begin{equation}\label{eq:moments}
  \mu_k = \int_0^1 x^k \rho(x)\,dx
  = \frac{16\pi^3}{k+4} + \frac{3\pi^2}{k+3} + \frac{2\pi}{k+2}.
\end{equation}
In particular $\mu_0 = \muzero$ and $\mu_1 = \muone$.

The $\mathbb{Z}_3$ clock generator $T: \JO\otimes\C \to \JO\otimes\C$
acts by $T(X)_{ij} = \omega^{i-j}X_{ij}$, giving the eigenspace decomposition
(Addendum~45, Proposition~7.2):
\begin{equation}\label{eq:Z3split}
  \JO\otimes\C = V_1 \oplus V_\omega \oplus V_{\omega^2},
  \qquad \dim V_1 = 3,\ \dim V_\omega = 8,\ \dim V_{\omega^2} = 16.
\end{equation}
The diagonal subspace $V_1 = \{\mathrm{diag}(\alpha_1,\alpha_2,\alpha_3)\}$ contains
the generation energy anchors.  The top quark sits in $V_1$; the charm in $V_{\omega^2}$.

%==========================================================================
\section{The Jordan triple product intertwiner}
\label{sec:intertwiner}
%==========================================================================

\subsection{Setup}

Addendum~45 (OP-B$'$) proposed as structural target the map
\begin{equation}
  \Phi: V_{\omega^2} \to V_1,
  \qquad \Phi(y) = \{e_3, y, e_3\},
\end{equation}
where $\{X,Y,Z\} = (X \circ Y)\circ Z + (Z\circ Y)\circ X - (X\circ Z)\circ Y$
is the Jordan triple product on $\JO$, and $e_3 = \mathrm{diag}(0,0,1) \in V_1$
is the third Jordan frame idempotent.

We compute $\Phi$ explicitly using the Peirce calculus of $\JO$.

\subsection{Peirce rules}

For diagonal idempotents $e_i = E_{ii}$ and off-diagonal elements
$E_{jk}(q)$ (matrix with octonion $q \in \OO$ at position $(j,k)$,
$\bar{q}$ at $(k,j)$, zero elsewhere), the Jordan product satisfies:
\begin{align}
  e_i \circ E_{jk}(q) &= \tfrac{1}{2}(\delta_{ij}+\delta_{ik})\,E_{jk}(q), \label{eq:peirce1}\\
  e_i \circ e_i &= e_i, \quad e_i \circ e_j = 0\quad (i\neq j). \label{eq:peirce2}
\end{align}
For $e_3 = \mathrm{diag}(0,0,1)$ specifically:
$e_3 \circ E_{12}(q) = 0$ (neither index is 3),
$e_3 \circ E_{13}(q) = \frac{1}{2}E_{13}(q)$,
$e_3 \circ E_{23}(q) = \frac{1}{2}E_{23}(q)$.

\subsection{Computing $\Phi$ on $V_{\omega^2}$}

The eigenspace $V_{\omega^2}$ consists of the $x_{12}$ and $x_{23}$
off-diagonal blocks (eigenvalue $\omega^2$ under $T$).

\textbf{Case 1}: $y = E_{12}(q) \in V_{\omega^2}$.
\begin{align}
  e_3 \circ y &= e_3 \circ E_{12}(q) = 0, \nonumber\\
  e_3 \circ (e_3 \circ y) &= 0, \nonumber\\
  e_3^2 \circ y &= e_3 \circ y = 0, \nonumber\\
  \Phi(y) &= 2\,e_3\circ(e_3\circ y) - e_3^2\circ y = 0. \label{eq:phi12}
\end{align}

\textbf{Case 2}: $y = E_{23}(q) \in V_{\omega^2}$.
\begin{align}
  e_3 \circ y &= \tfrac{1}{2}\,E_{23}(q), \nonumber\\
  e_3 \circ (e_3 \circ y) &= \tfrac{1}{2}\,e_3\circ E_{23}(q) = \tfrac{1}{4}\,E_{23}(q), \nonumber\\
  e_3^2 \circ y &= e_3 \circ y = \tfrac{1}{2}\,E_{23}(q), \nonumber\\
  \Phi(y) &= 2\cdot\tfrac{1}{4}\,E_{23}(q) - \tfrac{1}{2}\,E_{23}(q) = 0. \label{eq:phi23}
\end{align}

\begin{proposition}[$\Phi \equiv 0$]\label{prop:phi-zero}
The Jordan triple product map $\Phi(y) = \{e_3, y, e_3\}$ vanishes identically
on $V_{\omega^2}$.  In particular, $\|\Phi\|^2 = 0 \neq \muzero$.
\end{proposition}

\begin{proof}
Cases 1 and 2 cover all of $V_{\omega^2} = \langle E_{12}(q), E_{23}(q) : q\in\OO\rangle$.
\end{proof}

\begin{remark}[$T$-equivariance obstructs the intertwiner]
The Jordan triple product satisfies
$T(\{X,Y,Z\}) = \{T(X),T(Y),T(Z)\}$ whenever $T$ is a $\JO$-automorphism.
Since $T(e_3) = e_3$ and $T(y) = \omega^2 y$ for $y \in V_{\omega^2}$,
we get $T(\Phi(y)) = \{e_3,\omega^2 y,e_3\} = \omega^2\Phi(y)$,
so any nonzero value of $\Phi(y)$ would lie in $V_{\omega^2}$, not $V_1$.
The map cannot be a $V_{\omega^2}\to V_1$ intertwiner by equivariance alone;
OP-B$'$ as stated in Addendum~45 is algebraically impossible.
The vanishing of $\Phi$ is therefore forced, not accidental.
\end{remark}

%==========================================================================
\section{Conjecture~B as a $J_3(\mathbb{O})$ trace identity}
\label{sec:trace-B}
%==========================================================================

\subsection{The sector element}

\begin{definition}[Sector element]\label{def:Xsec}
The \emph{sector element} of $\JO$ is
\begin{equation}\label{eq:Xsec}
  \Xsec = \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}
Its three eigenvalues are the sector energies from Addendum~36:
$E_e = \pi$ (edge sector $S^1$),
$E_b = \pi^2$ (boundary sector $S^3$),
$E_B = 4\pi^3$ (bulk sector $B^4$),
equal to the integrals of the three Peirce-sector terms of $\rho(x)$
over $[0,1]$.
\end{definition}

\begin{proposition}[Sector trace identity]\label{prop:sector-trace}
\begin{equation}\label{eq:sector-trace}
  \Tr_{\JO}(\Xsec) = \pi + \pi^2 + 4\pi^3 = \muzero = \alpha^{-1}.
\end{equation}
This is an exact algebraic identity.
\end{proposition}

\begin{proof}
The TOE fine-structure constant is defined as $\muzero = \int_0^1 \rho(x)\,dx$
(Addendum~36, Definition~2.1).  Evaluating:
\begin{equation}
  \int_0^1 \rho(x)\,dx
  = 16\pi^3\cdot\tfrac{1}{4} + 3\pi^2\cdot\tfrac{1}{3} + 2\pi\cdot\tfrac{1}{2}
  = 4\pi^3 + \pi^2 + \pi = \Tr(\Xsec).
\end{equation}
Each term is the integral of the corresponding Peirce-sector density:
$\int_0^1 16\pi^3 x^3\,dx = 4\pi^3 = E_B$, and so on.
\end{proof}

\subsection{Conjecture~B reformulated}

\begin{proposition}[Conjecture~B $\Leftrightarrow$ trace identity]\label{prop:B-trace}
The top/charm energy gap formula $E_t - E_c = \ln(\muzero)/\MU$
(Conjecture~B, Addendum~45) is equivalent to
\begin{equation}\label{eq:B-trace}
  \frac{m_t}{m_c} = \Tr_{\JO}(\Xsec) = \muzero = \alpha^{-1}.
\end{equation}
\end{proposition}

\begin{proof}
By the TOE mass formula $m/m_e = \exp(\MU(E-\pi))$ (Addendum~36):
\begin{equation}
  \frac{m_t}{m_c}
  = \frac{\exp(\MU(E_t-\pi))}{\exp(\MU(E_c-\pi))}
  = \exp\bigl(\MU(E_t - E_c)\bigr)
  = \exp\bigl(\MU \cdot \tfrac{\ln\muzero}{\MU}\bigr)
  = \exp(\ln\muzero) = \muzero.
\end{equation}
By Proposition~\ref{prop:sector-trace}, $\muzero = \Tr(\Xsec)$.
\end{proof}

\begin{remark}[Interpretation]
Conjecture~B says the top-to-charm mass ratio equals the $\JO$ trace of the
sector element: the total spectral weight of the monad density
$\rho$, expressed as the $\JO$ trace norm.  The structural content is:
\emph{the generation-3 to generation-2 quark mass ratio equals the
normalization of the TOE monad density,} which is the $\JO$ trace of
the canonical element $\Xsec \in V_1$.

The top sits in $V_1$ (eigenvalue 1 under $T$) and the charm in
$V_{\omega^2}$ (eigenvalue $\omega^2$).  For the mass ratio to equal
$\Tr(\Xsec)$, the $\mathbb{Z}_3$-sector separation must translate
to a mass multiplicative factor equal to the $V_1$-trace of the sector
element.  This is the non-trivial structural claim.
\end{remark}

\subsection{Numerical verification}

\begin{equation}
  \muzero = 137.036, \qquad
  \left(\frac{m_t}{m_c}\right)_{\mathrm{PDG}} = \frac{172760}{1275} = 135.50.
\end{equation}
Discrepancy: $+1.1\%$ in the ratio ($+2.0\%$ in $m_t$ as reported in
Addendum~45).  The residual is consistent with a one-loop QCD running
correction from the geometric scale to the $m_t$ pole mass.

%==========================================================================
\section{Conjecture~D as a $J_3(\mathbb{O})$ trace identity}
\label{sec:trace-D}
%==========================================================================

\subsection{The first-moment element}

\begin{definition}[First-moment element]\label{def:X1}
The \emph{first-moment element} of $\JO$ is
\begin{equation}\label{eq:X1}
  \Xone = \tfrac{2\pi}{3}\,e_1 + \tfrac{3\pi^2}{4}\,e_2 + \tfrac{16\pi^3}{5}\,e_3
         = \mathrm{diag}\!\left(\tfrac{2\pi}{3},\,\tfrac{3\pi^2}{4},\,\tfrac{16\pi^3}{5}\right)
         \in V_1.
\end{equation}
Its three eigenvalues are the first-moment integrals of each Peirce sector:
\begin{equation}
  \int_0^1 x\cdot 2\pi x\,dx = \tfrac{2\pi}{3},\quad
  \int_0^1 x\cdot 3\pi^2 x^2\,dx = \tfrac{3\pi^2}{4},\quad
  \int_0^1 x\cdot 16\pi^3 x^3\,dx = \tfrac{16\pi^3}{5}.
\end{equation}
\end{definition}

\begin{proposition}[First-moment trace identity]\label{prop:moment-trace}
\begin{equation}\label{eq:mu1-trace}
  \Tr_{\JO}(\Xone) = \tfrac{2\pi}{3} + \tfrac{3\pi^2}{4} + \tfrac{16\pi^3}{5}
  = \muone = \int_0^1 x\,\rho(x)\,dx.
\end{equation}
\end{proposition}

\begin{proof}
Direct integration: $\int_0^1 x\rho(x)\,dx = 16\pi^3/5 + 3\pi^2/4 + 2\pi/3 = \muone$.
Each term equals the corresponding eigenvalue of $\Xone$.
\end{proof}

\begin{proposition}[Conjecture~D $\Leftrightarrow$ trace identity]\label{prop:D-trace}
The down quark energy formula $E_d = \ln(\muone)/\MU$ (Conjecture~D) is equivalent to
\begin{equation}\label{eq:D-trace}
  E_d = \frac{\ln\bigl(\Tr_{\JO}(\Xone)\bigr)}{\MU}.
\end{equation}
\end{proposition}

\begin{proof}
Immediate from Proposition~\ref{prop:moment-trace}: $\muone = \Tr(\Xone)$.
\end{proof}

\subsection{The Peirce center matrix and factorization}

\begin{definition}[Peirce center matrix]\label{def:K}
The \emph{Peirce center matrix} is
\begin{equation}\label{eq:K}
  K = \tfrac{2}{3}\,e_1 + \tfrac{3}{4}\,e_2 + \tfrac{4}{5}\,e_3
    = \mathrm{diag}\!\left(\tfrac{2}{3},\,\tfrac{3}{4},\,\tfrac{4}{5}\right) \in V_1.
\end{equation}
Its eigenvalues are $K_i = n_i/(n_i+1)$ where $n_i \in \{1,2,3\}$ are
the Peirce sector powers (edge $n=1$, boundary $n=2$, bulk $n=3$).
\end{definition}

\begin{proposition}[Factorization]\label{prop:factorization}
$\Xone = \Xsec \circ K$
in the $\JO$ Jordan product (componentwise multiplication on $V_1 \cong \R^3$).
\end{proposition}

\begin{proof}
$(E_e\cdot\tfrac{2}{3},\; E_b\cdot\tfrac{3}{4},\; E_B\cdot\tfrac{4}{5})
= (\tfrac{2\pi}{3},\tfrac{3\pi^2}{4},\tfrac{16\pi^3}{5}) = \Xone$.
\end{proof}

\begin{remark}[Peirce center interpretation]
The value $K_i = n_i/(n_i+1)$ is the expected value of the uniform distribution
on $[0,1]$ weighted by $x^{n_i}$:
$\int_0^1 x\cdot(n_i+1)x^{n_i}\,dx = n_i/(n_i+1)$.
Equivalently, $K_i$ is the center of mass of the $i$-th sector density
$\rho_i(x) \propto x^{n_i}$ on $[0,1]$.  The matrix $K$ is a natural
$G_2$-invariant element of $V_1$: since $G_2$ acts trivially on $V_1$
(the $\mathbb{Z}_3$ fixed locus), every diagonal element is $G_2$-fixed.
The specific eigenvalues $2/3, 3/4, 4/5$ are determined by the sector
powers $n_i = 1, 2, 3$ alone.
\end{remark}

%==========================================================================
\section{The $G_0$--$G_1$ link}
\label{sec:G0G1}
%==========================================================================

\subsection{Moment-logarithm spectrum}

\begin{definition}[$G_k$ spectrum]\label{def:Gk}
$G_k = \ln(\mu_k)/\MU$ where $\mu_k = \int_0^1 x^k\rho(x)\,dx
= 16\pi^3/(k+4) + 3\pi^2/(k+3) + 2\pi/(k+2)$.
\end{definition}

The first six values are given in Table~\ref{tab:Gk}.
Conjecture~B asserts $E_t - E_c = G_0$.
Conjecture~D asserts $E_d = G_1$.

\begin{table}[ht]
\centering
\caption{Moment-logarithm spectrum.}
\label{tab:Gk}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{ccc}
\toprule
$k$ & $\mu_k$ & $G_k$ \\
\midrule
0 & $137.036 = \muzero$ & $6.2019$ \\
1 & $108.717 = \muone$  & $5.9101$ \\
2 & $90.176$            & $5.6744$ \\
3 & $77.063$            & $5.4764$ \\
4 & $67.290$            & $5.3054$ \\
5 & $59.721$            & $5.1550$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Exact link}

\begin{theorem}[$G_0$--$G_1$ link]\label{thm:G01}
\begin{equation}\label{eq:G01}
  G_0 - G_1
  = \frac{\ln(\muzero/\muone)}{\MU}
  = -\frac{\ln\MU}{\MU}
  = \frac{\ln\bigl(\Tr(\Xsec)/\Tr(\Xone)\bigr)}{\MU}.
\end{equation}
This is an exact algebraic identity.  Numerically: $G_0 - G_1 = 0.2918$.
\end{theorem}

\begin{proof}
$G_0 - G_1 = (\ln\mu_0 - \ln\mu_1)/\MU = \ln(\mu_0/\mu_1)/\MU
= \ln(1/\MU)/\MU = -\ln(\MU)/\MU$.
The trace forms follow from Propositions~\ref{prop:sector-trace}
and~\ref{prop:moment-trace}.
\end{proof}

\begin{corollary}[Reduction of OP-D to OP-B]\label{cor:OPD-from-OPB}
If Conjecture~B is structural (i.e., $m_t/m_c = \Tr(\Xsec)$ has a
$\JO$-intrinsic derivation), then Conjecture~D is structural if and only if
$-\ln(\MU)/\MU$ admits a $\JO$-intrinsic derivation.
\end{corollary}

\begin{proof}
$E_d = G_1 = G_0 - (G_0 - G_1) = G_0 + \ln(\MU)/\MU$.
If $G_0$ is structural and $\ln(\MU)/\MU$ is a $\JO$ invariant, then $G_1 = E_d$
is structural.
\end{proof}

\subsection{The Peirce compression ratio}

The ratio $\MU = \Tr(\Xone)/\Tr(\Xsec) = \Tr(\Xsec \circ K)/\Tr(\Xsec)$
is the \emph{Peirce compression ratio}: the fraction of total spectral
weight retained when the sector element is weighted by the center matrix $K$.

\begin{equation}
  \MU
  = \frac{\pi\cdot\tfrac{2}{3} + \pi^2\cdot\tfrac{3}{4} + 4\pi^3\cdot\tfrac{4}{5}}
         {\pi + \pi^2 + 4\pi^3}
  = \frac{\muone}{\muzero} = 0.7933.
\end{equation}

The quantity $-\ln(\MU)/\MU = \ln(\muzero/\muone)/\MU$ is the Bregman divergence
\begin{equation}
  -\frac{\ln\MU}{\MU}
  = \frac{\muzero}{\muone}\,\ln\!\left(\frac{\muzero}{\muone}\right),
\end{equation}
or equivalently the self-information of $\MU$ under the $\rho$-distribution
scaled by $\muzero/\muone$.  Its identification as a $\JO$ intrinsic invariant
(rather than merely a function of the two traces) is the remaining open problem
for Conjecture~D.

%==========================================================================
\section{What structural would mean}
\label{sec:closing}
%==========================================================================

\subsection{Sharpened open problems}

\begin{description}
  \item[OP-B (final form)]
    Conjecture~B is structural if and only if there exists a $\JO$-intrinsic
    argument deriving
    \begin{equation}\label{eq:OPB}
      \frac{m_t}{m_c} = \Tr_{\JO}(\Xsec)
    \end{equation}
    from the $\mathbb{Z}_3$ sector structure and quark assignments,
    without using the mass formula in the forward direction.

  \item[OP-D (final form)]
    Given OP-B, Conjecture~D is structural if and only if
    $-\ln(\MU)/\MU = \ln(\Tr(\Xsec)/\Tr(\Xone))/\MU$
    admits a $\JO$-intrinsic interpretation---for example as
    a Casimir eigenvalue, a Killing form inner product, or an
    automorphism-group spectral gap.
\end{description}

\subsection{Three routes toward OP-B}

\textbf{Route 1: Characteristic polynomial.}
The characteristic polynomial of $\Xsec$ in $\JO$ is
\begin{equation}
  \chi_{\Xsec}(\lambda)
  = \lambda^3 - \muzero\,\lambda^2 + (\pi^3+4\pi^4+4\pi^5)\,\lambda - 4\pi^6.
\end{equation}
The leading coefficient is $\Tr(\Xsec) = \muzero$.  A structural proof via
this route would show that $m_t/m_c$ equals the leading (trace) coefficient
of the characteristic polynomial of the $\JO$ generation mass matrix,
by identifying $\Xsec$ as that matrix from first principles.

\textbf{Route 2: $G_2$ Killing form.}
The $G_2$ Killing form restricted to $V_1 \cong \R^3$ is proportional to
the $\JO$ trace form: $B(X,Y) = c\,\Tr(X\circ Y)$.  A structural proof via
this route would show that the top/charm mass ratio equals the $G_2$ Killing
norm of $\Xsec$ (with $c = 1/\Tr(\Xsec)$ setting the unit), effectively
deriving $\Tr(\Xsec) = \muzero$ from the Killing-form normalization of $G_2$.

\textbf{Route 3: $\JO$ spectral measure (most natural).}
Treat $\rho(x)$ as the spectral density of $\Xsec$ under a $\JO$ probability
measure.  The normalization of this measure is $\int_0^1 \rho(x)\,dx = \muzero = \Tr(\Xsec)$,
which is the trace formula for the $\JO$ spectral measure.  A structural
proof via this route would show:
\begin{enumerate}[noitemsep, label=(\alph*)]
  \item the charm quark has unit spectral weight in the $\JO$
    density (i.e., it saturates the $V_{\omega^2}$ sector exactly),
  \item the top quark's mass relative to a unit-weight reference
    equals the normalization of the $\JO$ spectral measure of $\Xsec$.
\end{enumerate}
Item (a) would follow from Conjecture~A (charm = non-bulk sector integral,
already structural per Addendum~45, Proposition~3.1): the charm sits at
$E_c = \pi^2+\pi$, which is the sub-bulk integral of $\rho$.  Item (b)
requires identifying the top as the element whose mass ratio to a unit-weight
reference equals $\Tr(\Xsec)$.

Route~3 is the most structurally motivated because it connects to the
already-proven Conjecture~A: the charm's structural position as the
non-bulk sector integral provides the normalization anchor, and the full
normalization $\muzero = \Tr(\Xsec)$ is the natural extension.

\subsection{Minimal checklist for closure}

For Conjecture~B:
\begin{enumerate}[label=(\roman*)]
  \item \textit{$\Xsec$ is the canonical $\JO$ mass element.}
    Derive $e_i \leftrightarrow E_i$ (sector energies as Jordan frame
    eigenvalues) from $G_2$-orbit structure, not postulation.
    Addendum~36 establishes the sector energies; the Jordan frame bijection
    should follow from the triality action on $\JO$.
  \item \textit{$\Tr(\Xsec)$ is a $G_2$-invariant.}
    Already satisfied: the $\JO$ trace form is $G_2$-invariant, and
    $\Xsec \in V_1$ is a fixed point of the $G_2$-action on $V_1$.
  \item \textit{$m_t/m_c = \Tr(\Xsec)$ without forward use of $E_t$.}
    Non-trivial.  The $\mathbb{Z}_3$-sector separation places top in $V_1$
    and charm in $V_{\omega^2}$; the mass ratio must equal the $V_1$-trace
    of $\Xsec$ via the $\JO$ spectral measure or Killing form.
  \item \textit{Scheme compatibility.}
    The $+1.1\%$ discrepancy is accounted for by QCD running; no new
    structural input required (one-loop estimate gives $\sim 1$--$2\%$).
\end{enumerate}

For Conjecture~D (additionally):
\begin{enumerate}[label=(\roman*), resume]
  \item \textit{$-\ln(\MU)/\MU$ is a $\JO$ Casimir or eigenvalue.}
    This is the additional requirement beyond OP-B.  The quantity is
    the Bregman divergence of $\Tr(\Xone)$ from $\Tr(\Xsec)$ scaled
    by $1/\MU$.  A natural candidate is the $G_2$-invariant inner product
    of $\log(\Xsec/\Xone)$ with $\Xsec^{-1}$ under the $\JO$ trace form,
    but no identification is currently established.
\end{enumerate}

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

The main results are:

\begin{enumerate}[noitemsep]
  \item \textbf{The Jordan triple product intertwiner $\Phi = \{e_3,\cdot,e_3\}$
    vanishes on $V_{\omega^2}$} (Proposition~\ref{prop:phi-zero}).
    This is forced by $T$-equivariance (Remark after the proposition),
    not just Peirce arithmetic: the map was never a $V_{\omega^2}\to V_1$
    intertwiner.  OP-B$'$ of Addendum~45 is closed as algebraically impossible.

  \item \textbf{Conjecture~B $\Leftrightarrow$ $m_t/m_c = \Tr_{\JO}(\Xsec)$}
    (Proposition~\ref{prop:B-trace}), where $\Xsec = \mathrm{diag}(\pi,\pi^2,4\pi^3)$
    and $\Tr(\Xsec) = \muzero = \alpha^{-1}$ exactly
    (Proposition~\ref{prop:sector-trace}).

  \item \textbf{Conjecture~D $\Leftrightarrow$ $E_d = \ln(\Tr_{\JO}(\Xone))/\MU$}
    (Proposition~\ref{prop:D-trace}), where $\Xone = \mathrm{diag}(2\pi/3, 3\pi^2/4, 16\pi^3/5)$
    and $\Tr(\Xone) = \muone$ exactly
    (Proposition~\ref{prop:moment-trace}).

  \item \textbf{The factorization $\Xone = \Xsec \circ K$}
    (Proposition~\ref{prop:factorization}) identifies the Peirce center
    matrix $K = \mathrm{diag}(2/3,3/4,4/5)$ as the natural $\JO$ object
    relating the two conjectures.  Its eigenvalues $n/(n+1)$ are the
    sector-power centers, fixed by the $G_2$-invariant Peirce decomposition.

  \item \textbf{The $G_0$--$G_1$ link $G_0 - G_1 = -\ln(\MU)/\MU$ is exact}
    (Theorem~\ref{thm:G01}).  It reads in $\JO$ language as
    $G_0 - G_1 = \ln(\Tr(\Xsec)/\Tr(\Xone))/\MU$.
    Corollary~\ref{cor:OPD-from-OPB}: OP-D reduces to OP-B plus identifying
    $-\ln(\MU)/\MU$ as a $\JO$ intrinsic invariant.

  \item \textbf{Route~3 (spectral measure) is the most natural path to OP-B}:
    treat $\rho$ as the $\JO$ spectral density of $\Xsec$; then $\Tr(\Xsec)$
    is its normalization, and Conjecture~A (already structural) supplies the
    charm anchor.
\end{enumerate}

\begin{table}[ht]
\centering
\caption{Updated status of Conjectures~B and~D after this addendum.}
\label{tab:status}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{p{3cm}p{5.5cm}p{5.5cm}}
\toprule
 & \textbf{Conjecture~B} & \textbf{Conjecture~D} \\
\midrule
Statement
  & $m_t/m_c = \muzero = \alpha^{-1}$
  & $E_d = \ln(\muone)/\MU$ \\
$\JO$ form
  & $m_t/m_c = \Tr(\Xsec)$
  & $E_d = \ln(\Tr(\Xone))/\MU$ \\
Key element
  & $\Xsec = \mathrm{diag}(\pi,\pi^2,4\pi^3) \in V_1$
  & $\Xone = \mathrm{diag}(\frac{2\pi}{3},\frac{3\pi^2}{4},\frac{16\pi^3}{5}) \in V_1$ \\
Trace identity
  & $\Tr(\Xsec) = \muzero$ (exact)
  & $\Tr(\Xone) = \muone$ (exact) \\
Mass/energy error
  & $+1.1\%$ ratio; $+2.0\%$ $m_t$
  & $-2.2\%$ $m_d$ \\
Previous target
  & $\|\{e_3,\cdot,e_3\}\|^2 = \muzero$ (OP-B$'$)
  & $G_0$--$G_1$ link \\
New status
  & OP-B$'$ closed (intertwiner $\equiv 0$); replaced by trace target
  & Reduces to OP-B + $-\ln(\MU)/\MU$ \\
Remaining input
  & $\mathbb{Z}_3$-spectral normalization of $m_t/m_c$
  & $-\ln(\MU)/\MU$ as $\JO$ Casimir \\
\bottomrule
\end{tabular}
\end{table}

\begin{thebibliography}{99}

\bibitem{P02} L.~F.\ Vlegels, \emph{Paper~II: The Fine-Structure Constant from $S^3$ Geometry}, TOE Corpus, 2024.

\bibitem{P17} L.~F.\ Vlegels, \emph{Paper~XVII: Mass Formula from Moment Hierarchy}, TOE Corpus, 2024.

\bibitem{P29} L.~F.\ Vlegels, \emph{Paper~XXIX: Isolation of Global Minima in Exact Architectures}, TOE Corpus, 2025.

\bibitem{P36} L.~F.\ Vlegels, \emph{Addendum~36: Sector-Energy Mass Formula---Charged Leptons}, TOE Corpus, 2026.

\bibitem{P38} L.~F.\ Vlegels, \emph{Addendum~38: Quark Sector and Neutrino Structure from $\JO$ Geometry}, TOE Corpus, 2026.

\bibitem{P41} L.~F.\ Vlegels, \emph{Addendum~41: Strange and Up Quark Energies from the Fano--Octonion Structure}, TOE Corpus, 2026.

\bibitem{P43} L.~F.\ Vlegels, \emph{Addendum~43: CFP--Noether Derivation}, TOE Corpus, 2026.

\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{PDG24} R.~L.\ Workman et al.\ (Particle Data Group), \emph{Review of Particle Physics}, Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022}, 083C01 (2022), updated 2024.

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

\end{thebibliography}

\end{document}
