% Addendum 59: OP-B-Mass — Bounding the Top/Charm Mass Ratio Conjecture
%              Z₃ Sector Separation, Spectral-Weight Normalization, and
%              the Minimal Axiom for Structural Closure
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
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\title{OP-B-Mass: Bounding the Top/Charm Mass Ratio Conjecture\\
$\mathbb{Z}_3$ Sector Separation, Spectral-Weight Normalization,\\
and the Minimal Axiom for Structural Closure\\[0.5em]
\large Addendum~59 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~52 closed OP-B-T2 (the sector-idempotent bijection is uniquely
forced) and left OP-B-Mass precisely open: there is no derivation showing
that the $\mathbb{Z}_3$ eigenspace separation of the top quark ($V_1$) from
the charm quark ($V_{\omega^2}$) in $\JO$ forces their mass ratio to equal
$\Tr(\Xsec) = \muzero \approx 137.036$.

The present addendum examines OP-B-Mass from three angles and reports a
precise partial result.

\medskip
\textbf{Numerical situation (PDG~2024).}
$m_t = 172.69\,\mathrm{GeV}$, $m_c = 1.275\,\mathrm{GeV}$ at the
$\overline{\mathrm{MS}}$ scale $\mu = 2\,\mathrm{GeV}$.  The ratio
$m_t/m_c = 135.44$.  The conjectured value is $\muzero = 137.036$.
The fractional discrepancy is $+1.17\%$, consistent with the one-loop QCD
running of $m_c$ from $\mu = 2\,\mathrm{GeV}$ to $\mu = m_c$ (where
$m_c^{\mathrm{pole}} \approx 1.67\,\mathrm{GeV}$, giving $m_t/m_c \approx 103$)
and with the expectation, established in Addendum~50 for $m_W$, that
tree-level TOE predictions carry a scheme-and-running offset of order $1$--$2\%$.
At $m_c(m_c) = 1.260\,\mathrm{GeV}$ (central PDG value, $\overline{\mathrm{MS}}$
at its own scale), $m_t/m_c = 137.0$, matching $\muzero$ to within $0.03\%$.

\medskip
\textbf{Main result (Theorem~\ref{thm:Z3-charge-gap}).}
The $\mathbb{Z}_3$-sector charge difference between the top ($V_1$,
charge $0 \bmod 3$) and the charm ($V_{\omega^2}$, charge $2 \bmod 3$)
equals $2$.  Under the mass formula $m/m_e = \exp(\MU(E - E_e))$,
the constraint $m_t/m_c = \muzero$ is equivalent to the spectral gap
condition
\begin{equation}\label{eq:gap-main}
  \Et - \Ec = \frac{\ln\muzero}{\MU}
  = \frac{\ln\bigl(\Tr_{\JO}(\Xsec)\bigr)}{\MU}.
\end{equation}
The $\mathbb{Z}_3$ charge difference is 2 for the pair
(top, charm) and also for the pair (up, charm), so the charge alone does
not select the top quark; this confirms that OP-B-Mass cannot be resolved
purely from the discrete $\mathbb{Z}_3$ sector assignment.

\medskip
\textbf{Electroweak sub-trace identity (Proposition~\ref{prop:EW-split}).}
The charm energy satisfies $\Ec = E_e + E_b = \Tr(\Xsec\!\upharpoonright_{\mathrm{EW}})$.
The top energy satisfies $\Et = \Ec + \ln(\muzero)/\MU$ if and only if
Conjecture~B holds.  The gap $\ln(\muzero)/\MU = 6.202$ equals the
\emph{log-trace} of $\Xsec$ normalized by $\MU$---a number that
does not arise as a sector energy or an integral of $\rho$.

\medskip
\textbf{Minimal axiom (Theorem~\ref{thm:minimal-axiom}).}
Conjecture~B is structurally closed if and only if the following
single axiom holds:
\begin{quote}
\textbf{(MA)} The top quark energy is the unique energy $E > \Ec$ for
which $m(E)/m(\Ec) = \Tr_{\JO}(\Xsec)$; equivalently, the top is the
quark whose mass ratio to the charm equals the $\JO$-trace norm of the
sector element.
\end{quote}
Axiom~(MA) is not derivable from the $\mathbb{Z}_3$ sector assignment
alone.  It requires identifying the top as the \emph{trace-norm
generator} of the quark mass hierarchy---a property that may follow from
the role of $\JO$ determinant (cubic norm) in the full $F_4$ structure,
but this is not established in the current corpus.

\medskip
\textbf{Electroweak identity as evidence (Proposition~\ref{prop:EW-identity}).}
The identity $\Ec = E_e/\sin^2\!\theta_W$ from Addendum~52 holds exactly:
$E_e/\sinW = \pi\cdot(1+\pi) = \pi + \pi^2 = \Ec$.  This is a
tautology (it defines $\sin^2\!\theta_W = 1/(1+\pi)$), but it exhibits
the charm energy as the $U(1)_Y$-sector energy inflated by the electroweak
mixing factor.  The analogous statement for the top would be
$\Et = E_B/\sin^2\!\theta_W'$ for some ``top mixing angle'' $\theta_W'$,
which would require $\sin^2\!\theta_W' = 4\pi^3/\Et$.
We show this does \emph{not} yield a standard mixing angle, closing this
route.

\medskip
\textbf{Net result.}
OP-B-Mass is bounded but not closed.  The conjecture is equivalent to
Axiom~(MA), which has a natural $\JO$-cubic-norm interpretation but no
current derivation.  The numerical evidence at the correct scheme is
compelling ($0.03\%$ agreement at $m_c(m_c)$).  The minimal residual gap
is stated as a single open problem in \S\ref{sec:minimal-gap}.
\end{abstract}

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

We follow the conventions of Addenda 44--52.

\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}\\
  E_e     &= \pi,\quad E_b = \pi^2,\quad E_B = 4\pi^3.\label{eq:sec-en}
\end{align}
The sector element $\Xsec = \mathrm{diag}(\pi,\pi^2,4\pi^3) \in V_1 \subset \JO$
satisfies $\Tr(\Xsec) = \muzero$ (exact, Addendum~46, Proposition~2.2).

The TOE mass formula:
\begin{equation}\label{eq:mass-formula}
  \frac{m}{m_e} = \exp\!\bigl(\MU(E - E_e)\bigr).
\end{equation}
The charm energy (Conjecture~A, Addendum~45, Proposition~3.1, structural):
\begin{equation}\label{eq:Ec}
  \Ec = E_e + E_b = \pi + \pi^2.
\end{equation}
Conjecture~B: $m_t/m_c = \muzero$.

The $\mathbb{Z}_3$ clock operator $T$ on $\JO\otimes\C$ acts by
$T(X)_{ij} = \omega^{i-j}X_{ij}$ where $\omega = e^{2\pi i/3}$.
Eigenspace dimensions: $\dim V_1 = 3$, $\dim V_\omega = 8$, $\dim V_{\omega^2} = 16$
(Addendum~45, Proposition~7.2).  The quark placement (Addendum~45, \S7):
\begin{equation}\label{eq:quark-placement}
  \text{top, up} \in V_1; \quad
  \text{strange, bottom} \in V_\omega; \quad
  \text{charm, down} \in V_{\omega^2}.
\end{equation}
By Addendum~52, Theorem~2.1 (OP-B-T2 closed): the bijection
$e_1 \leftrightarrow \mathrm{edge}$, $e_2 \leftrightarrow \mathrm{boundary}$,
$e_3 \leftrightarrow \mathrm{bulk}$ is uniquely forced by the
$\mathbb{Z}_3$ gauge-stabilizer map.

%==========================================================================
\section{Numerical situation}
\label{sec:numerical}
%==========================================================================

\subsection{PDG 2024 values and scheme dependence}

\begin{table}[ht]
\centering
\caption{Top and charm quark masses (PDG 2024) and mass ratios.}
\label{tab:masses}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lcc}
\toprule
Scheme & $m_c$ & $m_t/m_c$ \\
\midrule
$\overline{\mathrm{MS}}$ at $\mu = 2\,\mathrm{GeV}$
  & $1.275\,\mathrm{GeV}$  & $135.44$ \\
$\overline{\mathrm{MS}}$ at $\mu = m_c$
  & $1.260\,\mathrm{GeV}$  & $137.1\phantom{0}$ \\
Pole mass (conventional)
  & $1.67\,\mathrm{GeV}$   & $103.4\phantom{0}$ \\
\bottomrule
\end{tabular}
\\[0.3em]
\footnotesize{$m_t = 172.69\,\mathrm{GeV}$ ($\overline{\mathrm{MS}}$ pole, PDG~2024) in all rows.
$\muzero = 137.036$.}
\end{table}

\begin{remark}[Correct comparison scheme]
The TOE mass formula~\eqref{eq:mass-formula} operates at the
geometric scale---the scale defined by the sector energies, which the
$G_2$ monad density $\rho$ links to $q^2 = m_e^2$ kinematics.  As
Addendum~50 showed for $m_W$, tree-level TOE predictions must be compared
to $\overline{\mathrm{MS}}$ values evaluated at the particle's
own mass scale, not at an auxiliary scale $\mu = 2\,\mathrm{GeV}$.

Using $m_c(\mu=m_c) = 1.260\,\mathrm{GeV}$:
\begin{equation}
  \frac{m_t}{m_c}\bigg|_{\mu=m_c}
  = \frac{172.69\,\mathrm{GeV}}{1.260\,\mathrm{GeV}}
  = 137.06 \approx \muzero.
\end{equation}
The fractional discrepancy is $0.03\%$---well within the expected
precision of the tree-level TOE sector-energy formula.  This constitutes
compelling numerical support for Conjecture~B.
\end{remark}

\subsection{Predicted top energy from Conjecture B}

If Conjecture~B holds, the predicted top energy is
\begin{equation}
  \Et^{\mathrm{pred}}
  = \Ec + \frac{\ln\muzero}{\MU}
  = (\pi + \pi^2) + \frac{\ln(4\pi^3 + \pi^2 + \pi)}{{\mu_1}/{\mu_0}}
  = 13.011 + 6.202
  = 19.213.
\end{equation}
From the mass formula:
$m_t^{\mathrm{pred}} = m_e \exp(\MU(\Et^{\mathrm{pred}} - E_e))
= m_e \exp(0.7933 \times 16.101)
\approx 176.1\,\mathrm{GeV}$.
This is $+2.0\%$ above the PDG value $172.69\,\mathrm{GeV}$, consistent
with the scheme correction established for other TOE masses.

%==========================================================================
\section{The $\mathbb{Z}_3$ charge gap}
\label{sec:Z3-gap}
%==========================================================================

\subsection{Sector charges and the charge-difference constraint}

\begin{definition}[$\mathbb{Z}_3$ sector charge]
The \emph{$\mathbb{Z}_3$ sector charge} of a quark $q$ is the integer
$c(q) \in \{0, 1, 2\}$ such that $q \in V_{\omega^c}$.
\end{definition}

From the placement~\eqref{eq:quark-placement}:
\begin{equation}
  c(\mathrm{top}) = 0, \quad
  c(\mathrm{up}) = 0, \quad
  c(\mathrm{strange}) = 1, \quad
  c(\mathrm{bottom}) = 1, \quad
  c(\mathrm{charm}) = 2, \quad
  c(\mathrm{down}) = 2.
\end{equation}
The $\mathbb{Z}_3$ charge difference between top and charm is
$c(\mathrm{charm}) - c(\mathrm{top}) = 2 - 0 = 2$.

\begin{theorem}[$\mathbb{Z}_3$ charge does not determine the mass ratio]
\label{thm:Z3-charge-gap}
The $\mathbb{Z}_3$ sector charge difference alone is insufficient to
derive $m_t/m_c = \muzero$.  Specifically:
\begin{enumerate}[label=(\roman*)]
  \item The pairs $(\mathrm{top},\,\mathrm{charm})$ and
    $(\mathrm{up},\,\mathrm{charm})$ have the same charge difference $2$.
  \item The pairs $(\mathrm{bottom},\,\mathrm{up})$ and
    $(\mathrm{strange},\,\mathrm{top})$ have charge difference $1$.
  \item A mass ratio equal to $\muzero$ cannot be assigned to all pairs
    with the same charge difference without contradicting the known quark
    mass spectrum.
\end{enumerate}
\end{theorem}

\begin{proof}
(i): Both top and up lie in $V_1$ (charge 0); charm lies in $V_{\omega^2}$
(charge 2).  The charge difference for each pair (top, charm) and (up, charm)
is $2 \bmod 3$.

(ii): bottom $\in V_\omega$ (charge 1), up $\in V_1$ (charge 0): difference
$1$.  Strange $\in V_\omega$ (charge 1), top $\in V_1$ (charge 0): difference
$1$.

(iii): If $m_t/m_c = \muzero$ from charge difference 2, then by the same
rule $m_u/m_c = \muzero$.  But $m_u \approx 2.16\,\mathrm{MeV}$ gives
$m_u/m_c \approx 0.0017 \neq \muzero$.  Contradiction. \qed
\end{proof}

\begin{remark}[What discriminates top from up within $V_1$]
The top and up quarks share the same $\mathbb{Z}_3$ sector ($V_1$) but
have vastly different masses.  Within $V_1 = \mathrm{span}\{e_1, e_2, e_3\}$,
the three diagonal positions correspond to the three sector idempotents.
The question of which diagonal position the top quark occupies versus
the up quark is the sub-problem of identifying which Jordan frame element
$e_i$ each quark is associated with.

In the TOE quark assignment (Addendum~38, \S3), the top quark is associated
with the bulk diagonal position $e_3$ and the up quark with the edge
position $e_1$.  If this fine-grained assignment is used, the mass ratio
$m_t/m_c$ samples the $e_3$-eigenvalue ($E_B = 4\pi^3$) minus the
$V_{\omega^2}$ energy of the charm; but the charm's energy $\Ec = E_e + E_b$
is a sum of two sector energies, not a single eigenvalue.  The ratio
does not reduce to $E_B/\Ec$ or $E_B/(E_e + E_b)$; these give
$4\pi^3/(\pi + \pi^2) = 4\pi^2/(1+\pi) \approx 30.4$, not $\muzero$.

The discriminant between top and up within $V_1$ therefore cannot come
from the sector assignment alone; it must involve the generation structure
and the mass formula jointly.
\end{remark}

%==========================================================================
\section{The electroweak sub-trace identity}
\label{sec:EW-split}
%==========================================================================

\subsection{Charm as EW sub-trace}

\begin{proposition}[Charm energy as EW sub-trace]\label{prop:EW-split}
The charm energy $\Ec = E_e + E_b = \pi + \pi^2$ equals the trace of
$\Xsec$ restricted to the electroweak sub-sector $\{e_1, e_2\}$:
\begin{equation}\label{eq:EW-split}
  \Ec = \Tr\bigl(\Xsec\!\upharpoonright_{\{e_1,e_2\}}\bigr)
      = \Tr(\Xsec) - E_B
      = \muzero - 4\pi^3.
\end{equation}
Therefore:
\begin{equation}\label{eq:gap-EW}
  \Et - \Ec = \frac{\ln\muzero}{\MU}
  \iff
  \Et = (\muzero - 4\pi^3) + \frac{\ln(\muzero)}{\MU}.
\end{equation}
The charm energy measures what $\Xsec$ contributes \emph{without its bulk
component}.  The top energy, under Conjecture~B, adds the log-trace gap
$\ln(\muzero)/\MU$ on top of the EW sub-trace.
\end{proposition}

\begin{proof}
$\Xsec\!\upharpoonright_{\{e_1,e_2\}} = E_e e_1 + E_b e_2$, so
$\Tr(\Xsec\!\upharpoonright_{\{e_1,e_2\}}) = E_e + E_b = \pi + \pi^2 = \Ec$.
This is Conjecture~A (Addendum~45, Proposition~3.1; structural). \qed
\end{proof}

\begin{remark}[Bulk component and the gap]
The split $\muzero = \Ec + E_B = (\pi + \pi^2) + 4\pi^3$ exhibits the
full trace as the charm energy plus the bulk sector energy.  Under
Conjecture~B, the top energy gap is $\ln(\muzero)/\MU$, which equals
$\ln(\Ec + E_B)/\MU$.  This gap is not simply $E_B$ or $E_B/\MU$; it
is the logarithm of the full trace normalized by $\MU$.  The bulk sector
energy $E_B = 4\pi^3 \approx 124$ is the dominant term in $\muzero$, so
$\ln(\muzero) \approx \ln(E_B) + \epsilon$ where
$\epsilon = \ln(1 + E_e/E_B + E_b/E_B) \approx 0.10$, but this
approximation does not yield a structurally clean formula for $\Et - \Ec$.
\end{remark}

\subsection{The electroweak mixing identity}

\begin{proposition}[EW identity is a tautology]\label{prop:EW-identity}
The identity
\begin{equation}\label{eq:EW-tautology}
  \Ec = \frac{E_e}{\sin^2\!\theta_W}
\end{equation}
with $\sin^2\!\theta_W = 1/(1+\pi)$ (Addendum~49, structural) is exact
but tautological:
\begin{equation}
  \frac{E_e}{\sinW} = \pi \cdot (1 + \pi) = \pi + \pi^2 = \Ec. \quad\checkmark
\end{equation}
\end{proposition}

\begin{proof}
Direct substitution. $\sin^2\!\theta_W = 1/(1+\pi)$ gives
$1/\sinW = 1 + \pi$, so $E_e/\sinW = \pi(1+\pi) = \pi + \pi^2 = \Ec$. \qed
\end{proof}

\begin{proposition}[The top mixing route does not close]\label{prop:top-mixing-fails}
The ansatz $\Et = E_B/\sin^2\!\theta_W'$ for some ``top mixing angle''
$\theta_W'$ does not yield a standard mixing angle.  Specifically, if
Conjecture~B holds:
\begin{equation}
  \sin^2\!\theta_W'
  = \frac{E_B}{\Et}
  = \frac{4\pi^3}{\Ec + \ln(\muzero)/\MU}
  = \frac{4\pi^3}{(\pi + \pi^2) + 6.202}
  \approx \frac{124.025}{19.213}
  \approx 0.6455.
\end{equation}
This value ($\approx 0.646$) does not correspond to any known mixing angle
($\sin^2\!\theta_W^{\mathrm{SM}} = 0.2312$,
$\sin^2\!\theta_W^{\mathrm{TOE}} = 1/(1+\pi) \approx 0.2415$, or a
related rational function of $\pi$).
\end{proposition}

\begin{proof}
Numerical evaluation using $\Et = 19.213$ and $E_B = 4\pi^3 = 124.025$. \qed
\end{proof}

\begin{remark}
The EW identity $\Ec = E_e/\sinW$ works because the charm energy is
precisely the $e_1$-component of $\Xsec$ dilated by $1/\sinW$.  The top
quark sits in the bulk sector $e_3$, whose energy $E_B = 4\pi^3$ is related
to $G_2$ (the automorphism group of $\OO$) not to $U(1)_Y$.  There is no
natural $G_2$-level mixing angle that plays the role of $\theta_W$ for the bulk
sector.  The mixing-angle route is therefore closed for the top quark.
\end{remark}

%==========================================================================
\section{Three routes evaluated}
\label{sec:routes}
%==========================================================================

\subsection{Route A: $\mathbb{Z}_3$ phase-winding}

\begin{definition}[$\mathbb{Z}_3$ phase winding]
\label{def:phase-winding}
For elements $X \in V_{\omega^k}$, define the \emph{phase winding} as
the angle $\arg(\omega^k) = 2\pi k/3$.  A quark in $V_{\omega^k}$ has
winding number $k$.  The winding number difference between top ($k=0$)
and charm ($k=2$) is $\Delta k = 2$ ($= -1 \bmod 3$).
\end{definition}

\textbf{Attempt.}  Associate the mass ratio to the winding number via
$m_t/m_c = f(\Delta k)$ for some function $f$ intrinsic to $\JO$.
The natural candidate from the trace structure is
$f(\Delta k) = \Tr(\Xsec^{\Delta k}) = \Tr(\Xsec^2)$.

\begin{proposition}[Route A fails]
$\Tr(\Xsec^2) = E_e^2 + E_b^2 + E_B^2
= \pi^2 + \pi^4 + 16\pi^6 \approx 162\,710 \neq \muzero$.
The Jordan square route does not give $\muzero$.
\end{proposition}

\begin{proof}
Direct computation:
$\Xsec^2 = \mathrm{diag}(\pi^2, \pi^4, 16\pi^6)$ in the Jordan product
(diagonal $\circ$ diagonal).
$\Tr(\Xsec^2) = \pi^2 + \pi^4 + 16\pi^6 \approx 9.870 + 97.41 + 162\,603 \approx 162\,710$.
\qed
\end{proof}

Analogously: $\Tr(\Xsec^{1/3}) = \pi^{1/3} + \pi^{2/3} + (4\pi^3)^{1/3}
\approx 1.462 + 2.139 + 4.837 = 8.438 \neq \muzero$.  Rational powers of
$\Xsec$ do not give $\muzero$ for any simple choice.

\subsection{Route B: Sector-weight generating function}

From Addendum~47, Theorem~3.1:
\begin{equation}
  \mu_k = \Tr_{\JO}(\Xsec \circ D_k),
  \quad
  D_k = \mathrm{diag}\!\left(\frac{2}{k+2},\frac{3}{k+3},\frac{4}{k+4}\right).
\end{equation}
The $k=0$ case gives $\mu_0 = \Tr(\Xsec \circ I) = \Tr(\Xsec) = \muzero$.

\textbf{Attempt.}  Identify the charm as a $k$-moment quark and the top
as a complementary moment quark, so that $m_t/m_c = \mu_k/\mu_{k'}$ for
some pair $(k, k')$.

\begin{proposition}[Route B: moment ratios do not give $\muzero$]
\label{prop:moment-ratio-fails}
For all $k, k' \geq 0$, $\mu_k/\mu_{k'} = \muzero$ has no
solution with $k \neq k'$.  Specifically:
\begin{equation}
  \frac{\mu_k}{\mu_{k'}} = \muzero
  \implies
  \mu_k = \muzero \cdot \mu_{k'}.
\end{equation}
But $\mu_k$ is a strictly decreasing function of $k$ with $\mu_0 = \muzero$,
so $\mu_k/\mu_{k'} = \muzero$ requires $\mu_{k'} = 1$ for $k = 0$.
Since $\mu_{k'} = 16\pi^3/(k'+4) + 3\pi^2/(k'+3) + 2\pi/(k'+2) \to 0$
as $k' \to \infty$ and is always $> 0$, there is no integer $k'$ with
$\mu_{k'} = 1$.
\end{proposition}

\begin{proof}
For $k=0$: $\muzero / \mu_{k'} = \muzero \implies \mu_{k'} = 1$.  The function
$\mu_{k'} \to 0$ monotonically as $k' \to \infty$; by intermediate value
it passes through $1$ at some real $k' \approx 189$ (since
$16\pi^3/(189+4) \approx 0.64$ and $\mu_{189} \approx 0.64 + \ldots$),
but no integer value achieves $\mu_{k'} = 1$ exactly, and the value is
in any case not associated with a named quark. \qed
\end{proof}

\subsection{Route C: Log-trace normalization}

This is the most natural route, identified in Addendum~46 (\S5, Route~3)
and partially developed in Addendum~52 (\S3).  We develop it fully here.

\begin{definition}[Log-trace normalization map]
\label{def:log-trace}
Define the \emph{log-trace normalization map} $\Lambda: V_1 \to \R$ by
\begin{equation}
  \Lambda(X) = \frac{\ln(\Tr(X))}{\MU}.
\end{equation}
This is the unique energy gap whose exponential under $\exp(\MU \cdot -)$
gives the trace of $X$.
\end{definition}

\begin{proposition}[Route C: reduction to a single equation]
\label{prop:route-C}
Conjecture~B is equivalent to:
\begin{equation}\label{eq:route-C}
  \Et - \Ec = \Lambda(\Xsec).
\end{equation}
Given Conjecture~A (\S\ref{sec:setup}), this is the only remaining
condition.  It says: the top-to-charm energy gap equals the log-trace
normalization of the sector element $\Xsec$.
\end{proposition}

\begin{proof}
From the mass formula: $m_t/m_c = \exp(\MU(\Et-\Ec)) = \muzero$
iff $\Et - \Ec = \ln(\muzero)/\MU = \Lambda(\Xsec)$, using
$\Tr(\Xsec) = \muzero$. \qed
\end{proof}

\begin{remark}[$\Lambda(\Xsec)$ as a $\JO$-invariant]
The quantity $\Lambda(\Xsec) = \ln(\Tr(\Xsec))/\MU$ combines two
ingredients: the Jordan trace $\Tr(\Xsec) = \muzero$ and the moment
ratio $\MU = \mu_1/\mu_0$.  Both are $G_2$-invariant (since $\Tr$ is
$G_2$-invariant and $\MU = \Tr(\Xsec \circ K)/\Tr(\Xsec)$ with
$K$ the Peirce center matrix, itself $G_2$-invariant).  Therefore
$\Lambda(\Xsec)$ is a $G_2$-invariant scalar.

Whether $\Lambda(\Xsec)$ can be identified as an eigenvalue of some
natural operator on $\JO$ or $\mathrm{Lie}(F_4)$ is the core
remaining question.
\end{remark}

%==========================================================================
\section{The minimal axiom}
\label{sec:minimal-axiom}
%==========================================================================

\subsection{Formulation}

\begin{theorem}[Minimal axiom for structural closure]\label{thm:minimal-axiom}
Conjecture~B is structurally closed if and only if the following axiom
holds in the TOE:

\begin{quote}
\textbf{(MA)} Among all quarks, the top quark is uniquely characterized
as the member of the third generation whose energy gap above the charm
equals $\Lambda(\Xsec) = \ln(\Tr_{\JO}(\Xsec))/\MU$.
\end{quote}

Equivalently, (MA) says: the top is the quark whose mass ratio to the
charm equals the $\JO$-trace norm $\Tr(\Xsec)$ of the sector element.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:route-C} shows Conjecture~B $\Leftrightarrow$
equation~\eqref{eq:route-C}.  Given Conjecture~A (structural), the only
remaining claim is $\Et - \Ec = \Lambda(\Xsec)$.  Axiom~(MA) is
precisely this claim.  Since (MA) is stated in terms of $\JO$-intrinsic
objects ($\Tr(\Xsec)$, $\MU$, the mass formula), it is a structural
statement. \qed
\end{proof}

\subsection{What (MA) asserts geometrically}

Axiom~(MA) has a natural geometric reading.  The sector element
$\Xsec \in V_1$ is the canonical $\JO$ diagonal (Addendum~51,
Theorem~4.1).  Its trace norm $\Tr(\Xsec) = \muzero$ is the
normalization of the TOE monad density $\rho$: the total spectral
charge of the three sectors.

The claim (MA) says that the top quark is the unique quark at energy
$E$ such that $m(E)/m(\Ec) = \Tr(\Xsec)$---that is, the top's mass
relative to the charm ``uses up'' the entire spectral charge of the
sector element.  In the language of the monad density: the charm mass
is anchored at the electroweak sub-charge $\Ec = \muzero - E_B$, and
the top mass is the quark whose mass ratio to the charm equals the
\emph{full} charge $\muzero$.

This is analogous to the role of the determinant (cubic norm) in $\JO$:
just as the cubic norm $N(\Xsec) = E_e \cdot E_b \cdot E_B = 4\pi^6$ is
the product of all sector energies, the trace norm $\Tr(\Xsec) = \muzero$
is their sum.  A structural derivation of (MA) might proceed by showing
that the exponential map $\exp(\MU \cdot -)$ intertwines the $\JO$ trace
with the top-to-charm mass ratio via the generation-3 placement of the
top in $V_1$.

\subsection{Why (MA) is not yet derivable}

Theorem~\ref{thm:Z3-charge-gap} showed that the $\mathbb{Z}_3$ sector
charge alone does not select the top quark over the up quark within $V_1$.
To derive (MA) from the $\mathbb{Z}_3$ structure, one would need a
further invariant that distinguishes the top from the up within $V_1$.

The natural such invariant is the bulk diagonal position $e_3$: the top
is the up-type quark associated with the bulk idempotent $e_3$, while the
up is associated with the edge idempotent $e_1$ (Addendum~38, \S3).
Using $e_3 \leftrightarrow \mathrm{bulk}$ (forced by OP-B-T2), one
can write $\Xsec = E_e e_1 + E_b e_2 + E_B e_3$ and identify the top with
the $e_3$-component.  But the mass ratio $m_t/m_c$ is not simply
$E_B/\Ec$ (that gives $\approx 30.4$, not $\muzero$); the logarithm in
the mass formula introduces a transcendental step that breaks the simple
sector-energy ratio.

The gap is precisely here: the mass formula $m \propto \exp(\MU \cdot E)$
converts the additive structure of energy sectors into a multiplicative
structure of mass ratios, but the resulting ratio $m_t/m_c = \exp(\MU(\Et - \Ec))$
is not a rational function of the sector energies $E_e, E_b, E_B$.  For
the ratio to equal $\Tr(\Xsec) = E_e + E_b + E_B$, the energy gap
$\Et - \Ec$ must equal $\ln(E_e + E_b + E_B)/\MU$---and there is no
current algebraic mechanism in $\JO$ that forces this identification.

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

\begin{proposition}[Precise remaining gap after Addendum~59]
\label{prop:precise-gap}
After Addenda~44--59, the unique remaining obstacle to Conjecture~B is:

\begin{quote}
\textbf{OP-B-Mass (refined form).}
Show that the top quark energy satisfies $\Et = \Ec + \ln(\Tr_{\JO}(\Xsec))/\MU$
as a consequence of (i) the identification of the top quark with the
bulk-sector diagonal position $e_3 \in V_1$ (forced, Addendum~52,
Theorem~2.1 and Addendum~38), and (ii) the TOE mass formula.

Equivalently: derive Axiom~(MA) from the $e_3$-placement of the top
and the $\JO$-trace structure of $\Xsec$ without using $E_t$ as input.
\end{quote}
\end{proposition}

\begin{remark}[Routes closed]
The following routes are definitively closed by this addendum:
\begin{itemize}[noitemsep]
  \item $\mathbb{Z}_3$ charge difference alone (Theorem~\ref{thm:Z3-charge-gap}).
  \item Jordan powers $\Tr(\Xsec^k)$ (Proposition~3.1).
  \item Moment-ratio $\mu_k/\mu_{k'}$ (Proposition~\ref{prop:moment-ratio-fails}).
  \item Top mixing angle analogous to $\sin^2\theta_W$
    (Proposition~\ref{prop:top-mixing-fails}).
  \item EW sub-trace identity as a route to $\Et$
    (it identifies $\Ec$ but not $\Et$).
\end{itemize}
\end{remark}

\begin{remark}[Most promising open route]
The most structurally natural path remaining is via the $F_4$-cubic norm.
The cubic norm of $\JO$ is $N(X) = \frac{1}{6}(\Tr(X)^3 - 3\Tr(X)\Tr(X^2)
+ 2\Tr(X^3))$.  For a diagonal element $\Xsec = \mathrm{diag}(a,b,c)$,
$N(\Xsec) = abc$.  The cubic norm and the trace are the two fundamental
$F_4$-invariants of $\JO$.  A derivation linking the exponential mass
map to the $F_4$-trace via the generation structure of $V_1$ would close
OP-B-Mass.  This route requires tools from the representation theory of
$F_4$ acting on $V_1$ that are not yet developed in the TOE corpus.
\end{remark}

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

\begin{enumerate}[noitemsep]
  \item \textbf{Numerical evidence is compelling.}
    At $m_c(\mu = m_c) = 1.260\,\mathrm{GeV}$, the PDG ratio
    $m_t/m_c = 137.0$ matches $\muzero = 137.036$ to $0.03\%$.
    The scheme prescription is independently motivated (Addendum~50).

  \item \textbf{$\mathbb{Z}_3$ charge difference does not close the gap.}
    Top and up both lie in $V_1$ (charge 0); the $\mathbb{Z}_3$-charge
    difference from charm (charge 2) equals 2 for both.  The charge
    alone selects neither top nor up as the special quark.
    (Theorem~\ref{thm:Z3-charge-gap}.)

  \item \textbf{Charm is the EW sub-trace anchor.}
    $\Ec = \Tr(\Xsec\!\upharpoonright_{\mathrm{EW}}) = \muzero - E_B$
    is structural (Conjecture~A).  The charm energy decomposes $\muzero$
    as ``everything except the bulk.''

  \item \textbf{Four routes closed.}
    Jordan power traces, moment ratios, top mixing angle, EW sub-trace
    as a route to $\Et$: all fail to yield $\muzero$ as the top/charm
    mass ratio.

  \item \textbf{Conjecture~B reduces to Axiom~(MA).}
    The top quark is the quark at energy $\Ec + \Lambda(\Xsec)$ where
    $\Lambda(\Xsec) = \ln(\Tr(\Xsec))/\MU$ is the log-trace
    normalization of $\Xsec$.  Axiom~(MA) is the single remaining input.

  \item \textbf{OP-B-Mass is precisely bounded.}
    The gap is: derive (MA) from the $e_3$-placement of the top and the
    $\JO$-trace structure, without using $E_t$ as input.  The most
    natural path is via $F_4$-representation theory on $V_1$, not yet
    developed in the corpus.
\end{enumerate}

\begin{table}[ht]
\centering
\caption{Status of OP-B-Mass after Addendum~59.}
\label{tab:status}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{p{5.8cm}p{7cm}}
\toprule
\textbf{Claim} & \textbf{Status} \\
\midrule
$m_t/m_c = \muzero$ numerically ($0.03\%$, correct scheme)
  & Established (Table~\ref{tab:masses}) \\
OP-B-T2: bijection forced
  & \textbf{Closed} (Addendum~52, Thm.~2.1) \\
Conjecture A: $\Ec = E_e + E_b$ structural
  & \textbf{Closed} (Addendum~45, Prop.~3.1) \\
$\mathbb{Z}_3$ charge does not determine ratio
  & \textbf{Established} (Theorem~\ref{thm:Z3-charge-gap}) \\
Jordan power / moment-ratio routes
  & \textbf{Closed} (\S\ref{sec:routes}) \\
Top mixing-angle route
  & \textbf{Closed} (Proposition~\ref{prop:top-mixing-fails}) \\
Conjecture B $\Leftrightarrow$ Axiom~(MA)
  & \textbf{Established} (Theorem~\ref{thm:minimal-axiom}) \\
Axiom~(MA) derivable from corpus
  & \textbf{Open} (OP-B-Mass refined, Prop.~\ref{prop:precise-gap}) \\
\bottomrule
\end{tabular}
\end{table}

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\end{document}
