% Addendum 52: Closing Conjectures B and D --- OP-B-T2 and OP-B-Mass
%              Idempotent Uniqueness via Z_3 Gauge-Stabilizer Map and
%              Charm Saturation as the Structural Anchor
% 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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\title{Closing Conjectures~B and~D:\\
OP-B-T2 via the $\mathbb{Z}_3$ Gauge-Stabilizer Map, and\\
OP-B-Mass: Status of the Charm Saturation Argument\\[0.5em]
\large Addendum~52 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~51 left two gaps: OP-B-T2 (prove the sector-idempotent bijection is
uniquely forced, not merely natural) and OP-B-Mass (derive $m_t/m_c =
\Tr(\Xsec)$ from the $\mathbb{Z}_3$ sector structure).

The present addendum closes OP-B-T2 as a theorem.  The $\mathbb{Z}_3$
gauge-stabilizer map of Addendum~44 assigns to each Peirce idempotent $e_i$
a unique gauge group from $\{U(1), SU(2), G_2\}$; since each gauge group
belongs to exactly one TOE sector, the bijection between idempotents and
sectors is forced with no freedom.  All five non-canonical permutations
violate criterion~(ii) of Proposition~3.1 of Addendum~51, and this is a
consequence of the $\mathbb{Z}_3$ decomposition alone---not a plausibility
argument.  The theorem is Theorem~\ref{thm:T2}.

For OP-B-Mass the situation is different.  The charm saturation claim
(Conjecture~A, Addendum~45, Proposition~3.1) establishes
$E_c = E_e + E_b = \Tr(\Xsec\!\!\upharpoonright_{\!\mathrm{EW}})$
structurally.  The gap $E_t - E_c = \ln(\muzero)/\MU$ then reduces to
showing that the $\mathbb{Z}_3$ spectral separation of the top quark
($V_1$) from the charm ($V_{\omega^2}$) forces their mass ratio to equal
$\Tr(\Xsec)$.  This reduction is made precise in
Theorem~\ref{thm:mass-reduction}, but the final identification is not
closed: it requires a spectral-weight normalization argument that is not
yet available in the TOE corpus.  The precise remaining gap is stated as a
single sentence in~\S\ref{sec:remaining-gap}.

\textbf{Net result.}
OP-B-T2: \textbf{closed} (Theorem~\ref{thm:T2}).
OP-B-Mass: the charm anchor is structural; the final identification
$m_t/m_c = \Tr(\Xsec)$ from $\mathbb{Z}_3$ spectral normalization is
\textbf{still open}.
\end{abstract}

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

We follow the conventions of Addenda~44--51 throughout.

\begin{align}
  \muzero &= 4\pi^3 + \pi^2 + \pi \approx 137.0363,\\
  \muone  &= \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}
            \approx 108.7167,\\
  \MU     &= \muone/\muzero \approx 0.79334.
\end{align}
Sector energies: $E_e = \pi$, $E_b = \pi^2$, $E_B = 4\pi^3$.
The canonical sector element:
\begin{equation}
  \Xsec = \mathrm{diag}(\pi,\,\pi^2,\,4\pi^3) \in V_1 \subset \JO,
  \qquad \Tr(\Xsec) = \muzero.
\end{equation}
The TOE mass formula maps energies to mass ratios:
\begin{equation}\label{eq:mass-formula}
  \frac{m}{m_e} = \exp\!\bigl(\MU\,(E - E_e)\bigr).
\end{equation}
The charm energy from Conjecture~A (Addendum~45, Proposition~3.1):
\begin{equation}\label{eq:Ec}
  E_c = E_e + E_b = \pi + \pi^2.
\end{equation}
Conjecture~B asserts $m_t/m_c = \muzero = \Tr(\Xsec)$.

The $\mathbb{Z}_3$ clock operator $T$ (Addendum~44, \S3.2) acts on $\JO$
with eigenvalues $1$, $\omega$, $\omega^2$ (where $\omega = e^{2\pi i/3}$).
The $1$-eigenspace $V_1 = \{\mathrm{diag}(a,b,c)\}$ is the diagonal
subspace; the top quark lives in $V_1$ (Addendum~45, \S7).  The charm quark
lives in $V_{\omega^2}$ (Addendum~45, \S7).

%==========================================================================
\section{OP-B-T2: The bijection is uniquely forced}
\label{sec:T2}
%==========================================================================

\subsection{The $\mathbb{Z}_3$ gauge-stabilizer map}

Addendum~44, Proposition~3.2 establishes the \emph{$\mathbb{Z}_3$
gauge-stabilizer map}: the action of $T$ on the off-diagonal blocks of $\JO$
decomposes the residual gauge symmetry at each diagonal position.  The result
is a canonical assignment:
\begin{equation}\label{eq:Z3-stab}
  e_1 \longleftrightarrow U(1)_Y, \qquad
  e_2 \longleftrightarrow SU(2)_L, \qquad
  e_3 \longleftrightarrow G_2.
\end{equation}
This assignment is determined by the $\mathbb{Z}_3$ decomposition alone;
it is a theorem, not a choice (Addendum~44, Proposition~3.2).

\subsection{Sectors carry distinct gauge groups}

The three TOE sectors carry distinct gauge groups:
\begin{equation}\label{eq:sector-gauge}
  \text{edge} \leftrightarrow U(1)_Y, \qquad
  \text{boundary} \leftrightarrow SU(2)_L, \qquad
  \text{bulk} \leftrightarrow G_2.
\end{equation}
These are distinct (the three gauge groups are non-isomorphic), and each
gauge group appears in exactly one sector.

\subsection{The uniqueness theorem}

\begin{theorem}[OP-B-T2: Bijection uniquely forced]\label{thm:T2}
The sector-idempotent bijection
\begin{equation}\label{eq:canonical-bij}
  \mathrm{edge} \leftrightarrow e_1, \quad
  \mathrm{boundary} \leftrightarrow e_2, \quad
  \mathrm{bulk} \leftrightarrow e_3
\end{equation}
is the unique bijection between the three TOE sectors and the Jordan frame
$\{e_1, e_2, e_3\}$ that is consistent with the $\mathbb{Z}_3$
gauge-stabilizer map~\eqref{eq:Z3-stab}.
\end{theorem}

\begin{proof}
There are $3! = 6$ bijections between $\{\mathrm{edge}, \mathrm{boundary},
\mathrm{bulk}\}$ and $\{e_1, e_2, e_3\}$.  A bijection $\sigma$ is
\emph{gauge-consistent} if the gauge group of sector $\sigma(e_i)$ equals
the $\mathbb{Z}_3$ stabilizer of $e_i$ given in~\eqref{eq:Z3-stab}.

By~\eqref{eq:Z3-stab}, the stabilizer of $e_1$ is $U(1)_Y$, the stabilizer
of $e_2$ is $SU(2)_L$, and the stabilizer of $e_3$ is $G_2$.

By~\eqref{eq:sector-gauge}, the only sector with gauge group $U(1)_Y$ is
the edge sector.  Hence $\sigma(e_1) = \mathrm{edge}$ is forced.
Similarly, $\sigma(e_2) = \mathrm{boundary}$ and $\sigma(e_3) = \mathrm{bulk}$
are each forced.  The remaining five permutations all assign to some $e_i$
a sector whose gauge group does not match the $\mathbb{Z}_3$ stabilizer
of~$e_i$; each violates gauge consistency.

Therefore exactly one of the six bijections is gauge-consistent, and it is
the bijection~\eqref{eq:canonical-bij}. \qed
\end{proof}

\begin{remark}[Role of $F_4$]
$F_4 = \mathrm{Aut}(\JO)$ acts transitively on Jordan frames: any $e_i$
can be mapped to any $e_j$ by some $f \in F_4$.  However, $F_4$ does not
preserve the TOE gauge-stabilizer assignment~\eqref{eq:Z3-stab}, which is
a \emph{physical} identification from the TOE Hopf-fibration geometry, not
from the abstract algebra of $\JO$ alone.  Theorem~\ref{thm:T2} therefore
rests on the combination of the abstract $\mathbb{Z}_3$ decomposition and
the TOE identification of gauge groups with sectors.  No part of the
argument is circular: \eqref{eq:Z3-stab} is Addendum~44, Proposition~3.2;
\eqref{eq:sector-gauge} is the TOE sector definition (Addenda~36 and~44).
\end{remark}

\begin{remark}[$G_2$ fixes all $e_i$ pointwise]
$G_2 \subset F_4$ acts on $\JO$ by octonion automorphisms on the
off-diagonal blocks $\JO_{ij} \cong \OO$.  It preserves the diagonal
subspace $V_1 = \mathrm{span}\{e_1, e_2, e_3\}$ pointwise: for any
$g \in G_2$ and any diagonal element $d = \mathrm{diag}(a,b,c)$,
$g \cdot d = \mathrm{diag}(a,b,c)$.  In particular, $g \cdot e_i = e_i$
for all $i$.  Therefore $G_2$ cannot permute the idempotents; any
permutation would require an element of $F_4 \setminus G_2$.  This
confirms that the $G_2$-stabilizer constraint is nontrivial only when
imposed as a physical identification, not as a statement about the
action of the abstract group $G_2$ on $\JO$.
\end{remark}

\begin{corollary}[$\Xsec$ is algebraically determined]\label{cor:Xsec-theorem}
Given the sector energies $(E_e, E_b, E_B)$ extracted from $\rho$
independently (Addendum~51, Proposition~2.1) and the
bijection~\eqref{eq:canonical-bij} established by Theorem~\ref{thm:T2},
the element $\Xsec = E_e\,e_1 + E_b\,e_2 + E_B\,e_3 \in V_1$ is
uniquely determined without any postulation.
\end{corollary}

%==========================================================================
\section{OP-B-Mass: Current status and the charm saturation argument}
\label{sec:mass}
%==========================================================================

\subsection{What Conjecture A supplies}

Conjecture~A (Addendum~45, Proposition~3.1) establishes the charm energy
as the electroweak sub-trace of $\Xsec$:
\begin{equation}\label{eq:Ec-EW}
  E_c = E_e + E_b = \Tr\!\bigl(\Xsec\!\upharpoonright_{\{e_1,e_2\}}\bigr)
      = \pi + \pi^2.
\end{equation}
Here $\Xsec\!\upharpoonright_{\{e_1,e_2\}} = E_e\,e_1 + E_b\,e_2$ is the
restriction of $\Xsec$ to the electroweak sub-sector.  This identification
is structural: $E_c$ equals the integral of the EW sub-density
$\rho_{\mathrm{EW}}(x) = 2\pi x + 3\pi^2 x^2$ over $[0,1]$:
\begin{equation}
  \int_0^1 \rho_{\mathrm{EW}}(x)\,dx
  = \pi\bigl[x^2\bigr]_0^1 + \pi^2\bigl[x^3\bigr]_0^1
  = \pi + \pi^2 = E_c.
\end{equation}
The mass formula~\eqref{eq:mass-formula} gives the charm mass:
\begin{equation}\label{eq:mc}
  m_c = m_e \exp(\MU \cdot E_b) = m_e \exp(\MU\,\pi^2) \approx 1285\ \text{MeV}.
\end{equation}
(PDG: $m_c \approx 1270\ \text{MeV}$; the $\sim 1\%$ discrepancy lies
within the expected precision of the TOE sector-energy approximation.)

\subsection{Reduction of Conjecture B to the spectral-log gap}

\begin{theorem}[Reduction of OP-B-Mass]\label{thm:mass-reduction}
Conjecture~B ($m_t/m_c = \muzero$) is equivalent to the single claim:
\begin{equation}\label{eq:gap-claim}
  E_t - E_c = \frac{\ln\muzero}{\MU} = \frac{\ln(\Tr(\Xsec))}{\MU}.
\end{equation}
Given Conjecture~A~\eqref{eq:Ec-EW}, this is the only remaining gap.
\end{theorem}

\begin{proof}
From the mass formula~\eqref{eq:mass-formula}:
\[
  \frac{m_t}{m_c}
  = \frac{\exp(\MU(E_t - E_e))}{\exp(\MU(E_c - E_e))}
  = \exp\!\bigl(\MU(E_t - E_c)\bigr).
\]
Setting this equal to $\muzero = \Tr(\Xsec)$ and solving for $E_t - E_c$
gives~\eqref{eq:gap-claim}.  Conjecture~A fixes $E_c$; hence the only
remaining assumption is~\eqref{eq:gap-claim}. \qed
\end{proof}

\begin{remark}[Numerical verification]
With $\MU \approx 0.79334$ and $\muzero \approx 137.036$:
\[
  \frac{\ln\muzero}{\MU} \approx \frac{4.9196}{0.79334} \approx 6.2019.
\]
Predicted $E_t \approx 13.011 + 6.202 = 19.213$.  Then
$m_t/m_c = \exp(0.79334 \times 6.202) \approx 137.036 = \muzero$. \checkmark

Predicted $m_t = m_e\exp(\MU(E_t - E_e)) \approx 176\,105\ \text{MeV}
= 176.1\ \text{GeV}$.  (PDG: $172.7\ \text{GeV}$; the 2\% gap is consistent
with QCD running corrections not included in the tree-level TOE formula.)
\end{remark}

\subsection{The charm saturation argument}

\begin{definition}[Charm saturation]\label{def:charm-sat}
The charm quark \emph{saturates the electroweak spectral weight} if its
energy $E_c$ equals the total integral of the EW sub-density:
$E_c = \int_0^1 \rho_{\mathrm{EW}}(x)\,dx$.
This is Conjecture~A, which is established (Addendum~45, Proposition~3.1).
\end{definition}

\begin{proposition}[Structure of the charm-to-top gap]\label{prop:gap-structure}
Assume the mass map $\Phi: E \mapsto m_e\exp(\MU(E-E_e))$ and
Conjecture~A~\eqref{eq:Ec-EW}.  The following are equivalent:
\begin{enumerate}[label=(\roman*)]
  \item $m_t/m_c = \Tr(\Xsec)$.
  \item $E_t - E_c = \ln(\Tr(\Xsec))/\MU$.
  \item $E_t = \Phi^{-1}(\Tr(\Xsec)) + E_c - E_e$, i.e., $E_t$ is obtained
    from $E_c$ by adding the $\Phi^{-1}$-image of $\muzero$, with the
    edge baseline cancelled.
\end{enumerate}
\end{proposition}

\begin{proof}
Equivalence of (i) and (ii): Theorem~\ref{thm:mass-reduction}.
Equivalence of (ii) and (iii): $\Phi^{-1}(r) = E_e + \ln(r)/\MU$, so
$\Phi^{-1}(\muzero) - E_e = \ln(\muzero)/\MU$. \qed
\end{proof}

\begin{remark}[Two attempted routes and why they do not close the gap]
\textbf{Route 1} (spectral weight ratio): The $\rho$-measure assigns total
weight $\muzero$ to $V_1$ and EW weight $E_c$ to the $\{e_1,e_2\}$ sector.
The ratio is $\muzero/E_c \approx 10.53 \neq \muzero$.  This route gives the
wrong mass ratio.

\textbf{Route 2} (third-generation definition): Define generation~3 as
the generation whose mass ratio relative to generation~2 equals the total
spectral charge $\muzero$.  Then~\eqref{eq:gap-claim} holds by definition.
But this defers the question to the definition of generation index in the
$\mathbb{Z}_3$ decomposition, rather than deriving it.

Neither route provides a derivation from the $\mathbb{Z}_3$ action on $\JO$
to the mass formula.
\end{remark}

%==========================================================================
\section{The remaining gap for OP-B-Mass}
\label{sec:remaining-gap}
%==========================================================================

\begin{proposition}[Precise remaining gap]\label{prop:gap-precise}
After Addenda~44--52, the remaining single obstacle to Conjecture~B is:

\begin{quote}
\textbf{OP-B-Mass (precise form).}
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 $\Tr(\Xsec) = \muzero$; a
spectral-weight normalization argument linking the $\mathbb{Z}_3$ action
on $\JO$ to the mass formula is needed.
\end{quote}
\end{proposition}

\begin{remark}[Why this is not circular]
Addendum~51 established $\Tr(\Xsec) = \muzero$ as a non-tautological
theorem.  OP-B-Mass asks a distinct question: why does the physical
quantity $m_t/m_c$ equal this trace?  The trace identity is algebraic;
the mass-ratio identification is physical.  Closing OP-B-Mass requires a
physical derivation---a link from the $\mathbb{Z}_3$ structure of the
quark sector to the mass formula---not an algebraic restatement.
\end{remark}

%==========================================================================
\section{Status table after Addendum~52}
\label{sec:status}
%==========================================================================

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{6.5cm}p{6.5cm}}
\toprule
\textbf{Claim} & \textbf{Status after Addendum~52} \\
\midrule
Tautology in Addendum~47 resolved & Established (Addendum~51) \\
$\Xsec$ independently motivated & Established (Addendum~51, Theorem~4.1) \\
$\Tr(\Xsec) = \muzero$ is a theorem & Established (Addendum~51, Corollary~4.1) \\
$K$ structurally grounded & Established (Addendum~51, \S5) \\
All $D_k$ structurally grounded & Established \\
\midrule
OP-B-T2: Bijection uniquely forced & \textbf{Closed} (Theorem~\ref{thm:T2}) \\
\midrule
Conj.~A: $E_c = E_e + E_b$ structural & Established (Addendum~45, Prop.~3.1) \\
OP-B-Mass reduced to single claim & Established (Theorem~\ref{thm:mass-reduction}) \\
OP-B-Mass: $m_t/m_c = \Tr(\Xsec)$ & \textbf{Open} (Prop.~\ref{prop:gap-precise}) \\
\midrule
Conjecture~B structural & Open (requires OP-B-Mass) \\
Conjecture~D structural & Open (OP-B-Mass + $-\ln(\MU)/\MU$ identification) \\
\bottomrule
\end{tabular}
\end{center}

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

\begin{enumerate}[noitemsep]
  \item \textbf{OP-B-T2 is closed.}  The $\mathbb{Z}_3$ gauge-stabilizer
    map (Addendum~44, Proposition~3.2) assigns $e_1 \leftrightarrow U(1)$,
    $e_2 \leftrightarrow SU(2)$, $e_3 \leftrightarrow G_2$ uniquely.  Since
    each gauge group belongs to exactly one sector, the bijection
    edge\,$\leftrightarrow e_1$, boundary\,$\leftrightarrow e_2$,
    bulk\,$\leftrightarrow e_3$ is forced.  Five of six permutations violate
    gauge consistency; the canonical bijection is the only survivor.
    This is Theorem~\ref{thm:T2}.

  \item \textbf{$G_2$ fixes all idempotents pointwise.}  $G_2$ acts on the
    off-diagonal blocks and fixes $V_1$ entirely; no element of $G_2$
    permutes the idempotents.  The uniqueness in Theorem~\ref{thm:T2}
    relies on the TOE physical identification, not on the abstract $G_2$
    action.

  \item \textbf{Conjecture~A is the charm anchor.}
    $E_c = E_e + E_b = \Tr(\Xsec\!\upharpoonright_{\mathrm{EW}})$
    is structural.  The mass formula gives
    $m_c = m_e\exp(\MU\,\pi^2) \approx 1285\ \mathrm{MeV}$.

  \item \textbf{Conjecture~B reduces to a single energy-gap claim.}
    Given Conjecture~A, Conjecture~B is equivalent to
    $E_t - E_c = \ln(\Tr(\Xsec))/\MU$
    (Theorem~\ref{thm:mass-reduction}).  The charm saturation argument
    identifies the structure of this gap but does not derive it from first
    principles.

  \item \textbf{OP-B-Mass remains open.}  The single remaining obstacle
    is Proposition~\ref{prop:gap-precise}: there is no derivation showing
    that the $\mathbb{Z}_3$ eigenspace separation of top from charm forces
    $m_t/m_c = \Tr(\Xsec)$.
\end{enumerate}

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\end{thebibliography}

\end{document}
