% Addendum 40: J_3(O) Spectral Table and PMNS Corrections
% L. F. Vlegels, Independent Researcher

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\title{$J_3(\mathbb{O})$ Spectral Table, PMNS Corrections,\\and Higgs VEV Candidate\\[0.5em]
\large Addendum 40 to the TOE Corpus}

\author{L. F. Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
We complete three tasks that follow directly from the $\JO$ geometry established
in Addenda~37--39.
\textbf{(i) Spectral table.}  The 27-dimensional Albert algebra $\JO$ decomposes
under $\mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$ into
27 slots corresponding to three generations of fermions, the gauge bosons, and the
Higgs doublet.  We tabulate all 27 slots with their SM quantum numbers and the
known TOE energy anchors from Addenda~36--39.
\textbf{(ii) PMNS phase 3.5.}
The tribimaximal baseline of Addendum~38 (Proposition~4.1) has three deviations
from the PDG mixing angles.  We propose three conjectures, all derived from the
Cabibbo angle $\thetaC = \pi/\dim G_2 = \pi/14$ (Conjecture~C of Addendum~38)
and the $G_2$ dimension:
$\sin^2\theta_{13} = \tfrac{4}{9}\sin^2(\pi/14)$ at $+0.26\%$
(\textbf{Conjecture~F}),
$\sin^2\theta_{23} = \tfrac{15}{28}$ at $-1.88\%$ (\textbf{Conjecture~G}),
and $\sin^2\theta_{12} = \cos^2(\pi/14)/3$ at $+3.20\%$ (\textbf{Conjecture~H}).
\textbf{(iii) VEV candidate (OP1).}
The Higgs vacuum expectation value satisfies $v = e \cdot M_Z$ to $0.67\%$,
where $e = \exp(1)$ and $M_Z = 91.188$\,GeV.  We record this as a candidate formula
pending a structural derivation from the $\JO$ fold-map spectral problem.
\end{abstract}

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

Addendum~37 derived the gauge group $\mathrm{SU}(3)_c \subset G_2 = \mathrm{Aut}(\OO)$,
$\mathrm{SU}(2)_L$ as the left isometry of $S^3$, and $\mathrm{U}(1)_Y$ as the
structure group of the Hopf fiber.
Addendum~38 identified quark energies (Conjectures~A--E) and the tribimaximal
neutrino mixing baseline from the $\mathbb{Z}_3$ symmetry of the $\JO$ diagonal.
Addendum~39 identified the Higgs doublet as the $\mathrm{SU}(3)_c$-singlet
subspace $V_H = \C e_0 \oplus \C e_7$ of the Cayley--Dickson decomposition
$\C \otimes \OO = \mathbf{1} \oplus \mathbf{1} \oplus \mathbf{3} \oplus \bar{\mathbf{3}}$.

Throughout we use the TOE constants
\begin{equation}
  \muzero = 4\pi^3 + \pi^2 + \pi \approx 137.036,\qquad
  \muone  = \frac{16\pi^3}{5} + \frac{3\pi^2}{4} + \frac{2\pi}{3} \approx 108.717,\qquad
  \mathrm{MU} = \frac{\muone}{\muzero} \approx 0.7933,
\end{equation}
and the mass formula $m/m_e = \exp\bigl(\mathrm{MU}(E - \pi)\bigr)$
with $m_e = 0.511$\,MeV (Paper~17).
The Cabibbo angle from Addendum~38 Conjecture~C is
\begin{equation}
  \thetaC = \frac{\pi}{\dim G_2} = \frac{\pi}{14},\qquad
  \sin\thetaC = \sin\!\tfrac{\pi}{14} \approx 0.22252.
\end{equation}

%==========================================================================
\section{The \texorpdfstring{$J_3(\mathbb{O})$}{J3(O)} spectral table}
\label{sec:spectral}

\subsection{Decomposition structure}

An element of $\JO$ has the form
\begin{equation}
  X = \begin{pmatrix} \alpha_1 & x_{12} & x_{13} \\ \bar{x}_{12} & \alpha_2 & x_{23} \\
  \bar{x}_{13} & \bar{x}_{23} & \alpha_3 \end{pmatrix},
  \qquad \alpha_i \in \R,\; x_{ij} \in \OO.
\end{equation}
The three diagonal entries $\alpha_1, \alpha_2, \alpha_3$ are $\mathbb{Z}_3$-symmetric
generation labels (Addendum~38, Section~3).  Each off-diagonal entry $x_{ij} \in \OO$
has 8 real components, labelled by the octonion basis
$\{e_0, e_1, e_2, e_3, e_4, e_5, e_6, e_7\}$ where $e_0 = 1$ is the real unit
and $e_7$ is the $G_2$-stabilized imaginary unit.

Under the decomposition $\C \otimes \OO \cong \mathbf{1} \oplus \mathbf{1} \oplus
\mathbf{3} \oplus \bar{\mathbf{3}}$ (Addendum~39, Theorem~2.1):
\begin{itemize}[noitemsep]
  \item $e_0$: $\mathrm{SU}(3)_c$ singlet, $\mathrm{SU}(2)_L$ singlet, $Y = -1$
        (Higgs or gauge boson slot).
  \item $e_7$: $\mathrm{SU}(3)_c$ singlet, $\mathrm{SU}(2)_L$ singlet, $Y = -1/3$
        (CKM/Cabibbo mixing direction).
  \item $e_1, e_2, e_3$: colour triplet $\mathbf{3}$, $\mathrm{SU}(2)_L$ doublet
        component, $Y = +1/6$ (up-type quarks).
  \item $e_4, e_5, e_6$: colour anti-triplet $\bar{\mathbf{3}}$, $\mathrm{SU}(2)_L$
        singlet, $Y = -1/3$ (down-type quarks).
\end{itemize}

\subsection{The 27-slot table}

Table~\ref{tab:spectral} lists all 27 degrees of freedom of $\JO$, together with
the SM particle or role identified with each slot, the known TOE energy anchor,
and the predicted mass against the PDG value.

\begin{table}[ht]
\centering
\small
\setlength{\tabcolsep}{4pt}
\caption{$\JO$ spectral table.  $E$ = TOE energy parameter, $m_\mathrm{calc}$ from
$m = m_e \exp\!\bigl(\mathrm{MU}(E-\pi)\bigr)$.
Slots with ``---'' have no clean TOE formula yet.}
\label{tab:spectral}
\begin{tabular}{rlllrrrr}
\toprule
\# & Slot & $\mathrm{SU}(3)_c$ & SM particle & $E$-formula & $m_\mathrm{calc}$ & PDG & err \\
\midrule
\multicolumn{8}{l}{\textit{Diagonal --- generation labels}} \\
1 & $\alpha_1$ & singlet & electron & $\pi$ & 0.511\,MeV & 0.511\,MeV & exact \\
2 & $\alpha_2$ & singlet & muon & $\pi^2$ & 106.3\,MeV & 105.7\,MeV & $+0.60\%$ \\
3 & $\alpha_3$ & singlet & tau & $E_\tau$ & 1777\,MeV & 1777\,MeV & $+0.08\%$ \\[3pt]
\multicolumn{8}{l}{\textit{Off-diagonal block $x_{12}$ --- first/second generation}} \\
4  & $x_{12} \cdot e_0$ & singlet & $\nu_e/e_L$ doublet\rlap{$^*$} & --- & --- & ---       & --- \\
5  & $x_{12} \cdot e_7$ & singlet & $d$--$s$ Cabibbo dir.\ & $\ln(\muone)/\mathrm{MU}$ & 4.60\,MeV & 4.67\,MeV & $-1.6\%$ \\
6  & $x_{12} \cdot e_1$ & $\mathbf{3}$ & $u_r$ & --- & --- & 2.16\,MeV & --- \\
7  & $x_{12} \cdot e_2$ & $\mathbf{3}$ & $u_g$ & --- & --- & 2.16\,MeV & --- \\
8  & $x_{12} \cdot e_3$ & $\mathbf{3}$ & $u_b$ & --- & --- & 2.16\,MeV & --- \\
9  & $x_{12} \cdot e_4$ & $\bar{\mathbf{3}}$ & $d_r$ & $\ln(\muone)/\mathrm{MU}$ & 4.60\,MeV & 4.67\,MeV & $-1.6\%$ \\
10 & $x_{12} \cdot e_5$ & $\bar{\mathbf{3}}$ & $d_g$ & $\ln(\muone)/\mathrm{MU}$ & 4.60\,MeV & 4.67\,MeV & $-1.6\%$ \\
11 & $x_{12} \cdot e_6$ & $\bar{\mathbf{3}}$ & $d_b$ & $\ln(\muone)/\mathrm{MU}$ & 4.60\,MeV & 4.67\,MeV & $-1.6\%$ \\[3pt]
\multicolumn{8}{l}{\textit{Off-diagonal block $x_{13}$ --- first/third generation}} \\
12 & $x_{13} \cdot e_0$ & singlet & $\nu_\tau/\tau_L$ doublet\rlap{$^*$} & --- & --- & --- & --- \\
13 & $x_{13} \cdot e_7$ & singlet & $d$--$b$ CKM dir.\ & $\ln(\muone)/\mathrm{MU}$ & 4.60\,MeV & 4.67\,MeV & $-1.6\%$ \\
14 & $x_{13} \cdot e_1$ & $\mathbf{3}$ & $c_r$ & $\pi^2 + \pi$ & 1285\,MeV & 1270\,MeV & $+1.2\%$ \\
15 & $x_{13} \cdot e_2$ & $\mathbf{3}$ & $c_g$ & $\pi^2 + \pi$ & 1285\,MeV & 1270\,MeV & $+1.2\%$ \\
16 & $x_{13} \cdot e_3$ & $\mathbf{3}$ & $c_b$ & $\pi^2 + \pi$ & 1285\,MeV & 1270\,MeV & $+1.2\%$ \\
17 & $x_{13} \cdot e_4$ & $\bar{\mathbf{3}}$ & $s_r$ & --- & --- & 93.4\,MeV & --- \\
18 & $x_{13} \cdot e_5$ & $\bar{\mathbf{3}}$ & $s_g$ & --- & --- & 93.4\,MeV & --- \\
19 & $x_{13} \cdot e_6$ & $\bar{\mathbf{3}}$ & $s_b$ & --- & --- & 93.4\,MeV & --- \\[3pt]
\multicolumn{8}{l}{\textit{Off-diagonal block $x_{23}$ --- second/third generation}} \\
20 & $x_{23} \cdot e_0$ & singlet & Higgs\,/\,VEV & OP1 cand.\ & 246\,GeV & 246\,GeV & $(-0.67\%)$ \\
21 & $x_{23} \cdot e_7$ & singlet & $s$--$b$ CKM dir.\ & --- & --- & --- & --- \\
22 & $x_{23} \cdot e_1$ & $\mathbf{3}$ & $t_r$ & $\pi^2+\pi+\ln(\muzero)/\mathrm{MU}$ & 176\,GeV & 173\,GeV & $+1.9\%$ \\
23 & $x_{23} \cdot e_2$ & $\mathbf{3}$ & $t_g$ & $\pi^2+\pi+\ln(\muzero)/\mathrm{MU}$ & 176\,GeV & 173\,GeV & $+1.9\%$ \\
24 & $x_{23} \cdot e_3$ & $\mathbf{3}$ & $t_b$ & $\pi^2+\pi+\ln(\muzero)/\mathrm{MU}$ & 176\,GeV & 173\,GeV & $+1.9\%$ \\
25 & $x_{23} \cdot e_4$ & $\bar{\mathbf{3}}$ & $b_r$ & $\pi^{7/3}$ & 4040\,MeV & 4180\,MeV & $-3.3\%$ \\
26 & $x_{23} \cdot e_5$ & $\bar{\mathbf{3}}$ & $b_g$ & $\pi^{7/3}$ & 4040\,MeV & 4180\,MeV & $-3.3\%$ \\
27 & $x_{23} \cdot e_6$ & $\bar{\mathbf{3}}$ & $b_b$ & $\pi^{7/3}$ & 4040\,MeV & 4180\,MeV & $-3.3\%$ \\
\bottomrule
\end{tabular}
\end{table}

\smallskip
\noindent$^*$\,\textit{Corrected per Addendum~48 (Corollary~5.2):}
Slots~4 and~12 are lepton doublet matter slots in the $E_6$ matter $\mathbf{27}$,
not gauge boson slots.  $W^\pm$ and $Z^0$ reside in the $E_6$ adjoint
$\mathbf{78}$---specifically the $(1,8,1)$ piece under trinification---and
\emph{cannot} appear as entries of the matter $\mathbf{27}$.
The earlier labels ``$W$\,/\,$\nu_e$'' and ``$Z$\,/\,$\nu_\tau$'' were heuristic;
the correct identification is $\nu_e/e_L$ (first/second generation lepton doublet,
$Y=-1$) and $\nu_\tau/\tau_L$ (first/third generation lepton doublet, $Y=-1$).
PDG mass entries for $M_W$ and $M_Z$ are removed accordingly; gauge boson masses
require the adjoint $\mathbf{78}$ sector (open problem, Addendum~48 Section~6).

\begin{remark}
The 27 slots account for all SM matter content: three charged leptons (diagonal),
nine up-type quarks ($\mathbf{3}$ entries in each block), nine down-type quarks
($\bar{\mathbf{3}}$ entries), three CKM/Cabibbo directions ($e_7$ entries),
two lepton doublets (slots~4 and~12, $e_0$ entries in the quark/lepton blocks),
and the Higgs/VEV slot (slot~20, $e_0$ entry in the Higgs block).
Neutrinos do not appear as independent $\JO$ slots in this decomposition;
they arise as the $\mathrm{SU}(2)_L$ doublet partners of the charged leptons,
consistent with the Type~1 mass gap of Addendum~38 Proposition~3.2.
$W^\pm$ and $Z^0$ gauge bosons are \emph{not} matter-27 slots; see Addendum~48.
\end{remark}

\subsection{Derivation status of occupied slots}

Of the 27 slots, 16 carry known TOE energy formulas (slots 1--3, 5, 9--11, 13--16,
22--27), 4 carry candidate formulas (slot 2 at $+0.60\%$, slot 20 as OP1, slots
25--27 from Fano conjecture), and 7 remain open (slots 4, 6--8, 12, 17--19, 21).
The open slots correspond to the $u$ quark, $s$ quark, the two lepton doublets
(slots~4 and~12; corrected per Addendum~48 --- these are not $W$/$Z$ slots),
and two CKM off-diagonal directions.
$W^\pm$ and $Z^0$ masses are outside the matter $\mathbf{27}$ and require the
$E_6$ adjoint $\mathbf{78}$ (see Addendum~48, Section~6).

%==========================================================================
\section{Phase 3.5: PMNS mixing angles from Cabibbo and \texorpdfstring{$G_2$}{G2}}
\label{sec:pmns}

\subsection{TBM baseline and its deviations}

Addendum~38 Proposition~4.1 derived the tribimaximal (TBM) mixing baseline
\begin{equation}
  \sin^2\theta_{12}^\mathrm{TBM} = \tfrac{1}{3},\quad
  \sin^2\theta_{23}^\mathrm{TBM} = \tfrac{1}{2},\quad
  \theta_{13}^\mathrm{TBM} = 0
\end{equation}
from the $\mathbb{Z}_3$ symmetry of the $\JO$ diagonal via
$\pi_1(L(3,1)) \cong \mathbb{Z}_3$.  Against the PDG 2024 values
($\sin^2\theta_{12} = 0.307$, $\sin^2\theta_{23} = 0.546$,
$\sin^2\theta_{13} = 0.02195$), the TBM deviations are $+8.6\%$, $-8.4\%$,
and $-100\%$ respectively.

The TBM corrections arise from the off-diagonal $\JO$ phases under $G_2$ action.
Two inputs are available from the corpus: the Cabibbo angle
$\thetaC = \pi/14$ (Addendum~38 Conj.~C, $-0.80\%$ from $|V_{us}|$)
and the $G_2$ dimension $\dim G_2 = 14$.

\subsection{Reactor angle (Conjecture F)}

\begin{conjecture}[PMNS reactor angle]
\label{conj:theta13}
\begin{equation}
  \sin^2\theta_{13} = \frac{4}{9}\sin^2\!\frac{\pi}{14}
  = \frac{4}{9}\sin^2\thetaC.
\end{equation}
\end{conjecture}

\noindent
\textit{Numerical check.}
$\tfrac{4}{9}\sin^2(\pi/14) = \tfrac{4}{9} \times 0.049516 = 0.022007$.
PDG: $0.02195$.  Error: $+0.26\%$.

\smallskip\noindent
\textit{Structural interpretation.}
The factor $2/3$ in $\sin\theta_{13} = \tfrac{2}{3}\sin\thetaC$ equals the
hypercharge of the up-type quarks, $Y_u = +2/3$.  In the $\JO$ off-diagonal
blocks, the Cabibbo rotation lives in the $e_7$ direction (quark sector);
the reactor angle $\theta_{13}$ couples the lepton singlet ($e_0$ slot) to the
quark triplet.  The hypercharge suppression factor $Y_u = 2/3$ converts the
quark-sector Cabibbo rotation to the lepton-sector reactor angle, realising
a form of quark--lepton unification through the $\JO$ structure.

Equivalently, $\sin^2\theta_{13}/\sin^2\thetaC = 4/9 = Y_u^2$, so the
reactor angle is the Cabibbo angle attenuated by the square of the up-quark
hypercharge.

\subsection{Atmospheric mixing (Conjecture G)}

\begin{conjecture}[PMNS atmospheric angle]
\label{conj:theta23}
\begin{equation}
  \sin^2\theta_{23} = \frac{15}{28} = \frac{1}{2} + \frac{1}{2\dim G_2}.
\end{equation}
\end{conjecture}

\noindent
\textit{Numerical check.}
$15/28 \approx 0.53571$.  PDG: $0.546$.  Error: $-1.88\%$.

\smallskip\noindent
\textit{Structural interpretation.}
The TBM baseline $\sin^2\theta_{23} = 1/2$ is maximal mixing.
The $G_2$ correction $+1/(2\dim G_2) = +1/28$ is the first-order symmetry
breaking from $G_2 \supset \mathrm{SU}(3)_c$ (Addendum~37, Theorem~4.1).
Geometrically, $G_2$ has 14 generators; the $\mathrm{SU}(3)$ subalgebra uses 8,
leaving $14-8=6$ generators that mix the $\mathbf{3}$ and $\bar{\mathbf{3}}$
directions.  The correction $1/(2 \times 14) = 1/28$ reflects the density
of these cross-sector generators relative to the full $G_2$ algebra.

\begin{remark}
The sign of the correction ($+$, lifting $\theta_{23}$ above maximal) is
consistent with the PDG central value, which exceeds $45^\circ$.  The error of
$-1.88\%$ is larger than Conjecture~F but still within two standard deviations
of the current experimental uncertainty on $\sin^2\theta_{23}$.
\end{remark}

\subsection{Solar mixing (Conjecture H)}

\begin{conjecture}[PMNS solar angle]
\label{conj:theta12}
\begin{equation}
  \sin^2\theta_{12} = \frac{\cos^2(\pi/14)}{3} = \frac{1 - \sin^2\thetaC}{3}.
\end{equation}
\end{conjecture}

\noindent
\textit{Numerical check.}
$\cos^2(\pi/14)/3 = (1 - 0.049516)/3 = 0.31683$.
PDG: $0.307$.  Error: $+3.20\%$.

\smallskip\noindent
\textit{Structural interpretation.}
The TBM solar angle $1/3$ is the uniform distribution of the first-generation
lepton amplitude across three neutrino mass eigenstates ($\mathbb{Z}_3$ symmetry).
The Cabibbo correction reduces this to $\cos^2(\thetaC)/3$: the Cabibbo rotation
in the quark sector ``borrows'' a fraction $\sin^2\thetaC$ of the mixing amplitude,
leaving $(1 - \sin^2\thetaC)/3$ in the lepton sector.  This is a specific
realisation of quark--lepton complementarity
($\theta_{12} + \thetaC \approx \pi/4$) within the $\JO$ framework.

\begin{remark}
The $3.20\%$ error is larger than Conjectures~F and~G.  The $\theta_{12}$
prediction should be treated as a weaker conjecture pending a direct derivation
from the $G_2$ action on $\JO$ off-diagonal phases.  Nevertheless, the formula
$\sin^2\theta_{12} = (1-\sin^2\thetaC)/3$ has no free parameters: it uses
only the Cabibbo angle from Addendum~38 Conjecture~C and the TBM $1/3$ factor.
\end{remark}

\subsection{Summary of Phase 3.5 conjectures}

\begin{table}[ht]
\centering
\caption{PMNS mixing angle conjectures (Phase 3.5).
All formulas use $\thetaC = \pi/14$ from Addendum~38 Conj.~C and
$\dim G_2 = 14$ from Addendum~37 Theorem~4.1.}
\label{tab:pmns}
\begin{tabular}{lllrrrr}
\toprule
Label & Angle & Formula & Predicted $\sin^2$ & PDG & Error & Status \\
\midrule
Conj.\ F & $\theta_{13}$ & $(4/9)\sin^2(\pi/14)$ & $0.02201$ & $0.02195$ & $+0.26\%$ & strong \\
Conj.\ G & $\theta_{23}$ & $15/28$ & $0.53571$ & $0.546$ & $-1.88\%$ & moderate \\
Conj.\ H & $\theta_{12}$ & $\cos^2(\pi/14)/3$ & $0.31683$ & $0.307$ & $+3.20\%$ & weak \\
\midrule
TBM & $\theta_{12}$ & $1/3$ & $0.33333$ & $0.307$ & $+8.6\%$ & baseline \\
TBM & $\theta_{23}$ & $1/2$ & $0.50000$ & $0.546$ & $-8.4\%$ & baseline \\
TBM & $\theta_{13}$ & $0$ & $0$ & $0.02195$ & $-100\%$ & baseline \\
\bottomrule
\end{tabular}
\end{table}

All three conjectures improve significantly over the TBM baseline without
introducing any free parameters.  Conjecture~F achieves sub-percent agreement
and is the highest-confidence result of this addendum.

%==========================================================================
\section{OP1: Higgs VEV candidate}
\label{sec:vev}

Addendum~39 identified the Higgs doublet slot as $V_H = \C e_0 \oplus \C e_7$
(slot~20 in Table~\ref{tab:spectral}) but left the vacuum expectation value $v$
as Open Problem~1 (OP1).  The slot~20 energy is
\begin{equation}
  E(v) = \pi + \frac{\ln(v/m_e)}{\mathrm{MU}} \approx 19.636.
\end{equation}

\begin{conjecture}[Higgs VEV, OP1 candidate]
\label{conj:vev}
\begin{equation}
  v = e \cdot M_Z,
\end{equation}
where $e = \exp(1) \approx 2.7183$ and $M_Z = 91.188$\,\emph{GeV}.
\end{conjecture}

\noindent
\textit{Numerical check.}
$e \times 91.188 = 247.87$\,GeV.  PDG: $v = 246.22$\,GeV.  Error: $-0.67\%$.

\smallskip\noindent
\textit{Structural interpretation.}
In the TOE, $e = \exp(1)$ is the natural scale factor for first moments
(Paper~17: the first-moment coupling is $\mu_1$, computed from $\rho(x)$ via
$\mu_1 = \int_0^1 x\rho(x)\,dx$).  Slot~12 ($x_{13}\cdot e_0$) is the $\nu_\tau/\tau_L$ lepton doublet slot
(corrected per Addendum~48; the earlier label ``$Z$'' was heuristic).
The $e_0$ direction in block $x_{13}$ is the same geometric direction as the
Higgs/VEV slot (slot~20, $x_{23}\cdot e_0$), separated by one off-diagonal block.
A shift by one $\JO$ block in the $e_0$ direction by the Euler scale factor $e$
would yield $E_v = E(\mathrm{slot\,12}) + 1/\mathrm{MU} \approx 19.644$,
which is close to but not exactly $E(v) = 19.636$.

The formula $v = e \cdot M_Z$ is therefore recorded as a \emph{numerical candidate},
not a structural derivation.  A proof would require identifying the mechanism
by which the $e_0$ Higgs slot acquires a VEV scaled by $\exp(1)$ relative to
the $Z$ mass --- the fold-map spectral problem referenced in Addendum~39 OP1.

\begin{remark}
The three candidates investigated for OP1 in the search (Addendum~39,
Open Problem~1) were: (a) $v = e \cdot M_Z$ at $-0.67\%$;
(b) $v$ from a $\JO$ spectral eigenvalue (not yet identified); and
(c) $v = M_Z/\cos\theta_W$ at tree level, which with $\sin^2\theta_W = 1/(1+\pi)$
(Addendum~37 Conjecture~5.3) gives $v_\mathrm{tree} = M_Z/\sqrt{1-1/(1+\pi)} = M_Z\sqrt{(1+\pi)/\pi}
= 91.188 \times 2.032 = 185.3$\,GeV --- well outside the observed value,
ruling this route out at tree level.
\end{remark}

%==========================================================================
\section{Open problems}
\label{sec:open}

The spectral table and Phase~3.5 work leaves the following problems open.

\paragraph{OP1 (VEV).}
Derive $v$ from the $\JO$ fold-map spectral problem rather than from the
numerical coincidence $v \approx e \cdot M_Z$.

\paragraph{OP2 ($u$ quark).}
No clean TOE formula found for the $u$-quark energy $E_u \approx 4.959$.
The formula $E_u = \pi^2/2 = 4.935$ gives $-0.49\%$ but lacks a structural
derivation.  The $u$-quark slot (6--8) is the up-type quark in the $x_{12}$
block; its energy should be expressible in terms of $\pi$, $\muzero$, $\muone$
without free parameters.

\paragraph{OP3 ($s$ quark).}
The $s$-quark energy $E_s \approx 9.707$ has no clean formula.
The strange quark sits in slots 17--19 ($\bar{\mathbf{3}}$ in block $x_{13}$),
the down-type companion of the charm quark.  A formula relating $E_s$ to $E_d$
and $E_b$ via the Fano ratio $\pi^{N_{\mathrm{Im}}/N_c}$ is a natural target.

\paragraph{OP4 ($W$ and $Z$ masses --- adjoint sector).}
\textit{Corrected per Addendum~48:} $W^\pm$ and $Z^0$ are not slots in the
matter $\mathbf{27}$.  Slot~4 ($x_{12}\cdot e_0$) and slot~12 ($x_{13}\cdot e_0$)
are lepton doublet matter slots with $Y=-1$, not gauge boson slots.
$M_W$ and $M_Z$ must be derived from the $E_6$ adjoint $\mathbf{78}$, specifically
the $(1,8,1)$ piece under trinification (Addendum~48, Theorem~5.1 and Section~6).
The earlier observation that $M_W = M_Z\cos\theta_W$ at $-1.2\%$ via
$\sin^2\theta_W = 1/(1+\pi)$ (Addendum~37 Conjecture~5.3) remains numerically
valid as a cross-check, but the \emph{derivation} of the mass scale requires the
adjoint sector, not the matter-27 spectral table.

\paragraph{OP5 (exact PMNS angles).}
Conjectures~G and~H have errors of $-1.88\%$ and $+3.20\%$.  The exact
$\theta_{23}$ and $\theta_{12}$ values require a complete treatment of the
$G_2$ action on $\JO$ off-diagonal phases (Phase~3.5 completion, pending
Phase~4 completion of the full $\JO$ spectrum).

\paragraph{OP6 (muon $0.60\%$ residual).}
As noted in Addendum~38 Proposition~2.1, the muon energy $E_\mu = \pi^2$
(cruciform identification on $C_2$) overshoots by $0.60\%$ because MU uses
bulk-inclusive $\mu_1$, while the muon propagates on the boundary-only $C_2
\subset S^3$.  Deriving $\mathrm{MU}^{(S^3)}$ for propagation restricted to
$S^3$ would close this gap.

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

This addendum completes three intermediate goals of the TOE SM-completion
programme:

\begin{enumerate}[noitemsep]
  \item \textbf{Spectral table} (Section~\ref{sec:spectral}): the full 27-slot
  $\JO$ decomposition is tabulated with SM quantum numbers.  Of 27 slots,
  16 carry derived TOE energy anchors.  Seven slots (u quark, s quark,
  two lepton doublets in slots~4 and~12, two CKM directions) remain open.
  $W^\pm$/$Z^0$ masses are outside the matter $\mathbf{27}$ (Addendum~48).

  \item \textbf{PMNS Phase 3.5} (Section~\ref{sec:pmns}): three conjectures
  derived from the Cabibbo angle $\thetaC = \pi/14$ and $\dim G_2 = 14$:
  Conjecture~F ($\theta_{13}$, $+0.26\%$, strong), Conjecture~G ($\theta_{23}$,
  $-1.88\%$, moderate), and Conjecture~H ($\theta_{12}$, $+3.20\%$, weak).
  All improve on the TBM baseline.

  \item \textbf{VEV candidate} (Section~\ref{sec:vev}): $v = e \cdot M_Z$ at
  $-0.67\%$, recorded as a numerical candidate for OP1 pending structural derivation.
\end{enumerate}

The dominant open problem remains the derivation of the Higgs VEV from the
$\JO$ fold-map spectral problem, and the derivation of $W^\pm$/$Z^0$ masses
from the $E_6$ adjoint $\mathbf{78}$ sector (Addendum~48).
Once the gauge boson masses are derived from the adjoint sector, the full SM
spectrum will be accounted for by the TOE corpus.

%==========================================================================
\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{P28} L.~F.\ Vlegels, \emph{Paper~XXVIII: Lorentz Factor from Dual-Observer
  $S^3$ Phase Dynamics}, TOE Corpus, 2024.
\bibitem{P36} L.~F.\ Vlegels, \emph{Addendum~36: Sector-Energy Mass Formula ---
  Charged Leptons}, TOE Corpus, 2026.
\bibitem{P37} L.~F.\ Vlegels, \emph{Addendum~37: Gauge Group from $(B^4, S^3,
  \OO)$ Geometry}, TOE Corpus, 2026.
\bibitem{P38} L.~F.\ Vlegels, \emph{Addendum~38: Quark Sector and Neutrino
  Structure from $J_3(\OO)$ Geometry}, TOE Corpus, 2026.
\bibitem{P39} L.~F.\ Vlegels, \emph{Addendum~39: Higgs Sector Identification
  from $\C \otimes \OO$ Decomposition}, TOE Corpus, 2026.
\bibitem{P48} L.~F.\ Vlegels, \emph{Addendum~48: $E_6$ Covariance of $J_3(\OO)$:
  Hypercharge Forcing, Spectral Table Map, and the $W/Z$ Gap}, TOE Corpus, 2026.
\end{thebibliography}

\end{document}
