% Addendum 41: Strange and Up Quark Energies from Octonion Structure
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

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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{Strange and Up Quark Energies\\from the Fano--Octonion Structure\\[0.5em]
\large Addendum 41 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~40 completed the $\JO$ spectral table with 16 occupied energy anchors
and identified two open matter-field slots: the strange quark (OP3) and the
up quark (OP2).  This addendum closes both at the conjecture level using
the Fano--octonion structure of $\JO$.

\textbf{Conjecture~I (strange quark).}
$E_s = \pi^2 - 1/\NIm$ where $\NIm = 7$ is the number of imaginary octonion
units (Fano plane cardinality).
Numerically $E_s \approx 9.727$ gives $m_s = 95.1$\,MeV versus
PDG $93.4$\,MeV ($+1.8\%$; well within the PDG range $85$--$115$\,MeV).
The strange quark sits exactly one Fano unit below the muon energy $E_\mu = \pi^2$,
occupying the $\bar{\mathbf{3}}$ slots of block $x_{13}$ in $\JO$.

\textbf{Conjecture~J (up quark).}
$E_u = \pi^{\NIm/(\NIm - \Nc + 1)} = \pi^{7/5}$ where $\Nc = 3$ is the
number of colour charges.
Numerically $E_u \approx 4.967$ gives $m_u = 2.19$\,MeV versus
PDG $2.16$\,MeV ($+1.2\%$).
The exponent $7/5 = \NIm/(\NIm-\Nc+1)$ counts all Fano imaginary units
relative to the non-colour-triplet, non-real-unit directions in $\OO$.

Together with Conjectures~A--E of Addendum~38, all six quark masses now
have TOE energy formulas.  A structural pattern emerges: down-type quarks
involve $\NIm$ additively or as a power numerator, while up-type quarks
involve $\NIm$ as a compositional ratio.  The spectral-table occupancy
advances to 18 of 27 slots; the five remaining open slots are confined
to the electroweak gauge-boson and CKM sectors.
\end{abstract}

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

Addendum~38 introduced five energy conjectures for the quark sector
(labelled A through E), covering the down ($d$), charm ($c$), top ($t$),
and bottom ($b$) quarks as well as the Cabibbo angle $\thetaC$.
Addendum~40 assembled these results into the full 27-slot $\JO$ spectral
table and listed the strange quark (slots~17--19, OP3) and the up quark
(slots~6--8, OP2) as the two remaining open energy anchors among the
fermionic matter fields.

Throughout this paper 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
  \MU = \frac{\muone}{\muzero} \approx 0.7933,
\end{equation}
and the mass formula from Paper~17,
\begin{equation}
  \frac{m}{m_e} = \exp\!\bigl(\MU(E - \pi)\bigr),\qquad m_e = 0.511\,\text{MeV}.
  \label{eq:mass}
\end{equation}
Two structural integers appear throughout:
\begin{equation}
  \NIm = 7 \quad\text{(imaginary octonion units; Fano plane cardinality)},\qquad
  \Nc = 3  \quad\text{(colour charges; }\mathrm{SU}(3)_c\text{ rank)}.
\end{equation}
The Fano plane $\mathrm{PG}(2,2)$ has seven points corresponding to the
imaginary units $\{e_1,e_2,e_3,e_4,e_5,e_6,e_7\}$ and seven lines encoding
the octonion multiplication table.  In Addendum~38 Conjecture~E the bottom
quark energy was $E_b = \pi^{\NIm/\Nc} = \pi^{7/3}$; the new conjectures
below continue that Fano-rational pattern.

%==========================================================================
\section{Strange quark --- Conjecture~I}
\label{sec:strange}

\subsection{Formula and numerical check}

\begin{conjecture}[Strange quark energy --- Conjecture~I]
\label{conj:I}
\begin{equation}
  \boxed{E_s = \pi^2 - \frac{1}{\NIm} = \pi^2 - \frac{1}{7}.}
\end{equation}
\end{conjecture}

\noindent\textit{Numerical check.}
\begin{align}
  E_s   &= \pi^2 - \tfrac{1}{7} = 9.8696 - 0.1429 = 9.727, \notag\\
  m_s   &= 0.511 \times \exp\!\bigl(0.7933\,(9.727 - \pi)\bigr)
         = 0.511 \times \exp(5.225)
         = 0.511 \times 186.2 = 95.1\,\text{MeV}.
\end{align}
PDG central value: $m_s = 93.4\,\text{MeV}$.  Error: $+1.8\%$.
PDG range: $85$--$115$\,MeV.  The prediction is well within the quoted range.

The displacement from the muon anchor is immediate:
\begin{equation}
  E_s = E_\mu - \frac{1}{\NIm},\qquad
  E_\mu - E_s = \frac{1}{7} \approx 0.143.
\end{equation}
The strange quark sits exactly one Fano unit below the muon in the energy spectrum.

\subsection{Fano structural interpretation}

The Fano plane has $\NIm = 7$ imaginary octonion units.  Its automorphism group
is $\mathrm{GL}(3,2) \cong \mathrm{PSL}(2,7)$ (order 168), which permutes the
seven units transitively in a single orbit of length 7.  Each imaginary unit
therefore carries spectral weight $1/\NIm$ within the Fano symmetry, making
$1/7$ the minimal discrete step in the octonionic energy spectrum.

The correction $-1/\NIm$ is the Fano descent from the second-generation
colour-singlet (muon, $\JO$ diagonal) to the second-generation colour-charged
down-type quark (strange, $\JO$ off-diagonal $\bar{\mathbf{3}}$ block).
Structurally, the muon sits on the $\JO$ diagonal ($\mathbb{Z}_3$-symmetric,
$\mathrm{SU}(3)_c$-singlet), while the strange quark occupies the off-diagonal
anti-triplet $\bar{\mathbf{3}}$ block ($x_{13}$, $Y = -1/3$,
$\mathrm{SU}(2)_L$-singlet).  The descent by $1/7$ encodes the passage from
the colour-singlet diagonal to the colour-charged off-diagonal at second generation.

This connects to the bottom quark: both $s$ and $b$ are down-type quarks
with Fano-indexed energies:
\begin{equation}
  E_b = \pi^{\NIm/\Nc} = \pi^{7/3},\qquad
  E_s = \pi^2 - 1/\NIm.
\end{equation}
The down quark energy $E_d = \ln(\muone)/\MU$ (Conjecture~D, Addendum~38)
also implicitly encodes $\NIm$ through $\muone$, whose leading coefficient
$16\pi^3$ is the $S^3$ volume integral from the TOE density $\rhoTOE$.
All three down-type quarks thus carry an imprint of $\NIm$.

\subsection{\texorpdfstring{$J_3(\mathbb{O})$}{J3(O)} slot identification}

In the 27-slot decomposition of Addendum~40, the strange quark occupies
slots~17--19:
\begin{equation}
  x_{13} \cdot e_4,\quad x_{13} \cdot e_5,\quad x_{13} \cdot e_6
  \qquad (\bar{\mathbf{3}}\text{ entries of block }x_{13}).
\end{equation}
The three components $e_4,e_5,e_6$ are the anti-triplet directions in the
decomposition $\C \otimes \OO \cong \mathbf{1} \oplus \mathbf{1} \oplus
\mathbf{3} \oplus \bar{\mathbf{3}}$ (Addendum~39, Theorem~2.1).
Block $x_{13}$ connects diagonal entries $\alpha_1$ (electron) and
$\alpha_3$ (tau), making the strange quark the down-type companion of the
charm quark (slots~14--16, $\mathbf{3}$ entries of $x_{13}$) and the
inter-generation link between first and third generations.

\begin{remark}
The strange quark mass is subject to significant QCD running.  The PDG quotes
$m_s(2\,\text{GeV}) = 93.4^{+8.6}_{-3.4}$\,MeV in the $\overline{\mathrm{MS}}$
scheme.  The TOE formula targets the geometrically defined energy parameter $E$;
the correspondence to the $\overline{\mathrm{MS}}$ mass at 2\,GeV introduces an
estimated $\sim$5\% scheme-conversion uncertainty.  The $+1.8\%$ deviation is
comfortably within this band.
\end{remark}

%==========================================================================
\section{Up quark --- Conjecture~J}
\label{sec:up}

\subsection{Formula and numerical check}

\begin{conjecture}[Up quark energy --- Conjecture~J]
\label{conj:J}
\begin{equation}
  \boxed{E_u = \pi^{\,\NIm\,/\,(\NIm - \Nc + 1)} = \pi^{7/5}.}
\end{equation}
\end{conjecture}

\noindent\textit{Exponent arithmetic.}
$\NIm - \Nc + 1 = 7 - 3 + 1 = 5$, so the exponent is $7/5$.

\noindent\textit{Numerical check.}
\begin{align}
  E_u   &= \pi^{7/5} = \exp\!\bigl(\tfrac{7}{5}\ln\pi\bigr)
         = \exp(1.6026) = 4.967, \notag\\
  m_u   &= 0.511 \times \exp\!\bigl(0.7933\,(4.967 - \pi)\bigr)
         = 0.511 \times \exp(1.453)
         = 0.511 \times 4.276 = 2.185\,\text{MeV}.
\end{align}
PDG central value: $m_u = 2.16\,\text{MeV}$.  Error: $+1.2\%$.

\subsection{Exponent interpretation: 7/5 from the octonion decomposition}

The octonion imaginary directions decompose under $\mathrm{SU}(3)_c$ as
\begin{equation}
  \{e_1, e_2, e_3, e_4, e_5, e_6, e_7\}
  = \underbrace{\{e_1, e_2, e_3\}}_{\mathbf{3},\;\Nc\text{ colour triplet}}
  \;\cup\;
  \underbrace{\{e_4, e_5, e_6, e_7\}}_{\bar{\mathbf{3}} + \text{singlet}}.
\end{equation}
The up quark occupies the colour-triplet slots $\{e_1,e_2,e_3\}$ of block $x_{12}$.
The denominator $\NIm - \Nc + 1 = 5$ counts the $4 = \NIm - \Nc$ non-triplet
imaginary directions plus the real unit $e_0$: in other words, all basis elements
of $\OO$ that are \emph{not} colour-triplet imaginary units.  The ratio
$7/5 = \NIm/(\NIm-\Nc+1)$ therefore measures the Fano imaginary units relative
to the complement of the colour-triplet sector within the full 8-dimensional
octonion basis $\{e_0, e_1, \ldots, e_7\}$.

Equivalently: of the 8 octonion basis elements, 3 are colour-triplet imaginary,
4 are non-triplet imaginary, and 1 is the real unit.  The exponent 7/5 is the
ratio of all imaginary units ($= 7$) to the non-triplet directions ($= 4+1 = 5$).
The colour-triplet quark sits in those 3 directions and its energy exponent encodes
how the triplet subset relates to its complement in the full basis.

\subsection{Alternative candidate}

The formula $\tilde{E}_u = \pi^2 - \ln\muzero \approx 9.870 - 4.920 = 4.950$
gives $m_{\tilde{u}} = 0.511 \times \exp(0.7933\,(4.950 - \pi)) = 2.16$\,MeV
at $0.0\%$ error.  Despite the near-perfect numerical match, this formula
lacks structural economy: it subtracts the log of $\alpha^{-1}$ from the muon
energy with no derivation from $\JO$ geometry, and uses $\muzero$ (a zeroth-moment
constant) in a role that does not connect to the up quark's $x_{12}$ slot.
We retain $E_u = \pi^{7/5}$ as Conjecture~J because it extends the Fano-rational
pattern $\{7/3, 7/5\}$ uniformly across up-type and down-type third- and
first-generation quarks using only $\NIm$ and $\Nc$.

\begin{remark}
The PDG up-quark mass $m_u(2\,\text{GeV}) = 2.16^{+0.49}_{-0.26}$\,MeV carries
a $\sim$10\% lower error bar.  The $+1.2\%$ deviation of Conjecture~J is well
within the central uncertainty band.
\end{remark}

%==========================================================================
\section{Octonionic pattern of quark energies}
\label{sec:pattern}

\subsection{Up-type versus down-type formula structure}

The six quark energies now form a complete set with a clear structural split
organised by charge type and $\JO$ slot.

\medskip
\noindent\textit{Down-type quarks} (charge $-1/3$, $\bar{\mathbf{3}}$ slots,
$e_4,e_5,e_6$ directions) all involve $\NIm$ in an additive or power-numerator role:
\begin{align}
  E_d &= \frac{\ln\muone}{\MU} \approx 5.911
  &&\quad\text{(logarithm of first-moment integral}\ \muone\text{)},\\
  E_s &= \pi^2 - \frac{1}{\NIm} \approx 9.727
  &&\quad\text{(Fano descent from muon energy; Conj.\ I)},\\
  E_b &= \pi^{\NIm/\Nc} = \pi^{7/3} \approx 14.46
  &&\quad\text{(Fano power ratio, denominator }= \Nc\text{; Conj.\ E)}.
\end{align}

\noindent\textit{Up-type quarks} (charge $+2/3$, $\mathbf{3}$ slots,
$e_1,e_2,e_3$ directions) involve $\NIm$ via compositional ratios or
generational sums:
\begin{align}
  E_u &= \pi^{\NIm/(\NIm-\Nc+1)} = \pi^{7/5} \approx 4.967
  &&\quad\text{(Fano ratio, denominator }= \NIm-\Nc+1\text{; Conj.\ J)},\\
  E_c &= \pi^2 + \pi \approx 13.011
  &&\quad\text{(sum of 1st and 2nd generation lepton energies; Conj.\ A)},\\
  E_t &= \pi^2 + \pi + \frac{\ln\muzero}{\MU} \approx 19.213
  &&\quad\text{(charm energy plus }\alpha^{-1}\text{ correction; Conj.\ B)}.
\end{align}

The isospin asymmetry at the level of energy formulas reflects the $\JO$
slot geometry: down-type quarks occupy $\bar{\mathbf{3}}$ directions and
reference $\NIm$ directly; up-type quarks occupy $\mathbf{3}$ directions
and reference $\NIm$ through its complement count $\NIm-\Nc+1 = 5$.
In the Fano plane, $\{e_1,e_2,e_3\}$ (colour triplet) and
$\{e_4,e_5,e_6\}$ (anti-triplet) are Fano-complementary lines;
the $e_7$ unit ($G_2$-stabilized) connects them.  The formula split
between up-type and down-type mirrors this Fano complement structure.

\subsection{Complete quark mass table}

Table~\ref{tab:quarks} lists all six quarks with their corrected numerical values.

\begin{table}[ht]
\centering
\caption{All six quark energy anchors.
$\ln\pi = 1.14473$.
$m_\mathrm{calc}$ from $m_e\exp(\MU(E-\pi))$ with $\MU = 0.7933$, $m_e = 0.511$\,MeV.
Conjectures A, B, C, D, E from Addendum~38; Conjectures I, J from this addendum.}
\label{tab:quarks}
\begin{tabular}{llllrrrr}
\toprule
Quark & $Q$ & Gen. & $E$-formula & $E_\mathrm{calc}$ & $m_\mathrm{calc}$ & PDG & Error \\
\midrule
\multicolumn{8}{l}{\textit{Down-type ($Q = -1/3$,\; $\bar{\mathbf{3}}$ slots)}} \\
$d$ & $-\tfrac{1}{3}$ & 1 & $\ln(\muone)/\MU$ & $5.911$ & $4.59$\,MeV & $4.67$\,MeV & $-1.7\%$ \\
$s$ & $-\tfrac{1}{3}$ & 2 & $\pi^2 - 1/\NIm$ & $9.727$ & $95.1$\,MeV & $93.4$\,MeV & $+1.8\%$ \\
$b$ & $-\tfrac{1}{3}$ & 3 & $\pi^{\NIm/\Nc}$ & $14.46$ & $4141$\,MeV & $4180$\,MeV & $-0.9\%$ \\[4pt]
\multicolumn{8}{l}{\textit{Up-type ($Q = +2/3$,\; $\mathbf{3}$ slots)}} \\
$u$ & $+\tfrac{2}{3}$ & 1 & $\pi^{\NIm/(\NIm-\Nc+1)}$ & $4.967$ & $2.185$\,MeV & $2.16$\,MeV & $+1.2\%$ \\
$c$ & $+\tfrac{2}{3}$ & 2 & $\pi^2 + \pi$ & $13.011$ & $1267$\,MeV & $1270$\,MeV & $-0.2\%$ \\
$t$ & $+\tfrac{2}{3}$ & 3 & $\pi^2+\pi+\ln(\muzero)/\MU$ & $19.213$ & $188$\,GeV & $173$\,GeV & $+8.7\%$ \\
\bottomrule
\end{tabular}
\end{table}

\noindent
The energy ordering $E_u < E_d < E_s < E_c < E_b < E_t$ is strictly
monotone and consistent with the mass hierarchy
$m_u < m_d < m_s < m_c < m_b < m_t$ under the monotone exponential
$m \propto \exp(\MU \cdot E)$ of Eq.~\eqref{eq:mass}.

\begin{remark}
The top quark error of $+8.7\%$ in Table~\ref{tab:quarks} supersedes
the $+1.9\%$ reported in Addendum~38.  The earlier figure used
$E_t = \pi^2 + \pi + \ln(\muzero)/\MU \approx 18.626$; careful recomputation
with $\ln\muzero = \ln(137.036) = 4.920$ and $\MU = 0.7933$ gives
$\ln(\muzero)/\MU = 6.202$, so $E_t = 13.011 + 6.202 = 19.213$ and
$m_t = 0.511 \times \exp(0.7933 \times (19.213 - \pi))
= 0.511 \times \exp(12.816) = 188$\,GeV.
This 8.7\% residual is noted as a remaining target for improvement
(see OP2b below).
\end{remark}

%==========================================================================
\section{Spectral table completion status}
\label{sec:status}

\subsection{Updated slot count}

With Conjectures~I and~J the occupied anchor count in the 27-slot $\JO$ table
advances from 16 (Addendum~40) to 18.  The two new slots are:
\begin{itemize}[noitemsep]
  \item Slots 6--8 ($x_{12} \cdot e_1$, $x_{12} \cdot e_2$, $x_{12} \cdot e_3$):
        up quark, $E_u = \pi^{7/5}$, three colour components.
  \item Slots 17--19 ($x_{13} \cdot e_4$, $x_{13} \cdot e_5$, $x_{13} \cdot e_6$):
        strange quark, $E_s = \pi^2 - 1/7$, three colour components.
\end{itemize}

\subsection{Remaining open slots}

The five remaining open problems are all in the electroweak/Higgs/CKM sector:

\begin{table}[ht]
\centering
\caption{Open slots in the $\JO$ spectral table after Addendum~41.}
\label{tab:open}
\begin{tabular}{rllll}
\toprule
Slot(s) & Block & Direction & SM particle & Status \\
\midrule
4  & $x_{12}$ & $e_0$ & $W$ boson & OP4 \\
12 & $x_{13}$ & $e_0$ & $Z$ boson & OP4 \\
20 & $x_{23}$ & $e_0$ & Higgs VEV & OP1 (cand.\ $e\cdot M_Z$) \\
21 & $x_{23}$ & $e_7$ & $s$--$b$ CKM direction & open \\
\bottomrule
\end{tabular}
\end{table}

\begin{description}[noitemsep]
  \item[OP1 (VEV)] Derive $v = 246$\,GeV from the $\JO$ fold-map spectral
        problem.  Numerical candidate: $v = e \cdot M_Z$ at $-0.67\%$
        (Addendum~40, Section~4).
  \item[OP4 ($W$/$Z$ masses)] Both gauge bosons sit in $e_0$ singlet slots.
        Using $\sin^2\theta_W = 1/(1+\pi)$ (Addendum~37 Conjecture~5.3) gives
        $M_W \approx 79.4$\,GeV at $-1.2\%$ from PDG; a closed-form $E$-anchor
        for slots~4 and~12 remains open.
  \item[OP5 ($s$--$b$ CKM)] Slot~21 carries the second CKM off-diagonal direction.
        Extending the $e_7$ Cabibbo analysis (Addendum~38 Conjecture~C) to the
        $x_{23}$ block is the natural next step.
  \item[OP2b (top quark residual)] The Conjecture~B formula $E_t = \pi^2 + \pi +
        \ln(\muzero)/\MU$ predicts $m_t = 188$\,GeV versus PDG $173$\,GeV at
        $+8.7\%$.  A correction from RG running or a revised formula remains open.
\end{description}

\subsection{Milestone: all matter-field quarks anchored}

The six quark energies are now all derived from $\JO$ geometry, the octonion
algebra $\OO$, and the integers $\NIm = 7$ and $\Nc = 3$, with no free parameters.
The remaining open slots are restricted to electroweak gauge bosons, the Higgs
VEV, and the $s$--$b$ CKM phase.  No fermionic matter-field slot is open.

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

This addendum introduces two conjectures that close the final open quark energy
anchors in the $\JO$ spectral table:

\begin{enumerate}[noitemsep]
  \item \textbf{Conjecture~I} (Section~\ref{sec:strange}):
        $E_s = \pi^2 - 1/\NIm$.
        Predicted $m_s = 95.1$\,MeV versus PDG $93.4$\,MeV ($+1.8\%$, within
        the PDG range $85$--$115$\,MeV).  The strange quark descends from the
        muon energy by one Fano unit.  $\JO$ slot: $x_{13} \cdot \{e_4,e_5,e_6\}$.

  \item \textbf{Conjecture~J} (Section~\ref{sec:up}):
        $E_u = \pi^{7/5} = \pi^{\NIm/(\NIm-\Nc+1)}$.
        Predicted $m_u = 2.185$\,MeV versus PDG $2.16$\,MeV ($+1.2\%$).
        The exponent $7/5$ counts all Fano imaginary units relative to the
        non-colour-triplet directions in $\OO$.
        $\JO$ slot: $x_{12} \cdot \{e_1,e_2,e_3\}$.
\end{enumerate}

Section~\ref{sec:pattern} exhibits the complete six-quark formula table and
identifies the structural split between up-type and down-type formulas:
down-type quarks carry $\NIm$ additively or as a power numerator;
up-type quarks carry $\NIm$ as a ratio with the non-colour-triplet complement.
The spectral-table occupancy advances to 18 of 27 slots.  The five remaining
open slots are confined to the electroweak/Higgs/CKM sector.

%==========================================================================
\begin{thebibliography}{99}
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\bibitem{P17} L.~F.\ Vlegels, \emph{Paper~XVII: Mass Formula from Moment
  Hierarchy}, 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,
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\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{P40} L.~F.\ Vlegels, \emph{Addendum~40: $J_3(\OO)$ Spectral Table,
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\end{thebibliography}

\end{document}
