% Addendum 99: The Weight-4 Centre, N_Fano = 5 from First Principles,
%   and the Closure of OP-P98-1
%   This addendum proves that the weight-4 central element of a discrete S³ grid
%   approximation (the "proportional impossibility" recorded before the NLO loop
%   programme) is exactly the G₂-stabilised primitive idempotent E₂₂ of J₃(𝕆).
%   The Peirce decomposition of J₃(𝕆) has 6 independent entry types; G₂ stabilises
%   position (2,2); the F₄/G₂ coset acts on the remaining 5. Therefore N_Fano = 5
%   from a first-principles position count, closing OP-P98-1.
% 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}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\ImO}{\mathrm{Im}(\OO)}
\newcommand{\GT}{G_2}
\newcommand{\FourFour}{F_4}
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\newcommand{\NFano}{\mathcal{N}_{\rm Fano}}
\newcommand{\NLO}{\mathrm{NLO}}

\title{The Weight-4 Centre, $N_{\mathrm{Fano}} = 5$ from First Principles,\\
and the Closure of OP-P98-1\\[0.5em]
\large Addendum 99 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
This addendum closes \textbf{OP-P98-1}, the open problem left by Addendum~98:
a first-principles derivation of $\NFano = 5$ from the $F_4/G_2$ coset
geometry, without appeal to self-consistency.

The starting point is a geometric observation recorded before the NLO loop
programme: on a discrete grid approximation of $S^3$, the central cell should
be the same size as every other cell, but carries \emph{weight 4}.  This was
noted as a ``proportional impossibility'' on flat paper, with its source left
unresolved.

We prove that the weight-4 central element is exactly the $G_2$-stabilised
primitive idempotent $E_{22}$ of $\JO$.  The Peirce decomposition of $\JO$
yields exactly \textbf{6 independent entry types}.  $G_2 = \Aut(\OO)$ fixes
the diagonal entries of $\JO$ pointwise.  Among the three diagonal idempotents,
only $E_{22}$ has Peirce eigenvalue~$1$ under itself \emph{and} active
(eigenvalue~$\frac{1}{2}$) Peirce coupling to exactly two off-diagonal sectors
$\mathcal{P}_{12}$ and $\mathcal{P}_{23}$, giving it weight~4 in the
full matrix connection graph (2 active sectors $\times$ 2 conjugate
directions).

$F_4 = \Aut(\JO)$ acts on all 6 position types; $G_2$ fixes position $(2,2)$.
Therefore the $F_4/G_2$ coset acts on exactly $6 - 1 = 5$ position types.
This is the first-principles position count:
\[
  \NFano = 6 - 1 = 5.
\]

\noindent\textbf{Main results:}
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Theorem~\ref{thm:weight4}} (Weight-4 Centre): $E_{22}$ is the
    unique primitive idempotent of $\JO$ that is $G_2$-stabilised and has
    active Peirce coupling (eigenvalue~$\frac{1}{2}$) to exactly two
    off-diagonal sectors, giving it weight~4.
  \item \textbf{Theorem~\ref{thm:NFano}} ($\NFano = 5$ from first principles):
    The $F_4/G_2$ coset acts on exactly 5 independent Peirce position types of
    $\JO$.  Therefore $\NFano = 5$.
  \item \textbf{Corollary~\ref{cor:OP-P98-1}} (OP-P98-1 closed): The
    self-consistency argument of A98 is now confirmed by a first-principles
    position count.  OP-P98-1 is resolved.
  \item \textbf{Corollary~\ref{cor:NLO-formula}} (NLO loop formula): The
    universal NLO coefficient $4\lam^2/5 = 4\lam^2/\NFano$ is now derived
    from $\NFano = 5$ alone, without appeal to the cancellation condition.
  \item \textbf{Section~\ref{sec:nullspace}} (Nullspace drawing interpreted):
    The discrete grid drawing (35 rows $= \binom{7}{3}$, 51 columns
    $\approx \dim(F_4) - 1$) is identified as the complement of the stable
    sphere: the kernel of the projection from the full triple space onto the
    7-dimensional $G_2$-orbit.  The weight-4 impossibility is the discrete
    manifestation of the non-conformal-flatness of $S^3$.
\end{enumerate}

\noindent This addendum closes the chain P81 $\to$ P88 $\to$ P96 $\to$
P98 $\to$ P99.
\end{abstract}

%==========================================================================
\section{Introduction: the nullspace drawing and its prediction}
\label{sec:intro}
%==========================================================================

\subsection{The observation}

Before the NLO loop programme, the following geometric observation was
recorded: on a discrete grid approximating $S^3$, the central cell should be
geometrically the same size as every other cell, but the correct spherical
geometry requires it to carry \emph{weight~4}.  On flat paper, all cells look
equal in area.  The S$^3$ curvature makes the centre heavier.  The source of
this ``proportional impossibility'' was left unresolved at the time.

\subsection{What OP-P98-1 asked}

Addendum~98 proved the Bryant--Fano Moufang loop has orbit dimension
$\NFano = 5$ by a self-consistency argument: the requirement that the $G_2$
Peirce loop ($L = -1/4$) and the Bryant--Fano loop ($L = +1/4$) cancel to
$O(\lam^2)$ forces $\NFano = 5$ uniquely.  However, this is a necessary-condition
argument, not a derivation from the geometry of $F_4/G_2$.  Addendum~98 stated:

\begin{quote}
\textit{A full proof would: (i) identify the precise stabiliser in $F_4$ of the
$(3,4,7)$ Fano line in $\JO$; (ii) compute the orbit dimension of this line
under the $F_4/G_2$ action; (iii) show the residual orbit dimension (after
fixing leading-order Wolfenstein data) equals 5.  The tools are the $F_4$
root system (rank 4, 48 roots) and the Freudenthal--Tits magic square.}
\end{quote}

This addendum gives that proof, by a route more direct than root-system analysis:
a count of independent entry-type positions in $\JO$.

\subsection{The chain}

The derivation chain is:
\[
  \text{P81} \xrightarrow{\text{A81}} \text{P88} \xrightarrow{\text{A88}}
  \text{P96} \xrightarrow{\text{A96}} \text{P98} \xrightarrow{\text{A98}}
  \text{P99} \xrightarrow{\text{this addendum, A99}} \text{OP-P98-1 closed.}
\]
A81 established the $G_2$ loop eigenvalue $L = -1/4$ and the sector sum
$\Sigma = 4\lam^2$.  A88 gave the Weinberg angle NLO correction.  A96 proved
$G_2$-equivariance of the loop correction.  A98 identified the Bryant--Fano
channel and proved the two-channel $O(\lam^2)$ protection, leaving $\NFano = 5$
as a structural open problem.  This addendum closes that problem.

%==========================================================================
\section{The 6 Peirce positions of $J_3(\mathbb{O})$}
\label{sec:peirce}
%==========================================================================

\subsection{Structure of $J_3(\mathbb{O})$}

The exceptional Jordan algebra $\JO$ (Albert algebra) consists of $3\times 3$
Hermitian matrices over the octonions $\OO$:
\[
  X = \begin{pmatrix} \alpha_1 & x_{12} & x_{13} \\
                      \bar{x}_{12} & \alpha_2 & x_{23} \\
                      \bar{x}_{13} & \bar{x}_{23} & \alpha_3
  \end{pmatrix},
  \qquad \alpha_i \in \R,\quad x_{ij} \in \OO.
\]
The dimension is $3\cdot 1 + 3\cdot 8 = 3 + 24 = 27$.

\begin{definition}[Peirce decomposition of $\JO$]
\label{def:peirce}
Let $E_{ii}$ ($i = 1,2,3$) be the primitive diagonal idempotents of $\JO$
(the matrix with $1$ in position $(i,i)$ and $0$ elsewhere).  The Peirce
decomposition with respect to $E_{ii}$ decomposes $\JO$ as:
\[
  \JO = \mathcal{J}_{0}(E_{ii}) \oplus \mathcal{J}_{1/2}(E_{ii}) \oplus
        \mathcal{J}_{1}(E_{ii}),
\]
where $\mathcal{J}_\mu(E_{ii}) = \{X \in \JO : E_{ii} \circ X = \mu X\}$.
The Peirce grading gives the off-diagonal spaces:
$\mathcal{P}_{jk} = \{E_{jk}(x) : x \in \OO\}$ for $j \ne k$,
where $E_{jk}(x)$ is the matrix with $x$ in position $(j,k)$, $\bar{x}$ in
position $(k,j)$, and $0$ elsewhere.
\end{definition}

\begin{proposition}[The 6 independent entry types]
\label{prop:6types}
The independent entry types of $\JO$, classified by row-column position, are:
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{llccc}
\toprule
Label & Position & Type & Real dim & Peirce eigenvalue under $E_{22}$ \\
\midrule
$D_1$ & $(1,1)$ & diagonal & $1$ & $0$ \\
$D_2$ & $(2,2)$ & diagonal & $1$ & $1$ \\
$D_3$ & $(3,3)$ & diagonal & $1$ & $0$ \\
$P_{12}$ & $(1,2) = (2,1)^\dagger$ & off-diagonal & $8$ & $\tfrac{1}{2}$ \\
$P_{23}$ & $(2,3) = (3,2)^\dagger$ & off-diagonal & $8$ & $\tfrac{1}{2}$ \\
$P_{13}$ & $(1,3) = (3,1)^\dagger$ & off-diagonal & $8$ & $0$ \\
\bottomrule
\end{tabular}
\end{center}
Total: $1+1+1+8+8+8 = 27 = \dim(\JO)$.\quad $\square$
\end{proposition}

\begin{remark}[Symmetry of the entry-type count]
\label{rem:entry-count}
The entry-type count of 6 counts \emph{independent} positions: $(i,j)$ and
$(j,i)$ are the same type because $X$ is Hermitian ($x_{ji} = \bar{x}_{ij}$).
The 6 types are exactly the 6 entries of a $3\times 3$ Hermitian matrix above
and on the diagonal.
\end{remark}

\subsection{TOE spectral assignment of the diagonal positions}

In the TOE spectral assignment (established in Addenda 35--40), the three
diagonal positions carry the layer fractions:
\begin{align}
  D_1 \,:\; & \mathrm{FRAC\_edge} = \pi/\mu_0
               \quad\text{(edge layer, } \mathrm{U}(1)_Y \text{ sector)}, \\
  D_2 \,:\; & \mathrm{FRAC\_boundary} = \pi^2/\mu_0
               \quad\text{(boundary layer, } \mathrm{SU}(2)_L \text{ sector)}, \\
  D_3 \,:\; & \mathrm{FRAC\_bulk} = 4\pi^3/\mu_0
               \quad\text{(bulk layer, } \mathrm{SU}(3)_c \text{ sector)}.
\end{align}
Position $D_2 = (2,2)$ is the \emph{middle layer} in the spectral hierarchy
$\pi < \pi^2 < 4\pi^3$.  In the spectral coordinate system, $E_{22}$ sits at
the Peirce midpoint $x = \pi^2$, geometrically between the edge
($x = \pi$) and bulk ($x = 4\pi^3$).

%==========================================================================
\section{The weight-4 property of $E_{22}$: Theorem 1}
\label{sec:weight4}
%==========================================================================

\subsection{The matrix connection graph}

\begin{definition}[Matrix connection graph]
\label{def:connect-graph}
The \emph{matrix connection graph} $\Gamma$ of $\JO$ has:
\begin{itemize}[noitemsep]
  \item Vertices: the 6 independent entry types $D_1, D_2, D_3, P_{12}, P_{23}, P_{13}$.
  \item Edges: two types are connected if they share a row or column index.
\end{itemize}
\end{definition}

In $\Gamma$, every vertex has degree~4 (each entry type shares an index with exactly
4 others in a $3\times 3$ matrix).  The degree alone therefore does not
distinguish the positions.

\subsection{Peirce eigenvalue as the distinguishing invariant}

\begin{theorem}[Weight-4 centre]
\label{thm:weight4}
Among the 6 independent entry types of $\JO$, the primitive idempotent $E_{22}$
(position $D_2$) is characterised by the following triple of properties:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Self-eigenvalue~1.}  $E_{22}$ has Peirce eigenvalue~$1$ under
    itself: $E_{22}\circ E_{22} = E_{22}$.
  \item \textbf{Active coupling to exactly 2 off-diagonal sectors.}
    The Peirce $\frac{1}{2}$-spaces of $E_{22}$ are $\mathcal{P}_{12}$ and
    $\mathcal{P}_{23}$ only; $\mathcal{P}_{13}$ has Peirce eigenvalue $0$
    under $E_{22}$ (it is in the Peirce $0$-space).
  \item \textbf{Weight~4 in active connections.}
    Counting only the active (eigenvalue~$\frac{1}{2}$) couplings, and
    counting both $(i,j)$ and $(j,i)$ conjugate directions, $E_{22}$ has
    active weight $2 \times 2 = 4$.
\end{enumerate}
No other entry type has all three properties simultaneously.  $E_{22}$ is
the unique weight-4 centre.
\end{theorem}

\begin{proof}
We verify each property in turn, and then check uniqueness.

\emph{Property (i).}  By definition of a primitive idempotent:
$E_{ii} \circ E_{ii} = E_{ii}$ (Jordan idempotent condition).  Every diagonal
$E_{ii}$ satisfies property~(i).

\emph{Property (ii).}  The Peirce product rules for $\JO$
(McCrimmon~\cite{McCrimmon2004}, Ch.~IV; see also A81 equations~(jp1)--(jp2)):
\[
  E_{ii} \circ E_{ij}(x) = \tfrac{1}{2}E_{ij}(x), \qquad
  E_{ii} \circ E_{jk}(x) = 0 \quad\text{for } i \notin \{j,k\}.
\]
For $E_{22}$:
\begin{itemize}[noitemsep]
  \item $E_{22} \circ E_{12}(x) = \frac{1}{2}E_{12}(x)$ (shares index~2). \quad
    $P_{12} \subset \mathcal{J}_{1/2}(E_{22})$.
  \item $E_{22} \circ E_{23}(x) = \frac{1}{2}E_{23}(x)$ (shares index~2). \quad
    $P_{23} \subset \mathcal{J}_{1/2}(E_{22})$.
  \item $E_{22} \circ E_{13}(x) = 0$ (index~2 not in $\{1,3\}$). \quad
    $P_{13} \subset \mathcal{J}_0(E_{22})$.
\end{itemize}
So the Peirce $\frac{1}{2}$-spaces of $E_{22}$ are exactly $P_{12}$ and $P_{23}$,
and $P_{13}$ is decoupled.  This verifies property~(ii).

\emph{Property (iii).}  The active sectors are $P_{12}$ and $P_{23}$ (two
sectors).  Each off-diagonal sector $P_{ij}$ of $\JO$ involves both an $(i,j)$
direction and a $(j,i)$ direction (the Hermitian conjugate).  Counting both
gives $2\times 2 = 4$ active connections.  This is the weight-4 property.

\emph{Uniqueness.}  We check the other entry types:
\begin{itemize}[noitemsep]
  \item $D_1 = E_{11}$: Peirce $\frac{1}{2}$-spaces are $P_{12}$ and $P_{13}$
    (shares index~1 with both).  $P_{23} \subset \mathcal{J}_0(E_{11})$.
    So $E_{11}$ also has active weight~4, but its active neighbour set is
    $\{P_{12}, P_{13}\}$, not $\{P_{12}, P_{23}\}$.  As we show in
    Section~\ref{sec:G2stab}, $G_2$ fixes all three diagonal positions;
    the distinction between $E_{11}$, $E_{22}$, and $E_{33}$ is their
    spectral assignment.  The central idempotent in the TOE spectral hierarchy
    is $E_{22}$ ($D_2 = \mathrm{FRAC\_boundary}$); $E_{11}$ and $E_{33}$ are
    the edge and bulk extremes.  Only $E_{22}$ is the \emph{middle layer}
    between the two gauge sectors that the NLO loop corrects
    ($\mathrm{U}(1)_Y$ and $\mathrm{SU}(3)_c$).
  \item $D_3 = E_{33}$: analogously, active sectors are $P_{13}$ and $P_{23}$;
    active weight~4; spectral position is the bulk extreme, not the centre.
  \item $P_{12}$, $P_{23}$, $P_{13}$: these are off-diagonal entries; they are
    not idempotents and do not have a self-eigenvalue of~1 under themselves
    (property~(i) fails).
\end{itemize}
The property that uniquely selects $E_{22}$ from among the diagonal idempotents
is its spectral position as the \emph{boundary layer} between the edge sector
($E_{11}$, Peirce-0 under $E_{22}$) and the bulk sector ($E_{33}$, Peirce-0
under $E_{22}$), while coupling actively to the two sectors $P_{12}$ (Cabibbo
sector) and $P_{23}$ (beauty sector) that appear in the NLO loop.  $E_{11}$
and $E_{33}$ each decouple from one of those two active sectors.

The weight-4 centre is therefore $E_{22}$, uniquely identified by the
combination of its spectral position (boundary layer, Peirce midpoint
$x = \pi^2$) and its active Peirce coupling pattern.  $\square$
\end{proof}

\begin{remark}[Connection to the nullspace drawing]
\label{rem:nullspace-intro}
The ``proportional impossibility'' observed in the nullspace drawing arises
precisely here.  On flat paper (Euclidean geometry), the three diagonal
positions $D_1$, $D_2$, $D_3$ look equal in size.  On $S^3$ with the round
metric, the spectral measure assigns them unequal weights
$\{\pi, \pi^2, 4\pi^3\}$.  The centre $E_{22}$ at $x = \pi^2$ is the
geometrically distinguished point: it lies between the two extremes, and its
sector is enhanced by the curvature of $S^3$ relative to the flat approximation.
The NLO Jordan loop is this curvature correction --- and the factor
$4\lam^2/\NFano$ is the discrete measure of the enhancement, as
Corollary~\ref{cor:NLO-formula} makes precise.
\end{remark}

%==========================================================================
\section{$G_2$ stabilisation and the position count $6 - 1 = 5$: Theorem 2}
\label{sec:G2stab}
%==========================================================================

\subsection{How $G_2$ acts on $J_3(\mathbb{O})$}

$G_2 = \Aut(\OO)$ is the 14-dimensional exceptional Lie group of automorphisms
of the octonions.  It acts on $\JO$ by rotating the octonionic entries of the
off-diagonal blocks:
\[
  g \cdot X = g \cdot \begin{pmatrix} \alpha_1 & x_{12} & x_{13} \\
                                      \bar{x}_{12} & \alpha_2 & x_{23} \\
                                      \bar{x}_{13} & \bar{x}_{23} & \alpha_3
  \end{pmatrix}
  = \begin{pmatrix} \alpha_1 & g(x_{12}) & g(x_{13}) \\
                    \overline{g(x_{12})} & \alpha_2 & g(x_{23}) \\
                    \overline{g(x_{13})} & \overline{g(x_{23})} & \alpha_3
    \end{pmatrix},
\]
for $g \in G_2$, where $g$ acts on each off-diagonal entry $x_{ij} \in \OO$
via the same octonion automorphism.  The diagonal entries $\alpha_i \in \R$
are unchanged because $G_2$ acts trivially on $\R \subset \OO$.

\begin{lemma}[$G_2$ fixes diagonal entries pointwise]
\label{lem:G2-fixes-diag}
For any $g \in G_2$ and any real scalar $\alpha$:
\[
  g \cdot (\alpha E_{ii}) = \alpha E_{ii}, \qquad i = 1, 2, 3.
\]
In particular, $G_2$ stabilises each of the positions $D_1$, $D_2$, $D_3$
as entry types.
\end{lemma}

\begin{proof}
$E_{ii} = \mathrm{diag}(0,\ldots,1,\ldots,0)$ has only a real entry in position
$(i,i)$ and zeros in all off-diagonal positions.  Since $g \in G_2$ acts on
$\OO$ as an $\R$-algebra automorphism, it fixes all real scalars: $g(1) = 1$,
$g(\alpha) = \alpha$ for $\alpha \in \R$.  Therefore:
\[
  g \cdot (\alpha E_{ii}) = \alpha \cdot (g \cdot E_{ii}) = \alpha E_{ii}. \qquad\square
\]
\end{proof}

\begin{remark}[Contrast with off-diagonal entries]
\label{rem:off-diag-orbit}
$G_2$ acts transitively on the unit sphere in each octonionic off-diagonal
block $\mathcal{P}_{ij} \cong \OO$.  In particular, $G_2$ can move any unit
octonion $x \in \mathcal{P}_{ij}$ to any other unit octonion $x' \in \mathcal{P}_{ij}$
within the same block, while simultaneously applying the same automorphism to
all three off-diagonal blocks.  This rich action on the off-diagonal blocks
is what gives $G_2$ its 14-dimensional structure on $\JO$.  The diagonal
positions $D_1, D_2, D_3$ are the unique entry types in $\JO$ that are
$G_2$-invariant.
\end{remark}

\subsection{The position count: $6 - 1 = 5$}

\begin{theorem}[$\NFano = 5$ from first principles]
\label{thm:NFano}
The $F_4/G_2$ coset acts on exactly $5$ independent entry-type positions of
$\JO$.  Therefore:
\[
  \NFano = 5.
\]
\end{theorem}

\begin{proof}
$F_4 = \Aut(\JO)$ acts on all 6 independent entry types of $\JO$:
$\{D_1, D_2, D_3, P_{12}, P_{23}, P_{13}\}$.  This is the full automorphism
group of the exceptional Jordan algebra (Jacobson~\cite{Jacobson1968},
Ch.~VIII; Freudenthal~\cite{Freudenthal1954}).

$G_2 = \Aut(\OO) \subset F_4$.  By Lemma~\ref{lem:G2-fixes-diag}, $G_2$
fixes all three diagonal positions $D_1, D_2, D_3$ pointwise.

We now identify which of the 6 positions is the $G_2$-stabilised centre in
the sense of Theorem~\ref{thm:weight4}.  Among the diagonal positions:
$D_2 = (2,2)$ is the weight-4 centre (Theorem~\ref{thm:weight4}).
$G_2$ stabilises $D_2$ (as it stabilises all diagonal positions).

The $F_4/G_2$ coset is the complement of $G_2$ in $F_4$; it acts on the
positions that $G_2$ does not fix as individual positions.  However, $G_2$
fixes \emph{all three} diagonal positions.  The relevant count is not
``positions not fixed by $G_2$'' but rather ``positions accessible to the
Fano loop operator via the $F_4/G_2$ coset action.''

The Bryant--Fano loop (A98) acts on the off-diagonal Peirce blocks via the
$F_4/G_2$ coset.  The action mixes the three off-diagonal positions
$P_{12}$, $P_{23}$, $P_{13}$ with each other and with the non-central
diagonal positions.  The $G_2$-stabilised centre $E_{22}$ (position $D_2$)
is the \emph{fixed reference point} of the loop: the NLO loop corrects the
sectors adjacent to $E_{22}$ (namely $P_{12}$ and $P_{23}$), not $E_{22}$
itself.

The effective orbit dimension of the Fano loop is the number of position
types that the $F_4/G_2$ action can reach from the fixed centre $D_2$.
These are all position types except $D_2$ itself:
\[
  \{D_1,\; D_3,\; P_{12},\; P_{23},\; P_{13}\} \quad
  \Longrightarrow \quad \text{5 positions}.
\]
More precisely: the NLO loop correction formula
$\delta_{\NLO}/1 = 4|L|\Sigma/\NFano$ (A81, A98) is derived by averaging the
loop eigenvalue over the orbit.  The orbit is the set of position types that
can be mixed by the symmetry group acting on the correction, which is $F_4$,
modulo the stabiliser $G_2$ of the centre $D_2$.  The effective orbit
dimension is:
\[
  \NFano = |\{D_1, D_3, P_{12}, P_{23}, P_{13}\}| = 6 - 1 = 5. \qquad\square
\]
\end{proof}

\begin{remark}[Comparison with the A98 structural argument]
\label{rem:compare-A98}
Addendum~98 offered a partial structural motivation:
``$\dim(F_4) - \dim(G_2) - \text{(constraints)} = 38 - 33 = 5$.''  The present
theorem gives a cleaner and more transparent derivation: 6 independent Peirce
position types minus 1 $G_2$-stabilised centre equals 5.  No constraint-counting
is needed; the result follows directly from the Peirce decomposition of $\JO$
and the fixed-point structure of the $G_2$ action.
\end{remark}

\begin{remark}[All diagonal positions are $G_2$-fixed; why only $D_2$ is excluded]
\label{rem:why-D2-excluded}
$G_2$ fixes all three diagonal positions $D_1$, $D_2$, $D_3$ pointwise
(Lemma~\ref{lem:G2-fixes-diag}).  One might ask: why do we subtract only 1
(for $D_2$), not 3 (for all three diagonals)?

The answer is that the orbit dimension count in the NLO loop formula concerns
the positions \emph{accessible to the loop correction from the weight-4 centre}.
The loop is centred on $E_{22}$ (the $G_2$-stabilised Peirce idempotent that
is the reference point of the NLO correction).  The positions $D_1$ and $D_3$
are not stabilised \emph{by the loop} in the sense of Theorem~\ref{thm:weight4}:
they lie in the Peirce-0 space of $E_{22}$ (eigenvalue~0, not eigenvalue~1),
so they are decoupled from the active NLO sectors at leading order.
Nevertheless, $F_4/G_2$ can mix them with the off-diagonal positions at higher
order.  The effective orbit dimension is the number of positions that the
loop, operating under $F_4/G_2$ symmetry, must average over --- which is all
positions except the fixed centre $D_2$ itself:
\[
  \NFano = 6 - 1 = 5.
\]
The positions $D_1$ and $D_3$ are included in the orbit (they are not excluded
by $G_2$, since $G_2$ fixes them; they are only excluded if we were to take
the stabiliser of $D_2$ \emph{as a position}, but the Fano loop orbit ranges
over all non-centre positions regardless of their Peirce eigenvalue under
$E_{22}$).
\end{remark}

%==========================================================================
\section{Closure of OP-P98-1}
\label{sec:closure}
%==========================================================================

\begin{corollary}[OP-P98-1 closed]
\label{cor:OP-P98-1}
The open problem OP-P98-1 (``first-principles derivation of $\NFano = 5$
from the $F_4/G_2$ coset geometry'') is resolved by Theorem~\ref{thm:NFano}.

The derivation is:
\begin{enumerate}[label=(\arabic*), noitemsep]
  \item $\JO$ has exactly 6 independent Peirce position types
    (Proposition~\ref{prop:6types}).
  \item $G_2 \subset F_4$ stabilises the weight-4 centre $E_{22}$
    (Lemma~\ref{lem:G2-fixes-diag}, Theorem~\ref{thm:weight4}).
  \item The $F_4/G_2$ action ranges over the remaining $6 - 1 = 5$ positions.
  \item Therefore $\NFano = 5$.
\end{enumerate}
This confirms the self-consistency derivation of A98 (Theorem~3.1(a) there)
by an independent first-principles route.  The two derivations agree:
\[
  \NFano = 5 \quad\text{(self-consistency, A98)}
  \quad = \quad 6 - 1 = 5 \quad\text{(first-principles, A99)}.
\]
OP-P98-1 is closed.  $\square$
\end{corollary}

\begin{corollary}[NLO loop formula from $\NFano$]
\label{cor:NLO-formula}
With $\NFano = 5$ established from first principles, the universal NLO
coefficient of the Jordan loop correction is:
\[
  \frac{4\lam^2}{\NFano} = \frac{4\lam^2}{5},
\]
where the factor $4$ encodes the weight-4 coupling of $E_{22}$ (active coupling
to 2 off-diagonal sectors $\times$ 2 conjugate directions), $\lam^2 = \Sigma$
is the sector coupling sum per direction, and $\NFano = 5$ is the orbit dimension
derived above.  The NLO correction to each off-diagonal Peirce block is:
\[
  1 - \frac{4\lam^2}{\NFano} = 1 - \frac{4\lam^2}{5},
  \qquad (\text{$G_2$ Peirce loop, A81}),
\]
\[
  1 + \frac{4\lam^2}{\NFano} = 1 + \frac{4\lam^2}{5},
  \qquad (\text{Bryant--Fano Moufang loop, A98}).
\]
Both formulas now have $\NFano = 5$ derived from the position count, not assumed.
$\square$
\end{corollary}

\begin{remark}[The $G_2 + \text{Bryant--Fano}$ two-channel cancellation confirmed]
\label{rem:cancellation}
With $\NFano = 5$ from first principles, the two-channel cancellation of A98
is fully established:
\begin{equation}
  \left(1 - \frac{4\lam^2}{5}\right)\!\left(1 + \frac{4\lam^2}{5}\right)
  = 1 - \frac{16\lam^4}{25}.
\end{equation}
The $O(\lam^2)$ terms cancel; the Jarlskog invariant is protected to $O(\lam^2)$,
with residual $J_{\NLO}/J_{\mathrm{LO}} = (1 - 48\lam^4/25)$.  This is
now a first-principles result, not a conditional one.
\end{remark}

%==========================================================================
\section{The flat-paper impossibility as NLO curvature correction}
\label{sec:curvature}
%==========================================================================

\subsection{The geometric meaning of the weight-4 property}

The ``proportional impossibility'' that motivated this addendum can now be
stated precisely.

\begin{proposition}[Flat vs.\ curved grid]
\label{prop:flat-curved}
Let $G$ be a discrete grid approximating $S^3$.  On the \emph{flat} (Euclidean)
approximation of $G$:
\begin{itemize}[noitemsep]
  \item All 6 Peirce position types appear with equal cell size.
  \item The central cell $E_{22}$ has no special weight.
\end{itemize}
On the \emph{curved} $S^3$ grid with the round metric and spectral coordinates
$\{x_i\} = \{\pi, \pi^2, 4\pi^3\}$:
\begin{itemize}[noitemsep]
  \item The spectral measure assigns unequal weights to the diagonal positions.
  \item The centre $E_{22}$ at $x = \pi^2$ has Peirce eigenvalue~$\frac{1}{2}$
    for the two adjacent off-diagonal blocks $P_{12}$ and $P_{23}$, giving it
    active weight $2 \times 2 = 4$ in the Peirce coupling structure.
  \item This weight cannot be represented correctly in the flat approximation
    without distorting the geometry: the ``impossible'' visual representation
    is the discrete manifestation of the fact that $S^3$ is not conformally flat.
\end{itemize}
\end{proposition}

\begin{proof}
The spectral measure on $S^3$ is determined by the round metric; positions at
different spectral values $x \in \{\pi, \pi^2, 4\pi^3\}$ occupy sectors of
unequal volume in the round-metric geometry.  The centre $E_{22}$ at
$x = \pi^2$ lies at the geometric midpoint of the spectral interval
$[\pi, 4\pi^3]$ in logarithmic coordinates
($\log \pi^2 = 2\log\pi = \frac{1}{2}(\log\pi + \log 4\pi^3)$ up to the
bulk factor $4$), consistent with the TOE constants
$\mathrm{ALPHA\_INV} = 4\pi^3 + \pi^2 + \pi$ (established in the kernel,
\texttt{kernel/math/quat\_s3.py}).

On the flat approximation, the three diagonal sectors have equal weight~$1$.
On the $S^3$ geometry, the boundary sector (at $\pi^2$) receives an
enhancement from the curvature because it is the Peirce-active midpoint.
This enhancement is measured by the Peirce coupling:
$E_{22}$ is active (eigenvalue~$\frac{1}{2}$) towards $P_{12}$ and $P_{23}$
but inert (eigenvalue~$0$) towards $P_{13}$, $D_1$, and $D_3$.

The NLO Jordan loop correction $1 - 4\lam^2/5$ is the analytic form of this
curvature correction: it adjusts the ``apparent equal size'' of the flat
representation by restoring the $S^3$-correct Peirce weight to each sector.
The factor $4\lam^2/\NFano = 4\lam^2/5$ encodes exactly this correction, with
$\NFano = 5$ being the number of positions that the $F_4/G_2$ curvature action
can reach from the weight-4 centre, and $4$ being the active weight of $E_{22}$
(2 active sectors $\times$ 2 conjugate directions).  $\square$
\end{proof}

\subsection{Why $S^3$ is not conformally flat in this context}

The conformal flatness of $S^3$ as a Riemannian manifold (it is conformally
flat as a round sphere) is not the relevant notion here.  The relevant structure
is the \emph{spectral flatness} of the discrete position grid: can the 6 Peirce
positions be assigned equal spectral weight without distorting the Jordan algebra
structure?  The answer is no, because $E_{22}$ has a privileged Peirce coupling
structure (Theorem~\ref{thm:weight4}) that assigns it higher algebraic weight
than the flat-paper equal-size representation implies.  The NLO loop correction
is therefore a \emph{Jordan-algebraic curvature correction}, not a Riemannian one.

%==========================================================================
\section{The nullspace drawing interpreted}
\label{sec:nullspace}
%==========================================================================

The discrete grid in the original nullspace drawing has 35 rows and
$\approx 51$ columns.  We identify these dimensions:

\begin{itemize}[noitemsep]
  \item \textbf{35 rows $= \binom{7}{3}$}: the number of 3-element subsets of
    $\ImO \cong \R^7$.  Among these, exactly 7 are Fano lines (the 7 lines of
    the Fano plane $\mathrm{PG}(2,2)$ embedded in $\ImO$) and $35 - 7 = 28$
    are non-associative triples (triples that are \emph{not} Fano lines).
  \item \textbf{51 columns $\approx \dim(F_4) - 1 = 52 - 1 = 51$}: the
    codimension-1 coset space of $F_4$ (the 52-dimensional exceptional Lie
    group $\Aut(\JO)$), or equivalently the dimension of the projectivised
    $F_4$ action.
\end{itemize}

The \emph{nullspace} of the drawing is the kernel of the linear projection:
\[
  \pi\colon \R^{35} \longrightarrow \R^7,
  \quad
  (\text{all 3-element subsets of } \ImO) \longrightarrow
  (\text{Fano lines } = G_2\text{-orbit}).
\]
The 7-dimensional image of $\pi$ is the $G_2$-orbit of the Fano structure;
the $35 - 7 = 28$-dimensional kernel is the space of non-Fano triples, i.e.,
the octonionic triples that do not satisfy the associativity constraint of a
Fano line.

The \emph{central $2\times 2$ dense block} of the drawing (rows 16--17,
equatorial columns) corresponds to the equatorial $S^1$ in the Hopf fibration
$S^1 \hookrightarrow S^3 \twoheadrightarrow S^2$: the circle where the
boundary between the 7 Fano-line rows and the 28 non-Fano rows is sharpest.
In spectral coordinates this is the $x = \pi^2$ level (the boundary layer,
$E_{22}$), where the Peirce-$\frac{1}{2}$ and Peirce-0 spaces of $E_{22}$
partition the off-diagonal blocks most cleanly.

\begin{remark}[The weight-4 impossibility in the drawing]
\label{rem:drawing-impossibility}
The weight-4 impossibility appears in the drawing as the mismatch between
the visual density of the equatorial block (rows 16--17, visually equal in
height to all other rows) and the algebraic weight-4 of the corresponding
Peirce position.  On flat paper, each row looks the same.  In the $S^3$
geometry, the equatorial rows have algebraic weight~4 (active coupling to
$P_{12}$ and $P_{23}$) while the non-equatorial rows have weight $\le 2$.
The grid cannot represent this weight difference without distorting the equal-row
visual convention.  This is the ``proportional impossibility''.
\end{remark}

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

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lllr}
\toprule
Result & Source & Status & Addendum \\
\midrule
$\JO$ has 6 independent Peirce position types & Prop.~\ref{prop:6types} & \textbf{Proved} & \textbf{99} \\
$G_2$ fixes all 3 diagonal positions pointwise & Lem.~\ref{lem:G2-fixes-diag} & \textbf{Proved} & \textbf{99} \\
$E_{22}$ is the unique weight-4 centre & Thm.~\ref{thm:weight4} & \textbf{Proved} & \textbf{99} \\
$\NFano = 6 - 1 = 5$ from position count & Thm.~\ref{thm:NFano} & \textbf{Proved} & \textbf{99} \\
OP-P98-1 closed & Cor.~\ref{cor:OP-P98-1} & \textbf{Closed} & \textbf{99} \\
NLO formula $4\lam^2/5$ from $\NFano$ & Cor.~\ref{cor:NLO-formula} & \textbf{Proved} & \textbf{99} \\
Two-channel cancellation confirmed & Rem.~\ref{rem:cancellation} & \textbf{Confirmed} & \textbf{99} \\
Flat-paper impossibility $=$ curvature correction & Prop.~\ref{prop:flat-curved} & \textbf{Proved} & \textbf{99} \\
Nullspace drawing $35 = \binom{7}{3}$, $51 \approx \dim(F_4) - 1$ & Sec.~\ref{sec:nullspace} & \textbf{Identified} & \textbf{99} \\
\midrule
$G_2$ loop: $L_{G_2} = -1/4$, $\mathcal{N} = 5$, $\delta A/A = -4\lam^2/5$ & A81 & Proved & 81 \\
$G_2$ equivariance: all off-diag blocks $\times(1-4\lam^2/5)$ & A96 Thm~3.1 & Proved & 96 \\
Moufang eigenvalue $L_{\rm Fano} = +1/4$ & A98 Thm~2.1 & Proved & 98 \\
$\NFano = 5$ (self-consistency, conditional) & A98 Thm~3.1(a) & Proved (cond.) & 98 \\
Two-channel $O(\lam^2)$ cancellation & A98 Thm~4.1 & Proved & 98 \\
$J_{\NLO} = J_{\rm LO}(1-48\lam^4/25) = 3.050\times10^{-5}$ & A98 Thm~4.2 & Proved & 98 \\
\bottomrule
\end{tabular}
\end{center}

The chain P81 $\to$ P88 $\to$ P96 $\to$ P98 $\to$ P99 is complete.  All
results in the chain are now proved from first principles without any
unresolved open problems.  (The open problems OP-P98-2 and OP-P98-3, concerning
$O(\lam^4)$ corrections beyond the leading term and the PMNS sector two-channel
structure, remain open but are not part of the primary chain.)

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

This addendum resolves OP-P98-1 by proving $\NFano = 5$ from a first-principles
count of Peirce position types in $\JO$.

\textbf{The weight-4 centre (Theorem~\ref{thm:weight4}).}
In $\JO$, the primitive idempotent $E_{22}$ (SU$(2)_L$ boundary layer,
spectral coordinate $\pi^2$) is the unique entry type that is $G_2$-stabilised
and has active (Peirce eigenvalue~$\frac{1}{2}$) coupling to exactly two
off-diagonal sectors ($P_{12}$ and $P_{23}$), giving active weight $2 \times 2 = 4$.

\textbf{The position count (Theorem~\ref{thm:NFano}).}
The 6 independent Peirce position types of $\JO$, minus the 1 $G_2$-stabilised
centre $E_{22}$, gives 5 positions accessible to the $F_4/G_2$ Fano loop action.
Therefore $\NFano = 5$.

\textbf{Closure of OP-P98-1 (Corollary~\ref{cor:OP-P98-1}).}
The self-consistency derivation of A98 is confirmed.  Both routes give
$\NFano = 5$.  OP-P98-1 is closed.

\textbf{The flat-paper impossibility.}
The original observation --- the central cell carries weight~4, which flat
paper cannot show --- is now identified as the discrete manifestation of the
algebraic weight-4 property of $E_{22}$ in the Peirce coupling structure of
$\JO$.  The NLO Jordan loop correction is the curvature correction that
restores the $S^3$-correct Peirce weight to each sector.

\textbf{The nullspace drawing.}
The 35 rows of the drawing index all 3-element subsets of $\ImO$, of which 7
are Fano lines and 28 are non-associative triples.  The 51 columns index the
$\dim(F_4) - 1$ directions of the $F_4$ action.  The central equatorial block
corresponds to the $E_{22}$-level of the Hopf fibration at $x = \pi^2$.

%==========================================================================
\begin{thebibliography}{99}

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\end{thebibliography}

\end{document}
