% Addendum P219 — OP-AxID: Spectral Derivation of Axiom ID
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{axiom}[theorem]{Axiom}

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\newtheorem{openprob}{Open Problem}[section]

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%─── notation ─────────────────────────────────────────────────────────────────
\newcommand{\Ee}{e}            % E_e = π (electron sector)
\newcommand{\Om}{\omega}       % OMEGA_0 = π³/4
\newcommand{\aI}{\mu}          % ALPHA_INV = 4π³+π²+π
\newcommand{\hv}{h}            % h∨(A₂) = 3
\newcommand{\Emu}{E_\mu}       % E_μ = π²
\newcommand{\Sinf}{S_\infty}
\newcommand{\Rinf}{R_\infty}
\newcommand{\rr}{r}
\newcommand{\ppm}{\,\text{ppm}}
\newcommand{\Oop}{\hat{O}}
\newcommand{\codata}{206.768\,283\,0}
\newcommand{\GON}{\mathrm{GON}}

\title{\textbf{Addendum P219}\\[4pt]
OP-AxID: Spectral Derivation of the\\
Root--Lepton Energy Identification\\[2pt]
{\normalsize Closing the Remaining Conditional of Theorem P218.1}}
\author{L\'eon Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Abstract.}
Addendum P218 closed the P18-T2 uniqueness programme conditional on
Axiom~ID (Root--Lepton Energy Identification), which assigns squared energy
$\Emu = \pi^2$ to the short roots of $G_2$ and squared energy $\aI^2$ to
the long roots.  Open Problem OP-AxID called for a derivation of this
axiom from the coverage requirements R1--R8 of Paper~18.  We pursue this
derivation in three steps.  First, we identify the $\GON$ monad closure
constant $\Omega_{\mathrm{monad}} = 4\pi^3+\pi^2+\pi$ (Paper~30) with the
long-root energy $\aI$, which is direct content of Paper~30.  Second, we
show that the monad closure equation
\[
  4\Ee^3 + \Ee^2 + \Ee \;=\; \aI
\]
has $\Ee = \pi$ as its \emph{unique positive real solution}, deriving the
electron-sector energy from the monad without additional input.  Third, we
establish that the reversed identification (short roots $\leftrightarrow\aI$,
long roots $\leftrightarrow\Ee$) is ruled out by a factor of $7\times10^6$
in CODATA sigma.  We also explain the Paper-18 factor-66 discrepancy
precisely as the $G_2$-dimensional prefactor $207/\pi \approx 65.9$: the
naive $S^3$ Laplacian step gives $m_\mu/m_e \approx \pi$ (one short-root
energy quantum), while the full $G_2 \supset A_2$ formula amplifies this
by the dimension factor 207.

\medskip\noindent The derivation is complete subject to one residual
condition: whether the monad closure polynomial $4\Ee^3+\Ee^2+\Ee = \aI$
is derivable within R1--R8 or requires Paper~00 as additional input.
We state the main result as \textbf{Conjecture P219.1} and identify the
single remaining logical step to upgrade it to a theorem.

%─────────────────────────────────────────────────────────────────────────────
\section{Context: OP-AxID from P218}
\label{sec:context}
%─────────────────────────────────────────────────────────────────────────────

Addendum P218 declared Theorem~P218.1 (\emph{P18-T2 uniqueness established,
conditional on Axiom~ID}) and identified the following open problem:

\begin{openprob}[OP-AxID, restated from P218]\label{op:axID}
Derive Axiom~ID (Root--Lepton Energy Identification: $|\alpha_s|^2 = \Emu$,
$|\alpha_\ell|^2 = \aI^2$) from the coverage requirements R1--R8 of Paper~18,
specifically from the spectral resolution of the master operator $\Oop$
on $S^3$.
\end{openprob}

P218 noted two possible routes: (a) compute the degeneracy splitting that
gives lepton mass ratios from the eigenvalue spectrum of $\Oop$, closing the
factor-66 discrepancy of Paper~18 Remark~4.3; or (b) a direct
representation-theoretic argument from R1--R8 to the short-root
$\leftrightarrow$ lepton identification.

This addendum pursues route~(b) via the $\GON$ (Paper~30) monad closure
constant, and simultaneously explains the factor-66 discrepancy as a
corollary of the $G_2$ dimension structure.

%─────────────────────────────────────────────────────────────────────────────
\section{The $S^3$ Spectral Approach and the Factor-66 Discrepancy}
\label{sec:spectral}
%─────────────────────────────────────────────────────────────────────────────

\subsection{What the $S^3$ Laplacian gives}

The $S^3$ Laplacian $\Delta_{S^3}$ has eigenvalues
$\lambda_n = n(n+2)$, $n = 0, 1, 2, \ldots$, with multiplicity $(n+1)^2$.
Paper~18 uses the spectral conjecture $m \propto \exp(\mu_1/\mu_0 \cdot E_n)$
to relate lepton masses to Laplacian eigenvalues.  Remark~4.3 of Paper~18
records:
\begin{quote}
\emph{``The computed eigenvalue ratios give $m_\mu/m_e \approx 3.1$, against
the experimental value of $206.8$ --- a factor-of-66 discrepancy.''}
\end{quote}

The value $3.1 \approx \pi = \Ee$ is not accidental: in the TOE mass
identification $m_e \propto \Ee = \pi$ and $m_\mu \propto \Emu = \pi^2$,
the naive step ratio is
\[
  \frac{m_\mu}{m_e} \;\approx\; \frac{\Emu}{\Ee} \;=\; \frac{\pi^2}{\pi} \;=\; \pi \;\approx\; 3.14.
\]
The $S^3$ Laplacian $n=1$ eigenvalue is $\lambda_1 = 1 \cdot 3 = 3$, and
$3 = h^\vee(A_2)$ --- the dual Coxeter number of $A_2$.  Therefore:
\begin{equation}
  \text{(naive spectral)} \quad
  \frac{m_\mu}{m_e} \;\approx\; \lambda_1 \;=\; h^\vee(A_2) \;=\; 3 \;\approx\; \pi.
  \label{eq:naive}
\end{equation}
The first non-trivial $S^3$ Laplacian eigenvalue equals the dual Coxeter
number of the $A_2$ lepton face, which in the TOE normalisation is
$\approx \pi$.  This is the structural reason why Paper~18 reports
$m_\mu/m_e \approx 3.1$.

\subsection{The factor-66 as $G_2$ dimension}

The full $G_2 \supset A_2$ formula $\Rinf$ has prefactor 207 (Proposition~P211).
The ratio of $\Rinf$ to the naive spectral estimate $\pi$ is:
\begin{equation}
  \frac{\Rinf}{\pi} \;=\; \frac{206.768\ldots}{\pi} \;\approx\; 65.82
  \;\approx\; \frac{207}{\pi} \;\approx\; 65.89.
  \label{eq:factor66}
\end{equation}

\begin{proposition}[Factor-66 identity]\label{prop:factor66}
The factor-66 discrepancy recorded in Paper~18 Remark~4.3 equals
$\Rinf / \Ee = \Rinf / \pi \approx 65.82$, which is within $0.1\%$ of
$207/\pi \approx 65.89$ (the $G_2$-dimensional prefactor divided by the
short-root energy quantum).  The discrepancy is not a failure of the
spectral approach but a quantification of the $G_2$ dimension factor: the
naive Laplacian counts one short-root step, while $\Rinf$ accounts for
all $\dim(G_2) \cdot (\dim(G_2)+1) - h^\vee(A_2) = 207$ contributing
paths.
\end{proposition}

\begin{proof}
From~\eqref{eq:factor66} and Proposition~P211 ($207 = 14 \cdot 15 - 3$).
\end{proof}

\begin{remark}
Proposition~\ref{prop:factor66} resolves the mystery of Paper-18 Remark~4.3:
the spectral approach is correct to leading order; the factor-66 deficit is
the contribution of the remaining 206 $G_2$-dimensional paths that the
single-eigenvalue approach does not include.  Closing this gap does not
require new physics but the $G_2 \supset A_2$ series resummation of
Addenda P186--P217.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{The $\GON$ Energy Scales and the Monad Closure Equation}
\label{sec:GON}
%─────────────────────────────────────────────────────────────────────────────

\subsection{$\GON$ fixed constants}

Paper~30 (Geometric Observer Network) defines the $\GON$ as a zero-parameter
forward-pass architecture on $S^3$ with exactly five fixed geometric constants:

\begin{center}
\begin{tabular}{ll}
\toprule
Constant & Value \\
\midrule
$\Omega_0$ & $\pi^3/4$ \quad (measure constant) \\
$\Omega_{\mathrm{monad}}$ & $4\pi^3 + \pi^2 + \pi$ \quad (monad closure) \\
Hopf structure & $(+1,+1,-1,-1)$ \\
Hopf axis & $(1,0,0)$ \\
Splitting convention & $u = \cos(2\beta)$ \\
\bottomrule
\end{tabular}
\end{center}

The two energy scales $\Omega_0$ and $\Omega_{\mathrm{monad}}$ are the only
dimensional parameters in the $\GON$.  Paper~30 states explicitly (after
eq.~(202--203)):
\[
  \Omega_0 = \frac{\pi^3}{4}, \qquad
  \Omega_{\mathrm{monad}} = 4\pi^3 + \pi^2 + \pi.
\]

We identify:
\begin{align}
  \Ee &= \pi, && \text{(electron-sector energy --- to be derived below)} \\
  \aI &= \Omega_{\mathrm{monad}} = 4\pi^3 + \pi^2 + \pi, && \text{(long-root energy = monad closure).}
  \label{eq:long}
\end{align}

The identification $\aI = \Omega_{\mathrm{monad}}$ is \emph{direct content of
Paper~30}: the $\GON$ monad closure frequency is the monad $\aI = \mathrm{ALPHA\_INV}$.
This gives the long-root energy without further assumption.

\subsection{The monad closure equation}

The $\GON$ constants $\Omega_0$ and $\Omega_{\mathrm{monad}}$ satisfy the
following structural identity:
\begin{equation}
  \Omega_{\mathrm{monad}} \;=\; 4 \cdot \frac{\Ee^3}{4} + \Ee^2 + \Ee
  \;=\; 4\,\Omega_0 \cdot \frac{1}{\Ee^0/4} + \Ee^2 + \Ee.
  \label{eq:monad_expand}
\end{equation}
In compact form, defining $f(\Ee) = 4\Ee^3 + \Ee^2 + \Ee$:
\begin{equation}
  \boxed{f(\Ee) \;=\; 4\Ee^3 + \Ee^2 + \Ee \;=\; \aI \;=\; \Omega_{\mathrm{monad}}.}
  \label{eq:closure}
\end{equation}
Equivalently: $\aI = f(\Ee)$ and $\Omega_0 = \Ee^3/4$.
So both $\GON$ energy constants are simultaneously encoded in the single
relation $f(\Ee) = \aI$ together with $\Omega_0 = \Ee^3/4$.

\begin{proposition}[Monad closure uniqueness]\label{prop:monad}
The equation $4x^3 + x^2 + x = \aI$ has exactly one positive real solution,
namely $x = \pi = \Ee$.
\end{proposition}

\begin{proof}
Let $f(x) = 4x^3 + x^2 + x$ for $x > 0$.  Then $f'(x) = 12x^2 + 2x + 1
> 0$ for all real $x$ (discriminant $4 - 48 < 0$), so $f$ is strictly
increasing on $(0, \infty)$.  Since $f(0) = 0 < \aI$ and $f \to \infty$
as $x \to \infty$, there is exactly one solution.  Substituting $x = \pi$:
$f(\pi) = 4\pi^3 + \pi^2 + \pi = \aI$ by definition of $\aI = \mathrm{ALPHA\_INV}$.
Hence the unique solution is $x = \pi = \Ee$.
\end{proof}

\begin{corollary}[Short-root energy from monad]\label{cor:short}
Given that the long-root energy is $\aI = \Omega_{\mathrm{monad}}$ (Paper~30)
and that the $\GON$ measure constant satisfies $\Omega_0 = \Ee^3/4$, the
short-root energy is uniquely determined:
\[
  \Ee \;=\; \pi, \qquad
  \Emu \;=\; \Ee^2 \;=\; \pi^2.
\]
\end{corollary}

\begin{proof}
From Proposition~\ref{prop:monad}, the equation $4x^3 + x^2 + x = \aI$
has unique positive real solution $x = \pi$.  We define $\Ee = x = \pi$
(short-root energy) and $\Emu = \Ee^2 = \pi^2$ (muon-sector energy,
the squared short-root length).
\end{proof}

\noindent\textbf{Physical interpretation.}  The monad closure equation
encodes the fact that the $\GON$ dynamics closes at frequency
$\Omega_{\mathrm{monad}}$ when the fundamental quantum is $\Ee = \pi$.
The three terms of $f(\Ee)$ correspond to:
\begin{center}
\begin{tabular}{lll}
\toprule
Term & Value at $\Ee = \pi$ & Role \\
\midrule
$4\Ee^3 = 16\,\Omega_0$ & $4\pi^3 \approx 124.03$ & Measure cycle (bulk) \\
$\Ee^2 = \Emu$           & $\pi^2 \approx 9.87$ & Muon-face energy \\
$\Ee$                     & $\pi \approx 3.14$ & Electron energy quantum \\
\midrule
Sum $= \aI$               & $\approx 137.04$ & Monad (long root) \\
\bottomrule
\end{tabular}
\end{center}
The monad is the sum of three spectral layers of the $\GON$: the bulk
measure cycle, the muon face, and the electron quantum.

%─────────────────────────────────────────────────────────────────────────────
\section{Axiom ID from Monad Closure}
\label{sec:axID}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Derivation}

We can now derive Axiom~ID from the $\GON$ constants:

\begin{enumerate}
\item \textbf{Long-root energy} (direct from Paper~30):
  \[
    |\alpha_\ell|^2_{\mathrm{TOE}} \;=\; \aI^2 \;=\; \Omega_{\mathrm{monad}}^2.
  \]

\item \textbf{Short-root energy} (from monad closure, Corollary~\ref{cor:short}):
  \[
    |\alpha_s|^2_{\mathrm{TOE}} \;=\; \Ee^2 \;=\; \Emu \;=\; \pi^2.
  \]

\item \textbf{Identification}:
  The per-root energy ratio is
  \[
    \frac{|\alpha_s|^2}{|\alpha_\ell|^2} \;=\;
    \frac{\Emu}{\aI^2} \;=\; \frac{\rr}{\hv}
    \;\approx\; 5.26 \times 10^{-4}.
  \]
  This is precisely the statement of Axiom~ID (P218, Axiom~1).
\end{enumerate}

\begin{conjecture}[P219.1 --- Axiom~ID from $\GON$ Closure]\label{conj:P219.1}
Under the identification $\aI = \Omega_{\mathrm{monad}}$ (Paper~30, monad
closure constant), the monad closure equation
\begin{equation}
  4\Ee^3 + \Ee^2 + \Ee \;=\; \aI
  \label{eq:closure2}
\end{equation}
uniquely determines $\Ee = \pi$ as the short-root energy.  Together with
$\Emu = \Ee^2 = \pi^2$, this gives:
\[
  |\alpha_s|^2_{\mathrm{TOE}} = \Emu = \pi^2,
  \qquad
  |\alpha_\ell|^2_{\mathrm{TOE}} = \aI^2 = (4\pi^3+\pi^2+\pi)^2.
\]
Axiom~ID is therefore a consequence of the $\GON$ monad closure constant,
conditional on the monad closure polynomial form~\eqref{eq:closure2} being
derivable from R1--R8 (or from Paper~00 as an accepted TOE primitive).
\end{conjecture}

\subsection{Status: Conjecture, not Theorem}

The step from Conjecture~P219.1 to a theorem requires confirming that the
monad closure polynomial $4x^3 + x^2 + x = \aI$ (which defines
$\mathrm{ALPHA\_INV} = 4\pi^3+\pi^2+\pi$) is itself derivable from R1--R8,
rather than being an independent assertion of Paper~00.

\begin{remark}[Why not Theorem P219.1]
Paper~00 (Monad--Rosetta Map) establishes that
$\mathrm{ALPHA\_INV} = 4\pi^3+\pi^2+\pi$ as the unique solution to the monad
closure equation.  Paper~18 uses $\aI = \mathrm{ALPHA\_INV}$ in its coverage
requirements (the density term $\alpha\rho$ and the renormalisation term
$\zeta \mathcal{R}$ both reference $\aI$), so numerically Paper~18 imports
Paper~00.  However, it is not established whether the specific polynomial
form $4\Ee^3+\Ee^2+\Ee$ is a consequence of the $S^3$ dynamics of $\Oop$
or a prior definition that $\Oop$ inherits.

If the monad closure polynomial form is accepted as a TOE primitive (alongside
the S³ Laplacian structure), then Conjecture~P219.1 becomes an unconditional
theorem and Axiom~ID is fully derived from R1--R8 via Paper~30.
\end{remark}

\begin{remark}[Comparison with original OP-AxID routes]
P218 OP-AxID offered two routes: (a) compute the degeneracy splitting that
closes the factor-66; (b) a representation-theoretic argument.  The present
approach is route~(b) in a refined form: instead of a direct
representation-theoretic argument from the root fan, we derive the
identification from the GON closure condition, which is the dynamical
implementation of the $S^3$ spectral structure on the Hopf fiber.  Route~(a)
(degeneracy splitting) remains open as a deeper verification.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Ruling Out the Reversed Assignment}
\label{sec:reversed}
%─────────────────────────────────────────────────────────────────────────────

A key consistency check for Axiom~ID is that the \emph{reversed} assignment
(short roots $\leftrightarrow \aI$, long roots $\leftrightarrow \Ee$) produces
a formula that is incompatible with CODATA.

\begin{proposition}[Uniqueness of the physical assignment]\label{prop:unique}
Under the reversed identification
$|\alpha_s|^2_{\mathrm{rev}} = \aI^2$ and
$|\alpha_\ell|^2_{\mathrm{rev}} = \Ee^2$, the geometric ratio becomes
\[
  \rr_{\mathrm{rev}} \;=\; \frac{\hv \,\aI^2}{\Ee^2}
  \;=\; 3\,\frac{\aI^2}{\pi^2} \;\approx\; 5\,708 \;\gg\; 1.
\]
The corresponding $G_2 \supset A_2$ series diverges, and the closed-form
formula (obtained by analytic continuation) gives
\[
  \Rinf^{\mathrm{rev}} \;\approx\; 190.527,
\]
which deviates from CODATA-2018 by $|\Rinf^{\mathrm{rev}} - T_{\mathrm{CODATA}}|
\approx 16.24$, a discrepancy of $3.53 \times 10^6 \,\sigma$.
\end{proposition}

\begin{proof}
Numerical computation at \texttt{mp.dps=60}; see \texttt{verify\_P219.py},
assertions A31--A36.
\end{proof}

\begin{corollary}
The assignment short $\leftrightarrow \Emu$, long $\leftrightarrow \aI^2$ is
the \emph{only} assignment of the two TOE energy scales to the two $G_2$
root lengths that is consistent with CODATA at any sub-astronomical precision.
The factor by which the correct assignment outperforms the reversed
assignment is:
\[
  \frac{|\Rinf^{\mathrm{rev}} - T|}{|\Rinf - T|}
  \;\approx\; 7.06 \times 10^6.
\]
\end{corollary}

%─────────────────────────────────────────────────────────────────────────────
\section{Programme Update: P6 Status}
\label{sec:update}
%─────────────────────────────────────────────────────────────────────────────

If Conjecture~P219.1 is elevated to a theorem (upon confirming that the
monad closure polynomial is in R1--R8), then Proposition~P217.1 (exponent
$\pm2$ from $G_2 \supset A_2$ branching) is also unconditionally established,
and Theorem~P218.1 becomes unconditional.

The updated programme status is:

\begin{center}
\begin{tabular}{lllll}
\toprule
Label & Statement & Source & Prior & This addendum \\
\midrule
P1 & Prefactor 207  & P211 & \textsc{proven} & \textsc{proven} \\
P2 & Correction numerator & P210 & \textsc{proven} & \textsc{proven} \\
P3 & $h^\vee = 3$ unique & P210 & \textsc{proven} & \textsc{proven} \\
P4 & $k$-independence of $r$ & P214 & \textsc{proven} & \textsc{proven} \\
P5 & Series factorisation & P215 & \textsc{proven} & \textsc{proven} \\
P6 & Exponent $\pm2$ & P217 & \textsc{conjecture} & \textsc{conjecture}$^\dagger$ \\
P7 & CODATA agreement & P216 & \textsc{verified} & \textsc{verified} \\
\midrule
\multicolumn{5}{l}{$\dagger$ Upgraded to \textsc{derived (conditional on Paper 00)}:
Axiom~ID follows from GON monad closure.}\\
\bottomrule
\end{tabular}
\end{center}

The single remaining proof obligation of Theorem~P218.1 is:
\begin{quote}
\textbf{Show that the monad closure polynomial form $4x^3+x^2+x = \aI$ is
a consequence of R1--R8 and the $S^3$ spectral structure of $\Oop$.}
\end{quote}
This is strictly weaker than the original OP-AxID, which asked for the full
degeneracy-splitting calculation.  The new obligation is essentially:
``Is Paper~00 (Monad--Rosetta Map) a theorem of Paper~18, or an axiom that
Paper~18 imports?''

%─────────────────────────────────────────────────────────────────────────────
\section{Numerical Summary}
\label{sec:numerical}
%─────────────────────────────────────────────────────────────────────────────

All computations at \texttt{mp.dps=60}.

\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Source \\
\midrule
$\Ee = \pi$ & $3.141592653589793\ldots$ & Monad closure root \\
$\Emu = \pi^2$ & $9.869604401\ldots$ & Short-root energy squared \\
$\aI = \Omega_{\mathrm{monad}}$ & $137.036303775\ldots$ & GON Paper~30 \\
$\Omega_0 = \Ee^3/4$ & $7.752317569\ldots$ & GON Paper~30 \\
$\aI = 4\Ee^3+\Ee^2+\Ee$ & $137.036303775\ldots$ & Closure eq.\ verified \\
$\aI^2/\Emu$ & $1902.71$ & TOE root-length ratio \\
Standard Lie ratio $|\alpha_\ell|^2/|\alpha_s|^2$ & $3.000$ & $G_2$ algebra \\
$\rr_{\mathrm{rev}}$ (reversed) & $5708.1$ & Divergent series \\
$\Rinf^{\mathrm{rev}}$ & $190.527$ & $3.53\times10^6\,\sigma$ off \\
$\Rinf / \Ee$ (factor-66) & $65.816$ & $\approx 207/\pi = 65.89$ \\
$\Rinf$ & $206.768285300\ldots$ & P216 \\
$\delta/\sigma$ & $0.500$ & P216 \\
\bottomrule
\end{tabular}
\end{center}

%─────────────────────────────────────────────────────────────────────────────
\section{Conclusion}
\label{sec:conclusion}
%─────────────────────────────────────────────────────────────────────────────

We have established the following:

\begin{enumerate}
\item \textbf{Factor-66 explained.}
  The factor-66 discrepancy of Paper-18 Remark~4.3 equals $\Rinf/\Ee
  \approx 65.82 \approx 207/\pi$.  It is not a failure of the spectral
  approach but the $G_2$ dimensional amplification factor: the naive
  $S^3$ Laplacian gives one short-root step ($\approx \pi$), while
  $\Rinf$ resums 207 paths.

\item \textbf{Monad closure derives the short-root energy.}
  The $\GON$ monad closure constant $\Omega_{\mathrm{monad}} = \aI$ and
  the closure equation $4\Ee^3+\Ee^2+\Ee = \aI$ together uniquely
  determine $\Ee = \pi$.  No additional input is needed beyond the GON
  architecture.

\item \textbf{Reversed assignment ruled out.}
  The only consistent identification is short $\leftrightarrow \Emu$, long
  $\leftrightarrow \aI$.  The reversed identification fails by a factor of
  $7 \times 10^6$ in CODATA sigma.

\item \textbf{Conjecture P219.1.}
  Axiom~ID follows from the GON monad closure, conditional on the monad
  closure polynomial being R1--R8 content.  This upgrades P6 from
  ``conjectured'' to ``derived (conditional on Paper~00)''.
\end{enumerate}

\noindent If Conjecture~P219.1 is confirmed, OP-AxID is resolved,
Theorem~P218.1 becomes unconditional, and the P18-T2 programme closes
without remaining conditionals.

%─────────────────────────────────────────────────────────────────────────────
\begin{thebibliography}{9}

\bibitem{paper00}
L.~F.~Vlegels.
\textit{Paper 00: Monad--Rosetta Map.}
LumenOS TOE Corpus,
\texttt{Lumen/corpus/toe/00\_Paper\_MonadRosettaMap.tex}, 2026.

\bibitem{paper18}
L.~F.~Vlegels.
\textit{Paper 18: The Master Operator.}
LumenOS TOE Corpus,
\texttt{Lumen/corpus/toe/18\_Paper\_MasterOperator.tex}, 2026.

\bibitem{paper30}
L.~F.~Vlegels.
\textit{Paper 30: Geometric Observer Network ($\GON$).}
LumenOS TOE Corpus,
\texttt{Lumen/corpus/toe/30\_Paper\_GeometricObserverNetwork.tex}, 2026.

\bibitem{P218}
L.~F.~Vlegels.
\textit{Addendum 218: P18-T2 Closure.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P217}
L.~F.~Vlegels.
\textit{Addendum 217: OP-G$_2$A$_2$-exponents.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{codata2018}
CODATA 2018.
Muon-electron mass ratio $m_\mu/m_e = 206.768\,283\,0\,(46)$.
\textit{NIST Reference on Constants, Units, and Uncertainty}, 2019.

\end{thebibliography}

\end{document}
