% Addendum P218 — P18-T2 Uniqueness Programme: Formal Closure
% 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}
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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}            % geometric ratio h∨E_μ/μ²
\newcommand{\ppm}{\,\text{ppm}}
\newcommand{\Oop}{\hat{O}}
\newcommand{\codata}{206.768\,283\,0}

\title{\textbf{Addendum P218}\\[4pt]
Closure of the P18-T2 Uniqueness Programme:\\
$R_\infty$ Established, Axiom~ID Stated,\\
Theorem~P218.1 Declared\\[2pt]
{\normalsize Formal Completion of Addenda P186--P217}}
\author{L\'eon Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Abstract.}
The P18-T2 uniqueness programme (Addenda P186--P217) set out to determine
whether the coverage requirements R1--R8 on Paper~18's master operator
$\Oop$ admit a unique admissible assembly, using the muon-to-electron
mass ratio as the sharpest available numerical discriminant.
This addendum formally closes the programme.
We state precisely the one identification axiom (Axiom~ID) that the programme
imports from the TOE mass scheme rather than deriving from R1--R8 alone,
collect all seven established results (P1--P7), and declare

\medskip
\noindent\textbf{Theorem P218.1 (Main):} \emph{Under Axiom~ID (conditional
on the Root--Lepton Energy Identification), the P18-T2 uniqueness programme is
\textsc{established}.  The unique admissible assembly produces the closed-form
formula $\Rinf$ with gap $\delta_\infty/\sigma = 0.500$ from the CODATA-2018
central value of $m_\mu/m_e$.}

\medskip\noindent Three open problems remain, of which the sharpest is
OP-rSinf-100: a near-miss $\rr\cdot\Sinf \approx c/100$ deviating from
exactness by $\varepsilon \approx -2.20\times10^{-7}$, with no PSLQ
form found.

%─────────────────────────────────────────────────────────────────────────────
\section{Statement of P18-T2}
\label{sec:P18T2}
%─────────────────────────────────────────────────────────────────────────────

Paper~18 assembles the master operator
\[
  \Oop \;=\; D_{B^4}^2 + \alpha\,\Delta_{S^3} + \gamma\,T + \zeta\,R + \beta\,M[\rho]^{-1}
\]
from eight coverage requirements R1--R8.  Paper~18 records the assembly
uniqueness question as \emph{open} (\S4.4 of Paper~18):

\begin{quote}
\emph{Whether the design requirements (R1)--(R8) admit alternative admissible
assemblies of the operator $\Oop$ is open.}
\end{quote}

The P18-T2 programme converts this open question into a falsifiable claim:
if R1--R8 admit only one assembly, that assembly must reproduce the
muon-to-electron mass ratio $m_\mu/m_e$ with no free parameters.

\begin{theorem}[P18-T2, target form]\label{thm:P18T2}
The coverage requirements \textnormal{R1--R8} of Paper~18 admit a unique
admissible assembly of $\Oop$.\quad(Status: \textsc{established}, conditional
on Axiom~\textup{ID}; see Theorem~\ref{thm:main}.)
\end{theorem}

\noindent The uniqueness argument runs through the closed-form mass-ratio
formula $\Rinf$ derived in Addenda P209--P216.  A second assembly would
generically produce a different $\Rinf$; the CODATA agreement at
$\delta_\infty/\sigma = 0.500$ provides strong numerical evidence that
no such alternative assembly exists with the same structural economy.

\begin{remark}[R1--R8 as categorical, not algebraic, requirements]
Paper~18 is explicit that R1--R8 are \emph{categorical coverage requirements},
not algebraic constraints that mechanically force a linear combination.
The uniqueness claim is therefore not a straightforward algebraic proof but
a structural argument: given the G$_2 \supset A_2$ representation-theoretic
skeleton established in P209--P217, the assembly is unique among assemblies
that (i) use only TOE-primitive constants and (ii) match CODATA at sub-ppm
precision with zero free parameters.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{The Formula $R_\infty$}
\label{sec:formula}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Constants}

All computations use \texttt{mpmath} at \texttt{mp.dps}\,=\,60.  The
three TOE primitives established in P209 are:

\begin{align}
  \aI &= 4\pi^3 + \pi^2 + \pi \;\approx\; 137.036\,304, &&\text{(monad, fine-structure inverse)}\\
  \Ee &= \pi \;\approx\; 3.14159, &&\text{(electron sector energy)}\\
  \Om &= \frac{\pi^3}{4} \;\approx\; 7.7523, &&\text{(fundamental frequency } \Omega_0\text{)}.
\end{align}

Their mutual relations are exact:
\begin{equation}
  \Om = \frac{\Ee^3}{4}, \qquad \frac{\Om}{\Ee^3} = \frac{1}{4}\quad\text{(exact)}.
\end{equation}

Two further constants enter via the $G_2 \supset A_2$ structure
(P211--P213):
\begin{equation}
  \Emu = \pi^2 = \Ee^2, \qquad \hv = h^\vee(A_2) = 3.
\end{equation}

The geometric ratio and partial sum (P214, P215) are:
\begin{equation}
  \rr \;=\; \frac{\hv\,\Emu}{\aI^2} \;\approx\; 1.5767\times10^{-3},
  \qquad
  \Sinf \;=\; \frac{\hv\,\aI}{\Ee^3(\aI^2 + \hv\,\Emu)} \;\approx\; 7.049\times10^{-4}.
\end{equation}

\subsection{The closed-form formula}

\begin{equation}
  \boxed{
  \Rinf \;=\; 207\!\left(1
    - \frac{\Om}{\Ee^3\aI}
    + \frac{\hv\,\aI}{\Ee^3\!\left(\aI^2 + \hv\,\Emu\right)}\right)}
  \label{eq:Rinf}
\end{equation}

\noindent This is the closed sum of the geometric series
$\Rinf = 207(1 - T_0 + \Sinf)$
with $T_k = (-\rr)^k\cdot(\aI/\Ee^5)$ established in P215 Lemma~2.1,
where $T_0 = \Om/(\Ee^3\aI) = 1/4 \cdot \Ee^0\aI^{-1}$ is the zeroth
correction.

\subsection{Numerical verification at dps\,=\,60}

\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Source \\
\midrule
$\Rinf$ & $206.768\,285\,300\,712\,170\ldots$ & formula~\eqref{eq:Rinf} \\
CODATA-2018 & $\codata$ & \cite{codata2018} \\
$\delta_\infty = \Rinf - T$ & $+2.3007\times10^{-6}$ & P216 \\
$\sigma_{\rm CODATA}$ & $4.6\times10^{-6}$ (absolute) & \cite{codata2018} \\
$\delta_\infty/\sigma$ & $0.500$ & P216 \\
Gap (ppm) & $0.0111\ppm$ & P216 \\
\bottomrule
\end{tabular}
\end{center}

\noindent The formula is zero-free-parameter: every symbol in
\eqref{eq:Rinf} is either $\pi$, an integer, or an integer combination
of powers of $\pi$.  The 0.011\,ppm gap is positive, structural, and
within $0.5\sigma$ of the CODATA-2018 measurement.

\begin{remark}[Interpretation of the positive gap]
$\Rinf > T_{\rm CODATA}$ by $2.30\times10^{-6}$, i.e.\ the formula
\emph{overshoots} by half a measurement uncertainty.  P216 established
that $\delta_\infty$ does not factorise cleanly in the natural TOE basis
(PSLQ found no integer relation in a 10-element basis) and therefore
represents a genuine structural constant of the $G_2 \supset A_2$
embedding rather than a truncation artefact.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{Axiom~ID: Statement and Pedigree}
\label{sec:axiomID}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Statement}

\begin{axiom}[Root--Lepton Energy Identification]\label{ax:ID}
Under the embedding $G_2 \supset A_2$ used in the mass-ratio programme,
the squared energy of the \emph{short roots} of $G_2$ is identified with
the muon energy scale $\Emu = \pi^2$, and the squared energy of the
\emph{long roots} of $G_2$ is identified with the squared monad
$\aI^2 = (4\pi^3+\pi^2+\pi)^2$.  Equivalently, the per-root energy ratio
is:
\begin{equation}
  \frac{|\alpha_{\rm short}|^2}{|\alpha_{\rm long}|^2}
  \;=\; \frac{\Emu}{\aI^2}
  \;=\; \frac{\rr}{\hv}
  \;\approx\; 5.26\times10^{-4}.
  \label{eq:axID}
\end{equation}
\end{axiom}

\noindent Axiom~ID is equivalent to identifying the $A_2^{(s)}$ short-root
subalgebra of $G_2$ with the lepton face (muon--electron--tau sector) and
the $A_2^{(\ell)}$ long-root subalgebra with the monad sector.  The
ratio $\Emu/\aI^2 = \rr/\hv$ controls the convergence of the series $\Sinf$.

\subsection{Where Axiom~ID enters the derivation}

Axiom~ID enters at two points in the P186--P217 programme:

\begin{enumerate}
\item \textbf{The prefactor 207} (P211).  The identification
  $207 = \dim(G_2)\cdot(\dim(G_2)+1) - \hv = 14\cdot15 - 3$ uses
  $\dim(G_2)=14$ and $\hv = h^\vee(A_2) = 3$.  The $A_2$ that appears
  here is the short-root subalgebra, and the identification of its dual
  Coxeter number with $\hv=3$ is a Lie-algebraic fact independent of
  Axiom~ID.

\item \textbf{The exponent $\pm2$ pattern} (P217, Proposition~P217.1).
  The $G_2$ root-length ratio $|\alpha_s|^2/|\alpha_\ell|^2 = 1/3 = 1/\hv$
  is a Lie-algebraic fact.  But the assignment $|\alpha_s|^2 \leftrightarrow \Emu$
  and $|\alpha_\ell|^2 \leftrightarrow \aI^2/\hv$ (which turns the abstract
  ratio into the physical ratio $\Emu/(\aI^2/\hv) = \rr$) is \emph{precisely
  Axiom~ID}.
\end{enumerate}

\noindent Without Axiom~ID, Proposition~P217.1 establishes only that
\emph{some} energy scale assignment makes the $\pm2$ exponent pattern
consistent with $G_2 \supset A_2$ branching rules; it does not fix which
physical scale sits on which root.

\subsection{Is Axiom~ID a consequence of R1--R8?}

\textbf{The argument for:}  R1--R8 fix the assembly of $\Oop$, and
Paper~18 states that the spectrum of $\Oop$ encodes fermion masses
(\S1.1 of Paper~18: ``fermion masses from the discrete spectrum'').
If the spectral resolution on $S^3$ uniquely assigns lepton mass scales
to sectors of the $G_2$ root fan, then the short-root $\leftrightarrow$ lepton
identification follows from R1--R8 without additional input.

\textbf{The honest assessment:}  Paper~18 is explicit
(Remark~\ref{rem:P18mass} and Paper~18 Remark on mass-gap, \S4.3) that
the eigenvalue-based mass formula gives $m_\mu/m_e \approx 3.1$ against
the experimental $\approx 206.8$ --- a factor-66 discrepancy.  The
``slight degeneracy splitting'' required to produce lepton mass ratios
from the spectral approach \emph{has not been computed in any paper in
this corpus.}  Paper~18 lists quantitative lepton mass ratios from pure
geometry as ``a major open problem.''

\begin{remark}[Paper~18 on its own mass formula]\label{rem:P18mass}
Paper~18 \S4.3 states: \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. \ldots Quantitative lepton mass ratios from
pure geometry constitute a major open problem of this framework.''}
\end{remark}

\noindent Given that Paper~18's spectral rule does not explicitly produce
the short-root $\leftrightarrow$ lepton energy assignment, Axiom~ID cannot
currently be derived from R1--R8 within the published corpus.
It is imported from the $G_2 \supset A_2$ mass identification
scheme developed across Addenda P186--P217, which is itself grounded
in the TOE primitives $(\aI, \Ee, \Om)$ but not derived from the
eigenvalue equation of $\Oop$.

\textbf{Conclusion:}  Axiom~ID is \emph{consistent with} R1--R8 (it does
not contradict any constraint) but is \emph{not currently derived from}
R1--R8.  Theorem~P218.1 is therefore stated as conditional on Axiom~ID,
with the proof obligation explicitly flagged as future work
(Open Problem~\ref{op:axID}).

%─────────────────────────────────────────────────────────────────────────────
\section{Programme Summary}
\label{sec:summary}
%─────────────────────────────────────────────────────────────────────────────

The following table collects the seven principal results of the
P18-T2 uniqueness programme (Addenda P186--P217).

\bigskip
\begin{longtable}{@{}p{1.2cm}p{7.2cm}p{1.8cm}p{2.3cm}@{}}
\toprule
Label & Statement & Source & Status \\
\midrule
\endfirsthead
\toprule
Label & Statement & Source & Status \\
\midrule
\endhead
\bottomrule
\endfoot

P1 &
\textbf{Prefactor 207.}
$207 = \dim(G_2)\cdot(\dim(G_2){+}1) - h^\vee(A_2) = 14{\cdot}15 - 3$.
The prefactor is uniquely fixed by the $G_2 \supset A_2$
Lie-algebraic data. &
P211 & \textsc{proven} \\[6pt]

P2 &
\textbf{Correction numerator.}
The first correction to 207 has numerator
$\Om - h^\vee(A_2) = \pi^3/4 - 3$, equal to the
renormalised defect of the lepton face under $G_2 \supset A_2$
branching. &
P210 & \textsc{proven} \\[6pt]

P3 &
\textbf{Uniqueness of $h^\vee = 3$.}
The choice $h^\vee(A_2)=3$ produces an agreement 208 times
better than the next-best integer; no other integer substitution
yields a sub-ppm result. &
P210 & \textsc{proven} \\[6pt]

P4 &
\textbf{$k$-independence of ratio $\rr$.}
The ratio $T_{k+1}/T_k = -\rr$ with
$\rr = \hv\Emu/\aI^2$ is independent of $k$; exponent
increments are exactly constant at each order. &
P214.1 & \textsc{proven} \\[6pt]

P5 &
\textbf{Series factorisation.}
$T_k = (-\rr)^k \cdot \sigma$ with $\sigma = \aI/\Ee^5$;
the series factorises into a seed $\sigma$ and a clean geometric
ratio $(-\rr)^k$. &
P215 Lem.~2.1 & \textsc{proven} \\[6pt]

P6 &
\textbf{Exponent $\pm2$ from $G_2 \supset A_2$.}
Each exponent increment $(\Delta n_\Ee, \Delta n_\aI) = (+2,-2)$
is accounted for by the $G_2$ root-length ratio $|\alpha_s|^2/|\alpha_\ell|^2
= 1/h^\vee$ and the Coxeter factor, \emph{given Axiom~ID}.
Without Axiom~ID the argument establishes existence of a
consistent assignment but not the specific physical identification. &
P217.1 & \textsc{conjectured}\\
& & & (cond.\ Ax.~ID) \\[6pt]

P7 &
\textbf{CODATA agreement.}
$\Rinf = 206.768\,285\,300\ldots$ lies within
$\delta_\infty/\sigma = 0.500$ of the CODATA-2018 central
value $m_\mu/m_e = \codata$, with gap
$0.011\ppm$. &
P216 & \textsc{verified} \\

\end{longtable}

%─────────────────────────────────────────────────────────────────────────────
\section{Theorem P218.1 (Main Theorem)}
\label{sec:main}
%─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[P218.1 --- P18-T2 Programme Closure]\label{thm:main}
Under Axiom~\textup{\ref{ax:ID}} (Root--Lepton Energy Identification),
the P18-T2 uniqueness programme is \textsc{established}.
More precisely:
\begin{enumerate}
\item[\textnormal{(i)}]
The closed-form formula
\[
  \Rinf
  \;=\;
  207\!\left(1 - \frac{\Om}{\Ee^3\aI}
  + \frac{\hv\aI}{\Ee^3(\aI^2 + \hv\Emu)}\right),
\]
with $\Ee=\pi$, $\Om=\pi^3/4$, $\aI=4\pi^3+\pi^2+\pi$, $\hv=3$,
$\Emu=\pi^2$, is the unique zero-free-parameter formula consistent
with $G_2 \supset A_2$ structure at sub-ppm precision.

\item[\textnormal{(ii)}]
Every constant in $\Rinf$ is a TOE primitive or integer: no free
parameter appears.

\item[\textnormal{(iii)}]
$|\Rinf - T_{\rm CODATA}|/\sigma_{\rm CODATA} = 0.500 < 1$.
The formula lies within one measurement uncertainty of CODATA-2018.

\item[\textnormal{(iv)}]
The seven results P1--P7 collectively account for every structural
element of $\Rinf$: prefactor (P1), first correction numerator (P2),
uniqueness of $\hv$ (P3), $k$-independent ratio (P4), series
factorisation (P5), exponent source (P6, conditional), and CODATA
agreement (P7).

\item[\textnormal{(v)}]
Axiom~\textup{\ref{ax:ID}} is \emph{consistent with} R1--R8 and is not
contradicted by any constraint in Paper~18.  Whether it follows from
R1--R8 is an open proof obligation (Open Problem~\ref{op:axID}).
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Part~(i) follows from the uniqueness argument: the series
$207\sum_{k=0}^\infty(-\rr)^k T_0$ is uniquely determined by P1--P5
given Axiom~ID; its closed sum is~$\Rinf$ (P213, P215).

Part~(ii) is immediate from the definitions: $\pi$ is TOE-primitive
(P209); $\hv=3$ is the dual Coxeter number of $A_2$, a Lie-algebraic
integer; $207 = 14\cdot15-3$ is fixed by $\dim(G_2)$ and $\hv$ (P211).

Part~(iii) is a numerical computation at \texttt{mp.dps}=60 confirmed
by \texttt{verify\_P218.py} (P7 / P216).

Part~(iv) is the content of the programme summary in Section~\ref{sec:summary}.

Part~(v) follows from Section~\ref{sec:axiomID}: the argument for and
against is laid out there; the conditional declaration follows.
\end{proof}

\begin{corollary}[Uniqueness of assembly]
If Axiom~\textup{\ref{ax:ID}} is established as a consequence of R1--R8,
then Theorem~\ref{thm:P18T2} holds unconditionally: R1--R8 admit a
unique admissible assembly of $\Oop$.
\end{corollary}

%─────────────────────────────────────────────────────────────────────────────
\section{Remaining Open Problems}
\label{sec:open}
%─────────────────────────────────────────────────────────────────────────────

Three open problems survive the closure of the programme.

\begin{openprob}[OP-AxID: Derivation of Axiom~ID from R1--R8]\label{op:axID}
Show that the Root--Lepton Energy Identification (Axiom~\ref{ax:ID})
follows from the coverage requirements R1--R8 of Paper~18, specifically
from the spectral resolution of the master operator $\Oop$ on $S^3$.
This requires either (a) computing the degeneracy splitting that gives
lepton mass ratios from the eigenvalue spectrum of $\Oop$, closing the
factor-66 discrepancy noted in Paper~18 Remark~4.3; or (b) a direct
representation-theoretic argument showing that the only $G_2 \supset A_2$
sub-algebra assignment consistent with R1--R8 is the short-root
$\leftrightarrow$ lepton identification.
Resolution would upgrade P6 from \textsc{conjectured} to
\textsc{derived} and Theorem~P218.1 from conditional to unconditional.
\end{openprob}

\begin{openprob}[OP-rSinf-100: Near-miss $\rr\cdot\Sinf \approx c/100$]
\label{op:rSinf}
Let $c = \delta_\infty/207$ and $\varepsilon = c/(\rr\cdot\Sinf) - 1/100$.
Numerically,
\[
  \varepsilon \;\approx\; -2.204\times10^{-7}.
\]
PSLQ at \texttt{mp.dps}=60 finds no integer relation for $\varepsilon$
in a 7-element natural basis.  The near-miss $\rr\cdot\Sinf \approx c/100$
is therefore either a deep identity at higher order or a numerical
coincidence.  Resolve which, or find the exact algebraic form of
$\varepsilon$.
\end{openprob}

\begin{openprob}[OP-G2A2-lattice: Adjacency resolvent mismatch $\times\,97.7$]
\label{op:lattice}
The unweighted $A_2$ adjacency resolvent evaluated at the TOE energy
scale $\lambda = \aI/\Ee$ gives a value $\approx 97.7$ times larger
than $\Sinf$ (P215, Section~5).  A weight-lattice or holonomy integral
over the $A_2$ face that produces $\Sinf$ exactly from the adjacency
data of the $G_2/A_2$ coset has not been found.  Identifying this
mechanism would close OP-G2A2-lattice and potentially provide an
independent derivation of $\Sinf$ from Lie-algebraic data alone.
\end{openprob}

\begin{openprob}[OP-Exactness: Future measurement]
\label{op:exact}
$\Rinf$ overestimates CODATA-2018 by $2.30\times10^{-6}$ ($0.011\ppm$).
The CODATA-2018 uncertainty is $\sigma = 4.6\times10^{-6}$, so the
discrepancy is $0.5\sigma$.  A future measurement of $m_\mu/m_e$ with
$\sigma < 10^{-6}$ would either confirm that $\Rinf$ is the exact
theoretical value (if the measurement moves toward $\Rinf$) or falsify
the formula (if the gap exceeds $3\sigma$).  The AMM and LEPTON-G
programmes are the most likely candidates for such precision.
\end{openprob}

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

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

\bibitem{paper29}
L.~F.~Vlegels.
\textit{Paper 29: [Global Minima and Exposure Mode].}
LumenOS TOE Corpus, 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.

\bibitem{P213}
L.~F.~Vlegels.
\textit{Addendum 213: $G_2$--$A_2$ Unified Mass-Ratio Theorem.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P215}
L.~F.~Vlegels.
\textit{Addendum 215: OP-G$_2$A$_2$-lattice.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P216}
L.~F.~Vlegels.
\textit{Addendum 216: OP-delta-inf.}
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.

\end{thebibliography}

\end{document}
