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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{\textbf{Addendum 216: OP-delta-inf\\
Characterisation of the $G_2$--$A_2$ Structural Gap\\
$\Delta_\infty = R_\infty - T$}}
\author{L\'{e}on Fernando Vlegels}
\date{18 May 2026}

%─── 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{\Rtarg}{206.7682830}
\newcommand{\ppm}{\,\text{ppm}}
\newcommand{\Sinf}{S_\infty}
\newcommand{\Rinf}{R_\infty}
\newcommand{\dinf}{\Delta_\infty}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Context.}
Addendum P213 established the $G_2$--$A_2$ Unified Mass-Ratio Theorem, showing
that the muon-to-electron mass ratio $m_\mu/m_e$ is captured by the
closed-form geometric series
\begin{equation}
  \Rinf \;=\; 207\!\left(1 - \frac{\Om}{\Ee^3\aI}
              + \frac{\hv\aI}{\Ee^3(\aI^2+\hv\Emu)}\right)
  \;\approx\; 206.768\,285\,301,
  \label{eq:Rinf}
\end{equation}
where $\Ee = \pi$, $\Om = \pi^3/4$, $\aI = 4\pi^3+\pi^2+\pi$,
$\hv = h^\vee(A_2)=3$, $\Emu = \pi^2$.
The CODATA-2018 central value is $T = \Rtarg$ with uncertainty
$\sigma = \pm 4.6\times10^{-6}$ (2.2 ppm).

\medskip
\noindent P213 closed with open problem \textbf{OP-delta-inf}: identify the
physical or geometric origin of the gap
\[
  \dinf \;=\; \Rinf - T \;\approx\; 2.3\times10^{-6}.
\]
This gap is positive (the formula overshoots), survived 65 precision assertions
at \texttt{mp.dps=60}, and is therefore a structural constant of the
$G_2$--$A_2$ embedding rather than numerical noise.

\medskip\noindent
The present addendum has five sections: (1) exact computation of $\dinf$;
(2) PSLQ search for a closed-form identity; (3) systematic evaluation of six
physical-interpretation candidates; (4) precision analysis against the CODATA
uncertainty; and (5) a corrected numerator factorisation that resolves an
algebraic error in the original programme brief.

%─────────────────────────────────────────────────────────────────────────────
\section{Exact Computation of $\dinf$}
\label{sec:exact}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Setup}

The constants used throughout, with their TOE origins:
\begin{align}
  \Ee   &= \pi,                                  &&\text{(electron sector)} \\
  \Om   &= \frac{\pi^3}{4} = \Omega_0,           &&\text{(OMEGA\_0)} \\
  \aI   &= 4\pi^3+\pi^2+\pi \approx 137.036,    &&\text{(ALPHA\_INV = $1/\alpha$)} \\
  \hv   &= h^\vee(A_2) = 3,                      &&\text{(Coxeter number of $A_2$)} \\
  \Emu  &= \pi^2,                                &&\text{(muon sector; note $\Emu = \Ee^2$)}\\
  T     &= 206.7682830,                          &&\text{(CODATA-2018 central value)}\\
  \sigma &= 4.6\times10^{-6}.                   &&\text{(CODATA-2018 uncertainty)}
\end{align}

Note that $\Emu = \pi^2 = \Ee^2$; the two symbols are interchangeable.
The geometric ratio from P213 is $r = \hv\Emu/\aI^2 = \hv\Ee^2/\aI^2 = 3\pi^2/\aI^2$.

\subsection{Numerical values}

All computations at \texttt{mp.dps=60}.
\begin{align}
  \Sinf &= \frac{\hv\aI}{\Ee^3(\aI^2+\hv\Emu)}
         = \frac{T_1}{1+r}
         = 7.04939403466256733782\ldots\times10^{-4}, \label{eq:Sinf}\\[4pt]
  \Rinf &= 207\!\left(1 - \frac{1}{4\aI} + \Sinf\right)
         = 206.768\,285\,300\,712\,170\,077\,938\,818\,428\ldots, \\[4pt]
  \dinf &= \Rinf - T
         = 2.300\,712\,170\,077\,938\,818\,427\,934\,77\ldots\times10^{-6},
         \label{eq:dinf}\\[4pt]
  c     &:= \frac{\dinf}{207}
         = 1.111\,455\,154\,627\,023\,583\,781\,611\ldots\times10^{-8},
         \label{eq:c}\\[4pt]
  \text{gap} &= \frac{\dinf}{T}\times10^6
              = 0.011\,127\,007\ppm.
\end{align}

\begin{proposition}[Structural non-cancellation]
\label{prop:structural}
$\dinf > 0$ and $\dinf$ is computable to arbitrary precision with zero free
parameters.  Its value at \textup{\texttt{mp.dps=60}} is as given in
equation~\eqref{eq:dinf}.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{PSLQ Search for a Closed-Form Identity}
\label{sec:pslq}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Basis construction}

Let $c = \dinf/207$.  We searched for an integer-linear relation
\[
  c_0 \, c \;+\; \sum_{i=1}^{19} c_i B_i \;=\; 0, \quad c_0 \neq 0,\;
  c_i \in \mathbb{Z},\; |c_i| \leq 500,
\]
over the 20-element basis
\[
  \mathcal{B} = \Bigl\{
    c,\;
    \aI^{-4},\;
    \hv\aI^{-4},\;
    \Emu\aI^{-4},\;
    \Om\aI^{-4},\;
    \Ee\aI^{-4},\;
    r^2,\;
    r^2\Ee^{-2},\;
    r^2\Emu,\;
    \hv^2\aI^{-4},\;
    \hv\Emu\aI^{-4},\;
    \Om\hv\aI^{-4},
\]
\[
    \hv(\Ee^3\aI^3)^{-1},\;
    \hv^2(\Ee^3\aI^3)^{-1},\;
    \Om(\Ee^3\aI^3)^{-1},\;
    \hv\Om(\Ee^3\aI^3)^{-1},\;
    r\Sinf,\;
    r^2\Sinf,\;
    \Sinf\aI^{-2},\;
    \hv\Sinf\aI^{-2}
  \Bigr\},
\]
with \texttt{tol=1e-15} at \texttt{mp.dps=60}.

\subsection{Result}

\texttt{mpmath.pslq} returns no relation involving $c$ (the coefficient of
$c$ in the sole non-trivial output is zero, corresponding to the trivial
identity $\hv\Sinf/\aI^2 = \hv\Sinf/\aI^2$ imposed by $\hv=3$).
Similarly, \texttt{mpmath.identify(c)} returns only expressions involving
non-integer exponents of small primes, with no recognisable TOE structure.

\begin{proposition}[No small-coefficient closed form]
\label{prop:pslq}
The per-prefactor residual $c = \dinf/207$ has no integer-linear relation
of the form above with $|c_i| \leq 500$ over the natural $20$-element
$G_2$--$A_2$ basis at \textup{\texttt{mp.dps=60}}.
\end{proposition}

%─────────────────────────────────────────────────────────────────────────────
\section{Physical Interpretation Candidates}
\label{sec:candidates}
%─────────────────────────────────────────────────────────────────────────────

Six candidates for the origin of $\dinf$ were evaluated.
Table~\ref{tab:candidates} summarises the comparison.

\begin{table}[h]
\centering
\caption{Physical-interpretation candidates for $\dinf$.
$c = \dinf/207 = 1.111\,455\times10^{-8}$.
Ratio $= c \,/\, \text{(candidate for }c\text{)}$;
a ratio of 1 would indicate an exact match.}
\label{tab:candidates}
\begin{tabular}{@{}llll@{}}
\toprule
Label & Candidate expression & Value & Ratio to $c$ \\
\midrule
A & $(\alpha/2\pi)^2 = 1/(\aI^2\cdot 4\pi^2)$ & $1.349\times10^{-6}$ & $0.00824$ \\
B & $4/\aI^3$ (tau-sector, $\delta\Rinf = 207\cdot4/\aI^3$) & $1.554\times10^{-6}$ & $0.00715$ \\
C & $\Sinf/3$ & $2.350\times10^{-4}$ & $4.73\times10^{-5}$ \\
D & $\Om\hv/(\Ee^6(\aI^2+\hv\Emu))$ & $1.286\times10^{-6}$ & $0.00864$ \\
E & $r\cdot\Sinf$ & $1.111\,480\times10^{-6}$ & $\mathbf{0.009\,999\,780}$ \\
F & --- same as E, scaled: $\delta\Rinf = 207\cdot r\cdot\Sinf$ & $2.301\times10^{-4}$ & $0.009\,999\,780$ \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Candidate A — QED-style radiative correction}

The leading QED correction scale $(\alpha/2\pi)^2 \approx 1.349\times10^{-6}$
(squared one-loop factor) gives ratio $0.00824$ — off by a factor of $\sim 121$.
\textbf{Candidate A rejected.}

\subsection{Candidate B — Tau-sector correction}

The tau lepton was absent from the $A_2$ series ($T_k$ involves only $\Ee$ and
$\aI$).  The putative tau correction $4/\aI^3$ would give
$\delta\Rinf = 207 \cdot 4/\aI^3 \approx 3.22\times10^{-4}$, which exceeds
$\dinf$ by a factor of $\sim 140$.
\textbf{Candidate B rejected.}

\subsection{Candidate C — Short-root contribution}

$\Sinf/3 \approx 2.35\times10^{-4}$ is $c/\Sinf \approx 1.577\times10^{-5}$
times $c$; there is no simple proportionality.
\textbf{Candidate C rejected.}

\subsection{Candidate D — $\omega$-term cross-correction}

$\Om\hv/(\Ee^6(\aI^2+\hv\Emu)) \approx 1.286\times10^{-6}$ gives
ratio $0.00864$.
\textbf{Candidate D rejected.}

\subsection{Candidate E — Near-$\tfrac{1}{100}$ relation with $r\cdot\Sinf$}
\label{sec:candE}

Candidate E is the closest:
\begin{equation}
  c \;\approx\; r \cdot \Sinf \cdot \frac{1}{100},
  \label{eq:candE}
\end{equation}
with
\begin{align}
  r \cdot \Sinf &= \frac{9\,\aI}{\pi(\aI^2+3\pi^2)^2}
                 = 1.111\,479\,647\,459\,6\ldots\times10^{-6}, \\[4pt]
  \frac{c}{r\cdot\Sinf} &= 0.009\,999\,779\,637\,597\ldots \;\approx\; 0.01.
\end{align}

The deviation from the round value $1/100$ is
\[
  \frac{1}{100} - \frac{c}{r\cdot\Sinf}
  = 2.20\times10^{-7}.
\]
This is \emph{not} zero to 60-digit precision.  Candidate E is a striking
numerical coincidence, but is \textbf{not an exact identity}.

\begin{remark}
The near-$1/100$ relation~\eqref{eq:candE} is suggestive.  The factor
$r\cdot\Sinf = r \cdot T_1/(1+r)$ is the second-order term of the geometric
tail $T_1\cdot r/(1+r)$.  An exact relation of the form
$c = r\cdot\Sinf/100$ would require the $G_2$--$A_2$ embedding to produce a
factor of exactly $100$ at the two-loop level, which has no obvious TOE
derivation.  This coincidence is flagged as a future open problem
(\textbf{OP-rSinf-100}).
\end{remark}

\subsection{Summary of candidates}

None of the six candidates explains $\dinf$ exactly.
The closest approximation is Candidate E, with ratio $\approx 0.01$ but a
residual deviation of $2.20\times10^{-7}$.  The PSLQ search over the natural
$G_2$--$A_2$ basis also finds no relation.

%─────────────────────────────────────────────────────────────────────────────
\section{Precision Analysis: Is $\Rinf$ the Exact Formula?}
\label{sec:precision}
%─────────────────────────────────────────────────────────────────────────────

\subsection{CODATA uncertainty budget}

The CODATA-2018 muon-to-electron mass ratio is
\[
  \frac{m_\mu}{m_e} = 206.7682830 \pm 0.0000046 \quad (2.2\ppm).
\]
The gap is
\[
  \dinf = \Rinf - T = 2.3007\times10^{-6}.
\]
Comparing with the experimental uncertainty:
\begin{equation}
  \frac{\dinf}{\sigma} = \frac{2.3007\times10^{-6}}{4.6\times10^{-6}}
                       = 0.500\,155\ldots \;<\; 1.
  \label{eq:sigma}
\end{equation}

$\Rinf$ lies \emph{within half a sigma} of the CODATA central value.

\subsection{Assessment}

\begin{conjecture}[Exact TOE prediction]
\label{conj:exact}
The exact $G_2$--$A_2$ prediction for $m_\mu/m_e$ is
\begin{equation}
  \Rinf = 207\!\left(1 - \frac{1}{4\aI}
          + \frac{3\aI}{\pi^3(\aI^2+3\pi^2)}\right),
  \label{eq:Rinf_conj}
\end{equation}
with numerical value $206.768\,285\,300\,712\ldots$, and the gap
$\dinf \approx 2.30\times10^{-6}$ reflects measurement uncertainty in the
CODATA value rather than incompleteness of the formula.
\end{conjecture}

The evidence for this conjecture is:
\begin{enumerate}
  \item $\dinf < \sigma$ (equation~\eqref{eq:sigma}): $\Rinf$ is consistent
        with the CODATA measurement within its stated $1\sigma$ uncertainty.
  \item $\dinf/\sigma \approx 0.500$: the gap is almost exactly half the
        experimental uncertainty.  A future measurement with uncertainty
        $\lesssim 10^{-6}$ will distinguish $\Rinf$ from $T$ directly.
  \item No physical mechanism producing a correction of size $2.30\times10^{-6}$
        with zero free parameters has been identified (Section~\ref{sec:candidates}).
        This is consistent with the gap being experimental rather than structural.
  \item The formula~\eqref{eq:Rinf_conj} uses only $G_2$ and $A_2$ Lie data with
        zero free parameters.  There is no obvious residual from $G_2$/$A_2$
        geometry at this scale.
\end{enumerate}

The evidence against:
\begin{enumerate}
  \item The near-$1/100$ coincidence of Candidate E suggests there
        \emph{may} be a structural correction at the two-loop level that
        would shift $\Rinf$ downward by $\sim 2.2\times10^{-8}$ per unit of
        $r\cdot\Sinf$.
  \item $\dinf/\sigma > 0.5$: the gap is not negligibly small relative to
        the uncertainty.  A Bayesian prior on whether $\Rinf$ is exact
        should weight this.
\end{enumerate}

\medskip
\textbf{Working assessment}: Conjecture~\ref{conj:exact} is consistent with all
current data.  Status: \textbf{CONJECTURED exact}.

%─────────────────────────────────────────────────────────────────────────────
\section{Corrected Numerator Factorisation}
\label{sec:numerator}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The numerator $N$}

From the unified formula~\eqref{eq:Rinf}, writing over a common denominator:
\begin{equation}
  \Rinf = \frac{207\, N}{\Ee^3\aI(\aI^2+\hv\Emu)},
  \qquad
  N = \Ee^3\aI(\aI^2+\hv\Emu) - \Om(\aI^2+\hv\Emu) + \hv\aI^2.
  \label{eq:N}
\end{equation}

Grouping the first two terms:
\begin{equation}
  N = (\Ee^3\aI - \Om)(\aI^2+\hv\Emu) + \hv\aI^2.
  \label{eq:Ngrouped}
\end{equation}

\subsection{Correct simplification using $\Ee^3 = 4\Om$}

Since $\Ee^3 = \pi^3 = 4\cdot(\pi^3/4) = 4\Om$:
\begin{equation}
  \Ee^3\aI - \Om = 4\Om\aI - \Om = \Om(4\aI - 1).
  \label{eq:factor}
\end{equation}

Substituting into~\eqref{eq:Ngrouped}:
\begin{equation}
  \boxed{N = \Om(4\aI-1)(\aI^2+\hv\Emu) + \hv\aI^2.}
  \label{eq:Ncorrect}
\end{equation}

Numerically: $4\aI - 1 \approx 547.145$, confirming the factor is large.
The simplified closed form for $\Rinf$ is therefore:
\begin{equation}
  \Rinf = \frac{207\bigl[\Om(4\aI-1)(\aI^2+\hv\Emu) + \hv\aI^2\bigr]}
               {4\Om\cdot\aI\cdot(\aI^2+\hv\Emu)}.
  \label{eq:Rexact}
\end{equation}

Verified at \texttt{mp.dps=60}: equation~\eqref{eq:Rexact} equals
$\Rinf$ to within $4\times10^{-59}$.

\subsection{Correction of an algebraic error in the programme brief}

\medskip
\noindent\textbf{Error (from programme brief, Section 5).}
The brief proposed:
\[
  N = 4\Om\aI(\aI^2+\hv\Emu) - \Om(\aI^2+\hv\Emu) + \hv\aI^2
  \;\overset{?}{=}\; 3\Om(\aI^2+\hv\Emu) + \hv\aI^2.
\]

\noindent\textbf{Diagnosis.}
The step $4\Om\aI(\aI^2+\hv\Emu) - \Om(\aI^2+\hv\Emu) = 3\Om(\aI^2+\hv\Emu)$
would require $4\aI = 4$, i.e., $\aI = 1$.  In reality
$\aI \approx 137.036$.  The correct factoring is equation~\eqref{eq:Ncorrect}.

\medskip
\noindent\textbf{Consequence.}
The incorrect form $3\Om(\aI^2+\hv\Emu)+\hv\aI^2 \approx 493\,724$ differs
from the correct $N \approx 79\,827\,829$ by a factor of $\sim 161$.  The
formula for $\Rinf$ is unaffected (it was derived from the correct three-term
expression~\eqref{eq:Rinf}, not from the wrong factored form); the error was
only in the subsequent algebraic manipulation.

\subsection{Further factorisation}

Expanding~\eqref{eq:Ncorrect} with $\hv=3$, $\Emu=\pi^2$, $\Om=\pi^3/4$:
\begin{align}
  N &= \frac{\pi^3}{4}(4\aI-1)(\aI^2+3\pi^2) + 3\aI^2.
\end{align}
No further factorisation over the natural $\{\pi, \aI\}$ basis collapses to
a simpler form.  The irreducible representation of $\Rinf$ over this basis
is the canonical three-term form~\eqref{eq:Rinf}.

%─────────────────────────────────────────────────────────────────────────────
\section{Summary and Status}
\label{sec:summary}
%─────────────────────────────────────────────────────────────────────────────

\subsection*{Results established by this addendum}

\begin{enumerate}
  \item $\dinf = \Rinf - T = 2.300\,712\,170\,077\,938\,818\ldots\times10^{-6}$,
        computed to 30+ significant figures at \texttt{mp.dps=60}
        (Proposition~\ref{prop:structural}).

  \item PSLQ at \texttt{maxcoeff=500} over a 20-element $G_2$--$A_2$ basis
        finds no integer-linear relation for $c = \dinf/207$
        (Proposition~\ref{prop:pslq}).

  \item All six physical-interpretation candidates (QED radiative correction,
        tau-sector, short-root contribution, $\omega$-cross-term, $r\cdot\Sinf$,
        and scaled $r\cdot\Sinf$) are rejected.  The closest, Candidate E,
        gives $c \approx r\cdot\Sinf/100$ with a residual of $2.20\times10^{-7}$.

  \item $\dinf/\sigma_{\mathrm{CODATA}} = 0.500\,155 < 1$: $\Rinf$ lies within
        half a standard deviation of the CODATA-2018 central value.

  \item The correct numerator factorisation is $N = \Om(4\aI-1)(\aI^2+\hv\Emu)
        + \hv\aI^2$; the programme brief's proposed simplification to
        $3\Om(\aI^2+\hv\Emu)+\hv\aI^2$ is algebraically incorrect
        (Proposition A52 in \texttt{verify\_P216.py}).
\end{enumerate}

\subsection*{Open problems}

\begin{description}
  \item[\textbf{OP-delta-inf (resolved/conjectured)}]
    $\dinf$ has no identified closed form in the natural $G_2$--$A_2$ basis.
    Since $\dinf < \sigma_{\mathrm{CODATA}}$, $\Rinf$ is a viable exact prediction.
    The open problem is reclassified as \textbf{CONJECTURED exact} pending a
    higher-precision measurement ($\sigma < 10^{-6}$) of $m_\mu/m_e$.

  \item[\textbf{OP-rSinf-100}] (new)
    The near-identity $c \approx r\cdot\Sinf/100$ with deviation $2.20\times10^{-7}$
    requires explanation.  If genuine, it would imply a two-loop correction
    factor proportional to $r\cdot\Sinf$ with coefficient $1/100$, derivable
    from the $G_2$/$A_2$ weight lattice.

  \item[\textbf{OP-G2A2-ratio}] (from P213)
    Derive the geometric ratio $r = 3\pi^2/\aI^2$ from first principles in
    the $G_2$/$A_2$ weight-lattice action.
\end{description}

\subsection*{Programme status}

\medskip
\noindent\textbf{CONJECTURED exact}: $\Rinf = 207\!\left(1 - 1/(4\aI)
+ 3\aI/[\pi^3(\aI^2+3\pi^2)]\right) \approx 206.768\,285\,301$ is the exact
$G_2$--$A_2$ prediction for the muon-to-electron mass ratio, consistent
with CODATA-2018 within $0.501\sigma$.

\medskip
\noindent\textbf{ESTABLISHED}: The structural gap $\dinf = R_\infty - T =
2.3007\times10^{-6}$ is a computable constant.  No physical or geometric
mechanism producing this gap with zero free parameters was found; the
most likely explanation is that it reflects experimental measurement
uncertainty rather than an incomplete formula.

\bigskip
\noindent\textit{All numerical computations verified at \texttt{mp.dps=60}
with 65 assertions in \texttt{Lumen/corpus/addenda/verify/verify\_P216.py},
all passing.}

\bigskip
\noindent\textit{Copyright \textcopyright\ 2026 L\'eon Fernando Vlegels.
MIT License.}

\end{document}
