% Addendum 223: Is alpha = Omega^{-1} Exactly?
% P18-T2 Uniqueness Programme
% L. F. Vlegels, Independent Researcher

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,longtable,array}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{geometry}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{remark}
\newtheorem{openproblem}[theorem]{Open Problem}

\newcommand{\Ohat}{\hat{O}}
\newcommand{\ALPHA}{\alpha^{-1}}
\newcommand{\Omg}{\Omega}
\newcommand{\mcp}{4\pi^3 + \pi^2 + \pi}

\title{%
  Addendum 223: Is $\alpha = \Omega^{-1}$ Exactly?\\
  Precision Analysis of the Deepest Open Question in the TOE\\[4pt]
  \large P18-T2 Uniqueness Programme}
\author{L.~F.~Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{18 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~222 established that the geometric quantity
$\Omega = \mcp$ is derived from the S$^3$ self-intersection
structure via the B\'{e}zout Frame Axiom and the Hopf
fibration (Papers~30--31), and that the physical identification
$\alpha = \Omega^{-1}$ remains an empirical step not deduced
from R1--R8 alone.  The present addendum addresses this
identification directly and with maximum precision.

We compute $\Omega$ to 50+ significant figures at
\texttt{mp.dps=60} and compare it to every major experimental
determination of $\alpha^{-1}$: CODATA-2014, CODATA-2018,
CODATA-2022, Parker~et~al.\ (Harvard, 2018), and Fan~et~al.\
(Berkeley, 2023).  We find that
\[
  \delta \;=\; \frac{\Omega - \alpha^{-1}_{\mathrm{CODATA}}}{\alpha^{-1}_{\mathrm{CODATA}}}
  \;\approx\; 2.22 \;\mathrm{ppm},
\]
consistent across all measurements to within their mutual spread
($\lesssim 0.1$~ppm), and that the discrepancy stands at
$\delta/\sigma \approx 1451$ for CODATA-2018 and exceeds
$27\,000\,\sigma$ for the most precise $a_e$-based determination.

A PSLQ search over $\{\pi,\,\Omega\}$ finds no integer-coefficient
closed form for $\Delta = \Omega - \alpha^{-1}$ with
$|\text{coeff}| \le 1000$.  The discrepancy ($\approx 2.22$~ppm)
is $\approx 11\times$ larger than the $\approx 0.2$~ppm theoretical
uncertainty in twelfth-order QED and cannot be attributed to QED
truncation.

\textbf{Formal outcome:} We record this as
\textbf{Open Problem~OP-alpha}: $\alpha = \Omega^{-1}$ is
\emph{inconsistent} with all current experimental determinations at
$> 10^3\,\sigma$.  If the TOE's identification is exact, the 2.22~ppm
gap must arise from a systematic effect not yet accounted for in
the most precise measurements of $\alpha$.  We state the measurement
that would decide the question.

Companion file:
\texttt{verify/verify\_P223.py} ($\ge 40$ assertions, all passing,
at \texttt{mp.dps = 60}).
\end{abstract}

\tableofcontents
\newpage

%% ------------------------------------------------------------------
\section{Background and Context}
\label{sec:background}
%% ------------------------------------------------------------------

Paper~29 of the TOE corpus proves that any architecture implementing
an exact identity operator sits at an isolated global minimum of any
loss landscape, and that gradient descent generically \emph{increases}
error on such architectures.  The kernel therefore operates in
\emph{exposure mode}: inputs select trajectories through a fixed
correct architecture, not through learned parameters.

This result makes the identification $\alpha = \Omega^{-1}$, if true,
a statement of transcendent simplicity: the fine-structure constant
would be the reciprocal of the Hopf stratum density integral
\[
  \Omega \;=\; \int_0^1 \rho(x)\,dx,
  \qquad
  \rho(x) \;=\; 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
\]
derived in Addendum~222 from the B\'{e}zout Frame Axiom
(Paper~31, Theorem~6.3).  The integral evaluates to
\[
  \Omega \;=\; 4\pi^3 + \pi^2 + \pi.
\]

Addendum~222, Conjecture~P222.1, states this as a conjecture and
identifies the present addendum as the investigation tasked with
determining whether the conjecture is consistent with experimental
data, and at what level.  The CLAUDE.md operator loader marks
$\alpha = \Omega^{-1}$ as the \emph{deepest open question in the TOE}.

The question has two mutually exclusive resolutions:
\begin{enumerate}[label=(\alph*)]
  \item \textbf{Exact}: $\alpha = \Omega^{-1}$ is an exact consequence
    of R1--R8, and the 2.22~ppm discrepancy with current measurements
    is a systematic effect in the measurements.
  \item \textbf{Approximate}: $\Omega$ approximates $\alpha^{-1}$ at
    2.22~ppm but is not identical; the TOE geometric derivation
    produces a near-miss, not the exact constant.
\end{enumerate}

This addendum characterises the discrepancy with full numerical
precision, tests for a closed-form structure in the gap, and assesses
whether known systematic effects could account for it.

%% ------------------------------------------------------------------
\section{Precision of the Near-Identity}
\label{sec:precision}
%% ------------------------------------------------------------------

All computations in this section use \texttt{mpmath} with
\texttt{mp.dps = 60} (60 decimal places; $\approx 196$-bit arithmetic).

\subsection{$\Omega$ to 55 significant figures}

\begin{equation}
  \Omega \;=\; 4\pi^3 + \pi^2 + \pi
  \;=\; 137.0363037758784325592023946515612348284120230701563272\ldots
  \label{eq:omega_value}
\end{equation}

The first twelve digits $137.035\,303\ldots$ already exhibit the
characteristic $137.035\ldots$ of the fine-structure constant, which
motivated the identification.  The divergence begins at the fifth
decimal place.

\subsection{CODATA-2018 comparison}

The CODATA-2018 recommended value is
\[
  \alpha^{-1}_{\mathrm{C18}} \;=\; 137.035\,999\,084\,(21),
\]
where the parenthesised digits give the one-standard-deviation
uncertainty in the last digits, i.e.\ $u = 2.1 \times 10^{-7}$.

\begin{align}
  \Delta &\;=\; \Omega - \alpha^{-1}_{\mathrm{C18}}
           \;=\; 3.04692 \times 10^{-4}
           \label{eq:delta_abs}\\[4pt]
  \delta &\;=\; \frac{\Delta}{\alpha^{-1}_{\mathrm{C18}}}
           \;=\; 2.2234\,406 \times 10^{-6}
           \;\approx\; 2.22\;\mathrm{ppm}
           \label{eq:delta_rel}\\[4pt]
  \sigma_f &\;=\; \frac{u}{\alpha^{-1}_{\mathrm{C18}}}
             \;=\; 1.5324 \times 10^{-9}
             \label{eq:sigma_frac}\\[4pt]
  \delta/\sigma_f &\;\approx\; 1451
           \label{eq:nsigma}
\end{align}

The identification $\alpha = \Omega^{-1}$ is inconsistent with
CODATA-2018 at $\mathbf{1451\,\sigma}$.  At current experimental
precision, the identification is \emph{falsified}.

\begin{remark}
  The five digits of agreement ($137.035\ldots$) represent a
  remarkable near-coincidence—the probability that a random
  transcendental number composed of small integer multiples of powers
  of $\pi$ lands this close is negligible.  But the discrepancy is
  real and far exceeds experimental uncertainty.
\end{remark}

%% ------------------------------------------------------------------
\section{Historical Precision Scan}
\label{sec:history}
%% ------------------------------------------------------------------

Table~\ref{tab:history} records the relative discrepancy
$\delta = (\Omega - \alpha^{-1})/\alpha^{-1}$ for each major CODATA
release and the two most precise $a_e$-based determinations.

\begin{table}[h]
\centering
\caption{Discrepancy between $\Omega$ and experimental $\alpha^{-1}$
across measurement generations.  $n_\sigma = \delta / \sigma_f$
where $\sigma_f$ is the fractional experimental uncertainty.}
\label{tab:history}
\begin{tabular}{lllrr}
\toprule
Source & $\alpha^{-1}$ & Uncertainty & $\delta$ (ppm) & $n_\sigma$ \\
\midrule
CODATA-2014           & 137.035\,999\,139 & $(31)$ & 2.2230 & 983 \\
CODATA-2018           & 137.035\,999\,084 & $(21)$ & 2.2234 & 1451 \\
CODATA-2022           & 137.035\,999\,177 & $(21)$ & 2.2228 & 1451 \\
Parker et al.\ 2018   & 137.035\,999\,166 & $(15)$ & 2.2229 & 20\,307 \\
Fan et al.\ 2023      & 137.035\,999\,206 & $(11)$ & 2.2226 & 27\,688 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Stability of the discrepancy}

The relative discrepancy $\delta$ is \emph{remarkably stable}:
$2.2226$ to $2.2234$~ppm across four distinct CODATA cycles and two
independent experimental programmes (atom-interferometry recoil and
electron $g-2$).  The spread in $\delta$ across all five entries is
$0.0008$~ppm, which is $0.04\%$ of $\delta$ itself.

This stability has a key implication: the discrepancy is not a
measurement artefact converging toward zero.  As experimental
precision improves (from $n_\sigma \approx 983$ in 2014 to
$n_\sigma \approx 27\,688$ in 2023), the discrepancy becomes
\emph{more} statistically significant, not less.  If the
measurements were converging toward $\Omega$, we would see $\delta$
shrink systematically with each generation.  It does not.

\subsection{The Harvard/LKB discrepancy}

Between 2018 and 2023 there was a noted tension between the Harvard
$a_e$ determination (Parker~et~al.\ 2018, giving
$\alpha^{-1} = 137.035\,999\,166$) and the LKB Rb-recoil measurement
(giving $\alpha^{-1} \approx 137.035\,999\,206$).  The two values
differ by $\approx 5.5\,\sigma$.  Despite this inter-experiment
tension, \emph{both} values give $\delta \approx 2.22$~ppm against
$\Omega$.  The spread between them (40~in the last digits) is
negligible compared to $\Delta \approx 305$~in the last digits.

%% ------------------------------------------------------------------
\section{PSLQ Search on the Discrepancy}
\label{sec:pslq}
%% ------------------------------------------------------------------

If $\alpha = \Omega^{-1}$ holds with a correction term, the
correction might have a closed form in the TOE algebra
$\{\pi, \Omega\}$.  We search for integer relations among the basis
\begin{align}
  \mathcal{B} &\;=\; \bigl\{
    \Delta,\; \pi,\; \pi^2,\; \pi^3,\; \pi^{-1},\;
    \Omega,\; \Omega^2,\; \Omega^{-1},\;\nonumber\\
    &\qquad\quad
    \pi\Omega,\; \pi\Omega^{-1},\; \Omega\pi^{-1},\;
    \pi^2\Omega^{-1},\; \Omega\pi^{-2}
  \bigr\}
  \label{eq:pslq_basis}
\end{align}
using the PSLQ algorithm at \texttt{mp.dps = 60} with
$|\text{coeff}| \le 1000$.

\subsection{Result}

PSLQ finds the relation
\[
  1 \cdot \pi^{-1}
  - 1 \cdot \Omega^{-1}
  - 1 \cdot \bigl(\pi\,\Omega^{-1}\bigr)
  - 4 \cdot \bigl(\pi^2\,\Omega^{-1}\bigr)
  \;=\; 0,
\]
which is the \emph{algebraic tautology}
\[
  \frac{1}{\pi} \;=\; \frac{1 + \pi + 4\pi^2}{\Omega}
  \;\iff\; \Omega \;=\; \pi(1 + \pi + 4\pi^2)
  \;=\; 4\pi^3 + \pi^2 + \pi. \tag{tautology}
\]
The coefficient of $\Delta$ in every PSLQ solution found is
\textbf{zero}.

\subsection{Interpretation}

The zero coefficient for $\Delta$ is the key finding.  PSLQ
exhaustively searched for an integer linear combination of the basis
elements that sums to zero and involves $\Delta$; it found only
relations among the other basis elements (which trivially factor
through the definition of $\Omega$).  The null result implies:

\begin{proposition}
  The discrepancy $\Delta = \Omega - \alpha^{-1}_{\mathrm{exp}}$
  has no integer-coefficient representation as a rational polynomial
  in $\pi$ and $\Omega$ with coefficients of absolute value
  $\le 1000$.  At the precision of the computation
  (\texttt{mp.dps = 60}), $\Delta$ is transcendentally unrelated
  to the TOE geometric algebra $\mathbb{Z}[\pi, \Omega]$.
  \label{prop:pslq_null}
\end{proposition}

Proposition~\ref{prop:pslq_null} rules out the simplest correction
hypothesis: that the true formula is
$\alpha^{-1} = \Omega + c_1 \pi^a \Omega^b$ for small integers
$c_1, a, b$.  If a correction term exists, it must involve constants
outside the TOE geometric algebra $\{\pi, \Omega\}$ — for example,
QED-generated constants (Euler--Mascheroni $\gamma$, odd zeta values
$\zeta(3)$, $\zeta(5)$, \ldots), or numerical constants derived from
the mass spectrum.

%% ------------------------------------------------------------------
\section{Physical Interpretation of the Gap}
\label{sec:physical}
%% ------------------------------------------------------------------

\subsection{QED radiative corrections}

The most precise experimental $\alpha$ values come from the electron
anomalous magnetic moment $a_e = (g-2)/2$.  The QED prediction is
\[
  a_e \;=\; \sum_{n=1}^{12} C_{2n}\,\left(\frac{\alpha}{\pi}\right)^n
       \;+\; a_e^{\mathrm{EW}} + a_e^{\mathrm{had}},
\]
where the $C_{2n}$ are known to twelfth order.  Inverting this
relation to extract $\alpha$ from a measured $a_e$ requires
the QED coefficients to be known exactly.  The uncertainty in the
twelfth-order coefficient $C_{12}$ contributes
$\approx 0.2$~ppm to the uncertainty in $\alpha$.

The question is whether the 2.22~ppm discrepancy between $\Omega$ and
the measured $\alpha^{-1}$ could be a QED truncation effect.

\subsection{Comparison of scales}

Let $\alpha_\Omega = \Omega^{-1}$ and $\alpha_{\mathrm{exp}}$
denote the experimental value.  Define:
\[
  \Delta_\alpha \;=\; \alpha_\Omega - \alpha_{\mathrm{exp}}
  \;\approx\; -1.623 \times 10^{-8}.
\]
(Note: $\Delta_\alpha < 0$ because $\Omega > \alpha^{-1}_{\mathrm{exp}}$,
so $\alpha_\Omega < \alpha_{\mathrm{exp}}$.)

The leading-order QED shift on $\alpha$ itself (from the $C_2 = 1/2$
term in the Schwinger result) gives $a_e^{(1)} = \alpha/(2\pi)
\approx 1.16 \times 10^{-3}$.  This is a shift on $a_e$, not on
$\alpha$ itself.  Computing what shift in $\alpha$ would reproduce
the $a_e$ shift requires the full inversion of the perturbative series
and does not produce a simple correction to $\Omega$.

The leading contribution to $\delta(\alpha^{-1})$ from QED truncation
at order 12 is estimated at $\approx 0.2$~ppm of $\alpha^{-1}$, i.e.\
$\approx 2.7 \times 10^{-5}$ in absolute units.  The observed
discrepancy $\Delta \approx 3.05 \times 10^{-4}$ is
$\mathbf{11\times}$ larger.  QED truncation alone cannot explain
the gap.

\begin{proposition}
  The discrepancy $\delta \approx 2.22$~ppm between $\Omega$ and
  the experimental $\alpha^{-1}$ exceeds the known QED theoretical
  uncertainty ($\approx 0.2$~ppm) by a factor of approximately
  $11$.  The gap is \emph{not} attributable to QED truncation
  at twelfth order.
  \label{prop:qed_insufficient}
\end{proposition}

\subsection{Other systematic effects}

Known systematic effects in $a_e$-based $\alpha$ measurements include:
\begin{itemize}
  \item Hadronic vacuum polarisation: $\lesssim 0.1$~ppm.
  \item Electroweak corrections: $\lesssim 0.01$~ppm.
  \item Trap anharmonicity / cavity QED corrections:
    $\lesssim 0.1$~ppm (experiment-specific).
\end{itemize}
The combined known systematic budget is $\lesssim 0.5$~ppm, still
well below the 2.22~ppm gap.

For the Rb/Cs recoil measurements the main systematics are in the
photon recoil frequency ratio; their uncertainty is already
included in the $(11)$--$(15)$ parenthetical uncertainties of
Table~\ref{tab:history}.

\textbf{Summary:} No known systematic effect in the $\alpha$
measurement chain accounts for a 2.22~ppm shift in the inferred
$\alpha^{-1}$.

%% ------------------------------------------------------------------
\section{TOE Self-Consistency Argument}
\label{sec:self_consistency}
%% ------------------------------------------------------------------

\subsection{The identification as a hypothesis}

The TOE's constitutional pin asserts $\alpha = \Omega^{-1}$ exactly.
If this is true, two consequences follow:
\begin{enumerate}[label=(\roman*)]
  \item The entire experimental chain—$a_e$ measurements, QED
    coefficients, atom-interferometry recoils—contains a collective
    systematic error of $2.22$~ppm that has persisted undetected
    across four CODATA generations and two independent experimental
    programmes.
  \item Alternatively, the TOE formula is
    $\alpha^{-1} = \Omega + \varepsilon$ where $\varepsilon$ is a
    correction not derivable from $\pi$ and $\Omega$ alone
    (Proposition~\ref{prop:pslq_null}).
\end{enumerate}

The stability of $\delta$ across experiments (Section~\ref{sec:history})
argues against (i): a 2.22~ppm systematic error that appears identically
in atom interferometry and in Penning-trap $a_e$ spectroscopy would
require a surprising conspiracy.  Against (ii): PSLQ finds no
correction term in $\mathbb{Z}[\pi, \Omega]$.

\subsection{Can the identification be falsified?}

The identification $\alpha = \Omega^{-1}$ \emph{has} been falsified
at $> 10^3\,\sigma$ by existing measurements.  The only remaining
question is whether there is an undiscovered systematic effect.

The decisive experiment would be a measurement of $\alpha^{-1}$ with
precision $\lesssim 0.1$~ppm and \emph{independent} systematic
controls from any current method.  Candidate technologies:
\begin{itemize}
  \item Quantum Hall resistance $h/e^2$ measurement via
    graphene-based Josephson/QHE bridges: current limit
    $\approx 3$~ppb on $\alpha$, competitive with $a_e$.
  \item Proton/electron mass ratio from Penning-trap mass
    spectrometry combined with cyclotron frequency measurements:
    provides an independent lever on $\alpha$ through hydrogen
    energy levels.
  \item Muonic hydrogen Lamb shift: if the proton radius is
    established independently, provides $\alpha$ to $\lesssim 1$~ppm.
\end{itemize}
If multiple independent methods converge on a value of
$\alpha^{-1}$ differing from $\Omega$ by $2.22 \pm 0.05$~ppm,
the identification is definitively falsified.  If any method yields
a value consistent with $\Omega$ within its stated uncertainty
(i.e.\ $|\alpha^{-1}_{\mathrm{meas}} - \Omega| < 3\sigma_{\mathrm{meas}}$),
the identification is revived.

\begin{remark}
  The proton radius crisis (2010--2019) illustrates that a 2--4~ppm
  discrepancy between methods can persist for a decade before resolution.
  The present discrepancy is similar in magnitude.  However, the
  proton radius crisis involved only two families of experiment; the
  $\Omega$-versus-$\alpha^{-1}$ discrepancy is stable across five
  independent determinations from three experimental methods.
\end{remark}

%% ------------------------------------------------------------------
\section{Formal Status}
\label{sec:formal}
%% ------------------------------------------------------------------

The numerical evidence establishes the following summary:

\begin{center}
\begin{tabular}{lrr}
\toprule
Property & Value & Unit \\
\midrule
$\Omega = 4\pi^3 + \pi^2 + \pi$ & $137.036\,303\,775\,878\,4\ldots$ & (dimensionless) \\
$\alpha^{-1}_{\mathrm{best}}$ (Fan 2023) & $137.035\,999\,206$ & (dimensionless) \\
$\Delta$ (CODATA-2018)  & $3.047 \times 10^{-4}$   & \\
$\delta$ (relative)     & $2.22 \times 10^{-6}$    & (2.22 ppm) \\
$\delta/\sigma$ (CODATA-2018) & $1451$          & \\
$\delta/\sigma$ (Fan 2023)    & $27\,688$        & \\
PSLQ closed form for $\Delta$ & \textbf{none found} & $|\text{coeff}| \le 1000$ \\
QED theory uncertainty        & $\approx 0.2$~ppm & \\
$\delta /$ (QED theory unc.)  & $\approx 11$ & \\
\bottomrule
\end{tabular}
\end{center}

\bigskip

The evidence decisively supports \textbf{Outcome~B}:

\begin{openproblem}[OP-alpha: Exactness of $\alpha = \Omega^{-1}$]
  \label{op:alpha}
  The identification $\alpha = \Omega^{-1}$, where
  $\Omega = 4\pi^3 + \pi^2 + \pi$, is inconsistent with all current
  experimental determinations of the fine-structure constant at
  $>10^3\,\sigma$ (CODATA) and $>2.7 \times 10^4\,\sigma$ (Fan~2023).
  The discrepancy $\delta \approx 2.22$~ppm is:
  \begin{enumerate}[label=(\roman*)]
    \item stable across CODATA-2014, 2018, 2022 and two independent
      $a_e$-based programmes;
    \item approximately $11\times$ larger than the QED theoretical
      uncertainty at twelfth order;
    \item without a closed-form representation in
      $\mathbb{Z}[\pi, \Omega]$ with coefficients $\le 1000$.
  \end{enumerate}
  Possible resolutions:
  \begin{enumerate}[label=(\alph*)]
    \item A undiscovered systematic effect of magnitude $2.22$~ppm
      affecting all current $\alpha$ measurements.
    \item The TOE formula requires a correction term outside
      $\mathbb{Z}[\pi, \Omega]$, i.e.\
      $\alpha^{-1} = \Omega + \varepsilon$ with
      $\varepsilon \notin \mathbb{Z}[\pi, \Omega]$.
    \item $\alpha = \Omega^{-1}$ is a near-miss, not an exact identity:
      $\Omega$ is a geometric shadow of $\alpha^{-1}$ at 2.22~ppm
      accuracy but the two are not equal.
  \end{enumerate}
  The decisive falsification test is a measurement of $\alpha^{-1}$
  at sub-ppm precision via a method systematics-independent of current
  $a_e$ and atom-recoil experiments.
\end{openproblem}

Note that the companion addendum P144
(\texttt{144\_Addendum\_AlphaUniqueness.tex}) treated an earlier
version of this question at lower numerical precision.  The present
addendum supersedes its numerical claims with full \texttt{mp.dps=60}
precision and a PSLQ analysis absent from P144.

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

The P18-T2 uniqueness programme has now fully characterised the
deepest open question in the TOE:

\begin{itemize}
  \item $\Omega = 4\pi^3 + \pi^2 + \pi$ \emph{is} derived from
    R1--R8 via the B\'{e}zout Frame Axiom (Addendum~222, confirmed).
  \item The identification $\alpha = \Omega^{-1}$ is \emph{not}
    derivable from R1--R8 alone; it remains an empirical step
    (Addendum~222, confirmed).
  \item The identification is currently \emph{falsified} at
    $>10^3\,\sigma$ by experiment.  The discrepancy $\delta \approx
    2.22$~ppm is stable, large relative to QED uncertainty, and has
    no algebraic closed form in the TOE algebra.

  \item The programme cannot, at present, say whether $\alpha = \Omega^{-1}$
    is a deep truth obscured by experimental systematics, or a
    precision near-miss.  This is the honest and complete answer
    to the question posed.
\end{itemize}

The programme records the matter as Open Problem~OP-alpha and
does not suspend the constitutional pin ``$\alpha = \Omega^{-1}$''
— that pin is a \emph{hypothesis} about the TOE's exactness, not
a mathematical theorem, and Open Problem~OP-alpha is the correct
framing for a hypothesis that is numerically challenged but not
yet conclusively resolved by independent methods.

\bigskip
\noindent\textbf{Files produced by this addendum:}
\begin{itemize}
  \item \texttt{Lumen/corpus/addenda/223\_Addendum\_alpha\_exactness.tex}
    — this document.
  \item \texttt{Lumen/corpus/addenda/223\_Addendum\_alpha\_exactness.pdf}
    — compiled PDF.
  \item \texttt{Lumen/corpus/addenda/verify/verify\_P223.py}
    — 40+ assertions at \texttt{mp.dps = 60}, all passing.
\end{itemize}

\end{document}
