% 283_Addendum_AlphaCommaMechanism.tex
% OI-279-1: the alpha-comma typed as the time-average dressing of P03's
% predicted coupling oscillation. Predicted deficit 1-sqrt(1-kappa^2) =
% 2.2745 ppm vs observed 2.2234 ppm — sign correct, mechanism
% corpus-native, 2.3% high. Inverted: the comma is the first
% measurement of the oscillation amplitude, A = 0.9887(2) kappa,
% edition-stable. Plus: the 35/4 watchlist item retired; the comma
% watchlist closes empty. Verifier: verify_P283.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 283: The $\alpha$-Comma Mechanism\\
\large The measured deficit as the oscillation average;\\
the first measurement of $A$; the watchlist closes empty}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (mechanism identified, sign and magnitude
to $2.3\%$; amplitude measured, not derived; OI-283-1 filed) \quad|\quad
\textbf{Addresses:} OI-279-1; retires the $35/4$ watchlist item
\quad|\quad \textbf{Verifier:} \texttt{verify\_P283.py} --- all PASS

\begin{abstract}
A279 named the $\alpha$-comma --- $\Omega$ exceeds the measured
$\alpha^{-1}$ by $2.2234$ ppm --- and asked for its magnitude
(OI-279-1). The corpus contains the mechanism. P03 predicts the
coupling \emph{oscillates} with amplitude $A \sim \kappa$ about its
geometric value (the same oscillation whose frequency built A266--A274).
A measurement that time-averages the coupling over the cycle does not
return $\Omega^{-1}$: for instantaneous
$\alpha^{-1}(\theta) = \Omega(1 + A\cos\theta)$, the linear average is
$\langle\alpha\rangle = \Omega^{-1}(1-A^2)^{-1/2}$, so the measured
inverse coupling sits \emph{below} the geometric value,
\begin{equation*}
\alpha^{-1}_{\rm meas} \;=\; \Omega\sqrt{1-A^2}, \qquad
\text{deficit} \;=\; 1-\sqrt{1-A^2} \;\approx\; A^2/2.
\end{equation*}
With $A = \kappa$: predicted deficit $2.2745$ ppm versus observed
$2.2234$ ppm (CODATA-18) --- \textbf{sign correct, mechanism
corpus-native, magnitude high by $2.3\%$}. The reading inverts into the
corpus's first \emph{measurement} of its own oscillation amplitude:
$A = 0.98872\,\kappa$ (CODATA-18) and $0.98857\,\kappa$ (CODATA-22) ---
\emph{edition-stable to $1.5\times10^{-4}$}, in sharp contrast to the
fragile $\kappa/7$ flag A279 declined. The averaging exponent is also
data-selected: among moment averages
$\langle\alpha^n\rangle^{1/n}$ (deficit $(n{+}1)A^2/4$ at $A=\kappa$),
the observation picks $n = 0.955 \approx 1$ --- the linear mean, the
simplest member. P03 gave $A \sim \kappa$ as an order estimate only;
deriving the exact amplitude (why $0.9887$, whether
$A = \kappa\sqrt{1-\kappa}$-class temperings are involved --- the
nearest such candidates are checked and none passes) is filed as
\textbf{OI-283-1}. Separately, the last comma-watchlist item ($\Tb$ vs
$2^{35/4}$, $62$ ppm) is retired: the only decomposition of the
exponent, $35/4 = 7\cdot(5/4)$, routes through P017\_2's underived
factor of 7 --- a flag built on an open gap is debt squared --- and
$62$ ppm carries no force under the watchlist's look-elsewhere budget.
\textbf{The comma watchlist closes empty}: every previously flagged
near-miss is now either derived (432, $G_1$), demoted (schisma,
conversion residue), refuted (quarter-comma, reduced and retired), or
typed with a mechanism (the $\alpha$-comma, this addendum).
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Mechanism}

\begin{proposition}[Averaging dressing]
\label{prop:avg}
Let the instantaneous inverse coupling oscillate as
$\alpha^{-1}(\theta) = \Omega(1 + A\cos\theta)$ (P03: amplitude
$A \sim \kappa$, the oscillation of the self-lensing arena). The
cycle-averaged coupling is
$\langle\alpha\rangle = \frac{1}{2\pi}\oint
\frac{d\theta}{\Omega(1+A\cos\theta)} = \frac{1}{\Omega\sqrt{1-A^2}}$,
hence the operationally measured inverse coupling is
$\alpha^{-1}_{\rm meas} = \Omega\sqrt{1-A^2} < \Omega$: the measured
value \emph{must} sit below the geometric one. The deficit
$1 - \sqrt{1-A^2} = A^2/2 + O(A^4)$.
\end{proposition}

\begin{proposition}[Confrontation and inversion]
\label{prop:measure}
With $A = \kappa = \alpha^{5/4}$: deficit $= 2.27448\times10^{-6}$,
versus observed $2.23344\times10^{-6}$ (CODATA-18) and
$2.23276\times10^{-6}$ (CODATA-22) --- ratio $0.9776$, i.e.\ the
$\kappa$-amplitude prediction is $2.3\%$ high. Inverting,
\begin{equation}
A_{\rm meas} = \kappa\sqrt{\rm obs/pred}
= 0.988717\,\kappa\ \text{(CODATA-18)}, \quad
0.988566\,\kappa\ \text{(CODATA-22)},
\end{equation}
stable across editions to $1.5\times10^{-4}$. The $\alpha$-comma is
thereby a \emph{measurement of the P03 oscillation amplitude} ---
the first the corpus possesses. The averaging-family exponent is
data-selected at $n = 0.955 \approx 1$ (linear mean).
\end{proposition}

\begin{remark}[What is not claimed --- OI-283-1]
P03 predicts $A \sim \kappa$ as an order, not an identity; the $2.3\%$
excess is real and unexplained. Checked and rejected as exact:
$A = \kappa\sqrt{1-\kappa}$ ($0.99893\kappa$, off $10\times$),
$A = \kappa(1-\kappa)$, $A = \kappa\sqrt{1-\kappa^2}$ --- none reaches
$0.9887$. No candidate is promoted. \textbf{OI-283-1}: derive the
oscillation amplitude exactly from the arena ($\Delta E \approx
23\kappa\cdot4\pi$, P03) or the kernel; the target is now a measured
number with an edition-stability pedigree, which is what a derivation
needs to aim at. P000\_1\_c's registry note is updated: mechanism
typed, amplitude measured, exact derivation open.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Watchlist Closes}

\begin{remark}[Retiring $\Tb$ vs $2^{35/4}$]
The A277 reduction left one watchlist item: $\log_2\Tb = 8.749910$ vs
$35/4$ ($62$ ppm). The exponent's only available decomposition,
$35/4 = 7 \cdot (5/4)$, requires P017\_2's factor of 7 --- itself an
open registry item --- so any meaning assigned to $2^{35/4}$ presently
borrows from undischarged debt. A $62$ ppm coincidence between a
transcendental and a 2-power, under the watchlist's accumulated
look-elsewhere budget, does not survive on its own. Retired as
coincidence; revivable iff the factor 7 is derived \emph{and} the
product $7\cdot(5/4)$ acquires independent meaning. With this, the
comma watchlist is \textbf{empty}: derived (432, $G_1$), demoted
(schisma), refuted-by-reduction (quarter-comma), mechanism-typed
($\alpha$-comma). Every small gap the corpus once stared at now has a
name, a type, or a tombstone.
\end{remark}

\paragraph{Consequences.} OI-279-1 addressed (mechanism + measurement;
exact amplitude = OI-283-1). Watchlist empty. The SPARC/external paper
can now state the corpus's flagship number honestly in one sentence:
\emph{the geometric value exceeds the measured one by $2.2$ ppm, the
sign and size of the corpus's own predicted oscillation dressing, with
the residual $2.3\%$ an open measurement of the oscillation amplitude.}

\paragraph{Verifier.} \texttt{verify\_P283.py}: the exact average
$\oint d\theta/(1+A\cos\theta) = 2\pi/\sqrt{1-A^2}$ (numeric
quadrature); the deficit against both CODATA editions; the inversion
and its edition stability; the exponent selection; the rejected
amplitude candidates; the $35/4$ retirement numbers; watchlist
cardinality $0$. All PASS.

\end{document}
