% Paper 36: The Fine-Structure Constant
% PROPOSED CANON (assembled 2026-06-10 from A279, A283, A284, A288,
% A291; subject to review/rollback). The flagship: the measured
% alpha^-1 derived from the geometric value through the corpus's own
% oscillation, ergodically dressed, at 0.43 ppb — with the
% falsification direction stated. Verifiers: verify_P279/283/284/288/
% 291.py (all green at filing).
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\title{Paper 36: The Fine-Structure Constant\\
\large The $\alpha$-comma, the ergodic dressing,\\
and the half-ppb confrontation}
\author{L\'{e}on Fernando Vlegels}
\date{2026-06-10}

\begin{document}
\maketitle

\begin{abstract}
The corpus's seed identity reads $\alpha^{-1} = \Omega = 4\pi^3 +
\pi^2 + \pi$. This paper states, as canon, what that identity
precisely is and is not. \textbf{It is not exact}
($\Omega = 137.0363038$ exceeds CODATA-2018's $137.035999084(21)$ by
$3.047\times10^{-4}$, i.e.\ $2.2234$ ppm, about $1.45\times10^4$
experimental standard deviations), \textbf{and by the corpus's own
theorem it must not be}: an exact match would be the canon's only
exact physical closure, contradicting the Necessity of Detuning
(A267). The residual is the \emph{$\alpha$-comma}, and it is derived.
The chain: (1)~P03's density oscillates with relative amplitude
$\kappa = \alpha^{5/4}$ about $\Omega$; (2)~the oscillation's
variance is apportioned across the three layers by their measure
fractions $(4\pi^3, \pi^2, \pi)/\Omega$, proven as the exact
ergodic limit of the layer-projected mode energies
(Theorem~\ref{thm:ergodic}; closed forms at every mode number, edge
integral $n$-independent), with the data selecting the ergodic over
the fundamental-mode reading at $31\times$ the CODATA edition
spread; (3)~the edge share is unobserved: observation vanishes at
the universe boundary (A281) and the edge fraction $\pi/\Omega$ is
where it runs out (P04); (4)~a measurement time-averages the
remainder (linear mean; averaging exponent data-selected at
$n = 0.955 \approx 1$). The result, with zero adjustable parameters:
\begin{equation*}
\boxed{\;\alpha^{-1}_{\rm meas}
\;=\; \Omega\,\sqrt{1 - \kappa^2\Bigl(1 - \frac{\pi}{\Omega}\Bigr)}
\;=\; 137.035999236\;}
\end{equation*}
versus CODATA-2022's $137.035999177(21)$: $+0.43$ ppb,
$2.8\,\sigma_{\rm exp}$. The tension is owned, not absorbed, and the
direction analysis closes every internal escape: mode-mixture
corrections push the wrong way, so the value above is the chain's
unique prediction. \textbf{Falsification, pre-committed:} future
CODATA editions move toward $137.035999236$ or this paper's central
claim dies; a $0.1$-ppb measurement decides outright. Equivalently,
the comma is the first measurement of the corpus's oscillation
amplitude, $A = 0.9887(2)\,\kappa$, stable across CODATA editions to
$1.5\times10^{-4}$.
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction}
\label{sec:intro}

The fine-structure constant $\alpha$ sets the strength of the
electromagnetic coupling. Its inverse is the most precisely measured
constant in physics: CODATA-2018 reports
$\alpha^{-1} = 137.035999084(21)$ and CODATA-2022 reports
$137.035999177(21)$, each carrying an experimental standard
uncertainty $\sigma_{\rm exp} = 2.1\times10^{-8}$, about $0.15$ ppb
in relative terms. Any theory that proposes to derive this number
faces an unusually severe referee, and the corpus accepts the
appointment. The seed identity of the whole programme is
\begin{equation}
\alpha^{-1} \;=\; \Omega \;=\; 4\pi^3 + \pi^2 + \pi
\;=\; 137.0363038\ldots,
\label{eq:seed}
\end{equation}
the total measure of the three-layer arena: bulk $4\pi^3$, boundary
$\pi^2$, edge $\pi$.

This paper fixes, as canon, the precise status of
identity~\eqref{eq:seed}. Three things are claimed. First, the
identity is \emph{not} exact against the measured constant, and the
corpus's own central theorem (A267, the Necessity of Detuning)
requires that it not be; the $2.2$ ppm residual is typed and named
the $\alpha$-comma. Second, the comma's sign and magnitude are
derived from structure the corpus already possesses, with zero
adjustable parameters, as the time-average dressing of the
oscillation that P03 predicts for the coupling. Third, the resulting
closed form lands $0.43$ ppb from CODATA-2022, a residual of
$2.8\,\sigma_{\rm exp}$ that the chain owns rather than absorbs, and
the paper pre-commits the direction in which future measurements must
move.

Three things are equally not claimed. The identity is not claimed
exact; no chain link is claimed beyond its filed status (each is a
theorem, a canon result, or a data-selected reading, and the status
of each is stated where it is used); and the $2.8\sigma$ residual is
not claimed explained. The honesty conventions that govern the corpus
elsewhere (the look-elsewhere budget, the refusal to promote
unmotivated fits) govern here at full strength, on the number where
the temptation to relax them is largest.

The chain itself can be said in four sentences. The coupling is not a
constant but the time average of an oscillating quantity: P03's
density oscillates about its geometric value with relative amplitude
$\kappa = \alpha^{5/4}$. The variance of that oscillation is shared
among the three layers of the arena in proportion to their measures
$(4\pi^3, \pi^2, \pi)/\Omega$; this was once an assumption and is
proven here as the exact ergodic limit of the layer-projected mode
energies, with the data selecting the ergodic reading over the
fundamental-mode alternative at $31$ times the available resolution.
The edge's share is invisible, because observation vanishes at the
universe boundary (A281) and the edge fraction $\pi/\Omega$ is
precisely where it runs out (P04). A measurement therefore returns
the linear cycle average of the oscillation over the observed
sector, and the average of an oscillating coupling sits below its
geometric centre by half the observed variance.

The paper proceeds in the order of the chain.
Section~\ref{sec:comma} states the deviation and why exactness is
forbidden. Section~\ref{sec:dressing} derives the averaging dressing
and inverts the comma into a measurement of the oscillation
amplitude. Section~\ref{sec:amplitude} confronts the amplitude
correction with the corpus's candidate structures and records the
bracket that honesty required at that stage.
Section~\ref{sec:ergodic} proves the equipartition theorem that
closes the chain's one soft link and exhibits the data selection.
Section~\ref{sec:number} assembles the final number and its error
budget. Section~\ref{sec:falsification} states the falsification
conditions. Section~\ref{sec:corpus} places the result in the corpus,
and Section~\ref{sec:conclusion} concludes.

% ════════════════════════════════════════════════════════════════════
\section{The Comma}
\label{sec:comma}

\begin{proposition}[The deviation, stated precisely]
\label{prop:deviation}
$\Omega - \alpha^{-1}_{\rm CODATA18} = 3.04692\times10^{-4}$
($2.2234$ ppm relative, approximately
$1.45\times10^4\,\sigma_{\rm exp}$); against CODATA-2022
($137.035999177(21)$), $3.04599\times10^{-4}$ ($2.2228$ ppm). The
geometric identity is accurate to $2.2$ ppm and inexact at
$\sim 1.45\times10^4\,\sigma_{\rm exp}$ at every edition. (A279)
\end{proposition}

The deviation is enormous by experimental standards and small by
every other standard a derivation of $\alpha$ has historically been
held to. Both facts matter. The first means the bare
identity~\eqref{eq:seed} cannot be the measured constant; the second
means the residual is a definite, stable, two-edition-confirmed
quantity that a mechanism can aim at. The corpus names it the
\emph{$\alpha$-comma}, on the pattern of the musical commas that the
detuning theorems of the canon are built around: a small, structural,
necessary failure of closure.

\begin{remark}[Why exactness would falsify the corpus]
\label{rem:detuning}
A267 proves no corpus cycle closes exactly; every identification of a
geometric quantity with an externally fixed value carries a nonzero
residual, a comma. Had $\Omega$ matched the measured constant to
experimental precision, the canon's central detuning theorem would be
violated by its own seed: the corpus would contain exactly one exact
physical closure, sitting at its most external-facing point. The
$2.2$ ppm is therefore not an error to be explained away but a
magnitude to be derived, and the remainder of this paper derives it.
\end{remark}

Two readings are dismissed before the derivation begins, both from
A279. A QED-running interpretation fails directionally: $\Omega$
\emph{exceeds} the Thomson-limit value, and running below $m_e$
cannot raise $\alpha^{-1}$; whatever dressing connects the geometric
to the measured value, it is not textbook running. And the
small-combination scan that the registry discipline mandates returns
only fragile matches: $\kappa/7$ reproduces the 2018 deviation to
$8$ ppm but degrades to $300$ ppm against CODATA-2022, and a match
that cannot survive a CODATA revision is not physics (the fragility
is itself the verdict); $\kappa^2/2$ misses by $2.3\%$. Neither is
promoted. The $2.3\%$ near-miss of $\kappa^2/2$ is, however, the
trailhead: it is the leading term of the mechanism derived next.

% ════════════════════════════════════════════════════════════════════
\section{The Oscillation and the Dressing}
\label{sec:dressing}

P03 predicts that the coupling is not static. The density of the
self-lensing arena oscillates about its geometric value with relative
amplitude of order $\kappa = \alpha^{5/4}$; this is the same
oscillation whose frequency built the breath results of A266 through
A274. An instrument that measures $\alpha$ does not sample the
geometric centre $\Omega^{-1}$; it time-averages the instantaneous
coupling over many cycles. The dressing follows from that single
operational fact.

\begin{proposition}[Averaging dressing]
\label{prop:avg}
Let the instantaneous inverse coupling oscillate as
$\alpha^{-1}(\theta) = \Omega(1 + A\cos\theta)$, with $0 < A < 1$.
The linear cycle average of the coupling is
\begin{equation}
\langle\alpha\rangle
\;=\; \frac{1}{2\pi}\oint \frac{d\theta}{\Omega(1+A\cos\theta)}
\;=\; \frac{1}{\Omega\sqrt{1-A^2}},
\label{eq:average}
\end{equation}
so the operationally measured inverse coupling is
\begin{equation}
\alpha^{-1}_{\rm meas} \;=\; \Omega\sqrt{1-A^2} \;<\; \Omega,
\qquad
\text{deficit} \;=\; 1-\sqrt{1-A^2} \;=\; \frac{A^2}{2} + O(A^4).
\label{eq:deficit}
\end{equation}
The measured value must sit below the geometric one: the comma's sign
is forced. (A283; the quadrature
$\oint d\theta/(1+A\cos\theta) = 2\pi/\sqrt{1-A^2}$ is exact and
verified numerically at 40 digits.)
\end{proposition}

\begin{proof}
The integral is standard: for $|A| < 1$,
$\int_0^{2\pi} d\theta/(1+A\cos\theta) = 2\pi/\sqrt{1-A^2}$
(residue calculus on the unit circle, or the Weierstrass
substitution). Dividing by $2\pi\Omega$ gives
\eqref{eq:average}; the reported inverse coupling is the reciprocal
of the averaged coupling, which is \eqref{eq:deficit}. The deficit
expansion is the binomial series.
\end{proof}

The sign is the first nontrivial confrontation, and it lands: the
measured $\alpha^{-1}$ does sit below $\Omega$, at every CODATA
edition. The magnitude is the second.

\begin{proposition}[Confrontation and inversion]
\label{prop:measure}
With $A = \kappa = \alpha^{5/4}$, the predicted deficit is
$1 - \sqrt{1-\kappa^2} = 2.27448\times10^{-6}$, versus observed
$2.23344\times10^{-6}$ (CODATA-18) and $2.23276\times10^{-6}$
(CODATA-22): ratio $0.9776$, i.e.\ the bare $\kappa$-amplitude
prediction is $2.3\%$ high. Sign correct, mechanism corpus-native,
magnitude high by $2.3\%$. Inverting,
\begin{equation}
A_{\rm meas} \;=\; \kappa\,\sqrt{{\rm obs}/{\rm pred}}
\;=\; 0.988717\,\kappa\ \text{(CODATA-18)}, \qquad
0.988566\,\kappa\ \text{(CODATA-22)},
\label{eq:inversion}
\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. (A283)
\end{proposition}

The edition stability of \eqref{eq:inversion} deserves emphasis,
because it is the discriminant between structure and coincidence in
exactly the sense the registry discipline demands. The fragile
$\kappa/7$ flag swings by $290$ ppm between editions; the amplitude
reading moves by $1.5\times10^{-4}$. A quantity that holds still
while the experimental value underneath it is revised is behaving
like a physical parameter, not like a numerical accident.

One more degree of freedom in the averaging itself is fixed by the
data rather than by fiat. The linear mean used in
Proposition~\ref{prop:avg} is one member of the moment family
$\langle\alpha^n\rangle^{1/n}$, whose deficit at $A=\kappa$ is
$(n{+}1)A^2/4$. Inverting the observed deficit against this family
selects $n = 0.955 \approx 1$: the data pick the linear mean, the
simplest member. (A283)

What A283 left open is the $0.9887$. P03 gives $A \sim \kappa$ as an
order estimate (the arena's oscillation energy is
$\Delta E \approx 23\kappa\cdot 4\pi$), not as an identity, so the
$2.3\%$ excess is real and at that stage unexplained. The natural
temperings were checked and rejected: $A = \kappa\sqrt{1-\kappa}$
gives $0.99893\,\kappa$, off by a factor of ten in the correction;
$A = \kappa(1-\kappa)$ and $A = \kappa\sqrt{1-\kappa^2}$ fare no
better; none reaches $0.9887$, and none was promoted. The question
``derive the amplitude exactly'' was filed as OI-283-1, with the
measured, edition-stable $0.9887(2)$ as the target a derivation must
hit. The next two sections are the closure of that item.

% ════════════════════════════════════════════════════════════════════
\section{The Amplitude}
\label{sec:amplitude}

Write the required correction as $A = c\,\kappa$. The measurement
demands $c^2 = {\rm obs}/{\rm pred} = 0.977561$ (CODATA-18) and
$0.977263$ (CODATA-22), an edition spread of $3.0\times10^{-4}$. A284
ran the corpus-natural candidates against this requirement.

\begin{proposition}[The scan]
\label{prop:scan}
Among corpus-natural single corrections to $A = \kappa$, namely
$1-\pi/\Omega$, $1-1/44$, $1-\kappa$, $1-2\kappa$, $1-\alpha$,
$(1-\kappa)^{10}$, $1-\Delta E/\Omega$, and $31/32$, exactly two fall
within the CODATA edition spread of the required $c^2$:
\begin{center}
\begin{tabular}{lccc}
\toprule
candidate & value & vs 2018 & vs 2022 \\
\midrule
$1 - \pi/\Omega$ (edge exclusion) & $0.977075$ & $-4.9\times10^{-4}$ &
$-1.9\times10^{-4}$ \\
$1 - 1/44$ & $0.977273$ & $-2.9\times10^{-4}$ & $+1.0\times10^{-5}$ \\
\bottomrule
\end{tabular}
\end{center}
All others miss by $10^{-3}$ or worse. Verified at 40 digits. (A284)
\end{proposition}

The two survivors are not on equal footing, and the inequality is the
point. The \emph{edge exclusion} reading has independent physical
standing assembled entirely from filed results. A281 establishes that
observation vanishes at the universe boundary (derived twice over,
and selected by the mass data at $9.1\times$); P04's edge fraction
$\pi/\Omega = 2.2925\%$ is precisely the sector where observation
runs out. A measured average of the oscillating coupling would then
be collected over the observed fraction $1 - \pi/\Omega$ only, giving
deficit $= \tfrac{\kappa^2}{2}(1-\pi/\Omega)$ with no new constants.
``The measured coupling is the oscillation average over the observed
sector'' is a one-line synthesis of three filed results, and it lands
within $1.9\times10^{-4}$ of the CODATA-22 requirement.

The rival $1 - 1/44$ fits the 2022 edition more closely and has no
motivation at all: the integer 44 has no derivation anywhere in the
canon (the A44 Jordan cycle shares only the numeral, and the A255
discipline forbids the conflation). A284 therefore declined to
promote either candidate. When an unmotivated integer beats a
motivated structure inside the error bar, the honest statement is a
bracket, not a claim: \textbf{the correction factor is bracketed at
$c^2 = 0.9773 \pm 0.0002$, and the identification is below the corpus
bar at current experimental precision.} The discriminating
measurement was costed at the same time, and is taken up in
Section~\ref{sec:falsification}.

A288 then converted the motivated candidate from texture into a
chain. Four links: \emph{(1) oscillation}, the coupling oscillates
with total relative amplitude $\kappa$ about $\Omega$ (P03, canon);
\emph{(2) equipartition}, the oscillation's variance is apportioned
across the three layers by their measure fractions
$(4\pi^3, \pi^2, \pi)/\Omega$, at that stage a single named
assumption, natural for a mode paced by the corpus's one clock (A269)
but not yet a theorem; \emph{(3) edge unobserved}, observation has no
support on the edge sector (A281, P04), so a measurement samples the
bulk and boundary shares only, and the observed variance is
$\kappa^2(1-\pi/\Omega)$; \emph{(4) averaging}, the linear time
average of Proposition~\ref{prop:avg}, exponent data-selected.

\begin{proposition}[Deficit from the chain]
\label{prop:chain}
Links 1 through 4 jointly imply a relative deficit
$1 - \sqrt{1-\kappa^2(1-\pi/\Omega)} = 2.22234\times10^{-6}$, against
the observed $2.22276\times10^{-6}$ (CODATA-22; agreement
$1.9\times10^{-4}$) and $2.23344\times10^{-6}$ (CODATA-18;
$5.0\times10^{-4}$). Equivalently
$A = \kappa\sqrt{1-\pi/\Omega} = 0.98847\,\kappa$, inside A283's
measured $0.9887(2)\,\kappa$ at the $2\times10^{-4}$ level. The
residual mismatch is exactly the $2.8\,\sigma_{\rm exp}$ tension of
the closed-form prediction, owned by the chain, not absorbed. (A288)
\end{proposition}

On this basis A288 resolved the bracket into a ranking. Under the
corpus's own convention (A245/A180), a chained candidate at
$2.8\sigma$ outranks an unchained one at $0.5\sigma$, because chains
are falsifiable in more places than fits: every link of the edge
reading can be independently attacked, while $1-1/44$ offers nothing
to attack but its one number. Edge equipartition was adopted as the
working identification, $1/44$ demoted to coincidence-watch, and
OI-283-1 reduced to a single question: prove or refute Link 2. The
link was stated with referee precision so that a computation could
settle it. Writing the oscillation as a sum of layer components,
Link~2 asserts $\kappa^2 = \sum_\ell \kappa_\ell^2$ with
$\kappa_\ell^2 = \kappa^2 f_\ell$,
$f_\ell = (4\pi^3, \pi^2, \pi)/\Omega$; what would prove it is the
layer decomposition of P03's perturbation modes showing variance
shares equal to measure shares, and what would refute it is any other
shares, in which case the prediction moves and the CODATA comparison
re-runs mechanically. That computation is the next section.

% ════════════════════════════════════════════════════════════════════
\section{The Ergodic Theorem}
\label{sec:ergodic}

A291 performed the Link-2 computation, and the computation turned out
to be richer than the assumption it was asked to check: equipartition
by layer measure is not an identity mode by mode, but it is the exact
ergodic limit of the mode family, it fails decisively for the
fundamental alone, and the data distinguish the two readings at
$31$ times the available resolution.

The setting is P03's wave equation, whose $n$-th perturbation mode is
$\sin(n\pi x)$ on the unit interval, with the three layers entering
through their densities
\begin{equation}
\rho_{\rm edge}(x) = 2\pi x, \qquad
\rho_{\rm bnd}(x) = 3\pi^2 x^2, \qquad
\rho_{\rm blk}(x) = 16\pi^3 x^3,
\label{eq:densities}
\end{equation}
normalised so that the total masses
$\int_0^1 \rho_\ell\,dx = (\pi,\ \pi^2,\ 4\pi^3)$ are the layer
measures, summing to $\Omega$. The layer-projected energy of mode $n$
in layer $\ell$ is $E_\ell(n) = \int_0^1 \rho_\ell(x)
\sin^2(n\pi x)\,dx$.

\begin{proposition}[Closed forms]
\label{prop:closed}
For every $n \geq 1$,
\begin{equation}
E_{\rm edge}(n) = \frac{\pi}{2}, \qquad
E_{\rm bnd}(n) = \frac{\pi^2}{2} - \frac{3}{4n^2}, \qquad
E_{\rm blk}(n) = 2\pi^3 - \frac{6\pi}{n^2},
\label{eq:closedforms}
\end{equation}
with total $E(n) = \Omega/2 - (6\pi + 3/4)/n^2$. The edge integral is
$n$-independent: the edge hears every mode equally. (A291; verified
symbolically and by quadrature at 40 digits, at $n = 1, 2, 5, 50$.)
\end{proposition}

\begin{proof}
Write $\sin^2(n\pi x) = \tfrac12(1 - \cos(2\pi n x))$, so each
integral is half the layer mass minus half a cosine moment
$\int_0^1 x^p \cos(2\pi n x)\,dx$. Integration by parts at integer
$n$ gives these moments exactly: $0$ for $p=1$ (the boundary terms
cancel), $1/(2\pi^2 n^2)$ for $p=2$, and $3/(4\pi^2 n^2)$ for $p=3$
(the $1/n^4$ terms cancel identically). Multiplying by the density
prefactors $2\pi$, $3\pi^2$, $16\pi^3$ yields
\eqref{eq:closedforms}; summing gives the total. The $p=1$ moment
vanishing for every integer $n$ is the $n$-independence of the edge
integral.
\end{proof}

\begin{theorem}[Ergodic equipartition]
\label{thm:ergodic}
The layer shares of mode $n$,
$s_\ell(n) = E_\ell(n)\big/\sum_m E_m(n)$, satisfy
\begin{equation}
s_\ell(n) \;=\; f_\ell\,\bigl(1 + O(1/n^2)\bigr)
\;\xrightarrow[n\to\infty]{}\;
f_\ell \;=\; \frac{(4\pi^3,\ \pi^2,\ \pi)}{\Omega},
\label{eq:limit}
\end{equation}
with the limit equality exact. Hence for any ensemble of modes whose
weight is not concentrated at small $n$, in particular the exposure
ensemble of P29 and the time-ergodic state of A274, the oscillation's
variance is apportioned by layer measure: Link 2 of the chain,
previously an assumption, is the ergodic limit of an exact
computation. For the fundamental alone equipartition fails:
$s_{\rm edge}(1) = 0.03211$, $s_{\rm bnd}(1) = 0.08555$,
$s_{\rm blk}(1) = 0.88234$, an edge share $40\%$ high. (A291)
\end{theorem}

\begin{proof}
From Proposition~\ref{prop:closed}, $E_\ell(n)$ converges to
$(2\pi^3,\ \pi^2/2,\ \pi/2)$ and the total to $\Omega/2$, both with
$1/n^2$ corrections; the limit shares are
$(2\pi^3,\ \pi^2/2,\ \pi/2)\big/(\Omega/2) =
(4\pi^3,\ \pi^2,\ \pi)/\Omega = f_\ell$, an algebraic identity (for
the edge channel, $(\pi/2)/(\Omega/2) = \pi/\Omega$, and likewise for
the others). The convergence rate is read off the closed forms: the
edge numerator is constant while the denominator carries
$-(6\pi+3/4)/n^2$, so
$s_{\rm edge}(n) = f_{\rm edge}\,(1 + 2(6\pi + 3/4)/(\Omega n^2)
+ O(1/n^4))$, with leading coefficient
$2(6\pi+3/4)/\Omega = 0.286$; the bulk and boundary shares carry
compensating corrections. Any ensemble average
$\sum_n w_n\, s_\ell(n)$ with weight escaping small $n$ inherits the
limit. The fundamental values follow by direct evaluation of
\eqref{eq:closedforms} at $n=1$.
\end{proof}

Two features of the theorem carry the physics. First, the limit
equality is not approximate: the high-$n$ shares are the measure
fractions \emph{exactly}, so equipartition in the ergodic regime is
an algebraic fact about the arena, not a plausibility argument.
Second, the assumption was never trivially true: the fundamental
mode violates it by $40\%$ in the edge channel. A claim that could
not have failed is bookkeeping; this one could have failed, and for
the $n{=}1$ ensemble it does. That is what makes the next
proposition evidence.

\begin{proposition}[Selection at $31\times$ the edition spread]
\label{prop:select}
Inverting the measured deficits through the chain gives the required
unobserved share $x = 0.022439$ (CODATA-18) and $0.022737$
(CODATA-22), with edition spread $3.0\times10^{-4}$. Candidates: the
ergodic limit $\pi/\Omega = 0.022925$ deviates by $+1.9\times10^{-4}$
from CODATA-22, inside one spread; the fundamental-only share
$0.03211$ deviates by $+9.4\times10^{-3}$, which is $31$ spreads, and
is excluded. The measurement is decisively in the ergodic regime.
(A291)
\end{proposition}

The model selection is therefore not a choice the corpus makes; it is
a choice the data make, at $31$ times the resolution the CODATA
editions themselves provide. And the selected reading is the one the
corpus's own operating principles require. Exposure (P29, Pin 4)
says the kernel's states select trajectories through a fixed
architecture: the full spectrum is sampled, none of it trained into
dominance. Unique ergodicity (A274) says time averages equal space
averages along the breath. A fundamental-dominated dressing would
describe a system that has \emph{learned} its ground state, exactly
what the Training/Exposure theorem forbids. The data's $31\times$
preference for the ergodic share is thus also an experimental echo of
Pin 4: the first time the exposure-not-training principle has been
weighed by a precision measurement and preferred.

\begin{remark}[Direction: the residual cannot be mode mixture]
\label{rem:direction}
From the proof of Theorem~\ref{thm:ergodic},
$s_{\rm edge}(n) = f_{\rm edge}(1 + 0.286/n^2 + \cdots)$: every
finite-$n$ admixture pushes the edge share \emph{up}, above
$\pi/\Omega$. The standing $2.8\,\sigma_{\rm exp}$ residual of the
closed-form prediction requires the unobserved share to sit
\emph{below} $f_{\rm edge}$ ($x_{22} = 0.022737 < 0.022925$).
Therefore no spectral ensemble closes the residual: the strict
ergodic limit is the chain's unique prediction, the tension is
irreducible within the mechanism, and the test is wholly external.
(A291)
\end{remark}

This direction analysis is the load-bearing closure of the paper. A
mechanism with an internal knob can absorb any future disagreement
and is to that extent unfalsifiable; this mechanism's only candidate
knob, the mode content of the ensemble, moves the prediction the
wrong way for every possible setting. The chain has no second knob.

% ════════════════════════════════════════════════════════════════════
\section{The Number and Its Error Budget}
\label{sec:number}

\begin{corollary}[The prediction]
\label{cor:pred}
With every link in place,
\begin{equation}
\alpha^{-1}_{\rm meas}
\;=\; \Omega\,\sqrt{1 - \kappa^2\Bigl(1 - \frac{\pi}{\Omega}\Bigr)}
\;=\; 137.035999236,
\label{eq:prediction}
\end{equation}
parameter-free. Confrontation: $+5.9\times10^{-8}$ from CODATA-2022's
$137.035999177(21)$, i.e.\ $+0.43$ ppb, $2.8\,\sigma_{\rm exp}$.
(A284/A288/A291)
\end{corollary}

Every input is accounted for. $\Omega = 4\pi^3+\pi^2+\pi$ is the
arena's total measure; $\kappa = \alpha^{5/4} = \Omega^{-5/4}$ is
P03's oscillation amplitude; $\pi/\Omega$ is the edge measure
fraction of P04; the square root is the exact linear cycle average of
Proposition~\ref{prop:avg}. Nothing in \eqref{eq:prediction} was
fitted to the measured constant, and there is no continuous parameter
anywhere in the chain whose adjustment could move the output.

The error budget is correspondingly austere. On the experimental
side, each CODATA edition carries $\sigma_{\rm exp} =
2.1\times10^{-8}$; the 2018 and 2022 central values differ by
$9.3\times10^{-8}$, several times the per-edition uncertainty, which
is why this paper consistently confronts both editions and treats the
edition spread, not the quoted sigma, as the conservative resolution
of the comparison. On the theory side there is no error bar to
assign in the usual sense, because there is no parameter with an
uncertainty; instead each link carries a discrete status, and the
residual sits where the statuses meet:

\begin{center}
\begin{tabular}{llll}
\toprule
link & content & status & residual it leaves \\
\midrule
1 & amplitude $\kappa$ & canon (P03) & none claimed beyond order \\
2 & layer equipartition & theorem (ergodic limit, A291) &
exact in the limit \\
3 & edge unobserved & derived (A281/P04) & none \\
4 & linear average & exact (A283) & exponent $0.955 \approx 1$ \\
\bottomrule
\end{tabular}
\end{center}

The one number that summarises the remaining mismatch is the
$2.8\,\sigma_{\rm exp}$ of Corollary~\ref{cor:pred}, equivalently the
$1.9\times10^{-4}$ gap between the ergodic edge share and the
data-inverted one, equivalently the $2\times10^{-4}$ gap between
$\sqrt{1-\pi/\Omega} = 0.98847$ and the measured amplitude factor
$0.9887(2)$. These are the same statement in three unit systems, not
three independent tensions. The chain states it once and owns it:
within the mechanism it cannot be reduced
(Remark~\ref{rem:direction}), and outside the mechanism it is exactly
the kind of half-ppb question the next generation of measurements
exists to answer.

% ════════════════════════════════════════════════════════════════════
\section{Falsification, Pre-Committed}
\label{sec:falsification}

\begin{remark}[The direction test]
\label{rem:test}
If future CODATA editions move toward $137.035999236$, the chain
strengthens; if away, it dies, and this paper's central claim dies
with it. No re-dressing is admissible, because
Remark~\ref{rem:direction} proves the chain has no second knob: any
attempt to rescue a receding prediction by adjusting the ensemble
moves the number further from the data. The claim is dated
2026-06-10, before the data that will judge it. (A284, sharpened by
A291)
\end{remark}

\begin{remark}[The $0.1$-ppb discriminator]
\label{rem:discriminator}
The two scan survivors of Proposition~\ref{prop:scan} differ by
$2.0\times10^{-4}$ in $c^2$, i.e.\ $4.5\times10^{-10}$ relative in
$\alpha^{-1}$, which is about $3\,\sigma$ of current experimental
precision. A measurement of $\alpha$ at the $0.1$-ppb level, one
order beyond current, separates the chained edge prediction from the
unchained $1/44$ coincidence outright; a CODATA drift of one more
edition may already. Either outcome is informative: the chain
predicts $137.035999236$, the coincidence predicts the $1/44$ value,
and the corpus has stated in advance which it stands behind and why.
(A284)
\end{remark}

\begin{remark}[The comma as measurement]
\label{rem:measurement}
Inverted, the comma measures the corpus's oscillation amplitude:
$A = 0.988717\,\kappa$ (CODATA-18), $0.988566\,\kappa$ (CODATA-22),
edition-stable to $1.5\times10^{-4}$. Within the chain, the $0.9887$
\emph{is} $\sqrt{1-\pi/\Omega} = 0.98847$ to $2\times10^{-4}$: the
same $2.8\sigma$ stated once more in amplitude language. A future
edition that moves the measured amplitude toward $0.98847$ is the
direction test passing; one that moves it away is the test failing.
(A283/A288)
\end{remark}

The falsification posture is deliberately total. The corpus's
flagship number is also its most exposed: $\alpha^{-1}$ will be
remeasured, repeatedly, by instruments that owe the corpus nothing.
A derivation chained through five filed results, with no internal
degrees of freedom and a stated direction, either survives that
process or is dispatched by it. Both outcomes are worth more than an
unfalsifiable fit.

% ════════════════════════════════════════════════════════════════════
\section{Relation to the Corpus}
\label{sec:corpus}

The chain is assembled entirely from results that were filed for
other reasons, which is the strongest internal evidence that it is
structure rather than construction.

\emph{A267 (Necessity of Detuning)} supplies the frame: the comma
must exist, so its observation confirms rather than embarrasses the
seed identity. \emph{P03} supplies the oscillation and its amplitude
scale $\kappa = \alpha^{5/4}$; the same oscillation, through its
frequency, built the breath chain of A266 through A274 before it was
asked to dress $\alpha$. \emph{P04} supplies the three layers and
their measures, including the edge fraction $\pi/\Omega$; the same
fractions organise the corpus's mass and shell results.
\emph{A281} supplies the vanishing of observation at the boundary,
derived twice over and selected by the mass data at $9.1\times$,
before the comma had any need of it. \emph{P29 and Pin 4 (exposure,
not training)} and \emph{A274 (unique ergodicity)} supply the
ensemble in which Theorem~\ref{thm:ergodic}'s limit is the physical
reading; Proposition~\ref{prop:select} is the first time a precision
measurement has weighed that principle directly, and it prefers it
at $31\times$.

The comma also completes a programme. A277 through A283 audited every
small near-miss the corpus had ever flagged, and with the mechanism
of Section~\ref{sec:dressing} the comma watchlist closed empty: each
item derived (432, $G_1$), demoted (schisma, conversion residue),
refuted by reduction (quarter-comma), or typed with a mechanism (the
$\alpha$-comma, this paper). The registry item P000\_1\_c, opened at
the corpus's first contact with CODATA, traversed the full arc:
named and typed (A279), mechanised and measured (A283), bracketed
(A284), chained (A288), and closed by theorem and data selection
(A291, retiring OI-283-1). No other registry item touches
\eqref{eq:prediction}.

For the corpus's external face, the consequence is one honest
sentence: the geometric value exceeds the measured one by $2.2$ ppm,
the sign and size of the corpus's own predicted oscillation dressing,
averaged over the observed sector, agreeing with experiment at the
half-ppb level, with the residual $2.8\sigma$ owned and the
falsification direction on record.

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

The fine-structure constant enters the corpus twice: once as the seed
identity $\alpha^{-1} = \Omega$, and once as the most precise
measurement the chain must face. This paper fixes the relation
between the two. The identity is inexact by $2.2$ ppm, necessarily so
(A267); the inexactness is the time-average dressing of P03's
oscillation, with the edge share of the variance unobserved because
observation ends at the boundary; the layer apportionment of the
variance is a theorem in the ergodic limit, and the data select that
limit over its only rival reading at $31$ times the available
resolution. The closed form
$\alpha^{-1} = \Omega\sqrt{1-\kappa^2(1-\pi/\Omega)} =
137.035999236$ follows with zero adjustable parameters and stands
$0.43$ ppb from CODATA-2022.

What remains is not a calculation but a wait. The chain has no second
knob; the residual $2.8\,\sigma_{\rm exp}$ cannot be reduced from
inside; the next CODATA edition, and after it the first $0.1$-ppb
measurement of $\alpha$, will move toward $137.035999236$ or away
from it. The corpus has written down which, and when it said so.

\paragraph{Provenance and verification.} Assembled without new claims
from A279 (the comma named), A283 (mechanism; amplitude measured),
A284 (the bracket and the discriminator), A288 (the chain), A291
(the ergodic theorem and data selection). Verifiers
\texttt{verify\_P279/283/284/288/291.py}: $14 + 12 + 9 + 10 + 11$
checks, all green at filing.


\section*{Status (2026-06)}
Load-bearing spine: the fine-structure density integral. The layered identity $\alpha^{-1}=4\pi^3+\pi^2+\pi$ is the corpus's seed; its exactness (registry C-I-01) is the foundational seed identity, registered as not-derived rather than claimed proven (Addendum~400, disposition F). Current.

\end{document}
