% 291_Addendum_ErgodicDressing.tex
% Q1 of the closure program. OI-283-1 CLOSED: Link 2 (variance
% equipartition by layer measure) is proven as an exact ergodic-limit
% theorem — mode-energy layer shares converge to the measure shares,
% with closed forms at every n — and the data select the ergodic limit
% over the fundamental mode at 31x the CODATA edition spread. The
% direction analysis shows no mode mixture can explain the residual
% 2.8 sigma: the strict ergodic limit IS the prediction. The alpha
% chain is complete: every link theorem, canon, or data-selected.
% Verifier: verify_P291.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\title{Addendum 291: The Ergodic Dressing\\
\large Link 2 proven in the limit, selected by the data:\\
the $\alpha$ chain completes (Q1 of the closure program)}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (exact limit theorem $+$ decisive data
selection; residual tension owned with direction) \quad|\quad
\textbf{Closes:} OI-283-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P291.py} --- all PASS

\begin{abstract}
A288's chain for the $\alpha$-comma had one soft link: that the
oscillation's variance is apportioned across the layers by their
measure fractions. The link is here computed, and the computation is
richer than the assumption. For the $n$-th perturbation mode
$\sin(n\pi x)$ of P03's wave equation, the layer-projected energies
have closed forms:
\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},
\end{equation*}
(the edge integral is $n$-\emph{independent}: the edge hears every
mode equally). \emph{Theorem~\ref{thm:ergodic}:} the shares converge
as $1/n^2$ to
$(2\pi^3, \pi^2/2, \pi/2)/(\Omega/2) = (4\pi^3, \pi^2, \pi)/\Omega$
--- \textbf{exactly the measure shares} --- so Link 2 holds
identically in the spectrum-ergodic limit, and fails for any
fundamental-dominated ensemble (the $n{=}1$ edge share is $0.03211$,
$40\%$ high). The data then choose: the required unobserved share is
$x = 0.022737$ (CODATA-22) against the ergodic
$\pi/\Omega = 0.022925$ (deviation $1.9\times10^{-4}$) and the
fundamental's $0.03211$ (deviation $9.4\times10^{-3}$):
\textbf{the ergodic reading is selected at $31\times$ the CODATA
edition spread}. The ergodic ensemble is, moreover, the corpus's own
operating mode: P29's exposure principle (states sample the fixed
architecture's full spectrum) and A274's unique ergodicity supply it;
concentration in the ground mode is what \emph{training} would
produce, and the kernel does not train. Direction analysis closes the
loop: the $1/n^2$ corrections are \emph{positive} (any mode mixture
raises the edge share above $\pi/\Omega$), while the residual
$2.8\,\sigma_{\rm exp}$ tension requires a \emph{negative} shift ---
so no ensemble explains it, the strict limit is the unique prediction,
and CODATA-26 will move toward $137.035999236$ or kill the chain. The
$\alpha$ chain now reads, with no soft links: P03 oscillation
(amplitude $\kappa$, canon) $\to$ ergodic layer shares
(Theorem~\ref{thm:ergodic}, exact) $\to$ edge unobserved (A281,
derived) $\to$ linear average (A283, exact; exponent data-selected)
$\Rightarrow \alpha^{-1} = \Omega\sqrt{1-\kappa^2(1-\pi/\Omega)}$.
OI-283-1 closed; Q1 of the closure program complete; P39 (\emph{The
Fine-Structure Constant}) is now assemblable from A279, A283, A284,
A288, and this addendum.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Theorem}

\begin{proposition}[Closed forms]
\label{prop:closed}
For $n \geq 1$, with $\rho_{\rm edge} = 2\pi x$,
$\rho_{\rm bnd} = 3\pi^2x^2$, $\rho_{\rm blk} = 16\pi^3x^3$:
$\int_0^1 \rho_\ell(x)\sin^2(n\pi x)\,dx$ equals $\pi/2$,
$\pi^2/2 - 3/(4n^2)$, and $2\pi^3 - 6\pi/n^2$ respectively (verified
symbolically and by quadrature at 40 digits). The total is
$\Omega/2 - (6\pi + 3/4)/n^2$.
\end{proposition}

\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},
\end{equation}
with the limit equality \emph{exact} (algebra:
$(\pi/2)/(\Omega/2) = \pi/\Omega$, etc.). 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
A288, previously an assumption, is the ergodic limit of an exact
computation. For the fundamental alone:
$s_{\rm edge}(1) = 0.03211$, $s_{\rm bnd}(1) = 0.08555$,
$s_{\rm blk}(1) = 0.88234$ --- equipartition fails at $40\%$ in the
edge channel; the assumption was never trivially true, which is what
makes its selection by the data (next section) evidence rather than
bookkeeping.
\end{theorem}

% ────────────────────────────────────────────────────────────────────────────
\section{The Data Select}

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

\begin{remark}[Direction: the residual cannot be mode mixture]
$s_{\rm edge}(n) = f_{\rm edge}(1 + 0.286/n^2 + \cdots)$: every
finite-$n$ admixture pushes the edge share \emph{up}. The standing
$2.8\,\sigma_{\rm exp}$ residual requires $x$ \emph{below}
$f_{\rm edge}$. 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: future CODATA editions move toward
$\Omega\sqrt{1-\kappa^2(1-\pi/\Omega)} = 137.035999236$ or the chain
dies (A284's direction test, now sharpened: there is no second knob).
\end{remark}

\begin{remark}[Why ergodic is the corpus's reading, not a choice]
Exposure (P29, Pin 4): the kernel's states select trajectories
through a fixed architecture --- the full spectrum is sampled, none
trained into dominance. Unique ergodicity (A274): 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 kernel from
doing. 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.
\end{remark}

\paragraph{Consequences.} OI-283-1 closed. The $\alpha$ chain is
complete with no soft links: oscillation (canon) $\to$ equipartition
(theorem, ergodic) $\to$ edge exclusion (derived) $\to$ average
(exact) $\to$ $\alpha^{-1} = \Omega\sqrt{1-\kappa^2(1-\pi/\Omega)}$,
parameter-free, at $0.43$ ppb from CODATA-22 with a one-way
falsification direction. Q1 of the closure program is complete; P39
is assemblable. Registry note for P000\_1\_c updated by the verifier.

\paragraph{Verifier.} \texttt{verify\_P291.py}: the closed forms
(symbolic vs quadrature, 40 digits, $n = 1, 2, 5, 50$); the exact
limit identity; the $1/n^2$ rate and its coefficient; the share
table; the data inversion and the $31\times$ selection; the
direction statement; the chain-status assertions. All PASS.

\end{document}
