% 288_Addendum_VarianceWeight.tex
% OI-283-1: the averaging weight derived as layer equipartition — the
% oscillation variance is apportioned by layer measure, the edge's
% share is unobserved (A281), so the measured variance is
% kappa^2 (1 - pi/Omega). The edge candidate of A284 acquires a
% four-link canon chain; 1/44 has none and is demoted. The conditional
% prediction stands at 0.43 ppb with one soft link named.
% Verifier: verify_P288.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\usepackage{booktabs}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{Addendum 288: The Variance Weight\\
\large Layer equipartition derives the edge factor;\\
the $\alpha$-comma chain completes to one soft link}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (four-link chain, three links canon, one
named assumption; rival demoted) \quad|\quad
\textbf{Addresses:} OI-283-1 (reduced to the equipartition link)
\quad|\quad \textbf{Verifier:} \texttt{verify\_P288.py} --- all PASS

\begin{abstract}
A284 left the amplitude correction bracketed between the motivated
$1-\pi/\Omega$ and the unmotivated $1-1/44$. The motivated candidate
is here upgraded from ``texture'' to a four-link chain, of which
three links are filed canon and the fourth is a single named
assumption. \emph{Link 1 (oscillation):} the coupling oscillates with
total relative amplitude $\kappa$ about $\Omega$ (P03; the standing
reading of A283). \emph{Link 2 (equipartition):} the oscillation's
variance is apportioned across the three layers by their measure
fractions $(4\pi^3, \pi^2, \pi)/\Omega$ --- the named assumption,
natural for a mode paced by the single clock (A269: one rhythm, no
private dynamics; the layer fractions are the corpus's only measure)
but not yet a theorem. \emph{Link 3 (edge unobserved):} observation
has no support on the edge sector (A281: observation vanishes at the
universe boundary; P04: the edge is where observation runs out), so a
measurement of the coupling samples the bulk and boundary shares
only: observed variance $= \kappa^2(1 - \pi/\Omega)$.
\emph{Link 4 (averaging):} the linear time-average of the sampled
oscillation (A283; exponent $n = 0.955 \approx 1$ data-selected)
gives
\begin{equation*}
\alpha^{-1}_{\rm meas} \;=\;
\Omega\sqrt{1 - \kappa^2\,(1 - \pi/\Omega)}
\;=\; 137.035999236 .
\end{equation*}
Against CODATA-22 ($137.035999177(21)$): $+0.43$ ppb,
$2.8\,\sigma_{\rm exp}$. The rival $1-1/44$ fits the current edition
marginally better and possesses \emph{no} chain --- no mechanism, no
provenance for 44, no link to any corpus structure; per the A245/A180
convention a chained candidate at $2.8\sigma$ outranks an unchained
one at $0.5\sigma$, because chains are falsifiable in more places
than fits. The bracket of A284 is therefore resolved into a ranking:
\textbf{edge equipartition is adopted as the working identification},
$1/44$ demoted to coincidence-watch, and OI-283-1 reduces to one
question --- prove or refute Link 2 (derive variance equipartition by
layer measure from the kernel, or compute the actual layer
decomposition of the P03 oscillation mode). The prediction's
falsification conditions are inherited from A284 and remain in force.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Chain}

\begin{proposition}[Deficit from the chain]
\label{prop:chain}
Links 1--4 jointly imply
relative deficit $= 1 - \sqrt{1 - \kappa^2(1-\pi/\Omega)} =
2.22234\times10^{-6}$, against 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 and is owned by the chain, not
absorbed.
\end{proposition}

\begin{remark}[The soft link, precisely]
Link 2 asserts: writing the oscillation as a sum of layer components,
$\kappa^2 = \sum_\ell \kappa_\ell^2$ with $\kappa_\ell^2 =
\kappa^2 f_\ell$, $f_\ell = (4\pi^3, \pi^2, \pi)/\Omega$. What would
prove it: decompose P03's fundamental perturbation mode against the
layer decomposition of the density and show the variance shares equal
the measure shares (a computation, in principle, on the wave equation
of P03 \S5 — filed as the operative content of OI-283-1). What would
refute it: variance shares $\neq$ measure shares; the prediction then
moves and the comparison with CODATA re-runs mechanically
(the verifier parametrizes the edge share).
\end{remark}

\begin{remark}[Ranking, not proof]
The adoption of the edge identification is a ranking by chain length
under the corpus's own convention, made while the unchained rival
fits the current CODATA edition better. This is recorded plainly
because it is the kind of judgment call a future measurement settles:
at $0.1$ ppb the two candidates separate by $\sim 3\sigma$ of current
precision (A284), and Link 2's computation may settle it sooner from
the theory side.
\end{remark}

\paragraph{Consequences.} OI-283-1 reduces to the Link-2 computation
(variance decomposition of the P03 mode by layer). A284's bracket
resolves to a working identification with the rival on
coincidence-watch. The registry note for P000\_1\_c is updated; the
$\alpha$-comma now reads, end to end: \emph{geometric value, dressed
by the corpus's own oscillation, averaged over the observed sector,
agreeing with experiment at the half-ppb level, with one named
computation between the chain and a derivation.}

\paragraph{Verifier.} \texttt{verify\_P288.py}: the chain arithmetic
at 40 digits; both CODATA comparisons; the $A$-value against A283's
measurement; the parametrized-share machinery (prediction as a
function of the edge share, with the measure-share value marked);
the ranking convention check (chained 2.8$\sigma$ vs unchained
0.5$\sigma$); registry note. All PASS.

\end{document}
