% 265_Addendum_OverlapCoupling.tex
% TASK_032 / OI-264-1: the coupling alpha^3/pi^2 derived. It was never a
% boundary charge: alpha^3/pi^2 = alpha / T_breath^2. The MOND acceleration
% is one kernel coupling times the breath-paced unit acceleration.
% Verifier: Lumen/corpus/addenda/verify/verify_P265.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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

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

\newcommand{\Lh}{L_h}
\newcommand{\Tb}{T_{\rm breath}}
\newcommand{\azero}{a_0}
\newcommand{\TbSI}{T_{\rm b}^{\rm SI}}

\title{Addendum 265: The Overlap Coupling Derived\\
\large $\alpha^3/\pi^2 = \alpha/\Tb^2$: the galactic acceleration scale\\
as one kernel coupling per breath-squared}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A$-$ (derived from closed canon commitments modulo
one named inference) \quad|\quad \textbf{Task:} TASK\_032 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P265.py} --- all PASS

\begin{abstract}
A264 left the amplitude law's coupling data-selected: $X = \alpha^3/\pi^2$
in $\azero = X c^2/\Lh$. This addendum derives it, and corrects this
thread's own working hypothesis in the process. The decomposition that
closes is not layer-traversal or boundary-charge ($\pi^2 \neq
Q[SU(2)]$ here); it is the exact identity
\[
  \frac{\alpha^3}{\pi^2} \;=\; \frac{\alpha}{\Tb^2},
  \qquad \Tb = \pi\alpha^{-1},
\]
so the law reads $\azero = \alpha \cdot c^2/(\Tb^2 \Lh)$: \emph{the
galactic acceleration scale is one kernel coupling times the system's
native acceleration unit --- one ruler-length per breath, per breath.}
The chain: (i) the breath is the unique clock (P14/A239/A262, Pin~3);
(ii) $\Lh$ is the unique ruler (A260 unit reduction; A264 measurement);
(iii) the front speed converts ruler to clock (A263), so the native
acceleration is $a_{\rm breath} = c^2/(\Tb^2\Lh)$; (iv) the hidden
sector's equilibrium response to the visible field is linear in $G M_b$
(the law's own form) and a linear response through the kernel carries
exactly one factor of the unique scalar coupling $\lambda = \alpha$
(SR3.1/P191: $\lambda\mu_0 = 1$ forces $\lambda = \alpha$). Hence
$\azero = \alpha\, a_{\rm breath}$, numerically
$1.1726\times10^{-10}\,$m/s$^2$ at $\Lh = 0.978$ kpc against the
gas-calibrated $1.172\times10^{-10}$ (A264). Consequences: the amplitude
law upgrades from data-selected candidate to derived-modulo-one-inference;
$\Lh$ is now \emph{defined} by $\azero$ with a derived coefficient
($\Lh = \alpha c^2/(\Tb^2 \azero) = 0.956$--$0.979$ kpc); and the breath
acquires a conditional SI value, $\Tb \approx 1.37$ Myr, giving the fold
clock its calibration (A260's reduction, conditionally closed). The named
inference --- breath-paced single-$\alpha$ linear response --- is filed
as OI-265-1 for kernel-level derivation.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Identity and the Typed Correction}

\begin{proposition}[$\pi^2$ is breath-squared]
\label{prop:identity}
Exactly, at any precision:
$\alpha^3/\pi^2 = \alpha\,(\alpha/\pi)^2 = \alpha/\Tb^2$ with
$\Tb = \pi\alpha^{-1}$ (P14/A239). The working readings carried by
TASK\_032 (one $\alpha$ per ontological layer; $\pi^2$ as the $SU(2)$
Noether charge of A43) are hereby \emph{retired}: the two factors of
$\alpha$ accompanying the kernel coupling are the breath's, and the
$\pi^2$ is the square of its $\pi$. The coupling was never about layers;
it is about \emph{time}.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{The Derivation}

\begin{theorem}[$\azero = \alpha\, c^2/(\Tb^2\Lh)$]
\label{thm:main}
Under the canon's three unit commitments --- breath as unique clock
(Pin~3, A262 Prop.~2.1), $\Lh$ as unique ruler (A260 Thm.~2.1, measured
in A264), front speed as ruler/clock conversion (A263) --- the native
acceleration of any breath-paced subsystem is
$a_{\rm breath} = \Lh/(\Tb\tau_u)^2 = c^2/(\Tb^2\Lh)$ with
$\tau_u = \Lh/c$. The hidden sector's equilibrium speed obeys
$v_\infty^4 = \azero\, G M_b$ (A264, linear in the visible source), and a
response linear in the source acquires exactly one factor of the kernel's
unique scalar coupling $\lambda = \alpha$ (SR3.1: the linear equation
$\lambda\mu_0 = 1$ has the unique solution $\lambda = \alpha$; A218b
forecloses alternative assemblies). Therefore
\[
  \boxed{\;\azero \;=\; \alpha\,\frac{c^2}{\Tb^2\,\Lh}
  \;=\; \frac{\alpha^3}{\pi^2}\,\frac{c^2}{\Lh}\;}
\]
Numerically $1.1726\times10^{-10}$ m/s$^2$ at $\Lh = 0.978$ kpc ---
within $0.05\%$ of A264's gas-calibrated value (consistency by
construction of $\Lh$; the content is the derived \emph{coefficient}).
\end{theorem}

\begin{remark}[What is inference, what is theorem]
Steps (i)--(iii) are closed canon. Step (iv) --- \emph{breath-paced
single-$\alpha$ linear response} --- is an inference: it asserts that the
hidden sector's virialization against the visible field is (a) paced by
the breath (Pin~3 applied to hidden observers; natural, not yet a
kernel theorem) and (b) first order in the kernel coupling (supported by
the law's strict linearity in $M_b$ across three decades of SPARC mass,
A264 Thm.~2.2(i)). Filed as \textbf{OI-265-1}. Grade: Case A$-$.

\emph{Upgraded (2026-06-10, same day; A268).} Half (b) is now derived:
$\rho$ is Brjuno (KAM-linearizable) and the mean-field response gain of
the breath map is exactly unity, so the equilibrium coefficient is
$\alpha$ alone with no dynamical factor. Grade: \textbf{derived modulo
Pin~3}; the residue --- inference (a), breath-pacing as kernel theorem
--- is re-filed as OI-268-1.

\emph{Completed (2026-06-10, same day; A269).} Inference (a) is now a
spectral theorem: the breathing spectrum descends from the charge-blind
radial density (A261 separability), so every charge sector shares one
clock --- Pin~3 derived. The chain is \textbf{fully derived}; sole
remaining freedom is the unit $L_h$.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Consequences}

\begin{corollary}[The unit, defined]
$\Lh \equiv \alpha c^2/(\Tb^2 \azero)$: the canon's missing length is now
\emph{defined} by the measured galactic acceleration with a derived
coefficient --- $0.9785$ kpc at the gas calibration, $0.9557$ at the
empirical $1.2\times10^{-10}$. The unit lock (A264 Rem.~3.1) tightens
from ``four appearances, one measurement'' to ``four appearances, one
definition.''
\end{corollary}

\begin{corollary}[The breath in SI]
\label{cor:breathSI}
With $\tau_u = \Lh/c \approx 3.19\times10^{3}$ yr:
$\Tb \approx 1.37$ Myr. The fold clock's calibration (A260's reduction)
is conditionally closed; every breath-counted statement in the canon
acquires a tentative physical duration. Descriptive note, unexplored:
the conformal age of the universe contains $\approx 3.4\times10^{4}$
breaths.
\end{corollary}

\paragraph{Open items.} \textbf{OI-265-1}: derive breath-paced
single-$\alpha$ linear response from kernel exposure dynamics (the
formal home is plausibly holomorphic: the chart $u = z_2/z_1$ is a
holomorphic coordinate on $\mathbb{CP}^1$ and breath iteration acts on
it by M\"obius maps --- see TASK\_034). \textbf{OI-265-2}: the
$3.4\times10^4$ breath count, coincidence audit only if it ever
intersects another thread.

\paragraph{Verifier.} \texttt{verify\_P265.py} --- identity at 50
digits, the derived $\azero$ against A264's calibration, the inverse
definition of $\Lh$, the SI breath; all PASS.

\paragraph{Upgrade note (added 2026-06-10, same day).} OI-265-1 is
closed by A275 and this addendum upgrades \textbf{Case A$-$ $\to$
Case A} under one flagged reading. In chart-natural units the gain is an
exact identity: $a_0 (\TbSI)^2/\Lh = \alpha$ --- the universal
acceleration is $\alpha$ chart-lengths per breath$^2$ --- and the
coupling channel is symmetry-forced (the unique modulus deformation of
the breath map, A274/A275). The flagged reading: that baryons enter via
that channel is the v4 closure hypothesis, data-confronted in A264.
Additionally, $\tau_u = \Lh/c$ of Corollary~\ref{cor:breathSI} is
identified by A273/A275 as the system unit itself:
$\tau_u = \tau_1^{\rm SI}$, hence $c = \Tb$ in chart velocities. See
A273, A274, A275.

\end{document}
