% 269_Addendum_BreathPacingTheorem.tex
% OI-268-1 CLOSED: breath-pacing as a spectral theorem. The hidden sector
% shares the visible sector's clock because the breathing spectrum descends
% from the charge-blind radial density. Pin 3 upgrades from constitution to
% theorem; the a0 chain is fully derived modulo the unit.
% Verifier: verify_P269.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
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\newcommand{\Tb}{T_{\rm breath}}
\newcommand{\Qhat}{\hat{Q}}

\title{Addendum 269: Breath-Pacing as a Spectral Theorem\\
\large Why no subsystem can generate its own rhythm,\\
and the completion of the $a_0$ chain}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (OI-268-1 closed; Pin~3 derived; A265 chain
complete modulo the unit) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P269.py} --- all PASS

\begin{abstract}
A268 left one link between the galactic acceleration scale and pure
canon: inference (a) of A265 --- that the hidden sector is
\emph{breath-paced} --- held constitutionally (Pin~3) but not as kernel
theorem. This addendum derives it. The argument is spectral and rests on
A261's commutation table. The breath descends from the breathing
spectrum of the radial density: P03 linearizes perturbations of
$\rho(x)$ to obtain $\omega_n = n\pi\sqrt{1-\kappa}$, and P28 lifts the
fundamental through the $SU(2)/SO(3)$ covering to the arc-speed
$\|\Omega\| = 2\omega_1 \approx 2\pi$. The radial ODE contains no Hopf
charge: by A261, every radial operator commutes with $\Qhat$, so the
kernel Hamiltonian \emph{separates}, ${H = H_{\rm rad} \oplus H_{\rm
fiber}}$ with $H_{\rm fiber} = -c_f\Qhat^2$. Fiber charge shifts sector
energies but cancels in the \emph{spacing} of radial breathing levels:
the fundamental $\omega_1$ is identical in every charge sector ---
verified symbolically and numerically ($q$-independence exact).
Therefore the hidden (charged) sector oscillates, accumulates phase, and
closes its winding cycle on exactly the visible sector's period: there
is one clock because there is one radial density and one operator
(A218b). \textbf{Pin~3 --- ``no subsystem generates its own rhythm'' ---
upgrades from constitutional to derived.} Corollary: A265's inference
(a) closes, completing the chain
$a_0 = \alpha\,c^2/(\Tb^2 L_h) = \alpha^3 c^2/(\pi^2 L_h)$ as a
\emph{pure-canon theorem with zero undetermined dimensionless constants}
--- the only remaining freedom is the unit length $L_h$, which every
theory must carry and which rotation curves have measured at
$\approx 0.98$ kpc. Inherited residue: P28 TBS 28\_2 (the
$\sqrt{1-\kappa}$ factor, $0.107\%$, with A266's schisma near-miss
attached to exactly this number) now sits at the foundation of the
chain and is promoted to the thread's principal open refinement
(OI-269-1).
\end{abstract}

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

\begin{theorem}[Breath-pacing]
\label{thm:pacing}
Let the hidden sector be the Hopf-charged subspace of the canonical
$\hat{O}$ (A261), which is the unique admissible assembly (A218b). Then
its phase dynamics is paced by the same fundamental as the visible
sector's:
\begin{enumerate}
  \item \emph{(One operator)} There is no second generator: assembly
        uniqueness forecloses sector-specific dynamics; both charge
        sectors evolve under restrictions of the same $\hat{O}$.
  \item \emph{(Charge-blind density)} The breathing spectrum
        $\omega_n = n\pi\sqrt{1-\kappa}$ (P03) is obtained by
        linearizing the \emph{radial} density $\rho(x)$; by A261
        Thm.~1.1 every radial operator commutes with $\Qhat$, so the
        linearized problem is identical in every charge sector.
  \item \emph{(Separability)} $H = H_{\rm rad} \oplus H_{\rm fiber}$,
        $H_{\rm fiber} = -c_f \Qhat^2$: fiber charge adds a constant
        offset per sector and cancels in level spacings. The breathing
        fundamental --- hence the winding period, hence $\Tb$ --- is
        $q$-independent, exactly.
\end{enumerate}
One density, one operator, one clock. A subsystem \emph{cannot} generate
its own rhythm because every rhythm in the kernel descends from the
charge-blind radial spectrum: Pin~3 is a theorem.
\end{theorem}

\begin{corollary}[The $a_0$ chain, complete]
\label{cor:chain}
A265 inference (a) closes; with A268's gain-unity result (inference (b))
the chain is fully derived:
\[
  a_0 \;=\; \alpha\,\frac{c^2}{\Tb^2 L_h} \;=\;
  \frac{\alpha^3}{\pi^2}\,\frac{c^2}{L_h}
  \;=\; 1.1726\times10^{-10}\ \mathrm{m/s^2}\ \text{at } L_h = 0.978\ \mathrm{kpc},
\]
with \emph{zero} undetermined dimensionless constants. The single
dimensional freedom, $L_h$, is the unit any theory must carry --- here
measured by SPARC (A264) and quadruply locked (B$^4$ scale $\equiv$
fold-clock unit $\equiv$ amplitude length). Within the canon's own
standards this is the strongest derivational chain it possesses: from
the monad $\Omega = 4\pi^3+\pi^2+\pi$ to a number on a telescope, every
link now a theorem, every theorem with a verifier.
\end{corollary}

\begin{remark}[The residue moves to the foundation --- OI-269-1]
The lift $\|\Omega\| = 2\pi\sqrt{1-\kappa}$ carries P28's TBS 28\_2: a
$0.107\%$ deviation from exactly $2\pi$, previously cosmetic, now
foundational --- it propagates (at second order, hence harmlessly today)
into $\Tb$ and thus $a_0$. A266 flagged the schisma
($0.113\%$) as a near-miss to \emph{exactly this number}. Whether the
$\sqrt{1-\kappa}$ factor is exact physics (the breath genuinely detuned
from $2\pi$ --- another comma), or cancels (P28's open question),
becomes the chain's principal refinement. If it is real, the corpus's
two flagged near-misses (G1/quarter-comma; 28\_2/schisma) may be two
rungs of one temperament --- TASK\_035 chain 3 inherits this as its
sharpest target.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P269.py}: $\omega_1$ exact;
$q$-independence of the spacing across sectors; separability grounds
(A261 commutators); the lift residue $0.107\%$; the complete chain
re-derivation of $a_0$; zero-free-dimensionless audit; all PASS.

\end{document}
