% 267_Addendum_RotationNumberBreath.tex
% TASK_034/035 joint result: the rotation number of the breath map, the
% Necessity of Detuning, and the derivation of 432 as the nearest just-
% lattice point. Rotation numbers and commas are one formalism.
% Verifier: verify_P267.py.
% 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{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 267: The Rotation Number of the Breath\\
\large Necessity of detuning, the derivation of 432,\\
and the breath's intrinsic scale system}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (two theorems) $+$ structural ladder $+$ one
open handoff \quad|\quad \textbf{Tasks:} TASK\_034 (rotation layer),
TASK\_035 (chains 1--2) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P267.py} --- all PASS

\begin{abstract}
The breath acts on the Hopf fiber as a rigid rotation by $\Tb$ windings
per tick; as a circle map its rotation number is
$\rho = \Tb \bmod 1 = \pi\alpha^{-1} - 430 = 0.5122452\ldots$
\emph{Theorem 1 (Necessity of Detuning):} $\rho$ is irrational ($\pi$ is
transcendental), so no breath orbit ever closes: every identification of
a corpus cycle with an exact integer or rational chord carries a
necessarily nonzero residual. The corpus's small gaps are not errors
awaiting better derivations; they are invariants of an irrational
rotation --- commas. \emph{Theorem 2 (Derivation of 432):} among
3-smooth integers (the just-intonation lattice $2^a 3^b$), the nearest
to $\Tb$ is exactly $432 = 2^4 3^3$; P14's Vedic correspondence is
thereby derived rather than observed, and $G_1$ (A266) is the comma of
that nearest chord. \emph{The ladder:} the continued fraction of $\rho$,
$[0;1,1,19,1,10,\ldots]$, yields the breath's intrinsic near-closure
cycles --- $q = 2$ breaths (comma $0.0245$: successive breaths are
near-antipodal, a half-turn tritone structure consonant with the mirror
configuration of P28), then $q = 41$ (comma $0.0020$), then deeper
levels --- the system's own ``scale,'' analogous to the twelve-tone
system arising from the continued fraction of $\log_2(3/2)$. In
conditional SI (A265): the 2-breath near-cycle is $\approx 2.7$ Myr, the
41-breath near-closure $\approx 56$ Myr. \emph{Handoff:} by
small-denominator theory the system's response to periodic forcing is
amplified as $1/\mathrm{comma}$ at the near-closure cycles; whether the
kernel coupling $\alpha$ emerges as the susceptibility gain at a
distinguished rung (TASK\_034's Koenigs question, OI-265-1) is open and
explicitly \emph{not} claimed here.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Breath as a Circle Map}

\begin{theorem}[Necessity of Detuning]
\label{thm:detuning}
The breath map on the Hopf fiber is the rigid rotation
$\theta \mapsto \theta + \rho \pmod 1$ with
$\rho = \Tb \bmod 1 = \pi(4\pi^3 + \pi^2 + \pi) - 430$, irrational by
the transcendence of $\pi$. Hence: (i) no orbit is periodic --- no
corpus cycle built on breath counting closes exactly; (ii) every
rational/integer chord assignment carries a nonzero comma; (iii) orbits
are equidistributed (Weyl), so the system revisits every phase
neighbourhood without repetition --- the dynamical substrate of the
corpus's ``moves toward regions not visited yet.'' Detuning is not a
defect of the derivations; it is a theorem of the dynamics.
\end{theorem}

\begin{theorem}[Derivation of 432]
\label{thm:432}
Let $S_3 = \{2^a 3^b\}$ be the 3-smooth (just-intonation) lattice. The
nearest element of $S_3$ to $\Tb = 430.5122\ldots$ is $432 = 2^4 3^3$
(neighbours: $384$ and $486$, both farther by an order of magnitude in
distance). Therefore P14's 432 correspondence is the statement ``the
breath's nearest just chord is $2^4 3^3$,'' a derived property of $\Tb$,
and $G_1 = 432/\Tb - 1$ (A266) is the comma of that chord. The
historical resonance (Vedic 432) is thereby decoupled from the
mathematics: any culture tuning to small-prime ratios near this period
finds 432.
\end{theorem}

% ────────────────────────────────────────────────────────────────────────────
\section{The Intrinsic Scale System}

The continued fraction $\rho = [0;1,1,19,1,10,1,12,8,4,\ldots]$ gives
the hierarchy of best rational approximations $p/q$ --- the cycle
lengths at which the breath nearly closes:

\medskip
\begin{center}
\begin{tabular}{rrl}
\toprule
$q$ (breaths) & comma $\|q\rho\|$ & conditional SI (A265) \\
\midrule
2   & 0.0245  & $\approx 2.7$ Myr \\
41  & 0.0020  & $\approx 56$ Myr \\
449 & (next rung) & $\approx 0.6$ Gyr \\
\bottomrule
\end{tabular}
\end{center}
\medskip

\begin{remark}[Near-tritone structure]
$\rho \approx \tfrac12 + 0.0122$: each breath advances the fiber by
almost exactly a half-turn. Successive breaths are near-antipodal ---
the dynamics is a slowly precessing alternation, consonant with the
mirror configuration ($\Omega^{(-)} = -\Omega^{(+)}$) that P28 makes
canonical and with the Oscillate sector's role in the Wheel. The
2-breath near-cycle is the system's deepest periodicity; everything
slower is precession of it.
\end{remark}

\begin{remark}[Susceptibility handoff --- open]
For a circle map under periodic perturbation, response at cycle $q$ is
amplified as $1/\|q\rho\|$ (small denominators). The breath's
susceptibility ladder is therefore fixed by the table above. Whether
$\alpha$ appears as the gain at a distinguished rung --- which would
close OI-265-1 and upgrade A265 to Case A --- is the TASK\_034 numerical
target. Not claimed.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P267.py}: $\rho$ exact;
convergents and commas (corrected recurrence, monotone decreasing along
convergents); the 3-smooth scan; near-tritone bound; SI conversions;
all PASS.

\end{document}
