% 274_Addendum_BreathMapCP1.tex
% TASK_034 close-out: the breath map typed as an elliptic Mobius
% transformation on CP^1. Multiplier from A273's winding count (the base
% rotates by the q=2 comma); Julia set empty (Fatou/Julia dictionary
% formally retired); unique ergodicity makes the halo profile the unique
% invariant measure; Brjuno multiplier; lens Z3 commutes; OI-265-1
% sharpened to a named missing operator.
% Verifier: verify_P274.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{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}
\newcommand{\CP}{\mathbb{CP}^1}

\title{Addendum 274: The Breath Map on $\mathbb{CP}^1$\\
\large Elliptic M\"obius typing, unique ergodicity of the halo,\\
and the multiplier frame for OI-265-1}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A$-$ (foundations typed; two theorems; one target
closed negatively-with-replacement; OI-265-1 sharpened, not closed)
\quad|\quad \textbf{Task:} TASK\_034 (final target) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P274.py} --- all PASS

\begin{abstract}
TASK\_034's remaining target is discharged: the breath, acting on the
Hopf base $S^2 \cong \CP$, is typed as a discrete holomorphic dynamical
system, and every dynamical consequence available without new physics is
extracted. In the chart $u = z_2/z_1$ with the Hopf poles (A263's rest
and null configurations) at $0$ and $\infty$, one breath acts as the
M\"obius transformation $B(u) = \lambda u$ with multiplier
$\lambda = e^{2\pi i \rho_*}$, where $\rho_* = 2\Tb \bmod 1 =
2\rho - 1 = 0.0244904\ldots$ is fixed by A273's winding count: \emph{the
base rotates per breath by exactly the $q=2$ comma} (A267), while the
fiber carries $\rho = 0.5122452$ --- the two registers of A273 are the
fiber and base rotation numbers of one bundle map.
\emph{Theorem~\ref{thm:elliptic}:} $B$ is elliptic with irrational
rotation number, hence: Julia set empty, Fatou set all of $\CP$, $B$
conjugate to a rigid rotation, no periodic orbits --- the Necessity of
Detuning (A267) holds at the $\CP$ level. This formally retires
TASK\_034's Fatou/Julia$\,=\,$canonical/dark dictionary (an elliptic
breath has no Julia set to be dark); the replacement was already filed
(A268 Arnold tongues, A272 kernel typing).
\emph{Theorem~\ref{thm:ergodic}:} $B$ preserves each latitude circle
$|u| = \tan\beta$ and is uniquely ergodic on it (Weyl); the
hidden-branch halo profile $\rho_H \propto \sin^2(2\beta)/r^2$ (v4
kernel, A264) is a $B$-invariant function whose angular factor on each
invariant circle is \emph{the unique} $B$-invariant probability measure.
The halo is not a configuration that happens to be stationary; it is the
only stationary angular distribution the breath dynamics admits.
The multiplier is Brjuno (sum $\approx 0.287$), placing perturbations of
$B$ in the Siegel regime. The lens $\mathbb{Z}_3$ (A255) commutes with
$B$. OI-265-1 is sharpened, not closed: the unperturbed multiplier is
\emph{exactly} unimodular, so the conjecture
``$\alpha = $ per-breath linear-response gain'' is entirely a statement
about the derivative of $|\lambda|$ under a baryonic coupling operator
that the corpus does not yet define; the missing object is named.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Map}

\begin{proposition}[The breath map and its multiplier]
\label{prop:map}
Let $u = z_2/z_1$ be the holomorphic coordinate on the Hopf base
$\CP$, with the rotation axis fixed by the Hopf splitting (P28); the
fixed points $u = 0, \infty$ are A263's rest and null configurations.
The dual-observer flow $q(t) = \exp(\hat\Omega\,\|\Omega\| t/2)$
(P28, $\|\Omega\| = 2\omega_1$) induces on $u$ a rigid rotation; over
one breath (duration $\Tb\,\tau_1$ in P03 time, A273) the accumulated
SO(3) rotation angle is $\|\Omega\| \cdot \Tb\,\tau_1 = 4\pi\Tb$
exactly ($\kappa$-free, A273 Thm~2.1). The breath map is therefore the
M\"obius transformation
\begin{equation}
B(u) \;=\; \lambda\,u, \qquad
\lambda \;=\; e^{2\pi i \rho_*}, \qquad
\rho_* \;=\; 2\Tb \bmod 1 \;=\; 2\rho - 1 \;=\; 0.0244904\ldots,
\end{equation}
with $|\lambda| = 1$ exactly. The base rotation number $\rho_*$ equals
A267's $q = 2$ comma $|2\rho - 1|$, and the fiber rotation number is
$\rho = \Tb \bmod 1$ (A267, unconditional per A273): the two registers
of A273 are the fiber and base components of one bundle map. The
factor-2 bookkeeping is pinned by P28's lifting remark
($\|\Omega\| = 2\omega_1$); the register identity
$\rho_* = 2\rho - 1$ is convention-independent given A273's count of
$2\Tb$ SU(2) windings per breath.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Classification and Its Consequences}

\begin{theorem}[Elliptic, irrational, Julia-free]
\label{thm:elliptic}
$B$ is an elliptic M\"obius transformation
($|{\rm tr}| = 2|\cos(\pi\rho_*)| = 1.9941 < 2$) with irrational
rotation number ($\rho_* = 8\pi^4 + 2\pi^3 + 2\pi^2 - 861$ is a
nonconstant rational-coefficient polynomial in the transcendental $\pi$,
hence irrational --- the A267 argument). Consequently, by the standard
classification of M\"obius dynamics: (i)~the Julia set of $B$ is empty
and the Fatou set is all of $\CP$; (ii)~$B$ is conjugate in
${\rm PSL}(2,\mathbb{C})$ to the rigid rotation by $2\pi\rho_*$;
(iii)~$B$ has no periodic orbits other than the two fixed points ---
$\lambda$ is not a root of unity --- and every other orbit is dense in
its invariant circle. The Necessity of Detuning holds verbatim at the
$\CP$ level.
\end{theorem}

\begin{corollary}[Fatou/Julia dictionary retired]
\label{cor:retire}
TASK\_034's third target proposed canonical/dark $=$ Fatou/Julia for the
breath map. For the actual (elliptic) breath map the Julia set is empty:
under that dictionary nothing would be dark, contradicting A264's
measured halo and A272's never-empty corollary. The proposal is closed
negatively; its replacement is already on file --- the locked/drifting
dichotomy of Arnold tongues (A268) and the kernel typing of dark (A272).
This completes, at the properly-typed level, the correction begun when
A268 replaced the Fatou/Julia framing by tongues.
\end{corollary}

\begin{theorem}[Unique ergodicity; the halo as the unique invariant measure]
\label{thm:ergodic}
$B$ preserves each latitude circle $C_\beta = \{|u| = \tan\beta\}$, and
its restriction to each $C_\beta$ is an irrational rigid rotation, hence
uniquely ergodic (Weyl): the normalized arc-length measure is the
\emph{only} $B$-invariant probability measure on $C_\beta$, and every
orbit equidistributes. The hidden-branch halo profile
$\rho_H(r) \propto \sin^2(2\beta)/r^2$ with $\tan\beta = r/L_h$ (v4
kernel; confronted with SPARC in A264) depends on $u$ through $|u|$
only, hence is $B$-invariant; combined with unique ergodicity, the halo
is the push-forward through the chart of the unique invariant angular
measure on each circle. Stationarity of the halo is not an equilibrium
assumption: it is unique ergodicity of the breath map. Any stationary
angular distribution other than the uniform one is dynamically
impossible.
\end{theorem}

\begin{remark}[Brjuno multiplier; Siegel regime]
The continued fraction of $\rho_*$ is $[0;40,1,4,1,25,4,9,\ldots]$ with
convergent denominators $40, 41, 204, 245, 6329, \ldots$ and Brjuno sum
$\approx 0.287 < \infty$. Perturbations of $B$ that keep the multiplier
on the unit circle therefore lie in the Siegel linearization regime at
both fixed points. The $q = 41$ convergent names a $41$-breath
near-closure of the base --- in conditional SI (A265),
$\approx 56$ Myr --- the same scale as A267's fiber $q = 41$ rung: both
registers single out the 56-Myr cycle as the system's most robust
intrinsic timescale.
\end{remark}

\begin{remark}[Lens $\mathbb{Z}_3$]
The lens action $u \mapsto e^{2\pi i/3} u$ (A255) is a rotation about
the same axis, hence commutes with $B$: the $\mathbb{Z}_3$ symmetry is a
symmetry of the iteration, as TASK\_034 anticipated. Per A255
discipline, no identification with triality or the A44 Jordan cycle is
made.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Multiplier Frame for OI-265-1}

\begin{proposition}[Unperturbed gain is exactly zero]
\label{prop:gain}
$|\lambda| = 1$ exactly (Proposition~\ref{prop:map}: the exponent is
$\kappa$-free and purely imaginary). Hence the conjectured reading of
A265 --- $\alpha$ as the per-breath linear-response gain of the hidden
chart --- is entirely perturbative: it is a statement about
$\partial |\lambda'| / \partial s$ at $s = 0$, where $\lambda'(s)$ is
the multiplier of a baryon-perturbed breath map $B_s$ and $s$
parametrizes the source ($GM_b/c^2 L_h$ or the corpus-preferred
dimensionless form). The corpus defines no coupling operator taking the
baryonic potential into a deformation of $B$; without it the derivative
is not computable, and per the PSLQ-honesty and Butlerian conventions no
candidate is promoted here.
\end{proposition}

\paragraph{OI-265-1, sharpened (final form).} Define the baryonic
coupling operator on the breath map --- the map $s \mapsto B_s$ induced
by a baryon source on the Hopf base --- and compute
$\partial|\lambda'|/\partial s$ at the fixed points. Outcome grades:
$= \alpha$ (A265 upgrades A$-$ $\to$ A, the kernel coupling acquires its
dynamical meaning); $\neq \alpha$ (the gain reading of A265 is retired;
the identity $\alpha^3/\pi^2 = \alpha/\Tb^2$ stands regardless ---
A271's robustness theorem); not derivable (name the missing theory).
This is the whole remaining content of OI-265-1; the kinematic frame is
now fixed and verifier-pinned.

\paragraph{Consequences.} TASK\_034 closes: rotation layer (A267),
susceptibility (A268), 0.43 withdrawn (A270), ladder gate lifted
(A273), $\CP$ typing and target-3 retirement (this addendum); OI-265-1
remains open in sharpened form as the task's single surviving question.
ToENet dictionary unchanged (locked/drifting per A268; kernel typing per
A272), now with the base/fiber split explicit: fiber $\rho = 0.5122$,
base $\rho_* = 0.0245$.

\paragraph{Verifier.} \texttt{verify\_P274.py}: the multiplier and
register identity; ellipticity ($|{\rm tr}| < 2$); irrationality
structure; non-root-of-unity via convergent denominators; latitude
invariance and halo $B$-invariance (50 digits); Weyl equidistribution
(numeric, $N = 10^5$); Brjuno sum; lens commutation; the exact
unimodularity underlying Proposition~\ref{prop:gain}; the 56-Myr
register coincidence. All PASS.

\end{document}
