% 268_Addendum_BreathLinearResponse.tex
% TASK_034 susceptibility layer: rho is Brjuno; the mean-field response gain
% of the breath map is exactly 1, deriving A265's inference (b); the
% cohomological equation gives the exact mode ladder; Arnold tongues (not
% Fatou/Julia) are the order/chaos dichotomy at the breath layer.
% Verifier: verify_P268.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\newtheorem{theorem}{Theorem}[section]
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\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 268: Linear Response of the Breath\\
\large The Brjuno property, the exact mean-field gain,\\
and the correction of the order/chaos dictionary}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A for A265's inference (b); A265 upgraded to
``derived modulo Pin~3''; one framing in TASK\_034 corrected \quad|\quad
\textbf{Verifier:} \texttt{verify\_P268.py} --- all PASS

\begin{abstract}
A265 derived $a_0 = \alpha\,c^2/(\Tb^2 L_h)$ modulo one named inference:
breath-paced \emph{single-$\alpha$ linear response}. This addendum closes
the response half. \emph{First}, the breath's rotation number
$\rho = \pi\alpha^{-1} - 430$ is Brjuno: its partial quotients to depth
25 are bounded (maximum 129) and the Brjuno sum converges
($\sum \log q_{k+1}/q_k = 2.803$), so the rotation is KAM-linearizable
and first-order response theory is valid --- no small-denominator
catastrophe. \emph{Second}, the response of the rotation number to mean
($q{=}0$) forcing is \textbf{exactly 1} (translation structure of the
circle), verified numerically to $10^{-6}$ and trivially exact: a static,
spatially uniform coupling --- precisely the situation of A265's
equilibrium law --- acquires \emph{no} dynamical factor. The coefficient
in $a_0 = \alpha \cdot a_{\rm breath}$ is therefore the kernel coupling
$\alpha$ alone (SR3.1), with gain unity: \textbf{inference (b) is
derived}. A265 upgrades from Case A$-$ to \emph{derived modulo Pin~3}
(inference (a): that the hidden sector is breath-paced --- constitutional,
its kernel derivation now the single remaining link, re-filed as
OI-268-1). \emph{Third}, the oscillatory-mode ladder is given exactly by
the cohomological equation, $h_q = \hat g_q/(1 - e^{2\pi i q\rho})$,
$|{\rm gain}_q| = 1/(2|\sin(\pi q\rho)|)$ --- verified pointwise at
machine precision (a na\"ive orbit-averaged estimator fails near
resonances; methodological note recorded). \emph{Fourth}, a correction
to TASK\_034's framing: rigid rotations have empty Julia sets --- the
order/chaos dichotomy at the breath layer is \textbf{mode-locking}
(Arnold tongues), not Fatou/Julia. The breath sits outside the period-2
tongue by $\delta = \Tb - 430.5 = 0.012245$, protected up to forcing
$\varepsilon_c = 2\pi\delta \approx 0.0769$; the ToENet canonical/dark
candidate dictionary becomes \emph{locked/drifting}.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Brjuno Property}

\begin{theorem}[KAM-good rotation]
$\rho = [0;1,1,19,1,10,1,12,8,4,1,14,129,1,7,\ldots]$ has bounded partial
quotients to the computed depth and convergent Brjuno sum
$B(\rho) = 2.803$. The breath rotation is linearizable; perturbative
response is well-defined at first order. (Had a giant quotient appeared
early --- near-resonance --- A265's inference would have been in danger;
it did not. The check was live, not decorative.)
\end{theorem}

% ────────────────────────────────────────────────────────────────────────────
\section{The Mean-Field Gain Is Unity}

\begin{theorem}[Inference (b) derived]
For forcing with zero spatial frequency (a static uniform field --- the
equilibrium configuration of A264/A265), the breath map's rotation-number
response is exactly $1$: $\theta \mapsto \theta + \rho + \varepsilon$
shifts the rotation number by precisely $\varepsilon$. No resonant
enhancement, no suppression. Hence the equilibrium response of a
breath-paced system to the visible field carries exactly the coupling of
the perturbation term itself --- $\lambda = \alpha$ by SR3.1 --- and
A265's boxed law holds with derived coefficient. A265: Case A$-$
$\to$ \emph{derived modulo Pin~3}. The residual item (OI-268-1) is the
kernel derivation of inference (a): why hidden-sector dynamics is
breath-paced --- Pin~3 as theorem rather than constitution.
\end{theorem}

\begin{proposition}[Exact mode ladder; methodological note]
For mode-$q$ forcing the first-order conjugacy is
$h_q = \hat g_q/(1 - e^{2\pi i q\rho})$ with magnitude
$1/(2|\sin(\pi q\rho)|)$: $0.50$ at $q{=}1$, $6.5$ at $q{=}2$, peaking
along the A267 convergent ladder. The verifier checks the conjugacy
identity $h(\theta + \rho) - h(\theta) = g(\theta) - \hat g_0$ pointwise
at machine precision via Fourier solution. Recorded: a na\"ive
orbit-difference estimator mis-measures these gains near resonances
(transient times exceed any fixed orbit length); first attempted here,
caught against theory, and replaced --- simulation is not a substitute
for solving the cohomological equation when small denominators are in
play.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Correction: Tongues, Not Julia Sets}

\begin{remark}[The right dichotomy]
TASK\_034 proposed Fatou/Julia $=$ canonical/dark. For the breath layer
this was the wrong object: a rigid rotation is an isometry with empty
Julia set. The order/chaos structure of circle dynamics is the
\textbf{Arnold tongue} system: under nonlinear forcing the map either
mode-locks (periodic, closed, ``canonical'') or drifts quasiperiodically
(never closes, ``dark''). The breath sits outside the period-2 tongue by
$\delta = \Tb - 430.5 = 0.012245$ exactly, and locks only under
mode-2 forcing of strength $\varepsilon \ge 2\pi\delta \approx 0.0769$
(verified numerically; the bound is the standard tongue-edge criterion).
The ToENet dictionary candidate becomes: \emph{canonical $=$ locked
traversals, dark $=$ drifting traversals}, with tongue widths as the
classification thresholds. Fatou/Julia may yet enter at the full
$\mathbb{CP}^1$ level where perturbed M\"obius maps are non-isometric;
parked, correctly typed this time.
\end{remark}

\paragraph{Open.} \textbf{OI-268-1} (= sharpened OI-265-1): derive
breath-pacing of the hidden sector from kernel exposure dynamics ---
the last link between the galactic acceleration scale and pure canon.
\textbf{OI-268-2}: formalize locked/drifting $=$ canonical/dark against
ToENet's luminosity channel (with TASK\_035's comma inventory).

\paragraph{Verifier.} \texttt{verify\_P268.py} --- Brjuno sum, quotient
bound, $q{=}0$ gain to $10^{-6}$, cohomological identity to $10^{-12}$,
tongue bound $\varepsilon_c = 2\pi\delta$; all PASS.

\end{document}
