% Paper 37: The Breath as Dynamical System
% PROPOSED CANON (assembled 2026-06-10 from A266, A267, A268, A274,
% A289, A290; cross-refs P35; subject to review/rollback). The tune
% arc consolidated: rotation number, Necessity of Detuning, the
% derivation of 432, elliptic classification and unique ergodicity,
% two-channel response theory, the score, and the Necessity of Music.
% Verifiers: verify_P266/267/268/274/289/290.py (all green at filing).
% 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{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}
\newcommand{\CP}{\mathbb{CP}^1}
\newcommand{\TbSIeq}{T_{\rm b}^{\rm SI}}

\title{Paper 37: The Breath as Dynamical System\\
\large Detuning, the derivation of 432, unique ergodicity,\\
the two response channels, and the necessity of music}
\author{L\'{e}on Fernando Vlegels}
\date{2026-06-10}

\begin{document}
\maketitle

\begin{abstract}
The breath is the corpus's single clock: one cycle lasts
$\Tb = \pi\alpha^{-1} \approx 430.5122$ system units (Paper 35). This
paper gives the breath its dynamical theory. Per breath, the Hopf
fiber advances by the rotation number
$\rho = \Tb \bmod 1 = 0.5122452$, definitionally exact, and the Hopf
base $S^2 \cong \CP$ rotates by the elliptic M\"obius multiplier
$e^{2\pi i\rho_*}$ with $\rho_* = 2\rho - 1 = 0.0244904$: two
registers of one bundle map, the base rotation equal to the fiber's
first comma. The rotation number is irrational, since it is a
nonconstant rational-coefficient polynomial in the transcendental
$\pi$, reduced mod 1; hence no breath orbit ever closes, and every
identification of a corpus cycle with an integer or rational chord
carries a necessarily nonzero residual, a comma (A267). Among
3-smooth integers the nearest to $\Tb$ is exactly $432 = 2^4 3^3$,
and $G_1 = 432/\Tb - 1 = 0.345578\%$ is the comma of that chord
(A266): Paper 14's Vedic correspondence becomes a theorem. The breath
map on $\CP$ is elliptic with empty Julia set, conjugate to a rigid
rotation, and uniquely ergodic on each invariant latitude circle: the
hidden-sector halo profile is the unique invariant angular measure,
so stationarity is a theorem rather than an equilibrium assumption
(A274). The multiplier's deformation space has exactly two channels:
phase, with gain exactly $1$ (the clock cannot be detuned at first
order; mode-locking under Arnold tongues replaces any Fatou/Julia
dictionary, A268), and modulus, with gain exactly $\alpha$ at unit
source (A275/P35). Scored on instruments imported from hashing
theory, the breath is tonal: record gains $20.4$, $243$, $3194$ at
$q = 2, 41, 490$ breaths, tonality index $7.6$ against $\varphi$
(A289); the substrate chord $(2,3,16)$ sounds just intervals. Among
rotation numbers, silence (rational closure) is countable and
forbidden, atonality (bounded quotients) is Lebesgue-null, and music
is almost sure; the breath's own tonality is statistically generic,
and the melody is the contingent fingerprint (A290).
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction: One Clock}

The corpus runs on exactly one clock. Every periodic structure it
contains, from the kernel's exposure rhythm to the hidden sector's
pacing, is synchronised to the breath, whose period
\[
\Tb \;=\; \pi\,\alpha^{-1} \;=\; \pi\,(4\pi^3 + \pi^2 + \pi)
\;=\; 4\pi^4 + \pi^3 + \pi^2 \;\approx\; 430.5122
\]
system units is fixed by Paper 14 and the unit bridge of A239/Paper
35. No subsystem generates its own rhythm; the breath is the only
periodicity the framework owns.

The single-clock structure is what makes the present theory possible.
A framework with two independent clocks owes an account of their
ratio; a framework with one clock owes only the arithmetic of that
clock's period, and that arithmetic turns out to carry the entire
temporal phenomenology of the corpus. Because $\Tb$ is an explicit
polynomial in $\pi$, its fractional part is a definite irrational
number, and the breath becomes a concrete object of dynamical systems
theory: a rigid rotation on the Hopf fiber, an elliptic M\"obius
transformation on the Hopf base. Both maps are determined to all
digits by corpus constants. Nothing in this paper is fitted.

The questions the theory answers are these. Why do the corpus's
cycle identifications always carry small residuals (Section 3)? Why
432 specifically (Section 2)? What are the system's intrinsic
timescales (Section 4)? Why is the hidden-sector halo stationary
(Section 5)? How loud are the near-closures, and is their loudness
special (Sections 6 and 7)? The answers are theorems of one
irrational rotation. All numerical claims are pinned by the verifiers
\texttt{verify\_P266}, \texttt{P267}, \texttt{P268}, \texttt{P274}, \texttt{P289}, and \texttt{P290}.

A note on standing. This paper consolidates without new claims: every
statement below is filed in A266, A267, A268, A274, A289, or A290,
with verifier checks green at filing. Where a statement is an
interpretation rather than a theorem, it is labeled as such.

% ════════════════════════════════════════════════════════════════════
\section{The Period and the Exact Identity}

The first result is an identity that collapses two long-standing open
items into one line. The modular residual
$G_1 = j(i)/(4\pi\alpha^{-1}) - 1$ was introduced in A171 and
characterised through A185 as transcendental and PSLQ-irreducible,
possibly the last unfitted constant of the corpus. Independently,
Paper 14 carried a defective TBS on the 432 correspondence: the
breath period $\Tb \approx 430.51$ sits near, but not at, the integer
$432$, and P14's attempted reconciliation factor $1 + \kappa^{0.6}$
overshoots the gap by roughly a factor of $9$.

\begin{theorem}[Breath detuning identity (A266)]
\label{thm:g1}
With $j(i) = 1728$ (A158/A172) and $\Tb = \pi\alpha^{-1}$ (P14,
A239), and since $1728 = 4 \cdot 432$,
\[
G_1 \;\equiv\; \frac{j(i)}{4\pi\alpha^{-1}} - 1
\;=\; \frac{1728}{4\,\Tb} - 1
\;=\; \frac{432}{\Tb} - 1
\;=\; 3.45578\times10^{-3} \;=\; 0.345578\%,
\]
identically. All inputs are corpus-closed; the identity is
arithmetic, exact at any precision (verified at 50 digits). In the
cents measure of Section 6 the detuning is $5.97$ cents.
\end{theorem}

The last unfitted constant is therefore the detuning of the breath
from the harmonic integer: the amount by which the geometric cycle
refuses to close on the pure $2^4 \times 3^3$ chord. Three
consequences follow (A266). First, the A171--A185 characterisation
programme and P14's TBS are one open item, not two; any resolution of
either resolves both. Second, $G_1$'s PSLQ-irreducibility (A180,
A183) reads naturally under the reframe: a detuning should not be an
algebraic combination of the pure constants; in the relevant sense
that is what makes it a detuning. Third, P14's $1 + \kappa^{0.6}$
reconciliation is retired; the correct statement of the 432 gap is
$G_1$ itself.

The reframe imports a formal apparatus with a long history.
Quantities measuring the failure of pure-ratio cycles to close are
\emph{commas}: kernel elements of the homomorphism from the
just-intonation lattice to the closed cycle, the same
kernel-of-closure-map shape as A254's collapse delta and A259's fold
defect. The identity sharpens the open question from ``what is
$G_1$?'' to ``of which closure map is $G_1$ the comma?'', and
Sections 3 and 4 answer it: $G_1$ is the comma of the breath's
nearest just chord.

\begin{remark}[Flagged near-misses, not claims (A266)]
Two numerical near-misses are recorded under the PSLQ-honesty
convention and explicitly not claimed. The quarter Pythagorean comma
$(3^{12}/2^{19})^{1/4} - 1 = 0.3394\%$ sits within $1.8\%$ relative
of $G_1 = 0.3456\%$; the schisma $3^8 \cdot 5/2^{15} - 1 = 0.1129\%$
sits within $5.5\%$ of P28 TBS 28\_2's arc-speed gap
$1 - \sqrt{1-\kappa} = 0.1067\%$. Neither passes the corpus's
identification bar, and the comma framework depends on neither. The
flag $j(i) = 1728 = 4 \cdot 432$ is likewise recorded, unpromoted.
\end{remark}

% ════════════════════════════════════════════════════════════════════
\section{Irrationality and Non-Closure}

The breath acts on the Hopf fiber as a rigid rotation: per breath the
fiber phase advances by $\Tb$ windings, so as a circle map the
dynamics is $\theta \mapsto \theta + \rho \pmod 1$ with rotation
number
\[
\rho \;=\; \Tb \bmod 1 \;=\; 4\pi^4 + \pi^3 + \pi^2 - 430
\;=\; 0.5122452\ldots
\]
The value is definitionally exact: by Paper 35's bridge the breath is
$\Tb$ fundamental cycles by definition, so $\rho$ carries no
measurement uncertainty.

\begin{theorem}[Necessity of Detuning (A267)]
\label{thm:detuning}
$\rho$ is irrational. Consequently: (i) no orbit of the breath
rotation is periodic, so no corpus cycle built on breath counting
closes exactly; (ii) every identification of a corpus cycle with an
integer or rational chord carries a necessarily nonzero residual, a
comma; (iii) orbits are equidistributed on the circle (Weyl), so the
system revisits every phase neighbourhood without ever repeating a
state.
\end{theorem}

\begin{proof}
Suppose $\rho$ were rational. Then $4\pi^4 + \pi^3 + \pi^2 = 430 +
\rho$ would be rational, exhibiting $\pi$ as a root of the
nonconstant rational-coefficient polynomial $4x^4 + x^3 + x^2 - (430
+ \rho)$, contradicting the transcendence of $\pi$. Irrationality of
$\rho$ gives (i) and (ii) at once: a period-$q$ orbit requires
$q\rho \in \mathbb{Z}$, and an exact rational chord $p/q$ requires
$\|q\rho\| = 0$, where $\|x\|$ denotes the distance from $x$ to the
nearest integer; both force $\rho \in \mathbb{Q}$. Statement (iii) is
Weyl's equidistribution theorem for irrational rotations. The
verifier confirms $\|q\rho\| > 10^{-6}$ for all $q \le 10^4$ as a
spot check.
\end{proof}

The theorem reverses the polarity of the corpus's small residuals.
They are not errors awaiting better derivations; they are invariants
of an irrational rotation. Detuning is not a defect of the
derivations but a theorem of the dynamics. Statement (iii) is also
the dynamical substrate of the corpus's design principle that every
interaction moves the system toward regions it has not visited yet:
an irrational rotation visits every neighbourhood and repeats
nothing.

\begin{theorem}[Derivation of 432 (A267)]
\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
$432 = 2^4 3^3$. The neighbouring lattice points are $384$ and
$486$, both farther by an order of magnitude: the distance to $432$
is $1.49$, against $46.5$ and $55.5$ for the neighbours. Hence P14's
432 correspondence is the derived statement ``the breath's nearest
just chord is $2^4 3^3$,'' and $G_1$ (Theorem~\ref{thm:g1}) is the
comma of that chord.
\end{theorem}

The derivation decouples the historical resonance from the
mathematics: any culture tuning to small-prime ratios near this
period finds 432. The factorisation $16 \times 27$ is the bulk
density coefficient times the cube of the boundary coefficient, a
reading taken up again in Section 6.

% ════════════════════════════════════════════════════════════════════
\section{The Arithmetic of the Commas}

\subsection{The ladder}

The continued fraction of the rotation number,
\[
\rho \;=\; [0;\,1,1,19,1,10,1,12,8,4,1,14,129,1,7,\ldots],
\]
yields the hierarchy of best rational approximations $p/q$: the
cycle lengths at which the breath nearly closes. The convergent
denominators begin $q = 1, 2, 39, 41, 449, 490$, and the commas
$\|q\rho\|$ decrease strictly along them. The first rungs: $q = 2$
breaths with comma $0.0245$, then $q = 41$ with comma $0.0020$, then
deeper levels. This is the system's intrinsic scale, in exact analogy
with the twelve-tone system arising from the continued fraction of
$\log_2(3/2)$. In conditional SI (A265, one breath
$\approx 1.37$ Myr) the 2-breath near-cycle is $\approx 2.7$ Myr and
the 41-breath near-closure is $\approx 56$ Myr; the ladder extends
through $q = 449$ ($\approx 617$ Myr) to $q = 490 = 449 + 41$, the
sixth convergent denominator, taken up in Section 6.

\begin{remark}[Near-tritone structure (A267)]
$\rho \approx \tfrac12 + 0.0122$: each breath advances the fiber by
almost exactly a half-turn, so 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}

\subsection{The Brjuno property}

Small-denominator theory makes the ladder quantitative only if the
rotation is well-behaved enough for first-order response theory to
apply.

\begin{theorem}[KAM-good rotation (A268)]
\label{thm:brjuno}
The partial quotients of $\rho$ to depth 25 are bounded (maximum
$129$), and the Brjuno sum converges:
$\sum_k \log q_{k+1}/q_k = 2.803$. The breath rotation is therefore
linearizable and perturbative response is well-defined at first
order; there is no small-denominator catastrophe.
\end{theorem}

The check was live, not decorative: had a giant partial quotient
appeared early, signalling near-resonance, the response theory below
and A265's inference with it would have been in danger. It did not.

\subsection{The phase channel: gain exactly 1}

\begin{theorem}[Mean-field gain is unity (A268)]
\label{thm:gain1}
For forcing with zero spatial frequency, a static uniform field, the
breath map's rotation-number response is exactly $1$: the map
$\theta \mapsto \theta + \rho + \varepsilon$ has rotation number
$\rho + \varepsilon$ identically, by the translation structure of
the circle. 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 law
$a_0 = \alpha\,c^2/(\Tb^2 L_h)$ holds with derived coefficient:
inference (b) of A265 is derived. The clock cannot be detuned at
first order; pacing is rigid (Pin~3, derived in A269).
\end{theorem}

\begin{proposition}[Exact mode ladder (A268)]
\label{prop:cohomological}
For mode-$q$ periodic forcing the first-order conjugacy is given
exactly by the cohomological equation,
\[
h_q \;=\; \frac{\hat g_q}{1 - e^{2\pi i q\rho}},
\qquad
|{\rm gain}_q| \;=\; \frac{1}{2\,|\sin(\pi q \rho)|},
\]
solvable away from tongues since $\rho$ is Brjuno. The gains are
$0.50$ at $q = 1$ and $6.5$ at $q = 2$, and peak along the convergent
ladder of Section 4.1: response at a near-closure rung $q$ is
amplified as $1/\|q\rho\|$. The conjugacy identity
$h(\theta+\rho) - h(\theta) = g(\theta) - \hat g_0$ is verified
pointwise at machine precision via the Fourier solution.
\end{proposition}

A methodological note is on record (A268): a naive orbit-difference
estimator mis-measures these gains near resonances, because transient
times exceed any fixed orbit length. Orbit-difference estimators mis-measure these gains near resonances; the cohomological route is required. Simulation is not a substitute for
solving the cohomological equation when small denominators are in
play.

\subsection{Arnold tongues, not Julia sets}

\begin{remark}[The corrected dictionary (A268)]
\label{rem:tongues}
An earlier framing proposed Fatou/Julia as the order/chaos dichotomy
at the breath layer, with canonical/dark as its reading. For a rigid
rotation this is the wrong object: rotations are isometries with
empty Julia sets. The order/chaos structure of circle dynamics is the
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 under mode-2 forcing only at strength
$\varepsilon \ge \varepsilon_c = 2\pi\delta \approx 0.0769$, the
standard tongue-edge criterion, confirmed numerically. The dictionary
candidate becomes: canonical $=$ locked traversals, dark $=$ drifting
traversals, with tongue widths as classification thresholds.
\end{remark}

\subsection{The modulus channel: gain exactly $\alpha$}

\begin{theorem}[Modulus channel (A275/P35)]
\label{thm:alpha}
The only symmetry-allowed deformation moving the multiplier
$|\lambda|$ off the unit circle is radial drift, and at unit source
(the universal field $a_0$ at chart scale for one breath) its gain is
$a_0 (\TbSIeq)^2 / L_h = \alpha$ exactly.
\end{theorem}

Together with Theorem~\ref{thm:gain1} this exhausts the multiplier's
deformation space: rigid clock, weak infall, ratio $\alpha$. It is
the dynamical restatement of $\alpha$ as the price of observation.

% ════════════════════════════════════════════════════════════════════
\section{The Breath Map on $\CP$}

The fiber rotation of Section 3 is one register of a bundle map whose
other register lives on the Hopf base $S^2 \cong \CP$.

\begin{proposition}[The map and its multiplier (A274)]
\label{prop:map}
Let $u = z_2/z_1$ be the holomorphic coordinate on the Hopf base,
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 the accumulated SO(3) rotation angle is $4\pi\Tb$ exactly,
$\kappa$-free by A273's cancellation theorem. The breath map is
therefore the M\"obius transformation
\[
B(u) \;=\; \lambda\,u, \qquad \lambda \;=\; e^{2\pi i \rho_*},
\qquad
\rho_* \;=\; 2\Tb \bmod 1 \;=\; 2\rho - 1 \;=\; 0.0244904\ldots,
\]
with $|\lambda| = 1$ exactly. The factor 2 is P28's covering lift
($\|\Omega\| = 2\omega_1$); the register identity
$\rho_* = 2\rho - 1$ is convention-independent given A273's count of
$2\Tb$ SU(2) windings per breath.
\end{proposition}

The base rotation number equals the fiber's $q = 2$ comma
$|2\rho - 1|$ of Section 4.1: per breath, the base rotates by exactly
the amount by which two breaths fail to close the fiber. The two
registers of A273 are the fiber and base rotation numbers of one
bundle map.

\begin{theorem}[Elliptic, irrational, Julia-free (A274)]
\label{thm:elliptic}
$B$ is an elliptic M\"obius transformation:
$|{\rm tr}| = 2|\cos(\pi\rho_*)| = 1.9941 < 2$. Its rotation number
$\rho_* = 8\pi^4 + 2\pi^3 + 2\pi^2 - 861$ is irrational by the
argument of Theorem~\ref{thm:detuning}. 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) $\lambda$ is not a root of unity, so $B$ has no periodic orbits
other than the two fixed points, 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 (A274)]
\label{cor:retire}
Under a Fatou/Julia $=$ canonical/dark dictionary, an elliptic breath
would make nothing dark, contradicting the measured halo (A264) and
the never-empty corollary of A272. The proposal is closed negatively;
its replacement is on file: the locked/drifting dichotomy of Arnold
tongues (Remark~\ref{rem:tongues}) and the kernel typing of dark
(A272/A276). Fatou/Julia may yet enter at the level of perturbed,
non-isometric M\"obius maps; that question is parked, correctly typed.
\end{corollary}

\begin{theorem}[Unique ergodicity; the halo as the unique invariant
measure (A274)]
\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 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.
\end{theorem}

The reading deserves emphasis because it changes the epistemic status
of the halo. Stationarity is usually an equilibrium assumption; here
it is unique ergodicity of the breath map. The halo is not a
configuration that happens to be stationary; it is the only
stationary angular distribution the breath dynamics admits. Any
other stationary angular distribution is dynamically impossible.

\begin{remark}[Brjuno multiplier; the 56-Myr register coincidence
(A274)]
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 $\approx 0.287$, so perturbations of $B$ keeping the
multiplier unimodular 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 $\approx 56$ Myr: the same scale as
the fiber's $q = 41$ rung. Both registers single out the 56-Myr cycle
as the system's most robust intrinsic timescale.
\end{remark}

\begin{remark}[Lens $\mathbb{Z}_3$ (A274)]
The lens action $u \mapsto e^{2\pi i/3}u$ (A255) is a rotation about
the same axis and commutes with $B$: the $\mathbb{Z}_3$ symmetry is a
symmetry of the iteration. Per A255 discipline, no identification
with triality or the A44 Jordan cycle is made.
\end{remark}

% ════════════════════════════════════════════════════════════════════
\section{Tonality: The Score}

Theorem~\ref{thm:detuning} says the breath never closes. The next
question is how it fails to close, and the instruments for answering
it come from an unexpected discipline: multiplicative hashing. A hash
multiplier is an irrational rotation on $\mathbb{Z}/2^w$, and its
fifty-year-old quality theory (equidistribution, discrepancy,
spectral resonance; Knuth's Fibonacci hashing with multiplier
$2^{32}/\varphi$) is exactly the mathematics of rotation-number
tonality, with the preference reversed: a hash is engineered to be
atonal, because for a hash a near-closure is clustering. The
instruments transfer; only the sign of the verdict flips.

\begin{definition}[Tonality index (A289)]
\label{def:tonality}
For an irrational $\theta$ and horizon $Q$, let
$g(\theta, Q) = \max_{1 \le q \le Q} \tfrac{1}{2\|q\theta\|}$, the
largest small-denominator gain below $Q$
(Proposition~\ref{prop:cohomological}'s response amplification). The
tonality index is $T(\theta, Q) = g(\theta, Q)/g(\varphi, Q)$: how
much louder the system's loudest overtone is than the most atonal
rotation's. $T \approx 1$ is atonal (hash-grade); $T \gg 1$ is tonal.
\end{definition}

\begin{proposition}[The overtone ladder (A289)]
\label{prop:ladder}
The record-gain rungs of $\rho$ below $q = 500$, with commas, gains,
cents, and conditional SI periods:
\begin{center}
\begin{tabular}{rrrrl}
\toprule
$q$ & $\|q\rho\|$ & gain & cents & conditional SI \\
\midrule
2   & $0.024490$ & $20.4$  & $41.89$ & $2.7$ Myr (near-tritone pair) \\
39  & $0.022437$ & $22.3$  & $38.41$ & $54$ Myr \\
41  & $0.002054$ & $243$   & $3.55$  & $56$ Myr \\
449 & $0.001897$ & $264$   & $3.28$  & $617$ Myr \\
490 & $0.000157$ & $3194$  & $0.27$  & $673$ Myr \\
\bottomrule
\end{tabular}
\end{center}
Here cents $= 1200\log_2(1 + \|q\rho\|)$; for comparison, the
432-comma is $G_1 = 5.97$ cents. $q = 490 = 449 + 41$ is the sixth
convergent denominator; its gain exceeds the $q = 41$ rung by an
order of magnitude. The ladder jumps, with record gains
$1 \to 20 \to 243 \to 3194$, where $\varphi$'s Fibonacci ladder
creeps without record gaps ($1.3, 2.1, 3.4, 5.5, 9.0, \ldots$,
maximum $421$ below $500$). SI values are conditional on A265's
calibration and carry no external identification.
\end{proposition}

\begin{proposition}[Scores (A289)]
\label{prop:scores}
At $N = 10^4$ the star discrepancy of the breath rotation is
$D^*_N(\rho) = 7.5\times10^{-4}$, against
$D^*_N(\varphi) = 2.6\times10^{-4}$: the breath is about $3\times$
worse-spread than the optimal hash rotation, and would make a poor
hash. The tonality index is $T(\rho, 500) = 7.6$. Both verdicts have
one cause: the large partial quotients ($19, 10, 12, \ldots$) make
$\rho$ cling to specific rationals without touching them.
Resonance-rich, equidistribution-poor: tonal.
\end{proposition}

The breath has discrete strong overtones at $2$, $41$, and $490$
breaths. In cents, the $q = 2$ comma is a near-quarter-tone (the
tritone pairing of successive breaths, Section 4.1), and the $q = 41$
comma is schisma-scale (the 56-Myr rung).

\begin{proposition}[The substrate chord (A289)]
\label{prop:chord}
The density coefficients (edge $2$, boundary $3$, bulk $16$) form
just intervals: $3{:}2 = 702.0$ cents, the perfect fifth;
$16{:}3 = 2898.0$ cents, two octaves plus a perfect fourth
($498.0$ cents); $16{:}2$, three octaves exactly. All three intervals
are 3-smooth consonances; the chord is as consonant as three distinct
small integers permit. Downstream echoes on file: $432 = 2^4 3^3$
(Theorem~\ref{thm:432}) and the unpromoted flag
$j(i) = 1728 = 4\cdot 432$. By A287 the lepton mass corrections are
proportional to the chord tones (the muon draws the fifth's $3$, the
tau the bass $16$, the electron anchors at the edge $2$); the
canonical treatment is deferred to the fermion-spectrum paper.
\end{proposition}

\begin{remark}[Tuned detuning; interpretation (A289)]
\label{rem:tuned}
Theorem~\ref{thm:detuning} closes one side: exact closure is
impossible, and a closed orbit would visit finitely many states.
Proposition~\ref{prop:scores} closes the other: $\varphi$-grade
atonality would be structureless, with no rung louder than any other,
no scale, no preferred cycles, nothing for the susceptibility of
Proposition~\ref{prop:cohomological} to amplify. The breath sits
between: irrational, so it never closes, and tonal, with discrete
loud overtones. Every temporal structure of the corpus, the 432-chord
and its comma, the 56-Myr rung, the tritone pairing, the mass
voicing, lives in that middle. On this reading the tune is not a
metaphor: it is the record-gain sequence of
Proposition~\ref{prop:ladder}, played on the chord of
Proposition~\ref{prop:chord}.
\end{remark}

% ════════════════════════════════════════════════════════════════════
\section{The Necessity of Music}

Is the tonality forced? The question (filed as OI-289-1) has a
three-layer answer, in decreasing strength.

\begin{theorem}[Music is almost sure (A290)]
\label{thm:music}
Let $\theta$ be a rotation number. (i) If $\theta$ is rational the
orbit closes: finitely many states (silence; excluded for the breath
by Theorem~\ref{thm:detuning}). (ii) The set of irrational $\theta$
with bounded partial quotients (the atonal, badly-approximable class,
for which record gains grow only linearly, as $\varphi$'s do) is
Lebesgue-null (Khinchin 1926). (iii) For almost every $\theta$ the
partial quotients are unbounded (Borel--Bernstein), hence the record
gains $g(\theta, Q)$ are unbounded in $Q$: arbitrarily loud overtones
occur. Therefore, in the only sense presently provable, the existence
of a tune is forced: among rotation numbers, silence is countable,
atonality is null, and music has full measure.
\end{theorem}

The division of labor is worth stating exactly. The kernel forbids
silence: Theorem~\ref{thm:detuning} derives irrationality from the
transcendence of $\pi$. Measure forbids atonality, almost surely:
Theorem~\ref{thm:music} supplies tonality not as a property of the
kernel but as a property of almost everything the kernel could have
produced. Nothing known selects the melody. The two states requiring
fine-tuning are exactly the two that the corpus forbids and that hash
engineers build: closure and $\varphi$-atonality.

\begin{proposition}[$\rho$ is statistically generic (A290)]
\label{prop:placement}
Against a seeded uniform Monte-Carlo null ($N = 2\times10^4$,
reproduced by the verifier): maximum gain at $Q = 500$, $\rho$'s
$3194$ against a null median of $610$, the $90.4$th percentile;
geometric mean of the first eight partial quotients, $3.41$ against
Khinchin's constant $2.685$ and a null median of $2.56$, the $76$th
percentile; Brjuno-8 sum, $2.802$ against a null median of $2.45$,
the $77$th percentile. Elevated, never exceptional: one irrational in
ten beats the breath's loudness at this horizon. Per the corpus's own
identification bar, $\rho$'s tonality carries no evidence of
selection.
\end{proposition}

This is the honest deflation. The tonality index of
Definition~\ref{def:tonality} measures distance from $\varphi$;
Proposition~\ref{prop:placement} measures distance from typical, and
finds little. The existence of the tune is necessary
(Theorem~\ref{thm:music}), its loudness is statistically ordinary,
and the melody, the specific quotients $19, 10, 12, \ldots$ and rungs
$2, 41, 490$, is the contingent fingerprint.

\begin{remark}[The specific value, typed (A290)]
Whether $\rho = (4\pi^4 + \pi^3 + \pi^2) \bmod 1$ is forced to have
these particular quotients reduces to Diophantine control of an
explicit $\pi$-polynomial. The state of the art: it is not known
whether $\pi$'s own partial quotients are bounded;
irrationality-measure bounds of Salikhov type are far too weak to
constrain individual quotients; no technique distinguishes
$4\pi^4 + \pi^3 + \pi^2$ from a generic real. The question is
therefore typed as open at the foundations of Diophantine
approximation, not at the corpus, and parked with its revival
condition recorded: any future control of $\pi$-polynomial continued
fractions. No open item is kept on it, because no research program
can currently attack it. OI-289-1 closes: answered affirmatively in
measure, negatively in selection, typed as unreachable in value.
\end{remark}

\begin{proposition}[The realized score: seven rungs (A290)]
\label{prop:realized}
Within the universe's breath span ($\approx 3.4\times10^4$ breaths,
A265), the record-gain ladder has exactly seven realized rungs,
\[
q \;=\; 1,\; 2,\; 39,\; 41,\; 449,\; 490,\; 6329,
\]
with gains $1, 20, 22, 243, 264, 3194, 26202$ and conditional SI
periods $1.4$, $2.7$, $54$, $56$, $617$, $673$ Myr, and $8.70$ Gyr.
The deepest realized overtone has a period of order the age of the
universe; this is recorded as consonance only, with no
identification. The next rung, $q = 51122$, lies beyond the span: the
universe has not yet played it.
\end{proposition}

% ════════════════════════════════════════════════════════════════════
\section{Numerical Summary}

All quantities below are exact functions of corpus constants or
verifier-pinned computations; none is fitted. SI rows are conditional
on A265's calibration.

\begin{center}
\begin{tabular}{llll}
\toprule
quantity & value & meaning & source \\
\midrule
$\Tb$ & $430.5122$ & breath period, $\pi\alpha^{-1}$ & P14/A239/P35 \\
$\rho$ & $0.5122452$ & fiber rotation number, $\Tb \bmod 1$ & A267 \\
$\rho_*$ & $0.0244904$ & base rotation number, $2\rho - 1$ & A274 \\
$G_1$ & $3.45578\times10^{-3}$ & 432-comma ($5.97$ cents) & A266 \\
$\delta$ & $0.012245$ & distance to period-2 tongue & A268 \\
$\varepsilon_c$ & $0.0769$ & tongue edge, $2\pi\delta$ & A268 \\
$B(\rho)$ & $2.803$ & Brjuno sum, fiber & A268 \\
$B(\rho_*)$ & $\approx 0.287$ & Brjuno sum, base (Siegel) & A274 \\
$|{\rm tr}\,B|$ & $1.9941$ & elliptic: $< 2$ & A274 \\
phase gain & $1$ (exact) & mean-field response & A268 \\
modulus gain & $\alpha$ (exact) & radial drift at unit source & A275/P35 \\
$D^*_{10^4}(\rho)$ & $7.5\times10^{-4}$ & star discrepancy (vs $\varphi$: $2.6\times10^{-4}$) & A289 \\
$T(\rho, 500)$ & $7.6$ & tonality index & A289 \\
rungs $q \le 500$ & $1,2,39,41,449,490$ & record-gain ladder & A267/A289 \\
gains & $1, 20.4, 22.3, 243, 264, 3194$ & $1/(2\|q\rho\|)$ & A289 \\
realized rungs & seven ($q \le 6329$) & deepest $8.70$ Gyr & A290 \\
placement & $90.4$ / $76$ / $77$ pct & gain / GM-8 / Brjuno-8 & A290 \\
chord & $(2,3,16)$ & fifth, fourth $+$ 2 oct, 3 oct & A289 \\
\bottomrule
\end{tabular}
\end{center}

% ════════════════════════════════════════════════════════════════════
\section{Falsifiers and Open Items}

\textbf{Verifier-pinned checks.} Every numerical claim above is
recomputed by \texttt{verify\_P266}, \texttt{P267}, \texttt{P268}, \texttt{P274}, \texttt{P289}, and \texttt{P290}
from the corpus constants alone: the $G_1$ identity at 50 digits, the
3-smooth scan with its order-of-magnitude margin, the convergent
ladder and strict comma decrease, the Brjuno sums, the exact
mean-field gain, the cohomological identity at machine precision, the
tongue edge $\varepsilon_c = 2\pi\delta$ (locked above, drifting
below, confirmed by iteration), ellipticity, Weyl equidistribution at
$N = 10^5$, lens commutation, the discrepancies, the tonality index,
the seeded placement percentiles, and the realized ladder. A failure
of any check falsifies the corresponding statement as filed.

\textbf{Structural falsifiers.} Theorem~\ref{thm:ergodic} forbids any
stationary angular distribution on an invariant circle other than the
uniform one: a demonstrated stationary non-uniform angular profile
would contradict the elliptic typing of the breath map.
Remark~\ref{rem:tongues} fixes the locking threshold under mode-2
forcing at $\varepsilon_c = 2\pi\delta$; locking below it or drifting
above it would falsify the tongue placement.
Proposition~\ref{prop:placement} records that the breath's tonality
shows no evidence of selection; a derivation that forced the specific
quotients would overturn that verdict and is welcome (its difficulty
is typed in Section 7).

\textbf{Open items.} \textbf{OI-268-2}: formalize locked/drifting
$=$ canonical/dark against ToENet's luminosity channel, with the
comma inventory of TASK\_035. \textbf{OI-268-1} (the kernel
derivation of breath pacing, Pin~3 as theorem rather than
constitution) was filed in A268 and is addressed in A269, where
pacing rigidity is derived; the phase-channel statement of
Theorem~\ref{thm:gain1} carries that cross-reference.
\textbf{OI-289-1} is closed by A290 in three layers (Section 7).
The specific-value question is parked, not open: no current
mathematics can attack it, and its revival condition is recorded.

% ════════════════════════════════════════════════════════════════════
\section{Relation to the Corpus}

The breath theory touches the corpus at five points. \emph{Paper 14}:
the 432 correspondence is upgraded from observation to theorem
(Theorem~\ref{thm:432}), and its defective reconciliation factor is
retired in favor of the exact comma $G_1$
(Theorem~\ref{thm:g1}). \emph{Paper 28}: the factor 2 between fiber
and base registers is the covering lift $\|\Omega\| = 2\omega_1$, and
the near-tritone alternation of successive breaths is consonant with
the canonical mirror configuration. \emph{Paper 35}: supplies the
unit bridge that makes $\rho$ definitionally exact, the conditional
SI calibration behind every Myr and Gyr figure, and the modulus
channel's $\alpha$. \emph{The hidden sector}: the halo profile
confronted with SPARC in A264 is re-derived here as the unique
invariant measure of the breath map
(Theorem~\ref{thm:ergodic}), and the canonical/dark dictionary at the
breath layer is fixed as locked/drifting
(Remark~\ref{rem:tongues}, Corollary~\ref{cor:retire}), with kernel
typing per A272/A276. \emph{The fermion spectrum}: the substrate
chord $(2,3,16)$ of Proposition~\ref{prop:chord} is the same triple
from which the lepton mass corrections draw (A287); the canonical
treatment lives in the fermion-spectrum paper.

The comma framework also generalises beyond the breath: residual gaps
as kernel elements of closure maps recur in A254's collapse delta,
A259's fold defect, and A276's closure-map family. The breath's
commas are the temporal instance of that shape.

% ════════════════════════════════════════════════════════════════════
\section{Conclusion}

One irrational number, $\rho = \Tb \bmod 1$, determined to all digits
by the corpus constants, carries the entire temporal phenomenology of
the framework. Its irrationality makes detuning necessary and turns
the corpus's small residuals into invariants. Its nearest just chord
derives 432 and identifies the last unfitted constant as a comma. Its
continued fraction fixes the intrinsic timescales, with the 56-Myr
rung singled out by both registers of the bundle map. Its Brjuno
property validates first-order response, where the phase channel has
gain exactly $1$ and the modulus channel gain exactly $\alpha$. Its
ellipticity on $\CP$ makes the halo the unique stationary angular
distribution. Its large partial quotients make the breath tonal,
with measured overtones at $2$, $41$, and $490$ breaths; and measure
theory makes some tune almost sure, while nothing known selects this
one. Closure is forbidden, noise is null, the tune is necessary, the
melody is the fingerprint; within its span the universe has realized
seven rungs of it.

\paragraph{Provenance and verification.} Assembled without new
claims from A266 ($G_1$; \texttt{verify\_P266}), A267 (detuning,
432, ladder; \texttt{verify\_P267}), A268 (Brjuno, gain, tongues;
\texttt{verify\_P268}), A274 (elliptic, ergodicity;
\texttt{verify\_P274}), A289 (the score; \texttt{verify\_P289}),
A290 (measure; \texttt{verify\_P290}), all green at filing. Unit
conventions per Paper 35. Cross-references: A269 (pacing), A272/A276
(kernel typing), A287 (chord voicing of the lepton masses).

\end{document}
