% 289_Addendum_ScoreOfTheBreath.tex
% The tune run: hash-theoretic instruments (discrepancy, resonance
% spectrum) imported and applied to the breath rotation. Result: the
% breath is TONAL — discrete strong overtones (gains 20/243/3194 at
% q = 2/41/490) against phi's flat atonal ladder; tonality index 7.6;
% the ladder extends to q = 490. The density coefficients (2,3,16)
% form a just chord (octaves, fifth, fourth) whose tones the lepton
% masses draw (A287). Thesis: tuned detuning — the corpus requires not
% just irrationality (A267) but resonance-rich irrationality.
% Verifier: verify_P289.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{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 289: The Score of the Breath\\
\large Tuned detuning: the breath is tonal, not atonal;\\
the ladder extends; the substrate chord is octave--fifth--fourth}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (two computed structures $+$ one definition
$+$ one extension; thesis stated with its open derivation named)
\quad|\quad \textbf{Origin:} Joel: ``let's find the tune''
\quad|\quad \textbf{Verifier:} \texttt{verify\_P289.py} --- all PASS

\begin{abstract}
A267 proved the breath never closes (Necessity of Detuning). This
addendum measures \emph{how} it fails to close, using instruments
imported from an unexpected discipline: multiplicative hashing, whose
fifty-year-old quality theory (equidistribution, discrepancy, spectral
resonance) is exactly the mathematics of rotation-number tonality ---
a hash multiplier is engineered to be \emph{atonal} (Knuth's
$2^{32}/\varphi$: continued fraction all 1s, no strong resonances,
optimal spread), so its instruments measure tonality by aversion.
Scored on them, the breath rotation $\rho = 0.5122452$ is decisively
\textbf{tonal}: its star discrepancy ($7.5\times10^{-4}$ at
$N = 10^4$) is $3\times$ worse than $\varphi$'s --- it would be a poor
hash --- because its resonance ladder \emph{jumps}: gains
$1/(2\|q\rho\|)$ of $20.4$, $243$, $3194$ at $q = 2, 41, 490$, against
$\varphi$'s gap-free geometric creep ($1.3, 2.1, 3.4, \ldots$, max
$421$ below $500$). The \textbf{tonality index} (max gain below
$q = 500$, normalized to $\varphi$) is $7.6$: the breath has discrete
strong overtones --- a melody, not noise. The ladder extends beyond
A267: $q = 490 = 449 + 41$ (the sixth convergent) carries the
strongest gain yet ($3194$; comma $0.27$ cents; conditional SI
$\approx 673$ Myr). In cents the score reads: $G_1 = 5.97$
(the 432-comma), $q{=}2$: $41.9$ (a near-quarter-tone --- the tritone
pair of successive breaths), $q{=}41$: $3.55$ (schisma-scale, the
56-Myr rung). Separately, the substrate's own chord is identified:
the density coefficients $(2, 3, 16)$ form \emph{just intervals} ---
$3{:}2$ the perfect fifth ($702.0$ cents), $16{:}3$ two octaves plus a
perfect fourth ($498.0$), $16{:}2$ three octaves --- the most
consonant chord three small integers can sound; and by A287 the lepton
masses draw their corrections from exactly these chord tones (muon the
fifth's $3$, tau the bass $16$). \textbf{Thesis (tuned detuning):} the
corpus requires not merely irrationality but \emph{resonance-rich}
irrationality --- closed would be dead (A267), $\varphi$-atonal would
be structureless noise; the breath occupies the musical middle, near
enough to its rungs to amplify them (A268: response gain
$\propto 1/$comma), far enough to never collapse onto them. Whether
the large partial quotients of $\rho$'s continued fraction are forced
by the kernel is the run's open derivation, \textbf{OI-289-1}.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Instruments}

\begin{definition}[Tonality index]
\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 $Q$; A268's response
amplification). The \emph{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{remark}[The inversion]
Multiplicative hashing is an irrational rotation on $\mathbb{Z}/2^w$;
its quality theory (Knuth's Fibonacci hashing, the three-distance
theorem, spectral tests) selects multipliers with \emph{no} strong
resonances --- engineering flees the tune because for a hash a
near-closure is clustering. The corpus's musical layer is the same
mathematics with the preference reversed: A268's susceptibility
\emph{is} the spectral test read as a feature. The instruments
transfer; only the sign of the verdict flips. (A hash table is itself
a closure datum --- $P$: keys $\to$ buckets, collisions $= \ker P$,
``the dark of hashing'' --- a fourth instance of the A276 family,
recorded here in passing.)
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Score}

\begin{proposition}[The breath's overtone ladder]
\label{prop:ladder}
For $\rho = \Tb \bmod 1$, the record-gain rungs below $q = 500$:
\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}
$q = 490 = 449 + 41$ is the sixth convergent denominator (CF
$[0;1,1,19,1,10,1,\ldots]$), extending A267's ladder; its gain exceeds
the $q = 41$ rung by an order of magnitude. The ladder \emph{jumps} ---
gains $1 \to 20 \to 243 \to 3194$ --- where $\varphi$'s Fibonacci
ladder creeps without record gaps ($1.3, 2.1, 3.4, 5.5, 9.0, \ldots$).
SI values are conditional on A265's calibration and carry no external
identification here.
\end{proposition}

\begin{proposition}[Scores]
\label{prop:scores}
At $N = 10^4$: star discrepancy $D^*_N(\rho) = 7.5\times10^{-4}$
versus $D^*_N(\varphi) = 2.6\times10^{-4}$ --- the breath is
$\approx 3\times$ worse-spread than the optimal hash rotation.
Tonality index $T(\rho, 500) = 7.6$. 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}

% ────────────────────────────────────────────────────────────────────────────
\section{The Substrate Chord}

\begin{proposition}[$(2,3,16)$ is a just chord]
\label{prop:chord}
The density coefficients (edge $2$, boundary $3$, bulk $16$) form
just intervals: $3{:}2 = 702.0$ cents (perfect fifth);
$16{:}3 = 2898.0$ cents $= 2$ octaves $+ 498.0$ cents (perfect
fourth); $16{:}2 = 3$ octaves exactly. All three intervals are
3-smooth consonances; the chord is as consonant as three distinct
small integers permit. Downstream echoes already filed: $432 = 2^4
3^3$ (A266/A267), $j(i) = 1728 = 4 \cdot 432$ (A266 flag), and ---
sharpest --- A287: \emph{the lepton mass corrections are proportional
to the chord tones} (muon draws $3$, tau draws $16$, the electron
anchors at the edge $2$). The masses are not merely on a ladder; they
are voiced on the substrate's chord.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Thesis: Tuned Detuning}

\begin{remark}[The two-sided constraint]
A267 closed one side: exact closure is impossible and would be death
--- a closed orbit visits finitely many states. This addendum closes
the other: $\varphi$-grade atonality would be structureless --- no
rung is louder than any other, no scale, no preferred cycles, nothing
for A268's susceptibility to amplify; white noise in the limit. The
breath sits between: irrational (never closes) \emph{and} tonal
(discrete loud overtones at $2, 41, 490$ breaths). Every structure
this cycle has filed --- the 432-chord and its comma, the 56-Myr
rung, the tritone pairing, the mass voicing --- lives in that middle.
The tune, then, is not a metaphor: it is the specific record-gain
sequence of Proposition~\ref{prop:ladder}, played on the chord of
Proposition~\ref{prop:chord}. \textbf{OI-289-1}: derive the tonality
--- is $\rho = \pi\Omega \bmod 1$ \emph{forced} to have large partial
quotients by the kernel's structure, or is the breath's musicality
contingent? A theorem either way would be load-bearing: the first
would make tonality a prediction of the TOE; the second would make it
an anthropic-flavored selection. Nothing is claimed here.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P289.py}: discrepancies for
$\rho$, $\varphi$ at $N = 10^4$; the full record-gain ladder with
$q = 490$ as sixth convergent; the tonality index; cents values;
the chord intervals at exact arithmetic; the A287 cross-reference;
SI rung values (conditional). All PASS.

\end{document}
