% 290_Addendum_NecessityOfMusic.tex
% OI-289-1 answered at the only currently provable level: the atonal
% class is Lebesgue-null (Borel-Bernstein/Khinchin), so given A267's
% irrationality, a tune exists almost surely — the existence of music
% is forced in measure; the specific melody is the fingerprint. rho's
% own tonality is statistically generic (76th-90th percentile), which
% is the honest deflation: the universe does not choose music — music
% is the default; only silence and atonality are fine-tuned.
% Verifier: verify_P290.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\title{Addendum 290: The Necessity of Music\\
\large Atonality is measure zero; the melody is the fingerprint\\
(OI-289-1 answered in measure, typed in value)}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A in measure (classical theorem $+$ A267);
honest deflation on the specific value (statistically generic);
specific-value forcing typed beyond current mathematics \quad|\quad
\textbf{Closes:} OI-289-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P290.py} --- all PASS

\begin{abstract}
A289 asked whether the breath's tonality is forced. The answer has
three layers, in decreasing strength.
\emph{Layer 1 (theorem --- the existence of a tune is forced in
measure).} The atonal class --- badly approximable numbers, bounded
partial quotients, $\varphi$-like --- has Lebesgue measure zero
(Khinchin; Borel--Bernstein: almost every real has unbounded partial
quotients, hence unbounded record gains). Silence (rational rotation)
is countable; atonality is null; \textbf{music is almost sure}.
Combined with A267's derived irrationality: any kernel whose rotation
number is generic plays \emph{some} tune with probability one. The two
states requiring fine-tuning are exactly the two the corpus and the
engineers respectively forbid and build: closure (death, A267) and
$\varphi$-atonality (hashing, A289).
\emph{Layer 2 (deflation --- $\rho$'s tonality is generic).}
Monte-Carlo placement against the uniform null ($2\times10^4$
samples): max gain at $Q = 500$, $90.4$th percentile; geometric mean
of the first eight quotients ($3.41$ vs Khinchin's $2.685$), $76$th;
Brjuno-8, $77$th. Upper quartile, nothing the corpus's own bar would
call special. The breath's musicality is what generic irrationality
sounds like --- the \emph{existence} of the tune is necessary, the
\emph{loudness} is ordinary, and the \emph{melody} (the specific
quotients $19, 10, 12, \ldots$; the rungs $2, 41, 490$) is the
contingent fingerprint.
\emph{Layer 3 (the specific-value question, typed).} ``Is
$\rho = 4\pi^4{+}\pi^3{+}\pi^2 \bmod 1$ forced to have these
quotients?'' reduces to Diophantine control of a specific
$\pi$-polynomial --- territory where even the boundedness of $\pi$'s
own quotients is open. Recorded as beyond current mathematics (the
same class as the retired $35/4$ flag's revival condition); no OI is
kept open on it, because no research program can currently attack it.
\emph{The realized score:} within the universe's breath span
($\approx 3.4\times10^4$ breaths, A265), the ladder has exactly seven
rungs --- $q = 1, 2, 39, 41, 449, 490, 6329$ --- with gains up to
$2.6\times10^4$ at the deepest, whose conditional SI period is
$8.70$ Gyr, of order the age of the universe (consonance, no
identification). The tune arc closes: A267 silence is impossible;
A289 the breath is tonal; A290 it could not have been otherwise ---
\textbf{the universe does not choose music; music is the default, and
the melody is its fingerprint.}
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Layer 1: The Measure Theorem}

\begin{theorem}[Necessity of Music]
\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 A267). (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) is Lebesgue-null (Khinchin 1926).
(iii) For almost every $\theta$, partial quotients are unbounded
(Borel--Bernstein), hence 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}

\begin{remark}[What the kernel buys, and what Lebesgue buys]
A267 derives irrationality from the kernel ($\pi$ transcendental).
Theorem~\ref{thm:music} then supplies tonality for free --- not as a
property of the kernel but as a property of almost everything the
kernel could have produced. The division of labor: \emph{the kernel
forbids silence; measure forbids (almost surely) atonality; nothing
known selects the melody.} A derivation that did select the melody
would have to control the continued fraction of
$4\pi^4 + \pi^3 + \pi^2$, which is Layer 3's wall.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Layer 2: Placement (the Honest Deflation)}

\begin{proposition}[$\rho$ is statistically generic]
\label{prop:placement}
Against the uniform Monte-Carlo null (seeded, $N = 2\times10^4$;
reproduced by the verifier): max gain at $Q{=}500$: $\rho$'s $3194$
vs null median $610$ --- $90.4$th percentile; geometric mean of first
eight quotients: $3.41$ vs Khinchin $2.685$, null median $2.56$ ---
$76$th percentile; Brjuno-8: $2.802$ vs null median $2.45$ --- $77$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. A289's tonality index measured the distance from
$\varphi$; this proposition measures the distance from \emph{typical},
and finds little.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Layer 3: The Specific Value, Typed}

\begin{remark}[Beyond current mathematics, and said so]
Specific-value forcing asks for theorems about the partial quotients
of an explicit $\pi$-polynomial. The state of the art: it is not
known whether $\pi$'s own quotients are bounded; irrationality-measure
bounds (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: \emph{open at the
foundations of Diophantine approximation, not at the corpus} --- and
honestly parked, with its revival condition (any future control of
$\pi$-polynomial continued fractions) recorded. OI-289-1 closes:
answered affirmatively in measure (Theorem~\ref{thm:music}),
negatively in selection (Proposition~\ref{prop:placement}), and
typed-as-unreachable in value.
\end{remark}

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

\begin{proposition}[Seven rungs in the universe's span]
Within $q \leq 3.4\times10^4$ (the conformal breath count, A265), the
record-gain ladder is exactly
$q = 1, 2, 39, 41, 449, 490, 6329$, with gains
$1, 20, 22, 243, 264, 3194, 26202$ and conditional SI periods
$1.4$, $2.7$, $54$, $56$, $617$, $673$ Myr, $8.70$ Gyr. The deepest
realized overtone has a period of order the age of the universe;
recorded as consonance only. The next rung ($q = 51122$) lies beyond
the span: the universe has not yet played it.
\end{proposition}

\paragraph{Consequences.} OI-289-1 closed (three layers). The tune
arc (A266--A268, A271, A273--A274, A287, A289, this addendum)
completes with its thesis in final form: \emph{closure is forbidden,
noise is null, the tune is necessary, the melody is the fingerprint
--- and the universe is seven notes into it.}

\paragraph{Verifier.} \texttt{verify\_P290.py}: seeded Monte-Carlo
percentiles (all three measures); empirical Borel--Bernstein
demonstration on the null sample (unbounded-quotient fraction); the
realized ladder with gains and SI; the next-unplayed rung; classical
citations as structural checks. All PASS.

\end{document}
