% 266_Addendum_BreathDetuning.tex
% The G1 identity: the "last unfitted constant" equals the breath's detuning
% from 432, exactly. Two open items (A171-A185 G1 residual; P14 432-TBS)
% collapse into one. Comma framework flagged as the formal target.
% Origin: Joel's structural intuition "music happens between the three maps"
% (2026-06-10). Verifier: verify_P266.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}

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

\title{Addendum 266: The Breath Detuning Identity\\
\large $G_1 = 432/\Tb - 1$, exactly: the last unfitted constant\\
is the monad's temperament}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (exact identity; two open items unified) with
flagged near-miss candidates \quad|\quad
\textbf{Verifier:} \texttt{verify\_P266.py} --- all PASS

\begin{abstract}
Two long-standing open items collapse into one line. The modular residual
$G_1 = j(i)/(4\pi\alpha^{-1}) - 1$ (A171; characterised through A185 as
transcendental, PSLQ-irreducible, ``possibly the last unfitted constant'')
and P14's defective TBS on the 432 correspondence
($\Tb = \pi\alpha^{-1} \approx 430.51$ vs the integer $432$) are the same
number: since $j(i) = 1728 = 4 \cdot 432$,
\[
  \boxed{\;G_1 \;=\; \frac{432}{\Tb} - 1 \;=\; 0.345578\%\;}
\]
identically. The last unfitted constant is the \emph{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. The connection
went unseen because the two items lived on different maps: $G_1$ on the
mathematics map (modular forms, CM points), the 432 on the
consciousness-adjacent one (P14's cosmological rhythm); the identity sits
between them, where --- per the originating intuition --- the music
happens. The reframe imports a 2{,}500-year-old formal apparatus:
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. Two near-misses are flagged under the PSLQ-honesty convention,
\emph{not} claimed: $G_1$ vs the quarter Pythagorean comma
($0.3456\%$ vs $0.3394\%$, ratio $1.018$) and P28's TBS 28\_2 arc-speed
gap vs the schisma ($0.1067\%$ vs $0.1129\%$, ratio $0.945$). The
formalization target --- the corpus's residual gaps as the temperament of
the monad --- is TASK\_035.
\end{abstract}

\section{The Identity}

\begin{theorem}[Breath detuning]
With $j(i) = 1728$ (A158/A172) and $\Tb = \pi\alpha^{-1}$ (P14, A239):
$G_1 \equiv j(i)/(4\pi\alpha^{-1}) - 1 = 1728/(4\Tb) - 1 = 432/\Tb - 1$.
All quantities are corpus-closed; the identity is arithmetic. Numerically
$G_1 = 3.45578\times10^{-3}$ at any precision.
\end{theorem}

\begin{remark}[What this changes]
(i) A171--A185's characterisation programme and P14's TBS are one open
item, not two; any resolution of either resolves both. (ii) $G_1$'s
PSLQ-irreducibility (A180, A183) now reads naturally: a detuning
\emph{should not} be an algebraic combination of the pure constants ---
commas never are, in the relevant sense; that is what makes them commas.
(iii) P14's defective reconciliation attempt ($1 + \kappa^{0.6}$, off by
a factor $\sim$9) is retired; the correct statement of the 432 gap is
$G_1$ itself. (iv) The open question sharpens from ``what is $G_1$?'' to
``of which closure map is $G_1$ the comma?'' --- with the breath cycle as
the evident candidate and TASK\_034's rotation-number machinery
(elliptic M\"obius iteration) as the likely formal setting: temperament
commas and rotation numbers of circle maps are the same mathematics.
\end{remark}

\begin{remark}[Flagged near-misses --- not claims]
Quarter Pythagorean comma $\,(3^{12}/2^{19})^{1/4} - 1 = 0.3394\%$ vs
$G_1 = 0.3456\%$: off by $1.8\%$ relative. Schisma
$3^8\cdot5/2^{15} - 1 = 0.1129\%$ vs P28 TBS 28\_2's
$1 - \sqrt{1-\kappa} = 0.1067\%$ ($\kappa = \alpha^{5/4}$): off by
$5.5\%$. Recorded per A245/A180 convention; neither passes the corpus's
bar for identification. The \emph{framework} (residuals as commas) does
not depend on either match.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P266.py}: the identity at 50
digits; $j(i) = 4\cdot432$; the retirement check on P14's
$1+\kappa^{0.6}$; both near-miss ratios; all PASS.

\end{document}
