% 271_Addendum_TemperamentAudit.tex
% TBS 28_2 audited: the sqrt(1-kappa) question is precisely a missing unit
% bridge between P03's oscillation time and P14's system units. Branch table
% computed; a0 chain, 432, and G1 proven branch-independent; coherence leans
% tempered. Verifier: verify_P271.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{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 271: The Temperament Audit\\
\large $\sqrt{1-\kappa}$: exact physics or comma? Diagnosis, branch table,\\
and the robustness of the chain}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (question sharpened to a unit bridge; both
branches computed; chain proven branch-independent; tempered reading
argued, not proven) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P271.py} --- all PASS

\begin{abstract}
OI-269-1 asked whether the covering-lift factor $\sqrt{1-\kappa}$ (P28
TBS 28\_2, a $0.107\%$ deviation of the arc-speed from $2\pi$) is exact
physics or an artifact. The audit sharpens the question to its actual
content: \textbf{the corpus carries two time normalizations} --- P03's
density-oscillation time (in which $\omega_1 = \pi\sqrt{1-\kappa}$) and
P14's system units (in which $\Tb = \pi\alpha^{-1}$) --- \emph{with no
documented conversion between them}. The dimensionless,
normalization-free quantity is windings-per-breath, and it takes
different values under the two candidate bridges: Branch~A (P14 unit
$\equiv$ arc-radian time): $\Tb$ windings, $\rho_A = 0.5122$, ladder
$q = 2, 41, 449$, near-tritone; Branch~B (P14 unit $\equiv$ P03 time):
$\Tb\sqrt{1-\kappa}$ windings, $\rho_B = 0.0529$, CF $[0;18,1,9,\ldots]$,
ladder $q = 18, 19, 189, 208$, Brjuno sum $0.523$. \textbf{Robustness
theorem:} the $a_0$ chain (A265/A269), the 432 derivation, and the $G_1$
identity (A266/A267) use $\Tb = \pi\alpha^{-1}$ definitionally and are
\emph{identical in both branches} --- the corpus's strongest chain is
immune to its deepest unresolved normalization. Both branches are Brjuno
(A268's response theory holds in either). What is branch-dependent:
A267's scale ladder and the tritone reading. Coherence argument, flagged
as argument: A267's Necessity of Detuning makes an \emph{exactly}-$2\pi$
arc the only untempered closure in the canon --- suspicious by the
canon's own theorem --- and the deviation sits $5.5\%$ from the schisma
(A266 flag). The tempered Branch~B is adopted as \emph{working
hypothesis}; decision requires deriving the P03$\leftrightarrow$P14 unit
bridge, filed as \textbf{OI-271-1} (and now the precise, final form of
the old TBS 28\_2).
\end{abstract}

\section{The Branch Table}

\begin{center}
\begin{tabular}{lcc}
\toprule
 & Branch A ($2\pi$ exact) & Branch B (tempered) \\
\midrule
windings/breath & $430.5122$ & $430.0529$ \\
$\rho$ & $0.51225$ & $0.05290$ \\
CF & $[0;1,1,19,1,10,\ldots]$ & $[0;18,1,9,1,1,\ldots]$ \\
ladder $q$ & $2, 41, 449$ & $18, 19, 189, 208$ \\
Brjuno sum & $2.803$ & $0.523$ \\
structure & near-tritone (period 2) & near-$1/19$ (period 19) \\
ladder in SI & $2.7, 56$ Myr & $25, 26, 260$ Myr \\
\bottomrule
\end{tabular}
\end{center}

\begin{theorem}[Branch independence of the chain]
$a_0 = \alpha^3 c^2/(\pi^2 L_h)$ (A265/A269), $432 = $ nearest 3-smooth
integer to $\pi\alpha^{-1}$ (A267 Thm~1.2), and $G_1 = 432/\Tb - 1$
(A266) involve only $\Tb = \pi\alpha^{-1}$ as a defined corpus constant
and contain no winding count: they are invariant across the branch
table. (Branch~B's winding count $430.053$ also has nearest 3-smooth
432, so even the winding-referred reading of the 432 theorem survives.)
A268's Brjuno-based response theory holds in both branches.
\end{theorem}

\begin{remark}[Coherence, and what would decide]
For Branch A, the arc closes on exactly $2\pi$ --- the unique exact
closure in a corpus whose own A267 proves closures do not happen; for
Branch B, the arc is tempered like everything else, by an amount $5.5\%$
from the schisma (flagged near-miss, A266). Coherence favors B; coherence
is not proof. The decider is the unit bridge: a derivation of P14's
system unit from P03's oscillation time (or an independent physical
determination of windings-per-breath). \textbf{OI-271-1} replaces and
retires TBS 28\_2 as the precise statement of what was always missing.
A267's ladder results are hereby marked \emph{branch-conditional}; its
Necessity-of-Detuning and 432 theorems are unconditional.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P271.py}: both branch computations
(CFs, ladders, Brjuno sums), the branch-independence assertions, the
432-invariance in both readings; all PASS.

\paragraph{Note (added 2026-06-10, same day).} OI-271-1 is closed by
A273: the unit bridge is \emph{derived} from P14's Breathing-Period
theorem (the system unit is the tempered fundamental period
$\tau_1 = 2/\sqrt{1-\kappa}$), and the working hypothesis adopted above
--- Branch B --- is \textbf{overturned}: no register yields the count
$\Tb\sqrt{1-\kappa}$, and the density register carries exactly $\Tb$
windings per breath (Branch A's number, with corrected derivation). The
coherence argument's suspicion of an exact $2\pi$ closure is dissolved
rather than confirmed: the exactness is definitional (a unit
conversion), not dynamical, so the Necessity of Detuning is not
violated. The branch table above remains correct as a table of
\emph{readings}; column B is now known to be unphysical. The robustness
theorem is unaffected. TBS 28\_2 is retired. See A273.

\end{document}
