% 273_Addendum_UnitBridge.tex
% OI-271-1 closed: the P03<->P14 unit bridge is derived from P14's own
% Breathing-Period theorem. The system unit is the tempered fundamental
% period; all per-breath counts are kappa-free by cancellation; Branch B
% is excluded; A267's ladder becomes unconditional; TBS 28_2 retired.
% Verifier: verify_P273.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 273: The Unit Bridge\\
\large The P03 $\leftrightarrow$ P14 conversion derived; Branch B excluded;\\
the breath ladder unconditional; TBS 28\_2 retired}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (bridge derived from canon text; exact
cancellation theorem; branch decided) \quad|\quad
\textbf{Closes:} OI-271-1; retires TBS 28\_2 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P273.py} --- all PASS

\begin{abstract}
A271 sharpened TBS 28\_2 to a missing unit bridge: the corpus carries
P03's oscillation time (in which $\omega_1 = \pi\sqrt{1-\kappa}$) and
P14's system units (in which $\Tb = \pi\alpha^{-1}$) with no documented
conversion, leaving the windings-per-breath count two-valued (Branch A:
$\Tb$; Branch B: $\Tb\sqrt{1-\kappa}$). The bridge does not need to be
chosen; it is already derived in the canon. P14's Breathing-Period
theorem defines the breath by $\omega_{\rm breath} =
\omega_1/(\pi\alpha^{-1})$ --- the system unit is the \emph{fundamental
oscillation period} $\tau_1 = 2\pi/\omega_1 = 2/\sqrt{1-\kappa}$ in
P03 time (Proposition~\ref{prop:bridge}). Two consequences follow
exactly. (1)~\emph{Cancellation} (Theorem~\ref{thm:cancel}): every
per-breath count of an $\omega_1$-paced rate is $\kappa$-free, because
the rate carries $\omega_1$ and the unit carries $\omega_1^{-1}$; in
particular the density register completes exactly $\Tb$ cycles per
breath and the dual-observer arc is exactly $4\pi$ per system unit. TBS
28\_2 thereby dissolves: $\|\Omega\| = 2\pi\sqrt{1-\kappa}$ per P03 time
\emph{and} exactly $4\pi$ per system unit are the same speed in two
units; nothing vanishes, nothing is broken, and the exactness is a unit
conversion, not a dynamical closure --- the Necessity of Detuning (A267)
is untouched. (2)~\emph{Branch decision}
(Theorem~\ref{thm:branch}): Branch B's identification (P14 unit $\equiv$
P03 time) forces a breath of $\Tb/\tau_1 \approx 215.03$ system units,
contradicting the Breathing-Period theorem; Branch B is excluded by the
canon text. Branch A's \emph{number} is confirmed at the density
register --- $\rho = \Tb \bmod 1 = 0.5122452$, definitionally exact ---
though its stated identification (``arc-radian time'') is replaced by
the derived bridge. A267's ladder ($q = 2, 41, 449$) and tritone reading
are hereby \emph{unconditional}. A271's working hypothesis (tempered
Branch B) is overturned; recorded per convention. Two closing
identities: the SU(2) double-cover register carries $2\Tb$ windings per
breath with rotation number $2\rho - 1 = 0.0244904$ --- exactly A267's
$q=2$ comma (counting in the double cover \emph{is} observing breath
pairs); and $1 - \sqrt{1-\kappa} = \kappa/2 + O(\kappa^2)$ is retyped as
a conversion residue, demoting A266's schisma near-miss flag.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Bridge}

\begin{proposition}[The system unit is the fundamental period]
\label{prop:bridge}
P14, Theorem (Breathing Period), defines the breath by
$\omega_{\rm breath} = \omega_1/(\pi\alpha^{-1})$ with
$\omega_1 = \pi\sqrt{1-\kappa}$, ``measured in fundamental oscillation
units.'' Hence the P14 system unit is one period of the tempered
fundamental mode:
\begin{equation}
1\ \text{system unit} \;=\; \tau_1 \;=\; \frac{2\pi}{\omega_1}
\;=\; \frac{2}{\sqrt{1-\kappa}} \;=\; 2.0021362\ldots
\ \text{(P03 time)},
\qquad \tau_1\sqrt{1-\kappa} = 2 \ \text{exactly}.
\end{equation}
The conversion is \emph{tempered} (it carries $\sqrt{1-\kappa}$) and it
is \emph{derived}, not chosen: it is read off the canon's own definition
of the breath. The breath duration in P03 time is
$\Tb\,\tau_1 = 2\pi\alpha^{-1}/\sqrt{1-\kappa} = 861.944\ldots$
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Cancellation}

\begin{theorem}[Per-breath counts are $\kappa$-free]
\label{thm:cancel}
Let $r$ be any rate proportional to $\omega_1$ (a quantity paced by the
fundamental mode). Its count per system unit, $r\,\tau_1 =
r \cdot 2\pi/\omega_1$, and hence per breath, is independent of
$\kappa$: the temper in the rate and the temper in the unit cancel
identically. Instances: (i)~density-register cycles per breath
$= (\omega_1/2\pi)\cdot \Tb\,\tau_1 = \Tb$ exactly; (ii)~dual-observer
arc per system unit $= \|\Omega\|\,\tau_1 = 2\omega_1 \cdot 2\pi/\omega_1
= 4\pi$ exactly (P28 lifting, $\|\Omega\| = 2\omega_1$); (iii)~SU(2)
windings per breath $= 2\Tb$ exactly.
\end{theorem}

\begin{proof}
Immediate from Proposition~\ref{prop:bridge}: $r\,\tau_1 = 2\pi r/\omega_1$
and $r \propto \omega_1$. The instances substitute $r = \omega_1/2\pi$
and $r = 2\omega_1/2\pi$. Verified to 50 digits (verify\_P273, S2).
\end{proof}

\begin{corollary}[TBS 28\_2 retired]
\label{cor:tbs}
The dual-observer arc-speed is $2\pi\sqrt{1-\kappa}$ per unit P03 time
\emph{and} exactly $4\pi$ per system unit. P28's derivation of
$\|\Omega\| = 2\pi$ (in P28's clock, which is $\omega_1$-paced) and the
lifting computation $2\pi\sqrt{1-\kappa}$ (in P03 coordinate time) are
the same speed expressed in two units; the $0.107\%$ ``deviation'' was a
unit mismatch, not a physical discrepancy. Neither horn of TBS 28\_2
holds: the $\sqrt{1-\kappa}$ correction does not vanish (it is real in
P03 time), and no corrected formula is needed (both formulas are
correct). The TBS is retired by conversion. Note the exactness of
$4\pi$ per unit is \emph{definitional} --- the same class of exactness
as the fundamental period being one unit --- and therefore does not
constitute an exact dynamical closure; A267's Necessity of Detuning is
not violated. The dynamical detunings ($\rho$ irrational, $G_1 \neq 0$)
stand exactly where they were.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{Branch Decision}

\begin{theorem}[Branch B excluded; the density register carries $\Tb$]
\label{thm:branch}
Under the derived bridge: (i)~Branch B's identification (P14 unit
$\equiv$ P03 time) implies a breath of $\Tb$ P03-time units
$= \Tb/\tau_1 = 215.026\ldots$ system units, contradicting P14's
Breathing-Period theorem ($\Tb$ fundamental cycles per breath); no
register yields the count $\Tb\sqrt{1-\kappa}$. Branch B is excluded by
canon text. (ii)~The density register completes exactly $\Tb$ cycles
per breath (Theorem~\ref{thm:cancel}(i)), so the breath map's rotation
number is
\begin{equation}
\rho \;=\; \Tb \bmod 1 \;=\; \pi\alpha^{-1} - 430 \;=\; 0.5122452\ldots
\end{equation}
definitionally exact --- Branch A's number, with its derivation
corrected (the unit is the tempered fundamental period, not an abstract
arc-radian time).
\end{theorem}

\begin{corollary}[A267 unconditional]
A267's continued-fraction ladder ($q = 2, 41, 449$), the near-tritone
reading of successive breaths, and the conditional-SI rungs ($\approx
2.7$, $56$ Myr, conditional only on A265's SI calibration) are hereby
unconditional: the branch-conditionality marker of A271 is lifted.
A271's robustness theorem is unchanged (the $a_0$ chain, 432, and $G_1$
were branch-invariant and remain so --- now also branch-decided), and
its working hypothesis (Branch B) is overturned; A271 carries a note to
this effect per the same-day convention.
\end{corollary}

\begin{remark}[The double-cover shadow is the $q=2$ rung]
\label{rem:shadow}
The SU(2) register carries $2\Tb$ windings per breath
(Theorem~\ref{thm:cancel}(iii)); its rotation number is
$2\Tb \bmod 1 = 2\rho - 1 = 0.0244904\ldots$ --- \emph{exactly} A267's
$q = 2$ comma $|2\rho - 1|$. Counting in the double cover is observing
breath pairs: the covering map sends the breath map to its own first
near-closure, and the shadow's convergent denominators
($40, 41, 204, 245, \ldots$) contain A267's $q = 41$ rung. No new
dynamics; one identity locking A267's ladder to P28's lifting remark.
\end{remark}

\begin{remark}[Retyping $1-\sqrt{1-\kappa}$; schisma flag demoted]
\label{rem:schisma}
$1 - \sqrt{1-\kappa} = \kappa/2 + O(\kappa^2) = 0.0010670\ldots$ is now
typed: it is the per-unit \emph{conversion residue} between the two
clocks ($\tau_1/2 - 1$), not the comma of any closure map --- it
measures the difference of two tick lengths, not a failure to close.
A266's flagged near-miss against the schisma
($3^8 \cdot 5/2^{15} - 1 = 0.0011292$, ratio $0.945$) thereby loses its
candidate referent and is demoted from flagged near-miss to numerical
coincidence, unless future work gives the conversion residue a
closure-map reading. A266's quarter-comma/$G_1$ flag is unaffected.
\end{remark}

\paragraph{Consequences.} OI-271-1 closed (Case A). TBS 28\_2 retired in
its final form --- the canon's oldest unresolved normalization was a
missing unit conversion, now derived. A267 unconditional; TASK\_034's
branch-conditional gate lifted. The working dictionary for ToENet and
Kairos fixes $\rho = 0.5122452$ (density register) with SU(2) shadow
$0.0244904$.

\paragraph{Verifier.} \texttt{verify\_P273.py}: the bridge identity
$\tau_1\sqrt{1-\kappa} = 2$ at 50 digits; all three cancellation
instances; the Branch-B contradiction factor; $\rho$ and the recomputed
ladder; the shadow identity $2\Tb \bmod 1 = 2\rho - 1$ and its
convergents; the $\kappa/2$ expansion bound; the demotion ratio. All
PASS.

\end{document}
