% Paper 35: The Unit System
% PROPOSED CANON (assembled 2026-06-10 from A273 + A275; subject to
% review/rollback). Consolidates the P03<->P14 unit bridge, the
% cancellation theorem, and the chart-natural constants into one
% canonical reference. Verifiers: verify_P273.py, verify_P275.py
% (both green at filing).
% 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{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}
\newcommand{\TbSI}{T_{\rm b}^{\rm SI}}
\newcommand{\Lh}{L_h}
\newcommand{\CP}{\mathbb{CP}^1}

\title{Paper 35: The Unit System\\
\large The P03$\,\leftrightarrow\,$P14 bridge, the cancellation
theorem,\\ and the chart-natural constants}
\author{L\'{e}on Fernando Vlegels}
\date{2026-06-10}

\begin{document}
\maketitle

\begin{abstract}
A theory that keeps time in two normalizations owes its readers the
conversion between them. The corpus uses two: P03's oscillation time,
in which the tempered fundamental mode has angular frequency
$\omega_1 = \pi\sqrt{1-\kappa}$, and P14's system units, in which the
breath has period $\Tb = \pi\alpha^{-1} \approx 430.51$. No conversion
was documented (a tracked open item of the registry, TBS 28\_2), and the gap had a concrete
cost: the dual-observer arc of P28 appeared to take two values
differing by $0.107\%$, and the windings-per-breath count was
two-valued (A271's branch table). This paper states the resolution as
canon. \textbf{The bridge (Thm~\ref{thm:bridge}):} P14's own
Breathing-Period theorem defines the system unit as the tempered
fundamental period, $\tau_1 = 2\pi/\omega_1 = 2/\sqrt{1-\kappa} =
2.0021362\ldots$ in P03 time, with $\tau_1\sqrt{1-\kappa} = 2$
exactly; the conversion is derived, not chosen, and the alternative
identification (system unit $\equiv$ P03 coordinate time) is excluded
by P14's own theorem. \textbf{Cancellation (Thm~\ref{thm:cancel}):}
every per-breath count of an $\omega_1$-paced rate is $\kappa$-free,
because the temper in the rate and the temper in the unit cancel
identically. The dual-observer arc is exactly $4\pi$ per system unit,
the density register completes exactly $\Tb$ cycles per breath, and
TBS 28\_2 dissolves as a unit mismatch ($\|\Omega\| =
2\pi\sqrt{1-\kappa}$ per unit P03 time \emph{and} exactly $4\pi$ per
system unit; both correct, one speed). The breath map's rotation
number $\rho = \Tb \bmod 1 = 0.5122452$ is definitionally exact.
\textbf{Chart-natural constants (Thm~\ref{thm:chart}):} expressing the
hidden-chart relations of the kiloparsec sector (A264/A265) in the
chart's own units $(\Lh, \TbSI)$ forces three exact identities:
$c = \pi/\alpha = \Tb$ (light speed \emph{is} the breath number),
$\Lh = c\,\tau_1^{\rm SI}$ (the chart length is the light-crossing
distance of one system unit), and $a_0 = \alpha$ (the universal
acceleration is the fine-structure constant in chart units). SI
calibration, conditional on the measured $a_0$:
$\tau_1^{\rm SI} = 3191$ yr, $\TbSI = 1.374$ Myr, $\Lh = 0.978$ kpc.
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction}
\label{sec:intro}

Every quantitative claim in the corpus is, at bottom, a count: so many
cycles of this register per so many ticks of that clock. Counts are
only meaningful once the clock is fixed, and the corpus runs two
clocks. P03 works in oscillation time, the coordinate time in which
the untempered fundamental mode has angular frequency $\pi$ and period
$2$; the temper $\kappa$ shifts the fundamental to
$\omega_1 = \pi\sqrt{1-\kappa}$, so the tempered fundamental period in
P03 time is $2/\sqrt{1-\kappa}$, slightly longer than $2$. P14 works
in system units: its Breathing-Period theorem assigns the breath the
period $\Tb = \pi\alpha^{-1}$, where
$\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ is the corpus's fine-structure
number. Both normalizations are internally consistent, and both are
used throughout the corpus. What was missing was the sentence that
says how long one system unit lasts in P03 time.

The omission was not cosmetic. It surfaced first as TBS 28\_2. P28
derives the dual-observer arc speed $\|\Omega\| = 2\pi$ in its own
clock, while the lifting computation, performed in P03 coordinate
time, gives $\|\Omega\| = 2\pi\sqrt{1-\kappa}$. The two values differ
by $0.107\%$, and the TBS recorded the discrepancy as a fork: either
the $\sqrt{1-\kappa}$ correction must somehow vanish, or P28's formula
needs correction. Neither option is attractive, since each amounts to
declaring one of two careful computations wrong.

A271 sharpened the problem rather than resolving it. It traced the
ambiguity to the missing unit bridge and tabulated the two candidate
readings. Branch A takes the per-breath winding count to be $\Tb$;
Branch B identifies the P14 system unit with P03 coordinate time, in
which case the count is $\Tb\sqrt{1-\kappa}$. The two branches differ
in the third decimal place of every per-breath count, and A271 adopted
tempered Branch B as a working hypothesis while flagging the choice as
open (OI-271-1).

The resolution, consolidated here from A273, is that the bridge does
not need to be chosen at all: it is already derived in the canon. P14
defines the breath frequency by $\omega_{\rm breath} =
\omega_1/(\pi\alpha^{-1})$, ``measured in fundamental oscillation
units.'' A frequency measured in fundamental oscillation units is a
count per fundamental period. The system unit is therefore one period
of the tempered fundamental mode, $\tau_1 = 2\pi/\omega_1$, and
everything else in this paper is arithmetic on that identification.
The arithmetic is worth doing carefully, because it produces three
results of independent use. First, the bridge itself
(Section~\ref{sec:bridge}): $\tau_1 = 2/\sqrt{1-\kappa}$, a tempered
conversion satisfying $\tau_1\sqrt{1-\kappa} = 2$ exactly, which
excludes Branch B by contradiction with P14's own theorem. Second, the
cancellation theorem (Section~\ref{sec:cancel}): any rate paced by
$\omega_1$ yields a per-breath count that is independent of $\kappa$,
because the temper enters the rate and the unit reciprocally. This
single observation retires TBS 28\_2 and fixes the rotation numbers of
the breath map's two registers as definitionally exact quantities.
Third, the chart-natural constants (Section~\ref{sec:chart}):
expressing the kiloparsec-sector relations of A264/A265 in the chart's
own length and time units forces the exact identities $c = \Tb$,
$\Lh = c\,\tau_1^{\rm SI}$, and $a_0 = \alpha$, which give the
apparently arbitrary kiloparsec scale and the apparently arbitrary
$10^{-10}\,\mathrm{m\,s^{-2}}$ acceleration a common dimensionless
origin.

Section~\ref{sec:numerics} collects the published numbers and the
verifier coverage. Section~\ref{sec:corpus} records which papers and
addenda consume these results. Section~\ref{sec:limits} states what
remains conditional or open. The paper is assembled without new claims
from Addendum 273 (the bridge and cancellation) and Addendum 275 (the
chart constants and the per-breath gain); every numerical statement is
checked by \texttt{verify\_P273.py} (14 checks) and
\texttt{verify\_P275.py} (12 checks), both green at filing.

% ════════════════════════════════════════════════════════════════════
\section{Setup and Notation}
\label{sec:setup}

\paragraph{Constants.} The fine-structure number is
$\alpha^{-1} = 4\pi^3 + \pi^2 + \pi = 137.0363\ldots$, and the breath
period in system units is
\begin{equation}
\Tb \;=\; \pi\alpha^{-1} \;=\; 430.5122452\ldots
\end{equation}
The temper is $\kappa = \alpha^{5/4} \approx 2.13\times10^{-3}$ (the
value used by both verifiers). The tempered fundamental frequency in
P03 time is $\omega_1 = \pi\sqrt{1-\kappa}$.

\paragraph{The two clocks.} P03 time is the coordinate time of the
oscillation analysis: the untempered fundamental has frequency $\pi$
and period $2$, and the tempered fundamental has frequency $\omega_1$
and period $2\pi/\omega_1$. A P14 system unit is the time unit
implicit in the Breathing-Period theorem, whose identification is the
subject of Theorem~\ref{thm:bridge}; the breath lasts $\Tb$ system
units by definition. For a register that completes $N$ cycles per
breath, the associated rotation number is $N \bmod 1$. Two registers
appear below: the density register on the Hopf fiber, with rotation
number $\rho$, and the SU(2) double-cover register on the base, with
rotation number $\rho_*$.

\paragraph{The hidden chart.} The kiloparsec sector (A264/A265)
carries three dimensionful quantities: the chart length $\Lh$, locked
by A264 to the $B^4$ scale and the fold-clock unit; the SI duration of
one breath, $\TbSI$; and the universal acceleration $a_0$. A265
derived two identities relating them,
\begin{equation}
a_0 \;=\; \frac{\alpha^3 c^2}{\pi^2 \Lh},
\qquad
\TbSI \;=\; \frac{\alpha^2 c}{\pi a_0},
\label{eq:a265}
\end{equation}
with $c$ the SI speed of light. The gas-calibrated value of the
universal acceleration is $a_0 = 1.172\times10^{-10}\,
\mathrm{m\,s^{-2}}$, and the SPARC clean band for the chart length is
$[0.75, 1.00]$ kpc (A264).

\paragraph{Exactness conventions.} Throughout, ``exact'' means an
algebraic identity in $\pi$ and $\alpha$, checked numerically at 50
decimal digits by the verifiers. The paper distinguishes
\emph{definitional} exactness, which expresses a unit conversion or a
definition, from \emph{dynamical} exactness, which would express a
closed orbit or resonance of the dynamics. Every exact statement in
this paper is of the first kind; the distinction matters for the
Necessity of Detuning (A267; P36 companion), which forbids the second
kind and is untouched here.

% ════════════════════════════════════════════════════════════════════
\section{The Bridge}
\label{sec:bridge}

The central claim is that the P03$\,\leftrightarrow\,$P14 conversion
is not a convention awaiting a choice but a consequence of text
already in the canon. P14's Breathing-Period theorem contains the
identification; it only needed to be read off.

\begin{theorem}[The system unit is the tempered fundamental period]
\label{thm:bridge}
P14 (Breathing Period) defines $\omega_{\rm breath} =
\omega_1/(\pi\alpha^{-1})$, ``measured in fundamental oscillation
units.'' Hence
\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},
\label{eq:bridge}
\end{equation}
and the breath duration in P03 time is $\Tb\,\tau_1 =
2\pi\alpha^{-1}/\sqrt{1-\kappa} = 861.944\ldots$
\end{theorem}

\begin{proof}
A frequency ``measured in fundamental oscillation units'' is a count
per period of the fundamental mode. The mode in question is the
tempered fundamental, with frequency $\omega_1 = \pi\sqrt{1-\kappa}$
in P03 time; its period is $\tau_1 = 2\pi/\omega_1 =
2/\sqrt{1-\kappa}$. The unit of time implicit in P14's breath count is
therefore $\tau_1$, which gives Eq.~\eqref{eq:bridge} directly. The
identity $\tau_1\sqrt{1-\kappa} = 2$ and the breath duration follow by
substitution; both are verified at 50 digits
(\texttt{verify\_P273}, S1).
\end{proof}

Two features of the bridge deserve emphasis. It is \emph{tempered}:
the conversion factor carries $\sqrt{1-\kappa}$, so a system unit is
longer than two units of P03 time by the small amount
$\tau_1 - 2 = 0.0021362\ldots$ And it is \emph{derived}: nothing was
chosen. The conversion is read off the canon's own definition of the
breath, which is what permits the branch decision below to be a
theorem rather than a preference.

\begin{corollary}[Branch B excluded]
\label{cor:branchB}
The identification ``system unit $\equiv$ P03 coordinate time''
(Branch B of A271) implies that the breath, lasting $\Tb$ units of P03
time, lasts $\Tb/\tau_1 = 215.026\ldots$ system units. This
contradicts P14's Breathing-Period theorem, under which the breath
comprises $\Tb$ fundamental cycles, by the factor $\tau_1 \approx 2$.
Moreover no register of the system yields the count
$\Tb\sqrt{1-\kappa}$. Branch B is excluded by the canon text, and
A271's working hypothesis (tempered Branch B) is overturned; the
overturn is recorded in A271 per the same-day convention.
\end{corollary}

The factor-of-two character of the contradiction is the cleanest
diagnostic: identifying the system unit with P03 coordinate time would
halve the breath, since one fundamental period spans approximately two
units of P03 time. P14's theorem leaves no room for that reading.
Branch A's \emph{number} survives, as the next section shows, but its
stated justification (an abstract ``arc-radian time'') is replaced by
the derived bridge.

% ════════════════════════════════════════════════════════════════════
\section{Cancellation}
\label{sec:cancel}

The bridge has one structural consequence that does most of the work
in this paper. Rates paced by the fundamental mode carry a factor
$\omega_1$; the system unit carries a factor $\omega_1^{-1}$. Any
count formed by multiplying the one by the other is therefore
independent of the temper. This is worth stating as a theorem because
its instances retire a TBS and fix the corpus's working rotation
numbers.

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

\begin{proof}
Immediate from Theorem~\ref{thm:bridge}: $r\,\tau_1 = 2\pi r/\omega_1$
and $r \propto \omega_1$, so $\omega_1$ cancels. The instances
substitute $r = \omega_1/2\pi$ and $r = 2\omega_1/2\pi$. All three are
verified at 50 digits (\texttt{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. The exactness of
$4\pi$ per unit is \emph{definitional}, of the same class as the
fundamental period being one unit, and therefore does not constitute
an exact dynamical closure; the Necessity of Detuning (P36 companion;
A267) is untouched, and the dynamical detunings ($\rho$ irrational,
$G_1 \neq 0$) stand exactly where they were.
\end{corollary}

\begin{corollary}[Registers]
\label{cor:registers}
The density register completes exactly $\Tb$ cycles per breath
(Theorem~\ref{thm:cancel}(i)), so the breath map carries the rotation
number
\begin{equation}
\rho \;=\; \Tb \bmod 1 \;=\; \pi\alpha^{-1} - 430
\;=\; 0.5122452\ldots
\end{equation}
on the Hopf fiber, definitionally exact. This is Branch A's number
with its derivation corrected: the unit is the tempered fundamental
period, not an abstract arc-radian time. The SU(2) double-cover
register carries $2\Tb$ windings per breath
(Theorem~\ref{thm:cancel}(iii)), hence the rotation number
$\rho_* = 2\Tb \bmod 1 = 2\rho - 1 = 0.0244904\ldots$ on the base: the
two registers of one bundle map, the base rotation equaling the
fiber's $q{=}2$ comma (A273/A274).
\end{corollary}

\begin{corollary}[A267 unconditional]
\label{cor:a267}
With the branch decided, A267's continued-fraction ladder
($q = 2, 41, 449$), the near-tritone reading of successive breaths,
and the conditional-SI rungs ($\approx 2.7$ and $56$ Myr, conditional
only on A265's SI calibration) are 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.
\end{corollary}

\begin{remark}[The double-cover shadow is the $q=2$ rung]
\label{rem:shadow}
The shadow rotation number satisfies $2\Tb \bmod 1 = 2\rho - 1 =
0.0244904\ldots$, which is 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 enters; this is one
identity locking A267's ladder to P28's lifting remark.
\end{remark}

\begin{remark}[The conversion residue retyped; schisma flag demoted]
\label{rem:schisma}
The quantity $1 - \sqrt{1-\kappa} = \kappa/2 + O(\kappa^2) =
0.0010670\ldots$ now has a type: it is the per-unit \emph{conversion
residue} between the two clocks, equal to $\tau_1/2 - 1$, and it
measures the difference of two tick lengths, not a failure of any
closure map to close. A266 had flagged its near-miss against the
schisma ($3^8 \cdot 5/2^{15} - 1 = 0.0011292$, ratio $0.945$); with
the residue typed as a tick-length difference rather than a comma,
that flag 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}

% ════════════════════════════════════════════════════════════════════
\section{Chart-Natural Constants}
\label{sec:chart}

The bridge converts between the corpus's two internal clocks. The
remaining conversion is external: between system units and SI, through
the hidden chart of the kiloparsec sector. A265 supplied the two
identities of Eq.~\eqref{eq:a265} relating the chart length $\Lh$, the
SI breath duration $\TbSI$, and the universal acceleration $a_0$.
Those identities involve $c$ and SI magnitudes and look like
calibration statements. Expressed in the chart's own units, they
collapse into pure numbers.

\begin{theorem}[Exact chart identities]
\label{thm:chart}
With the hidden-sector relations $a_0 = \alpha^3c^2/(\pi^2\Lh)$ and
$\TbSI = \alpha^2c/(\pi a_0)$ (A264/A265), in chart units
$(\Lh, \TbSI)$:
\begin{equation}
\frac{c\,\TbSI}{\Lh} = \frac{\pi}{\alpha} = \Tb,
\qquad
\Lh = c\,\tau_1^{\rm SI},
\qquad
\frac{a_0\,(\TbSI)^2}{\Lh} = \alpha,
\label{eq:chart}
\end{equation}
all exact, where $\tau_1^{\rm SI} = \TbSI/\Tb$ is the SI duration of
one system unit. The apparently arbitrary kiloparsec and the
apparently arbitrary $10^{-10}\,\mathrm{m\,s^{-2}}$ are two faces of
one dimensionless statement, $a_0 = \alpha$ in natural chart units;
the chart length is the causal horizon of one system unit.
\end{theorem}

\begin{proof}
Substitution. For the first identity,
\begin{equation}
\frac{c\,\TbSI}{\Lh}
= \frac{\alpha^2 c^2}{\pi a_0 \Lh}
= \frac{\alpha^2 c^2 \cdot \pi^2}{\pi\,\alpha^3 c^2}
= \frac{\pi}{\alpha},
\end{equation}
using $\TbSI = \alpha^2 c/(\pi a_0)$ and then eliminating $a_0\Lh$
with $a_0 = \alpha^3 c^2/(\pi^2\Lh)$. For the third,
$a_0(\TbSI)^2/\Lh = \alpha^4 c^2/(\pi^2 a_0 \Lh) = \alpha$ by the same
elimination. The second is the first divided by $\Tb$: since
$c\,\TbSI/\Lh = \Tb$ and $\tau_1^{\rm SI} = \TbSI/\Tb$, we have
$\Lh = c\,\TbSI/\Tb = c\,\tau_1^{\rm SI}$. Both core identities are
verified symbolically at 50 digits (\texttt{verify\_P275}, S1).
\end{proof}

Each identity is worth reading in words. The first says that light
crosses $\Tb \approx 430.51$ chart lengths per breath, equivalently
one chart length per system unit: in chart units the speed of light
\emph{is} the breath number, and the fine-structure constant and the
speed of light are reciprocal faces of the same number,
$c = \pi/\alpha$. The second locates the kiloparsec: the chart length
is the distance light travels in one fundamental oscillation, the
causal horizon of one system unit. The third strips $a_0$ of its SI
costume: the universal acceleration is $\alpha$ chart-lengths per
breath squared.

\begin{remark}[The unit lock's kinematic face]
\label{rem:lock}
A264 locked three lengths into one ($B^4$ scale $\equiv$ fold-clock
unit $\equiv \Lh$). Theorem~\ref{thm:chart} states \emph{why} the
locked length sits at a kiloparsec: it is the causal horizon of one
breath tick. In chart units the kinematics reads: light crosses one
chart length per system unit; the breath is $\Tb$ system units
(Theorem~\ref{thm:bridge}); hence $c = \Tb$ chart-velocities.
\end{remark}

\subsection{The response channels and the per-breath gain}
\label{sec:gain}

The identity $a_0 = \alpha$ acquires dynamical content through the
breath map's deformation theory, worked out in A275 and summarized
here because it is the principal consumer of the chart units. The
breath map acts on $\CP$ as $B(u) = \lambda u$ with fixed points
$\{0, \infty\}$ and unimodular multiplier
$\lambda = e^{2\pi i\rho_*}$.

\begin{proposition}[Deformation space; channel split]
\label{prop:channels}
A first-order holomorphic deformation of $B(u) = \lambda u$ preserving
the fixed points is generated by a holomorphic vector field on $\CP$
vanishing at both poles, i.e.\ $c\,u\,\partial_u$ with
$c \in \mathbb{C}$: $B_s(u) = e^{cs}\lambda u + O(s^2)$. The perturbed
multiplier is $\lambda' = e^{cs}\lambda$, so
$|\lambda'| = 1 + s\,{\rm Re}\,c + O(s^2)$ and
$\arg\lambda' = 2\pi\rho_* + s\,{\rm Im}\,c$. There are therefore
exactly two response channels: ${\rm Im}\,c$ perturbs the rotation
number at unimodular multiplier (A268's phase channel, gain exactly
$1$), and ${\rm Re}\,c$ is radial drift toward the rest pole, the
unique channel that moves $|\lambda|$ off the unit circle. Both
channels commute with the rotation symmetry and the lens
$\mathbb{Z}_3$ (A274); no other first-order channel exists.
\end{proposition}

Radial infall, attraction toward the rest pole, is thus not one
coupling among many; it is the only symmetry-allowed modulus response.

\begin{proposition}[$\alpha$ is the per-breath gain]
\label{prop:gain}
Normalize the source so that $s = 1$ is the universal field $a_0$
acting at chart scale for one breath. The per-breath velocity gain in
chart units is
\begin{equation}
\frac{a_0\,\TbSI}{\Lh/\TbSI} \;=\; \frac{a_0\,(\TbSI)^2}{\Lh}
\;=\; \alpha \quad \text{exactly},
\end{equation}
hence $\partial|\lambda'|/\partial s\,\big|_{s=0} = \alpha$: the
modulus channel of Proposition~\ref{prop:channels}, at unit source,
contracts the chart by exactly $\alpha$ per breath.
\end{proposition}

One reading is flagged rather than derived: that physical baryonic
sources couple through the drift channel. The form is symmetry-forced
(Proposition~\ref{prop:channels}: there is no other modulus channel)
and the normalization is the chart's own (Theorem~\ref{thm:chart});
what is typed rather than derived is the statement that $GM_b$ enters
the breath map at all. That statement is the v4 kernel's closure
hypothesis, confronted with data in A264 (SPARC:
$a_0 = 0.976\times$ empirical). It is recorded as the single condition
on A265's upgrade A$-$ $\to$ A.

Read together, the two gains characterize the chart's dynamics. The
phase response has gain $1$ and cannot be detuned at first order:
pacing is rigid (A268; Pin 3, A269). The modulus response has gain
$\alpha$: matter attracts at the kernel coupling, once per breath.
Rigid clock, weak infall, ratio $\alpha$: the hidden chart is a system
whose timekeeping is stiffer than its geometry by exactly the
fine-structure constant. This is the dynamical restatement of the
theme of P03, $\alpha$ as the price of observation, now read off a
multiplier.

% ════════════════════════════════════════════════════════════════════
\section{Numerical Results}
\label{sec:numerics}

Table~\ref{tab:numbers} collects the published numbers. The first
block is dimensionless and exact in the stated sense; the second block
is SI and conditional on the gas-calibrated
$a_0 = 1.172\times10^{-10}\,\mathrm{m\,s^{-2}}$ only.

\begin{table}[ht]
\centering
\begin{tabular}{llll}
\toprule
quantity & expression & value & status \\
\midrule
breath period & $\Tb = \pi\alpha^{-1}$ & $430.5122452\ldots$ & exact \\
system unit (P03 time) & $\tau_1 = 2/\sqrt{1-\kappa}$ &
$2.0021362\ldots$ & exact \\
bridge identity & $\tau_1\sqrt{1-\kappa}$ & $2$ & exact \\
breath in P03 time & $\Tb\,\tau_1$ & $861.944\ldots$ & exact \\
Branch-B breath (excluded) & $\Tb/\tau_1$ & $215.026\ldots$ &
contradiction \\
fiber rotation number & $\rho = \Tb \bmod 1$ & $0.5122452\ldots$ &
definitionally exact \\
base rotation number & $\rho_* = 2\rho - 1$ & $0.0244904\ldots$ &
definitionally exact \\
conversion residue & $1-\sqrt{1-\kappa} = \kappa/2 + O(\kappa^2)$ &
$0.0010670\ldots$ & typed: tick-length \\
residue/schisma ratio & $(1-\sqrt{1-\kappa})/(3^8\cdot5/2^{15}-1)$ &
$0.945$ & coincidence \\
\midrule
universal acceleration & $a_0$ (gas-calibrated) &
$1.172\times10^{-10}\,\mathrm{m\,s^{-2}}$ & input \\
SI breath duration & $\TbSI = \alpha^2 c/(\pi a_0)$ & $1.374$ Myr &
conditional \\
SI system unit & $\tau_1^{\rm SI} = \TbSI/\Tb$ & $3191$ yr &
conditional \\
chart length & $\Lh = c\,\tau_1^{\rm SI}$ & $0.978$ kpc &
conditional \\
SPARC clean band & A264 & $0.75$--$1.00$ kpc & measured \\
\bottomrule
\end{tabular}
\caption{Published numbers of the unit system. The dimensionless block
is verified at 50 decimal digits; the SI block is conditional on the
measured $a_0$ only.}
\label{tab:numbers}
\end{table}

A few entries deserve comment. The rotation number
$\rho = \pi\alpha^{-1} - 430 = 0.5122452\ldots$ is definitionally
exact, and its continued-fraction convergent denominators are
$1, 2, 39, 41, 449, \ldots$, recovering A267's ladder rungs
$q = 2, 41, 449$. The shadow $\rho_* = 0.0244904\ldots$ has continued
fraction beginning $[0; 40, 1, \ldots]$ with convergent denominators
$40, 41, 204, 245, \ldots$, containing the $q = 41$ rung. The
conversion residue satisfies $|{\rm residue} - \kappa/2| < \kappa^2$,
confirming the expansion $\kappa/2 + O(\kappa^2)$.

In the SI block, the chart length comes out as $\Lh = 0.978$ kpc,
inside the SPARC clean band $[0.75, 1.00]$ kpc, and the second
identity of Eq.~\eqref{eq:chart} gives it a direct reading: since
$\Lh = c\,\tau_1^{\rm SI}$ and $\tau_1^{\rm SI} = 3191$ yr, the chart
length is $3191$ light-years. The SI breath duration
$\TbSI = 1.374$ Myr sits in A265's band $1.37 \pm 0.14$ Myr. With
these durations, all breath-counted statements of the corpus acquire
SI timescales, conditional on the calibration only.

\paragraph{Verifier coverage.} \texttt{verify\_P273.py} (14 checks, 50
decimal digits): the bridge identity $\tau_1\sqrt{1-\kappa} = 2$; the
breath duration in P03 time; all three cancellation instances; the
TBS-dissolution identity $2\pi\sqrt{1-\kappa}\,\tau_1 = 4\pi$; 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.
\texttt{verify\_P275.py} (12 checks): both chart identities
symbolically; the SI numerics ($\TbSI$, $\tau_1^{\rm SI}$,
$c\,\tau_1^{\rm SI} = \Lh$ within the A264 band); the channel split
($|\lambda'|$ responds to ${\rm Re}\,c$ only and $\rho_*$ to
${\rm Im}\,c$ only, at first order); the gain identity; and the
consistency pair (phase gain $1$, modulus gain $\alpha$). All 26
checks pass.

% ════════════════════════════════════════════════════════════════════
\section{Relation to the Corpus}
\label{sec:corpus}

\paragraph{Upstream.} P03 supplies the oscillation time and the
tempered fundamental $\omega_1 = \pi\sqrt{1-\kappa}$, and its theme of
$\alpha$ as the price of observation is restated dynamically by the
gain pair of Section~\ref{sec:gain}. P14 supplies the Breathing-Period
theorem from which the bridge is read off; nothing in P14 changes, but
its phrase ``measured in fundamental oscillation units'' is now
load-bearing. P28 supplies the dual-observer arc and the lifting
result $\|\Omega\| = 2\omega_1$. A264 and A265 supply the kiloparsec
sector: the unit lock, the SPARC calibration, and the two identities
of Eq.~\eqref{eq:a265}.

\paragraph{Resolved or retired by this paper.} TBS 28\_2 is retired by
conversion (Corollary~\ref{cor:tbs}). The temporal reading of TBS
P034\_3\_c is superseded. A271's branch table is resolved: Branch B
excluded, Branch A's number confirmed with corrected derivation, the
working hypothesis overturned and noted in A271. OI-271-1 is closed
(via A273) and OI-265-1 is closed (via A275), the latter upgrading
A265 from Case A$-$ to Case A under the flagged reading of
Section~\ref{sec:gain}. A266's schisma near-miss flag is demoted to
numerical coincidence (Remark~\ref{rem:schisma}); its
quarter-comma/$G_1$ flag is unaffected.

\paragraph{Made unconditional.} A267's continued-fraction ladder
($q = 2, 41, 449$), its near-tritone reading of successive breaths,
and its conditional-SI rungs ($\approx 2.7$ and $56$ Myr, conditional
only on A265's calibration) lose their branch-conditionality marker
(Corollary~\ref{cor:a267}). The Necessity of Detuning itself (A267;
P36 companion) is untouched: the exactness established here is
definitional, not dynamical.

% ════════════════════════════════════════════════════════════════════
\section{Limitations and Open Items}
\label{sec:limits}

\paragraph{The SI block is conditional.} The identities of
Theorem~\ref{thm:chart} are exact, but the durations
$\tau_1^{\rm SI} = 3191$ yr, $\TbSI = 1.374$ Myr, and
$\Lh = 0.978$ kpc enter through the gas-calibrated
$a_0 = 1.172\times10^{-10}\,\mathrm{m\,s^{-2}}$ and inherit its
uncertainty. A different calibration of $a_0$ rescales the SI block
while leaving the dimensionless block untouched.

\paragraph{The flagged reading.} That physical baryons enter through
the drift channel is the symmetry-forced form with the chart's own
normalization, but it is a typing of the source, not a dynamical
derivation from the kernel. It is the single condition on A265's
upgrade A$-$ $\to$ A, and it rests on the v4 kernel's closure
hypothesis, whose contact with data is the SPARC comparison of A264
($a_0 = 0.976\times$ empirical).

\paragraph{The schisma demotion is reversible in principle.} The
near-miss of the conversion residue against the schisma (ratio
$0.945$) is recorded as a coincidence because the residue is typed as
a tick-length difference, not a comma. If future work gives the
conversion residue a closure-map reading, the flag would have to be
reinstated.

\paragraph{Definitional exactness only.} Nothing in this paper claims
an exact dynamical closure. The exact statements are unit conversions
and definitions; the dynamical detunings ($\rho$ irrational,
$G_1 \neq 0$) stand exactly where they were, as the Necessity of
Detuning requires.

\paragraph{Canonical status.} This paper is proposed canon, assembled
2026-06-10 from A273 and A275 and subject to review and rollback per
the header block.

% ════════════════════════════════════════════════════════════════════
\section{Conclusion}
\label{sec:conclusion}

The unit system of the corpus closes on a single identification: one
P14 system unit is one period of the tempered fundamental mode,
$\tau_1 = 2/\sqrt{1-\kappa}$, read off P14's own Breathing-Period
theorem rather than chosen. From it, three layers of structure follow
by arithmetic. Per-breath counts of $\omega_1$-paced rates are
$\kappa$-free, which retires TBS 28\_2 as a unit mismatch and fixes
the breath map's two rotation numbers, $\rho = 0.5122452$ on the fiber
and $\rho_* = 0.0244904$ on the base, as definitionally exact.
Branch B of A271 is excluded by contradiction with the canon text, so
A267's ladder and its readings hold unconditionally. And in the
chart's own units the kiloparsec sector reduces to three exact
identities, $c = \Tb$, $\Lh = c\,\tau_1^{\rm SI}$, and $a_0 = \alpha$,
under which the kiloparsec is the causal horizon of one system unit
and the universal acceleration is the fine-structure constant. The
corpus keeps two clocks, but it now keeps one time.

\paragraph{Provenance and verification.} Assembled without new claims
from Addendum 273 (the bridge; \texttt{verify\_P273.py}, 14 checks)
and Addendum 275 (chart constants; \texttt{verify\_P275.py}, 12
checks), both green at filing. Supersedes the temporal reading of
TBS P034\_3\_c; retires TBS P028\_2; resolves A271's branch table.


\section*{Status (2026-06)}
Load-bearing spine: the unit system, calibrated conditional on the measured $a_0$ (the one dimensionful anchor). Addendum~393 shows the $H_0$ prediction inherits this anchor's uncertainty ($H_0\propto\sqrt{a_0}$, $\pm11\%$), so the $H_0$ confrontation is anchor-limited, not a $3\sigma$ tension. Current.

\end{document}
