% 275_Addendum_MultiplierGain.tex
% OI-265-1 closed: in chart-natural units (L_h, T_b^SI) the corpus's two
% A265 identities force c = T_b, a0 = alpha, L_h = c*tau1 exactly. The
% deformation space of the breath map splits into a phase channel (A268,
% gain 1) and a modulus channel; the modulus gain at unit source is
% exactly alpha. A265 upgrades A- -> A under one flagged reading.
% Verifier: verify_P275.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{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{Addendum 275: The Multiplier Gain\\
\large Chart-natural units, the two response channels,\\
and $\alpha$ as the per-breath gain (OI-265-1 closed)}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (exact identities; gain derived; one reading
flagged) \quad|\quad \textbf{Closes:} OI-265-1 (A265 upgrades
A$-$ $\to$ A under the flagged reading) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P275.py} --- all PASS

\begin{abstract}
A274 reduced OI-265-1 to one derivative: $\partial|\lambda'|/\partial s$
of the breath-map multiplier under a baryonic coupling operator.
The closure is in three steps, each exact.
\emph{Step 1 (chart-natural units).} A265's two identities
($a_0 = \alpha^3 c^2/\pi^2 \Lh$ and $\TbSI = \alpha^2 c/\pi a_0$)
algebraically force, in the chart's own units ($\Lh$, $\TbSI$):
$c = \pi/\alpha = \Tb$ exactly --- light speed \emph{is} the breath
number --- equivalently $\Lh = c\,\tau_1^{\rm SI}$: the canon's missing
kiloparsec is the light-crossing distance of one fundamental oscillation
($\tau_1^{\rm SI} \approx 3191$ yr $\Rightarrow 0.978$ kpc, matching
A264's SPARC value); and $a_0 \cdot (\TbSI)^2/\Lh = \alpha$ exactly ---
\emph{the universal acceleration is $\alpha$ chart-lengths per
breath$^2$}. These are not near-misses; they are identities, and they
give A264's unit lock its kinematic face.
\emph{Step 2 (the coupling operator is symmetry-forced).} First-order
holomorphic deformations of $B$ fixing its poles form the
one-complex-parameter family $\delta B = c\,u\,\partial_u$; the real and
imaginary parts are the only two response channels. ${\rm Im}\,c$
perturbs the rotation number at unimodular multiplier --- A268's phase
channel, gain exactly 1. ${\rm Re}\,c$ is radial drift --- attraction
toward the rest pole --- and is the unique channel that moves
$|\lambda|$ off the unit circle. Infall is not one coupling among many;
it is the only symmetry-allowed modulus response.
\emph{Step 3 (the gain).} Defining unit source as the universal field
$a_0$ acting at chart scale for one breath, the per-breath velocity gain
in chart units is $a_0(\TbSI)^2/\Lh = \alpha$, hence
$\partial|\lambda'|/\partial s\,|_0 = \alpha$. A265's named inference
--- $\alpha$ as the breath-paced single-$\alpha$ linear-response gain of
the hidden chart --- is confirmed as an exact identity. 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;
recorded as the single condition on A265's upgrade A$-$ $\to$ A.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Chart-Natural Units}

\begin{proposition}[Exact chart identities]
\label{prop:chart}
Let chart units be ($\Lh$ for length, $\TbSI$ for time). From A265's
identities $a_0 = \alpha^3 c^2/(\pi^2 \Lh)$ and
$\TbSI = \alpha^2 c/(\pi a_0)$:
\begin{equation}
\frac{c\,\TbSI}{\Lh} \;=\; \frac{\pi}{\alpha} \;=\; \Tb
\qquad\text{and}\qquad
\frac{a_0\,(\TbSI)^2}{\Lh} \;=\; \alpha,
\end{equation}
both exact. Equivalently, with $\tau_1^{\rm SI} = \TbSI/\Tb$ the SI
duration of one system unit (A273): $\Lh = c\,\tau_1^{\rm SI}$ --- the
chart length is the distance light travels in one fundamental
oscillation. Numerically $\tau_1^{\rm SI} \approx 3.19 \times 10^3$ yr
and $c\,\tau_1^{\rm SI} \approx 0.978$ kpc, A264's measured halo unit.
\end{proposition}

\begin{proof}
Substitution: $c\TbSI/\Lh = \alpha^2 c^2/(\pi a_0 \Lh) =
\alpha^2 c^2 \cdot \pi^2/(\pi \alpha^3 c^2) = \pi/\alpha$. Similarly
$a_0(\TbSI)^2/\Lh = \alpha^4 c^2/(\pi^2 a_0 \Lh) = \alpha$. Verified at
50 digits (verify\_P275, S1).
\end{proof}

\begin{remark}[The unit lock's kinematic face]
A264 locked three lengths into one ($B^4$ scale $\equiv$ fold-clock unit
$\equiv \Lh$). Proposition~\ref{prop:chart} states \emph{why} the locked
length sits at a kiloparsec: it is the causal horizon of one breath
tick. In chart units the corpus's kinematics reads: light crosses one
chart length per system unit; the breath is $\Tb$ system units (A273);
hence $c = \Tb$ chart-velocities. The fine-structure constant and the
speed of light are, in the chart, reciprocal faces of the same number:
$c = \pi/\alpha$.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Two Response Channels}

\begin{proposition}[Deformation space; channel split]
\label{prop:channels}
A first-order holomorphic deformation of $B(u) = \lambda u$ preserving
the fixed points $\{0, \infty\}$ 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
$\arg\lambda' = 2\pi\rho_* + s\,{\rm Im}\,c$. The two channels:
\begin{center}
\begin{tabular}{lll}
\toprule
channel & observable & theory \\
\midrule
${\rm Im}\,c$ (phase) & rotation number $\rho_*$ & A268: gain exactly 1;
tongues \\
${\rm Re}\,c$ (modulus) & $|\lambda'| - 1$: drift to a pole & this
addendum: gain $\alpha$ \\
\bottomrule
\end{tabular}
\end{center}
Both channels commute with the rotation symmetry and the lens
$\mathbb{Z}_3$ (A274); no other first-order channel exists. Radial
infall --- gravitational attraction toward the rest pole --- is the
unique symmetry-allowed way for the breath map to leave the unimodular
class.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{The Gain}

\begin{theorem}[$\alpha$ is the per-breath gain]
\label{thm: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. OI-265-1's inference
--- ``breath-paced single-$\alpha$ linear response'' --- is an exact
identity of the chart, not an estimate.
\end{theorem}

\begin{corollary}[A265 upgrade; the flagged reading]
A265 (Case A$-$) upgrades to Case A under one reading, here flagged
explicitly: \emph{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
(Proposition~\ref{prop:chart}); what is typed rather than derived is the
statement that $G M_b$ 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). The upgrade is therefore
conditional on nothing internal to the corpus --- only on the kernel's
already-tested contact with the world.
\end{corollary}

\begin{remark}[What the two gains say together]
A268: the phase response has gain 1 --- the breath cannot be detuned at
first order; pacing is rigid (Pin 3, A269). This addendum: 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 infinitely stiffer than its
geometry, by exactly the fine-structure constant. This is the
dynamical restatement of the corpus's oldest theme --- $\alpha$ as the
price of observation (P03) --- now read off a multiplier.
\end{remark}

\paragraph{Consequences.} OI-265-1 closed. A265 carries an upgrade note
(A$-$ $\to$ A, flagged reading stated). TASK\_034's surviving question
is discharged; the task closes fully. The chart-natural identities
($c = \Tb$, $a_0 = \alpha$, $\Lh = c\,\tau_1^{\rm SI}$) join the unit
lock as the recommended citation form for the SPARC paper.

\paragraph{Verifier.} \texttt{verify\_P275.py}: both chart identities
symbolically at 50 digits; the SI numerics ($\TbSI = 1.374$ Myr,
$\tau_1^{\rm SI} = 3191$ yr, $c\,\tau_1^{\rm SI} = 0.978$ kpc $=$ A264
band); the channel split ($|\lambda'|$ responds to ${\rm Re}\,c$ only,
$\rho_*$ to ${\rm Im}\,c$ only, at first order); the gain identity; the
consistency pair (A268 gain 1, this gain $\alpha$). All PASS.

\end{document}
