\documentclass[11pt]{article}
% Addendum 392 (the derivation 391 asked for, run and ACCEPTED: the collapse/rebound ratio is ONE).
% P14 l.259: C*P=I does NOT fix amplitudes -> the ratio must come from the oscillation theory. That theory
% (kappa=alpha^(5/4), omega_1=pi*sqrt(1-kappa); breath lapse m=sqrt(1-u^2), u=cos2b) is magnitude-SYMMETRIC,
% carrying at most a ~0.1-0.2% asymmetry -- 33x too small for the 3.5-11% the H0 data demand. So the P14
% l.273 collapse/rebound asymmetry is CAUSAL (which phase drives which), NOT magnitudinal; 369's symmetric
% F=2pi STANDS, H0=75.8 is unmoved, and 391's rescue is KILLED cleanly. The fix lies deeper (chart/closure/
% data), confirming 388/389. One unmotivated factor (1-alpha^(5/8)->72.3) flagged as the trap, rejected.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P392.py (9/9 pass).
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\newcommand{\Tb}{T_{\mathrm{breath}}}

\title{Addendum 392: The Collapse/Rebound Ratio Is One\\
\large The derivation 391 called for, run and accepted: the breath's infall/rebound\\
asymmetry is causal, not magnitudinal --- so it does not rescue $H_0$\\[0.3em]
\normalsize (Settles the open derivation of Addendum 391; Papers 14, 13, 17, 30; Addendum 369)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~391 diagnosed the $H_0$ over-prediction ($H_0=2\alpha^2/\Tb=75.8$, $\sim\!3\sigma$ high) as a
possibly flattened breath asymmetry: 369 identifies the driven-collapse \emph{infall} gain $a_0$ with the
rebound-projection \emph{expansion} (de~Sitter) temperature as ``one temperature read two ways''
($F=2\pi$), while Paper~14 (line~273) calls those phases ``not symmetric in mechanism.'' 391 left the fix
\emph{open} and named the single derivation that would settle it: derive the collapse/rebound
\emph{magnitude} ratio from Paper~14's $C$ and $P$ operators, then read off $H_0$ and accept it. This
addendum runs that derivation, and the result is a clean negative. \textbf{The corpus forbids the shortcut
and supplies the substitute.} Paper~14 (line~259) states that $C\circ P=I$ ``is compositional, not
dynamical \ldots without by itself fixing rates, amplitudes, or periods''; the amplitudes are fixed
instead by the oscillation theory --- the parameter $\kappa=\alpha^{5/4}$ and frequency
$\omega_1=\pi\sqrt{1-\kappa}$ (Paper~13), realised as the breath map's lapse $m=\sqrt{1-u^2}$,
$u=\cos2\beta$ (Paper~30). \textbf{That structure is magnitude-symmetric.} The lapse is a mirror image
across the breath phase: the inward (collapse) and outward (projection) halves have equal amplitude, as
$x(t)=A\cos\omega t$ has equal in- and out-swing. The largest asymmetry the oscillation parameter can
carry is $\kappa$ itself, entering as $\sqrt{1-\kappa}$ ($-0.11\%$) or $1-\kappa$ ($-0.21\%$); either
leaves $H_0$ at $75.6$--$75.7$. \textbf{The data need $33\times$ more.} The required correction is $-3.5\%$
(SH0ES) to $-11\%$ (Planck); the available principled asymmetry, $\sim\!0.1\%$, is about $33\times$ too
small. \textbf{Verdict.} The Paper-14 asymmetry is \emph{causal} --- which phase is the driver and which
the rebound --- not \emph{magnitudinal}; the magnitudes are mirror-symmetric to within $\alpha^{5/4}\sim
0.2\%$. So 369's symmetric identification stands, $F=2\pi$ holds, and $H_0=75.8$ is unmoved by this
mechanism. 391's rescue is killed cleanly: the $H_0$ overshoot is \emph{not} in the infall/rebound ratio,
and the fix lies deeper --- the chart anchoring ($c=\pi/\alpha$, $\Tb$), the closure hypothesis (370), or
the data --- confirming 388/389. One unmotivated factor that lands near the window
($1-\alpha^{5/8}\to H_0=72.3$) is flagged as the numerology trap and rejected. All claims pinned by
\texttt{verify\_P392.py} (9/9).
\end{abstract}

\section{The corpus forbids the shortcut and names the substitute}
\label{sec:forbid}

\begin{proposition}[$C\circ P=I$ does not fix the ratio]
\label{prop:no-fix}
Paper~14, line~259: ``The identity $C\circ P=I$ is compositional, not dynamical: it constrains what the
composition of the two operators must yield, without by itself fixing rates, amplitudes, or periods. To
extract a period from it, additional structure must be supplied, and the structure available \ldots is the
oscillation theory \ldots a fundamental frequency $\omega_1=\pi\sqrt{1-\kappa}$ set by the oscillation
parameter $\kappa=\alpha^{5/4}$.'' So the collapse/rebound magnitude ratio cannot be read off the
operators $C$ and $P$ alone; whatever asymmetry exists in the magnitudes must enter through the oscillation
theory. The derivation 391 requested is therefore a question about $\kappa$ and the breath map, not about
$C$ and $P$ in isolation.
\end{proposition}

\section{The oscillation is magnitude-symmetric}
\label{sec:sym}

\begin{theorem}[The breath's inward and outward amplitudes are equal to within $\kappa$]
\label{thm:sym}
The oscillation theory realises the breath as the GON dynamics (Paper~30) with lapse $m=\sqrt{1-u^2}$ and
Hopf latitude $u=\cos2\beta$. As $\beta$ advances, $u$ traces a symmetric path and $m=|\sin2\beta|$ is a
mirror image across each turning point: the collapse half and the projection half have \emph{equal
amplitude}, exactly as a harmonic oscillation $x(t)=A\cos\omega_1 t$ has equal inward and outward swing.
The only quantity that distinguishes the dynamics from a perfectly symmetric oscillation is the parameter
$\kappa=\alpha^{5/4}\approx2.13\times10^{-3}$, which can enter an amplitude asymmetry at most as
$\sqrt{1-\kappa}$ (a $-0.11\%$ effect) or $1-\kappa$ ($-0.21\%$). Hence the collapse/rebound magnitude
ratio is $1$ to within $\alpha^{5/4}$.
\end{theorem}

\begin{corollary}[The asymmetry is far too small to move $H_0$]
\label{cor:small}
With $H_0\propto F$ and $F=2\pi r$ for an asymmetry factor $r$, the largest principled $r$ available is
$\sqrt{1-\kappa}=0.99893$, giving $H_0=75.72$; $1-\kappa$ gives $75.64$. The data require $r=0.965$ (SH0ES,
$-3.5\%$) down to $r=0.889$ (Planck, $-11\%$). The available asymmetry, $\sim\!0.1\%$, is about $33\times$
smaller than the smallest correction the data need. The mechanism cannot reach the window.
\end{corollary}

\section{Verdict: causal, not magnitudinal}
\label{sec:verdict}

\begin{theorem}[369's symmetric identification stands; 391's rescue is closed]
\label{thm:stands}
By Proposition~\ref{prop:no-fix} and Theorem~\ref{thm:sym}, Paper~14's ``not symmetric in mechanism''
(line~273) is a statement about which phase is the \emph{driver} ($C$, the continuous generator) and which
is the \emph{rebound} ($P$, ``the form that failure takes'') --- a \emph{causal} asymmetry. It is not a
magnitude asymmetry: the oscillation that fixes the amplitudes is mirror-symmetric to within
$\alpha^{5/4}$. Therefore 369's identification of $a_0$ (infall) with the expansion temperature at $F=2\pi$
is correct at the level of magnitudes, $H_0=75.8$ is unmoved, and the candidate rescue of Addendum~391 is
closed. The $H_0$ overshoot is not located in the collapse/rebound ratio.
\end{theorem}

\begin{remark}[The trap, flagged and rejected]
\label{rmk:trap}
One factor lands temptingly close: $1-\kappa^{1/2}=1-\alpha^{5/8}=0.9538$ gives $H_0=72.3$, $\sim\!1\sigma$
from SH0ES. But $\alpha^{5/8}$ is \emph{not} the oscillation parameter --- that is $\alpha^{5/4}$ --- and
the square root is inserted for no reason other than that it lands. Adopting it would be precisely the
$\delta_{CP}=240^\circ$/PRVE numerology the corpus has refused and registered (A344, P40): a coincidental
fit dressed as a derivation. It is recorded here to be \emph{rejected} unless an independent derivation of
the $5/8$ power is found. The discipline that produced this addendum's negative is the same discipline that
rejects this factor: the ratio is whatever the oscillation theory says it is, and the oscillation theory
says it is $1$.
\end{remark}

\begin{remark}[Where the $H_0$ fix must now be sought]
\label{rmk:deeper}
With the horizon \emph{shape} (389: forced $2\pi$), the family/CPT \emph{orbifold} (390: deficits
overshoot), and now the breath \emph{dynamics} (this addendum: ratio $=1$) all exhausted, the $H_0$
over-prediction is confirmed structural along every near-surface axis. The remaining candidates are the
ones 388/389 already named as deeper: the $a_0=cH_0/2\pi$ chart anchoring itself (the identifications
$c=\pi/\alpha$ and the breath-to-Myr calibration of $\Tb$), the closure hypothesis (370) on which the
whole bridge is conditional, or an upward revision of the measured $H_0$. This addendum does not choose
among them; it removes one, by derivation.
\end{remark}

\section*{Status}
This settles the open derivation of Addendum~391. Asked to derive the collapse/rebound magnitude ratio
from Paper~14's $C$ and $P$ operators and accept the result, the derivation returns $1$. Paper~14
(line~259) forbids reading the ratio from $C\circ P=I$ (which fixes no amplitudes) and routes it to the
oscillation theory ($\kappa=\alpha^{5/4}$, $\omega_1=\pi\sqrt{1-\kappa}$;
Proposition~\ref{prop:no-fix}); that theory, realised as the GON breath lapse $m=\sqrt{1-u^2}$, is
magnitude-symmetric, carrying at most an $\alpha^{5/4}\sim0.1$--$0.2\%$ asymmetry (Theorem~\ref{thm:sym}),
about $33\times$ too small for the $3.5$--$11\%$ the data demand (Corollary~\ref{cor:small}). So Paper~14's
``not symmetric in mechanism'' is causal, not magnitudinal; 369's symmetric $F=2\pi$ stands and $H_0=75.8$
is unmoved (Theorem~\ref{thm:stands}). 391's rescue is closed --- the overshoot is not in the
infall/rebound ratio --- and the fix lies deeper (the chart anchoring, the closure hypothesis~370, or the
data; Remark~\ref{rmk:deeper}), confirming 388/389. The tempting unmotivated factor
$1-\alpha^{5/8}\to72.3$ is flagged and rejected (Remark~\ref{rmk:trap}). An honest negative, derived and
accepted. No published number changes. Verifier: \texttt{verify\_P392.py}, 9/9 pass.

\end{document}
