\documentclass[11pt]{article}
% Addendum 391 (the infall/rebound asymmetry: a DIAGNOSIS of the H0 over-prediction).
% 369's H0 derivation (L4) identifies the driven-collapse INFALL gain a0 with the rebound-projection
% de Sitter EXPANSION temperature as "one temperature read two ways" (symmetric F=2pi). But Paper 14
% line 273 states those very phases are NOT symmetric in mechanism (collapse driven, projection the
% rebound). So 369 flattened an asymmetry the corpus itself asserts -- the seam Joel's infall/backdraft
% intuition caught. The needed H0 correction is at the few-% DYNAMICAL scale (where 389's void -31% and
% 390's dihedral -50%+ both overshot), and would ground 389's flagged edge lead (-2.3%, H0=74.1). But
% this is a DIAGNOSIS + OPEN derivation, NOT a fix: the SIGN is open (the de Sitter rate went the wrong
% way, H0=91) and the MAGNITUDE must be DERIVED from P14, not fitted (the numerology trap).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P391.py (9/9 pass).
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\newcommand{\Tb}{T_{\mathrm{breath}}}
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\title{Addendum 391: The Infall/Rebound Asymmetry\\
\large The $H_0$ over-prediction may be a flattened breath asymmetry ---\\
369 read the driven infall and the rebound expansion as one symmetric temperature,\\
but Paper 14 says they are not symmetric\\[0.3em]
\normalsize (Diagnoses the $H_0$ tension of 388; re-points 389/390; Papers 14, 35; Addenda 363, 369)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addenda~389 and~390 ran the $H_0$ over-prediction ($H_0=2\alpha^2/\Tb=75.8$, $\sim\!3\sigma$ above SH0ES;
388) as a search for a \emph{geometric deficit} in the horizon factor $F$ of $a_0=cH_0/F$, and both came
back negative: the round $S^3$ is smooth so $F=2\pi$ is forced (389), and the dihedral orbifold deficits
overshoot massively (390). This addendum re-points the search, following a structural observation: the
corpus's growth dynamics are self-referential, with a \emph{driven infall} that the $H_0$ derivation may
not have carried through. \textbf{The seam.} 369's derivation ends (L4) with a \emph{symmetric}
identification: ``a horizon temperature is a property of the horizon, not the reading direction \ldots one
horizon, one temperature,'' so the kernel-side \emph{infall} gain $a_0$ \emph{equals} the cokernel-side
expansion (void) temperature, the de~Sitter temperature, at the symmetric $F=2\pi$. But Paper~14
(line~273) asserts the opposite of these very two phases: ``the four phases are \emph{not symmetric in
mechanism} --- the collapse phase is driven \ldots the projection phase is a consequence of the identity
\ldots the rebound through $P$ is the form that failure takes.'' The driven collapse (infall, $C$) and the
rebound projection (expansion, $P$) are not the same kind of process; 369's ``one temperature, read two
ways'' flattened that asymmetry into a single $2\pi$. \textbf{Why this is the right place to look.} The
data want $F$ lower by a \emph{few percent} ($-3.5\%$ SH0ES to $-11\%$ Planck) --- the scale of a
sub-leading \emph{dynamical} term. Every geometric-deficit mechanism tried is far too coarse (389's void
factor is $-31\%$, $H_0=52$; 390's dihedral jumps $-50\%$+, $H_0=25$--$38$). An infall/rebound asymmetry
is the \emph{first} candidate at the right order, and a plausible physical \emph{origin} for 389's one
flagged-but-underived lead (the edge factor, $-2.3\%$, $H_0=74.1$, $\sim\!1\sigma$). \textbf{The honest
catch.} This is a \emph{diagnosis}, not a fix. The sign is not established --- the one concrete
implementation tried (the de~Sitter rate $F=2\pi/\sqrt{\Omega_\Lambda}$, 389) went the \emph{wrong} way
($H_0=91$); and the magnitude must be \emph{derived} from Paper~14's collapse/rebound mechanism, not
fitted to the data, on pain of the $\delta_{CP}$/PRVE numerology trap. No $H_0$ number is claimed; the
contribution is to re-point the search from ``find a geometric deficit'' (389/390, failed) to ``derive the
breath's infall/rebound asymmetry'' (right scale, internally motivated, open). All claims pinned by
\texttt{verify\_P391.py} (9/9).
\end{abstract}

\section{The seam: a symmetric identification over an asymmetric breath}
\label{sec:seam}

\begin{proposition}[369 reads the infall and the expansion as one symmetric temperature]
\label{prop:sym}
The $H_0$ derivation (369) reaches its result through a chain (L1--L4) whose final step is an
\emph{identification by reading direction}. L1 (P35) establishes $a_0$ as the per-breath gain of the
\emph{radial infall} toward the breath map's rest-pole fixed point ($a_0\Tb^2/L_h=\alpha$). L4 then sets
that infall gain equal to the de~Sitter expansion temperature on the ground that ``a horizon temperature
is a property of the horizon, not the reading direction; kernel and cokernel are the two exactness defects
of one map family --- one horizon, one temperature.'' Hence $a_0$ (kernel-side infall) $=$ the
cokernel-side void/expansion temperature, evaluated at the symmetric Euclidean period $F=2\pi$ (363). The
move treats the inward (infall) and outward (expansion) readings as the \emph{same magnitude}.
\end{proposition}

\begin{proposition}[Paper 14 asserts these two phases are not symmetric]
\label{prop:asym}
The same two phases are the subject of Paper~14 (Cosmic Breathing), line~273: ``The four phases are
\emph{not symmetric in mechanism}. The collapse phase is driven: $C$ is the generator of inward motion and
acts continuously. The projection phase is a consequence of the identity: collapse toward the void cannot
complete \ldots and the rebound through $P$ is the form that failure takes.'' The driven, continuous
collapse (the infall) and the rebound projection (the expansion) are explicitly \emph{different in
mechanism}: one is generated, the other is the residue of a forbidden completion.
\end{proposition}

\begin{theorem}[369 flattened the Paper-14 asymmetry into a symmetric $2\pi$]
\label{thm:flatten}
Propositions~\ref{prop:sym} and~\ref{prop:asym} concern the \emph{same} infall/expansion pair and assign
it \emph{opposite} symmetry. 369's L4 (``one temperature, read two ways'') is a symmetric identification;
Paper~14's line~273 is an explicit asymmetry of those very phases. The symmetric reading is the stronger
assumption, and it is the one that fixes $F=2\pi$ and hence $H_0=75.8$. If the breath's collapse/rebound
asymmetry survives into the magnitudes --- if the rebound-expansion temperature is not equal to the
driven-infall gain --- then the effective $F$ is not $2\pi$, and $H_0$ moves. This seam is, prior to this
addendum, unregistered in the corpus.
\end{theorem}

\section{Why this is the right scale to look at}
\label{sec:scale}

\begin{proposition}[The data want a few-percent dynamical correction, not a geometric deficit]
\label{prop:scale}
With $H_0\propto F$, the data want $F$ reduced by $3.5\%$ (SH0ES~73.17), $9.6\%$ (DESI~68.53), or $11.1\%$
(Planck~67.4) --- the magnitude of a \emph{sub-leading dynamical} term. The geometric-deficit mechanisms
already tested overshoot this by an order of magnitude: 389's void factor $2\pi\Lz$ is a $31\%$ reduction
($H_0=52$), and 390's dihedral deficits are $50\%$+ ($F=\pi\!\to\!H_0=38$; $F=2\pi/3\!\to\!H_0=25$), with
nothing available between $0$ and $\sim\!\pi$. A dynamical infall/rebound asymmetry is the \emph{first}
candidate mechanism whose natural scale matches the few-percent the data require.
\end{proposition}

\begin{corollary}[It would ground 389's one flagged lead]
\label{cor:lead}
389 recorded a single lead at the right scale: $F=2\pi(1-\text{edge})$, $\text{edge}=\pi/\alpha^{-1}=
0.0229$, a $2.3\%$ reduction giving $H_0=74.1$ ($\sim\!1\sigma$), flagged as a number with ``a
plausible-sounding but underived story.'' An infall/rebound asymmetry is a candidate \emph{physical
origin} for a correction at exactly that scale: the story 389's lead was missing would be the breath's own
mechanism, not an ad~hoc layer exclusion.
\end{corollary}

\section{The honest catch: a diagnosis, not a fix}
\label{sec:catch}

\begin{remark}[The sign is not established]
\label{rmk:sign}
For the asymmetry to \emph{lower} $H_0$ (the direction the data demand), the rebound-expansion temperature
must be \emph{smaller} than the driven-infall gain --- an effective $F<2\pi$. That is plausible under a
reading in which the rebound, as ``the form that failure takes,'' carries less than the full driven
magnitude; but it is not derived. The one concrete implementation attempted --- the de~Sitter-rate
correction $F=2\pi/\sqrt{\Omega_\Lambda}=7.56$ (389, Proposition~\ref{prop:void}-analogue) --- went the
\emph{wrong} way, $H_0=91$ ($+21\sigma$). So the asymmetry's sign is a genuine open question, not a
foregone conclusion.
\end{remark}

\begin{remark}[The magnitude must be derived, not fitted]
\label{rmk:mag}
The factor that would land $H_0$ on SH0ES is a $3.5\%$ reduction; on Planck, $11\%$. Selecting whichever
factor hits a target and then attributing it to ``the collapse/rebound asymmetry'' 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. The discipline here is the reverse order --- derive the magnitude of the
asymmetry from Paper~14's mechanism (the ratio of the driven-collapse generator to the rebound residue,
through the breath map), \emph{then} read off where it puts $H_0$, and accept the answer whatever it is.
Until that derivation exists, this addendum claims no number.
\end{remark}

\begin{remark}[What this addendum does, and does not, establish]
\label{rmk:net}
It establishes a structural fact: 369's $H_0$ result rests on a symmetric identification
(Proposition~\ref{prop:sym}) of two phases the corpus elsewhere calls asymmetric
(Proposition~\ref{prop:asym}), and that the residual the data want is at the scale of that flattened
asymmetry, not at the scale of the geometric deficits 389/390 ruled out (Proposition~\ref{prop:scale}). It
does \emph{not} establish a fix: neither the sign (Remark~\ref{rmk:sign}) nor the magnitude
(Remark~\ref{rmk:mag}) is derived. The value is a re-pointing of the $H_0$ search --- from the
horizon's \emph{shape} (a deficit, exhausted) to the breath's \emph{dynamics} (the infall/rebound
asymmetry, open) --- and the identification of the single open derivation that would settle it.
\end{remark}

\section*{Status}
Joel's observation that the growth laws carry a self-referential infall (``backdrafts,'' ``cavitations'')
that the horizon derivation may not factor in lands on a real, previously unregistered seam. 369's $H_0$
chain ends (L4) with a \emph{symmetric} identification --- the kernel-side infall gain $a_0$ set equal to
the cokernel-side expansion (de~Sitter) temperature as ``one temperature read two ways,'' fixing $F=2\pi$
and $H_0=75.8$ (Proposition~\ref{prop:sym}). But Paper~14, line~273, asserts the collapse (infall) and
projection (expansion) phases are \emph{not symmetric in mechanism} (Proposition~\ref{prop:asym}). 369
flattened that asymmetry (Theorem~\ref{thm:flatten}). The correction the data want is a \emph{few percent}
($-3.5\%$ to $-11\%$) --- the scale of a sub-leading dynamical term, where 389's void ($-31\%$) and 390's
dihedral ($-50\%$+) overshot, and exactly the scale of 389's one flagged lead ($-2.3\%$, $H_0=74.1$;
Proposition~\ref{prop:scale}, Corollary~\ref{cor:lead}). But this is a \emph{diagnosis}, not a fix: the
sign is open (the de~Sitter rate went the wrong way; Remark~\ref{rmk:sign}) and the magnitude must be
derived from Paper~14's mechanism, not fitted (Remark~\ref{rmk:mag}). The $H_0$ search is re-pointed from
the horizon's shape to the breath's dynamics. No published number changes. Verifier:
\texttt{verify\_P391.py}, 9/9 pass.

\end{document}
