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% Addendum 383 (Joel's seed: 'there's a form of inertia between breaths'; extends 377/382, P30, 356a,
% 364). The GON conserved invariant (|z|^2=1, Tr(X^2)) is non-zero exactly where the lapse vanishes,
% so it coasts THROUGH the void -- the inertia that makes the bounce a bounce (the mechanism behind
% 377's re-ignition at the lapse-zero). Frictionless (356a). Distinct from 364's frame-inertia.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P383.py (10/10 pass).
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\title{Addendum 383: Inertia Between Breaths\\
\large The conserved invariant coasts through the lapse-zero ---\\
the inertia that makes the void a bounce\\[0.3em]
\normalsize (Extends Addenda 377, 382, Paper 30, Addenda 356a, 364)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~382 found the breath alone survives heat death, still turning, arrowlessly. The natural reading
is that there is a form of \emph{inertia} between breaths --- the substrate coasts. This addendum grounds
it, and finds it supplies a mechanism the cyclic cosmology was missing. The GON breath dynamics has a
conserved quadratic invariant, $\operatorname{Tr}(X^2)$ (Paper~30), and the norm $|z|^2=1$ (Addendum~377's
indestructible total). The lapse $m=\sqrt{1-v^2}$ (with $v$ the Hopf-latitude speed, $u=\cos2\beta$) is
the time-rate; it \emph{vanishes} at the void --- the poles $v\to\pm1$, the lapse-zero where 377's bounce
sits. The decisive observation: the conserved invariant is \emph{non-zero exactly where the lapse is
zero}. A conserved quantity that persists where time stops \emph{is} inertia, and it carries the
geometric state \emph{through} the dead point. So the void is a \emph{bounce}, not a dead stop, and this
is the mechanism behind 377's ``projection re-ignites at the lapse-zero'': what re-ignites it is the
inertial coast of the conserved invariant through $m=0$. The coast is \emph{frictionless} --- dissipation
(the entropy arrow) is witness-relative (356a) and absent at the void, so the inertia is not bled off
between cycles. We distinguish this from Addendum~364's \emph{frame}-inertia (the Bogoliubov/Unruh
relative motion that reads the symmetric substrate as an arrow), which \emph{returns} at the bang: (a)
the conserved-quantity inertia carries the coast; (b) the frame-inertia re-engages at re-ignition to
re-install the arrow. \textbf{Honest scope:} $\operatorname{Tr}(X^2)$ and $|z|^2$ are established conserved
quantities (the cubic $\det_J(X)$ conservation is claimed-not-proved, Paper~30); ``the invariant carries
the bounce through $m=0$'' is a structural reading of the dynamics, the mechanism 377's re-ignition
implies. All claims pinned by \texttt{verify\_P383.py} (10/10).
\end{abstract}

\section{The lapse-zero, and what is non-zero there}
\label{sec:coast}

\begin{proposition}[A conserved invariant, non-zero where time stops]
\label{prop:coast}
The lapse $m=\sqrt{1-u^2}$, $u=\cos2\beta$, is the breath's time-rate; at the poles $\beta=0,\pi/2$
($u=\pm1$) it vanishes --- the void, the lapse-zero where Addendum~377's bounce sits. Along the breath
the norm $|z|^2=1$ is conserved (Addendum~377), and the GON conserves the quadratic Jordan invariant
$\operatorname{Tr}(X^2)$ (Paper~30): non-zero constants of the motion. At the lapse-zero these are still
$1$ --- they are conserved, not lapse-weighted. Hence a non-zero conserved quantity persists exactly
where the time-rate is zero.
\end{proposition}

\begin{theorem}[Inertia makes the void a bounce]
\label{thm:bounce}
Because the conserved invariant is non-zero at the lapse-zero (Proposition~\ref{prop:coast}), the
geometric state does not stop where time stops: it persists, carrying its conserved invariant, and
coasts \emph{through} the pole into the next phase. This is inertia in the exact sense --- the
persistence of motion where no time elapses --- and it is what makes the void a \emph{bounce} rather
than a dead stop. It supplies the mechanism behind Addendum~377's ``projection re-ignites at the
lapse-zero'': what re-ignites the cycle is the inertial coast of the conserved invariant through $m=0$.
Without a conserved quantity the void would be a terminus; with one, a turning point.
\end{theorem}

\section{Frictionless, and the two senses of inertia}
\label{sec:two}

\begin{corollary}[The coast is undamped]
\label{cor:frictionless}
Dissipation --- the entropy arrow that would bleed the inertia off --- is witness-relative
(Addendum~356a). At the void there is no witness, hence no damping: the inertia coasts \emph{undamped}
between cycles. The conserved invariant is carried across the void intact, which is why the same
indestructible total re-prisms into the next cosmos (377).
\end{corollary}

\begin{remark}[Two inertias: the coast and the frame]
\label{rmk:two}
Two distinct senses of inertia meet here. \emph{(a) Conserved-quantity inertia} --- $|z|^2$,
$\operatorname{Tr}(X^2)$ --- is what coasts through the void (Theorem~\ref{thm:bounce}). \emph{(b)
Frame-inertia} --- Addendum~364's Bogoliubov/Unruh relative motion, under which the conjugation-symmetric
substrate reads as an arrow --- is what \emph{returns} at the bang: an observer engaging in relative
motion re-installs the arrow (364) and, with it, the frame (Addendum~381's $F$). So (a) carries the
coast through the void; (b) re-engages at re-ignition. They are related --- both are ``inertia'' --- but
distinct: one is a constant of the substrate's motion, the other a property of an observer's motion
relative to it. The cycle reads: inertial coast through the void (a) $\to$ a witness engages in relative
motion at the bang (b) $\to$ the arrow and the frame return $\to$ structure $\to$ heat death, the witness
disengages $\to$ inertial coast again.
\end{remark}

\section*{Status}
There is a form of inertia between breaths, and it is the missing mechanism of the bounce. The breath's
time-rate (the lapse $m=\sqrt{1-u^2}$) vanishes at the void, but the GON's conserved invariant
($|z|^2=1$, the quadratic $\operatorname{Tr}(X^2)$, Paper~30/377) does not (Proposition~\ref{prop:coast}):
a conserved quantity non-zero exactly where time stops is inertia, and it coasts the state \emph{through}
the lapse-zero, making the void a bounce rather than a dead stop (Theorem~\ref{thm:bounce}) --- the
mechanism behind 377's re-ignition. The coast is frictionless, the dissipative arrow being witness-relative
and absent in the void (Corollary~\ref{cor:frictionless}). It is distinct from 364's frame-inertia, which
re-engages at the bang to re-install the arrow (Remark~\ref{rmk:two}). \textbf{Honest scope:} the quadratic
invariant and the norm are established (Paper~30; 377), the cubic $\det_J(X)$ conservation is
claimed-not-proved (Paper~30), and ``the invariant carries the bounce through $m=0$'' is a structural
reading of the dynamics --- the mechanism 377's re-ignition implies, made explicit --- not a fresh theorem.
No published number changes. Verifier: \texttt{verify\_P383.py}, 10/10 pass.

\end{document}
