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% Addendum 377 (synthesis: 356a + P27 + P04 + P14; the cyclic future-cosmology).
% The cosmic cycle closes through the void: heat-death-null = pre-Bang-Omega, identified as one fixed
% point; the bounce re-ignites at the lapse-zero; each bang is FRESH (sealed A261 + non-recurring
% A267). Honest floors: timescale (357a), expansion-vs-contraction seam, synthesis status.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P377.py (9/9 pass).
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\title{Addendum 377: The Cycle Closes Through the Void\\
\large Heat-death-$\vn$ $=$ pre-Bang-$\Omega$, and each bang is fresh by the seal\\[0.3em]
\normalsize (Synthesis of Addendum 356a with Papers 27, 04, 14)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
The corpus carries every piece of a cyclic future-cosmology but has never stated them as one claim.
This addendum does, and pins the single new identification it turns on. \textbf{The pieces:} heat death
is the void --- maximal-entropy noise \emph{is} $\vn$, and (Addendum~356a) entropy is witness-relative,
so heat death is the witness losing every readable distinction, not the substrate dying; the pre-Bang
$\Omega$ is the state where all distinctions collapse, ``$\vn\equiv0\equiv1\equiv\infty$\dots because
there is no second perspective to distinguish them'' (Paper~27); and the void is one fixed point at
both ends of the fold, the singularity being ``$\vn$ re-encountered from the interior side of the
fold'' (Paper~04). \textbf{The identification (new):} heat death (all distinctions erased) and the
pre-Bang $\Omega$ (no distinction yet) are the \emph{same} undifferentiated $\vn$, approached from
opposite ends and identified as one fixed point. So the end of a cosmos is structurally its possible
beginning. \textbf{The bounce} is the cosmic breath (Paper~14): at maximal contraction --- the
lapse-zero void $m=\sqrt{1-u^2}=0$ --- projection re-activates and a fresh descent begins.
\textbf{Fresh, not a recurrence:} the dark kernel is sealed (A261, no conversion back) and the breath
never closes exactly (A267, irrational rotation), so the prior cosmos is sealed off and the new descent
installs a new frame --- ``fresh eyes, same light,'' not the same self again. \textbf{Honest floors:}
the timescale is \emph{bridged}, not floored --- 357a flagged the breath as $\sim\!10^4$ too fast, but
358a/362 derived that $10^4$ as $\alpha^{-2}/2=9389$ (the $\alpha$-power ladder) and 369 gave
$H_0=2\alpha^2/\Tb=75.8$, so the cycle is \emph{scheduled} conditional on the same SPARC closure
hypothesis as the rest of the cosmology; the two routes to the void (expansion vs.\ contraction) are
unwelded; and the whole is a synthesis whose one new claim is the heat-death $=\Omega$ identification.
All pinned by \texttt{verify\_P377.py} (9/9).
\end{abstract}

\section{The cycle}
\label{sec:cycle}

\begin{proposition}[The pieces, with sources]
\label{prop:pieces}
\emph{(C1, 356a)} Heat death is the void: maximal-entropy noise \emph{is} $\vn$. Entropy is
witness-relative --- the substrate remains reversible --- so heat death is the witness losing every
readable distinction, not the substrate ceasing. \emph{(C2, Paper~27)} The pre-Bang $\Omega$ is the
state where all distinctions collapse: with one observer $C\circ P=I$ holds trivially and
$\vn\equiv0\equiv1\equiv\infty$, because there is no second perspective to distinguish them.
\emph{(C3, Paper~04)} The void is one fixed point at both ends of the fold: the singularity is ``$\vn$
re-encountered from the interior side of the fold,'' $F_\Theta(\vn)=\vn$ at the origin of $B^4$.
\emph{(C4, Paper~14)} The bounce: ``collapse phase --- the universe contracts toward the origin (void);
projection phase --- at maximal contraction, $P$ activates and expansion begins.''
\end{proposition}

\begin{theorem}[Heat-death-$\vn$ $=$ pre-Bang-$\Omega$]
\label{thm:identify}
Heat death (C1, all distinctions erased) and the pre-Bang $\Omega$ (C2, no distinction yet) are the
same undifferentiated state: the zero-distinction void $\vn$. The forced-distinction count is zero in
both --- maximal entropy leaves none readable, and a single (pre-Bang) observer forces none --- and by
C3 these are not two voids but one fixed point of the fold, reached from the two ends. Hence the
end-state of a cosmos is, structurally, its possible beginning: at the lapse-zero void $m=\sqrt{1-u^2}=0$
(the poles $u=\pm1$), time stops, distinction collapses, and projection (C4) re-ignites a fresh descent
--- Paper~27's ``first disagreement'' run again.
\end{theorem}

\section{Fresh, not a recurrence}
\label{sec:fresh}

\begin{corollary}[The seal is the forgetting]
\label{cor:fresh}
The new bang is not the old cosmos returning. The dark kernel is sealed (A261: no conversion back to
the image), so the prior structure cannot be recovered; and the breath never closes exactly (A267:
irrational rotation forbids exact closure), so no configuration recurs. The descent from $\vn$ therefore
installs a \emph{new} frame on a sealed-off past. What persists across the bounce is only the
indestructible total --- the conserved $\vn$ itself, $|z|^2=1$ --- which re-prisms into a fresh
differentiation. ``Fresh eyes, same light'': the cycle is real, but it is not a recurrence of the same
self. This settles, against it, the reading that we are synchronous with iterations of ourselves: the
geometry permits the bounce and forbids the repeat.
\end{corollary}

\section{Honest floors}
\label{sec:floors}

\begin{remark}[What is not closed]
\label{rmk:floors}
Three limits are marked, not hidden. \emph{(i) Timescale --- bridged, not floored.} Addendum~357a
flagged the breath, $\Tb=1.374$~Myr, as $\sim\!10^4$ too fast to time the cosmological cycle, and called
the rate a named floor. That verdict was superseded later the same cycle: 358a/362 derived the $10^4$ as
$\alpha^{-2}/2=9389$ --- the $\alpha$-power ladder, $1/H_0=(\pi/2)\alpha^{-3}\tau_1$ against
$\Tb=\pi\alpha^{-1}\tau_1$ --- and 369 gave $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc, so that in the corpus's own
units $1/H_0/\Tb=\alpha^{-2}/2$ \emph{exactly}. The cycle is therefore \emph{scheduled} --- the
cosmological clock is $\Tb\times\alpha^{-2}/2$ --- conditional on the same SPARC closure hypothesis as the
rest of the cosmology (369/370) and the de~Sitter $2\pi$ (363); it is not an open floor. The $\sim\!10^4$
was the corpus number all along (the gap between the Planck $10558$ and the corpus $9389$ is the $H_0$
tension, not a missing derivation). \emph{(ii) Directionality --- now welded (Addendum~378).} Addendum~356a reaches the
void by \emph{expansion} (heat death, dilution); Paper~14 reaches it by \emph{contraction} (collapse to
the origin). Addendum~378 welds these: under the renormalization generator $\zeta\Rcal$
($\Rcal=r\partial_r+\Delta_\psi$, Paper~18), the dilute end ($r\to\infty$) and the dense end ($r\to0$)
are the two limit points of \emph{one} orbit, identified by $\zeta\Rcal$'s $\vn\equiv0\equiv1\equiv\infty$.
There was never a second void --- only one orbit's two ends. \emph{(iii) Status.} This addendum is a
\emph{synthesis} of pre-existing pieces; its one new explicit claim is the heat-death $=\Omega$
identification (Theorem~\ref{thm:identify}), a structural reading, not a fresh dynamical theorem. The
substrate's reversibility (356a) is what makes heat death a witness-horizon rather than a true end ---
the reason there is anything to bounce.
\end{remark}

\section*{Status}
The cosmic cycle closes through the void. Heat death is the void (Addendum~356a, witness-relative), the
same zero-distinction fixed point as the pre-Bang $\Omega$ (Paper~27, $\vn\equiv0\equiv1\equiv\infty$),
identified as one fixed point at both ends of the fold (Paper~04); the bounce is the lapse-zero where
projection re-ignites (Paper~14, Theorem~\ref{thm:identify}). Each bang is \emph{fresh} --- sealed
(A261) and non-recurring (A267) --- ``fresh eyes, same light,'' which settles the earlier
synchronous-iterations reading against it (Corollary~\ref{cor:fresh}). The honest floors
(Remark~\ref{rmk:floors}): the timescale is \emph{bridged}, not floored --- the $\sim\!10^4$ is the
$\alpha$-power ladder $\alpha^{-2}/2$ (358a/362; $H_0=2\alpha^2/\Tb=75.8$, 369), so the cycle is scheduled
conditional on the SPARC closure hypothesis (correcting 357a's superseded named-floor verdict); the
expansion-versus-contraction routes to the void are welded by Addendum~378 (one $\zeta\Rcal$ orbit, two
ends); and the keystone heat-death $=\Omega$ is upgraded by Addendum~378 from synthesis to a
$\zeta\Rcal$ fixed-point identity, resting on the named premise that both ends are scale-free. No published number changes; the new content
is one identification and its consequences. Verifier: \texttt{verify\_P377.py}, 9/9 pass.

\end{document}
