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% Addendum 378 (solidifies 377's keystone; grounds in P18 zeta-R + 356a/365/P04/A272).
% The two voids are one zeta-R fixed point: heat-death-null (dilute end r->inf) and origin-null (dense
% end r->0) are the two limit points of ONE renormalization orbit, identified by zeta-R's
% null==0==1==infinity (P18). Welds 377's floor (ii); upgrades heat-death=Omega from synthesis to a
% fixed-point identity resting on one named premise (both ends scale-free).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P378.py (10/10 pass).
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\title{Addendum 378: The Two Voids Are One $\zeta\Rcal$ Fixed Point\\
\large Solidifying heat-death $=\Omega$ by the master operator's\\
own renormalization generator\\[0.3em]
\normalsize (Solidifies Addendum 377; grounds in Paper 18, Addenda 356a/365, Papers 04, A272)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~377 identified heat-death-$\vn$ (356a) with the pre-Bang $\Omega$ (Paper~27) as a synthesis ---
both are zero-distinction states --- and left two seams: the two routes to the void (expansion
$r\to\infty$ versus contraction $r\to0$) were unwelded (its floor~ii), and the identification was a
structural reading rather than a theorem. This addendum discharges both with an operator already inside
the master operator: the renormalization generator $\zeta\Rcal$, with $\Rcal=r\,\partial_r+\Delta_\psi$,
which Paper~18 states (line~306) ``implements the identity $\vn\equiv0\equiv1\equiv\infty$ under
renormalization flow,'' $e^{t\Rcal}\psi(r)=e^{t\Delta_\psi}\psi(e^tr)$. The argument is short. The
renormalization orbit of any $r_0>0$ under $r\mapsto e^tr$ is all of $(0,\infty)$, whose two limit
points are $r\to0$ and $r\to\infty$; so $0,1,\infty$ lie on one orbit. A scale-invariant observable is
constant ($r\,f'(r)=0\Rightarrow f'=0$), so no scale-free observable distinguishes the orbit's points or
its two ends --- this \emph{is} $\vn\equiv0\equiv1\equiv\infty$. The two voids are then the two ends of
\emph{one} orbit: heat-death-$\vn$ is the dilute end ($r\to\infty$), the fold's origin-$\vn$ (Paper~04)
the dense end ($r\to0$), and $\zeta\Rcal$ identifies them. \textbf{This welds floor~(ii)} --- there was
never a second void, only one orbit's two ends --- and \textbf{upgrades heat-death $=\Omega$ from
synthesis to a $\zeta\Rcal$ fixed-point identity}, resting on a single named premise: that both ends are
\emph{scale-free} (heat death by 356a/365, the origin by Paper~04). The honest residual is complete
scale-dissolution at heat death; the front-null typing inherits A272's candidate status. All claims
pinned by \texttt{verify\_P378.py} (10/10).
\end{abstract}

\section{The operator and the identity}
\label{sec:op}

\begin{proposition}[$\zeta\Rcal$ and $\vn\equiv0\equiv1\equiv\infty$, made precise]
\label{prop:identity}
Let $\Rcal=r\,\partial_r+\Delta_\psi$ be the renormalization generator of Paper~18, with flow
$e^{t\Rcal}\psi(r)=e^{t\Delta_\psi}\psi(e^tr)$. The dilation part has $r\,\partial_r(r^s)=s\,r^s$, so
$r^s$ are its eigenfunctions with eigenvalue the scaling dimension $s$. Two facts follow.
\emph{(1) One orbit.} The flow orbit of any $r_0>0$, namely $\{e^tr_0:t\in\mathbb{R}\}$, is all of
$(0,\infty)$, with limit points $r\to0$ ($t\to-\infty$) and $r\to\infty$ ($t\to+\infty$). Hence $0$,
$1$, and $\infty$ lie on a single orbit and its closure. \emph{(2) Scale-free $=$ constant.} A
scale-invariant observable, $f(e^tr)=f(r)$ for all $t$, satisfies $r\,f'(r)=0$, hence $f'\equiv0$ and
$f$ is constant; so $f(0^+)=f(1)=f(\infty)$. No scale-free observable distinguishes the orbit's points
or its two ends. Together these are Paper~18's $\vn\equiv0\equiv1\equiv\infty$.
\end{proposition}

\section{The two voids are one}
\label{sec:weld}

\begin{theorem}[Heat-death-$\vn$ and origin-$\vn$ are one $\zeta\Rcal$ fixed point]
\label{thm:onevoid}
Heat death is the \emph{dilute} void --- the $r\to\infty$ limit of the renormalization flow (scale
growing without bound, maximal entropy). The fold's origin (Paper~04) is the \emph{dense} void --- the
$r\to0$ limit. By Proposition~\ref{prop:identity} these are the two limit points of \emph{one} orbit,
and no scale-free observable separates them; $\zeta\Rcal$ identifies them. Therefore heat-death-$\vn$ and
origin-$\vn$ are the same void, the $\zeta\Rcal$ fixed-point locus where $\vn\equiv0\equiv1\equiv\infty$
--- which is the pre-Bang $\Omega$ of Paper~27 (the state where, with no second perspective,
$\vn\equiv0\equiv1\equiv\infty$). The synthesis of Addendum~377 is thus a $\zeta\Rcal$ fixed-point
identity.
\end{theorem}

\begin{corollary}[Floor (ii) is welded]
\label{cor:weld}
The two routes to the void that Addendum~377 left unwelded --- expansion ($r\to\infty$) and contraction
($r\to0$) --- are not two voids but the two ends of one $\zeta\Rcal$ orbit. There was never a second
void to weld; the renormalization generator already joins them. The directionality seam of 377 closes.
\end{corollary}

\section{The named premise, the two nulls, and the residual}
\label{sec:premise}

\begin{remark}[What the identity now rests on]
\label{rmk:premise}
The identification is not unconditional; it rests on one premise, now named and weak. Theorem~\ref{thm:onevoid}
identifies the two ends \emph{provided both are scale-free} --- carry no surviving characteristic scale
$r_0$. Heat death is scale-free: Addendum~356a (no distinctions, hence no scale, on a reversible
substrate) and Addendum~365 (the witness tower terminates, structure dissolves). The origin is
scale-free: Paper~04 (the symmetric fold fixed point). So the claim reduces from ``the two voids are
mysteriously the same'' to ``both ends carry no surviving scale,'' which $\zeta\Rcal$ then identifies.
The \emph{two nulls} meet here too: the front null (Nullity, the kernel of frame-installation --- A272's
flagged candidate) and the back null (dark, the kernel of the fold-projection --- A272 theorem) both sit
at the scale-free $\zeta\Rcal$ fixed point, so heat death reaches the re-bangable front null rather than a
sealed dead end (this inherits A272's ``candidate, worth future typing'' status for the front null). The
\emph{honest residual} is complete scale-dissolution at heat death: if a characteristic scale survived ---
a residual mass or length --- the two ends would not fully identify, and the bounce would be incomplete.
\end{remark}

\section*{Status}
The keystone of Addendum~377 --- heat-death-$\vn$ $=$ pre-Bang-$\Omega$ --- is solidified by the master
operator's own renormalization generator. With $\Rcal=r\,\partial_r+\Delta_\psi$ (Paper~18), the
renormalization orbit of any scale sweeps $(0,\infty)$ with limit points $0$ and $\infty$, and scale-free
observables are constant, so $\vn\equiv0\equiv1\equiv\infty$ (Proposition~\ref{prop:identity}, P18
line~306 made precise). The dilute void (heat death, $r\to\infty$, 356a) and the dense void (the origin,
$r\to0$, Paper~04) are then the two ends of \emph{one} $\zeta\Rcal$ orbit, identified by $\zeta\Rcal$
(Theorem~\ref{thm:onevoid}), which is the pre-Bang $\Omega$ (Paper~27). This \emph{welds} 377's floor~(ii)
(Corollary~\ref{cor:weld}: the two routes are one orbit's ends, not two voids) and \emph{upgrades}
heat-death $=\Omega$ from a four-paper synthesis to a $\zeta\Rcal$ fixed-point identity. It now rests on
(i) $\zeta\Rcal$ (Paper~18, solid), (ii) the named premise that both ends are scale-free (356a/365/Paper~04),
and (iii) the front-null typing (A272 candidate). The honest residual is complete scale-dissolution at
heat death (Remark~\ref{rmk:premise}). No published number changes; the new content is the operator-level
identification of the two voids and the welding of 377's directionality floor. Verifier:
\texttt{verify\_P378.py}, 10/10 pass.

\end{document}
