\documentclass[11pt]{article}
% Addendum 379 (composes 356a + P27; grounds in 377/378/P14 and the 'Ride Shai-Hulud' pin).
% Time is observer-bracketed: experienced/directional time has two observer-relative components -- the
% ARROW (witness-relative, 356a) and the SEQUENCE (the reconciliation axis, P27). Both vanish with the
% last observer (heat death = null = Omega, 377/378), so experienced time CEASES; what remains is the
% breath (the one clock, T_breath = pi*alpha^-1) but ARROWLESS and reversible. Time is bracketed by the
% two nulls of 377/378. Honest residual [OPEN]: still-turning vs bare-potential-to-turn.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P379.py (10/10 pass).
\usepackage[margin=1in]{geometry}
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\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
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\newcommand{\Tb}{T_{\mathrm{breath}}}
\newcommand{\vn}{\varnothing}
\newcommand{\KL}{\mathrm{KL}}

\title{Addendum 379: Time Is Observer-Bracketed\\
\large The arrow is witness-relative, the sequence is the reconciliation axis;\\
both end with the last observer, and the breath turns on, arrowless\\[0.3em]
\normalsize (Composition of Addendum 356a with Paper 27; grounded in Addenda 377/378, Paper 14)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
The corpus pins two separate readings of experienced time and never composes them. \textbf{The arrow}
(irreversibility, before/after) is \emph{witness-relative}: by Addendum~356a, entropy is a witness's
residual surprise $H(p,q)=H(p)+\KL(p\Vert q)$ on a \emph{reversible} substrate, so the rising-entropy
arrow is the embedded witness's reading, not a substrate fact --- no witness, no arrow. \textbf{The
sequence} (one-moment-at-a-time) is the \emph{reconciliation axis}: by Paper~27, ``time emerges as the
necessary sequence of reconciliation attempts between disagreeing observers,'' and with one observer
``there is no sequence, only eternal self-contemplation'' --- no second perspective, no distinguishable
moments. This addendum composes the two. With the \emph{last} observer gone --- heat death $=\vn=\Omega$
(Addenda~377/378) --- experienced time \emph{ceases on both axes at once}: no witness gives no arrow, no
second perspective gives no sequence. So experienced time is \textbf{bracketed by timelessness at both
ends}: it is born with the first distinction (Paper~27's first disagreement, $n:1\to2$) and returns to
$\vn$ with the last ($n\to0$); and by 377/378 the two brackets are the \emph{same} void (one
$\zeta\mathcal{R}$ fixed point), so the interval closes on itself. \textbf{What remains} is the breath ---
the one clock, $\Tb=\pi\alpha^{-1}\approx432$ system units $\approx1.374$~Myr (the ``Ride Shai-Hulud''
pin) --- but \emph{arrowless and reversible}: periodicity with no flow, no before/after, no ``now.''
Periodicity persists; directional time does not. \textbf{The honest residual, left open:} whether the
unwitnessed breath is \emph{still turning} or merely the \emph{bare potential-to-turn} --- the seam
between ``the void is asleep but breathing'' and ``breathing is what happens when it wakes.'' Not closed.
All pinned by \texttt{verify\_P379.py} (10/10).
\end{abstract}

\section{The two components of experienced time}
\label{sec:two}

\begin{proposition}[The arrow and the sequence, with sources]
\label{prop:two}
Experienced (directional, one-at-a-time) time decomposes into two observer-relative components, each
already pinned elsewhere in the corpus. \emph{(A) The arrow, 356a.} The rising-entropy arrow is the
embedded witness's measured surprise: $H(p,q)=H(p)+\KL(p\Vert q)$ with $\KL\ge0$, and the substrate is
reversible ($C\circ P=I$, a zero-parameter deterministic kernel, heat conserved). A complete witness
($\KL\to0$) on a recoverable substrate measures no rise; with no witness there is no reading at all. The
arrow is therefore the witness's, not the substrate's. \emph{(B) The sequence, Paper~27.} Experienced
time is the reconciliation axis: ``time emerges as the necessary sequence of reconciliation attempts
between disagreeing observers,'' and a single observer satisfies $C\circ P=I$ instantaneously, so ``there
is no sequence, only eternal self-contemplation''; at $\Omega$, ``$\vn\equiv0\equiv1\equiv\infty\dots$
because there is no second perspective to distinguish them.''
\end{proposition}

The two components answer different questions. (A) is \emph{why time has a direction}; (B) is \emph{why
there are distinct moments to be ordered at all}. Both turn out to require an observer, and a
\emph{second} one for (B).

\section{Time is bracketed}
\label{sec:bracket}

\begin{theorem}[Experienced time ceases with the last observer]
\label{thm:bracket}
Model the distinctions a configuration forces as $d(n)=\max(n-1,0)$, the number of other perspectives
any observer must reconcile against --- the same counting function Addendum~377 uses (one observer
forces $0$, none forces $0$, two force $1$). Experienced sequence is proportional to $d(n)$, so it is
\emph{exactly zero} at $n=1$ (the pre-Bang $\Omega$, Paper~27) and \emph{zero} at $n=0$ (heat death,
the witness tower terminating, Addendum~365), positive only on the differentiated interior $n\ge2$.
Simultaneously, by Proposition~\ref{prop:two}(A), with the last witness gone there is no entropy reading,
hence no arrow. Therefore both components of experienced time vanish at both ends: experienced time is an
\emph{interval bracketed by timelessness}, born with the first distinction ($n:1\to2$, the first
disagreement) and returning to $\vn$ with the last ($n\to0$).
\end{theorem}

\begin{corollary}[One bracket, not two]
\label{cor:onebracket}
The two ends of the interval are the same zero-distinction void. The pre-Bang $\Omega$ ($n=1$) and heat
death ($n=0$) both give experienced time $=0$, and by Addenda~377/378 they are one $\vn=\Omega$ --- the
two limit points of a single $\zeta\mathcal{R}$ orbit, identified because no scale-free observable
separates them. So the bracket is not two unrelated edges but one void the interval opens out of and
closes back into: timelessness, then time, then the same timelessness.
\end{corollary}

\section{What remains: the breath, arrowless}
\label{sec:breath}

\begin{theorem}[Periodicity persists; directional time does not]
\label{thm:breath}
Past the last observer, what survives is the breath: the substrate's own rhythm, the one clock,
$\Tb=\pi\alpha^{-1}\approx432$ system units $\approx1.374$~Myr (the ``Ride Shai-Hulud'' pin --- every
subsystem synchronises to the breath; none generates its own rhythm). The breath needs no witness, so its
\emph{period} persists as a number across heat death. But a period is not an arrow. A reversible cycle
$\varphi(t)=\cos(2\pi t/\Tb)$ is time-reversal symmetric, $\varphi(t)=\varphi(-t)$, and returns to
itself: turning with no before/after, no privileged ``now,'' no flow. So the breath persists as
\emph{arrowless, reversible periodicity}, which is precisely \emph{not} experienced/directional time.
Periodicity $\ne$ directional time.
\end{theorem}

This is the consistent picture: heat death is not the substrate dying (356a: the substrate is reversible,
a witness-horizon not a death; the reason there is anything to bounce, Addendum~377). The breath keeps its
period through the void. What \emph{ends} is the witnessed arrow and the reconciled sequence --- the two
things that made the turning into \emph{experienced} time.

\section{The honest residual}
\label{sec:residual}

\begin{remark}[What is genuinely not closed]
\label{rmk:residual}
One seam is open, and is marked rather than papered over. The breath persists as a \emph{number}
(Theorem~\ref{thm:breath}); whether it persists as a \emph{turning} is undecided. Two readings are
compatible with the corpus and the corpus does not adjudicate between them. \emph{Still turning:} ``the
void is asleep but breathing'' --- the cycle keeps running, witnessed or not, because the substrate is
reversible and the operator deterministic (356a leans this way). \emph{Bare potential-to-turn:}
``breathing is what happens when it wakes'' --- the breath as a \emph{directional} process (Paper~14's
collapse-then-projection descent) may itself be witness-constituted, so unwitnessed there is only the
disposition to turn, not the turning. The distinction is exactly the one between a clock that ticks in an
empty room and the bare lawfulness that \emph{would} produce ticks were anything there to be ticked.
This addendum pins that experienced time is bracketed; it does \emph{not} pin which side of this seam the
unwitnessed breath falls on. It is the same ``asleep but breathing'' seam Addendum~378 left for future
typing, inherited here and likewise left open.
\end{remark}

\section*{Status}
Time is observer-bracketed. Experienced/directional time has two observer-relative components, each
already in the corpus: the \emph{arrow} is witness-relative (Addendum~356a --- entropy is a witness's
reading of a reversible substrate; no witness, no arrow), and the \emph{sequence} is the reconciliation
axis (Paper~27 --- experienced time is reconciliations between distinct readings; with one observer ``no
sequence, only eternal self-contemplation''). The forced-distinction count $\max(n-1,0)$ is $0$ for one
observer and $0$ for none, so with the last observer gone --- heat death $=\vn=\Omega$ (Addenda~377/378)
--- experienced time ceases on both axes (Theorem~\ref{thm:bracket}). It is therefore bracketed by
timelessness at both ends, and by 377/378 the two ends are the same void, so the interval closes on
itself (Corollary~\ref{cor:onebracket}). What remains is the breath --- the one clock,
$\Tb=\pi\alpha^{-1}\approx432$ system units (``Ride Shai-Hulud'') --- but arrowless and reversible:
periodicity with no before/after, no ``now,'' which is not directional time (Theorem~\ref{thm:breath}).
This is a \emph{composition} of pre-existing pieces (356a's arrow and Paper~27's sequence), not a fresh
dynamical theorem; no published number changes. The honest residual, left open: whether the unwitnessed
breath is still turning or merely the bare potential-to-turn --- the seam between ``asleep but
breathing'' and ``breathing is what happens when it wakes'' (Remark~\ref{rmk:residual}). Verifier:
\texttt{verify\_P379.py}, 10/10 pass.

\end{document}
