\documentclass[11pt]{article}
% Addendum 382 (unifies + resolves 378's and 379's residuals; extends 356a/365/377/378/379, P07/P29).
% The breath is the one surviving scale. Every observer-relative scale dissolves to the zeta-R
% scale-free fixed point at heat death; the single survivor is T_breath = pi alpha^-1 (the substrate
% rhythm). So 378's complete-scale-dissolution = 'complete up to the breath', and that survivor IS
% 379's 'does the unwitnessed breath turn?' -- one residual. Typed: still-turning (substrate) +
% arrowless (356a); the deeper 'really happening' is a category boundary (370 family).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P382.py (10/10 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\mathrm{breath}}}
\newcommand{\vn}{\varnothing}
\newcommand{\Rcal}{\mathcal{R}}

\title{Addendum 382: The Breath Is the One Surviving Scale\\
\large 378's and 379's residuals are the same residual ---\\
the bounce is complete up to one substrate period\\[0.3em]
\normalsize (Unifies and resolves the residuals of Addenda 378, 379; extends 356a, 365, P07, P29)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
The cyclic-void cosmology left two residuals that look separate. Addendum~378: the two-voids identity
(heat-death-$\vn$ $=$ origin-$\vn$) is complete only if \emph{both} ends are scale-free --- if a
characteristic scale survived heat death, the bounce would be incomplete. Addendum~379: is the
unwitnessed breath ``still turning'' or merely ``bare potential-to-turn''? This addendum shows they are
\emph{one} residual, and resolves it. Under $\zeta\Rcal$ flow every scale-bearing mode (scaling
dimension $s\neq0$) leaves the fixed point; only the scale-free $s=0$ mode is fixed (378/381). The
corpus's scales split in two: the \emph{observer-relative} scales --- particle masses (Paper~07:
``mass $=$ the stabilization scale of nested self-referential \emph{observation}''), the galactic
$L_h$ --- are witness-relative (356a) on a witness tower that \emph{terminates} (365), so at heat death
(no observation) they flow to the scale-free fixed point and \emph{dissolve}; and the \emph{substrate}
rhythm, the breath period $\Tb=\pi\alpha^{-1}$, which by the only-clock pin (the breath is the one clock,
none generates its own rhythm) and Paper~29's exposure mode (the fixed architecture's rhythm, not
observer-generated) is \emph{not} a distinction observation makes, and does not dissolve. \textbf{So
complete scale-dissolution holds for everything except one named scale: the breath} --- the bounce is
complete up to the single substrate period $\Tb$ (378 closed, sharpened). \textbf{And whether that one
surviving scale ``still turns'' unwitnessed is exactly 379's question} --- so the two residuals unify.
Typed: the breath \emph{still turns} (periodicity is substrate, observer-independent) but
\emph{arrowlessly} (the arrow and the ``now'' are witness-relative, 356a); and whether unwitnessed
arrowless turning is ``really happening'' is a \emph{category boundary} (the 370 family), not a gap. All
claims pinned by \texttt{verify\_P382.py} (10/10).
\end{abstract}

\section{Two kinds of scale, two fates}
\label{sec:scales}

\begin{proposition}[Observer-relative scales dissolve; the substrate breath survives]
\label{prop:fates}
Under the renormalization flow $e^{t\Rcal}$, a mode of scaling dimension $s$ is scaled by $e^{ts}$; only
the $s=0$ scale-free mode is fixed (378/381). The corpus's scales fall in two kinds.
\emph{(i) Observer-relative scales.} Particle masses are observational --- Paper~07: ``particle masses
do not exist as intrinsic properties''; mass is the stabilization scale of nested self-referential
observation --- and the galactic $L_h$ likewise. By Addendum~356a these are witness-relative, and by
Addendum~365 the witness tower (Cayley--Dickson, $\mathbb R\to\mathbb C\to\mathbb H\to\mathbb O$)
\emph{terminates}, so its finite structure dissolves at maximal entropy. At heat death, with no
observation, these scales flow to the $s=0$ fixed point: they \emph{dissolve}.
\emph{(ii) The substrate rhythm.} The breath period $\Tb=\pi\alpha^{-1}$ is the substrate's own clock.
By the only-clock pin (the breath is the one clock; none generates its own rhythm) and Paper~29's
exposure mode (the breath is the rhythm of the \emph{fixed} architecture, not observer-generated), the
breath is not a distinction observation makes. It does not dissolve with the distinctions.
\end{proposition}

\section{The two residuals are one}
\label{sec:one}

\begin{theorem}[378 closes to ``complete up to the breath''; the survivor is 379]
\label{thm:one}
By Proposition~\ref{prop:fates}, complete scale-dissolution at heat death holds for every scale
\emph{except} one: the substrate breath $\Tb$. So Addendum~378's premise --- that both ends of the
two-voids identity are scale-free --- holds \emph{up to} the single period $\Tb$; the bounce is complete
up to one named substrate scale. And whether that one surviving scale ``still turns'' unwitnessed is
precisely the residual of Addendum~379. Hence \textbf{378's surviving-scale residual and 379's
breath-turning residual are the same residual}: the cosmology residual is one thing, the breath, not
three.
\end{theorem}

\begin{corollary}[The type: still turning, arrowlessly; the limit: a category boundary]
\label{cor:type}
The residual resolves at the structural level and is honestly bounded at the deepest. \emph{Periodicity}
--- the bare turning of $\Tb$ --- is observer-\emph{independent} (the only-clock pin, Paper~29), so the
unwitnessed breath \emph{still turns}. The \emph{arrow} and the \emph{now} are witness-\emph{relative}
(356a/P27), so they vanish with the last observer. Thus the unwitnessed breath is ``still turning,
\emph{arrowlessly}'' --- periodicity without flow, before/after, or now. The remaining question ---
whether unwitnessed, arrowless turning is ``really happening'' or has content --- is \emph{not} a
derivable gap: happening is constituted through observation, so one cannot ask, from within the frame,
whether something happens with no one to whom it happens. This is a \emph{category boundary}, of the
same family as Addendum~370's, and it is the corpus's one honestly-flagged limit on the cyclic
cosmology.
\end{corollary}

\section*{Status}
The cyclic-void cosmology's residual is one thing, not three. Under $\zeta\Rcal$ flow every
\emph{observer-relative} scale --- particle masses (Paper~07), the galactic $L_h$ --- dissolves to the
scale-free fixed point at heat death (356a witness-relativity $+$ 365 the terminating witness tower),
while the \emph{substrate} breath $\Tb=\pi\alpha^{-1}$ survives (the only-clock pin, Paper~29's exposure
mode). So Addendum~378's complete-scale-dissolution closes to ``complete \emph{up to} the breath''
(Theorem~\ref{thm:one}), and that single survivor is exactly Addendum~379's ``does the unwitnessed
breath turn?'' --- the two residuals unify. The type (Corollary~\ref{cor:type}): the breath still turns
(periodicity is substrate, observer-independent) but arrowlessly (the arrow/now are witness-relative,
356a); and whether unwitnessed arrowless turning is ``really happening'' is a category boundary (the 370
family), not a gap. The bounce is complete up to one named substrate period whose turning is the
corpus's one honest limit. No published number changes; the new content is the unification of two
residuals and their resolution. Verifier: \texttt{verify\_P382.py}, 10/10 pass.

\end{document}
