\documentclass[11pt]{article}
% Addendum 369 (closes the horizon program; extends P35/P38/P04/363/366).
% Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P369.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}

\title{Addendum 369: Closing the Horizon Program\\
\large $a_0=$ the horizon temperature is a theorem --- conditional on the one\\
posit the SPARC success already rests on\\[0.3em]
\normalsize (Addendum to Papers 35, 38, 04; closes the program of 363/366)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~366 reduced the cosmological bridge $H_0=2\alpha^2/\Tb$ to a single residual: the
\emph{identification} $a_0=$ the void horizon's temperature-as-acceleration, corpus-expressed (via
363's $2\pi$, P02's partition function, P38's void) but imported as the MOND-type hypothesis. This
addendum closes it to a theorem --- conditionally --- and finds the condition is a posit the corpus
already names. The chain: \textbf{(L1, P35)} $a_0$ is the per-breath gain of the \emph{unique}
symmetry-allowed modulus channel, radial infall toward the breath map's rest-pole fixed point
($a_0\Tb^2/L_h=\alpha$ exactly); so $a_0$ is structurally the fixed-point horizon's per-breath infall
acceleration. \textbf{(L2, P04/P38)} the fixed point is the fold's singularity $F_\Theta(\vn)=\vn$
(the interior endpoint of the event horizon $\mathcal{H}$ at $\Theta=\pi$), and dark matter is the
\emph{kernel} --- the fold's sector annihilated by observation; the fixed-point horizon is the
kernel-side horizon. \textbf{(L3, P38/A276)} the void $\Lambda_0=1-\pi^2/32$ is the cokernel horizon,
and kernel and cokernel are \emph{one} closure-map family read in two directions. \textbf{(L4)} a
horizon temperature is a property of the horizon, not the reading direction; kernel and cokernel are
the two exactness defects of one map family --- one horizon, one temperature. Hence $a_0$ (the
kernel-side infall gain) equals the cokernel-side void temperature, the de~Sitter temperature; with
363's $2\pi$, $a_0=cH_0/(2\pi)$, so $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc, derived. \textbf{The condition},
named by P35 itself: that physical sources ($GM_b$, hence $a_0$) couple through the breath map at all
is ``typed rather than derived --- the v4 kernel's closure hypothesis,'' confronted with SPARC
($a_0=0.976\times$ empirical). This is \emph{not} a new posit: it is the same single closure
hypothesis on which the corpus's hidden-branch/SPARC success rests. So $H_0=2\alpha^2/\Tb$ rests on
\emph{no more} than the SPARC bet --- the cosmological prediction and the galactic success are one
hypothesis. All claims pinned by \texttt{verify\_P369.py} (10/10).
\end{abstract}

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

\begin{proposition}[L1: $a_0$ is the fixed-point horizon's infall gain]
\label{prop:L1}
By Paper~35's deformation theory, the only symmetry-allowed modulus response of the breath map is
radial infall toward its rest-pole fixed point, and $a_0$ is the per-breath gain of that unique
channel: $a_0\Tb^2/L_h=\alpha$ exactly. So $a_0$ is, structurally, the per-breath infall acceleration
toward the fixed-point horizon.
\end{proposition}

\begin{proposition}[L2--L3: the two horizons are one map family]
\label{prop:L2L3}
The fixed point is the fold's singularity $F_\Theta(\vn)=\vn$ at the origin of $B^4$ (Paper~04, the
interior endpoint of the event horizon $\mathcal{H}$, reached at $\Theta=\pi$); dark matter is the
\emph{kernel}, the fold's sector annihilated by observation (Paper~38/A272). The void
$\Lambda_0=1-\pi^2/32$ is the \emph{cokernel} horizon (Paper~38/366). By A276 the kernel and cokernel
are one closure-map family read in two directions: ``the cokernel of $(P,C)$ is the kernel of the
opposite datum.'' The kernel-side (fixed-point) horizon and the cokernel-side (void) horizon are thus
two faces of one map family.
\end{proposition}

\begin{theorem}[L4: $a_0$ is the de Sitter temperature, conditionally]
\label{thm:L4}
A horizon temperature is a property of the horizon --- its Euclidean periodicity and surface gravity
--- not of the direction the closure map is read. Since kernel and cokernel are the two exactness
defects of one map family (one horizon), they carry one temperature. Hence $a_0$, the kernel-side
fixed-point infall gain (Proposition~\ref{prop:L1}), equals the cokernel-side void temperature, the
de~Sitter temperature (Proposition~\ref{prop:L2L3}). With the smoothness factor $2\pi$ of
Addendum~363, $a_0=cH_0/(2\pi)$, so
\[
  H_0=2\alpha^2/\Tb=75.8~\text{km/s/Mpc}.
\]
This holds provided $a_0$ is a physical horizon acceleration --- the condition of \S\ref{sec:floor}.
\end{theorem}

\section{The condition is the SPARC bet}
\label{sec:floor}

\begin{remark}[The v4 closure hypothesis]
\label{rmk:floor}
Proposition~\ref{prop:L1} makes $a_0$ the breath map's infall gain; for that to be a \emph{physical}
horizon acceleration, physical sources must couple to the breath map. Paper~35 flags exactly this as
the one undischarged premise: that $GM_b$ (hence $a_0$) ``enters the breath map at all'' is ``typed
rather than derived --- the v4 kernel's closure hypothesis,'' confronted with data in SPARC
($a_0=0.976\times$ empirical, $\sim\!2.4\%$). It is the single condition on Paper~35's $\mathrm{A}^-\!\to\mathrm{A}$
upgrade. This is \emph{not} a new posit introduced here: it is the same closure hypothesis on which
the corpus's hidden-branch galactic-rotation success already rests. So the cosmological prediction
$H_0=2\alpha^2/\Tb$ and the galactic SPARC success are one and the same hypothesis, confronted twice.
\end{remark}

\section*{Status}
The identification $a_0=$ the horizon temperature, which 366 imported as the MOND-type hypothesis, is
closed to a \textbf{theorem --- conditionally}. The chain is corpus-solid: $a_0$ is the unique
fixed-point infall channel's per-breath gain (Proposition~\ref{prop:L1}, Paper~35); the fixed point
is the fold/kernel horizon (Proposition~\ref{prop:L2L3}, Papers~04/38); the void is the cokernel
horizon and kernel and cokernel are one map family (A276); one family carries one horizon temperature
(Theorem~\ref{thm:L4}); so $a_0$ is the de~Sitter temperature and $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc is
derived. The single condition is the v4 closure hypothesis --- that $a_0/GM_b$ couples through the
breath map at all --- which Paper~35 types rather than derives and confronts with SPARC
($a_0=0.976\times$ empirical, Remark~\ref{rmk:floor}). Because that is the \emph{same} posit the
corpus's SPARC success rests on, $H_0=2\alpha^2/\Tb$ rests on no more than the SPARC bet: the
cosmological prediction and the galactic success are one hypothesis, confronted twice. \textbf{The
honest ceiling:} this is a theorem conditional on the corpus's single empirical closure hypothesis,
not an unconditional derivation; $a_0=\alpha$ remains the dimensionful anchor (Paper~35). Closing it
unconditionally means deriving the closure hypothesis itself --- that gravitational sources couple
through the breath map --- from the Butlerian/observer axioms, which is the open frontier. No new
objects; no published number changes. Verifier: \texttt{verify\_P369.py}, 10/10 pass.

\paragraph{What the program achieved.}
357a named the fold-rate floor; 358a/362 built the $\alpha$-power ladder $H_0=(\pi/2)\alpha^{-3}\tau_1$;
363 grounded the $2\pi$ as the $\Theta$-fold horizon smoothness; 366 assembled $a_0=cH_0/(2\pi)$ from
corpus structures; and 369 derives the assembly from the kernel/cokernel duality, reducing the whole
cosmological-bridge residual to the one closure hypothesis the corpus already stakes on SPARC. The
$H_0\approx75.8$ prediction --- leaning local/high --- is now exactly as well-founded as the corpus's
best empirical result.

\end{document}
