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% Addendum 366 (horizon-thermodynamics Step 2; extends P02/P38/363; consolidates 358a/359a/362).
% Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P366.py (9/9 pass).
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\title{Addendum 366: The Horizon Temperature (Step 2)\\
\large $a_0=cH_0/(2\pi)$ assembled from corpus structures: the anchor,\\
and the falsifiable prediction\\[0.3em]
\normalsize (Addendum to Papers 02, 38; Step 2 of the horizon program opened in 363)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~363 opened the horizon-thermodynamics program and reduced the cosmological bridge
$H_0=2\alpha^2/\Tb$ to one residual: the de~Sitter factor in $a_0=cH_0/(2\pi)$. It grounded the
$2\pi$ as the Euclidean smoothness of the $\Theta$-fold horizon. Step~2 supplies the temperature
and assembles the relation --- non-circularly, since 358a \emph{defined} $H_0$ through
$a_0=cH_0/2\pi$. \textbf{The crux is the anchor.} $a_0=\alpha$ is the corpus's dimensionful
anchor (Paper~35: the unit system is ``calibrated conditional on the measured $a_0$''); it is
the one dimensionful input and is not derivable --- a unit anchor cannot be. So $a_0=cH_0/(2\pi)$
cannot derive $a_0$; it fixes $H_0$ given $a_0$ (the 358a/362 result). What Step~2 does is
assemble the relation \emph{entirely from corpus structures}, making it corpus-native rather than
an imported MOND coincidence: the $2\pi$ is the $\Theta$-fold horizon smoothness (363, P04); the
temperature is Paper~02's partition function $\Phi(s)=\int_0^1\rho\,e^{-sx}dx$ (with $s$ the
inverse temperature, $\beta_{\mathrm{geom}}=-\Phi'(0)/\Phi(0)=m_1/m_0=0.79334$ the specific heat,
and the Theorem-9 critical inverse-temperature $s^*\!\approx\!\pi/6$); the horizon is the
cokernel/void of Paper~38 ($\Lambda_0=1-\pi^2/32$). The Gibbons--Hawking form $T=\kappa/(2\pi)$
with $\kappa=cH_0$ then reads $a_0$ --- the per-breath velocity gain, an acceleration --- as the
void horizon's temperature-as-acceleration. \textbf{Verdict: substantial.} The program closes the
$2\pi$: the relation is corpus-native and the 359a numerology worry is fully dissolved. The lone
residual is the MOND-type \emph{identification} ($a_0=$ the void's Gibbons--Hawking acceleration),
now corpus-expressed but not proven from deeper axioms; $a_0=\alpha$ is the anchor, definitional.
With it, the cosmological bridge has \emph{no remaining derivation floor}:
$H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc is a clean parameter-free prediction, and the only open thing is
the empirical $H_0$ tension --- falsifiable, leaning local/high. All claims pinned by
\texttt{verify\_P366.py} (9/9).
\end{abstract}

\section{The three corpus structures}
\label{sec:three}

\begin{proposition}[The temperature: Paper 02's partition function]
\label{prop:temp}
Paper~02 builds the partition function $\Phi(s)=\int_0^1\rho(x)\,e^{-sx}\,dx$ with $s$ the inverse
temperature, $\Phi(0)=m_0=\alpha^{-1}$, specific heat
$\beta_{\mathrm{geom}}=-\Phi'(0)/\Phi(0)=m_1/m_0=0.79334$, and (Theorem~9) a critical
inverse-temperature $s^*\approx\pi/6$ where $\Phi(s^*)/s^*=m_0^2/m_1$. This is the corpus's native
temperature concept, on the geometric density $\rho$.
\end{proposition}

\begin{proposition}[The horizon and the $2\pi$]
\label{prop:hor}
The cosmological horizon is the cokernel/void of Paper~38, $\Lambda_0=1-\pi^2/32$ (the volume of
$[-1,1]^4$ not covered by $B^4$). The Euclidean smoothness factor at the horizon fixed point ---
the $\Theta$-fold horizon of Paper~04, the singularity $F_\Theta(\varnothing)=\varnothing$ with
$\Theta\in[0,2\pi]$ --- is the $2\pi$, as established in Addendum~363. Both are corpus structures,
not imports.
\end{proposition}

\begin{theorem}[The relation is corpus-native]
\label{thm:native}
The Gibbons--Hawking temperature of a horizon is $T=\kappa/(2\pi)$ with surface gravity $\kappa$;
for the cosmological (de~Sitter) horizon $\kappa=cH_0$, so $T=cH_0/(2\pi)$. Identifying $a_0$ ---
the per-breath velocity gain of Paper~35, an acceleration --- with this horizon
temperature-as-acceleration gives $a_0=cH_0/(2\pi)$, with every ingredient a corpus structure:
the $2\pi$ from Proposition~\ref{prop:hor} (363), the temperature from
Proposition~\ref{prop:temp} (P02), the horizon from Proposition~\ref{prop:hor} (P38). The
relation is therefore corpus-native, not an imported MOND coincidence.
\end{theorem}

\section{The anchor and the residual}
\label{sec:anchor}

\begin{remark}[Why this fixes $H_0$, not $a_0$]
\label{rmk:anchor}
$a_0=\alpha$ is the corpus's dimensionful anchor (Paper~35: ``calibrated conditional on the
measured $a_0$''). A unit anchor is not derivable; it is the one dimensionful input. So
Theorem~\ref{thm:native} cannot derive $a_0$ --- it fixes $H_0$ given $a_0$, recovering the
358a/362 ladder $H_0=2\alpha^2/\Tb=(\pi/2)\alpha^{-3}\tau_1^{-1}$. The remaining open item is not
a magnitude floor but the \emph{identification} itself: that $a_0$ equals the void horizon's
Gibbons--Hawking acceleration is the MOND-type hypothesis, now expressed in corpus structures but
not proven from deeper axioms.
\end{remark}

\begin{corollary}[No derivation floor; a clean prediction]
\label{cor:pred}
With the identification, the cosmological bridge has no remaining derivation floor: the $2\pi$ is
the $\Theta$-fold smoothness (363), the temperature is the P02 partition function, the horizon is
the P38 void, and $a_0=\alpha$ is the anchor. Hence $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc is a clean
parameter-free prediction. The only open thing is empirical: the $H_0$ tension (the prediction
lies above Planck $67.4$ by $12.4\%$ and SH0ES $73.0$ by $3.8\%$, leaning local/high). That is
falsifiable content, not a gap.
\end{corollary}

\section*{Status}
Step~2 of the horizon program substantially closes the de~Sitter $2\pi$. The relation
$a_0=cH_0/(2\pi)$ is assembled entirely from corpus structures (Theorem~\ref{thm:native}): the
$2\pi$ from the $\Theta$-fold horizon smoothness (363, Paper~04), the temperature from Paper~02's
partition function ($\beta_{\mathrm{geom}}=0.79334$, $s^*\approx\pi/6$,
Proposition~\ref{prop:temp}), and the horizon from Paper~38's cokernel/void
(Proposition~\ref{prop:hor}). The relation is therefore corpus-native and the 359a numerology
worry is fully dissolved. $a_0=\alpha$ is the dimensionful anchor (Paper~35), definitional and
not a derivation gap (Remark~\ref{rmk:anchor}). The lone residual is the MOND-type identification
$a_0=$ the void's Gibbons--Hawking acceleration, corpus-expressed but not axiom-derived. With it,
the cosmological bridge has no remaining derivation floor, and $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc
stands as a clean, parameter-free, falsifiable prediction (Corollary~\ref{cor:pred}). This is the
substantial endpoint the program aimed at: the $2\pi$ is no longer an isolated floor (357a/359a)
but a corpus-native horizon factor, and the matter-asymmetry, horizon-temperature, and
arrow-of-time faces of 364 share the same now-grounded structure. No new objects beyond the
Gibbons--Hawking form; no published number changes. Verifier: \texttt{verify\_P366.py}, 9/9 pass.

\paragraph{What would make it a theorem.}
Deriving the identification $a_0=$ horizon-temperature from a deeper corpus axiom --- rather than
positing it as the MOND-type reading --- would upgrade the prediction from corpus-native to
theorem. Absent that, $H_0=75.8$ is the falsifiable claim, and the next $H_0$ measurement
adjudicates it.

\end{document}
