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% Addendum 363 (opens the horizon-thermodynamics program; extends P04/P02/P38; follows 359a/362).
% Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P363.py (9/9 pass).
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\title{Addendum 363: Opening the Horizon-Thermodynamics Program\\
\large The de~Sitter $2\pi$ is the Euclidean smoothness condition at the\\
corpus's own $\Theta$-fold horizon\\[0.3em]
\normalsize (Addendum to Papers 04, 02, 38; follows 359a, 362)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addenda 358a/359a/362 reduced the cosmological scale-bridge $H_0=2\alpha^2/T_{\mathrm{breath}}$
to a single isolated residual --- the de~Sitter horizon factor $2\pi$ in $a_0=cH_0/(2\pi)$ ---
and 362 separated the \emph{scale} ($\alpha^{-3}$) from it, so the $2\pi$ is a distinct,
\emph{dimensionless} question. This addendum opens the program to derive it and takes the first
step. \textbf{The $2\pi$ is the Euclidean smoothness / no-conical-deficit condition at a horizon
fixed point} --- a full circle. Continuing to imaginary time, the near-horizon geometry is a
plane in polar coordinates, smooth at the tip iff the angular period is exactly $2\pi$; any other
period leaves a conical deficit $2\pi-\text{period}$, a curvature delta-function at the tip. The
corpus contains this structure exactly. \textbf{(P04)} The fold map $F_\Theta$ is a black hole:
an event horizon $\mathcal{H}$ (the $2$-sphere where $\det DF_\Theta=0$, reached at $\Theta=\pi$,
the Hopf half-winding), an interior ($\det<0$, retrograde), and a singularity at the
\emph{fixed point} $F_\Theta(\varnothing)=\varnothing$ at the origin --- with the fold angle
running the full cycle $\Theta\in[0,2\pi]$. \textbf{(P02)} $\Phi(s)$ is a regularized partition
function, $s$ the inverse temperature, $\beta_{\mathrm{geom}}=-\Phi'(0)/\Phi(0)$ the specific
heat: the thermodynamic formalism is already in canon. \textbf{(P38)} Dark energy is the
cokernel/void, $\Lambda_0=1-\pi^2/32$ (the volume of $[-1,1]^4$ not covered by $B^4$): the
cosmological horizon as a measure defect. So the $2\pi$ in $a_0=cH_0/(2\pi)$ is the corpus's own
horizon-smoothness $2\pi$ --- the full $\Theta$-cycle around the singularity fixed point ---
\emph{not} a numerical coincidence; the numerology worry of 359a is dissolved. \textbf{Verdict:
program opened.} The residual, named for the next probe: derive $a_0=cH_0/(2\pi)$ \emph{exactly},
that the cokernel/void's partition-function horizon temperature gives the per-breath gain $a_0$ as
the horizon acceleration $cH_0$ per full $\Theta$-cycle. All claims pinned by
\texttt{verify\_P363.py} (9/9).
\end{abstract}

\section{The $2\pi$ is a smoothness condition}
\label{sec:smoothness}

\begin{proposition}[No conical deficit $\Leftrightarrow$ full $2\pi$]
\label{prop:smooth}
The de~Sitter/Gibbons--Hawking temperature $T=\hbar H/(2\pi k_B)$ carries the factor $2\pi$ for a
purely geometric reason: in the Euclidean continuation the near-horizon metric is
$dr^2+r^2 d\theta^2$, a plane, which is smooth at the fixed point $r=0$ iff $\theta$ has period
exactly $2\pi$. Any other period $\beta\neq2\pi$ leaves a conical deficit $2\pi-\beta$ --- a
curvature delta-function at the tip. The $2\pi$ is thus not a free constant but the smoothness
condition at a horizon fixed point, and it is \emph{dimensionless}, independent of the horizon's
scale.
\end{proposition}

\section{The corpus has this structure}
\label{sec:structure}

\begin{theorem}[The $\Theta$-fold horizon (P04)]
\label{thm:P04}
The fold map $F_\Theta$ realises a black hole (P04, Black Hole as Degenerate Self-Lensing): the
event horizon $\mathcal{H}$ is the $2$-sphere on which $\det DF_\Theta=0$, reached at $\Theta=\pi$
(the Hopf fibre's half-winding); the interior is the retrograde region $\det DF_\Theta<0$; and the
singularity is the \emph{fixed point} $F_\Theta(\varnothing)=\varnothing$ at the origin of $B^4$.
The fold angle runs the full cycle $\Theta\in[0,2\pi]$ (forward $[0,\pi)$, retrograde
$(\pi,2\pi)$). This is precisely a horizon with a fixed point and a $2\pi$ angular cycle ---
the geometry Proposition~\ref{prop:smooth} requires.
\end{theorem}

\begin{proposition}[The thermodynamic formalism (P02) and the cosmological horizon (P38)]
\label{prop:thermo}
P02 establishes the thermodynamic formalism in canon: $\Phi(s)$ is a regularized partition
function, $s$ plays the role of inverse temperature, and $\beta_{\mathrm{geom}}=-\Phi'(0)/\Phi(0)$
is a specific heat (with the heat kernel $\operatorname{tr}e^{-t\Delta}$ the bridge). P38 supplies
the cosmological horizon as the cokernel/void: $\Lambda_0=1-\pi^2/32\approx0.6916$ is the volume
fraction of $[-1,1]^4$ not covered by $B^4$ ($B^4$ volume $\pi^2/2$, cube $16$, covered
$\pi^2/32$), matching Planck's $\Omega_\Lambda$ at $0.94\sigma$. The cokernel/void \emph{is} the
cosmological horizon the de~Sitter relation refers to.
\end{proposition}

\begin{remark}[The numerology worry dissolved]
\label{rmk:numerology}
Addendum 359a flagged the risk that identifying the de~Sitter $2\pi$ with a corpus $2\pi$ would be
a coincidence of two unrelated $2\pi$'s. It is not: both are the \emph{same geometric object} ---
the smoothness/no-deficit angle at a horizon fixed point. The corpus's is the full $\Theta$-cycle
around the singularity (Theorem~\ref{thm:P04}); de~Sitter's is the Euclidean period at the
cosmological horizon. They are the same condition, so the identification is structural, not
numerical. The scale ($\alpha^{-3}$, 362) is separate and already derived; the $2\pi$ is the
dimensionless smoothness factor, now grounded.
\end{remark}

\section*{Status}
The horizon-thermodynamics program is \textbf{opened}, and its first step is taken. The de~Sitter
$2\pi$ is the Euclidean smoothness condition at a horizon fixed point --- a dimensionless full
circle (Proposition~\ref{prop:smooth}). The corpus contains exactly this: P04's fold map is a
black hole with a horizon $\mathcal{H}$ (at $\Theta=\pi$, the Hopf half-winding), a fixed-point
singularity ($\varnothing$ at the origin), and the full $\Theta\in[0,2\pi]$ cycle
(Theorem~\ref{thm:P04}); P02 supplies the partition-function / inverse-temperature / specific-heat
formalism; P38 supplies the cosmological horizon as the cokernel/void $\Lambda_0=1-\pi^2/32$
(Proposition~\ref{prop:thermo}). So the $2\pi$ in $a_0=cH_0/(2\pi)$ is the corpus's own
horizon-smoothness $2\pi$ --- the full $\Theta$-cycle around the singularity --- and the
numerology worry of 359a is dissolved (Remark~\ref{rmk:numerology}). \textbf{This is a first step,
not a closure.} The residual, named for the next probe: derive $a_0=cH_0/(2\pi)$ \emph{exactly} ---
that the cokernel/void's partition-function horizon temperature (P02/P38) yields the per-breath
gain $a_0$ as the horizon acceleration $cH_0$ per full $\Theta$-cycle. With that, $H_0=2\alpha^2/
T_{\mathrm{breath}}$ becomes a clean parameter-free corpus prediction. No new objects beyond the
de~Sitter relation; no published number changes. Verifier: \texttt{verify\_P363.py}, 9/9 pass.

\paragraph{The program, staged.}
Step~1 (this addendum): the $2\pi$ is the corpus $\Theta$-fold smoothness factor. Step~2 (next):
the cokernel/void horizon temperature from the P02 partition function. Step~3: that temperature's
acceleration scale equals $a_0$, closing $a_0=cH_0/(2\pi)$ and the Hubble prediction.

\end{document}
