\documentclass[11pt]{article}
% Addendum 389 (Target 1, run honestly: NEGATIVE). The horizon factor is forced to 2pi by the smooth
% round S^3, the data want F < 2pi (a conical deficit the round horizon lacks), so H0=75.8 is structural
% and the ~3 sigma tension is not a fixable factor. One post-hoc lead (edge-exclusion, 74.1) flagged,
% not claimed. Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P389.py (9/9 pass).
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\title{Addendum 389: Target 1 --- The Horizon Factor\\
\large Run honestly: $2\pi$ is forced by the smooth round $S^3$, so the\\
$H_0$ over-prediction is structural, not a fixable factor\\[0.3em]
\normalsize (Addresses the $H_0$ tension of Addendum 388; extends 363/369, P04)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
The 2026 scorecard (Addendum~388) found $H_0=2\alpha^2/T_{\mathrm{breath}}=75.8$~km/s/Mpc lying
$\sim\!3\sigma$ above SH0ES --- the corpus over-predicts $H_0$. The one data-facing knob is the horizon
factor $F$ in $a_0=cH_0/F$ (with $H_0\propto F$; the corpus posits $F=2\pi$, the de~Sitter / $\Theta$-fold
smoothness, Addendum~363). Target~1 asks whether the $\Theta$-fold/void structure gives $F<2\pi$,
lowering $H_0$ into the data window. Run honestly, it does not, and the negative is clean.
\textbf{The window.} $H_0\propto F$ gives a data window $F\in[5.587,6.066]$ (Planck to SH0ES); the posit
$2\pi=6.283$ sits \emph{above} it, so a rescue requires $F<2\pi$ --- a conical \emph{deficit} at the
horizon. \textbf{The obstruction.} The corpus's horizon is the round $S^3=\partial B^4$, and it is
smooth: the family quotient $L(3,1)=S^3/\mathbb Z_3$ is a \emph{free} action (no fixed points, no
orbifold cone), so the Euclidean smoothness period is exactly $2\pi$, no deficit. $F=2\pi$ is forced.
\textbf{The void is the wrong kind of defect.} $\Lz=1-\pi^2/32$ is a volume-corner residual
($\mathrm{vol}(B^4)/\mathrm{vol}(\text{cube})=\pi^2/32$ covered, $\Lz$ the corners not covered) ---
a defect of the embedding \emph{outside} the round $S^3$, not a conical deficit \emph{of} it; forcing it,
$F=2\pi\Lz=4.35$, gives $H_0=52$ ($-24\sigma$). The de~Sitter-rate correction $F=2\pi/\sqrt{\Omega_\Lambda}
=7.56$ gives $H_0=91$ ($+21\sigma$), the wrong direction. \textbf{Verdict:} $F=2\pi$ is forced, $H_0=75.8$
is the corpus's genuine prediction, and the $\sim\!3\sigma$ tension is \emph{structural}, not a fixable
factor. One post-hoc lead --- $F=2\pi(1-\text{edge})=6.14$, $H_0=74.1$, $\sim\!1\sigma$ from SH0ES --- is
flagged to ground or reject, not claimed. The honest fix lies deeper. All claims pinned by
\texttt{verify\_P389.py} (9/9).
\end{abstract}

\section{The window, and why a rescue needs a deficit}
\label{sec:window}

Since $a_0=cH_0/F$ and $a_0$ is anchored, $H_0\propto F$, and the posit $F=2\pi$ gives $H_0=75.8$. The
data require $F=2\pi\,(H_0^{\mathrm{data}}/75.8)$: SH0ES ($73.17$) wants $F=6.066$, DESI ($68.53$) wants
$5.681$, Planck ($67.4$) wants $5.587$. So the window is $F\in[5.587,6.066]$, entirely \emph{below}
$2\pi=6.283$. A rescue must therefore \emph{lower} $F$ below $2\pi$ --- which, since the $2\pi$ is the
Euclidean smoothness (no-conical-deficit) period of the horizon, means introducing a conical
\emph{deficit} (a period less than $2\pi$).

\section{The round horizon is smooth: $F=2\pi$ is forced}
\label{sec:forced}

\begin{theorem}[No conical deficit on the round $S^3$]
\label{thm:smooth}
The corpus's horizon is the round $S^3=\partial B^4$, and it is smooth. The family quotient
$L(3,1)=S^3/\mathbb Z_3$ acts \emph{freely} (no fixed points), so it introduces no orbifold cone; the
Euclidean smoothness period at the horizon is exactly $2\pi$, with no conical deficit. Hence $F=2\pi$ is
forced, and $H_0=75.8$. A smooth round horizon cannot carry $F<2\pi$ without a conical singularity it does
not have.
\end{theorem}

\begin{proposition}[The void is a volume residual, not an angle deficit]
\label{prop:void}
$\Lz=1-\pi^2/32$ is the corner residual of the embedding $B^4\hookrightarrow[-1,1]^4$:
$\mathrm{vol}(B^4)/\mathrm{vol}(\text{cube})=(\pi^2/2)/16=\pi^2/32$ is covered, and $\Lz$ is the corners
not covered. This is a defect of the embedding \emph{outside} the round $S^3$, not a conical (angle)
deficit \emph{of} the horizon. Forcing it as a factor gives $F=2\pi\Lz=4.35$, $H_0=52$ ($-24\sigma$):
unprincipled and wildly wrong. The de~Sitter-rate correction $F=2\pi/\sqrt{\Omega_\Lambda}=7.56$ gives
$H_0=91$ ($+21\sigma$), the wrong direction. Neither yields a principled $F$ in the window.
\end{proposition}

\section{Verdict and the one flagged lead}
\label{sec:verdict}

\begin{corollary}[Target 1 is negative; the tension is structural]
\label{cor:neg}
By Theorem~\ref{thm:smooth} and Proposition~\ref{prop:void}, the $\Theta$-fold/void structure offers no
principled horizon factor below $2\pi$. So $F=2\pi$ stands, $H_0=75.8$ is the corpus's genuine prediction,
and the $\sim\!3\sigma$ over-prediction against SH0ES is \emph{structural} --- a consequence of the
horizon's smoothness, not an adjustable coefficient. The bridge is not rescued at the horizon factor.
\end{corollary}

\begin{remark}[The one lead, flagged and not claimed]
\label{rmk:lead}
One candidate lands close: $F=2\pi(1-\text{edge})$ with $\text{edge}=\pi/\alpha^{-1}=0.0229$ gives
$F=6.14$, $H_0=74.1$, $\sim\!1.0\sigma$ from SH0ES. Its story would be that the Euclidean smoothness is
read over the \emph{observable} (bulk $+$ boundary) horizon, excluding the unobservable monadic edge
layer. But there is no derivation that the horizon angle loses a layer fraction; it is a number landing
near the window with a plausible-sounding but underived story. It is recorded here as a candidate to be
\emph{grounded or rejected} --- not as a rescue. Claiming it would be exactly the
$\delta_{CP}$/PRVE numerology trap the corpus refuses: a coincidental fit dressed as a derivation. If a
principled reason the horizon excludes the edge layer can be derived, this becomes a real fix; absent
that, it is a coincidence on the watchlist.
\end{remark}

\section*{Status}
Target~1, run honestly, is a negative. The horizon factor in $a_0=cH_0/F$ is forced to $2\pi$ by the
smooth round $S^3$ horizon ($L(3,1)=S^3/\mathbb Z_3$ free, no orbifold cone; Theorem~\ref{thm:smooth}),
so $H_0=75.8$ is the corpus's genuine prediction and the $\sim\!3\sigma$ tension against SH0ES is
structural, not a fixable coefficient (Corollary~\ref{cor:neg}). The data want $F<2\pi$ --- a conical
deficit the round horizon lacks --- and the void $\Lz$ is a volume-corner residual outside the horizon,
not an angle-deficit (forcing it gives $H_0=52$; Proposition~\ref{prop:void}). One post-hoc lead
(edge-exclusion, $H_0=74.1$, $\sim\!1\sigma$) is flagged to ground or reject, not claimed
(Remark~\ref{rmk:lead}). The honest fix lies deeper --- the $a_0=cH_0/2\pi$ relation itself, the chart
anchoring ($c=\pi/\alpha$, $T_{\mathrm{breath}}$), or the closure hypothesis (Addendum~370) --- or the
data must rise. The horizon factor does not rescue the bridge; it shows the $H_0$ tension is structural
and re-points the search. No published number changes. Verifier: \texttt{verify\_P389.py}, 9/9 pass.

\end{document}
