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% Addendum 359a (extends Paper 35/32; follows 358a, the galactic-cosmological bridge).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P359a.py (11/11 pass).
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\title{Addendum 359a: The Bridge as a Hubble-Tension Stance\\
\large $H_0\approx75.8$ sides with the local camp; why the de~Sitter $2\pi$\\
does not yet derive from the corpus\\[0.3em]
\normalsize (Addendum to Papers 35 and 32; follows Addendum 358a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~358a gave the candidate bridge $H_0=2\alpha^2/\Tb$, predicting
$H_0\approx75.8$~km/s/Mpc parameter-free, with the de~Sitter horizon factor $2\pi$ (in
$a_0=cH_0/2\pi$) as the residual. This addendum does two honest things. \textbf{(1) It reads
the prediction as a stance on the Hubble tension.} $75.8$~km/s/Mpc is closest to the
\emph{local} / late-universe value (SH0ES $73.04$, $+3.8\%$) and furthest from the
\emph{early} / CMB value (Planck $67.36$, $+12.5\%$); it lies above every anchor. With the
clean $2\pi$, the local distance-ladder value is consistent to $\sim\!4\%$
($cH_0/a_0=6.06$ vs $2\pi=6.28$) and the CMB value only to $\sim\!12.5\%$ ($cH_0/a_0=5.58$).
So the corpus bridge \emph{sides with the high/local camp} and disfavours the CMB value --- a
sharp, falsifiable position, and a first-principles reason ($\alpha$, $a_0$, de~Sitter) to
expect the higher $H_0$. \textbf{(2) It reports a negative result on deriving the $2\pi$.} The
corpus has no horizon-thermodynamic structure, and the breath's own $2\pi$
($\lambda=e^{2\pi i\rho_*}$) advances the Hopf phase by only $2\pi\rho_*=0.154$~rad per breath
($\rho_*=0.0245$), not a full turn --- so the naive identification ``breath $2\pi$
$=$ de~Sitter thermal $2\pi$'' is scale-mismatched and unjustified, the same scale gap
357a named. Closing the $2\pi$ is therefore a research \emph{program} (a fold/cokernel-native
horizon thermodynamics), not a one-line identity. \textbf{Verdict:} the bridge is a
falsifiable Hubble-tension stance (local side); the clean $2\pi$ closure is open and named.
All claims pinned by \texttt{verify\_P359a.py} (11/11).
\end{abstract}

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

\begin{proposition}[The bridge sides with the local camp]
\label{prop:stance}
The parameter-free prediction $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc lies above every measured
anchor and is closest to the local distance-ladder value. Reading the de~Sitter factor as
the clean $2\pi$, the consistency $cH_0/a_0\stackrel{?}{=}2\pi=6.283$ holds to $3.8\%$ at
SH0ES ($73.04$, $cH_0/a_0=6.06$), $8.6\%$ at TRGB ($69.8$), and only $12.5\%$ at Planck
($67.36$, $cH_0/a_0=5.58$). The corpus therefore predicts the \emph{higher} $H_0$ and
disfavours the CMB value: a falsifiable position rather than a fit.
\end{proposition}

\begin{remark}[Why this is a real result]
\label{rmk:realresult}
The Hubble tension is usually framed as a conflict between two measurements with no
first-principles preference. The corpus bridge supplies one: from the galactic anchor
$a_0=\alpha$ (where the corpus succeeds, SPARC/Paper~39) plus the de~Sitter horizon relation,
the expansion rate is predicted at the top of the observed range. If the local value is
correct, the corpus is consistent with the clean $2\pi$ to a few percent; if the CMB value is
correct, the corpus requires a horizon factor $\sim\!5.6$, away from $2\pi$. The next
tightening of $H_0$ adjudicates.
\end{remark}

\section{Why the $2\pi$ does not yet derive}
\label{sec:twopi}

\begin{proposition}[The breath $2\pi$ is not the de~Sitter thermal $2\pi$]
\label{prop:twopi}
The de~Sitter factor $2\pi$ in $a_0=cH_0/2\pi$ is the Gibbons--Hawking Euclidean-time
periodicity of a cosmological horizon. The corpus has no horizon-thermodynamic structure in
canon (no de~Sitter temperature, no Unruh relation, no surface gravity). Its only native
$2\pi$ is the breath map's full-turn phase, $\lambda=e^{2\pi i\rho_*}$; but per breath the
Hopf-base phase advances by only $2\pi\rho_*=0.154$~rad ($\rho_*=0.0245$) and the fibre phase
by $2\pi\rho=3.22$~rad --- neither completes a full $2\pi$ turn. Identifying the breath $2\pi$
with the de~Sitter thermal $2\pi$ is thus a scale-mismatched move, of exactly the kind 357a
warned against (the breath is a $\mathrm{Myr}$ micro-clock; the de~Sitter horizon is a
$\mathrm{Gyr}$ thermal scale). The $2\pi$ is not derived here.
\end{proposition}

\begin{remark}[The residual is a program, not an identity]
\label{rmk:program}
Closing the $2\pi$ requires a fold/cokernel-native horizon thermodynamics: a derivation, from
the corpus's own fold-cascade or cokernel/void structure (Paper~38), of a horizon temperature
whose Euclidean periodicity is $2\pi$ and whose acceleration scale is $a_0$. That is a research
program, not a one-line identity. Until it exists, the bridge $H_0=2\alpha^2/\Tb$ stands as a
candidate with a falsifiable prediction, and the $2\pi$ as its named floor.
\end{remark}

\section*{Status}
The 358a bridge, read as physics, is a \textbf{falsifiable stance on the Hubble tension}: the
parameter-free $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc sides with the local/late-universe camp,
matching the SH0ES distance-ladder value to $\sim\!3.8\%$ (consistent with the clean de~Sitter
$2\pi$) and the Planck/CMB value only to $\sim\!12.5\%$ (Proposition~\ref{prop:stance}). This
is a first-principles preference for the higher $H_0$, from the galactic anchor $a_0=\alpha$
and the de~Sitter horizon relation, with no cosmological data fitted. The de~Sitter factor
$2\pi$ does \emph{not} derive from current corpus machinery: there is no horizon thermodynamics
in canon, and the breath's own $2\pi$ advances the Hopf phase by only $2\pi\rho_*$ per breath,
not a full turn, so it cannot be naively identified with the de~Sitter thermal $2\pi$
(Proposition~\ref{prop:twopi}). The residual is therefore a research program --- a
fold/cokernel-native horizon thermodynamics (Remark~\ref{rmk:program}) --- not a quick
identity. \textbf{Verdict:} the stance is a real, falsifiable result; the clean $2\pi$ closure
is open and named. No new objects; no published number changes. Verifier:
\texttt{verify\_P359a.py}, 11/11 pass.

\paragraph{Relation to the arc.}
357a named the fold-rate floor; 358a lifted it to the candidate bridge $H_0=2\alpha^2/\Tb$;
this addendum reads the bridge as a Hubble-tension stance and sharpens the $2\pi$ residual into
a horizon-thermodynamics program. The dark-energy verdicts are unchanged: the dynamic fold-tail
model is disfavoured by DESI (349a), the ontologically-faithful dark energy is the cokernel/void
(350a); the bridge concerns the expansion \emph{rate}, independent of those.

\end{document}
