\documentclass[11pt]{article}
% Addendum 358a (extends Paper 35 unit system + Paper 32 cosmology; lifts the 357a fold-rate floor).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P358a.py (12/12 pass).
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\title{Addendum 358a: The Galactic-to-Cosmological Time-Bridge\\
\large $H_0 = 2\alpha^2/\Tb$, a parameter-free Hubble prediction, and\\
the lifting of the fold-rate floor\\[0.3em]
\normalsize (Addendum to Papers 35 and 32; lifts the floor of Addendum 357a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~357a recorded the fold-rate $\nu=\tfrac14 H_0$ as a named floor: the corpus unit
system (Paper~35) is calibrated at the \emph{galactic} scale --- the universal/MOND
acceleration $a_0=\alpha$ in chart units, with $L_h=0.978$~kpc the SPARC/Paper~39 scale ---
and its clock, the breath $\Tb=1.374$~Myr, is $\sim\!10^4$ too fast to clock a cosmological
fold, with no bridge to span the gap. This addendum supplies a \emph{candidate} bridge from
corpus machinery plus one standard relation, and finds that it makes a parameter-free Hubble
prediction. The two ingredients: (1)~the corpus $a_0$-chain (Paper~35, A264/A265), in chart
units $a_0=\alpha$, $c=\pi/\alpha$, $\Tb=\alpha^2 c/(\pi a_0)$, with $a_0$ the per-breath
velocity gain; and (2)~the de~Sitter/MOND horizon relation $a_0=cH_0/(2\pi)$. Combining,
\[
  H_0 \;=\; \frac{2\pi a_0}{c} \;=\; 2\alpha^2 \quad(\text{chart units}),
  \qquad H_0=\frac{2\alpha^2}{\Tb},\qquad \frac1{H_0}=\frac{\alpha^{-2}}{2}\ \text{breaths}.
\]
With $\Tb^{\mathrm{SI}}=1.374$~Myr this predicts $H_0\approx75.8$~km/s/Mpc with \emph{no
cosmological input}, landing in and just above the $H_0$-tension window ($+3.8\%$ vs SH0ES
$73.0$, $+12.4\%$ vs Planck $67.4$). The $357a$ obstruction --- ``$\sim\!10^4$ breaths per
fold'' --- becomes $\alpha^{-2}/2\approx9389$ breaths, a \emph{corpus number}.
\textbf{Verdict: candidate bridge, not closure.} The residual is the exact horizon factor:
the data prefer $cH_0/a_0\approx5.6$ against the clean $2\pi=6.28$ ($\sim\!12\%$, within the
$H_0$ tension and $a_0$ uncertainty), and a corpus \emph{derivation} of that $2\pi$ ---
mapping the de~Sitter Gibbons--Hawking $2\pi$ to the breath's own full-turn $2\pi$ (the Hopf
fibre period) --- is not delivered here. The $2\pi$ is de~Sitter-motivated, not fit to the
data; the $12\%$ is reported, not hidden. All claims pinned by \texttt{verify\_P358a.py}
(12/12).
\end{abstract}

\section{The floor, and the foothold}
\label{sec:floor}

Addendum~357a showed that the corpus has no cosmological clock: its absolute clock is the
breath ($\Tb=1.374$~Myr), $\sim\!10^4$ too fast to clock a fold, and its unit system is
anchored galactically. The anchor is precise, though, and it is the foothold. Paper~35 fixes
the dimensionful scale through the universal acceleration $a_0$, with the chart identities
\begin{equation}
  a_0=\alpha,\qquad c=\frac{\pi}{\alpha},\qquad \Tb=\frac{\alpha^2 c}{\pi a_0}
  \label{eq:chain}
\end{equation}
(Paper~35, A264/A265), where $a_0$ is the per-breath velocity gain ($a_0\Tb^2/L_h=\alpha$
exactly). The same $a_0$ that sets the galactic scale is, in standard cosmology, tied to the
cosmological expansion rate --- and that tie is the bridge.

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

\begin{proposition}[The de Sitter horizon relation]
\label{prop:desitter}
In a de~Sitter background the horizon acceleration is $c^2/R_H=cH_0$ (horizon radius
$R_H=c/H_0$), and the MOND acceleration scale is the horizon acceleration divided by the
Gibbons--Hawking factor $2\pi$:
\begin{equation}
  a_0 \;=\; \frac{cH_0}{2\pi}.
  \label{eq:desitter}
\end{equation}
This is the standard MOND--cosmology coincidence. It is taken here as an external input; its
corpus derivation is the residual (\S\ref{sec:residual}).
\end{proposition}

\begin{theorem}[The time-bridge and the Hubble prediction]
\label{thm:bridge}
Combining \eqref{eq:chain} and \eqref{eq:desitter},
\begin{equation}
  H_0=\frac{2\pi a_0}{c}=\frac{2\pi\,\alpha}{\pi/\alpha}=2\alpha^2\quad(\text{chart units}),
  \qquad\text{i.e.}\qquad H_0=\frac{2\alpha^2}{\Tb},\quad \frac1{H_0}=\frac{\alpha^{-2}}{2}\ \text{breaths}.
\end{equation}
With $\Tb^{\mathrm{SI}}=1.374$~Myr this is the parameter-free prediction
$H_0\approx75.8$~km/s/Mpc (Hubble time $\approx12.9$~Gyr), using no cosmological data --- only
$\alpha$, $a_0$, $c$, and the de~Sitter factor $2\pi$. The prediction lands in and just above
the measured window: $+3.8\%$ above SH0ES ($73.0$) and $+12.4\%$ above Planck ($67.4$).
\end{theorem}

\begin{corollary}[The fold-rate floor is lifted to a corpus number]
\label{cor:foldrate}
Under Theorem~\ref{thm:bridge} the fold-rate is $\nu=\tfrac14 H_0=\alpha^2/(2\Tb)$, and the
fold period (one fold per e-fold) is $1/H_0=\alpha^{-2}/2\approx9389$ breaths. The
``$\sim\!10^4$ breaths per fold'' obstruction of 357a is therefore not arbitrary: it is
$\alpha^{-2}/2$, the square of the corpus fine-structure number. The observed value
($\approx10560$ breaths) differs by the ratio $1.125$ --- the MOND-coincidence slack.
\end{corollary}

\section{The residual, and why this is not the seed-identity trap}
\label{sec:residual}

\begin{remark}[The exact horizon factor]
\label{rmk:residual}
What is not delivered is a corpus derivation of the factor $2\pi$ in \eqref{eq:desitter}. The
data prefer $cH_0/a_0\approx5.6$ (using the corpus $a_0=1.172\times10^{-10}$) against the
clean $2\pi=6.28$, a $\sim\!12\%$ gap that is within the present $H_0$ tension and the $a_0$
uncertainty. A corpus derivation would map the de~Sitter Gibbons--Hawking $2\pi$ --- a
horizon-temperature factor --- onto the corpus's own fundamental $2\pi$: the breath map's
full-turn multiplier $\lambda=e^{2\pi i\rho_*}$ and the Hopf fibre period. That mapping is the
open residual; with it, $H_0=2\alpha^2/\Tb$ would be a closed prediction.
\end{remark}

\begin{remark}[Discipline]
\label{rmk:discipline}
The corpus flags reverse-engineered constants as the seed-identity mode. This is not that: the
factor $2\pi$ is fixed by the de~Sitter horizon \emph{before} comparison, and the data prefer
$5.6$, so the $12\%$ is a reported miss, not a tuned hit. The bridge predicts $H_0$ high
(toward the local/SH0ES value); a future tightening of $H_0$ and $a_0$ either confirms the
clean $2\pi$ or measures the true factor and refutes it.
\end{remark}

\section*{Status}
The galactic-to-cosmological time-bridge of 357a's floor is supplied as a \textbf{candidate}:
the corpus $a_0$-chain (Paper~35) together with the de~Sitter horizon relation
$a_0=cH_0/(2\pi)$ give $H_0=2\alpha^2/\Tb$, a parameter-free prediction $H_0\approx75.8$~km/s/Mpc
that lands in and just above the measured $H_0$ window, and lifts the fold-rate floor to the
corpus number $1/H_0=\alpha^{-2}/2$ breaths (Theorem~\ref{thm:bridge},
Corollary~\ref{cor:foldrate}). \textbf{Verdict: candidate bridge, not closure.} The residual is
the exact horizon factor: the clean $2\pi$ exceeds the data factor $cH_0/a_0\approx5.6$ by
$\sim\!12\%$ (within the $H_0$ tension and $a_0$ uncertainty), and a corpus derivation of the
$2\pi$ --- identifying the de~Sitter Gibbons--Hawking factor with the breath's full-turn $2\pi$
and the Hopf fibre period --- is the open piece (Remark~\ref{rmk:residual}). This is the first
corpus statement to connect the galactic anchor ($a_0=\alpha$, the SPARC/Paper~39 scale) to the
cosmological expansion rate, and it does so with a falsifiable number rather than a fit. No new
objects beyond the de~Sitter relation; no published corpus number changes. Verifier:
\texttt{verify\_P358a.py}, 12/12 pass.

\paragraph{Relation to the cosmology arc.}
357a named the floor; this addendum lifts it to a candidate bridge with a falsifiable $H_0$.
The dynamic fold-tail model remains disfavoured by DESI (349a) and the ontologically-faithful
dark energy remains the cokernel/void (350a); the bridge here concerns the \emph{rate/clock},
not the dark-energy equation of state, and is independent of those verdicts.

\end{document}
