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% Addendum 357a (extends Paper 32; follows 349a, the fold-to-redshift map closure).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P357a.py (9/9 pass).
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\title{Addendum 357a: The Fold-Rate is a Named Floor\\
\large Why $\nu=\tfrac14 H_0$ does not derive: the missing\\
galactic-to-cosmological time-bridge\\[0.3em]
\normalsize (Addendum to Paper 32; sharpens the rate residual of Addendum 349a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~349a closed the fold-to-redshift map (Conjecture~32.2) in two halves: the
\emph{shape} derives (the Euler continuation $\chi(s)=1+\cos\pi s$ gives
$C(\xi)=4\xi+\sin(4\pi\xi)/\pi$, $\Lambda(\xi)/\Lz=4/C(\xi)$), while the \emph{rate} ---
the conversion from fold-progress to the scale factor --- was supplied by ``one fold per
e-fold'' ($dk/d\ln a=1$, i.e.\ $\nu=\tfrac14 H_0$ for $\xi=k/4$) and honestly flagged as
\emph{reverse-engineered} to the DESI fit, not derived. This addendum asks whether the rate
derives from corpus machinery. \textbf{It does not}, and the obstruction is quantified. The
two candidate handles both fail to fix an absolute rate. (H1)~Paper~32's ``time as
fold-progress'' remark gives $d\tau/dt=\sqrt{1-v^2}$ --- a dual-observer \emph{ratio}
(Paper~28), not the absolute fold-rate against cosmic expansion. (H2)~The unit system
(Paper~35) is calibrated at the \emph{galactic} scale: the chart length is $L_h=0.978$~kpc
(the SPARC / Paper~39 hidden-branch halo scale, with $L_h/R_{\mathrm{Hubble}}\sim2\times10^{-7}$),
and the only absolute clock, the breath $T_{\mathrm{breath}}=1.374$~Myr, is $\sim\!10^4$ too
\emph{fast} to clock a cosmological fold (one fold per e-fold needs a fold period of
$1/H_0\approx14.5$~Gyr $\approx1.06\times10^4$ breaths). The corpus provides no
galactic-to-cosmological time-bridge. \textbf{Verdict: the rate is a NAMED FLOOR}, not a
derivation-in-waiting; ``one fold per e-fold'' remains the natural \emph{dimensionless}
posit (the seed-identity mode the corpus flags about $\alpha$). This sharpens 349a --- the
rate's open status is now an irreducible floor with a stated reason --- and is fully
consistent with 349a's data verdict that the closed model is disfavoured by DESI. All claims
pinned by \texttt{verify\_P357a.py} (9/9).
\end{abstract}

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

Addendum~349a left exactly one ingredient of the fold-to-redshift map non-derived: the rate
$\nu$ in $d\xi/d\ln a=\nu/E(a)$. The value $\nu=\tfrac14 H_0$, equivalent to ``one boundary
fold completes per e-fold of expansion'' ($dk/d\ln a=1$ with $\xi=k/4$), reproduces
$w_0=-0.815$ against DESI's $-0.821$, but 349a flagged it as recognised from the fit and the
principle reverse-engineered to it. The honest question is whether the corpus fixes the rate
independently. There are two places it could come from: the proper-time identification of
Paper~32, and the unit system of Paper~35.

\section{H1: the proper-time remark fixes only a ratio}
\label{sec:H1}

\begin{proposition}[Dual-observer ratio, not absolute rate]
\label{prop:H1}
Paper~32's ``time as fold-progress'' remark identifies the inside-observer completed-fold
count with proper time $\tau$ and gives $d\tau/dt=\sqrt{1-v^2}$, with $v$ the interior/exterior
fold-rate differential of the Paper~28 dual-observer geometry. This is a statement about the
\emph{ratio} of two fold-counters (inside versus outside); it carries no absolute scale
relating the fold-count to the cosmic scale factor $a$. It therefore cannot fix $\nu$.
\end{proposition}

\section{H2: the unit system is calibrated galactically}
\label{sec:H2}

\begin{proposition}[The breath is the wrong scale by $\sim\!10^4$]
\label{prop:H2}
The corpus's only absolute clock is the breath, $T_{\mathrm{breath}}^{\mathrm{SI}}=1.374$~Myr
($\tau_1^{\mathrm{SI}}=3191$~yr), and its natural length is the chart length $L_h=0.978$~kpc
(Paper~35). One fold per e-fold requires a fold period equal to $1/H_0\approx14.5$~Gyr, which
is $\approx1.06\times10^4$ breaths. The breath is thus $\sim\!10^4$ times too fast to serve as
the fold clock. Moreover $L_h=0.978$~kpc is a galactic length --- it is the SPARC / Paper~39
hidden-branch halo scale --- with $L_h/R_{\mathrm{Hubble}}\sim2\times10^{-7}$. The unit bridge
of Paper~35 is fixed at the galactic scale and provides no cosmological time-bridge from which
$\nu$ could be read.
\end{proposition}

\begin{remark}[Why this is the right diagnosis]
\label{rmk:diagnosis}
The corpus is empirically anchored at the galactic scale: the one outside-facing data
confrontation that succeeds is the hidden branch against SPARC (Paper~39), at $L_h\approx1$~kpc.
The fold-cascade dark-energy dynamics live four orders of magnitude higher, at the Hubble scale.
The rate $\nu$ is precisely the conversion across that gap, and nothing in the corpus spans it.
This is not a failure of the fold model's geometry (the shape is derived, 349a); it is a missing
unit-bridge, of exactly the kind Paper~35 supplies at the galactic scale and does not at the
cosmological one.
\end{remark}

\section*{Status}
The fold-rate $\nu=\tfrac14 H_0$ does \textbf{not} derive from current corpus machinery. The
Paper~32 proper-time remark fixes only the dual-observer ratio $d\tau/dt=\sqrt{1-v^2}$
(Proposition~\ref{prop:H1}); the Paper~35 unit system is calibrated at the galactic scale
($L_h=0.978$~kpc, the SPARC scale) and its absolute clock (the breath, $1.374$~Myr) is
$\sim\!10^4$ too fast to clock a cosmological fold (Proposition~\ref{prop:H2}). \textbf{Verdict:
the rate is a NAMED FLOOR} --- a missing galactic-to-cosmological time-bridge --- not a
derivation-in-waiting. ``One fold per e-fold'' ($dk/d\ln a=1$) remains the natural
\emph{dimensionless} posit, the seed-identity mode the corpus flags about $\alpha^{-1}$. This
sharpens Addendum~349a: the rate's open status is now an irreducible floor with a quantified
reason, not an unfinished calculation. It is fully consistent with 349a's verdict that the
closed, fully-derived fold model is disfavoured by DESI~DR2 at $3$--$4.5\sigma$. No new objects
are introduced and no published number changes. Verifier: \texttt{verify\_P357a.py}, 9/9 pass.

\paragraph{What would close it.}
A first-principles galactic-to-cosmological time-bridge --- an extension of the Paper~35 unit
system to the Hubble scale, fixing the fold-clock rate against $H_0$ without reference to the
DESI fit --- would convert the floor into a derivation. Absent that, the rate is posited, and
the fold dark-energy model stands as a sharp, disfavoured, zero-parameter prediction.

\end{document}
