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% Addendum 345a (extends Paper 32, Dark Energy as Geometric Residual).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P345.py (19/19 pass).
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\newcommand{\Lz}{\Lambda_0}
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\title{Addendum 345a: Inflation and Dark Energy as One Fold Cascade\\
\large The DESI confrontation, the golden-ratio refutation,\\
and the near-closure bound\\[0.3em]
\normalsize (Addendum to Paper 32)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Paper~32 derives the static dark-energy residual $\Lz = 1 - \pi^2/32 \approx 0.6916$
and its fold-progress extension $\Lk = \Lz\cdot 4/\Ck$. This addendum makes one
synthesis and runs one confrontation. The synthesis: the fold cascade
$B^4 \!\to\! S^3 \!\to\! S^2 \!\to\! S^1 \!\to\! S^0$, whose dark-energy density
runs from $\Lambda \!=\! \infty$ at the pre-fold stage to $\Lz$ at completion,
is a \emph{single declining engine}; inflation is its violent early end and dark
energy its exhausted tail. This unifies three previously separate corpus
conjectures (Paper~20~\S7.2, Paper~27~\S6, Paper~5). The confrontation: against
DESI~DR2 (2025), which finds dark energy weakening today
($w_0 = -0.821\pm0.065$, $w_a = -0.76$, time variation preferred at
$2.8$--$4.2\sigma$), we establish three quantitative results. (i)~\textbf{Direction}:
the robust DESI signal --- dark energy modestly higher in the recent past,
declining now --- is the fold engine relaxing onto its floor. (ii)~\textbf{Magnitude
bound}: the integer fold step is a factor of $2$ ($+100\%$), but the observed
density swing is only $+7\%$; the step overshoots by $\sim\!14\times$, so the
mechanism is viable \emph{only} if ``now'' sits $\sim\!93\%$ through the final
fold --- a derived near-closure bound. (iii)~\textbf{Golden-ratio refutation}: a
golden self-similar fold rate, suggested by the nesting-sphere structure, is
\emph{excluded}: the breath's rotation number $\rho = (\pi\alpha^{-1})\bmod 1
= 0.5122$ is near-rational (continued fraction with large partial quotients,
near the tritone $\tfrac12$), the maximal opposite of the noble number; the
corpus's native self-similar ratio is $2$, and $\varphi$ belongs to the E8 sector
(Addenda~94/95), not the cascade. We also locate ``now'' away from full closure:
$w_0 > -1$ at $\gtrsim 3\sigma$ forces $x_{\mathrm{now}} < 1$. The
fraction-versus-density unit reconciliation is \emph{closed} (\S\ref{sec:floor2}):
the dynamic $\Lambda$ is the dark-energy density in present-critical units, not a
fraction, and the static residual is the present fraction; the two coincide today
by construction. Two floors remain open and are not crossed: the fold-to-redshift
map (Conjecture~32.2) and the deep-time join to the primordial epoch. All
numerical claims are pinned by \texttt{verify\_P345a.py} (19/19).
\end{abstract}

\section{The synthesis: one declining engine}
\label{sec:synthesis}

Three corpus statements describe the same object from different ends.
Paper~20~\S7.2 conjectures that ``the breathing dynamics $C\circ P = I$ at early
times drives inflation.'' Paper~27~\S6 reads the cosmic arc as the descent from
$\Omega$, with dark energy ``the overhead cost of maintaining consensus across
expanding space'' and the Big Bang as the first disagreement. Paper~5 lists
``inflation from geometric oscillation dynamics.'' Paper~32's fold-progress
table closes the loop quantitatively: the dark-energy density runs
\[
  \Lambda(k):\quad \infty \;\longrightarrow\; \infty \;\longrightarrow\;
  2\Lz \;\longrightarrow\; 2\Lz \;\longrightarrow\; \Lz
  \qquad (k = 0,1,2,3,4),
\]
from a divergent pre-fold value to the static floor $\Lz$.

\begin{proposition}[Cascade unification]
\label{prop:unification}
Inflation and dark energy are the two ends of the single fold cascade
$B^4 \to S^3 \to S^2 \to S^1 \to S^0$: the early, divergent stages
($\Lambda \gg \Lz$) are the inflationary epoch, and the final stage's relaxation
onto $\Lz$ is the present dark-energy era. No new object is introduced beyond the
Paper~32 \S5 fold-collapse machinery; the proposition names a single engine
where the corpus previously carried three separate conjectures.
\end{proposition}

The content of Proposition~\ref{prop:unification} is qualitative and inherits the
conjecture status of its three sources. What follows tests its one sharp,
observable consequence --- that the engine is \emph{declining} now --- against
current data.

\section{The DESI confrontation}
\label{sec:desi}

DESI~DR2 (CMB+DESI+SN, $w_0w_a$CDM) reports
$w_0 = -0.821 \pm 0.065$ and $w_a = -0.76^{+0.31}_{-0.26}$, a preference for
time-varying dark energy at $2.8$--$4.2\sigma$, with $w>-1$ today and $w<-1$ in
the recent past. The dark-energy density history for $w(a) = w_0 + w_a(1-a)$ is
\begin{equation}
  \frac{\rho_{\mathrm{DE}}(a)}{\rho_{\mathrm{DE},0}}
   = a^{-3(1+w_0+w_a)}\,\exp\!\bigl[-3w_a(1-a)\bigr].
  \label{eq:rhode}
\end{equation}

\paragraph{(i) Direction.}
Equation~\eqref{eq:rhode} with the DESI central values has
$\rho_{\mathrm{DE}}$ modestly higher in the recent past and declining toward
today. This is exactly the fold engine relaxing onto its floor. The apparent
rise on the far side --- $\rho_{\mathrm{DE}}$ small in the deep past --- is the
linear-$w(a)$ extrapolation beyond the range DESI constrains ($z\lesssim 2$) and
is the least robust part of the fit; the robust statement is the recent decline,
which is fold-faithful.

\paragraph{(ii) The magnitude bound and near-closure.}
The integer cascade changes $\Lambda$ by a factor of two across an active fold
($2\Lz \to \Lz$, i.e.\ $+100\%$). Equation~\eqref{eq:rhode} gives a gentle,
non-monotone density bump peaking at only $+7.1\%$ near $z \approx 0.31$. The
fold step therefore overshoots the observed swing by $\sim 14\times$.

\begin{theorem}[Near-closure bound]
\label{thm:closure}
If the present dark-energy evolution is the tail of the final fold, then the
visible residual amplitude ($\sim 7\%$) as a fraction of a full fold step
($100\%$) places ``now'' at
\[
  1 - \frac{\Delta\rho_{\mathrm{obs}}}{\Delta\rho_{\mathrm{fold}}}
   \;\approx\; 1 - \frac{0.071}{1.00}
   \;\approx\; 0.93,
\]
i.e.\ $\sim\!93\%$ through the final fold, with only its last $\sim\!7\%$ of
amplitude still unfolding. The factor-of-two fold step is reconciled with the
small observed evolution \emph{only} under near-closure; this is not an
assumption but a consequence of the magnitude mismatch.
\end{theorem}

\paragraph{(iii) The golden-ratio refutation.}
The fold sequence is literally nesting spheres, and the golden ratio is the
canonical fixed point of self-similar nesting, so a golden fold rate is the
natural conjecture. It is excluded by the corpus's own clock. The breath
rotation number is
\[
  \rho = (\pi\,\alpha^{-1}) \bmod 1 = 0.5122452,
  \qquad
  \rho = [0;\,1,1,19,1,10,1,12,8,4,\ldots].
\]
The large partial quotients ($19,\ldots,129$) make $\rho$ a \emph{near-rational}
sitting beside the tritone $\tfrac12$ (Paper~37: ``each breath advances the fibre
by almost exactly a half-turn''). The golden ratio is $\varphi = [0;1,1,1,1,\ldots]$,
all ones --- the noble, maximally irrational number. The breath is its opposite in
character. The corpus's native self-similar ratio is therefore $2$ (halving,
octave, tritone), and a golden ratio step ($+61.8\%$) still overshoots the
observed $+7\%$ by $\sim 9\times$. Where $\varphi$ does live in the corpus --- the
E8 sector (Addenda~94, 95) --- is structurally disjoint from the cosmological
cascade.

\section{The location of ``now''}
\label{sec:now}

At full closure the fold-progress function $f$ (with $\Lambda = \Lz f$, $f(1)=1$,
$f>1$ for $x<1$) attains its minimum, so $f'(1)=0$ and $w=-1$ exactly: no
evolution. DESI measures $w_0 = -0.821 \neq -1$ at $2.8$--$4.2\sigma$, a statement
about the \emph{shape} of $w(z)$ and more robust than the absolute density.

\begin{corollary}[Now is not closure]
\label{cor:now}
$w_0 > -1$ at $\gtrsim 3\sigma$ implies $f'(x_{\mathrm{now}}) \neq 0$, hence
$x_{\mathrm{now}} < 1$: the final fold is in progress, not complete. The static
identification of ``now'' with full closure ($k=4$, the boundary condition that
fixes the normalisation $4/\Ck$) is therefore a definitional placement, not a
derived location, and is contradicted at the DESI significance. The static floor
$\Lz$ is the future asymptote the engine is relaxing onto, not the present value.
\end{corollary}

Corollary~\ref{cor:now} and Theorem~\ref{thm:closure} agree: ``now'' is near, but
short of, full closure --- the descending tail of the last fold.

\section{Closing Floor 2: the fraction--density reconciliation}
\label{sec:floor2}

The static and dynamic uses of $\Lambda$ appear to conflict: the corner residual
$\Lz \approx 0.69$ is a volume \emph{fraction} (bounded by $1$ by construction)
identified with $\Omega_{\Lambda,0}$, yet the dynamic $\Lambda(2) = 2\Lz \approx
1.383$ exceeds $1$. The resolution is that the two are different cosmological
quantities, and the dynamic formula is already the correct one.

\begin{proposition}[Fraction--density reconciliation]
\label{prop:reconcile}
Write $\rho_{\mathrm{crit},0} = 3H_0^2/8\pi G$. Then:
\begin{enumerate}
\item the static residual is the present density fraction,
  $\Lz = \Omega_{\Lambda,0} = \rho_{\Lambda,0}/\rho_{\mathrm{crit},0} \le 1$;
\item the fold factor, normalised to the present, is the density evolution,
  \[
    \frac{\rho_\Lambda(k)}{\rho_{\Lambda,0}}
      = \frac{4/\Ck}{4/C(4)} = \frac{4}{\Ck}, \qquad C(4)=4;
  \]
\item hence the dynamic quantity is the dark-energy density in present-critical
  units, $\Lambda(k) = \Lz\cdot 4/\Ck = \rho_\Lambda(k)/\rho_{\mathrm{crit},0}$,
  which may exceed $1$ in the past without being a fraction.
\end{enumerate}
The boundary condition $\Lambda(4)=\Lz$ is then not an extra assumption but the
identity ``density in present units equals the present fraction,'' holding because
$\rho_{\mathrm{crit}}(\text{now}) = \rho_{\mathrm{crit},0}$.
\end{proposition}

\begin{corollary}[The fraction never exceeds unity]
\label{cor:bounded}
For any matter density $\rho_m>0$ the epoch fraction is
$\Omega_\Lambda(k) = \rho_\Lambda(k)/[\rho_\Lambda(k)+\rho_m+\cdots] \le 1$.
Numerically, the half-fold density $2\Lz=1.383$ in present units gives
$\Omega_\Lambda = 1.383/(1.383+\Omega_m(1+z)^3)$, which equals $0.815$ at the
$z=0$ floor and is smaller at the earlier, denser epoch where the half-fold
actually sits. The $>1$ values of $\Lambda(k)$ are densities; the corresponding
fractions are bounded automatically.
\end{corollary}

\begin{remark}[Consequence for the confrontation]
The reconciliation retroactively well-poses \S\ref{sec:desi}: the DESI quantity
$\rho_{\mathrm{DE}}(z)/\rho_{\mathrm{DE},0}$ and the corpus factor $4/\Ck$ are the
same kind of object --- dark-energy density normalised to the present --- so
\S\ref{sec:desi} was density-to-density throughout, not a category error.
Floor~2 is closed; it required no new object, only the correct identification of
the dynamic $\Lambda$ as a density rather than a fraction.
\end{remark}

\section{Remaining floors and falsifiers}
\label{sec:floors}

This addendum bounds the dynamic model and closes one of its floors
(\S\ref{sec:floor2}). Two floors remain, recorded but not crossed.

\begin{remark}[Floor 1: the fold-to-redshift map, Conjecture~32.2]
No closed-form $x\!\leftrightarrow\!z$ map exists, so no quantitative $\Lambda(z)$
curve can be laid over DESI's redshift bins. Every result here is a
\emph{direction} or a \emph{magnitude bound}; the verifier performs no fit. Closing
the map is what would upgrade this addendum to a confrontation paper.
\end{remark}

\begin{remark}[Floor 2: the deep-time join]
The pre-fold divergence $\Lambda(k\!\le\!1)=\infty$ joining to a physical
$\sim\!10^{-32}\,$s inflationary epoch is the $x\!\leftrightarrow\!z$ map at its
extreme. The \emph{direction} is corpus-faithful (Proposition~\ref{prop:unification},
Papers~20/27); the \emph{magnitude} is unwritten.
\end{remark}

\paragraph{Falsifiers.}
The picture is falsifiable on the data already arriving. (F1) Near-closure
predicts dark energy continues weakening toward $\Lz \approx 0.6916$ and then
\emph{halts}: $w \to -1$ from above, never crossing below. If future DESI/SN data
show $w$ continuing past $-1$ or oscillating, the fold tail is refuted. (F2) The
robust signal must remain a late-time \emph{decline}; if deep-$z$ reconstructions
firmly establish a genuine rise of $\rho_{\mathrm{DE}}$ from near zero (a true
overshoot), then a monotone fold descent is insufficient and an oscillatory
(ring-down) completion is required --- new structure beyond this addendum.
(F3)~The golden refutation is falsified only by a re-derivation of the breath
rotation number; $\rho = 0.5122$ is pinned by $\pi\alpha^{-1}$.

\section*{Status}
Synthesis (Proposition~\ref{prop:unification}) is conjecture-tier, inheriting
Papers~20/27/5. The DESI results (\S\ref{sec:desi}, Theorem~\ref{thm:closure},
Corollary~\ref{cor:now}) and the golden refutation are quantitative and verifier-
pinned. The three floors are open. No parameters are fit to cosmological data:
the fold step, the breath rotation number, and $\Lz$ are all fixed upstream; the
DESI values enter only as the measured target. Verifier:
\texttt{verify\_P345a.py}, 19/19 pass.

\paragraph{Data sources.}
DESI~DR2 dark-energy constraints (2025); Planck~2018 $\Omega_\Lambda$. See the
DESI~DR2 results release and the DESI~DR2 BAO dynamical-dark-energy analyses.

\end{document}
