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% Addendum 348a (extends Paper 32; consolidates the 345a/346a/347a line).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P348.py (26/26 pass).
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\newcommand{\Lz}{\Lambda_0}
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\title{Addendum 348a: Cosmology from the Fold Cascade\\
\large Inflation, dark energy, and the DESI/Planck confrontation\\[0.3em]
\normalsize (Addendum to Paper 32; consolidates the 345a--347a line)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
This addendum consolidates one cosmological line of the corpus. From the
fold-progress dark-energy cascade of Paper~32 it draws a single synthesis and
runs two confrontations. \textbf{Synthesis:} the 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 $\Lz=1-\pi^2/32$ at completion, is a
single declining engine --- inflation its violent early end, dark energy its
exhausted tail --- unifying three prior conjectures (Papers~20, 27, 5).
\textbf{Dark-energy confrontation:} against DESI~DR2 the robust signal (dark
energy weakening now) is the engine relaxing onto its floor; the integer fold
step ($\times2$) overshoots the observed $+7\%$ swing by $\sim\!14\times$, forcing
``now'' to $\sim\!93\%$ through the final fold (Theorem~\ref{thm:closure}); a
golden self-similar rate is refuted (the breath rotation number is near-rational,
not noble); and the fraction-versus-density unit question is closed
(Proposition~\ref{prop:reconcile}). \textbf{Inflation confrontation:} the
fold-to-redshift map does not close --- it reduces to one posit and is reframed as
vacuum-gated, with the matter era a dormant gap (\S\ref{sec:map}) --- but the
\emph{canonical} self-lensing potential of Paper~18,
$V_{\mathrm{self}}=(E_{\mathrm{self}}/m_0^2)(1-e^{-r/r_0})$, a saturating plateau,
yields $n_s = 1 - 2/N = 0.9649$ at $N=57$, an exact match to Planck, with
$r\sim10^{-8}$ (\S\ref{sec:inflation}). No quantities are fit. The dynamic-$\Lambda$
spine remains Conjecture~32.2; this is a dark-sector continuation, not a new
pillar. Verifier: \texttt{verify\_P348a.py} (26/26).
\end{abstract}

\tableofcontents

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

Three corpus statements describe one 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.''
Paper~5 lists ``inflation from geometric oscillation dynamics.'' Paper~32's
fold-progress table closes the loop quantitatively: with
$\chi(S^n)=1+(-1)^n$ and the cumulative Euler sum $\Ck=\sum_{i\le k}\chi(\partial
M_i)$,
\[
  \Lambda(k)=\Lz\cdot\frac{4}{\Ck}:\quad
  \infty,\ \infty,\ 2\Lz,\ 2\Lz,\ \Lz
  \qquad(k=0,1,2,3,4),\quad \Ck=(0,0,2,2,4).
\]

\begin{proposition}[Cascade unification]
\label{prop:unification}
Inflation and dark energy are the two ends of the single fold cascade: 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. The proposition
names one engine where the corpus carried three separate conjectures; it is
qualitative and inherits their conjecture status.
\end{proposition}

The static value is itself a zero-parameter geometric result: $\Lz=1-\pi^2/32=
0.6916$ is the fraction of the bounding $4$-cube not covered by the unit $4$-ball
(Paper~32; the Cantor-bridge addendum shows the cube is the pre-geometric
container of $\mathbb{R}^4$, so the choice is forced). It matches Planck's
$\Omega_\Lambda=0.6847\pm0.0073$ at $0.94\sigma$.

\section{The dark-energy confrontation}
\label{sec:desi}

DESI~DR2 (CMB+DESI+SN, $w_0w_a$CDM) reports $w_0=-0.821\pm0.065$,
$w_a=-0.76^{+0.31}_{-0.26}$, time variation preferred at $2.8$--$4.2\sigma$, with
$w>-1$ today and $w<-1$ in the recent past. The density history for $w(a)=w_0+
w_a(1-a)$ is $\rho_{\mathrm{DE}}(a)/\rho_{\mathrm{DE},0}=a^{-3(1+w_0+w_a)}
\exp[-3w_a(1-a)]$.

\paragraph{Direction.} This is modestly higher in the recent past and declining
toward today --- the engine relaxing onto its floor. The apparent deep-past rise
is the linear-$w(a)$ extrapolation beyond the DESI range ($z\lesssim2$) and is the
least robust part; the robust statement is the recent decline, which is
fold-faithful.

\paragraph{Magnitude and near-closure.} The integer cascade changes $\Lambda$ by
$\times2$ across an active fold ($+100\%$); the observed density bump peaks at only
$+7.1\%$ near $z\approx0.31$. The fold step overshoots by $\sim\!14\times$.

\begin{theorem}[Near-closure bound]
\label{thm:closure}
If the present evolution is the tail of the final fold, the visible residual
amplitude ($\sim7\%$) as a fraction of a full fold step ($100\%$) places ``now''
at $1-0.071/1.00\approx0.93$, i.e.\ $\sim\!93\%$ through the final fold. The
factor-of-two step is reconciled with the small observed evolution \emph{only}
under near-closure; this is a consequence of the magnitude mismatch, not an
assumption.
\end{theorem}

\paragraph{Golden-ratio refutation.} The cascade is nesting spheres, and the
golden ratio is the 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 $\rho=(\pi\alpha^{-1})\bmod1=0.5122$ has continued fraction
$[0;1,1,19,1,10,\dots]$ with large partial quotients --- a near-rational beside
the tritone $\tfrac12$ --- whereas $\varphi=[0;1,1,1,\dots]$ is the maximally
irrational noble number. The native self-similar ratio is $2$ (halving); $\varphi$
lives in the E8 sector (Addenda~94/95), not the cascade. A golden step ($+62\%$)
still overshoots the observed $+7\%$ by $\sim\!9\times$.

\paragraph{Location of ``now''.} At full closure the fold function $f$
($\Lambda=\Lz f$, $f(1)=1$, $f>1$ for $x<1$) is at its minimum, so $f'(1)=0$ and
$w=-1$: no evolution. DESI's $w_0=-0.821\neq-1$ at $\gtrsim3\sigma$ forces
$x_{\mathrm{now}}<1$. ``Now $=$ closure'' is a definitional placement, contradicted
by the data; $\Lz$ is the future asymptote, not the present value.

\section{The fraction--density reconciliation}
\label{sec:floor2}

The static $\Lz\approx0.69$ is a volume \emph{fraction} ($\le1$), yet the dynamic
$\Lambda(2)=2\Lz\approx1.383$ exceeds $1$. They are different quantities.

\begin{proposition}[Reconciliation]
\label{prop:reconcile}
With $\rho_{\mathrm{crit},0}=3H_0^2/8\pi G$: (i) the static residual is the present
fraction, $\Lz=\Omega_{\Lambda,0}=\rho_{\Lambda,0}/\rho_{\mathrm{crit},0}\le1$;
(ii) the fold factor normalised to the present is the density evolution,
$\rho_\Lambda(k)/\rho_{\Lambda,0}=(4/\Ck)/(4/C(4))=4/\Ck$; (iii) hence
$\Lambda(k)=\Lz\cdot4/\Ck=\rho_\Lambda(k)/\rho_{\mathrm{crit},0}$, a density in
present-critical units, which may exceed $1$ in the past without being a fraction.
The boundary condition $\Lambda(4)=\Lz$ is the identity ``present density $=$
present fraction.''
\end{proposition}

For any matter density the epoch fraction $\Omega_\Lambda=\rho_\Lambda/(\rho_\Lambda
+\rho_m+\cdots)\le1$ automatically; the half-fold density $1.383$ gives
$\Omega_\Lambda=0.815$ at the $z=0$ floor and less earlier. The confrontation of
\S\ref{sec:desi} was density-to-density throughout, so it is well-posed.

\section{The fold-to-redshift map: reduction and vacuum gating}
\label{sec:map}

A quantitative $\Lambda(z)$ curve requires a fold coordinate and a law to $a$. The
corpus offers three fold coordinates, never identified: \textbf{(A)} the
boundary-collapse index $k$; \textbf{(B)} the self-lensing $x$ in Paper~20's
$H^2\sim e^{-x/\kappa}$; \textbf{(C)} the Cantor-bridge growth phase
$\Theta(x)=4\pi^3x^4+\pi^2x^3+\pi x^2$, $\Theta(1)=\alpha^{-1}$.

\begin{proposition}[Non-closure and vacuum gating]
\label{prop:map}
(i) The map does not close from committed structure: (A), (B), (C) are unbridged
and no global law to $a$ is supplied; closing it requires an identification
A$=$B$=$C and a dynamical law, both new objects. (ii) The one dynamical link,
$H^2\sim e^{-x/\kappa}$, is vacuum-specific: in matter domination $H^2\propto
a^{-3}$ tracks matter, not the fold. Hence the fold-clock advances only when
vacuum-dominated --- inflation and the present dark-energy era --- and is dormant
through the matter era. The map is piecewise; the matter era is a gap, not a fold.
\end{proposition}

This reduces Conjecture~32.2 to (i)~a canonical identification A$=$B$=$C and
(ii)~one posit (that the inflation law governs both vacuum epochs). Under it,
near-closure sharpens to $\Delta x=\kappa\ln2\approx1.5\times10^{-3}$ of
fold-progress.

\section{The inflation sector from the canonical potential}
\label{sec:inflation}

Paper~18~\S3 fixes the self-lensing potential of the master operator as
\begin{equation}
  V_{\mathrm{self}}^{\mathrm{can}}(r)=\frac{E_{\mathrm{self}}}{m_0^2}
  \bigl(1-e^{-r/r_0}\bigr),\qquad r_0=\kappa=\alpha^{5/4}=0.00213,\quad
  E_{\mathrm{self}}=13.177.
  \label{eq:vself}
\end{equation}
This is a \emph{saturating plateau} (concave, $V''<0$), the observationally
favoured class --- not the convex decaying heuristic. Taking the inflaton to be the
canonically normalised radial field and $u=e^{-\phi/r_0}\ll1$ on the plateau:
$\epsilon\approx u^2/2r_0^2$, $\eta\approx-u/r_0^2$, and $N\approx r_0^2/u_*$, so
$u_*=r_0^2/N$ and $\eta_*=-1/N$.

\begin{theorem}[Plateau-attractor spectral index]
\label{thm:ns}
For \eqref{eq:vself}, $n_s=1+2\eta_*-6\epsilon_*=1-2/N$ to leading order on the
plateau, independent of $r_0$; the tensor-to-scalar ratio is $r=16\epsilon_*=
8r_0^2/N^2$.
\end{theorem}

\begin{corollary}[Planck match]
\label{cor:planck}
$n_s=1-2/N=0.9649$ at $N=57$ e-folds (standard window), an exact match to Planck
$0.9649\pm0.0042$; across $N\in[50,60]$, $n_s\in[0.960,0.967]$ (all
$\lesssim1.2\sigma$). With $r_0=\kappa$, $r=8r_0^2/N^2\approx1.1\times10^{-8}$ ---
effectively no primordial gravitational waves.
\end{corollary}

This is the universal value of Starobinsky, Higgs, and $\alpha$-attractor
inflation: the plateau shape, not the detailed $r_0$, fixes $n_s$. The naive
decaying heuristic gives $n_s\approx0.99$ ($\sim5\sigma$); using the correct
potential removes the tension entirely.

\section{Status, falsifiers, and what would earn Paper status}
\label{sec:status}

\textbf{Standing.} Two falsifiable confrontations now attach to the cascade: the
dark-energy near-closure (Theorem~\ref{thm:closure}) and the inflation index
$n_s=1-2/N$ (Corollary~\ref{cor:planck}). The fraction--density floor is closed
(Proposition~\ref{prop:reconcile}). No quantities are fit: $\Lz$, the fold step,
$\rho$, $\kappa$, $E_{\mathrm{self}}$, and the form \eqref{eq:vself} are fixed
upstream; DESI and Planck enter only as targets.

\textbf{Open.} The synthesis (Proposition~\ref{prop:unification}) is
conjecture-tier; the fold-to-redshift map is reduced, not closed
(Proposition~\ref{prop:map}); the dynamic-$\Lambda$ spine remains
Conjecture~32.2. This work is a continuation of the dark-sector line
(Papers~32/38/39), not a new pillar.

\textbf{Falsifiers.} (F1) Dark energy must keep weakening toward $\Lz\approx0.6916$
and halt ($w\to-1$ from above); $w$ crossing below or oscillating refutes the fold
tail. (F2) $n_s$ outside $[0.960,0.967]$ refutes the plateau prediction. (F3) A
tensor detection at $r\gtrsim10^{-3}$ refutes the $r\sim10^{-8}$ prediction.

\textbf{Path to Paper status.} A canonical identification A$=$B$=$C would close
Conjecture~32.2, turning the fold-clock from a posited into a \emph{derived}
cosmological coordinate, with the two confrontations following from first
principles. That --- new derived structure, not synthesis over an open conjecture
--- is what would re-cut this material as a standalone Paper.

\paragraph{Data sources.} DESI~DR2 (2025); Planck~2018 ($\Omega_\Lambda$, $n_s$,
$r$ with BICEP/Keck).

\end{document}
