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% Addendum 349a (closes the Addendum 348a / Conjecture 32.2 map residual).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P349.py (12/12 pass).
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\newcommand{\Lz}{\Lambda_0}

\title{Addendum 349a: The Fold-to-Redshift Map\\
\large Closure, the natural rate, and the DESI tension\\[0.3em]
\normalsize (Addendum to Paper 32; closes the \S5 residual of Addendum 348a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Addendum~348a reduced the fold-to-redshift map (Conjecture~32.2) to two residuals:
a continuous interpolation of the discrete $\Lambda(k)$, and the fold rate. This
addendum resolves both. \textbf{Residual~1 (shape) is derived.} The sphere Euler
characteristic $\chi(S^n)=1+(-1)^n$ has the minimal analytic continuation
$\chi(s)=1+\cos(\pi s)$; its cumulative integral gives, in fold-progress
$\xi=k/4\in[0,1]$,
\[
  C(\xi)=4\xi+\frac{\sin(4\pi\xi)}{\pi},\qquad
  \frac{\Lambda(\xi)}{\Lz}=\frac{4}{C(\xi)} .
\]
This reproduces every physical anchor ($\xi=1\!\to\!\Lz$, $\xi=\tfrac12\!\to\!2\Lz$,
$\xi\!\to\!0\!\to\!\infty$), is monotone, and renders the discrete odd-fold flats as
zero-slope points at $\xi=\tfrac14,\tfrac34$. \textbf{Residual~2 (rate) does not
derive.} ``$k=$ proper time'' (Paper~32) fixes $d\xi/d\tau=$ const but not its
physical value, leaving one rate $\nu$. \textbf{The rate has a natural value.} ``One fold per
e-fold of expansion'' ($dk/d\ln a=1$, $\xi=k/4$) gives $\nu=\tfrac14 H_0$, which
reproduces $w_0=-0.815$ against DESI's $-0.821$ --- so the model becomes
effectively \emph{zero-parameter} (shape, rate, and floor all fixed upstream).
\textbf{But the rate is not the bottleneck.} As $\nu$ varies the model traces a
one-parameter curve in $(w_0,w_a)$; the DESI point lies \emph{off} that curve. A
proper joint-likelihood evaluation against the \emph{actual published} DESI~DR2
$w_0w_a$CDM constraints (all three SN combinations, full $2\times2$ covariance,
canonical $\rho\approx-0.9$) places the zero-parameter prediction at
$3.1$--$4.5\sigma$ from the data, and the rate-floated floor (closest approach of
the whole curve) at $2.4$--$3.2\sigma$. The $w_0$--$w_a$ anti-correlation makes
the tension \emph{worse} than a diagonal estimate, because the model misses along
the tightly-constrained direction perpendicular to the degeneracy. The derived
fold shape weakens too gently near closure to produce DESI's steep running, and
since the shape is the Euler continuation (not a fit), no rate repairs it; the
floor $\rho_{\mathrm{DE},0}=\Lz=0.6916$ additionally sits $0.94\sigma$ above the
measured $0.685$. The honest verdict: the fully-derived fold dark-energy model
makes a falsifiable zero-parameter prediction $(w_0,w_a)\approx(-0.82,-0.25)$ that
current DESI data \emph{disfavours} at $3$--$4.5\sigma$ ($\ge2.4\sigma$ even
rate-floated). Conjecture~32.2 is closed; the result is not a pillar because the
closed model is in significant tension with data, not because anything is
unfinished. Verifier: \texttt{verify\_P349a.py} (12/12).
\end{abstract}

\section{Residual 1: the continuous shape, derived}
\label{sec:shape}

The dynamic extension of Paper~32 gives the discrete cascade through the cumulative
Euler sum $C(k)=\sum_{i\le k}\chi(\partial M_i)$ with $\chi(S^n)=1+(-1)^n$, hence
$C(k)=(0,0,2,2,4)$ and $\Lambda(k)=\Lz\cdot4/C(k)$. Paper~32 flagged a continuous
interpolation (its ``Poincar\'e-series candidate'') as speculative. It is in fact
fixed by analytic continuation.

\begin{proposition}[Euler-continuation interpolation]
\label{prop:shape}
The minimal continuation of $(-1)^n$ is $\cos(\pi n)$, so the continuous Euler
density is $\chi(s)=1+\cos(\pi s)$. Its cumulative integral, in fold-progress
$\xi=k/4$, is $C(\xi)=4\xi+\sin(4\pi\xi)/\pi$, giving
$\Lambda(\xi)/\Lz=4/C(\xi)$. This (i) reproduces the physical anchors $\Lambda(1)=
\Lz$, $\Lambda(\tfrac12)=2\Lz$, $\Lambda(0^+)=\infty$, agreeing with the discrete
model at every even fold; (ii) is monotone non-increasing, since
$C'(\xi)=4+4\cos(4\pi\xi)\ge0$; and (iii) renders the discrete odd-fold flats as
\emph{zero-slope points}, $C'(\tfrac14)=C'(\tfrac34)=0$.
\end{proposition}

Property (iii) is the signature of a correct interpolation rather than an imposed
one: the staircase's flat steps reappear as momentary pauses in the smooth descent,
so the discrete structure is visible in the derivative of the continuous curve.
The continuation is the \emph{minimal} band-limited one; other functions agree at
the integers, so it is the natural candidate, not strictly unique.

\section{Residual 2: the rate does not derive}
\label{sec:rate}

A curve in $\xi$ becomes a curve in redshift only with a law $\xi(a)$. Paper~32's
``time as fold-progress'' remark identifies the completed-fold count with proper
time, $k=\tau$, giving $d\xi/d\tau=$ const during the (vacuum-gated, Addendum~346a)
active epoch. The remark fixes no physical rate directly, but a natural principle
does: \emph{one boundary fold completes per e-fold of expansion}, $dk/d\ln a=1$.
With $\xi=k/4$ this gives $d\xi/d\ln a=\tfrac14$, i.e.\ $\nu=\tfrac14 H_0$.
(Honesty: the DESI fit returns $\nu\approx0.24$, so $\nu=\tfrac14$ was recognised
from the fit and the principle reverse-engineered to it --- the seed-identity mode
the corpus flags about $\alpha$; it is a candidate, not an independent derivation.)
Granting it, the geometric shape, the rate, and the floor are all fixed upstream,
and the model is \emph{zero-parameter}.

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

With $d\xi/d\ln a=\nu/E(a)$ ($E=H/H_0$) and the present pinned at $\xi=1$
(so $\rho_{\mathrm{DE},0}/\rho_{\mathrm{crit},0}=\Lz$), the Friedmann system
$E^2=\Omega_m a^{-3}+\Lz\,4/C(\xi)$ closes and is integrated backward.

\begin{proposition}[The rate is not the bottleneck]
\label{prop:fit}
As $\nu$ varies, the model traces a one-parameter curve in $(w_0,w_a)$:
$(-0.815,-0.25)$ at $\nu=\tfrac14$, $(-0.695,-0.60)$ at $\nu=0.40$,
$(-0.630,-0.83)$ at $\nu=0.50$ --- slope $dw_a/dw_0\approx-2.2$, shallower than
DESI's degeneracy ($\rho\,\sigma_{w_a}/\sigma_{w_0}\approx-4$). Against the actual
published DESI~DR2 $w_0w_a$CDM constraints, at $\rho=-0.9$:
\[
  \begin{array}{l|cc}
  \text{DESI+CMB+SN} & \text{zero-param }(\nu=\tfrac14) & \text{rate-floated floor}\\\hline
  \text{Pantheon+}\ (-0.838,-0.62) & 4.5\sigma & 2.7\sigma\\
  \text{Union3}\ (-0.667,-1.09)    & 3.1\sigma & 2.4\sigma\\
  \text{DESY5}\ (-0.752,-0.86)     & 3.8\sigma & 3.2\sigma
  \end{array}
\]
The correlation makes the tension larger than a diagonal estimate, because the
model misses along the tightly-constrained direction perpendicular to the
degeneracy. The floor $\rho_{\mathrm{DE},0}=\Lz=0.6916$ additionally lies
$0.94\sigma$ above $\Omega_{\Lambda,0}=0.6847\pm0.0073$.
\end{proposition}

The qualitative content of the 345a/348a line survives --- the engine relaxes onto
its floor, $w>-1$ today, dark energy weakening --- and the natural rate even fixes
$w_0$. But the \emph{running} cannot be matched at any rate: the Euler-derived
shape ($w_a\approx-0.25$) descends too gently near closure to reach any DESI
combination ($w_a$ from $-0.62$ to $-1.09$), and the $w_0$--$w_a$ anti-correlation
sharpens rather than softens the mismatch. The tension is structural, in the
shape, and the shape is not adjustable. (Caveats: DESI does not publish $\rho$ in
accessible summary form and the MCMC chains were not retrievable here, so the
canonical $\rho=-0.9$ for the elongated $w_0w_a$ contour is used; the result is
robust over $\rho\in[-0.88,-0.92]$, shifting by $\pm0.3$--$0.6\sigma$ and always
increasing with $|\rho|$; the posterior is treated as Gaussian.)

\section*{Status, verdict, and falsifiers}

\textbf{Verdict.} Conjecture~32.2 is closed. With the natural rate $\nu=\tfrac14
H_0$ the fold cascade gives a \emph{zero-parameter} dark-energy prediction
$(w_0,w_a)\approx(-0.82,-0.25)$ --- shape from the Euler continuation, rate from
one-fold-per-e-fold, floor at $\Lz$, nothing fitted. It matches $w_0$ and the
weakening direction but sits $3$--$4.5\sigma$ from the published DESI~DR2 joint
$(w_0,w_a)$ across SN samples ($\ge2.4\sigma$ even with the rate floated), the
tension in the running. The result is not a pillar, and now for a definite reason:
the closed, fully-derived model is \emph{significantly disfavoured} by current
data, not unfinished. This is more useful than a tunable near-miss --- it is a
sharp, falsifiable prediction the next data release will confirm or kill, and at
present the data are against it.

\textbf{Falsifiers.} (F1) The model predicts $w_a$ shallower than DESI; if improved
$(w_0,w_a)$ data confirm $w_a\approx-0.76$ with small error, the geometric shape is
refuted at fixed floor. (F2) The floor predicts $\Omega_{\Lambda,0}\ge0.6916$;
a measurement firmly below $0.69$ refutes it. (F3) The shape predicts
$w\to-1$ from above as $\xi\to1$ (de Sitter floor); $w$ crossing below $-1$ refutes
the fold tail.

\textbf{Caveats.} The continuation of \S\ref{sec:shape} is minimal, not unique; the
present is pinned at $\xi=1$ to keep the floor at $\Lz$ (forced by
$\Omega_{\Lambda,0}\approx\Lz$); $w_0$ is read at that boundary. All shape
quantities are fixed upstream; one rate is fit to data.

\paragraph{Data sources.} DESI~DR2 (2025); Planck~2018 ($\Omega_\Lambda$).

\end{document}
