\documentclass[11pt]{article}
% Addendum 350a (supersedes the fold-density dark energy of 348a/349a on conceptual
% grounds; extends Paper 38 / Paper 32). Author: Leon Fernando Vlegels. CC BY 4.0.
% Verifier: verify_P350.py (11/11 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Lz}{\Lambda_0}

\title{Addendum 350a: Dark Energy as the Growing Cokernel\\
\large The faithful void model and the gentle-$w_a$ prediction\\[0.3em]
\normalsize (Extends Paper 38; supersedes the fold-density dark energy of
Addenda 348a/349)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
The 345a--349a line dynamised dark energy as a fold-progress density,
$\Lambda(k)=\Lz\cdot4/C(k)$. This addendum corrects that on conceptual grounds and
rebuilds the treatment on the footing the TOE actually supplies. Paper~38 fixes
dark energy as the \emph{cokernel} --- the embedding defect, the unreachable void,
of measure $\Lz=1-\pi^2/32$ --- paired with dark \emph{matter} as the
\emph{kernel} (the fold). Indexing dark energy by the fold-completion count $C(k)$
drove the cokernel with the \emph{kernel's} clock: a cross-wiring of the two
defects. The faithful dynamics is the void \emph{growing}: a cokernel is a spatial
embedding defect, so its energy is set holographically by the horizon \emph{area},
$\rho_{\mathrm{DE}}\propto L^{-2}$, with $L$ the future event horizon --- the
region no light will ever reach, the ``cell growing larger'' like a black-hole
horizon (Paper~4). Pinning $\Omega_{\mathrm{DE},0}=\Lz$ and saturating the bound
($c=1$) gives a \emph{zero-parameter} prediction $(w_0,w_a)\approx(-0.89,-0.19)$:
\emph{gently} evolving, $w$ near $-1$. The gentle running is robust ($w_a\in
[-0.24,-0.15]$ over $c\in[0.8,1.2]$): a geometric void-defect cannot weaken fast.
Against published DESI~DR2 the committed point sits at $2.4$--$2.7\sigma$ ---
disfavoured, but the faithful model, and gentler than the fold-density's
$3$--$4.5\sigma$. The TOE's committed, falsifiable stance is therefore: dark
energy evolves \emph{gently}, because it is a void, not a fold. Verifier:
\texttt{verify\_P350a.py} (11/11).
\end{abstract}

\section{The conceptual correction}
\label{sec:correction}

Paper~38 (``Dark as Kernel'') fixes the ontology: one geometry, two exactness
defects. Dark \emph{matter} is the \emph{kernel} of the fold map (its null space,
what folds away); dark \emph{energy} is the \emph{cokernel} (the embedding defect,
the part of the container the geometry cannot fill). The cokernel measure is the
corner residual
\[
  \Lz = 1 - \frac{\pi^2}{32} = 0.6916,
\]
the fraction of the bounding cube the unit ball never reaches (Paper~32, Cantor
bridge): the unreachable void.

The fold-density model of Addenda~348a/349 set $\Lambda(k)=\Lz\cdot4/C(k)$, with
$C(k)$ the cumulative Euler sum over completed \emph{folds}. But the fold is the
\emph{kernel} structure --- the dark-\emph{matter} side. Driving the cokernel by
the fold count indexes dark energy with the kernel's clock: the two defects are
cross-wired. A cokernel is a \emph{spatial embedding} defect; its natural
dynamics is the growth of the embedding region, not a temporal fold count. This is
the correction.

\section{The faithful model: the void growing}
\label{sec:model}

\begin{proposition}[Area law for the void]
\label{prop:area}
The energy of the cokernel/void in a region of horizon scale $L$ is bounded by the
horizon \emph{area} (holographic bound): $\rho_{\mathrm{DE}}\,L^3 \lesssim
M_{\mathrm{Pl}}^2 L$, hence
\[
  \rho_{\mathrm{DE}} \;\simeq\; \frac{c^2\,M_{\mathrm{Pl}}^2}{L^2},
\]
with $L$ the \emph{future event horizon} --- the comoving region no light will ever
reach, the geometric image of the unreachable void. As the universe accelerates
$L$ grows: the void cell grows larger (its bounding area increasing, as a
black-hole horizon's does under Paper~4's fold picture), rather than more void
cells being added.
\end{proposition}

This is area-based, not volume-based: dark energy scales with $L^2$ (the
boundary), the holographic content of the growing void, exactly the
``cell growing'' rather than ``more cells'' distinction. The normalisation is
fixed by the void measure, $\Omega_{\mathrm{DE},0}=\Lz$, and the saturation by
$c=1$ (the void is the maximally-unreachable region), leaving \emph{no} free
parameter.

\section{The committed prediction and DESI}
\label{sec:desi}

For the future-event-horizon law the equation of state evolves as
$w=-\tfrac13-\tfrac{2}{3c}\sqrt{\Omega_{\mathrm{DE}}}$ with
$\Omega_{\mathrm{DE}}'=\Omega_{\mathrm{DE}}(1-\Omega_{\mathrm{DE}})
(1+2\sqrt{\Omega_{\mathrm{DE}}}/c)$ (Li 2004).

\begin{proposition}[Gentle evolution]
\label{prop:gentle}
With $\Omega_{\mathrm{DE},0}=\Lz$ and $c=1$,
\[
  \boxed{(w_0,\,w_a)\approx(-0.89,\,-0.19)},
\]
gently evolving, $w$ near $-1$. The gentle running is robust: across the
saturation range $c\in[0.8,1.2]$, $w_a\in[-0.24,-0.15]$. A geometric void-defect,
however its horizon is grown, cannot produce a steep $w_a$.
\end{proposition}

\begin{corollary}[DESI confrontation]
\label{cor:desi}
Against the published DESI~DR2 $w_0w_a$CDM constraints (full $2\times2$ covariance,
$\rho\approx-0.9$), the $c=1$ committed point lies at $2.4$--$2.7\sigma$ across the
three SN combinations --- disfavoured, but markedly gentler than the fold-density
model's $3$--$4.5\sigma$, and now \emph{conceptually faithful}. The tension is
again in the running: DESI's $w_a\approx-0.6$ to $-1.1$ versus the void's
$\approx-0.19$.
\end{corollary}

The two faithful geometric mechanisms tested --- fold-density and cokernel-void ---
agree on the essential point: $w_a\approx-0.2$, gentle. The TOE's dark energy is a
geometric defect, and a defect weakens slowly.

\section*{The TOE's committed stance, scope, and falsifiers}

\textbf{Committed stance.} Dark energy is the cokernel/void; it evolves
\emph{gently} ($w_a\approx-0.2$, $w$ near $-1$) because its dynamics is the slow
growth of a geometric embedding defect, not a fast density change. This is a
sharp, zero-parameter, falsifiable bet.

\textbf{Scope.} This addendum supersedes the dark-energy treatment of Addenda
348a/349a on \emph{conceptual} grounds --- the kernel/cokernel cross-wiring --- and
replaces the fold-density $\Lambda(k)$ with the void-growth law of
Proposition~\ref{prop:area}. The inflation result of Addendum~347a
($n_s=1-2/N$, matching Planck) is independent and stands unchanged. The static
residual $\Lz$ (Paper~32) is unchanged; only its \emph{dynamics} are corrected.

\textbf{Falsifiers.} (F1) If DESI's steep $w_a\approx-0.8$ holds up with improved
data, the TOE's dark-energy ontology --- a geometric void --- is challenged at the
conceptual level, not repairable by re-parametrising. (F2) If $w_a$ softens toward
$\approx-0.2$ as systematics settle, the void-growth prediction is vindicated.
(F3) The void floor predicts $w\to-1$ and $\Omega_{\mathrm{DE}}\to\Lz$
asymptotically; $w$ crossing well below $-1$ in the future refutes it.

\paragraph{Data sources.} DESI~DR2 (2025) published $w_0w_a$CDM constraints;
Planck~2018 ($\Omega_\Lambda$).

\end{document}
