% Addendum to Paper 32: Dynamic Lambda — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Formalises the fold-progress dark energy result: Lambda(k) = Lambda_0 * 4/C(k).
% Status: current canon; Paper 32's headline Lambda_0 = 1 - pi^2/32 is current canon, consolidated in Paper 38.
% Registry: no registry items — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P032.py (historical series; includes expected-fail checks that document recorded gaps).

\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}

\newcommand{\Lz}{\Lambda_0}
\newcommand{\Ck}{C(k)}
\newcommand{\Lk}{\Lambda(k)}
\newcommand{\chiS}{\chi(S^n)}

\title{Addendum to Paper 32:\\Fold-Progress Dark Energy: $\Lambda(k) = \Lambda_0 \cdot 4/C(k)$}
\author{L\'{e}on Fernando Vlegels}
\date{2026-04}

\begin{document}
\maketitle

\begin{abstract}
Paper 32 derives $\Lz = 1 - \pi^2/32$ as a static corner residual.
This addendum formalises the \emph{dynamic} extension: dark energy is not constant
but tracks the fold-completion state of the universe's $B^4 \to S^3 \to S^2 \to S^1 \to S^0$
boundary collapse sequence.
The key quantity is the cumulative Euler characteristic $\Ck = \sum_{i=1}^{k} \chi(\partial M_i)$
over the $k$ completed boundary folds.
The main result is $\Lk = \Lz \cdot 4/\Ck$ for $\Ck > 0$.
At full fold completion ($k=4$, current era) $\Lambda(4) = \Lz$.
At the half-fold state ($k=2$) the model predicts $\Lambda(2) = 2\Lz \approx 1.383$.
The derivation uses only $\chiS = 1 + (-1)^n$ and the Paper~32 \S5 fold-collapse machinery;
no new objects are introduced.
\end{abstract}

% ── §1  The Fold Sequence ──────────────────────────────────────────────────────
\section{The Fold Sequence and Its Euler Characteristics}

Paper~32 \S5 establishes the boundary collapse sequence
\[
  B^4 \;\longrightarrow\; S^3 \;\longrightarrow\; S^2 \;\longrightarrow\; S^1
  \;\longrightarrow\; S^0.
\]
The Euler characteristic of each manifold is given by the exact formula
\[
  \chi(S^n) = 1 + (-1)^n,
\]
so $\chi(S^n) = 2$ for even $n$ and $\chi(S^n) = 0$ for odd $n$.
$B^4$ is contractible: $\chi(B^4) = 1$.

\medskip
\noindent
There are exactly \textbf{four active boundary folds}:
$S^3 \to S^2$, $S^2 \to S^1$, $S^1 \to S^0$, and the collapse to a point.
We index them $k = 1, 2, 3, 4$, corresponding to the boundary manifolds
$\partial M_1 = S^3$, $\partial M_2 = S^2$, $\partial M_3 = S^1$, $\partial M_4 = S^0$.

\begin{table}[h!]
\centering
\begin{tabular}{ccccc}
\toprule
$k$ & Fold manifold $\partial M_k$ & $\chi(\partial M_k)$ & $\Ck$ & $\Lk / \Lz$ \\
\midrule
0 & (pre-fold) & -- & 0 & $+\infty$ \\
1 & $S^3$      & 0   & 0 & $+\infty$ \\
2 & $S^2$      & 2   & 2 & $2$       \\
3 & $S^1$      & 0   & 2 & $2$       \\
4 & $S^0$      & 2   & 4 & $1$       \\
\bottomrule
\end{tabular}
\caption{Cumulative Euler sum and dark energy ratio by fold state.}
\end{table}

% ── §2  Main Proposition ──────────────────────────────────────────────────────
\section{Main Result}

\begin{proposition}[Fold-progress dark energy]
\label{prop:dynamic-lambda}
Let $k \in \{0, 1, 2, 3, 4\}$ count the completed boundary folds in the
sequence $S^3 \to S^2 \to S^1 \to S^0$.
Define the cumulative Euler sum
\[
  \Ck \;=\; \sum_{i=1}^{k} \chi(\partial M_i),
\]
where the sum is zero for $k = 0$.
Then the fold-progress dark energy density parameter is
\[
  \boxed{\Lk \;=\; \Lz \cdot \frac{4}{\Ck}} \qquad \text{for } \Ck > 0,
\]
with $\Lambda(0) = \Lambda(1) = +\infty$ (pre-fold regime, $\Ck = 0$).
\end{proposition}

\begin{proof}
At full fold completion $k = 4$, all four boundary manifolds have collapsed and
$C(4) = \chi(S^3) + \chi(S^2) + \chi(S^1) + \chi(S^0) = 0 + 2 + 0 + 2 = 4$.
Requiring $\Lambda(4) = \Lz$ (the static value derived in Paper~32 \S3) fixes the
normalisation constant to~$4$, yielding $\Lk = \Lz \cdot 4/\Ck$.
\end{proof}

% ── §3  Corollary: the Half-fold Prediction ───────────────────────────────────
\section{Corollary: the Half-fold Prediction}

\begin{corollary}[Half-fold dark energy]
\label{cor:half-fold}
At the intermediate fold state $k = 2$ (folds $S^3 \to S^2$ completed,
$S^1$ and $S^0$ collapses still pending), the model predicts
\[
  \Lambda(2) \;=\; 2\Lz \;=\; 2\!\left(1 - \frac{\pi^2}{32}\right) \;\approx\; 1.383.
\]
This epoch (dark energy twice its present value) would correspond to an
intermediate cosmological era if the fold-to-redshift mapping can be established.
\end{corollary}

\noindent
\textbf{Status.}
Whether the fold-to-redshift mapping closes without introducing new mathematical
objects determines whether this is a Paper~32 addendum (no new objects) or a
new paper (additional structure required).
The current derivation is complete without new objects; the mapping step is recorded
as Conjecture~32.2 and left to a separate investigation.

% ── §4  Remark: Derivation Boundaries ────────────────────────────────────────
\section{Remark: Derivation Boundaries and Open Questions}

\begin{remark}
This addendum uses only:
\begin{enumerate}
  \item $\chiS = 1 + (-1)^n$, the standard Euler characteristic formula for spheres;
  \item The fold-collapse sequence $B^4 \to S^3 \to S^2 \to S^1 \to S^0$ established
        in Paper~32 \S5;
  \item The boundary condition $\Lambda(4) = \Lz$ from Paper~32 \S3 Theorem~1.
\end{enumerate}
No parameters are fit to cosmological data; the $\Lambda(2) = 2\Lz$ prediction is
a consequence of the integer Euler oscillation $(0, 2, 0, 2)$ alone.
\end{remark}

\begin{remark}[Hubble tension]
The $\varepsilon$ correction required to explain the Hubble tension is \emph{not}
claimed here.
Paper~32 \S6 Conjecture~6.2 records this as an open item; it requires deriving
$\varepsilon$ from first principles rather than fitting.
Any attempt to attribute the Hubble tension to the dynamic-$\Lambda$ mechanism
without that derivation would be premature.
\end{remark}

\begin{remark}[4-fold vs.\ 8-fold ambiguity]
The octonion count gives 8 total folds ($1$ real $+ 7$ imaginary units), whereas
only 4 boundary folds appear in the Euler sequence.
The resolution adopted here: $k$ counts \emph{active boundary fold events} in the
$B^4$ collapse, not octonion dimensions.
Under the identification $k_8 = 2 k_4$ the step structure is preserved and the
result remains a Paper~32 addendum.
\end{remark}

\end{document}
