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% Addendum 351a (a-series, dark-sector cosmology thread; foundational note).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P351a.py (11/11 pass).
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\title{Addendum 351a: Why Expansion\\
\large The boundary cascade from monad to self-similar floor\\[0.3em]
\normalsize (a-series, dark-sector thread; a foundational note, bolted to the
fold cascade at both ends)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
The 345a--350a line built the fold cascade and confronted it with data but took
the cascade's \emph{existence} as given. This note supplies the bootstrap: why a
monad must spawn at all, and why the spawning must run until it stops. The chain
is: a lone monad cannot hold a distinction (no witness), so existence forces a
\emph{second} (Paper~27); a distinction is two-sided, so it spawns its complement;
witnessing it requires an \emph{orthogonal} vantage, and the resulting
$90^\circ$ wedge closes into the four axes of $\hat O$; each boundary is itself a
region with a boundary, so boundaries \emph{cascade} (the fold sequence
$B^4\!\to\!S^3\!\to\!S^2\!\to\!S^1\!\to\!S^0$, driven by Fork); and the cascade can
terminate only where rescaling produces no new distinction --- self-similarity ---
which is simultaneously the operator's scale-invariant soft direction $\zeta$
(the b-thread's lone floor) and the cosmological de Sitter floor $\Lz=1-\pi^2/32$
(this thread's dark-energy asymptote). Expansion is therefore necessary on both
ends: it cannot fail to start (boundary necessity) and cannot fail to stop
(self-similarity is its only fixed point). The note is explicitly
foundational/interpretive; a status ledger marks each link as analytic, canonical,
or interpretive. Verifier: \texttt{verify\_P351a.py} (11/11).
\end{abstract}

\section{The spawn: why a monad does not stay one}
\label{sec:spawn}

\textbf{Witness necessity (canonical, Paper~27).} A single monad satisfies the
coherence identity $C\circ P = I$ trivially: with no second, there is nothing for
a distinction to be a distinction \emph{from}, and nothing to witness it. Paper~27
makes this the Level-$(-1)$ foundation --- existence requires witness, witness
requires two --- and reads the Big Bang as ``the first disagreement.'' This is the
answer to ``why would a single monad spawn'': it must, because being-distinct is
what existing means here, and being-distinct is impossible alone.

\textbf{Two-sidedness (analytic).} A boundary is two-sided by its nature: to posit
$A$ is to posit $\lnot A$; a one-sided distinction is incoherent. So the first
distinction does not add one thing, it spawns a complementary pair. This step is
near-forced, not assumed.

\section{The $90^\circ$ witness and the four-axis closure}
\label{sec:ninety}

\begin{proposition}[The wedge closes]
\label{prop:wedge}
A distinction cannot be witnessed from within itself; holding it requires a vantage
\emph{orthogonal} to it. So the spawned structure has two angular commitments: the
complement at $180^\circ$ (the other side) and the witness at $90^\circ$ (the
second observer's direction). Iterating the witness requirement,
$4\times90^\circ=360^\circ$: the wedge closes into \emph{four orthogonal axes},
identified with the four scalar coefficients of the master operator $\hat O$ ---
$\alpha$ (density), $\gamma$ (layer-cycle), $\beta$ (moment), $\zeta$
(renormalisation). This is the monad map: $\varnothing/0$ at the centre, four axes
unfolding, with $\zeta=\alpha^{5/4}$ the dependent fold off $\alpha$.
\end{proposition}

The $90^\circ$-as-witness identification is the load-bearing interpretive step ---
it is where orthogonality, and hence the four-fold axis structure, enters. It is
flagged as such in the ledger (\S\ref{sec:status}).

\section{The cascade and its geometric form}
\label{sec:cascade}

Each spawned side is itself a region, hence bounded, hence the site of a further
distinction: boundaries \emph{cascade}. Geometrically the two-sidedness fills the
pre-geometric container --- the cube $[-1,1]^4$, whose four axes carry opposite
faces ($\pm1$) and whose sixteen corners, in eight antipodal pairs, are the
octonion basis (Paper~32). The dimensional descent
\[
  B^4 \to S^3 \to S^2 \to S^1 \to S^0,
  \qquad \chi(S^n)=1+(-1)^n,
\]
is the cascade made literal; in Wheel terms it is \textbf{Fork} ($D^2_{B^4}$), the
spawning operator, running the descent. This is exactly the cascade the 345a--350a
line built the cosmology on; here it is derived as a necessity rather than posited.

\section{The termination: distinction closing on itself}
\label{sec:terminate}

The cascade has exactly one way to stop: reach a state where a further rescaling
produces no new distinction --- where the structure is \emph{self-similar}. That
condition appears, identically, at two scales the two threads study.

\begin{itemize}
\item \textbf{Operator level (b-thread).} The terminating direction is $\zeta$, the
  scaling/dilation generator (Perturb, $\zeta R$). Its symmetric part is the
  state-independent constant ($-2$, A332): the one \emph{soft} direction, where
  dilation does nothing. This is the lone non-floor gap the b-thread reduced the
  whole P18-T2 (iii) problem to --- and it is, in this reading, the place
  ``distinction closes on itself'' at the operator scale.
\item \textbf{Cosmic level (this thread).} The terminating state is the de Sitter
  floor $\Lz = 1-\pi^2/32 = 0.6916$, where $w\to-1$, expansion becomes
  self-similar, and the fold cascade halts (Addenda 349a/350a). The dark-energy
  asymptote \emph{is} the cosmological form of self-similar closure.
\end{itemize}

\begin{proposition}[Why expansion]
\label{prop:why}
Expansion is the boundary cascade running. It cannot fail to start (a distinction
spawns its complement and requires its witness --- boundary necessity) and cannot
fail to stop (self-similarity is the cascade's only fixed point). The arc from
\textbf{Fork} (spawn) to $\zeta/\Lz$ (close) is the fold cascade itself; the
dark-energy floor of this thread is the geometric end-state of the bootstrap.
\end{proposition}

\section{Status ledger}
\label{sec:status}

Stated plainly, so the foundational and the derived are not conflated:

\begin{center}
\begin{tabular}{ll}
\hline
\textbf{Link} & \textbf{Status} \\
\hline
Monad must spawn (witness necessity) & Canonical (Paper~27, Level $-1$) \\
A distinction is two-sided & Analytic \\
Witness is orthogonal ($90^\circ$); four-axis closure & Analytic --- pinned in Add.~352a \\
Opposite sides $\to$ cube $\to$ octonion corners & Canonical (Paper~32) \\
Cascade $=$ fold sequence (Fork) & Canonical (Paper~32; Wheel pin) \\
$\Lz=1-\pi^2/32$ as cube/ball residual & Theorem (Paper~32) \\
Cascade terminates \emph{because} distinction closes & Interpretive seam \\
\hline
\end{tabular}
\end{center}

\begin{remark}
This note derives no new physics. It supplies the bootstrap that makes the derived
cascade cohere --- why it exists (\S\ref{sec:spawn}--\ref{sec:cascade}) and why it
ends (\S\ref{sec:terminate}) --- and bolts onto both threads at the two points
where they already meet: Fork-spawn and $\zeta/\Lz$-close. The one step that needed
pinning, not asserting --- $90^\circ=$ witness (\S\ref{sec:ninety}), which carries
the orthogonality and the four-axis structure --- is pinned in Addendum~352a:
a clean distinction is orthogonal because perfect distinguishability \emph{is}
orthogonality (the Helstrom bound), leaving only the corpus's canonical
inner-product geometry as the foundation.
\end{remark}

\end{document}
