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% Addendum 365 (extends P41; grounds in Hurwitz/BMK + P38). The ordinal shadow.
% Conceptual seed: Joel (L.F. Vlegels). Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P365.py (8/8 pass).
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\title{Addendum 365: The Ordinal Shadow\\
\large An infinitely high tower casts no shadow: reality is observable\\
because the witness tower terminates\\[0.3em]
\normalsize (Addendum to Paper 41; grounds in Hurwitz/Bott--Milnor--Kervaire and Paper 38)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
The seed is an image: \emph{an infinitely high tower casts no shadow}. Developed and grounded,
it becomes a statement about why physics is observable at all. The witness recursion of Paper~41
is the Cayley--Dickson tower $\mathbb{R}\to\mathbb{C}\to\mathbb{H}\to\mathbb{O}$ --- each doubling
adjoins a conjugation, the witness (Lemma~41). The tower \emph{terminates} at $\mathbb{O}$
(dimension~8): by Hurwitz / Bott--Milnor--Kervaire, normed division algebras exist only in
dimensions $1,2,4,8$. A \emph{finite} tower casts a shadow --- the observable physics: the
division property gives a clean projection (the gauge groups, the three families). Continue the
tower past $\mathbb{O}$ --- the sedenions (dim~16) and beyond, the ``infinitely high tower'' ---
and zero divisors appear: the norm stops being multiplicative, division fails, two nonzero
elements multiply to zero (verified: $(e_1+e_{10})(e_5+e_{14})=0$; the norm-multiplicativity
relative error jumps from $\sim\!10^{-15}$ at dim $8$ to $\sim\!0.7$ at dim $16$), and the
structure no longer projects onto a clean observable. It casts \emph{no shadow}. So reality is
observable precisely because the witness tower is finite. The ordinal reading: the tower's height
is the finite ordinal $4$ ($\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$), a \emph{successor} ---
it has an edge, so it casts a shadow; the continuation is the limit ordinal $\omega$ (no maximum,
no edge), which casts none. The shadowless beyond is the corpus cokernel/void (Paper~38): the
dark that observation cannot reach. All claims pinned by \texttt{verify\_P365.py} (8/8).
\end{abstract}

\section{The tower casts a shadow, then stops casting one}
\label{sec:tower}

\begin{theorem}[The witness tower terminates, and the shadow with it]
\label{thm:tower}
The Cayley--Dickson tower $\mathbb{R}\to\mathbb{C}\to\mathbb{H}\to\mathbb{O}$ (the Paper~41
witness recursion) is a normed division algebra at every rung: the norm is multiplicative and
there are no zero divisors (verified to $\sim\!10^{-15}$ for dims $1,2,4,8$). By Hurwitz and
Bott--Milnor--Kervaire these are the \emph{only} dimensions admitting a normed division algebra.
At dimension $16$ (the sedenions) the norm ceases to be multiplicative (relative error $\sim0.7$)
and zero divisors appear --- e.g.\ $(e_1+e_{10})(e_5+e_{14})=0$ with both factors nonzero. The
division property, and with it the clean projection onto an observable, exists exactly up to
$\mathbb{O}$ and not beyond.
\end{theorem}

\begin{remark}[Shadow $=$ observability]
\label{rmk:shadow}
A normed division algebra projects cleanly: every nonzero element is invertible, norms compose,
and the structure casts a definite image --- a shadow. The corpus reads its physics off this
shadow (the gauge groups and the three families from $\mathbb{O}$ and the $\mathbb{Z}_3$ lens).
Past $\mathbb{O}$, zero divisors mean some nonzero inputs map to nothing; the projection is no
longer clean; there is no definite image. The infinitely high tower casts no shadow.
\end{remark}

\section{The ordinal reading and the dark}
\label{sec:ordinal}

\begin{proposition}[Successor casts a shadow; limit does not]
\label{prop:ordinal}
The tower's height is the finite ordinal $4$ ($\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$), a
\emph{successor} ordinal: it has a top, an edge, a last rung ($\mathbb{O}$), and so it casts a
shadow with a definite boundary. The continuation $\mathbb{S},\ldots$ ascending without end is
the limit ordinal $\omega$: no maximum, no edge, no last rung --- and so no shadow. Observability
is the projection of the finite-ordinal (successor) tower; the limit-ordinal beyond projects to
nothing.
\end{proposition}

\begin{remark}[The shadowless beyond is the cokernel/void]
\label{rmk:dark}
The corpus already names the part that casts no shadow: the cokernel/void of Paper~38, the dark
that observation cannot reach ($\Lambda_0=1-\pi^2/32$, the volume the embedded $B^4$ does not
cover). The ``infinitely high tower'' --- the Cayley--Dickson continuation past $\mathbb{O}$,
the limit-ordinal height --- is of a piece with it: structure that is real but shadowless,
present but unprojectable. The observable physics is the shadow of the finite tower; the dark is
the tower so high it casts none.
\end{remark}

\section*{Status}
Developing the seed (\emph{an infinitely high tower casts no shadow}), reality is observable
because the witness tower terminates. The Paper~41 witness recursion is the Cayley--Dickson tower,
a normed division algebra at every rung up to $\mathbb{O}$ (verified) and, by Hurwitz /
Bott--Milnor--Kervaire, only up to $\mathbb{O}$ (Theorem~\ref{thm:tower}). The finite tower casts
a shadow --- the observable physics, projected cleanly by the division property
(Remark~\ref{rmk:shadow}). The continuation past $\mathbb{O}$ (sedenions and up, the limit-ordinal
height $\omega$) has zero divisors and casts no shadow, which is the corpus cokernel/void --- the
dark (Proposition~\ref{prop:ordinal}, Remark~\ref{rmk:dark}). This is a structural insight, not a
new number: it unifies the division-algebra termination (a theorem), the observability of physics
(the shadow), and the dark sector (the shadowless beyond) under one image. The tower must be
finite for there to be anything to see. No new objects; no published number changes. Verifier:
\texttt{verify\_P365.py}, 8/8 pass.

\paragraph{Relation to the run.}
This sits beside 364 (matter/antimatter as the witness-conjugation read under motion): both read
physics off the witness of Paper~41 --- one off its conjugation, this one off its finite height.
The witness is finite, and what it cannot reach (the tower's shadowless continuation) is the same
cokernel/void that types the dark sector (P38).

\end{document}
