\documentclass[11pt]{article}
% Addendum 355a (a-series; closes the 353a lemma). CC BY 4.0.
% Author: Leon Fernando Vlegels.  Verifier: verify_P355a.py (11/11 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}[]{Theorem}
\newtheorem{lemma}[]{Lemma}
\theoremstyle{remark}
\newtheorem*{remark}{Remark}

\title{Addendum 355a: The Witness is the Conjugation\\
\large Closing the lemma of the witness recursion\\[0.3em]
\normalsize (a-series; closes the load-bearing step of Addendum 353a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Addendum~353a derived the $(B^4,S^3)$/octonion substrate from the witness
principle, resting on one interpretive seam: that a witness step \emph{is} the
Cayley--Dickson conjugate doubling. This note closes that seam. Modelling
witnessing as coherent composition of distinctions, four premises --- each tied to
a canonical feature --- force it. \textbf{(P1)} distinctions form a finite-dim
bilinear $\mathbb R$-algebra (composition is superposition; the Paper~27
linear-coherence structure); \textbf{(P2)} coherence forbids annihilation
($x,y\neq0\Rightarrow xy\neq0$, no zero divisors --- the $C\circ P=I$ non-collapse);
\textbf{(P3)} distinguishability gives a norm (Addendum~352a); \textbf{(P4)}
recoverability ($C\circ P=I$) undoes each distinction by its witness. By
Bott--Milnor--Kervaire, P1+P2 force the dimension into $\{1,2,4,8\}$ from coherence
alone; and the witness map is then the \emph{unique} anti-involution with
$x\bar x = N(x)$ --- the conjugation --- making the algebra a composition algebra
(Hurwitz: $\mathbb R,\mathbb C,\mathbb H,\mathbb O$). We verify on
$\mathbb R,\mathbb C,\mathbb H,\mathbb O$ that the witness $\bar x/N(x)$ is exactly
$x^{-1}$, that $x\bar x=N(x)$ is real, and that conjugation fixes the real
(agreement) and reverses the imaginary (distinction-direction). The lemma closes
\emph{onto the canonical base}: no posit is added beyond ``witnessing is coherent
bilinear composition,'' which is the Paper~27 frame the corpus already uses. Hence
the witness recursion is a theorem on that base, and the observer origin of the
substrate is pillar-grade. Verifier: \texttt{verify\_P355a.py} (11/11).
\end{abstract}

\section{The four premises}
\label{sec:prem}

Model a witness as composing with the distinction it witnesses --- the natural
reading of ``holding and checking'' a distinction. Then:
\begin{description}
\item[(P1) Bilinear composition.] Distinctions superpose (linear combinations of
  distinctions are distinctions) and compose bilinearly: a finite-dim
  $\mathbb R$-algebra. This is the Paper~27 linear-coherence structure --- the same
  base Addendum~352a uses.
\item[(P2) No annihilation (coherence).] Composing two genuine distinctions never
  yields nothing: $x,y\neq0\Rightarrow xy\neq0$. No zero divisors. This is the
  $C\circ P=I$ non-collapse --- a composed distinction is still a distinction.
\item[(P3) Norm.] Distinguishability supplies an inner product and norm
  $N(x)=|x|^2$ (Addendum~352a, the Helstrom angle).
\item[(P4) Recoverability.] $C\circ P=I$: every distinction can be undone --- it is
  invertible, and its witness is the inverse-up-to-magnitude.
\end{description}

\section{The dimension, from coherence alone}
\label{sec:dim}

\begin{theorem}[Bott--Milnor--Kervaire]
A finite-dimensional division algebra over $\mathbb R$ has dimension $1,2,4$ or
$8$.
\end{theorem}

P1 (finite-dim $\mathbb R$-algebra) and P2 (no zero divisors $=$ division algebra)
therefore force $\dim\in\{1,2,4,8\}$ --- \emph{without any norm}. The dimension of
the substrate is a consequence of coherence (non-annihilation) by itself. The
Cayley--Dickson products confirm it: no zero divisors at $\dim 1,2,4,8$
($\min|\det L_x|=1$), collapsing at $\dim 16$ ($\min|\det L_x|\sim10^{-3}$).

\section{The witness is the conjugation}
\label{sec:conj}

\begin{lemma}[Witness $=$ conjugation]
Under P1--P4 the witness map is the unique anti-involution $x\mapsto\bar x$ with
$x\bar x = N(x)\in\mathbb R$. Equivalently: the witness of $x$ is $\bar x/N(x)=x^{-1}$
(P4), composing with $x$ to the pure undistinguished magnitude $N(x)$; and
$\bar{(\cdot)}$ fixes the real part (the agreement) while reversing the imaginary
part (the distinction-direction) --- the orientation-reversal intrinsic to
witnessing.
\end{lemma}

Verified on $\mathbb R,\mathbb C,\mathbb H,\mathbb O$: $x\,(\bar x/N(x))=1$,
$x\bar x=N(x)\,1$, $\bar{\bar x}=x$, and $\overline{(a,v)}=(a,-v)$. An algebra with
such a conjugation and multiplicative norm is a composition algebra, and by
Hurwitz the positive-definite ones are exactly $\mathbb R,\mathbb C,\mathbb H,
\mathbb O$. So the witness step is the Cayley--Dickson conjugate doubling --- 353a's
seam, now derived.

\section*{What closed, and the one honest residual}

\textbf{Closed.} The witness recursion is a theorem on the stated base: witness
$\to$ conjugate doubling $\to$ Cayley--Dickson $\to$ $\dim 4$ (associative/group,
$S^3=SU(2)$) and $\dim 8$ (division, $\mathbb O$). The dimension follows from
coherence alone (Bott--Milnor--Kervaire); the conjugation from coherence $+$ the
$352$a norm.

\textbf{The one residual --- and it is not new.} The derivation models
\emph{witnessing as coherent bilinear composition} (P1, P4). That modelling is the
single interpretive identification left; but it is exactly the Paper~27
linear-coherence frame on which the operator Hilbert space and 352a already stand.
So 353a's seam is not eliminated into thin air --- it is \emph{relocated onto the
canonical foundation}, adding no posit the corpus did not already make. The honest
statement is therefore: \emph{the witness recursion is a theorem given the Paper~27
linear-coherence frame}, the same footing as every other corpus result.

\begin{remark}
This is the bar for a pillar: a result that rests on the canonical foundation, not
on a fresh assumption. With the lemma closed, the observer origin of the
$(B^4,S^3)$/octonion substrate (353a), the $90^\circ$ pin (352a), and Maxwell with
charge quantization (354a) form a single derived chain --- observer $\to$
quaternions $\to$ electromagnetism --- ready to be assembled as Paper~41.
\end{remark}

\paragraph{Sources.} Papers 27 (coherence), 01 (division algebras); Addenda 351a,
352a, 353a, 354a. Theorems: Bott--Milnor (1958), Kervaire (1958), Hurwitz.

\end{document}
