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% Addendum 352a (a-series; sharpens 351a). Author: Leon Fernando Vlegels. CC BY 4.0.
% Verifier: verify_P352a.py (8/8 pass).
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\title{Addendum 352a: The Orthogonality of Witness\\
\large Pinning the $90^\circ$ step of the bootstrap\\[0.3em]
\normalsize (a-series; sharpens Addendum 351a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Addendum~351a left exactly one load-bearing \emph{interpretive} posit: that the
witness of a distinction sits at $90^\circ$ to it (the step that carries
orthogonality, and with it the four-axis monad structure). This note pins it. In
any inner-product space, two states are \emph{perfectly distinguishable} if and
only if they are \emph{orthogonal} --- the Helstrom two-state discrimination bound,
$P_{\max}=\tfrac12(1+\sin\theta)$, equals $1$ only at $\theta=90^\circ$. A
\emph{clean} distinction (one side never confused with the other) therefore
\emph{forces} orthogonality; the $90^\circ$ is not a drawn right angle but the
distinguishability condition itself. The only remaining input --- that distinctions
live in an inner-product space --- is the corpus's own coherence geometry
(Paper~27's $S^3$, the operator Hilbert space, projective measurement), not a fresh
assumption. So $90^\circ=$ witness is downgraded from interpretive posit to
\emph{analytic fact on a canonical base}; the foundational seam relocates to
``why an inner-product structure,'' which is the Paper~27 witness-necessity it
already rests on. Verifier: \texttt{verify\_P352a.py} (8/8).
\end{abstract}

\section{Distinguishability is orthogonality}
\label{sec:disting}

A witness must tell A from $\lnot A$. Ask how well it can, in an inner-product
space. For two equiprobable pure states with overlap $|\langle A|\lnot A\rangle| =
\cos\theta$, the optimal single-shot discrimination success is the Helstrom bound
\[
  P_{\max}(\theta) = \tfrac12\bigl(1+\sin\theta\bigr).
\]

\begin{proposition}[A clean distinction is a right angle]
\label{prop:right}
$P_{\max}(\theta)=1$ (the two sides are \emph{never} confused) holds iff
$\sin\theta=1$, i.e.\ $\theta=90^\circ$, i.e.\ $\langle A|\lnot A\rangle=0$. For
$\theta<90^\circ$ the overlap is nonzero and the confusion probability
$1-P_{\max}>0$. Hence a clean distinction --- one whose two sides are perfectly
witnessable --- \emph{requires} its sides orthogonal. The $90^\circ$ is the
distinguishability condition, not an added geometric assumption.
\end{proposition}

This replaces 351a's interpretive step. There, ``the witness sits orthogonal'' was
asserted as the load-bearing posit; here it is the content of
Proposition~\ref{prop:right}: to witness cleanly \emph{is} to read an orthogonal
pair.

\section{The inner product is canonical, not new}
\label{sec:hilbert}

Proposition~\ref{prop:right} needs one structure: an inner product (a notion of
overlap, hence of distinguishability). The corpus already supplies it, and not as a
new posit:
\begin{itemize}
\item $S^3$ is the unique space supporting coherent multi-observer agreement
  (Paper~27) --- agreement \emph{is} an overlap structure;
\item the master operator $\hat O$ acts on a Hilbert space (Paper~18);
\item measurement is the orthogonal projection $\Pi_\downarrow$ (Paper~4).
\end{itemize}
So $90^\circ=$ witness reduces to ``a clean distinction in the corpus's own Hilbert
geometry'' $=$ distinguishability (analytic, \S\ref{sec:disting}) $+$ inner product
(canonical). The step is pinned.

\section{The four-axis closure, now derived not drawn}
\label{sec:four}

351a obtained the four monad axes from $4\times90^\circ$ closing a wedge --- a
drawn angle. With Proposition~\ref{prop:right} the same structure is forced
algebraically: $n$ \emph{mutually}-clean distinctions are $n$ \emph{mutually}
orthogonal directions ($\langle e_i|e_j\rangle=0$, $i\neq j$). The monad map's
right angles are the pairwise distinguishability of its axes, not a feature of the
drawing (which was exactly the projection-fidelity point that started this).

The \emph{count} four is not a separate posit either; it is the same argument
continued one step. Require the distinctions to \emph{compose without collapse} ---
combining two genuine distinctions never yields nothing, the no-zero-divisor form
of the coherence identity $C\circ P=I$. Orthogonal directions that compose without
zero divisors are precisely a \emph{normed division algebra}, and by Hurwitz those
exist only in dimensions $1,2,4,8$ ($\mathbb R,\mathbb C,\mathbb H,\mathbb O$). The
witnessing structure must moreover be associative (a consistent group of
compositions) and non-abelian (order of distinction matters); the unique
sphere-group meeting both is $S^3=$ unit quaternions $=SU(2)$ ($S^7$ is
parallelizable but non-associative, hence not a group; $S^0,S^1$ are abelian).
Hence the bulk is $B^4$, dimension four, and the four orthogonal axes are the four
quaternion components --- the four scalar coefficients of $\hat O$,
$\alpha,\gamma,\beta,\zeta$. This is the Paper~01 division-algebra derivation
(Hurwitz/Adams), reached here from distinguishability rather than assumed.

\section*{Status and honest residual}

\textbf{What is pinned.} 351a's lone interpretive load-bearer, $90^\circ=$ witness,
is downgraded to an \emph{analytic} fact (distinguishability $=$ orthogonality,
Proposition~\ref{prop:right}) resting on a \emph{canonical} base (the Paper~27
coherence Hilbert geometry). The four-axis closure follows from mutual
distinguishability rather than a drawn wedge.

\textbf{What remains a seam.} The residual is relocated, not removed: ``why an
inner-product / projective-measurement structure at all.'' The answer is the
Paper~27 coherence geometry --- coherent agreement among witnesses requires a
shared space with an overlap measure --- which itself rests on the
witness-necessity that opened the bootstrap. So the seam moves up to a single
canonical foundation already in play, rather than sitting as a free geometric
choice. No new physics; a foundation tidied.

\begin{remark}
With this, the bootstrap chain of 351a has no free interpretive load-bearer left:
witness-necessity (canonical), two-sidedness (analytic), $90^\circ$ (now analytic
via Proposition~\ref{prop:right}), the cascade (canonical), and the
$\zeta/\Lambda_0$ termination (derived/canonical). The one genuine foundation is
the Paper~27 inner-product coherence geometry, on which everything else now hangs
analytically.
\end{remark}

\end{document}
