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% Addendum 385 (the mirror-universe dig; extends 364 from C to P to CPT, grounds in P18/P06/356a).
% The fold's L(3,1)/L(3,2)=-L(3,1) orientation pair is a literal mirror pair (P18, Reidemeister-Franz),
% the flip being omega<->omega-bar = 364's C. Extending 364's frame-effect to parity: the mirror
% universe is the parity-CONJUGATE reading of the one fold, not a separate world -- dissolved like
% antimatter. CPT = C+P+T of one substrate. Suggestive: C,P share omega<->omega-bar (hint of CP).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P385.py (9/9 pass).
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\title{Addendum 385: Parity as a Frame-Effect\\
\large The mirror universe is the conjugate reading of the one fold ---\\
extending the matter-asymmetry mechanism from $C$ to $P$ to $CPT$\\[0.3em]
\normalsize (Extends Addendum 364; grounds in Papers 18, 06, Addendum 356a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Is the corpus's fold-handedness a parity / mirror-universe structure? It is, precisely, and the answer
mirrors Addendum~364's treatment of antimatter. \textbf{The structures.} Parity, in the corpus, is the
fold orientation: by Paper~18, the layer-cycle generator $T_{\rm cycle}$ gives $S^3/\langle
T_{\rm cycle}\rangle=L(3,1)$ (the three-fermion-families lens space, Paper~06) and $S^3/\langle
T_{\rm cycle}^2\rangle=L(3,2)$, and by Reidemeister--Franz $L(3,2)=-L(3,1)$ --- homeomorphic, opposite
orientation: a literal mirror pair, of which the corpus selects $L(3,1)$. The two orientations are
\emph{conjugates}: $T_{\rm cycle}=\omega=e^{2\pi i/3}$ and $T_{\rm cycle}^2=\omega^2=\bar\omega$, so the
orientation flip $L(3,1)\leftrightarrow L(3,2)$ \emph{is} the complex conjugation $\omega\leftrightarrow
\bar\omega$ --- the same conjugation Addendum~364 names charge conjugation $C$. \textbf{The extension.}
Addendum~364 showed the matter/antimatter asymmetry is a frame-effect: the vacuum is $C$-symmetric, and
the asymmetry is the inertial reading (``the asymmetry is the symmetry, under motion''). Apply this to
parity. The fold is parity-symmetric --- both orientations are readings of the one fold, related by
$\omega\leftrightarrow\bar\omega$ --- and an oriented observer reads one handedness ($L(3,1)$). So
\emph{parity violation is the oriented reading of a parity-symmetric fold}, the same frame-effect; and
the ``mirror universe'' ($L(3,2)$) is the parity-\emph{conjugate reading} of the one fold, \emph{not} a
separate parallel world --- dissolved exactly as 364 dissolves antimatter. Combined, $C$ (364) $+$ $P$
(orientation) $+$ $T$ (356a's witness-relative arrow) is the full conjugation of the one substrate: our
universe and the $CPT$-mirror (the ``anti-universe'') are conjugate readings of one thing, not two
worlds. \textbf{Honest scope:} the mirror pair and the frame-effect are grounded; extending to parity is
the new connection, and the specific weak-sector identification is not claimed. All claims pinned by
\texttt{verify\_P385.py} (9/9).
\end{abstract}

\section{The fold is handed; the two orientations are conjugates}
\label{sec:handed}

\begin{proposition}[The mirror pair, and the conjugation that flips it]
\label{prop:mirror}
By Paper~18, $T_{\rm cycle}$ is the layer-cycle ($\mathbb Z_3$) generator with
$S^3/\langle T_{\rm cycle}\rangle=L(3,1)$ and $S^3/\langle T_{\rm cycle}^2\rangle=L(3,2)$; by the
Reidemeister--Franz lens-space classification $L(3,2)=-L(3,1)$ (homeomorphic, opposite orientation), a
literal mirror pair. Paper~06 requires $L(3,1)$ for the three fermion families, so the corpus fold is
\emph{handed}. The two orientations are conjugates: with $\omega=e^{2\pi i/3}$,
$T_{\rm cycle}=\omega$ and $T_{\rm cycle}^2=\omega^2=\bar\omega$ (since $|\omega|=1$, $\omega^3=1$, so
$\omega^2=\omega^{-1}=\bar\omega$). Hence the orientation flip $L(3,1)\leftrightarrow L(3,2)$ is the
complex conjugation $\omega\leftrightarrow\bar\omega$ --- the same conjugation Addendum~364 identifies as
charge conjugation $C$ (the $\mathbb R\to\mathbb C$ conjugation, particle~$\leftrightarrow$~antiparticle).
\end{proposition}

\section{The mirror is the conjugate reading}
\label{sec:reading}

\begin{theorem}[Parity violation is a frame-effect; the mirror universe is the conjugate reading]
\label{thm:frame}
Addendum~364 established that the matter/antimatter asymmetry is a frame-effect: the inertial vacuum is
$C$-symmetric ($\text{matter}=\text{antimatter}$), and the asymmetry is the inertial reading. The same
holds for parity. The fold is parity-symmetric --- both orientations $L(3,1)$ and $L(3,2)$ are readings
of the one fold, related by $\omega\leftrightarrow\bar\omega$ (Proposition~\ref{prop:mirror}) --- and an
oriented (inertial) observer reads one handedness, $L(3,1)$. Therefore parity violation is the oriented
reading of a parity-symmetric fold, not a broken symmetry of the fold itself. The ``mirror universe,''
$L(3,2)$, is the parity-\emph{conjugate reading} of the one fold, not a separate parallel world ---
dissolved exactly as Addendum~364 dissolves antimatter (one vacuum, two readings). Combining the three
conjugations, $C$ (Addendum~364) $+$ $P$ (the orientation flip) $+$ $T$ (Addendum~356a's witness-relative
arrow) is the full $CPT$ conjugation of the one substrate: our universe and the $CPT$-mirror (the
``anti-universe'') are conjugate readings of one thing, not two worlds.
\end{theorem}

\begin{remark}[Suggestive: $C$ and $P$ share the conjugation]
\label{rmk:cp}
On the $\mathbb Z_3$ family sector the orientation flip and the charge conjugation are the \emph{same}
operation, $\omega\leftrightarrow\bar\omega$ (Proposition~\ref{prop:mirror}). That $C$ and $P$ share one
conjugation there is a hint of a $CP$ structure. It is recorded as suggestive only; it is \emph{not}
claimed as a derivation of the Standard Model's $CP$, and nothing here turns on it.
\end{remark}

\section*{Status}
The corpus's fold-handedness is a parity / mirror structure, and the mirror universe is dissolved the way
antimatter was. The fold's two orientations $L(3,1)$ and $L(3,2)=-L(3,1)$ are a literal mirror pair
(Paper~18, Reidemeister--Franz), the flip being the conjugation $\omega\leftrightarrow\bar\omega$ that
Addendum~364 calls $C$ (Proposition~\ref{prop:mirror}). Extending 364's frame-effect from $C$ to $P$
(Theorem~\ref{thm:frame}): the fold is parity-symmetric, the observed handedness is the oriented reading,
and the ``mirror universe'' ($L(3,2)$) is the parity-conjugate reading of the one fold --- not a separate
world, dissolved as antimatter is. With $T$ (356a) this gives $CPT$ as the full conjugation of one
substrate; our universe and the anti-universe are conjugate readings, not parallel worlds. A suggestive
$C$/$P$ conjugation-sharing is noted, not claimed (Remark~\ref{rmk:cp}). \textbf{Honest scope:} the mirror
pair and the frame-effect are grounded; extending to parity is the new connection; the specific
weak-sector parity-violation~$\to$~$L(3,1)$ identification is further work and is not claimed --- this
addendum establishes the \emph{structure} and the frame-effect reading, not a derivation of the Standard
Model's weak parity violation. No published number changes. Verifier: \texttt{verify\_P385.py}, 9/9 pass.

\end{document}
