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% Addendum 386 (picks up 385's CP flag; extends 364/385, A344, 356a).
% Z3 has one nontrivial automorphism (inversion), realizing BOTH C and P, so CP = identity on the Z3
% family sector -> CP-invariant, no CP phase. Confirms A344 (delta_CP refuted) + 364 (asymmetry = frame
% not CP); CPT reduces to T (356a). Honest tension: observed meson CP violation NOT captured (must live
% in the weak sector, 385's flag 2, open).
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P386.py (8/8 pass).
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\title{Addendum 386: $CP$ on the Family Sector\\
\large One $\mathbb Z_3$ automorphism realizes both $C$ and $P$, so $CP=\mathrm{id}$ ---\\
confirming the $\delta_{CP}$ refutation and the frame-effect, with one honest tension\\[0.3em]
\normalsize (Develops Addendum 385's flag; extends Addenda 364, A344, 356a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~385 flagged that on the $\mathbb Z_3$ family sector $C$ (charge conjugation, Addendum~364) and
$P$ (the fold orientation flip $L(3,1)\leftrightarrow L(3,2)$) are realized by the same operation, the
complex conjugation $\omega\leftrightarrow\bar\omega$ --- a hint of $CP$ structure, recorded but not
developed. This addendum develops it. The family symmetry is $\mathbb Z_3=\{1,\omega,\omega^2\}$,
$\omega=e^{2\pi i/3}$, and $\operatorname{Aut}(\mathbb Z_3)=\mathbb Z_2$: there is exactly \emph{one}
nontrivial automorphism, the inversion $\omega\mapsto\omega^2=\bar\omega$. Both corpus operations land on
it --- $P$ is the orientation flip $T_{\rm cycle}\leftrightarrow T_{\rm cycle}^2$, $C$ is complex
conjugation $\omega\leftrightarrow\bar\omega$, the same inversion --- so
\[
  CP \;=\; \text{inversion}\circ\text{inversion} \;=\; \mathrm{id}
\]
on the $\mathbb Z_3$ family sector: the family geometry is \textbf{$CP$-invariant}, with no $CP$ phase
available. This confirms two standing corpus results. It explains why the $\delta_{CP}\approx240^\circ$
prediction was \emph{refuted} (A344): there is no $CP$ phase to predict in the family sector. And it
\emph{is} Addendum~364's statement that the matter asymmetry is ``not real $CP$ violation, no $CP$ phase
generates it'' --- the family structure cannot supply $CP$ violation, so the asymmetry is the
frame-effect. With $CP=\mathrm{id}$, $CPT$ on the sector reduces to $T$ alone --- the witness-relative
arrow (356a). \textbf{The honest tension:} observed $CP$ violation ($K$ and $B$ mesons, real physics
since 1964) is \emph{not} captured by $CP=\mathrm{id}$ on the family geometry; if real --- it is --- it
must live outside the $\mathbb Z_3$ family sector, in the weak sector proper (Addendum~385's still-open
flag~2). This establishes the corpus's internal family-sector $CP$ structure; it does not account for
observed meson $CP$ violation, a real open gap. All claims pinned by \texttt{verify\_P386.py} (8/8).
\end{abstract}

\section{$CP=\mathrm{id}$ on the family sector}
\label{sec:cp}

\begin{proposition}[One automorphism, both $C$ and $P$]
\label{prop:one}
The family symmetry is $\mathbb Z_3=\{1,\omega,\omega^2\}$ with $\omega=e^{2\pi i/3}$
($\omega^3=1$, $\omega^2=\bar\omega=\omega^{-1}$). Its automorphism group is
$\operatorname{Aut}(\mathbb Z_3)=\mathbb Z_2$: the only nontrivial automorphism is the inversion
$\omega\mapsto\omega^2=\bar\omega$, an involution. Both corpus operations are this inversion:
$P$ (Addendum~385) is the fold orientation flip $T_{\rm cycle}\leftrightarrow T_{\rm cycle}^2$, i.e.\
$\omega\leftrightarrow\omega^2$; $C$ (Addendum~364) is the complex conjugation
$\omega\leftrightarrow\bar\omega=\omega\leftrightarrow\omega^2$ --- the same inversion.
\end{proposition}

\begin{theorem}[$CP$ is the identity on $\mathbb Z_3$]
\label{thm:cp}
Since $C$ and $P$ are the single nontrivial automorphism of $\mathbb Z_3$ (Proposition~\ref{prop:one}),
their composite is its square, and an involution squared is the identity:
$CP=\text{inversion}\circ\text{inversion}=\mathrm{id}$. The $\mathbb Z_3$ family sector is therefore
$CP$-invariant: it carries no $CP$ phase. With $CP=\mathrm{id}$ on the sector, $CPT$ acts there as $T$
alone --- the witness-relative arrow of Addendum~356a; the family substrate's full conjugation reduces
to the $T$ (frame) effect.
\end{theorem}

\section{What it confirms, and the tension it leaves}
\label{sec:confirm}

\begin{corollary}[Consistency with the $\delta_{CP}$ refutation and the frame-effect]
\label{cor:consistency}
$CP=\mathrm{id}$ on the family sector confirms two standing results. \emph{(i)} The corpus's
$\delta_{CP}\approx240^\circ$ prediction was refuted (A344, ``the $CP$ phase prediction fails on current
data''); Theorem~\ref{thm:cp} says there was no $CP$ phase to predict in the family geometry --- the
structural source the geometric $\delta_{CP}$ lacked. \emph{(ii)} Addendum~364 reads the matter
asymmetry as ``not real $CP$ violation, no $CP$ phase generates it'' (a frame-effect); $CP=\mathrm{id}$
on the sector \emph{is} that statement --- the family structure cannot supply $CP$ violation, so the
asymmetry must be the frame-effect, not $CP$.
\end{corollary}

\begin{remark}[The honest tension: observed $CP$ violation]
\label{rmk:tension}
Observed $CP$ violation --- the $K$- and $B$-meson results, real physics since 1964 --- is \emph{not}
captured by $CP=\mathrm{id}$ on the family geometry. If real, and it is, it must live outside the
$\mathbb Z_3$ family sector, in the weak sector proper (the still-open flag~2 of Addendum~385, the
weak-parity-violation~$\to$~$L(3,1)$ identification). So this addendum establishes the corpus's internal
family-sector $CP$ structure ($CP$-invariant), consistent with its own refuted $\delta_{CP}$ and its
frame-effect reading; it does \emph{not} account for observed meson $CP$ violation, which remains a real
open gap, not closed here. The result is a structural observation with consequences, honestly bounded,
not a derivation of Standard-Model $CP$.
\end{remark}

\section*{Status}
Picking up Addendum~385's $CP$ flag: on the $\mathbb Z_3$ family sector, $\operatorname{Aut}(\mathbb Z_3)
=\mathbb Z_2$ has a single nontrivial automorphism, the inversion $\omega\mapsto\omega^2=\bar\omega$, and
both $P$ (385's orientation flip) and $C$ (364's charge conjugation) are realized by it
(Proposition~\ref{prop:one}), so $CP=\text{inversion}^2=\mathrm{id}$ (Theorem~\ref{thm:cp}). The family
geometry is $CP$-invariant, carrying no $CP$ phase; with $CP=\mathrm{id}$, $CPT$ reduces to $T$ (356a).
This confirms the $\delta_{CP}$ refutation (A344: no phase to predict) and the frame-effect reading of
the matter asymmetry (364: not real $CP$ violation) (Corollary~\ref{cor:consistency}). \textbf{Honest
tension} (Remark~\ref{rmk:tension}): observed meson $CP$ violation is not captured by the family
geometry and must live in the weak sector (385's open flag~2); the internal structure is established, the
observed violation is a real open gap. No published number changes. Verifier: \texttt{verify\_P386.py},
8/8 pass.

\end{document}
