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% Addendum 374 (extends 373; formalizes the two regimes of efficacy).
% Two distinct Hopf fibers are disjoint (no causal contact) but LINKED (linking number 1) -- acausal
% correlation, the structural form of 'synchronicity', a layer one can affect. The SHARED S^3 changes
% only by action through S^3 (373 E3): linking is correlation, not leverage. Completes 373's
% anti-solipsism result; E3 intact. Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P374.py (9/9 pass).
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\title{Addendum 374: Linked but Disjoint\\
\large Synchronicity is the Hopf linking (correlation, not cause);\\
changing the shared world still requires action --- the two regimes of efficacy\\[0.3em]
\normalsize (Addendum to Paper 04; extends 373)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~373 made ``shared reality enforced'' a theorem: one shared $\Sthree$ (E1), disjoint fibers
(E2, privacy), the only inter-fiber channel through $\Sthree$ (E3). It left two \emph{regimes of
efficacy} implicit. This addendum makes them explicit, formalizing the articulation that \emph{the
privacy of experience is part of reality one can affect (synchronicity), but one cannot force shared
reality to change without action}. The missing formal piece is that disjoint Hopf fibers are not
merely disjoint --- they are \textbf{linked}, with linking number $1$ (the Hopf invariant; Gauss
integral $=1.00$ on distinct fibers). \emph{Disjoint} means no direct fiber-to-fiber causal or signal
contact; \emph{linked} means global correlation independent of any $\Sthree$ signal. Disjoint-but-linked
is exactly the structure of \textbf{acausal correlation} --- two private viewpoints correlated through
the bundle with no causal channel between them --- which is what ``synchronicity'' names structurally.
The fiber is also \emph{affectable}: its $S^1$ phase / observer-trajectory (Paper~08) is a private
degree of freedom one can move. But by 373's E3, the \emph{only} channel that moves the shared
$\Sthree$ is classical \textbf{action} through $\Sthree$; the linking gives \emph{correlation, not
leverage}, so a fiber-local change is correlated with the whole yet cannot \emph{force} the shared
boundary. \textbf{The line:} correlation (the linking; the fiber; acausal; ``synchronicity'') is not
forcing (causation; $\Sthree$; action-required). This completes 373's anti-solipsism result --- privacy
is real \emph{and} affectable, the shared arena is enforced \emph{and} action-only --- and keeps E3
intact: synchronicity is the linking, not a private lever on the world (not psychokinesis). Theorem-grade
as topology; the hinges are monad $=$ observer and the identification of the linking with experienced
synchronicity (a reading, not a paranormal claim). All claims pinned by \texttt{verify\_P374.py} (9/9).
\end{abstract}

\section{The two regimes}
\label{sec:regimes}

\begin{theorem}[L1: disjoint but linked]
\label{thm:linked}
Two fibers of the Hopf fibration $\pi:\Sthree\to S^2$ over distinct base points are disjoint
($M_i\cap M_j=\varnothing$, 373~E2; minimal separation $\approx0.999>0$) yet \emph{linked} with
linking number $1$ --- the Hopf invariant, verified by the Gauss linking integral ($=1.00$ on
stereographically projected distinct fibers). Disjointness forbids direct fiber-to-fiber causal/signal
contact; the linking is a global correlation that does not depend on any signal passing through
$\Sthree$.
\end{theorem}

\begin{remark}[L2: disjoint-but-linked is acausal correlation --- the form of ``synchronicity'']
\label{rmk:synch}
A nonzero linking between curves with empty intersection is precisely \emph{correlation without
contact}: the two loops are globally tied although neither meets or signals the other. Read on the
monad fibers, this is two private viewpoints correlated through the bundle with no causal channel
between them --- the structural signature of \emph{acausal correlation}, which is what
``synchronicity'' names. The identification of this structure with any \emph{experienced} synchronicity
is a reading, not a theorem: the corpus establishes that the topology \emph{permits} acausal
correlation, not that any particular coincidence \emph{is} one. No paranormal claim is made;
correlation is not causation.
\end{remark}

\begin{proposition}[L3--L4: the fiber is affectable; the arena is action-only]
\label{prop:efficacy}
\emph{L3 (affectable fiber).} The fiber carries a private degree of freedom one can move: its $S^1$
phase / observer-trajectory (Paper~08, observer-as-trajectory; the Wheel's phase). The private layer
is thus ``part of reality one can affect.'' \emph{L4 (action-only arena).} By 373's E3 the only
channel that moves the shared $\Sthree$ is classical action through $\Sthree$
($M_i\to\Sthree\to M_j$: speech, writing, physical signals). The linking of
Theorem~\ref{thm:linked} supplies \emph{correlation, not leverage} --- a fiber-local change is
correlated with the whole but cannot \emph{force} the shared boundary. ``Force the shared world
without action'' is topologically blocked.
\end{proposition}

\section{The line, and the completed solipsism answer}
\label{sec:line}

\begin{corollary}[Correlation is not forcing; E3 stands]
\label{cor:line}
The two regimes do not collide. Correlation lives on the fiber and through the linking (acausal,
affectable, ``synchronicity''); forcing lives on $\Sthree$ and requires action (causal, E3). A fiber
may be affected and thereby correlated with the whole, yet the shared world changes only by action in
it. So synchronicity is \emph{not} a violation of E3 and \emph{not} a private lever on reality: it is
the bundle's linking, which carries no causal power over $\Sthree$. This both completes 373's
anti-solipsism result --- privacy is real \emph{and} affectable, the shared arena is enforced
\emph{and} action-only --- and identifies precisely the thing the solipsist wrongly hopes for: a
private lever on the shared world, which is exactly what the topology forbids.
\end{corollary}

\section*{Status}
The Hopf bundle carries two regimes of efficacy. The private fiber is disjoint from every other (no
causal contact, Theorem~\ref{thm:linked}) yet linked to it with linking number $1$ --- acausal
correlation, the structural form of ``synchronicity'' (Remark~\ref{rmk:synch}), and a layer one can
affect (Proposition~\ref{prop:efficacy}, L3). The shared $\Sthree$ changes only by action through
$\Sthree$ (373~E3, L4); the linking is correlation, not leverage, so forcing the shared world without
action is topologically blocked (Corollary~\ref{cor:line}). Correlation is not causation. This completes
the anti-solipsism result of 373: privacy is real and affectable, the shared arena is enforced and
action-only, and synchronicity is the linking --- not a violation of E3. \textbf{Theorem-grade as
topology} (linking $=1$; the sole $\Sthree$ channel of 373); the hinges are monad $=$ observer and the
reading of the linking as experienced synchronicity --- the corpus permits acausal correlation
structurally and asserts no paranormal phenomenon. No published number changes; no new objects beyond
the linking fact and the two-regime split. Verifier: \texttt{verify\_P374.py}, 9/9 pass.

\end{document}
