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% Addendum 387 (studies the observer-pair seam; extends 373/374/377/385, P27/P38/P04).
% Case A (intra-fibre, antipodal -x=e^{i pi}x = the fold's Theta=pi void) = DARK ENERGY (cokernel
% Lambda_0=1-pi^2/32=0.69, matches observed). Case B (inter-fibre, two monads) = SYNCHRONICITY (the
% linking, 374). Different objects; fused only by the antipodality posit; data picks A.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P387.py (10/10 pass).
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\newcommand{\Lz}{\Lambda_0}

\title{Addendum 387: The Observer-Pair, Two Readings\\
\large The antipodal self-split is dark energy; the two-monad relation is\\
synchronicity --- different objects, fused only by a posit\\[0.3em]
\normalsize (Studies a seam in 373/374; extends 377, 385, Papers 27, 38, 04)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
A probe asked whether the ``observer pair'' --- whose relation was identified with dark energy --- is
\emph{(A)} one Hopf fibre's two antipodal poles, or \emph{(B)} two distinct fibres (two monads); the
corpus calls both ``dark energy'' (Paper~27's consensus overhead; Paper~38's cokernel $\Lz$) without
forcing them equal. Studying both resolves the seam: they are \emph{different objects}.
\textbf{Case~A (intra-fibre, antipodal).} The pair is $x$ and $-x$; on $S^3\subset\mathbb C^2$,
$-x=e^{i\pi}x$, so $-x$ is the global-phase-$\pi$ point on $x$'s \emph{own} Hopf fibre, and the phase
$\pi$ is the fold's half-winding $\Theta=\pi$ --- Paper~04's horizon, the void, Addendum~377's lapse-zero
bounce point. The residual of this self-split is the cokernel void $\Lz=1-\pi^2/32=0.6916$, which
\emph{matches observed dark energy} $\Omega_\Lambda\approx0.69$. \textbf{Case~B (inter-fibre, two
monads).} The pair is two distinct fibres (373/374): disjoint and linked with linking number one. Its
residual is the linking --- which Addendum~374 already identified as \emph{synchronicity} (acausal
correlation), not dark energy; the ``shared'' boundary fraction $\pi^2/\alpha^{-1}\approx0.072$ is not
$\Omega_\Lambda$ either. \textbf{Resolution.} Case~A is dark energy (the antipodal void, matching data);
Case~B is synchronicity (the linking). They coincide \emph{only} under the antipodality posit --- that
the consensus-pair \emph{is} its own antipode (``back to back'') --- which fuses Paper~27's consensus
(needs two observers) with Paper~38's cokernel (the antipodal void) and makes consensus
self-reconciliation across the void: dark energy as the cost of an observer agreeing with its own
antipode. The data select Case~A. All claims pinned by \texttt{verify\_P387.py} (10/10).
\end{abstract}

\section{Case A: the antipodal self-split is dark energy}
\label{sec:A}

\begin{proposition}[The antipode is the void-pole of the fold]
\label{prop:A}
On $S^3\subset\mathbb C^2$ the antipode of $x$ is $-x=e^{i\pi}x$: the point at global phase $\pi$ on
$x$'s own Hopf fibre. So $x$ and $-x$ are two poles of \emph{one} fibre, and the separating phase is
$\pi$, the fold's half-winding $\Theta=\pi$ --- the locus $\det(DF_\Theta)=0$ of Paper~04 (the event
horizon), the void, the lapse-zero at which Addendum~377 places the bounce. The antipode is the
void-pole of the fold; the antipodal self-split is the split between the bulk-facing and void-facing
poles of a single observer.
\end{proposition}

\begin{theorem}[Case A $=$ dark energy]
\label{thm:A}
The residual of the antipodal self-split is the cokernel void: the corner residual
$\Lz=1-\pi^2/32=0.6916$ (Paper~38/P32), the measure of what the embedded $B^4$ fails to cover. This
matches the observed dark-energy fraction $\Omega_\Lambda\approx0.69$ (Planck). So under the intra-fibre,
antipodal reading the observer-pair's residual \emph{is} dark energy, and it is the void at $\Theta=\pi$.
\end{theorem}

\section{Case B: the two-monad relation is synchronicity}
\label{sec:B}

\begin{theorem}[Case B $=$ the linking $=$ synchronicity, not dark energy]
\label{thm:B}
Under the inter-fibre reading the pair is two \emph{distinct} Hopf fibres (two monads, 373/374): disjoint
and linked, with linking number one. This is Paper~27's second-observer / consensus relation. But its
residual is the \emph{linking}, which Addendum~374 already identified as synchronicity --- acausal
correlation between distinct private viewpoints --- not the cosmological constant. The ``shared''
boundary fraction $\pi^2/\alpha^{-1}\approx0.072$ (Paper~27's consensus layer) is likewise not
$\Omega_\Lambda$. So Case~B's geometry yields a different object and a different number than Case~A.
\end{theorem}

\section{Resolution: different objects, fused only by a posit}
\label{sec:resolution}

\begin{corollary}[The seam resolves; the data select Case A]
\label{cor:resolve}
The two readings are not the same dark energy --- they are different objects. Case~A (antipodal) is dark
energy: the cokernel $\Lz=0.69$, the void, matching observation. Case~B (two distinct monads) is
synchronicity: the linking of Addendum~374. They coincide \emph{only} under the antipodality posit ---
that the consensus-pair is its own antipode, ``back to back.'' That posit is the extra assumption that
fuses Paper~27's consensus (which requires two observers) with Paper~38's cokernel (the antipodal void):
under it, consensus reduces to \emph{self}-reconciliation across the void, and dark energy is the cost of
an observer agreeing with its own antipode. Without the posit, the two part ways, and the data select
Case~A --- $\Lz=0.69$ is the observed dark energy; the two-monad linking is synchronicity, a different
thing.
\end{corollary}

\begin{remark}[Honest scope, and a tarpit avoided]
\label{rmk:scope}
Three flags. The antipodality (``back to back'') is a \emph{posit}, not a theorem --- it is the
assumption that makes Paper~27's consensus and Paper~38's cokernel the same dark energy; the corpus does
not force it, and this addendum marks it as the load-bearing choice. The originating ``four stable
points, one per tier'' framing (electron / carbon / water / observer) is a \emph{heuristic}, not canon
(the electron is Paper~07 and water is Paper~09, but the tiering is not a derived result); the strong
conclusion --- observer-relation $=$ dark energy $=\Lz$ --- is Paper~27 with Paper~38 and does not need
the tiering. Finally, a tarpit avoided: $S^3/\{\pm1\}=\mathbb{RP}^3$ is orientable (odd dimension), so
the back-to-back antipodal structure forces no non-orientable object --- no Klein bottle, no cross-cap;
the pair stays orientable.
\end{remark}

\section*{Status}
Studying both readings of the observer-pair resolves the seam by separating two objects the question had
fused. Case~A (intra-fibre, antipodal: $-x=e^{i\pi}x$, the global-phase-$\pi$ pole on one fibre $=$ the
fold's $\Theta=\pi$ void, Proposition~\ref{prop:A}) is dark energy --- the cokernel $\Lz=1-\pi^2/32=0.69$,
matching observed $\Omega_\Lambda$ (Theorem~\ref{thm:A}). Case~B (inter-fibre, two distinct monads,
disjoint and linked) is synchronicity --- the linking of Addendum~374 --- not the cosmological constant
(Theorem~\ref{thm:B}). They coincide only under the antipodality posit, which fuses Paper~27's consensus
with Paper~38's cokernel and reads consensus as self-reconciliation across the void; the data select
Case~A (Corollary~\ref{cor:resolve}). Honest scope (Remark~\ref{rmk:scope}): the antipodality is a posit,
the four-stable-points tiering a heuristic, and $\mathbb{RP}^3$ orientable. So the observer-relation \emph{is}
dark energy under the antipodal reading (and is the void at $\Theta=\pi$), while the two-monad relation is
the synchronicity-linking of 374 --- the two were never one dark energy. No published number changes
beyond confirming $\Lz=0.69$ against $\Omega_\Lambda$. Verifier: \texttt{verify\_P387.py}, 10/10 pass.

\end{document}
