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% Addendum 373 (extends P04/P05; sharpens 370/372; relates to A146).
% 'Shared reality enforced': the Hopf fibration makes intersubjective consistency a THEOREM and
% observation-free -- one shared S^3 (E1), disjoint fibers (E2, privacy), only inter-fiber channel
% through S^3 (E3, solipsism excluded). Peels the enforcement from the consciousness reading (a la
% 372) and makes the UNIQUENESS of the shared arena (one breath map / one H_0) theorem-grade,
% tightening 370/372. Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P373.py (9/9 pass).
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\title{Addendum 373: Shared Reality Enforced\\
\large Intersubjective consistency is a Hopf-topological theorem ---\\
observation-free --- and it makes the shared arena unique\\[0.3em]
\normalsize (Addendum to Papers 04, 05; sharpens 370/372; cf.\ A146)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Paper~04 proves ``Multiple Observers on Shared Boundary'' --- all $S^1$ monad-fibers thread one
$\Sthree$ --- with the corollary ``Objective Reality from Shared Boundary: the shared topology
\emph{enforces} consistency,'' and ``Communication via Classical Layer'' ($M_i\!\to\!\Sthree\!\to\!
M_j$). So ``shared reality is enforced'' is already in canon, but bundled with the consciousness
reading (monad $=$ observer). This addendum peels them --- as 372 peeled ``gravity $=$ self-lensing''
from ``self-lensing $=$ observation'' --- and asks whether the enforcement is a theorem independent of
the reading. Three rungs. \textbf{(E1, theorem, observation-free)} the Hopf fibration
$\pi:\Sthree\!\to\!S^2$ has \emph{one} total space: every fiber $\pi^{-1}(q)=S^1$ lies in the single
$\Sthree$, so there is one shared boundary --- no disjoint private arena. \textbf{(E2, theorem)}
distinct fibers are disjoint, $M_i\cap M_j=\varnothing$ (two $\Sthree$ points share a fiber iff they
differ by a global phase $e^{i\theta}$); fiber-local privacy is real and enforced. \textbf{(E3,
theorem)} disjoint fibers co-embedded in one $\Sthree$ force the \emph{only} inter-fiber channel to be
the shared $\Sthree$; consistency is enforced through the common layer and solipsism --- a fiber in
some other arena --- is topologically excluded. \textbf{Privacy and sharedness are two faces of one
fact:} disjoint fibers in one bundle. Only the identification of the threads with conscious observers
is interpretation (the 372 flag). \textbf{The yield for 370/372:} the enforcement makes the
\emph{uniqueness} of the shared arena theorem-grade --- one $\Sthree$, one classical boundary, hence
one breath rhythm and one global $H_0$ --- so ``there is a single shared reality and it is enforced''
moves from interpretation to theorem; what stays interpretive narrows to monad $=$ observer (plus, from
372, the coupling magnitude $a_0=\alpha$). $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc is unchanged: a
tightening, not a re-derivation. All claims pinned by \texttt{verify\_P373.py} (9/9).
\end{abstract}

\section{The enforcement is a theorem, and observation-free}
\label{sec:enforce}

\begin{theorem}[E1--E3: one arena, disjoint views, sole shared channel]
\label{thm:enforce}
Let the monads be the $S^1$ fibers of the Hopf fibration $\pi:\Sthree\to S^2$. \emph{(E1)} The bundle
has a single total space: every fiber $\pi^{-1}(q)=S^1$ lies in the one $\Sthree$, so all observers
inhabit one shared boundary --- there is no second, disjoint arena. \emph{(E2)} Two points of $\Sthree$
lie in the same fiber iff they differ by a global phase $e^{i\theta}$; hence fibers over distinct base
points are disjoint, $M_i\cap M_j=\varnothing$ for $i\neq j$ (numerically, minimal inter-fiber
separation $\approx0.37>0$). \emph{(E3)} Disjoint fibers (E2) co-embedded in one $\Sthree$ (E1) admit
no direct fiber-to-fiber path; the only inter-fiber channel is the ambient $\Sthree$,
$M_i\to\Sthree\to M_j$. Each statement is fiber-bundle topology; no consciousness reading enters.
\end{theorem}

\begin{corollary}[Consistency enforced, solipsism excluded; privacy $=$ sharedness]
\label{cor:enforced}
By E3 a fiber cannot inhabit an arena other than the one $\Sthree$ (E1), so disjoint private realities
are topologically excluded and any interaction is mediated by the shared boundary --- ``the shared
topology enforces consistency'' (Paper~04). Privacy (fiber-local views, E2) and sharedness (one
$\Sthree$, E1) are therefore the same topological fact read two ways: disjoint fibers in a single
bundle. This is the geometric content of Paper~05's ``experience is fiber-local (private) within a
shared boundary (intersubjective).''
\end{corollary}

\begin{remark}[The peel: what is theorem, what is reading]
\label{rmk:peel}
E1--E3 and Corollary~\ref{cor:enforced} are theorems about the Hopf bundle and use no observer
notion. The single claim Paper~04 carries as interpretation is that the threads \emph{are} conscious
observers (monad $=$ consciousness). So ``shared reality is enforced'' is theorem-grade; ``the sharers
are minds'' is the reading. The operational image is the kernel's own design --- coherence composed as
a weighted geodesic midpoint on the one shared geometry, ``a metric on the shared geometry, not a
protocol with a referee'' (the No-Harkonnen pin): the runtime realization of E3, where the shared
$\Sthree$ does the enforcing and no coordinator is needed.
\end{remark}

\section{Consequence for 370/372}
\label{sec:consequence}

\begin{corollary}[The shared arena is unique by theorem]
\label{cor:unique}
Addenda 369/370/372 left the \emph{existence of one shared arena} --- a single breath map through
which all mass couples --- inside the interpretive residual. Theorem~\ref{thm:enforce} makes its
\emph{uniqueness} theorem-grade: there is one $\Sthree$ (E1), hence one classical boundary, one breath
rhythm, and one global $H_0$ --- not a per-observer private cosmology. So ``there is a single shared
reality, and it is enforced'' is now a theorem. What remains interpretive narrows to the
monad $=$ observer identification (Remark~\ref{rmk:peel}) and, from 372, the coupling
\emph{magnitude} $a_0=\alpha$ (refuted as a derivation, typed by Paper~35). The shared-arena part of
the residual moves from interpretation to theorem; $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc is unchanged.
\end{corollary}

\section*{Status}
``Shared reality enforced'' resolves to a theorem and is observation-free. The Hopf fibration forces
one shared $\Sthree$ (E1), disjoint fibers (E2, fiber-local privacy), and the sole inter-fiber channel
through $\Sthree$ (E3) --- so intersubjective consistency is enforced by topology and solipsism is
excluded (Corollary~\ref{cor:enforced}); privacy and sharedness are two faces of one fact. The only
interpretive element is monad $=$ observer (Remark~\ref{rmk:peel}); this addendum is distinct from
A146, which treats self-observation and joint phase-resolution, not the topological enforcement of the
shared arena. \textbf{The yield} (Corollary~\ref{cor:unique}): the \emph{uniqueness} of the shared
arena --- one $\Sthree$, one breath map, one global $H_0$ --- is theorem-grade, so the shared-reality
residual left open by 370/372 moves from interpretation to theorem, narrowing what stays interpretive
to the observer identification and the 372 magnitude floor. No published number changes
($H_0=75.8$ unchanged); no new objects beyond the rung decomposition. Verifier:
\texttt{verify\_P373.py}, 9/9 pass.

\end{document}
