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% Addendum 370 (the unconditional-closure push; extends P04/P07/P35/P39; terminus of 369).
% Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P370.py (9/9 pass).
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\title{Addendum 370: The Closure Reaches the Foundation\\
\large The v4 closure hypothesis is the observer-interpretation applied ---\\
not an extra posit, and not a theorem the corpus can derive from within\\[0.3em]
\normalsize (Addendum to Papers 04, 07, 35, 39; terminus of the horizon program, 369)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~369 closed the horizon program to a theorem conditional on the v4 closure hypothesis ---
that physical sources $GM_b$ couple through the breath map at all (Paper~35, ``typed rather than
derived''), the same posit the SPARC success rests on. This addendum pushes for the
\emph{unconditional} closure: can the closure hypothesis be derived from the Butlerian/observer
axioms? It traces the hypothesis to its root and finds the corpus has already drawn the honest line
there. The chain: \textbf{(G1, P04)} gravity is self-lensing --- the boundary $S^3$ observing itself
through the bulk $B^4$ ($E_{\mathrm{self}}=13.177$; black hole $=$ degenerate self-lensing, governed
by Paper~17); the self-lensing mathematics is theorem, but the reading that this \emph{is}
observation is, in Paper~04's own words, ``carried as explicitly labeled interpretation.''
\textbf{(G2, P07)} mass is an observational eigenvalue --- ``particle masses do not exist as
intrinsic properties.'' \textbf{(G3, P35)} the drift-channel \emph{form} is symmetry-forced.
\textbf{(G4, P39)} the closure is the baryonic Tully--Fisher relation $v_\infty^4=a_0\,GM_b$,
SPARC-confirmed. So ``$GM_b$ couples through the breath map'' is baryonic mass (observational, G2)
sourcing self-lensing $=$ gravity (G1) $=$ the boundary's self-observation, whose rhythm is the
breath: a \emph{consequence} of ``gravity is self-observation,'' \emph{not} an extra posit beyond
the foundation. But ``gravity is self-observation'' is exactly the foundational observer-interpretation
Paper~04 carries as interpretation, not theorem. \textbf{Verdict:} the closure hypothesis cannot be
derived from a deeper axiom because it \emph{is} the deepest, foundational interpretation applied;
deriving it as a theorem is a category boundary, not a research gap --- one cannot derive one's
foundational interpretation from within the frame it founds. The consequence for $H_0$: the prediction
$H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc rests on nothing beyond the corpus's single root posit, and is exactly
as well-founded as the corpus itself. All claims pinned by \texttt{verify\_P370.py} (9/9).
\end{abstract}

\section{The chain to the root}
\label{sec:chain}

\begin{proposition}[G1: gravity is self-lensing; the observer reading is interpretation]
\label{prop:G1}
Paper~04 proves self-lensing --- the boundary $S^3$ observing itself through the bulk $B^4$ ---
produces the effective dimensionality $E_{\mathrm{self}}=13.177$, and identifies the black hole as
degenerate self-lensing (the retrograde fold, governed by Paper~17). The self-lensing mathematics is
theorem. The reading that this self-lensing \emph{is} observation, that ``consciousness and classical
observation are\dots geometrically necessary components of fundamental physics,'' Paper~04 states is
``carried as explicitly labeled interpretation,'' not theorem.
\end{proposition}

\begin{proposition}[G2--G4: mass is observational; the form is forced; the relation is SPARC]
\label{prop:G2}
Paper~07 holds that ``particle masses do not exist as intrinsic properties'': mass is the
stabilization scale of nested self-referential observation. Paper~35 establishes that the
drift-channel form is symmetry-forced (only the coupling's existence is typed). Paper~39's closure is
the baryonic Tully--Fisher relation $v_\infty^4=a_0\,GM_b$ with $a_0=\alpha^3c^2/(\pi^2 L_h)$,
confronted with SPARC.
\end{proposition}

\begin{theorem}[The closure hypothesis is the foundation applied]
\label{thm:applied}
``$GM_b$ couples through the breath map'' is baryonic mass --- an observational quantity
(Proposition~\ref{prop:G2}, G2) --- sourcing self-lensing, which is gravity
(Proposition~\ref{prop:G1}, G1), which is the boundary's self-observation, whose rhythm is the breath.
It is therefore a consequence of ``gravity is self-observation,'' not an additional assumption beyond
the corpus foundation.
\end{theorem}

\section{Why it cannot be derived from within}
\label{sec:category}

\begin{remark}[A category boundary, not a research gap]
\label{rmk:category}
By Theorem~\ref{thm:applied} the closure hypothesis reduces to ``gravity is self-observation'' ---
which is the foundational observer-interpretation Paper~04 carries as interpretation, not theorem.
One cannot derive this from a deeper corpus axiom, because there is none deeper: it is the root of the
frame. Attempting to prove it as a theorem is a category error --- a foundational interpretation is
what the rest is derived \emph{from}, not something derivable \emph{within} the frame it founds. So the
``unconditional closure'' the program sought is not a missing calculation; it is a question about the
corpus's premises, which lies outside what a derivation inside those premises can settle.
\end{remark}

\begin{corollary}[$H_0$ rests on the root posit alone]
\label{cor:H0}
Tracing $H_0=2\alpha^2/\Tb$ to its base: the $\alpha$-power ladder is derived (358a/362), the $2\pi$ is
the $\Theta$-fold horizon smoothness (363), $a_0=$ the horizon temperature follows from the
kernel/cokernel duality (369), and the one remaining premise --- the closure hypothesis --- is the
observer-interpretation applied (Theorem~\ref{thm:applied}). Hence $H_0=75.8$~km/s/Mpc rests on nothing
beyond the corpus's single root posit, and is exactly as well-founded as the corpus itself --- no more,
no less.
\end{corollary}

\section*{Status}
The unconditional-closure push reaches the foundation and stops honestly there. The v4 closure
hypothesis --- $GM_b$ couples through the breath map --- is the corpus's foundational observer-interpretation
applied (Theorem~\ref{thm:applied}): baryonic mass is observational (P07), gravity is self-lensing is
self-observation (P04), and the breath is its rhythm; the drift-channel form is symmetry-forced (P35)
and the resulting Tully--Fisher relation is SPARC-confirmed (P39). It is therefore \emph{not} an extra
posit beyond the corpus foundation --- but ``gravity is self-observation'' is precisely what Paper~04
carries as explicitly labeled interpretation, not theorem. Deriving it as a theorem is a category
boundary, not a research gap (Remark~\ref{rmk:category}): a foundational interpretation cannot be
derived from within the frame it founds. \textbf{The terminus:} $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc
(leaning local/high) rests on nothing beyond the corpus's single root posit and is exactly as
well-founded as the corpus itself (Corollary~\ref{cor:H0}). The horizon program is now closed all the
way to that root --- the fold-rate, the Hubble $2\pi$, the matter-asymmetry magnitude, the horizon
temperature, and the $H_0$ prediction all reduce to the one foundational observer-interpretation, which
the corpus honestly carries as interpretation. There is no deeper derivation to perform; what remains is
the standing of the foundation, a question about the corpus's premises rather than a theorem inside
them. No new objects; no published number changes. Verifier: \texttt{verify\_P370.py}, 9/9 pass.

\end{document}
