\documentclass[11pt]{article}
% Addendum 372 (sharpens 369/370; extends P04/P17/P35/P40).
% Peels 'gravity = self-lensing' from 'self-lensing = self-observation': the black-hole rung is a
% theorem and observation-free (P04), but the coupling MAGNITUDE is refuted (P17/P40) and typed
% (P35) -- so the closure hypothesis does not upgrade to a theorem via the physics half. The H0
% residual is re-located, not re-derived. Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P372.py (9/9 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\mathrm{breath}}}
\newcommand{\DF}{\det(DF_\Theta)}

\title{Addendum 372: The Physics Half of the Closure Hypothesis\\
\large How much of ``gravity $=$ self-lensing'' is a theorem once the\\
observation reading is peeled --- and what it does to the $H_0$ residual\\[0.3em]
\normalsize (Addendum to Papers 04, 17, 35, 40; sharpens 369/370)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~370 traced the v4 closure hypothesis --- that $GM_b$ couples through the breath map --- to
``gravity is self-observation,'' the foundational observer-interpretation Paper~04 carries as
not-theorem, and called the residual a category boundary. That framing bundled two separable claims:
the \emph{geometry} of gravity (gravity $=$ self-lensing $=$ the fold) and the \emph{consciousness}
reading (self-lensing $=$ observation). This addendum peels them and asks whether the physics half is
a theorem independent of the observation reading. It decomposes ``gravity $=$ self-lensing'' into
three rungs with distinct status. \textbf{(R1, P04, theorem, observation-free)} the black hole
\emph{is} degenerate self-lensing: the event horizon is the $2$-sphere where $\DF=0$ (the Hopf
half-winding $\Theta=\pi$), the interior is the retrograde $\DF<0$, the singularity is the fixed
point $F_\Theta(\varnothing)=\varnothing$ --- and this maps exactly onto Schwarzschild, the
$(1-2M/r)$ sign-change at $r=2M$ being the coordinate face of the Jacobian sign-change. No
consciousness reading enters; \emph{one rung of ``gravity $=$ self-lensing'' is theorem-grade and
fully peeled from the observer-interpretation.} \textbf{(R2, P17, structural)} gravity $=$ the $C$
(collapse) operator of the C$\circ$P breath; survives, qualitative. \textbf{(R3, P17, refuted)} the
quantitative coupling \emph{magnitude} $G/M_{\mathrm{Pl}}$ from the first moment $\mu_1$ via the
$3\pi/20$ spectral formula is \emph{refuted} in Paper~40 (P017\_2, P017\_3\_c: the radial Dirac
spectrum is not recovered on recomputation). \textbf{Consequence:} the closure hypothesis needs the
coupling \emph{magnitude} ($a_0$), which is R3 --- refuted as a derivation, typed by Paper~35 --- not
R1; so the physics half does \emph{not} upgrade the closure hypothesis to a theorem. \textbf{The
yield is a sharpening of 370:} the $H_0$ residual splits into an interpretive
\emph{existence-of-coupling} part (370's category boundary, where the observation reading genuinely
lives) and a named \emph{physics-floor magnitude} part ($a_0=\alpha$, P17-refuted / P35-typed) that
does \emph{not} require the consciousness reading. $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc is unchanged ---
this re-locates the residual, it does not move the number. All claims pinned by
\texttt{verify\_P372.py} (9/9).
\end{abstract}

\section{Three rungs of ``gravity $=$ self-lensing''}
\label{sec:rungs}

\begin{theorem}[R1: the black-hole rung is a theorem, and observation-free]
\label{thm:R1}
Paper~04 proves the black hole \emph{is} degenerate self-lensing: the forward fold
($\Theta\in[0,\pi)$) is a diffeomorphism with $\DF>0$; at the Hopf half-winding $\Theta=\pi$ the
outward projection and inward passage become indistinguishable and $\DF=0$, defining the event
horizon $\mathcal H$; for $\Theta\in(\pi,2\pi)$ the fold is retrograde, $\DF<0$, and the only fixed
point is the origin $\varnothing$ (the singularity). This is the coordinate content of Schwarzschild:
$(1-2M/r)$ changes sign at $r=2M$, the $r\leftrightarrow t$ swap being precisely the Jacobian
sign-change. The argument is entirely $\DF$ Jacobian-sign geometry; the consciousness reading does
not enter. Hence this rung of ``gravity $=$ self-lensing'' is theorem-grade and peeled from the
observer-interpretation.
\end{theorem}

\begin{proposition}[R2--R3: collapse survives, the magnitude is refuted]
\label{prop:R2R3}
\emph{R2 (structural, survives).} Paper~17's ``Gravity as Collapse'' identifies gravity with the $C$
(inward-fall) operator of the C$\circ$P breath (Paper~27); bulk propagation via $D_{B^4}$ transmits
it. Qualitative, not a derived coupling. \emph{R3 (refuted).} Paper~17's quantitative coupling
magnitude --- $G/M_{\mathrm{Pl}}$ from the first moment $\mu_1$ through the $3\pi/20$ spectral formula
--- is refuted in Paper~40: P017\_2 (the factor-of-seven inter-level gap) and P017\_3\_c (the radial
Dirac spectrum, ``refuted-as-stated,'' not recovered on recomputation). The geometric \emph{derivation
of the coupling magnitude} fails.
\end{proposition}

\section{Consequence and the sharpening of 370}
\label{sec:sharpen}

\begin{corollary}[The physics half does not upgrade the closure hypothesis]
\label{cor:noupgrade}
The closure hypothesis requires the coupling \emph{magnitude} $a_0$ --- the strength with which $GM_b$
sources the breath map. That magnitude is R3 (a refuted derivation, typed by Paper~35's ``v4 closure
hypothesis''), not R1 (which is the fold-\emph{geometry} of gravity). Therefore the theorem-grade,
observation-free content (R1) does not reach the coupling strength, and ``gravity $=$ self-lensing''
cannot discharge the closure hypothesis as a theorem.
\end{corollary}

\begin{remark}[The residual splits; 370 is sharpened, not overturned]
\label{rmk:split}
Addendum~370 attributed the whole $H_0$ residual to the observer-interpretation. The peel shows it
splits. \emph{(i) Existence of the coupling} --- that mass sources the fold at all --- is the
interpretive part, 370's category boundary, where the observation reading genuinely lives.
\emph{(ii) Magnitude of the coupling} --- $a_0=\alpha$, the strength --- is a \emph{named physics
floor}: Paper~17's geometric derivation of it is refuted (Paper~40) and Paper~35 types it. The
consciousness reading is \emph{not} required for (ii); it is a floor in the $a_0=\alpha$ family, the
same posture as 366/369. And by Theorem~\ref{thm:R1} the fold-geometry of gravity is theorem-grade
and observation-free. So 370's verdict survives for the existence-of-coupling, while the hardest
\emph{quantitative} residual is re-located from ``the observer-interpretation'' to ``a typed physics
coupling whose geometric derivation was refuted.''
\end{remark}

\section*{Status}
The frontier 370 surfaced --- whether ``gravity $=$ self-observation'' can move from interpretation to
theorem --- yields a real but bounded result. Peeling ``gravity $=$ self-lensing'' from
``self-lensing $=$ observation'': the black-hole rung R1 (Theorem~\ref{thm:R1}, $\DF=0=$ Schwarzschild
horizon) is theorem-grade \emph{and observation-free}; the collapse identification R2 survives
structurally; but the coupling magnitude R3 (Paper~17's $\mu_1$ / $3\pi/20$) is \emph{refuted}
(Paper~40, Proposition~\ref{prop:R2R3}). Because the closure hypothesis needs the magnitude
(Corollary~\ref{cor:noupgrade}), the physics half does \emph{not} upgrade it to a theorem.
\textbf{The honest yield} (Remark~\ref{rmk:split}): the $H_0$ residual splits into an interpretive
existence-of-coupling part (370's category boundary) and a named physics-floor magnitude part
($a_0=\alpha$, P17-refuted / P35-typed) that does not require the observer-interpretation --- with the
fold-geometry of gravity itself theorem-grade and observation-free. $H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc
is unchanged: this re-locates the residual, it does not move the number, and it manufactures no
closure. No published number changes; no new objects beyond the rung decomposition. Verifier:
\texttt{verify\_P372.py}, 9/9 pass.

\end{document}
