\documentclass[11pt]{article}
% Addendum 384 (pulls the inertia thread to the equivalence principle; assembles 383 + 372/P04/P17).
% Gravity is geometry (fold/C-collapse, P04/P17/372-R1, not a force), so a body coasts inertially (383)
% through it -- free-fall is the inertial coast (a geodesic, weightless); curvature mimics attraction
% with zero force; inertial mass = gravitational mass structurally (one invariant, one geometry).
% Honest: numerical equivalence is a floor (372 R3); corpus had no explicit EP -- new connection.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P384.py (8/8 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}

\title{Addendum 384: The Equivalence Principle\\
\large Gravity is geometry, so free-fall is the inertial coast ---\\
inertial mass $=$ gravitational mass, structurally\\[0.3em]
\normalsize (Assembles Addendum 383 with Papers 04, 17 and Addendum 372)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~383 gave inertia --- the conserved-invariant coast. The corpus separately has gravity as the
fold/self-lensing geometry, the $C$ (collapse) operator (Paper~04's ``Gravity as Collapse,'' Paper~17,
and Addendum~372's R1: the fold-geometry of gravity is theorem-grade and observation-free), i.e.\
gravity is geometry, \emph{not} a force. The corpus has these two pieces but never assembled them. This
addendum does, and the assembly is exactly the equivalence principle. \textbf{(1)} Gravity is geometry,
not force. \textbf{(2)} By 383 a body's natural motion is the inertial coast of its conserved invariant;
through the gravity-geometry that coast is the geodesic, so a freely-falling body coasts inertially and
feels \emph{no} force --- it is weightless. Free-fall \emph{is} inertial motion. \textbf{(3)} Curvature
mimics attraction with zero force: two geodesics parallel on a positively curved space (the boundary
$S^3$) converge --- on the $2$-sphere two meridians at separation $d_0\cos(\text{lat})\to0$ at the pole
--- so ``gravitational attraction'' is geodesic convergence, geometry not force. \textbf{(4)} The
invariant that coasts (inertial mass, 383) is the \emph{same} conserved invariant moving through the
\emph{same} fold geometry that \emph{is} gravity (P04/372): one invariant, one geometry, so inertial and
gravitational mass are the same object --- structurally equal. \textbf{(5)} The ``weight'' felt at rest
is the surface ($S^3$) pushing the body off its geodesic, a real normal force; gravity is what one feels
the \emph{absence} of in free-fall. \textbf{Honest scope:} the gravitational coupling magnitude is
refuted/typed (372 R3), so the \emph{numerical} equivalence (E\"otv\"os, $\sim\!10^{-13}$) is not
derived --- only the structural one; and the corpus had no explicit equivalence-principle treatment, so
this is a new connection, a structural reading. All claims pinned by \texttt{verify\_P384.py} (8/8).
\end{abstract}

\section{Free-fall is the inertial coast}
\label{sec:freefall}

\begin{proposition}[Gravity is geometry; free-fall is inertial]
\label{prop:freefall}
Gravity, in the corpus, is the fold/self-lensing geometry --- the $C$ (collapse) operator (Paper~04,
Paper~17), whose fold-geometry is theorem-grade and observation-free (Addendum~372, R1: the black hole
is degenerate self-lensing, mapping to Schwarzschild). There is no gravitational force term; the
``pull'' is the shape of the geometry. By Addendum~383 a body's natural motion is the inertial coast of
its conserved invariant. Through the gravity-geometry that coast is the geodesic: a freely-falling body
coasts inertially along the fold geometry and feels no force --- it is weightless. Free-fall is
therefore inertial motion, which is the core of the equivalence principle.
\end{proposition}

\begin{theorem}[Curvature mimics attraction with zero force]
\label{thm:focus}
On a positively curved space two geodesics that start parallel converge (geodesic focusing) with no
force between them. On the unit $2$-sphere, two meridians parallel at the equator have metric separation
$d_0\cos(\mathrm{lat})$, which decreases monotonically to $0$ at the pole: the bodies ``fall together''
by curvature alone. Since the corpus boundary $S^3$ is positively curved, ``gravitational attraction''
is geodesic convergence --- geometry, not force. The attraction is real and the force is absent
simultaneously, which is precisely the equivalence principle's geometric content.
\end{theorem}

\section{Inertial mass is gravitational mass}
\label{sec:masses}

\begin{corollary}[Structural equivalence of the two masses]
\label{cor:masses}
The invariant that coasts --- the inertial mass of Addendum~383 --- is the same conserved invariant
moving through the same fold geometry that \emph{is} gravity (Paper~04/372): the gravitational mass.
One invariant, one geometry: the two masses are the same object, hence equal. This is the equivalence of
inertial and gravitational mass, read structurally. Relatedly, the ``weight'' felt at rest on a surface
is the surface (the boundary $S^3$) pushing the body off its geodesic --- a real normal force --- so
weight is what one feels the absence of in free-fall, not gravity itself.
\end{corollary}

\begin{remark}[Honest scope: the magnitude floor and the new connection]
\label{rmk:scope}
Two limits. \emph{(i)} The gravitational coupling \emph{magnitude} is refuted/typed (Addendum~372, R3:
Paper~17's $\mu_1/3\pi/20$ derivation of $G/M_{\mathrm{Pl}}$ is refuted in Paper~40), so the
\emph{numerical} equivalence of inertial and gravitational mass --- the E\"otv\"os result at
$\sim\!10^{-13}$ --- is not derived. Only the \emph{structural} equivalence is: gravitational motion is
inertial motion because gravity is geometry. Form derived, magnitude a floor, the same posture as the
$H_0$ work. \emph{(ii)} The corpus has no explicit equivalence-principle, geodesic, or free-fall
treatment; this addendum assembles the principle from existing pieces (gravity is geometry, 372/P04;
inertia is the coast, 383). It is a structural reading, not a previously-stated corpus theorem.
\end{remark}

\section*{Status}
Pulling the inertia thread of Addendum~383 reaches the equivalence principle, which the corpus had the
pieces for but never assembled. Gravity is the fold-geometry / the $C$-collapse operator
(Paper~04/Paper~17/372-R1), not a force; by 383 a body coasts inertially through it, so free-fall is the
inertial coast --- a geodesic, weightless (Proposition~\ref{prop:freefall}). Curvature mimics attraction
with zero force: geodesics converge on the positively curved $S^3$ (Theorem~\ref{thm:focus}). The
invariant that coasts (inertial mass) is the same one the geometry bends (gravitational mass), so the two
are structurally equal (Corollary~\ref{cor:masses}); the weight felt at rest is the surface pushing the
body off its geodesic, not gravity. \textbf{Honest scope} (Remark~\ref{rmk:scope}): the numerical
equivalence is a floor (372's magnitude refutation), and the corpus had no explicit equivalence
principle --- this is the new connection, a structural reading. No published number changes. Verifier:
\texttt{verify\_P384.py}, 8/8 pass.

\end{document}
