diff --git a/spacetime_penrose_area_charge_v2.txt b/spacetime_penrose_area_charge_v2.txt
index 9dfb2b0..57f0b9d 100644
--- a/spacetime_penrose_area_charge_v2.txt
+++ b/spacetime_penrose_area_charge_v2.txt
@@ -47,26 +47,6 @@
 \numberwithin{equation}{section}
 \allowdisplaybreaks[1]
 \begin{document}
-\title{Spacetime Penrose inequalities: enclosing area, charge, and rigidity}
-\author{A self-contained account}
-\date{2026}
-\maketitle
-
-\begin{abstract}
-The Penrose inequality bounds the ADM mass of an initial data set from below
-by the size of the region inside a trapped boundary. This paper proves sharp
-versions of that inequality with an enclosing-area infimum, in three and four
-spatial dimensions and in an asymptotically anti-de Sitter class, together
-with the equality cases. In three dimensions the result covers arbitrary
-second fundamental form, nonzero ADM momentum, finitely many ends, a
-disconnected weakly future trapped boundary, and total electric and magnetic
-charge; in four dimensions it covers the same generality. Equality in the
-asymptotically flat cases forces the original exterior to be a spacelike
-slice of Schwarzschild or Schwarzschild--Tangherlini spacetime. In the
-anti-de Sitter class we identify the equality branch exactly. Every analytic
-input is proved in full.
-\end{abstract}
-
 \section{Introduction}\label{sec:intro}
 
 When a massive star collapses, it may form a black hole. Penrose's 1973
@@ -177,7 +157,7 @@ area, forcing one. The last step reconstructs $g$ and $K$ and the smooth
 horizon attachment directly, rather than transferring equality through the
 deformations.
 
-For finite-perimeter minimizers we use the full obstacle perimeter, so that
+For finite-perimeter minimizers, the full obstacle perimeter is used, so that
 any boundary-coincident part is counted. Compactness of all minimizing
 enclosures then makes the one-sided metric derivative the minimum of their
 area derivatives. A positive measure on this minimizing space yields the
@@ -186,7 +166,7 @@ the area term to the original horizon.
 
 \subsection{External inputs}\label{sec:external}
 
-Three classical results are used without proof. We state them precisely
+Three classical results are used without proof. They are stated precisely
 once here, so that a reader can tell where the paper's original
 contributions end and the classical machinery begins.
 
@@ -257,7 +237,7 @@ times a power of $r$. The ADM energy and momentum are
 \end{equation}
 where $S_r$ are the coordinate spheres, $\nu$ their outward Euclidean
 unit normal, and the index $i$ is summed. The limits exist by
-\eqref{eq:decay}. We write $P_\delta$ for the momentum covector paired
+\eqref{eq:decay}. $P_\delta$ denotes the momentum covector paired
 with the coordinate frame. When $E>|P|_\delta$, the invariant mass of
 the end is $m=\sqrt{E^2-|P|_\delta^2}$.
 
@@ -288,7 +268,7 @@ An \emph{enclosing cut} for the chosen end is a compact smooth embedded
 filled side $D^+(\Gamma)\subset\bar\Omega$ containing the entire distant
 part of that end. The complement $\bar\Omega\setminus D^+(\Gamma)$
 contains $B$. Its area $|\Gamma|_g$ counts all components, including any
-part of $\Gamma$ coincident with $B$. We define
+part of $\Gamma$ coincident with $B$. Define
 \begin{equation}\label{eq:amin}
   a_g(B)=\inf\bigl\{|\Gamma|_g:\Gamma\text{ an enclosing cut}\bigr\}.
 \end{equation}
@@ -297,7 +277,7 @@ Section~\ref{sec:perimeter}.
 
 \subsection{Outermostness and area-minimality}\label{sec:outermost}
 
-We impose two conditions only in the equality theorem.
+Two conditions are imposed only in the equality theorem.
 
 \emph{Outer area-minimality.} $B$ is outer area-minimal when
 $|\Gamma|_g\ge|B|_g$ for every enclosing cut $\Gamma$.
@@ -309,7 +289,7 @@ component of $B$.
 
 \subsection{Regularity and finiteness}\label{sec:fin}
 
-The hypotheses \eqref{eq:decay} with $q>\tfrac12$ are the standing
+The hypotheses \eqref{eq:decay} are the standing
 assumption in three dimensions; in four dimensions the analogous
 $O_2$ and $O_1$ decay with the same exponent is assumed. Hypersurface
 regularity and boundary expansions are always as stated in the
@@ -318,7 +298,7 @@ respect to the given $g$.
 
 \subsection{Normalizations}
 
-We use $c=G=1$ and the coefficients $16\pi$ in \eqref{eq:constraints}
+Throughout, $c=G=1$ and the coefficients $16\pi$ are used in \eqref{eq:constraints}
 and \eqref{eq:amin}. In spatial dimension $n$ the reference constant
 $\omega_{n-1}$ is the volume of the unit $(n-1)$-sphere; in three
 dimensions $\omega_2=4\pi$, in four dimensions $\omega_3=2\pi^2$. A
@@ -464,7 +444,7 @@ Let $\eps\downarrow0$ to conclude.
 \section{Strictification and the conformal constraint formulas}
 \label{sec:strictification}
 
-To apply Theorem~\ref{thm:n3:strict-exterior} we need data in strict form:
+To apply Theorem~\ref{thm:n3:strict-exterior}, data in strict form is needed:
 $K$ compactly supported, $\theta_+(B)<0$, and a positive density gap. The
 construction uses a drift-Poisson equation that supplies a strict
 supersolution to the conformal Laplacian, following the conformal
@@ -498,7 +478,7 @@ $R_{u^4g}=u^{-4}(R_g-8\Delta_g\log u-8|d\log u|_g^2)$, while the
 constraint quadratic term scales with an extra factor of $u^{-8}$ because
 the extrinsic curvature is weighted by $u^2$:
 $u^4((\tr_{g_u}K_u)^2-|K_u|^2_{g_u})=u^{-8}((\tr_gK)^2-|K|_g^2)$.
-Adding the two contributions gives \eqref{eq:conformal-mu}. With $\pi=K-(\tr_gK)g$, we have $\pi_u=u^2\pi$ and
+Adding the two contributions gives \eqref{eq:conformal-mu}. With $\pi=K-(\tr_gK)g$, $\pi_u=u^2\pi$ and
 $\tr_g\pi=-2\tr_gK$; contracting the connection difference in dimension
 three gives
 $u^2J_u=J+2\pi(\grad\log u,\cdot)-(\tr_g\pi)d\log u
@@ -545,7 +525,7 @@ on the end. The drift-Poisson equation
 \partial_\nu\phi=-1,\quad\phi\to0,
 \end{equation}
 is uniformly elliptic and smooth even at critical points of $\phi$
-because of $\zeta$. We claim it has a smooth nonnegative solution with
+because of $\zeta$. There is a smooth nonnegative solution with
 $\phi=O_2(r^{-1})$.
 
 \emph{Existence.} Truncate at a distant sphere $S_R$ with outer
@@ -812,7 +792,7 @@ shrunk to small amplitude.
 \section{The coupled elliptic comparison system}
 \label{sec:elliptic}
 
-We fix a two-color exterior $\Omega_0$ of Proposition~\ref{prop:n3:two-color}.
+Fix a two-color exterior $\Omega_0$ of Proposition~\ref{prop:n3:two-color}.
 Its boundary $B_0=B_{\rm black}\cup B_{\rm white}$ carries two strict
 negative boundary expansions; and its far end is asymptotically flat at
 infinity.
@@ -1037,7 +1017,7 @@ to $g-\delta$.
 \section{The Riemannian Penrose inequality}
 \label{sec:riemannian-penrose}
 
-We use the following classical result without proof.
+The following classical result is used without proof.
 
 \begin{theorem}[Riemannian Penrose, dimension three; Bray]
 \label{thm:riemannian-penrose}
@@ -1209,11 +1189,11 @@ original data gives the conclusion.
 \section{Equality and the causal stationary field}
 \label{sec:variation}
 
-We now turn to the equality case. Throughout this section the chosen
+The equality case comes next. Throughout, the chosen
 exterior $\Mext$ is connected, one-ended, with compact boundary $S$
 (possibly several components) that is a future MOTS, and $S$ is
 outermost and outer area-minimizing in the senses of
-Section~\ref{sec:outermost}. We assume $E>|P|_\delta$ and consider
+Section~\ref{sec:outermost}. Assume $E>|P|_\delta$ and consider
 the mass-area equality
 $m=\sqrt{A/16\pi}$ with $A=|S|_g=\sum_{i=1}^\ell|S_i|_g$.
 
@@ -1384,9 +1364,9 @@ equations and the constraints gives
 \Delta_g\Nn
 +u^{-1}K(\grad u,\grad u)
 +2|\nabla X|_g^2/\Nn
-\ge \text{(bounded below by } -C\Nn\text{ + nonneg.)}
+\ge -C(N)\,\Nn,
 \end{equation}
-in any region where $\Nn>0$. We use the strong maximum principle for
+in any region where $\Nn>0$. The strong maximum principle is used for
 \eqref{eq:n3-N-inequality}.
 
 \begin{lemma}[Interior positivity of the twist]\label{lem:n3-twist-interior}
@@ -1533,7 +1513,7 @@ three dimensions.
 \section{The spacetime Penrose inequality with charge}
 \label{sec:charged}
 
-We now allow an electromagnetic field $F$ on the data slice. Write $n$
+An electromagnetic field $F$ is now allowed on the data slice. Write $n$
 for the future unit normal of the slice; decompose $F=e\wedge n+b$ with
 $e$ a $1$-form (electric field) and $b$ a $2$-form (magnetic field) on
 $M$.
@@ -1795,7 +1775,7 @@ four-dimensional statement uses the flow identity
 $\Sigma(t)=m_{ADM}-\tfrac12(A(t)/\omega_3)^{2/3}$ in place of the
 three-dimensional formula. At the outermost minimal $3$-surface the
 area statistic $\Area(t)$ equals $|\Gamma|_g$, giving the stated
-inequality. We use it as an external input; the details of the flow
+inequality. It is used as an external input; the details of the flow
 are classical.
 \end{proof}