==== TABLE OF CONTENTS ====
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//00-setting.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//01-domains.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//02-system.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//03-estimates.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//05-end.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//degree.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-bounds.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-chapter.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-existence.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-frontiers.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-geometry.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-opening.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-regularity.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-system.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//four-transfer.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//introduction.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//maximal-method.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//neutral.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//paper.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//transition.tex
Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//references.bib
===========================

==== BEGIN Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//00-setting.tex ====
\section{A boundary deformation in three dimensions}\label{n3:intro:section}

The aim of this part is to replace an initial-data exterior by a
Riemannian exterior while keeping a genuine inner boundary. Three
quantities have to be controlled together: scalar curvature, the area
of every enclosing cut, and the ADM energy. The inner boundary creates
a flux term in the energy balance. We choose its geometry and the
boundary conditions so that this term is absorbed by a positive bulk
integral.

\subsection{Initial data, normals, and enclosing area}

We work in three spatial dimensions, with zero cosmological constant and
\(G=c=1\). For a Riemannian metric \(g\) and a symmetric covariant
two-tensor \(K\), define the constraint densities by
\begin{equation}\label{n3:intro:constraints}
 16\pi\mu=R_g+(\tr_gK)^2-|K|_g^2,\qquad
 8\pi J=\Div_g\bigl(K-(\tr_gK)g\bigr).
\end{equation}
Here \(J\) is a covector field. The dominant energy condition is
\begin{equation}\label{n3:intro:dec}
                         \mu\ge |J|_g.
\end{equation}
All initial data considered here are smooth, including at a compact
boundary when one is present.

On an asymptotically flat end with Euclidean coordinates \(x\) and
\(r=|x|\), our basic assumptions are
\begin{equation}\label{n3:intro:decay}
 g_{ij}-\delta_{ij}=O_2(r^{-q}),\qquad
 K_{ij}=O_1(r^{-1-q}),\qquad q>\tfrac12.
\end{equation}
The notation \(O_j(r^{-a})\) includes the corresponding coordinate
derivative bounds through order \(j\). We assume that \(\mu\) and
\(|J|_g\) are integrable and that the following ADM limits exist and are
finite:
\begin{align}
 E&=\frac1{16\pi}\lim_{r\to\infty}
   \int_{S_r}(\partial_jg_{ij}-\partial_i g_{jj})n^i
                    \dd A_\delta,\label{n3:intro:energy}\\
 P_i&=\frac1{8\pi}\lim_{r\to\infty}
   \int_{S_r}\bigl(K_{ij}-(\tr_gK)g_{ij}\bigr)n^j
                    \dd A_\delta.\label{n3:intro:momentum}
\end{align}
The normal and area form in these definitions are Euclidean. Whenever
\(E>|P|_\delta\), write
\begin{equation}\label{n3:intro:rest-mass}
                         m=\sqrt{E^2-|P|_\delta^2}.
\end{equation}

For a two-sided surface \(\Sigma\) with specified unit normal \(\nu\),
we use
\[
 H_\Sigma=\Div_\Sigma\nu,\qquad
 \theta_+(\Sigma)=H_\Sigma+\tr_\Sigma K.
\]
Thus Euclidean spheres have positive mean curvature for the normal
toward infinity. A future marginally outer trapped surface, abbreviated
MOTS, satisfies \(\theta_+=0\). A weakly future outer trapped surface
satisfies \(\theta_+\le0\). Our spacetime sign convention is
\begin{equation}\label{n3:intro:k-sign}
 K(Y,Z)=\mathbf g(\mathbf\nabla_Y n,Z),
\end{equation}
where \(n\) is the future unit timelike normal. Equivalently, in Gaussian
normal coordinates the spatial metric has normal derivative \(2K\).

\begin{definition}[Exterior and enclosing area]\label{n3:intro:exterior-class}
An exterior is a connected orientable smooth three-manifold \(\Omega\)
with nonempty compact smooth boundary, complete as a metric space with
its boundary included, and with one asymptotically flat coordinate end
whose complement is compact.
An enclosing cut is the entire compact smooth embedded two-sided
intrinsic boundary of a connected smooth end-containing domain $D$
closed in $\overline\Omega$, whose intrinsic interior lies in
$\operatorname{int}\Omega$. It separates the entire inner boundary
from the distant end. The inner side includes the original obstacle
and every bounded complementary pocket. All components of the cut
and every portion coincident with the inner boundary are counted.
Its normal points into $D$, toward the end. We put
\[
 a_g(\partial\Omega)=
  \inf\{|\Gamma|_g:\Gamma\text{ is an enclosing cut}\}.
\]
When using perimeter compactness, we take the closure of this class
among filled inner sets and include the area of any frontier coinciding
with the obstacle. Smooth outward approximation gives the same
infimum. Enclosing cuts and minimizing sets are taken with bounded
complementary pockets filled. Auxiliary perimeter arguments may use
competitors containing the inner obstacle before filling: filling a bounded
pocket deletes its frontier, cannot increase perimeter, and leaves the enclosing
infimum unchanged.
\end{definition}

The definition permits coincidence because first variations of the
minimum need not be realized by a cut lying strictly outside the
obstacle. It permits disconnected cuts because neither the minimizing
hull nor an intermediate trapped boundary need be connected. The smooth-cut and full-perimeter formulations have the same infimum;
we use the independent full-enclosure geometry of~\cite[Lemma~2.1]{RiemannianCompanion}.

\subsection{Prepared data and the geometric output}

The construction itself starts from the following explicit geometric
conditions. Its hypotheses do not require that the data have been
obtained by a particular end-replacement procedure.

\begin{definition}[Prepared three-dimensional data]\label{n3:prepared:data}
Let $(\Omega_0,g,K)$ be an exterior in
Definition~\ref{n3:intro:exterior-class}, with inner boundary $S_0$.
Suppose $K$ is compactly supported and, on its single coordinate end,
$g-\delta=O_j(r^{-1})$ for every fixed $j\ge0$. Assume the physical
constraint densities in Equation~\eqref{n3:intro:constraints} are
integrable, the ADM energy is $E>0$, and the ADM momentum is zero.
There are $0<\delta<1$, $c>0$, and a smooth radius extension
$\varrho\ge1$, equal to $r$ sufficiently far out, such that
\[
 R_g=O(r^{-3-\delta}),\qquad
 8\pi(\mu-|J|_g)\ge c\varrho^{-3-\delta}.
\]
The boundary is strictly future trapped with its normal into the
exterior: $\theta_+(S_0)<0$ on every component. Its full enclosing-area
infimum is denoted by $A_*=a_g(S_0)>0$.
\end{definition}

\begin{theorem}[Three-dimensional boundary deformation]
\label{n3:prepared:output}
For every prepared exterior $(\Omega_0,g,K)$ there is a connected
one-ended subexterior $\Omega\subset\Omega_0$ whose smooth compact
boundary $B$ separates $S_0$ from infinity, with the following property.
For each sufficiently small fixed $\epsilon>0$ and every sufficiently
large integer $N$, there is a smooth metric $\widehat g_{\epsilon,N}$
on $\overline\Omega$, complete with its boundary included, for which
\begin{align}
 R_{\widehat g_{\epsilon,N}}&\ge0,
 &H_{\partial\Omega}(\widehat g_{\epsilon,N})&<0,
 \label{n3:prepared:curvature}\\
 \widehat g_{\epsilon,N}&\ge e^{-4\epsilon}g,
 &E_{\widehat g_{\epsilon,N}}&\le E+\eta_{\epsilon,N},
 \label{n3:prepared:comparison}
\end{align}
where the boundary normal points into $\Omega$ and, at fixed
$\epsilon$, $\eta_{\epsilon,N}\to0$. More precisely, for constants
$C,a$ depending on the fixed data and $\epsilon$,
\[
 0\le\eta_{\epsilon,N}\le
 C(1+N)^a\bigl(e^{-\epsilon N}+e^{-\epsilon N/2}\bigr).
\]
The new end has $\widehat g_{\epsilon,N}-\delta=O_2(r^{-1})$ and
$R_{\widehat g_{\epsilon,N}}=O(r^{-3-\delta'})$ for some
$\delta'>0$. Every full enclosing cut $T$ in $\Omega$ satisfies
\[
 |T|_{\widehat g_{\epsilon,N}}\ge e^{-4\epsilon}A_*.
\]
\end{theorem}

The construction begins by selecting maximal black and white trapped
regions. Their outer collars prescribe opposite signs for the graph
slope, but the same negative mean-curvature condition for the final
metric. Section~\ref{n3:domains:section} proves the required geometry,
including the passage from weak supports to smooth enclosures.
Section~\ref{n3:deformation:section} specifies the trace and divergence
equations and excludes a lower conformal barrier. The estimates of
Section~\ref{n3:apriori:section} then control the solutions before
uniform ellipticity is available. Section~\ref{n3:existence:section}
supplies the boundary solve and degree argument. Finally,
Section~\ref{n3:limit:section} computes the end charge and proves
Theorem~\ref{n3:prepared:output}.

The algebraic inputs from~\cite[Propositions~4.1--4.2 and Lemma~4.3]{NumericalCompanion} are its scalar
identity, weighted coercivity, and the pointwise algebraic clause of
its floor lemma. Their
proofs are independent of its numerical conclusion. The boundary
construction here retains the signed-collar estimates, boundary flux,
homotopy, and degree that a filled construction does not require.

\subsection{Consequence for weakly decaying original data}

\begin{theorem}[Minimum-enclosing-area inequality]\label{n3:intro:exterior-inequality}
Let \((\Omega,g,K)\) be an exterior as in
Definition~\ref{n3:intro:exterior-class}. Assume
Equations~\eqref{n3:intro:constraints}--\eqref{n3:intro:decay}, integrability
of \(\mu\) and \(|J|_g\), and the finite ADM limits
\eqref{n3:intro:energy}--\eqref{n3:intro:momentum}. Orient \(\partial\Omega\)
by the normal into \(\Omega\), and suppose
\(\theta_+(\partial\Omega)\le0\). If \(E>|P|_\delta\), then
\begin{equation}\label{n3:intro:penrose}
          \sqrt{E^2-|P|_\delta^2}
               \ge \sqrt{\frac{a_g(\partial\Omega)}{16\pi}}.
\end{equation}
No extension of the data across \(\partial\Omega\) is required.
\end{theorem}


The last assertion will follow from the boundary construction and the
weak three-dimensional preparation theorem of~\cite[Proposition~7.13]{NumericalCompanion}.
That preparation supplies compact tensor support, strict trapping and
the weighted margin in Definition~\ref{n3:prepared:data}; its energies
converge to $\sqrt{E^2-|P|^2}$ and its full enclosing areas converge
to the original infimum. We will use only this preparation theorem,
not the companion's numerical inequality. In particular, the ADM
normalizations in Equations~\eqref{n3:intro:energy} and
\eqref{n3:intro:momentum} remain unchanged.
==== END Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//00-setting.tex ====

==== BEGIN Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//01-domains.tex ====
\section{Trapped domains, collars, and weak supports}
\label{n3:domains:section}

We need a boundary on which the graph height can be constant and its
slope has a definite sign. A future trapped face supplies one sign; a
past trapped face supplies the other. We first choose maximal regions
of these two types, then construct outward collars and compatible
height offsets. The original obstacle stays on their excluded side,
so every resulting enclosing cut still controls its full area.
Throughout, the data satisfy
Definition~\ref{n3:prepared:data}.  Write $S_0$ for their inner
boundary.  Thus $S_0$ is strictly future outer trapped, $K$ has compact
support, and the single end has the all-order $O(r^{-1})$ bounds stipulated
in that definition.  In particular, sufficiently large coordinate spheres
have positive outward mean curvature.  The function
\begin{equation}\label{n3:domains:margin}
 \mathfrak m(x)=8\pi\bigl(\mu(x)-|J(x)|_g\bigr)
\end{equation}
is positive, has a positive lower bound on every compact set, and has
the positive $r^{-3-\delta}$ lower bound on the end, with $0<\delta<1$.  Let $A_*$ denote the prepared data's
filled enclosing perimeter infimum.

\subsection{The bounding-region convention}

For a symmetric tensor $Q$ and a two-sided surface $\Sigma$ with a
specified unit normal $\nu$, put
\begin{equation}\label{n3:domains:expansion}
 \Theta_Q(\Sigma,\nu)
   =H_\Sigma(\nu)+\tr_\Sigma Q.
\end{equation}
For the boundary of a domain the specified normal always points
\emph{out of that domain}.  Notice both elementary identities
\begin{equation}\label{n3:domains:shift-reversal}
 \Theta_{Q-(c/2)g}=\Theta_Q-c,
 \qquad
 \Theta_{-Q}(\Sigma,-\nu)=-\Theta_Q(\Sigma,\nu).
\end{equation}
The factor $1/2$ in the first formula is the reciprocal of the surface
dimension.

We use the following geometric theorem.  Its boundary parts may be
disconnected.

\begin{theorem}[MOTS barriers and the total trapped region]
\label{n3:domains:region-theorem}
Let $(N,g,Q)$ be a smooth compact three-dimensional initial data set
with boundary $\Gamma_-\sqcup\Gamma_+$.  Assume $\Gamma_+$ is nonempty
and has positive expansion with its normal out of $N$.  The inner part
$\Gamma_-$ may be empty; when it is present, assume its expansion is
negative with its normal into $N$.

Consider all smooth domains whose inner boundary is $\Gamma_-$ and
whose remaining boundary $\Sigma$ lies in the interior of $N$ and
satisfies $\Theta_Q(\Sigma)\leq0$.  They avoid $\Gamma_+$, and their
normal on $\Sigma$ points out of the domain.  If their union is
nonempty, its closure has a smooth compact embedded stable MOTS
frontier in the interior, separating it from $\Gamma_+$.  This closure
is itself the closure of such a domain and contains every domain in
the defining class.  If $\Gamma_-$ is nonempty, the two strict barriers
also give a nonempty smooth embedded enclosing MOTS.
\end{theorem}

The total-region assertion is
\cite[Definitions 7.1--7.2 and Theorem 7.3]{AnderssonMetzger2009}; the
barrier assertion is Theorem 3.1 there, with stability furnished by
Theorem 4.1.  The preliminary conventions in Section 2 of that paper
allow an empty inner boundary.  That paper uses open regions;
taking their closures preserves the smooth frontier and its expansion,
giving the closed-region formulation above.  These results require no energy
condition on $Q$.  We therefore apply them to the shifted tensors in
Equation~\eqref{n3:domains:shift-reversal}, even though these tensors
need not satisfy the dominant energy condition.  A component of the
complement of a smooth trapped domain which meets no designated outer
boundary may be filled: its entire boundary is then deleted and the
remaining boundary has the same expansion.  The maximal region is
therefore filled in this relative sense.

We use compact inner fillings only to express enclosing competitors.
One may attach a smooth collar behind $S_0$, extend the one-sided
metric smoothly, and complete the inner side by a fixed compact
filling.  All competitors contain that filling and only their
boundaries in the original exterior are counted.  This introduces no
geometric condition behind $S_0$.  For perimeter minimization with
this smooth obstacle, the relevant regularity input is
\cite[Regularity Theorem 1.3(iii)]{HuiskenIlmanen2001}: the minimizing
boundary is $C^{1,1}$ and is smooth away from contact.  We retain the filled enclosing class and its smooth approximation
from the independent enclosure geometry of~\cite[Lemmas~2.1--2.2]{RiemannianCompanion}.  The trapped regions below are not
assumed to minimize perimeter.

\subsection{Choosing the two maximal regions}

Fix a large sphere $S_{R_0}$ outside $\supp K$ such that all coordinate
spheres with radius at least $R_0$ have positive outward mean
curvature.  A compact smooth domain with $\Theta_{\pm K}\leq c\leq0$
cannot reach this part of the end.  Indeed, at a point where its
boundary maximizes the coordinate radius, it lies inside the tangent
coordinate sphere and has the same outward normal.  The graph
second-derivative comparison gives
$H_{\partial D}\geq H_{S_r}>0$ there; also $K=0$ there.  This
contradicts its assumed expansion.  The same argument applies to an
enclosing domain with prescribed inner obstacle, since the maximum
radius is attained on its free boundary.  Consequently all the
negative-threshold regions used below lie in one fixed compact radial
range.  They may be constructed using any sufficiently distant
sphere as the outer barrier, without dependence on that choice.

For
\begin{equation}\label{n3:domains:black-interval}
 \max_{S_0}\Theta_K(S_0)0$.
For $c<0$ close to zero, let $\mathcal W_c$ be the closed total
trapped region of compact smooth domains in that complement with
\begin{equation}\label{n3:domains:white-condition}
 \Theta_{-K}=H-\tr_\Sigma K\leq c.
\end{equation}
There is no required inner boundary in this application of
Theorem~\ref{n3:domains:region-theorem}.  The shifted expansion of a
reversed black face is $-c_b-c>0$, and that of a distant sphere is
also positive.  Thus the theorem applies whenever the defining union
is nonempty.  When the union is empty we simply put
$\mathcal W_c=\varnothing$.  When nonempty, it has smooth stable
frontier of white expansion $c$ and lies a positive distance from all
black faces.  The family $\mathcal W_c$ is increasing, with the black
region held fixed.

To control limits of the deformation, we choose thresholds at which these
regions are right-continuous.  The following lemma includes empty regions.

\begin{lemma}[Right-continuous thresholds]\label{n3:domains:right-continuity}
Let $E_c$, $c$ in an interval, be an increasing family of closed
subsets of a fixed compact metric space.  Outside a countable set of
parameters,
\begin{equation}\label{n3:domains:hausdorff-right}
 E_{c_j}\longrightarrow E_c\quad\hbox{in Hausdorff distance whenever}
 \quad c_j\downarrow c.
\end{equation}
Here continuity at $E_c=\varnothing$ means that $E_{c'}$ is empty
for every sufficiently close $c'>c$.
\end{lemma}

\begin{proof}
Choose a countable dense set $\{x_i\}$ and a number $L$ larger than
the diameter of the compact space.  Set
$d_c(x)=\min\{L,\operatorname{dist}(x,E_c)\}$, with the value $L$
when $E_c$ is empty.  For each $i$, $c\mapsto d_c(x_i)$ is a
bounded nonincreasing real function, so its right discontinuities
form a countable set.  Avoid their countable union.  Then
$d_{c_j}(x_i)\to d_c(x_i)$ for all $i$ whenever $c_j\downarrow c$.
Every $d_c$ is $1$-Lipschitz.  A finite net chosen from the dense set
therefore upgrades this convergence to uniform convergence on the
compact space.

For nonempty $E_c$, if points $y_j\in E_{c_j}$ stayed a fixed
distance from $E_c$, uniform convergence evaluated at $y_j$ would
contradict $d_{c_j}(y_j)=0$.  The opposite Hausdorff inclusion follows
from $E_c\subset E_{c_j}$.  For empty $E_c$, uniform convergence to
$L$ is incompatible with any nonempty $E_{c_j}$, since its distance
function has a zero.  This proves the stated convention as well.
\end{proof}

Choose $c_b$ at such a right-continuity point of the black family,
then choose $c_w<0$ at such a point of the white family for this
fixed black region.  Both choices can be made arbitrarily close to
zero.  Write $\mathcal W=\mathcal W_{c_w}$, and let $\Omega$ be the
closure of the component of the complement of
$\mathcal B\cup\mathcal W$ which reaches the selected end.  Its
boundary $B$ is a finite disjoint union of complete components of
the two smooth frontiers.  Write
\begin{equation}\label{n3:domains:faces}
 B=B_b\sqcup B_w,\qquad
 H=H_B(\nu),\qquad P_B=\tr_BK,
\end{equation}
where $\nu$ points into $\Omega$.  Thus
\begin{equation}\label{n3:domains:face-expansions}
 H+P_B=c_b\quad\hbox{on }B_b,
 \qquad
 H-P_B=c_w\quad\hbox{on }B_w.
\end{equation}
The set $B_w$ may be empty, and $B_b$ need not contain every black
frontier component: a white component may cut a black face off from
the end.  We retain exactly the boundary of the component that reaches
infinity.  The compact, disjoint frontiers create no intersection corners.

The excluded side of $B$ contains $S_0$ and a collar of it.  Hence
every compact smooth enclosing cut $T$ in $\Omega$, with all its
components retained, is an enclosing competitor for the original
filled obstacle after adjoining the discarded regions to its inner
side.  It follows that
\begin{equation}\label{n3:domains:inherited-area}
 |T|_g\geq A_*.
\end{equation}
The same conclusion holds for perimeter competitors.  This is a
statement about inclusion of competitor classes; no minimizing
property of $B$ is used.

Every enclosing cut now carries the area lower bound $A_*$. The next
step is local: on each retained face we need a smooth outward family
whose expansion is strictly larger than its threshold. Stability
alone allows a zero principal eigenvalue, so that case requires the
maximality of the chosen region.

\subsection{Outward collars, including zero stability eigenvalue}

\begin{lemma}[Collars of a maximal frontier]\label{n3:domains:stable-collar}
Let $(N^d,g)$ be smooth, with $d\in\{3,4\}$ and a fixed smooth
symmetric tensor $Q$.  Let $\Sigma$ be a compact connected smooth
two-sided hypersurface without boundary, forming a component of
the frontier of a closed region $D$, with normal pointing out of
$D$.  Assume $\Theta_Q(\Sigma)=c$ and
stability for outward variations.  Assume also local
\emph{outward-replacement maximality}: in some collar of $\Sigma$
disjoint from the other frontier components and all designated
barrier faces, no positive outward normal graph can replace
$\Sigma$, leaving $D$ unchanged outside that collar, and have
expansion at most $c$ everywhere on the new component.
There is a smooth foliation $\Sigma_s$, $0\leq sc\quad(00$, take $v=s\varphi$.  Smooth dependence gives
$\Phi(s\varphi)=s\lambda\varphi+O(s^2)$ in $C^0$, which is
positive for all sufficiently small $s>0$.  The graph velocity is
positive as well.

Suppose $\lambda=0$.  The kernel of $L$ is spanned by $\varphi$,
and the adjoint kernel is spanned by a positive smooth function
$\varphi^*$.  These are the principal-eigenfunction facts for a
scalar elliptic operator on a compact connected closed manifold
\cite[Section~4, Lemma~4.1]{AnderssonMarsSimon2008}.  For
example, simplicity of the kernel also follows by writing a kernel
element as $v=\varphi w$ and applying the strong maximum principle
to the resulting equation for $w$, whose zeroth-order term vanishes.
Consider the augmented map
\[
 \mathcal G(v,a)
   =\left(\Phi(v)-a,\ \int_\Sigma v\,\dd A\right).
\]
Its derivative at $(0,0)$ is
\begin{equation}\label{n3:domains:augmented-linearization}
 (v,a)\longmapsto
 \left(Lv-a,\ \int_\Sigma v\,\dd A\right).
\end{equation}
This map is an isomorphism.  Indeed, to solve $Lv-a=f$ with
$\int v=b$, the Fredholm compatibility condition uniquely fixes
\[
 a=-\frac{\int_\Sigma\varphi^*f\,\dd A}
          {\int_\Sigma\varphi^*\,\dd A}.
\]
The equation for $v$ then has a solution; adding a multiple of
$\varphi$ imposes its prescribed integral uniquely.  Elliptic
estimates and this one-dimensional normalization give a bounded
inverse between the indicated H\"older spaces.

The implicit-function theorem consequently supplies smooth
$v(s),a(s)$ with $\mathcal G(v(s),a(s))=(0,s)$.  Differentiating at
zero and pairing with $\varphi^*$ gives $a'(0)=0$ and
\[
 v'(0)=\frac{\varphi}{\int_\Sigma\varphi\,\dd A}>0.
\]
After shortening the interval, all these graphs form an outward
foliation and have constant expansion $c+a(s)$.  If $a(s)\leq0$
at any positive $s$ in this interval, replace only this frontier
component by $\Sigma(v(s))$.  This graph strictly enlarges $D$
inside the chosen collar, leaves every other frontier component
and designated barrier fixed, and has expansion at most $c$ on
the replaced component.  This contradicts the assumed local
outward-replacement maximality.  Therefore $a(s)>0$ for every such
$s>0$.

In either case positive graph velocity and compactness give the
asserted bounds on $|\nabla s|$.  Smooth elliptic bootstrapping of
the graph equation gives smooth leaves in the zero-eigenvalue case.
\end{proof}

In the present application $d=3$.  The full black and white regions
are $\mathcal B_{c_b}$ and $\mathcal W_{c_w}$, with $Q=K$ and
$Q=-K$, respectively.  Theorem~\ref{n3:domains:region-theorem}
gives their smooth two-sided stable frontiers, whose retained
components have expansion $c_b$ and $c_w$.  Total-region maximality
verifies the local replacement hypothesis: a positive outward
replacement with expansion at most the threshold, leaving all
other frontier components and designated barriers fixed, would
be an admissible strictly larger trapped region.  For white faces
the entire replacement lies in the fixed black complement.
Thus the lemma applies separately to each retained face.
There are finitely many faces, so choose disjoint
collars in $\Omega$ and shorten them to have a common positive
parameter range when convenient.  Write $\nu_s=\nabla s/|\nabla s|$
for their outward leaf normal; it agrees with $\nu$ at $B$.

\subsection{From weak exterior supports to a smooth enclosure}

The sublevel limits used below need not have smooth frontiers.  We now
show how an expansion bound on their exterior supports produces a smooth
enclosure.  The argument applies to arbitrary closed sets, without
positive reach or a curvature bound on the supporting surfaces.

\begin{definition}\label{n3:domains:support-definition}
Let $(N^d,g)$ be a smooth Riemannian manifold, $d\in\{3,4\}$,
with a smooth symmetric tensor $Q$.  Let $D\subset N$ be closed,
and let $x\in\partial D$ be an interior point of $N$.
A smooth exterior support is a smooth function $\phi$ near $x$ with
$\phi(x)=0$, $\dd\phi(x)\ne0$, and
$D\subset\{\phi\leq0\}$ locally near $x$.  Its normal is
$n=\nabla\phi/|\nabla\phi|$.  Define
\begin{equation}\label{n3:domains:test-expansion}
 \mathcal E_Q[\phi](x)
   =\frac{(g^{ij}-n^in^j)\nabla_i\nabla_j\phi}{|\nabla\phi|}
       +\tr Q-Q(n,n).
\end{equation}
We say $D$ has exterior-support expansion at most $c$ if every such
support at every point of its frontier has
$\mathcal E_Q[\phi]\leq c$.
\end{definition}

For a smooth domain this is exactly the usual upper bound on its
outward expansion: at contact an exterior supporting graph has mean
curvature at most that of the enclosed smooth graph.  The definition
is unchanged by increasing smooth reparametrizations of $\phi$,
because the additional Hessian term is a multiple of
$\dd\phi\otimes\dd\phi$ and has zero tangential trace.

\begin{lemma}[Parallel neighborhoods and additive error]
\label{n3:domains:parallel-support}
Let $(N^d,g,Q)$ be smooth, with $d\in\{3,4\}$.  Suppose the
frontier of a closed set $D$ lies in a fixed compact interior
region and has exterior-support expansion at most $c$.  Put
$D_z=\{y:\operatorname{dist}(y,D)\leq z\}$.  For sufficiently small
$z>0$, all exterior supports of $D_z$ have expansion at most
\begin{equation}\label{n3:domains:parallel-estimate}
 c+Cz.
\end{equation}
The allowable radius and $C$ depend only on the compact ambient
geometry and $Q$, and not on the support, its second fundamental
form, or the regularity of $D$.
\end{lemma}

\begin{proof}
If $D=\varnothing$, use $\operatorname{dist}(y,D)=+\infty$;
then $D_z=\varnothing$ and the assertion is vacuous.  Otherwise,
take a support $\phi$ to $D_z$ at $x'$.  Since $x'\in\partial D_z$,
its distance from $D$ is $z$.  For sufficiently small radii the
relevant distance neighborhood of $\partial D$ lies in a fixed
compact interior neighborhood.  There is therefore a nearest point
$x\in D$ and a length-$z$ minimizing unit-speed geodesic
$\gamma:[0,z]\to N$ from $x$ to $x'$.  This nearest point lies on
$\partial D$: if it were interior to $D$, moving a short distance
along $\gamma$ toward $x'$ would decrease the distance.  The entire
segment lies in the fixed smooth compact neighborhood and is
shorter than its injectivity radius.  The ball
$\overline B_z(x)$ is contained in $D_z$ and touches its exterior
support at $x'$.  Its tangent plane there therefore agrees with that
of the support and
\begin{equation}\label{n3:domains:radial-normal}
 \frac{\nabla\phi(x')}{|\nabla\phi(x')|}=\dot\gamma(z).
\end{equation}
Let $P_t:T_xN\to T_{\gamma(t)}N$ denote parallel transport and put
$P=P_z$.

For every $X\in T_xN$ solve the Jacobi boundary problem
\begin{equation}\label{n3:domains:jacobi-endpoints}
 J_X''+\operatorname{Rm}(J_X,\dot\gamma)\dot\gamma=0,
 \qquad J_X(0)=X,\qquad J_X(z)=PX.
\end{equation}
Here primes denote covariant differentiation along $\gamma$.
The short interval and absence of conjugate points make this problem
uniquely solvable.  Its estimates can be seen without division by an
uncontrolled quantity.  In a parallel frame write
$j_X(t)=P_t^{-1}J_X(t)$ and $\mathcal R(t)$ for the curvature
coefficient.  The integral equation is
\[
 j_X(t)=X+tA_X-
       \int_0^t(t-s)\mathcal R(s)j_X(s)\,\dd s,
 \qquad
 A_X=\frac1z\int_0^z(z-s)\mathcal R(s)j_X(s)\,\dd s.
\]
It implies, by absorption for small $z$,
\begin{equation}\label{n3:domains:jacobi-estimates}
 \begin{aligned}
 \sup_{0\leq t\leq z}|j_X(t)|&\leq2|X|,
 \qquad |A_X|\leq Cz|X|,\\
 \sup_{0\leq t\leq z}|j_X(t)-X|&\leq Cz^2|X|.
 \end{aligned}
\end{equation}
The second line follows from the same integral equation and the
first-line bounds.
Also $\langle J_X,\dot\gamma\rangle$ is affine in $t$ and has
equal endpoint values.  Thus
$\langle A_X,\dot\gamma(0)\rangle=0$.

We may therefore prescribe the first jet of a smooth unit vector
field $V$ near $x$ by
\[
 V(x)=\dot\gamma(0),\qquad \nabla_XV(x)=A_X.
\]
To realize it with uniform second-jet bounds, choose a smooth local
orthonormal frame obtained by radial parallel transport from $x$.
In that frame take the affine coefficient vector with the prescribed
value and first derivatives, and normalize its length.  The
orthogonality just proved means normalization preserves its prescribed
first derivatives.  The frame has uniformly bounded derivatives on
a fixed small coordinate ball, and Equation~\eqref{n3:domains:jacobi-estimates}
therefore gives uniformly bounded second derivatives of $V$.
This construction uses the radial direction and the ambient
geometry, with no derivatives of the support $\phi$.

Define the local endpoint map
\[
 F_z(y)=\exp_y\bigl(zV(y)\bigr).
\]
Its differential at $x$ is the terminal value of the Jacobi field
with initial values $X,\nabla_XV(x)$.  By
Equation~\eqref{n3:domains:jacobi-endpoints}, it has the exact property
\begin{equation}\label{n3:domains:exact-endpoint-isometry}
 F_z(x)=x',\qquad (\dd F_z)_x=P,
 \qquad |(\nabla\dd F_z)_x|\leq Cz.
\end{equation}
The last bound follows by differentiating the
smooth map $(y,v,z)\mapsto\exp_y(zv)$ twice in $y$, including the
first and second derivatives of $V$.  At $z=0$ the map is the
identity and its covariant second derivative vanishes.  Its
$z$-derivative is uniformly bounded for the bounded jets just
constructed, giving the displayed $Cz$ bound.  The exact middle
identity, rather than a near-isometry estimate, is essential.

The map $F_z$ takes $D$ locally into $D_z$, since its defining
geodesic from any $y\in D$ has length $z$.  Consequently
$\psi=\phi\circ F_z$ is an exterior support to $D$ at $x$.
Its derivative is $P^*\dd\phi$, so its gradient length is exactly
$|\nabla\phi(x')|$ and, by
Equation~\eqref{n3:domains:radial-normal}, its normal is
$\dot\gamma(0)$.  The covariant chain rule gives
\[
 \Hess\psi(X,Y)
   =\Hess\phi(PX,PY)
        +\dd\phi\bigl((\nabla\dd F_z)(X,Y)\bigr)
 \quad\hbox{at }x.
\]
Trace over an orthonormal basis of $\dot\gamma(0)^\perp$ and divide
by the common gradient length.  The first term is exactly the
corresponding normalized tangential trace at $x'$, and the remaining
term has absolute value at most $Cz$.  Finally, the tensor trace
changes by at most $Cz$, by the $C^1$ bound on $Q$ and parallel
transport.  It follows that
\[
 \bigl|\mathcal E_Q[\psi](x)
             -\mathcal E_Q[\phi](x')\bigr|\leq Cz.
\]
The assumed support inequality at $x$ proves
Equation~\eqref{n3:domains:parallel-estimate}.  There is no term
containing $z|\Hess\phi|$.
\end{proof}

\begin{lemma}[A smooth enclosure from weak supports]
\label{n3:domains:weak-support}
Let $N$ be a smooth compact connected three-manifold with boundary
$\Gamma_-\sqcup\Gamma_+$, where $\Gamma_+$ is nonempty and
$\Gamma_-$ may be empty.  Let $D$ be a nonempty closed subset which
contains a relative collar of $\Gamma_-$ when that boundary is
prescribed, and stays a positive distance from $\Gamma_+$.  Suppose
its interior frontier has exterior-support expansion at most $c$
for a smooth tensor $Q$.

For every $c'>c$ such that
$\Theta_Q(\Gamma_+)>c'$ with its normal out of $N$, there is a
smooth bounding domain $E$ containing $D$ in its relative interior,
containing the required inner collar, and avoiding $\Gamma_+$,
whose free boundary satisfies
\begin{equation}\label{n3:domains:smooth-enclosure-expansion}
 \Theta_Q(\partial E\setminus\Gamma_-)=c'.
\end{equation}
Its boundary and the domain may be disconnected.  In particular $D$
is contained in the closed total trapped region at any such larger
threshold.  No regularity or positive-thickness assumption on $D$
is required.
\end{lemma}

\begin{proof}
All distance neighborhoods in this proof are relative to $N$.
Since $D$ contains a collar of $\Gamma_-$ and avoids $\Gamma_+$,
their new frontiers, at sufficiently small radii, lie in a compact
subset of the interior.  A minimizing segment ending on a new
frontier has its nearest point on the interior frontier of $D$;
it cannot start on $\Gamma_-$ because its covered collar has fixed
positive width.  Thus the proof of
Lemma~\ref{n3:domains:parallel-support} applies unchanged.  Choose
$\beta>0$ so small that its conclusion holds for $00$.  Moreover $D_z$ lies on the inner side of
$\Sigma$.  To see this directly, if $d_D(y)0.
\end{equation}
There is a smooth positive function $\rho$, equal to a positive
constant times $r^{-4}$ on the far end and satisfying symbol
derivative bounds there, such that, for every $h\in[b_-,b_+]$,
\begin{equation}\label{n3:domains:offset-smallness}
 |(h+C)\tr K|+|\nabla C|+\rho
      \leq\tfrac12\mathfrak m
 \quad\hbox{everywhere on }\Omega.
\end{equation}
The thresholds, collars, offset, and $\rho$ are fixed before the
deformation parameters are chosen.  After fixing the thresholds and
collars, the offset and its first derivatives can be made arbitrarily
small.
\end{proposition}

\begin{proof}
The large-sphere comparison above confines all negative-threshold
regions to a fixed compact set, independently of how close the
thresholds are to zero.  Choose a slightly larger compact set
$\mathcal K$ containing this range and $\supp K$ in its interior.
Let
\[
 m_0=\min_{\mathcal K}\mathfrak m>0,
 \qquad T_0=\sup_{\mathcal K}|\tr K|.
\]
First restrict both negative thresholds to an interval so close to
zero that
\[
 \max\{|c_b|,|c_w|\}\,T_00$ so that
$1-\kappa_Bs>0$ throughout that collar.  For a common amplitude
$a>0$, define the contribution of a black collar to be
$a\chi(s)(1-\kappa_Bs)$, and that of a white collar to be
$-a\chi(s)(1-\kappa_Bs)$.  Extend by zero outside the disjoint
collars and sum.  This produces a smooth function on the manifold
with boundary $\Omega$, supported in $\mathcal K$, with
$C_b=a$, $C_w=-a$, and $k_B=a\kappa_B$ on the respective collars.
These remain valid bounds when a type is absent.  Its $C^1$ norm
tends to zero with $a$, even though the collar widths have already
been fixed.

For $h\in[b_-,b_+]$ and $C\in[C_w,C_b]$,
\[
 |h+C|\leq\max\{|c_b|,|c_w|\}+C_b-C_w.
\]
Choose $a$ so small that
$(C_b-C_w)T_0+\|\nabla C\|_\infty2/\epsilon$.

\subsection{Variables and equations}
\label{n3:deformation:variables}

Choose a smooth nonincreasing $\vartheta:\R\to[0,1]$ equal to one on
$(-\infty,0]$ and zero on $[1,\infty)$. Define
\begin{equation}\label{n3:deformation:l-profile}
 \begin{gathered}
 p(t)=N\vartheta(Nt),\qquad
 l(t)=\exp\left(\int_0^t p(s)\dd s\right),\\
 L(t)=l(t)e^{2t},\qquad p_L=p+2.
 \end{gathered}
\end{equation}
Thus $l(0)=1$, $l(t)=e^{Nt}$ for $t\le0$, and $l$ is constant for
$t\ge1/N$. Derivatives of $p$ of every fixed order are polynomially bounded.
Set
\begin{equation}\label{n3:deformation:small-parameters}
 \ell=e^{-\epsilon N},\qquad \tau=\ell^{3/2}.
\end{equation}
Whenever $t\ge-\epsilon$, one has $\ell\le l(t)\le e$.

For a function $f$, set
\begin{equation}\label{n3:deformation:basic-variables}
 \begin{gathered}
 \sigma=|\nabla f|,\quad D=1+l^2\sigma^2,\quad d=D^{-1},\\
 a=l\sigma/\sqrt D,\quad v=a^2=1-d,\quad
 w=l\nabla f/\sqrt D,\quad \chi=4d/(4+pv),\\
 Z=t+\tfrac18\log D,\quad u=L\sqrt D,\quad U=l\sqrt D,\quad h=\tau f.
 \end{gathered}
\end{equation}
Derivatives and contractions refer to $g$ unless indicated otherwise.
At $\sigma>0$, write $e=\nabla f/\sigma$ and
$A_j=I+(j-1)e\otimes e$. The matrices used below have the direction-free
expressions
\begin{equation}\label{n3:deformation:axial-matrices}
 A_d=I-w\otimes w,\qquad
 A_\chi=I-\frac{p+4}{4+pv}w\otimes w.
\end{equation}
Vectors and covectors are identified with $g$.

The unknowns will be $(f,Z)$. For fixed $\nabla f$,
$\partial Z/\partial t=1+pv/4>0$, and its defining expression tends to
$-\infty$ and $+\infty$ at the two ends of the $t$-axis. It therefore
defines a unique smooth $t=t(Z,\nabla f)$ for all inputs, without a floor
assumption. The coefficients $t,d,v,w,\chi,u,U$ and the direction-free
matrices are smooth at $\nabla f=0$; occurrences of $\sigma$ and $a$
in the equations combine into smooth invariant expressions. One has
$0<\chi\le d\le1$ for finite inputs.
Define
\begin{equation}\label{n3:deformation:metrics}
 \bar g=g+l^2\dd f^2,\qquad \hat g=e^{4t}\bar g,\qquad
 H^f=\frac l{\sqrt D}\Hess f,\qquad S=K+H^f.
\end{equation}
The original trace equation is
\begin{equation}\label{n3:deformation:trace}
 \tr_{A_\chi}S=F,\qquad F=h+C,
\end{equation}
where $\tr_A S$ is contraction with the contravariant matrix $A$.
Along the continuation path we will specify modified right sides $F$.
The full flux is
\begin{equation}\label{n3:deformation:flux}
 V=uA_\chi\bigl(4\nabla Z+K(w,\cdot)\bigr).
\end{equation}
Here $f$ is the graph height, while $Z$ combines the conformal
variable $t$ with the logarithm of the graph slope. The numbers $d$
and $\chi$ record ellipticity in the gradient direction, and $u$
weights the divergence flux. The scaled height $h=\tau f$ makes the
prescribed trace strictly increasing in $f$.
At $S_R$ prescribe $f=Z=0$. Initially the inner Dirichlet values are
$h=b_-$ on $B_b$ and $h=b_+$ on $B_w$.

\subsection{The exact scalar-curvature identity}

The graph-curvature strategy originates in Jang's equation and the
Schoen--Yau positive-energy argument
\cite{Jang1978,SchoenYau1981}. Its generalized warped-graph form and
Penrose coupling were developed by Bray and Khuri
\cite{BrayKhuri2010,BrayKhuri2011}; see in particular Identity~9
of~\cite{BrayKhuri2011}. We use the independently proved algebraic identity of~\cite[Proposition~4.1]{NumericalCompanion},
with the normalization below. The prescribed trace $F=h+C$, nonzero source,
and floor penalty below define a different coupled system; the
identity does not import global solvability of a zero-divergence
coupling.

The next identity separates the original constraint densities, a
nonnegative quadratic term, and a divergence.  The second equation will
prescribe that divergence so that the deformed metric has nonnegative
scalar curvature.

\begin{proposition}[Deformation identity]\label{n3:deformation:identity}
Let $f,t$ be smooth and $F=\tr_{A_\chi}(K+H^f)$. At $\sigma>0$ decompose
$S$ relative to $\R e\oplus e^\perp$:
\[
 T=S|_{e^\perp\times e^\perp},\quad
 M=S(e,\cdot)|_{e^\perp},\quad q=S(e,e),\quad \tr T=F-\chi q.
\]
Then
\begin{equation}\label{n3:deformation:scalar-identity}
 \begin{split}
 \tfrac12e^{4t}\Scal_{\hat g}
 ={}&8\pi(\mu+J(w))+\mathcal T
       +w(F)-F\tr K-u^{-1}\Div V,
 \end{split}
\end{equation}
where
\begin{equation}\label{n3:deformation:quadratic-term}
 \begin{split}
 \mathcal T={}&\tfrac12|T^{\operatorname{tf}}|^2
 +(M+pa\nabla_\perp t)\cdot(M+(p+2)a\nabla_\perp t)\\
 &+4(p+1)|\dd t|_{A_d}^2
   +(\chi-\chi^2/4)q^2+2(p+1)\chi qa\,\partial_e t\\
 &-\chi Fq/2+3F^2/4.
 \end{split}
\end{equation}
After substituting $F=\tr_{A_\chi}S$, this expression extends smoothly
across $\sigma=0$.
\end{proposition}

\begin{proof}
Apply the scalar identity of~\cite[Proposition~4.1]{NumericalCompanion} in spatial
dimension three, so its surface dimension is $k=2$. To distinguish
the two conventions temporarily, add a subscript $\mathrm A$ to its
variables. The exact dictionary is
\begin{equation}\label{n3:deformation:normalization-dictionary}
 \begin{gathered}
 t_{\mathrm A}=2t,\quad Z_{\mathrm A}=2Z,\quad
 l_{\mathrm A}(s)=l(s/2),\quad p_{\mathrm A}=p/2,\\
 \mu_{\mathrm A}=8\pi\mu,\qquad J_{\mathrm A}=8\pi J,
 \qquad S_{\mathrm A}=S.
 \end{gathered}
\end{equation}
The quantities $D,d,w,a,v,H^f,F,U$ are identical. In particular,
\[
 \frac{2d}{2+p_{\mathrm A}v}=\frac{4d}{4+pv}=\chi,
 \quad u_{\mathrm A}=e^{t_{\mathrm A}}l\sqrt D=u,
 \quad 2\nabla Z_{\mathrm A}=4\nabla Z,
\]
and the companion metric $e^{2t_{\mathrm A}}\bar g$ is precisely
$\hat g=e^{4t}\bar g$. Thus the divergence term, trace prescription
and metric match without rescaling their flux. The physical density
conversion gives $8\pi(\mu+J(w))$; the ADM energy and momentum are
not changed by that conversion.

For the quadratic expression, the companion coefficient
$s_p=p_{\mathrm A}+(k-1)/2$ is $(p+1)/2$, and
$dt_{\mathrm A}=2dt$. Its transverse square is
$|M+(p+1)a\nabla_\perp t|^2$. Combining its residual
$-[a^2|\nabla_\perp t|^2]$ with that square gives
$(M+pa\nabla_\perp t)\cdot(M+(p+2)a\nabla_\perp t)$.
The remaining gradient contribution is
$4(p+1)|dt|_{A_d}^2$. Its normal terms become
\[
 (\chi-\chi^2/4)q^2+2(p+1)\chi qa\,\partial_e t
       -\chi Fq/2+3F^2/4.
\]
These substitutions give Equations~\eqref{n3:deformation:scalar-identity}
and~\eqref{n3:deformation:quadratic-term}. Smoothness at zero gradient
is part of the invariant identity: the temporary axis has not been
used to define its coefficients. The import is pointwise and
requires neither an energy condition nor a solved divergence equation.
\end{proof}

\begin{lemma}[Polynomial coercivity]\label{n3:deformation:coercivity}
The weighted coercivity theorem of~\cite[Proposition~4.2]{NumericalCompanion}, with
Equation~\eqref{n3:deformation:normalization-dictionary}, makes
$\mathcal T$ nonnegative. For $0\le p\le N$,
\begin{equation}\label{n3:deformation:coercive-estimate}
 |\dd t|_{A_d}^2+|T|^2+|M|^2+dq^2+F^2
 \le C(1+N)^A\mathcal T
\end{equation}
with universal $C,A$.
At $\dd t=0$ there is the stronger comparison, uniform in $p,d$,
\begin{equation}\label{n3:deformation:floor-norm}
 c\bigl(|T^{\operatorname{tf}}|^2+|M|^2+\chi q^2+F^2\bigr)
 \le\mathcal T
 \le C\bigl(|T^{\operatorname{tf}}|^2+|M|^2+\chi q^2+F^2\bigr).
\end{equation}
\end{lemma}

\begin{proof}
With $x=\partial_e t$ and $y=\nabla_\perp t$, exact completion gives
\begin{equation}\label{n3:deformation:square-completion}
 \begin{split}
 \mathcal T={}&\tfrac12|T^{\operatorname{tf}}|^2
 +|M+(p+1)ay|^2+[4(p+1)-v]|y|^2\\
 &+4(p+1)d\left(x+\frac{\chi aq}{4d}\right)^2\\
 &+\tfrac34\left(F-\frac{\chi q}{3}\right)^2
 +\frac{4d(9-d)}{3(4+pv)^2}q^2.
 \end{split}
\end{equation}
Completing first in $x$ leaves $3\chi q^2/(4+pv)$; completing next in
$F$ leaves $3\chi/(4+pv)-\chi^2/12=4d(9-d)/[3(4+pv)^2]$.
The latter coefficient is at least $32d/[3(4+N)^2]$.
It controls $dq^2$ polynomially. The $F$-square then controls $F^2$,
since $\chi^2q^2\le dq^2$.
The transverse gradient coefficient is at least three, and the shifted
$M$-square controls $M$ with an additional factor $(1+N)^2$.
The squared $d$-norm of the shift in the $x$-square is
$\chi^2vq^2/(16d)\le dq^2/16$, so it also controls $dx^2$.
Finally $|T|^2=|T^{\operatorname{tf}}|^2+(F-\chi q)^2/2$.
This proves \eqref{n3:deformation:coercive-estimate}.
If $\dd t=0$, the sole cross term is $-\chi Fq/2$.
Using $|\chi Fq|/2\le\chi q^2/4+\chi F^2/4$ and $0<\chi\le1$
proves \eqref{n3:deformation:floor-norm}.
\end{proof}

The scalar identity separates curvature from divergence. At the inner
boundary we must also relate this divergence flux to mean curvature.
Differentiating the definition of $Z$ gives, wherever $df\ne0$,
\begin{equation}\label{n3:deformation:Z-differential}
 4\,dt+\tfrac12\,d\log D
       =4\,dZ=(4+pv)\,dt+v\,d\log\sigma.
\end{equation}
This relation is smooth in its direction-free form and will be used
only in contractions that extend across $df=0$.

\begin{lemma}[Boundary identity]\label{n3:deformation:boundary-identity}
On a face where $f$ is constant,
\begin{equation}\label{n3:deformation:boundary}
 V_\nu/u=d(H+4\partial_\nu t)-H+w_\nu(F-P_B).
\end{equation}
Its mean curvature in $\hat g$, with normal into $\Omega$, is
$e^{-2t}\sqrt d\,(H+4\partial_\nu t)$.
\end{lemma}

\begin{proof}
The tangential Hessian is
$\Hess f|_{TB}=(\partial_\nu f)\operatorname{II}_g$.
Thus $\tr_\perp H^f=w_\nu H$ and
$\chi q=F-P_B-w_\nu H$.
The normal component of \eqref{n3:deformation:flux}, using
\eqref{n3:deformation:Z-differential}, is
\[
 V_\nu/u=4d\partial_\nu t+\chi w_\nu q
        =4d\partial_\nu t+w_\nu(F-P_B)-vH.
\]
The $\bar g$-normal is $\sqrt d\,\nu$. Its mean curvature is $\sqrt d\,H$,
because the tangential metric is unchanged and its normal first variation
is multiplied by $\sqrt d$. The conformal mean-curvature formula gives
the asserted mean curvature for $\hat g$.
\end{proof}

\subsection{The source, boundary condition, and continuation path}

We now prescribe the divergence and inner flux. For the original system,
the source is chosen to give the deformed metric nonnegative scalar
curvature. A penalty near the lower bound will exclude contact with that
bound along the continuation path.

Choose $3/4<\gamma<1$ and fixed positive smooth weights
\[
 \rho_0\asymp r^{-4},\qquad \rho_1\asymp r^{-2\gamma},
\]
with differentiated end bounds. Let $m:\R\to[0,1]$ be smooth, equal to one
on $(-\infty,0]$ and zero on $[1,\infty)$, and set
\[
 m_0(t)=m(N(t+\epsilon)),\qquad
 \mathcal P=C_N(\rho_0+v\rho_1).
\]
The positive $\delta_0$ is fixed independently of $N,R$ in
Proposition~\ref{n3:deformation:floor}; thereafter $C_N$ is a sufficiently
large polynomial constant. Define
\begin{equation}\label{n3:deformation:source}
 \Xi=\delta_0\mathcal T+\rho+\delta_0\tau a\sigma-m_0(t)\mathcal P.
\end{equation}
Choose a smooth face function $N_B>1+\sup_B|H|$.
The original second equation and its inner condition are
\begin{equation}\label{n3:deformation:original-second-equation}
 \Div V=u\Xi,\qquad V_\nu/u=-H+w_\nu(F-P_B)-N_Bd.
\end{equation}
Lemma~\ref{n3:deformation:boundary-identity} gives
$H+4\partial_\nu t=-N_B$.
Only $-\epsilon\le t\le-\epsilon+1/N$ is penalized in the admissible range.
Since $N>2/\epsilon$, this band has negative $t$, and
\begin{equation}\label{n3:deformation:penalty-band}
 \ell\le l\le e\ell,\qquad c\ell\le L\le C\ell
 \quad\hbox{on the admissible penalty band}.
\end{equation}
The latter constants can be uniform for $0<\epsilon<1$.

\begin{proposition}[Specified homotopy]\label{n3:deformation:homotopy}
There is a continuous piecewise smooth family of systems indexed by
$s\in[0,4]$, beginning at
\eqref{n3:deformation:trace}, \eqref{n3:deformation:original-second-equation}
and the original Dirichlet values, and ending at $f=0$ and the scalar
problem with zero source and zero inner conormal flux.
Throughout its first three stages the source is
\eqref{n3:deformation:source}, with $\mathcal T$ evaluated using the actual
trace $F$. Each trace equation has right-side derivative at least one
in $h$, with $x,Z,\nabla f$ fixed.
\end{proposition}

\begin{proof}
Fix a smooth $j:\R\to[0,1]$ which vanishes for $t\le0$ and equals one
for $t\ge1$. For $0\le\lambda\le1$ put
\[
 V_\lambda=uA_\chi(4\nabla Z+\lambda K(w,\cdot)),\qquad
 \mathcal B=-H+w_\nu(F-P_B)-N_Bd.
\]
Before the last stage use the boundary condition
\begin{equation}\label{n3:deformation:homotopy-boundary}
 \begin{split}
 (V_\lambda)_\nu/u
 ={}&[1-(1-\lambda)j(t)]\\
 &\cdot[\mathcal B-(1-\lambda)(A_\chi K(w,\cdot))_\nu].
 \end{split}
\end{equation}
Outer values remain $f=Z=0$ throughout.

We specify two smooth collar modifications. Let
$\nu_s=\nabla s/|\nabla s|$ be the outward unit normal to a collar leaf;
the leaf coordinate $s$ here is local and is separate from the homotopy
parameter. Choose disjoint collar cutoffs equal to one at the faces.
Take $\eta>0$ so small that $b_-+3\eta1/2$ and equal to one at $1$, and a nondecreasing
height cutoff equal to zero below $b_+-2\eta$ and one above $b_+-\eta$.
Decrease $\eta$ to separate these ranges. Their sum $D_0$ is smooth,
bounded, supported in the collars, and nondecreasing in $h$.
It is nonpositive for $h\le b_-+\eta$, nonnegative for
$h\ge b_+-\eta$, and vanishes on black outward and white inward
directions, for every length $0\le|w|\le1$.

Choose the magnitudes larger than $2\sup_B|H|+1$.
Equation~\eqref{n3:deformation:face-thresholds} then gives, for
$F=h+C+D_0$ at the assigned boundary height $b=b_\pm$,
\begin{equation}\label{n3:deformation:directional-barriers}
 F(x,+\nu,b)\ge P_B+H,\qquad F(x,-\nu,b)\le P_B-H.
\end{equation}
These are limiting directional evaluations at $a=1,\chi=0$.
The compatible direction has equality; $D_0$ corrects the other one.
Its derivatives on bounded height ranges are independent of $N$.
Choose a smooth compactly supported extension $b(x)$, constant equal
to $b_\pm$ on each smaller collar.
On these collars let $D_1(x,w)=A_1w\cdot\nu_s$, extended by a collar
cutoff, with $A_1>\sup_B|H|+1$.
Then $D_1(x,0)=0$ and $D_1(x,\pm\nu)=\pm A_1$ on $B$.

The four stages are the following.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item For $0\le s\le1$, set $\lambda=1-s$, keep $F=h+C$ and $h|_B=b$,
 and use $\Div V_\lambda=u\Xi$ and
 \eqref{n3:deformation:homotopy-boundary}.
 \item For $1\le s\le2$, set $\lambda=0$, $\alpha=s-1$, retain $h|_B=b$,
 and replace the trace right side by $F=h+C+\alpha D_0(x,w,h)$.
 Keep the same scalar equation and boundary prescription.
 \item For $2\le s\le3$, set $\lambda=0$, $\zeta=s-2$, prescribe
 $h|_B=(1-\zeta)b$, and use
 \begin{equation}\label{n3:deformation:interpolation-trace}
 \begin{split}
 F={}&h+(1-\zeta)[C+D_0(x,w,h+\zeta b(x))]\\
    &+\zeta[\tr_{A_\chi}K+D_1(x,w)].
 \end{split}
 \end{equation}
 Keep the same scalar equation and boundary prescription.
 At a face, $h+\zeta b=b$. The value at $w=\pm\nu$ is the convex
 combination of the preceding directional value and
 $P_B+D_1(x,\pm\nu)$, so \eqref{n3:deformation:directional-barriers} persists.
 \item For $3\le s\le4$, retain the trace problem at $\zeta=1$ and
 multiply both $\Xi$ and the right side of
 \eqref{n3:deformation:homotopy-boundary} by $\eta_s=4-s$.
\end{enumerate}
The prescriptions agree at the common endpoints.
Their $h$-derivatives are $1$, $1+\alpha\partial_hD_0$, or
$1+(1-\zeta)\partial_hD_0$, all at least one.

At $\zeta=1$ the trace equation is
$\tr_{A_\chi}H^f=h+D_1(x,w)$ with zero Dirichlet values.
At a positive interior maximum of $f$, one has $w=0$, $A_\chi=I$,
$D_1=0$, and $l\Delta f=\tau f>0$, a contradiction.
A negative minimum is excluded in the same way.
Thus $f=0$ and $t=Z$ in the last stage.
At $s=4$ the remaining equation is
$\Div(4L(Z)\nabla Z)=0$, with zero inner conormal flux and $Z=0$ on
$S_R$. Multiplication by $Z$ and integration give $Z=0$.
For the fixed-point construction in Section~\ref{n3:existence:section},
the final linear scalar problem has zero source and zero inner flux
for every input; consequently that last map has output identically zero.
\end{proof}

\subsection{Elementary height and end barriers}

\begin{lemma}[Height control]\label{n3:deformation:height}
Every smooth trace solution in Proposition~\ref{n3:deformation:homotopy}
satisfies $|h|\le C_0$, independently of $N,R$, the homotopy parameter,
and the input $Z$. Its actual right side satisfies $|F|\le C_F$
with the same independence.
\end{lemma}

\begin{proof}
The collar modifications vanish at $w=0$.
At an interior extremum the trace equation reads either
$\tr K+l\Delta f=h+C$ or
$\tr K+l\Delta f=h+(1-\zeta)C+\zeta\tr K$.
The maximum and minimum inequalities bound $h$ by fixed bounds for
$|\tr K|+|C|$. Boundary values range between $b_-,0,b_+$.
This proves the height bound without using the floor.
Boundedness of $D_0,D_1$ and $|\tr_{A_\chi}K|\le3|K|$ proves the
assertion for $F$.
\end{proof}

\begin{lemma}[End height and outer boundary]\label{n3:deformation:end-height}
There are $r_0,C>0$, independent of $N,R$ and the homotopy parameter, such
that for $R\ge2r_0$,
\begin{equation}\label{n3:deformation:end-height-estimate}
 |h|\le C(r^{-\gamma}-R^{-\gamma})\quad(r_0\le r\le R).
\end{equation}
In particular,
\begin{equation}\label{n3:deformation:outer-gradient}
 |\nabla f|\le C R^{-1-\gamma}/\tau\quad\hbox{on }S_R.
\end{equation}
For fixed $\epsilon,N$, a sufficiently large lower bound for $R$ gives
$t>-\epsilon$ on $S_R$.
\end{lemma}

\begin{proof}
Take $r_0$ outside the supports of $K,C,D_0,D_1$.
There $F=h$ in the first three stages, and $h=0$ in the last one.
For $\psi=r^{-\gamma}$, with axis $\nabla\psi/|\nabla\psi|$,
the controlled end gives, uniformly for $0<\chi\le1$,
\[
 \tr_{A_\chi}\Hess\psi
 =\gamma((\gamma+1)\chi-2)r^{-\gamma-2}
      +O(r^{-\gamma-3})<0
\]
after enlarging $r_0$.
At a comparison contact, coefficients use the given $Z$ and the common
gradient of test and solution. The positive factor $l/(\tau\sqrt D)$
preserves this sign. Height monotonicity therefore makes
$C(\psi-R^{-\gamma})$ an upper barrier and its negative a lower barrier.
Choose $C$ to dominate $C_0$ on $r=r_0$ for every $R\ge2r_0$.
Differentiating at the common outer value gives
\eqref{n3:deformation:outer-gradient}; tangential derivatives there vanish.

At $S_R$, $Z=0$. The monotonicity of its defining expression implies
$t>-\epsilon$ if $\ell^2|\nabla f|^20$ and polynomial constants $C_N$ such
that, at $\dd t=0$,
\begin{equation}\label{n3:deformation:floor-algebra-general}
 \mathcal T-\mathcal S(H^f)
 \ge c_*\mathcal T-C_N(F^2+|K|^2).
\end{equation}
If $K=0$, the more useful uniform estimate is
\begin{equation}\label{n3:deformation:floor-algebra-end}
 \mathcal T-\mathcal S(H^f)\ge c_*\mathcal T-C_N vF^2.
\end{equation}
\end{lemma}

\begin{proof}
Consider the graph of $f$ in the Riemannian product with warped fiber
$(\Omega\times\R,g+l^2\dd b_0^2)$. Its induced metric is $\bar g$.
At $\dd t=0$ one also has $\dd l=0$, and its second fundamental form,
up to orientation, is $H^f$. Write $E=l^{-1}\partial_{b_0}$.
The graph unit normal is $\sqrt d\,E-ae$. The ambient curvature formulas
at this point are
\[
 \begin{gathered}
 \Scal_{\mathrm{amb}}=\Scal_g-2\Delta l/l,\\
 \Ric_{\mathrm{amb}}|_{T\Omega}=\Ric_g-\Hess l/l,\qquad
 \Ric_{\mathrm{amb}}(E,E)=-\Delta l/l,
 \end{gathered}
\]
with zero mixed Ricci terms. Hence the ambient contribution to half the
graph scalar curvature is
\[
 \tfrac12\Scal_g-v\Ric_g(e,e)
      -v\,\tr_\perp\Hess l/l.
\]
The last term is nonpositive: at $\dd t=0$,
$\Hess l=lp\,\Hess t$ and $p\ge0$.
The second-form contribution is
$[(\tr_{\bar g}H^f)^2-|H^f|_{\bar g}^2]/2$.
Since the inverse graph metric is $A_d$, its expansion is exactly
\eqref{n3:deformation:gauss-quadratic}.
Finally the conformal contribution at this point is
$-4\bar\Delta t\le0$. This proves \eqref{n3:deformation:floor-gauss}.

The pointwise algebraic clause of~\cite[Lemma~4.3]{NumericalCompanion} applies
under Equation~\eqref{n3:deformation:normalization-dictionary}. We also
record its three-dimensional calculation, because the boundary
homotopy below needs its precise degenerate normal coefficient. First evaluate $\mathcal S$ on
$S=K+H^f$, whose trace in the transverse plane is $F-\chi q$.
Subtracting \eqref{n3:deformation:gauss-quadratic} from
\eqref{n3:deformation:quadratic-term} at $\dd t=0$ gives
\begin{equation}\label{n3:deformation:floor-difference}
 \begin{split}
 \mathcal T-\mathcal S(S)
 ={}&|T^{\operatorname{tf}}|^2+(1+d)|M|^2\\
 &+\chi(1+d-\chi/2)q^2-dFq+F^2/2.
 \end{split}
\end{equation}
The normal coefficient is at least $\chi$, while
$|dFq|\le\chi q^2/2+d^2F^2/(2\chi)$.
Because
\[
 d^2/\chi=d(4+pv)/4\le1+N/4,
\]
Equation~\eqref{n3:deformation:floor-norm} implies
$\mathcal T-\mathcal S(S)\ge c\mathcal T-C(1+N)F^2$
with universal $c>0$.

To replace $S$ by $H^f=S-K$, polarize
\eqref{n3:deformation:gauss-quadratic}. The difference is bounded by
\[
 C|K|(|T^{\operatorname{tf}}|+|M|+|F|+d|q|)+C|K|^2.
\]
Indeed the only extra normal product is $dq\,\tr_\perp K$;
all remaining products involve transverse entries, $dM$, or
$\tr_\perp S=F-\chi q$.
By \eqref{n3:deformation:floor-norm} and $d/\sqrt\chi\le C\sqrt{1+N}$,
this bound is at most
$C\sqrt{1+N}|K|\sqrt{\mathcal T}+C|K|^2$.
Young's inequality absorbs a fixed small fraction of $\mathcal T$
and costs only $C(1+N)|K|^2$. This proves
\eqref{n3:deformation:floor-algebra-general}.

If $(1+p)v\le\eta_*$, then $d$ and $\chi$ are within $C\eta_*$
of one. At $d=\chi=1$ the normal block in
\eqref{n3:deformation:floor-difference} is
$3q^2/2-Fq+F^2/2$; its symmetric matrix has smallest eigenvalue
$(2-\sqrt2)/2>0$.
Choose $\eta_*>0$ small enough that its coefficients remain within
half this spectral gap. The transverse coefficients are already
bounded below. For $K=0$, Equation~\eqref{n3:deformation:floor-norm}
therefore gives $\mathcal T-\mathcal S(H^f)\ge c_*\mathcal T$
in this regime.
In the remaining regime $v>\eta_* /(1+p)$, use
\eqref{n3:deformation:floor-algebra-general} with $K=0$ and absorb
$(1+N)F^2$ into $C(1+N)^2vF^2$.
This proves \eqref{n3:deformation:floor-algebra-end}.
All formulas extend to $\sigma=0$ by their invariant expressions;
there $d=\chi=1$ and the full quadratic expressions are independent of
the temporary choice of axis.
\end{proof}

\begin{lemma}[Homotopy errors at a floor contact]
\label{n3:deformation:floor-errors}
There is a fixed compact set $\mathcal K\subset\overline\Omega$ containing
the supports of $K,C,b,D_0,D_1$ such that, at an interior minimum
$t=-\epsilon$ of a smooth solution in the first three stages,
\begin{equation}\label{n3:deformation:floor-divergence-lower}
 \Div V_\lambda/u
 \ge 2\delta_0\mathcal T+\tau a\sigma
      -C(1+N)^A
       [\mathbf1_{\mathcal K}+v(\rho_0+\rho_1)].
\end{equation}
Here $\delta_0>0$ can be fixed sufficiently small independently of
$N,R$ and the homotopy parameter.
\end{lemma}

\begin{proof}
At this contact $\dd t=0$ and $p=N$ is locally constant as a function
of $t$ near its value $-\epsilon$. Differentiating $w$ in an orthonormal
frame gives
\begin{equation}\label{n3:deformation:floor-w-derivative}
 \nabla_iw_j=H^f_{ij}-w_jH^f_{ik}w^k.
\end{equation}
In the $e$-frame the normal vector slot is multiplied by $d$.
Equations \eqref{n3:deformation:floor-norm} and $d/\sqrt\chi\le C\sqrt{1+N}$
therefore show
\begin{equation}\label{n3:deformation:floor-w-control}
 |\nabla w|\le C(1+N)^A(|K|+\sqrt{\mathcal T}).
\end{equation}
No bound for the uncontracted normal Hessian is asserted.

Each first-three-stage right side can be written as a smooth function
$F(x,w,h,p)$. Its $h$-derivative is at least one.
On the bounded height range supplied by Lemma~\ref{n3:deformation:height},
its explicit $x$- and $w$-derivatives are polynomially bounded and
compactly supported, apart from the derivative of the term $h$.
This follows directly from the fixed collar cutoffs and
$A_\chi=I-(p+4)w\otimes w/(4+pv)$, whose denominator is at least four.
For example its first $w$-derivatives on $|w|\le1$ are bounded by
$C(1+N)^2$, and the other explicit modifications have bounds independent
of $N$. Combining these bounds with
\eqref{n3:deformation:floor-w-control} and Young's inequality therefore
produces a fixed finite power of $1+N$.
Since $\nabla p=0$ at the contact and $w(h)=\tau a\sigma$, the chain rule
and \eqref{n3:deformation:floor-w-control} give, for every fixed $\eta>0$,
\begin{equation}\label{n3:deformation:floor-F-control}
 w(F)\ge\tau a\sigma-\eta\mathcal T
                 -C_\eta(1+N)^A\mathbf1_{\mathcal K}.
\end{equation}
The dependence $D_0(x,w,h+\zeta b(x))$ only adds the fixed compact
derivative of $b$ and does not change this conclusion.

The omitted flux is $(1-\lambda)uA_\chi K(w,\cdot)$.
Let $B_K=A_\chi K(w,\cdot)$. Its derivative is bounded by a polynomial
constant times $1+\sqrt{\mathcal T}$ on $\mathcal K$, by
\eqref{n3:deformation:floor-w-control}; derivatives of $p$ vanish.
The weight term requires the contraction rather than a bound for
$|\nabla\log u|$. At the contact,
\[
 \nabla\log u=H^f(w,\cdot),\qquad
 \langle\nabla\log u,B_K\rangle
 =a^2\bigl(\chi H^f_{ee}K_{ee}
                    +H^f_{e\perp}\cdot K_{e\perp}\bigr).
\]
This involves only $\chi H^f_{ee}$ and $H^f_{e\perp}$ and is controlled
by \eqref{n3:deformation:floor-norm}. It follows that
\begin{equation}\label{n3:deformation:floor-flux-control}
 |\Div(uB_K)|/u
 \le\eta\mathcal T+C_\eta(1+N)^A\mathbf1_{\mathcal K}.
\end{equation}

Combine the scalar identity with \eqref{n3:deformation:floor-gauss}.
The difference $8\pi\mu-\Scal_g/2$ equals
$((\tr K)^2-|K|^2)/2$ and is compactly supported; so is $J$.
The term $F\tr K$ is compactly bounded by Lemma~\ref{n3:deformation:height}.
On $\mathcal K$, Equation~\eqref{n3:deformation:floor-algebra-general}
therefore contributes $c_*\mathcal T-C(1+N)^A$.
Outside $\mathcal K$, $K=0$, $F=h$, and
\eqref{n3:deformation:floor-algebra-end} together with
Lemma~\ref{n3:deformation:end-height} gives
$c_*\mathcal T-C(1+N)^Av r^{-2\gamma}$.
The remaining Ricci term is bounded below by
$-C\mathbf1_{\mathcal K}-Cv r^{-3}$; enlarge $\mathcal K$ if necessary.
Both end weights are bounded by a multiple of $\rho_0+\rho_1$.
Finally subtract the omitted flux and use
\eqref{n3:deformation:floor-F-control}--\eqref{n3:deformation:floor-flux-control},
choosing their two $\eta$'s with sum at most $c_*/2$.
Fix $0<\delta_0<\min(c_*/4,1/4)$. Decreasing it slightly if necessary
gives \eqref{n3:deformation:floor-divergence-lower}.
\end{proof}

\begin{proposition}[Floor exclusion]\label{n3:deformation:floor}
Choose $\delta_0$ as in Lemma~\ref{n3:deformation:floor-errors}.
There is a polynomial choice $C_N=C(1+N)^A$ in the penalty such that,
for all sufficiently large $N$ and all
$R\ge R_{\mathrm{floor}}(N,\epsilon)$, no smooth solution anywhere along
Proposition~\ref{n3:deformation:homotopy} can satisfy $t\ge-\epsilon$
and attain $t=-\epsilon$.
In particular every such admissible solution has $t>-\epsilon$ on
$\overline{\Omega_R}$.
\end{proposition}

\begin{proof}
Choose $R_{\mathrm{floor}}$ to include the requirement of
Lemma~\ref{n3:deformation:end-height}. A floor contact cannot occur on $S_R$.
In the first three stages, an inner contact has $j(t)=0$.
Equation~\eqref{n3:deformation:homotopy-boundary} then cancels the omitted
flux on both sides and gives $V_\nu/u=\mathcal B$.
Equation~\eqref{n3:deformation:boundary} forces
$H+4\partial_\nu t=-N_B$, so $\partial_\nu t<0$.
This contradicts the nonnegative derivative into $\Omega_R$ at a
boundary minimum.

Consider an interior contact. Let
$E_N=C(1+N)^A[\mathbf1_{\mathcal K}+v(\rho_0+\rho_1)]$
denote the error in \eqref{n3:deformation:floor-divergence-lower}.
Because $\rho_0$ has a positive minimum on $\mathcal K$,
$\mathbf1_{\mathcal K}\le C_{\mathcal K}\rho_0$.
Also $v\rho_0\le\rho_0$, and $\rho/\rho_0$ is globally bounded.
We can therefore choose a polynomial $C_N$ so large that, pointwise
for all $0\le v\le1$,
\begin{equation}\label{n3:deformation:penalty-choice}
 \mathcal P\ge E_N+2\rho+\rho_0.
\end{equation}
At the floor $m_0=1$, so the scalar equation and the lower estimate give
\[
 \begin{split}
 0
 &\ge \delta_0\mathcal T+(1-\delta_0)\tau a\sigma
                     +\mathcal P-E_N-\rho\\
 &\ge \rho+\rho_0>0,
 \end{split}
\]
a contradiction. Enlarging $C_N$ below preserves this inequality and
preserves its polynomial dependence.

In the last stage $f=0$, $t=Z$, $u=L$, and $v=0$.
At an interior minimum,
$\mathcal T=(|K|^2+(\tr K)^2)/2$.
Increase $C_N$ polynomially if necessary so that
$\delta_0\mathcal T+\rho-C_N\rho_0<0$ at all such points.
For $\eta_s>0$, the scalar equation at a floor minimum has left side
$4L\Delta t\ge0$ and strictly negative right side.
At an inner floor minimum its flux is
$4\partial_\nu t=\eta_s(-H-N_B)<0$, also impossible.
If $\eta_s=0$, multiplication of
$\Div(4L(t)\nabla t)=0$ by $t$, with zero outer value and zero inner
flux, gives $t=0$.
Thus the last stage has no floor contact either.
\end{proof}

We have excluded floor contact for every smooth admissible solution
along the specified homotopy. Section~\ref{n3:apriori:section} supplies the
remaining a priori bounds, and Section~\ref{n3:existence:section} proves
existence. The choice of $\mathcal P$ uses only polynomial losses in $N$;
it is independent of all fixed-$N$ Schauder constants.
==== END Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//02-system.tex ====

==== BEGIN Boundary-graph-deformations-for-the-spacetime-Penrose-inequality//03-estimates.tex ====
\section{A priori estimates for the deformation}
\label{n3:apriori:section}

We bound the height, boundary slopes, and conformal variable before
using classical regularity. Fix the prepared exterior of
Proposition~\ref{n3:domains:prepared-domain}, its thresholds and collars,
and $\epsilon>0$. We use the variables and four stages of
Proposition~\ref{n3:deformation:homotopy}. A solution is smooth on a finite
truncation and satisfies $t\ge-\epsilon$, so the estimates include
putative floor-contact solutions. No Hessian bound or upper bound for
$Z$ is assumed.

A quantity denoted by $\Pi_N$
is bounded by $C(1+N)^A$, with $C,A$ independent of the truncation,
the homotopy parameter, and $N$.  It can increase at successive uses.
The fixed data, collars, and $\epsilon$ can enter $C,A$.  A constant
denoted by $C(N)$ can depend arbitrarily on these fixed data and on
$N,\ell,\tau$; it is still independent of the truncation and the
homotopy parameter.  Constants explicitly described as fixed do not
depend on $N$.  Recall that
\begin{equation}
 \ell=e^{-\epsilon N},\qquad \tau=\ell^{3/2},\qquad
 \ell\le l\le e,\qquad 0\le p\le N,
 \qquad \frac{4d}{4+N}\le\chi\le d.
 \label{n3:apriori:parameter-comparisons}
\end{equation}
In particular $\Pi_N\tau/\ell\to0$. This small factor will absorb
only terms whose constants are polynomial.

\subsection{Sublevels and exclusion of boundary layers}

The limiting height will be only lower semicontinuous.  We first show
how an expansion bound on its lower tests controls every exterior
support of a closed sublevel, including a plateau.  This local step
will also apply to the four-dimensional constructions.

\begin{lemma}[Exterior supports of a closed sublevel]
\label{n3:apriori:sublevel-support}
Let $U$ be an open subset of a smooth Riemannian manifold of dimension
$d\in\{3,4\}$, let $Q$ be a smooth symmetric tensor on $U$, and let
$u:U\to\R$ be finite and lower semicontinuous.  Fix $kk$, lower semicontinuity gives a neighborhood on which
$u\ge k+\delta$ for some $\delta>0$, and $v_j\to1$ uniformly there.

Let $\phi$ define an exterior support of $D_k$ at $x$:
$\phi(x)=0$, $\dd\phi(x)\ne0$, and $\phi\le0$ on $D_k$ near $x$.
Shrink to a small normal-coordinate ball compactly contained in $U$ on which
$|\phi|<1/2$.  Then $\phi$ is a lower test of $I_k$ at $x$.
Subtract a small positive multiple of the fourth power of the distance
from $x$ to make this contact strict without changing its two-jet;
write the resulting function as $\phi_0$.
Minimize $v_j-\phi_0$ on the closed ball.  The preceding relaxed-limit
description, the strict contact, and $v_j(x)\to0$ show that the
minimizers $x_j$ lie in its interior for large $j$, satisfy $x_j\to x$,
and have contact values $v_j(x_j)\to0$.  Thus $\phi_0+a_j$ is a
lower test of $v_j$ at $x_j$, where $a_j\to0$.

For each fixed large $j$, restrict to a neighborhood of $x_j$ on
which $\phi_0+a_j$ takes values in $(0,1)$.  The increasing function
$S_j^{-1}$ makes $S_j^{-1}(\phi_0+a_j)$ a smooth lower test of $u$.
Its contact height is
\[
 u(x_j)=k+j^{-1}
       +j^{-2}\log\frac{v_j(x_j)}{1-v_j(x_j)}
       \le k+j^{-1}0$,
\[
 \Hess(\theta\circ\phi_0)
  =\theta'\Hess\phi_0
       +\theta''\dd\phi_0\otimes\dd\phi_0,
 \qquad
 \frac{\tr_{(\dd\phi_0)^\perp}\Hess(\theta\circ\phi_0)}
 {|\dd(\theta\circ\phi_0)|}
 =\frac{\tr_{(\dd\phi_0)^\perp}\Hess\phi_0}{|\dd\phi_0|}.
\]
The hypothesis therefore gives
$\mathcal E_Q[\phi_0+a_j](x_j)\le q_0$.
Letting $j\to\infty$ yields $\mathcal E_Q[\phi](x)\le q_0$,
because $\phi_0$ and $\phi$ have the same two-jet at $x$.
\end{proof}

For notation in this section, let $\mathcal B_c$ be the full closed
maximal black region at threshold $c$, and let $\mathcal W_c$ be the
corresponding white region with $\mathcal B_{c_b}$ held fixed.  These
are the families used in Proposition~\ref{n3:domains:prepared-domain}.

\begin{lemma}[Uniform separation from the wrong height]
\label{n3:apriori:sublevels}
In the first two homotopy stages, the following holds.  If
$Q\subset\bar\Omega$ is compact and disjoint from $\mathcal B_{c_b}$,
there are $\eta_Q>0$ and $N_Q$ such that
\begin{equation}
 h\ge b_-+\eta_Q\quad\hbox{on }Q
 \label{n3:apriori:lower-height}
\end{equation}
for every $N\ge N_Q$, every sufficiently large allowed truncation,
and every admissible or floor-contact solution in those stages.
If $Q$ is disjoint from $\mathcal W_{c_w}$, the analogous conclusion is
$h\le b_+-\eta_Q$.  The compact sets are relative to the full closure:
they may contain faces of the opposite color.
\end{lemma}

\begin{proof}
We prove the lower assertion.  It is enough to consider an arbitrary
sequence $N_i\to\infty$, $R_i\to\infty$ and solutions in either of the
first two stages.  The elementary height and end estimates in
Lemmas~\ref{n3:deformation:height} and~\ref{n3:deformation:end-height} give
\begin{equation}
 |h_i|\le C_0,\qquad
 |h_i|\le C(r^{-\gamma}-R_i^{-\gamma})\quad\hbox{on the end},
 \qquad 3/4<\gamma<1.
 \label{n3:apriori:elementary-height}
\end{equation}
Near each white face extend $h_i$ to the other side by the constant
$b_+$.  Extend the fixed geometric coefficients smoothly there.
The extended functions are continuous.  Define the lower relaxed limit
\[
 \underline h(x)=\lim_{j\to\infty}
       \inf\{h_i(y):i\ge j,\ \operatorname{dist}(x,y)<1/j\}.
\]
It is lower semicontinuous, bounded, and satisfies the limiting end
bound.  No equicontinuity of $h_i$ is asserted.

Choose $b_-k'$.
Thus every sufficiently large $i$ uses the interior trace equation.

At these contacts,
\[
 \nabla f_i=\tau_i^{-1}\nabla\phi,
 \qquad
 d_i\le\frac{\tau_i^2}{\ell_i^2|d\phi(x_i)|^2}
          \longrightarrow0,
 \qquad a_i\longrightarrow1,
 \qquad \chi_i\longrightarrow0.
\]
Here $a_i=l_i|\nabla f_i|/\sqrt{D_i}$ is the deformation variable.
Positivity of $A_{\chi_i}$ and the Hessian
inequality at a lower contact imply
\[
 \tr_{A_{\chi_i}}K+
 \frac{l_i}{\tau_i\sqrt{D_i}}
                 \tr_{A_{\chi_i}}\Hess\phi
 \le F_i(x_i).
\]
In the second stage $D_0\le0$ at these values; in the first stage it
is absent.  Consequently $\limsup F_i(x_i)\le k'+C_b$.
Passing to the limit gives
\begin{equation}
 \frac{\tr_{(d\phi)^\perp}\Hess\phi}{|d\phi|}
           +\tr_{(d\phi)^\perp}K\le k'+C_b.
 \label{n3:apriori:relaxed-test}
\end{equation}
This is the expansion of the level surface of $\phi$, oriented toward
increasing $\phi$.  The only Hessian used in this passage is the
fixed test Hessian.  Removing the localization proves the assertion
for non-strict contacts as well.

Fix $b_-