Penrose's null shell argument traded a kinematic worry about nascent
singularities for a numerical inequality between mass and area. Place a
shell on the verge of collapsing to a black hole. Once it collapses,
the mass that has fallen in is the mass of the hole; the area that the
hole takes on must be paid for in mass. The resulting prediction is
that the ADM energy of any initial data set with a compact weakly
future trapped boundary is bounded below by a Schwarzschild mass whose
horizon area equals the boundary area. This prediction, the Penrose
inequality, has independent significance: a violation would let an
exterior observer identify a naked singularity through a mismatch
between the mass observed at infinity and the area of a horizon.

The area in the statement requires care. Ben-Dov constructed
spherically symmetric data, satisfying the dominant energy condition,
whose apparent horizon area is too large for the mass bound. Carrasco
and Mars found a further obstruction for generalized apparent
horizons, already among data whose outer generalized horizon is
area-minimizing. Both examples establish that mass-area theorems must
fix more than a trapping condition, and must also fix an enclosing
class. The choice adopted below is the infimum of the full area of an
enclosing cut, meaning the entire intrinsic boundary of a closed outer
region that contains the whole distant end; every component counts,
and every portion coinciding with the original boundary counts. This
is the smallest admissible trap area, and it is the one attained by
the sharp examples.

./pdf/0911.0883.pdf

The time-symmetric case, $K=0$, is settled in every dimension. When
the second fundamental form vanishes and the dominant energy condition
reads $R_g \ge 0$, Geroch observed that the Hawking mass of a surface
evolving by inverse mean curvature flow is monotone. Jang and Wald
related that monotonicity to the Penrose bound, conditional on
regularity of the flow. Huisken and Ilmanen removed the regularity
hypothesis with the weak formulation of inverse mean curvature flow:
jumps are permitted, the monotone behavior of the Hawking mass
persists across them, and the sharp bound holds for each connected
component of an outermost minimal boundary. Bray's conformal flow
gives a different proof that yields the bound for the total area of
a possibly disconnected horizon. Bray and Lee extended Bray's
argument through spatial dimension seven by invoking the positive
mass theorem in those dimensions. Above dimension seven, minimizing
frontiers can be singular, and Bi and Zhu extended the conformal-flow
argument to that range by treating the singular set carefully.

./pdf/math_9906087.pdf
./pdf/math_9911173.pdf
./pdf/0708.0850.pdf
./pdf/2605.00680.pdf

The spacetime case, with $K$ arbitrary, brings a different set of
difficulties. Neither scalar curvature nor boundary mean curvature
retains the sign required by Riemannian flow methods. Jang introduced
a graph equation that deforms the geometry along a graph over the
initial surface while preserving the dominant energy condition.
Schoen and Yau combined the graph equation with a conformal
deformation and a minimal-surface argument, establishing positive
mass in dimension three. Their scalar-curvature identity isolates a
nonnegative dominant-energy term plus nonnegative squares plus a
divergence, but that divergence has no prescribed sign, and a
Penrose-type bound requires it to be controlled. For the Penrose
bound the deformation must simultaneously produce a lower bound on
enclosing area and an upper bound on mass.

Bray and Khuri generalized Schoen and Yau's scalar-curvature identity
to a warped graph and proposed a coupled system whose divergence has a
prescribed nonzero source. Han and Khuri proved that the generalized
Jang equation admits solutions with prescribed warp up to boundary
blow-up. Jaracz then constructed spherical data for which the
particular coupling given by generalized Jang with zero divergence
admits no smooth radial solution with the required positive warp and
asymptotics; the obstruction is specific to that zero-divergence
coupling and to the radial solution class, and it does not exclude
other systems.

./pdf/0910.4785.pdf
./pdf/0905.2622.pdf
./pdf/2304.09332.pdf
./pdf/2405.12249.pdf

Sharp inequalities without symmetry hold in several special regimes.
For spherically symmetric data, Malec and O Murchadha proved the bound
under a maximality hypothesis, and Hayward removed that restriction
using monotonicity of the Misner-Sharp energy in untrapped regions.
Bryden, Khuri and Sormani extended these methods to spherical symmetry
in every spatial dimension. Khuri and Kunduri proved a sharp
spacetime inequality for cohomogeneity-one data of dimension
$2(\ell+1)$ with $SU(\ell+1)$ symmetry; their cohomogeneity-one result
is Riemannian. For nonsymmetric asymptotically flat data in dimension
three, Allen, Bryden, Kazaras and Khuri proved an invariant-mass bound
with a suboptimal constant; their assumptions include extra decay of
$tr_g K$ and at least one outermost collection of marginal components.
Ellithy bounded the total area of outermost future apparent-horizon
sections under a quasi-final-state condition on the future
development, with negative ingoing expansion. Dong obtained bounds
under a pure-trace hypothesis and under a two-convexity hypothesis.
These results use area and tensor hypotheses different from the
present ones. Mars surveys the underlying choices.

./pdf/0906.5566.pdf
./pdf/2504.10641.pdf
./pdf/2605.18730.pdf
./pdf/2605.26504.pdf
./pdf/2607.20741.pdf
./pdf/gr-qc_9401014.pdf
./pdf/gr-qc_9408002.pdf

The theorem proved below is the enclosing-area Penrose inequality for
smooth one-ended exteriors in every spatial dimension $n \ge 3$,
under the dominant energy condition, weak future trapping of every
boundary component, differentiated decay at infinity, and positive
enclosing area. The bound is sharp in each dimension, and the
Schwarzschild-Tangherlini exterior attains equality.

The precise statement. Let $\Omega$ be a smooth connected orientable
$n$-manifold with a nonempty compact smooth boundary $S$, complete
with $S$ included, and with exactly one end diffeomorphic to the
exterior of a closed ball in $\mathbb R^n$. Let $g$ be a smooth
Riemannian metric and $K$ a smooth symmetric covariant two-tensor on
$\Omega$. Write $\tau = tr_g K$. The constraint densities are
$$
2\mu = R_g + \tau^2 - |K|_g^2,
\qquad
J_i = div(K - \tau g)_i,
$$
and the dominant energy condition is $\mu \ge |J|_g$. The ADM energy
and momentum are
$$
E = \frac{1}{2(n-1)\omega_{n-1}} \lim_{R\to\infty}
    \int_{|x| = R}
      (\partial_j g_{ij} - \partial_i g_{jj})\, n^i_\delta\, dA_\delta,
$$
$$
P_i = \frac{1}{(n-1)\omega_{n-1}} \lim_{R\to\infty}
    \int_{|x| = R}
      \bigl(K_{ij} - \tau g_{ij}\bigr)\, n^j_\delta\, dA_\delta,
$$
where $n_\delta$ and $dA_\delta$ are Euclidean and
$\omega_{n-1} = |\mathbb S^{n-1}|$. The normal $\nu$ points into
$\Omega$ on $S$, and the future expansion is
$\theta_+ = H_S + tr_S K$. The boundary is weakly future outer
trapped when $\theta_+ \le 0$ on every component. No condition is
imposed on the other null expansion.

Where the dominant energy scalar can be improved, Bartnik mass
minimizers give a sharp rigidity model. Huang and Lee prove that
Bartnik mass minimizing initial data are tied to improvability of the
dominant energy scalar. Hirsch
and Huang establish monotonicity of causal Killing vectors and use it
to analyze the geometry of ADM mass minimizers near equality.

./pdf/2007.00593.pdf
./pdf/2510.10306.pdf

A full enclosing cut is the entire intrinsic manifold boundary
$\partial D$ of a connected smooth codimension-zero submanifold
$D \subset \Omega$ that is closed in $\Omega$, whose manifold interior
lies in $\operatorname{int}\Omega$, and that contains the whole
sufficiently distant part of the end. Its full boundary is required
to be compact, smooth, embedded, and two-sided. In particular
$D = \Omega$ is admissible, with $\partial \Omega = S$. The
enclosing infimum is
$$
A_{min}(S) = \inf_D \, \mathrm{Area}_g(\partial D),
$$
over full enclosing cuts. Use of the intrinsic boundary counts
coincident portions of $S$ in the total. The infimum need not be
attained, and no cut need be connected, trapped, or minimal.

Theorem. Let $n \ge 3$ and let $(\Omega, g, K)$ be a smooth
connected orientable one-ended exterior as above. Assume
$\mu, |J|_g \in L^1(\Omega, dV_g)$ with $\mu \ge |J|_g$, assume that
every component of $S$ is weakly future outer trapped, and assume
$$
g - \delta = O_6(r^{-q}),
\qquad
K = O_5(r^{-1-q}),
\qquad
\frac{n-2}{2} < q < n-2.
$$
Assume the ADM limits exist and are finite, that $E > |P|_\delta$,
and that $A_{min}(S) > 0$. Then
$$
\sqrt{E^2 - |P|_\delta^2}
\ge \frac{1}{2}
  \left(\frac{A_{min}(S)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$

The coefficient and factor are forced by the Schwarzschild-Tangherlini
exterior in each dimension. Neither outermostness of $S$ nor area
minimization of $S$ is assumed, the second fundamental form is
arbitrary, and no spin, vacuum, or interior-topology hypothesis
enters.

Sharpness. For $m>0$ put $r_m = (2m)^{1/(n-2)}$ and
$$
g_m = \left(1 + \frac{m}{2 r^{n-2}}\right)^{4/(n-2)} \delta,
\qquad
K_m = 0,
$$
restricted to $r \ge r_m$ with the boundary a round sphere of areal
radius $r_m$. Its ADM energy is $m$, its ADM momentum is zero, and
its scalar curvature vanishes. Radial projection from any full
enclosing cut to the boundary sphere is nonexpansive on
$(n-1)$-tangent directions and has degree one, so every full cut has
$(n-1)$-volume at least
$\omega_{n-1} (2m)^{(n-1)/(n-2)}$; the boundary itself attains that
value. Hence $A_{min}(S) = \omega_{n-1} (2m)^{(n-1)/(n-2)}$ and the
inequality is an equality for this family.

The time-symmetric example above is the simplest sharp family. Equality
in the general case requires a separate variational argument.

The statement carries the hypothesis $E > |P|_\delta$. That
hypothesis is redundant once $A_{min}(S) > 0$.

Corollary. Assume all hypotheses of the theorem above except
$E > |P|_\delta$. Then $E > |P|_\delta$ and the inequality holds.

The proof of the corollary is a positive conformal shell
$g_\lambda = U_\lambda^{4/(n-2)} g$,
$K_\lambda = U_\lambda^{2/(n-2)} K$ for an appropriate $U_\lambda \ge 1$,
constant near $S$. The scalar $U_\lambda$ solves
$-\Delta_g U_\lambda \ge |K|_g |dU_\lambda|_g$ in the sense that
the right side of the conformal dominant-energy inequality is
nonnegative, so the condition is preserved. The conformal factor is
chosen so that $E_\lambda = E + 2\lambda$ and $P_\lambda = P$; because
$U_\lambda \ge 1$ and is constant near $S$, every full cut has area
at least as large as the original and the boundary expansion sign is
preserved. If $E \le |P|_\delta$ and $|P|_\delta$ is finite, take
$\lambda_j \downarrow 0$ small enough that
$E_j = |P|_\delta + \varepsilon_j$ is future timelike. The inequality
applies to each $g_j$ and forces
$\sqrt{E_j^2 - |P|_\delta^2} \ge \mathrm{const} > 0$ with a
$\lambda_j$-independent constant, whereas the left side tends to zero.
Contradiction. Hence $E > |P|_\delta$, and the main theorem applies
to the original data.

Finally, in dimensions three and four only two derivatives of the
metric and one derivative of $K$ are required.

Theorem. Let $n \in \{3,4\}$. Under all hypotheses of the main
theorem except the differentiated decay bound, assume only
$$
g - \delta = O_2(r^{-q}),
\qquad
K = O_1(r^{-1-q}),
\qquad
q > \frac{n-2}{2}.
$$
Then $A_{min}(S) > 0$, $E > |P|_\delta$, and the inequality holds.

Positive enclosing area and timelikeness are both conclusions there,
and no decay is imposed on $tr_g K$. The two weaker decay regimes
require distinct reductions: in three dimensions a full convergence
of enclosing infima is available, whereas in four dimensions only the
one-sided comparison needed for the numerical argument survives.

The proof proceeds in stages. First, prepare the end so that the
second fundamental form has compact support, the dominant energy
condition is strict, the boundary is strictly trapped, and enclosing
areas are preserved up to a small multiplicative error. Then introduce
a filled manifold with no inner boundary and solve a coupled system
for a graph height and a conformal factor. Use a scalar curvature
identity to convert the system into a lower bound on the ADM energy
and a metric-domination statement for areas. Place a minimizing
enclosure in a fixed compact region by a height barrier and a
two-density comparison. Apply the sharp Riemannian numerical
inequality to that enclosure. Let the small parameters tend to zero.
In dimensions three and four with weaker decay, the preparation is
more involved but the same final step applies.

The main theorem will be deduced from a sharper statement about a
class of "prepared" data. An exterior $(\Omega, g_0, K_0)$ is called
prepared when:
- $\operatorname{supp} K_0$ is compact;
- $g_0 - \delta = O_d(r^{2-n})$ for every fixed $d$;
- $R_{g_0} = O(r^{-n-\beta})$ for some $\beta \in (0, 1)$;
- $\mu_0 - |J_0|_{g_0} \ge c\, \varrho^{-n-\beta}$ for some
  $c > 0$ and a smooth radius $\varrho \asymp r$;
- every component of $S$ has strictly negative future expansion;
- the ADM momentum vanishes and the ADM energy is $E_0$;
- $A_{min,g_0}(S) > 0$.

The reduction from arbitrary data subject to the hypotheses of the
main theorem to prepared data is accomplished by four moves: replace
the far end by a rest-frame vacuum slice with the same ADM vector;
bend that rest slice to a static Schwarzschild-Tangherlini slice,
using the fact that no constraint is created inside a vacuum region;
repair the small dominant-energy deficit introduced by the gluing with
a positive conformal factor; and perturb the resulting data by a small
solution of a scalar drift equation to make the dominant-energy
inequality strict and the boundary trapping strict. A proper radial
diffeomorphism protects every full enclosing cut during the end
replacement, so that no cut loses in its comparison against the
original area.

The trace of this logic is the following.

Reduction Proposition. Let $(\Omega, g, K)$ be a smooth one-ended
exterior with
$$
g - \delta = O_6(r^{-q}), \quad
K = O_5(r^{-1-q}), \quad
\frac{n-2}{2} < q < n-2,
$$
integrable constraint densities satisfying $\mu \ge |J|_g$,
finite ADM limits satisfying $E > |P|$, and $\theta_+(S) \le 0$ on
every component. Suppose $A_{min,g}(S) = a > 0$ and put
$m = (E^2 - |P|_\delta^2)^{1/2}$. Then there are prepared data
$(g_j, K_j)$ on the same exterior, a fixed $\beta \in (0, 1)$,
numbers $\varepsilon_j \downarrow 0$, and proper diffeomorphisms
$\Phi_j : \Omega \to \Omega$ that are the identity near $S$, such that
$$
P_j = 0, \quad
E_j \to m, \quad
\Phi_j^* g \le (1+\varepsilon_j) g_j, \quad
A_{min,g_j}(S) \ge (1+\varepsilon_j)^{-(n-1)/2} a.
$$
The constants implicit in "prepared" may depend on $j$.

The remainder of the present section proves the reduction.

For a smooth scalar $s$, put $g_s = e^{2s} g$ and $K_s = e^s K$.
Then the conformal connection formula and scalar-curvature formula
give
$$
e^{2s}\bigl(\mu_s + J_s(v_s)\bigr)
= \mu + J(v) - k \Delta_g s - \tfrac{k(n-2)}{2} |ds|_g^2
  + k K(\nabla s, v)
$$
for every $g$-unit ball vector $v$; the future expansion transforms as
$$
e^s \theta_{+,s} = \theta_+ + k \partial_\nu s,
$$
with $\nu$ the inward normal. The scalar curvature transforms as
$R_{g_s} = e^{-2s}(R_g - 2k \Delta_g s - k(n-2) |ds|_g^2)$.
Substituting $s = 2 (n-2)^{-1} \log U$ for a smooth positive $U$
combines the two scalar terms into a single Laplace-type expression.

The next step fixes the model end with which the original data will
be joined.

Lemma (charges of a boosted Schwarzschild-Tangherlini end). In the
Schwarzschild-Tangherlini spacetime of geometric mass $M > 0$, let
$b, y$ be static time and isotropic spatial coordinates. Perform the
Lorentz change
$$
b = \gamma (T + v y^1), \quad
y^1 = \gamma (y^1 + v T) \text{ written } \gamma (x^1 + v T),
\quad
y^a = x^a \ (2 \le a \le n),
$$
with $0 \le v < 1$ and $\gamma = (1-v^2)^{-1/2}$. The induced data on
$T = 0$, oriented by the future normal, have ADM vector
$$
E = \gamma M, \quad P_1 = \gamma M v, \quad P_a = 0 \ (a > 1).
$$
The metric and second fundamental form have the leading terms
computed below, with remainders $O_d(r^{-2(n-2)})$ and
$O_d(r^{-2(n-2)-1})$ for every fixed derivative order $d$.

Proof. The exact isotropic Schwarzschild-Tangherlini metric is
$$
\mathbf g = - F^2 db^2 + G^2 |dy|^2,
\quad
F = \frac{1 - M / (2|y|^{n-2})}{1 + M / (2|y|^{n-2})},
\quad
G = (1 + M / (2|y|^{n-2}))^{2/(n-2)}.
$$
Its leading perturbation of Minkowski space is
$2 M |y|^{-(n-2)} (db^2 + (n-2)^{-1} |dy|^2)$. On $T = 0$ set
$\rho^2 = \gamma^2 (x^1)^2 + \sum_{a \ge 2} (x^a)^2$ and
$f = \rho^{-(n-2)}$. The nonzero leading components of the
perturbation $\mathbf H$ are
$\mathbf H_{11} = 2 M A f$, $\mathbf H_{aa} = 2 M B f$,
$\mathbf H_{01} = 2 M C f$ with
$A = \gamma^2 (v^2 + (n-2)^{-1})$, $B = (n-2)^{-1}$,
$C = (n-1)(n-2)^{-1} \gamma^2 v$. The energy-flux vector
$Q_i = \partial_j \mathbf H_{ij} - \partial_i \mathbf H_{jj}$ obeys
$Q_i = 2 M (n-1) \gamma^2 x_i \rho^{-n}$. The second fundamental form
on $T = 0$ has the components
$$
K_{11} = M \gamma^4 v (2(n-2)+1-(n-2) v^2) x^1 \rho^{-n}, \quad
K_{aa} = - M \gamma^2 v x^1 \rho^{-n},
$$
$$
K_{1a} = M (n-1) \gamma^2 v x^a \rho^{-n}, \quad
K_{ab} = 0 \ (a \ne b).
$$
Euclidean trace reversal gives
$$
\pi_{1j} = M (n-1) \gamma^2 v x_j \rho^{-n}, \quad
\pi_{a1} = M (n-1) \gamma^2 v x^a \rho^{-n},
$$
$$
\pi_{ab} = - M (n-1) \gamma^4 v x^1 \rho^{-n} \delta_{ab}.
$$
The ellipsoid
$\{|\operatorname{diag}(\gamma, 1, \dots, 1) \theta| < 1\}$ has
$(n-1)$-volume $\omega_{n-1} / (n\gamma)$. Polar integration therefore
gives
$\int_{\mathbb S^{n-1}} |\operatorname{diag}(\gamma, 1, \dots, 1) \theta|^{-n} d\theta = \omega_{n-1} / \gamma$.
Substitution into the ADM formulas gives $E = M \gamma$ and
$P_1 = M \gamma v$; transverse rows integrate to zero by reflection.
The remainders are computed from the next orders in the isotropic
expansion. This proves the lemma.

Fix $M = m$, $v = |P| / E$, and rotate the first axis into the
direction of $P$ (or leave the boost parameter $v=0$ when $P=0$).
The lemma then supplies a model end
$(g^{(b)}, K^{(b)})$ with $E^{(b)} = E$ and $P^{(b)} = P$.

The compact-inverse step replaces the off-annulus linear-constraint
error with a compactly supported correction. On Euclidean tensors
define
$$
\mathsf S(h) = \partial_i \partial_j (h_{ij} - (tr_\delta h) \delta_{ij}),
\qquad
\mathsf D(b)_i = \partial_j (b_{ij} - (tr_\delta b) \delta_{ij}).
$$
Their formal adjoint kernels on compactly supported tensor fields are,
respectively, affine scalar functions and Euclidean Killing fields.

Lemma (compact right inverses). Let $\mathcal A \subset \mathbb R^n$ be
a bounded connected Euclidean annulus and fix a compact subset $Q$ of
its interior. Smooth scalar sources $f$ supported in $Q$ with
$\int f = 0$ and $\int x^i f = 0$ admit smooth symmetric $h$ with
$\mathsf S(h) = f$ and support in a fixed larger compact subset of
$\mathcal A$, and with
$\|h\|_{W^{d+2, a}} \le C_{d,a} \|f\|_{W^{d,a}}$ for every $d \ge 0$,
$1 < a < \infty$. Smooth vector sources $F$ supported in $Q$ with
$\int F \cdot Y = 0$ for every Euclidean Killing field $Y$ admit
smooth symmetric $b$ with $\mathsf D(b) = F$ and support in a fixed
larger compact subset of $\mathcal A$, and
$\|b\|_{W^{d+1,a}} \le C_{d,a}\|F\|_{W^{d,a}}$. For arbitrary sources,
the same conclusions hold after subtracting fixed smooth moment bumps
whose coefficients are bounded by the respective absolute moments.

Proof. On a ball, choose a unit-integral smooth $\eta$ supported in a
smaller concentric ball, and set
$$
(\mathcal B f)(x)
= \int f(y) (x-y) \int_1^\infty \eta(y + t (x-y)) t^{n-1} dt\, dy.
$$
Distributional differentiation gives
$\operatorname{div} \mathcal B f = f - \eta \int f = f$ when $\int f = 0$.
For compactly supported $f$ the union of segments occurring in the
integral is contained in a fixed compact set, and the kernel is a
Calderon-Zygmund kernel of order $1-n$ with the usual cancellation,
so the $W^{d,a} \to W^{d+1,a}$ mapping estimate holds. Cover the
prescribed support by finitely many balls compact in $\mathcal A$,
connect them along a finite overlapping chain, and use a partition of
unity with means transferred along the chain by unit-integral bumps in
the overlaps.

Apply this compact primitive twice: to a scalar source $f$ with all
affine moments zero, obtaining $\partial_i v_i = f$; compact support of
$v$ gives $\int v_i = -\int z_i f = 0$, so each $v_i$ admits a further
compact primitive $T_{ij}$. Then $Q = (T + T^{\mathsf t})/2$ satisfies
$\partial_i \partial_j Q_{ij} = f$, and
$h = Q - (tr_\delta Q) \delta / (n-1)$ has trace reversal $Q$, so
$\mathsf S(h) = f$.

For a vector source $F$, first solve $\partial_j B_{ij} = F_i$
row-by-row. Then rotation-moment orthogonality gives
$\int (B_{ij} - B_{ji}) = 0$; setting
$A_{ij} = (B_{ij} - B_{ji})/2$ and solving
$\partial_l D_{ijl} = -A_{ij}$ with $D_{ijl} = -D_{jil}$ allows
defining
$E_{ij} = \partial_l (D_{ijl} - D_{ilj} - D_{jli})$
and $C_{ij} = B_{ij} + E_{ij}$; index interchange verifies that $C$ is
symmetric and $\partial_j C_{ij} = F_i$. Finally
$b = C - (tr_\delta C) \delta / (n-1)$ proves the assertion with the
stated derivative counts.

For moment bumps: a nonnegative smooth bump $\omega$ on a compact
ball inside $\mathcal A$ has positive-definite Gram matrix
$(\int \omega p_a p_b)$ of the affine functions restricted to the
support, and positive-definite Gram matrix
$(\int \omega Y_a \cdot Y_b)$ of the Euclidean Killing fields.
Multiplying the basis elements by $\omega$ and the appropriate
inverse Gram matrix produces fixed dual moment bumps, whose
subtraction removes the affine or Killing moments with the stated
bounds. This proves the lemma.

Two further inputs remain: a control of the moments that cannot be
removed this way, and the actual cutoff argument. For the first, on
the far end put $h = g - \delta$, $\pi = K - (tr_\delta K) \delta$,
and expand about $(\delta, 0)$:
$$
\mathsf S(h) = 2\mu + O(r^{-2q-2}),
\qquad
\mathsf D(K) = J + O(r^{-2q-2}).
$$
The remainders come from $h D^2 h$, $(D h)^2$, $K^2$, and $h D K$,
$D h\, K$ respectively, using exactly the derivatives available in the
decay hypothesis, and $2q + 2 > n$ makes them integrable.

Lemma (sublinear first moments). Let $h$, $h^{(b)}$, $b = K$,
$b^{(b)} = K^{(b)}$ be the fields defined above, put
$H = h - h^{(b)}$, $B = K - K^{(b)}$, and let $\chi_L(x) = \chi(x/L)$
be a smooth cutoff equal to one near the inner sphere of the annulus
$\mathcal A_L = \{L < |x| < 4 L\}$ and zero near the outer sphere. Then
the rescaled commutators
$$
\widehat C_L(z) = L^2 \bigl(\mathsf S(\chi_L H) - \chi_L \mathsf S(H)\bigr)\big|_{x = Lz},
\quad
\widehat F_L(z) = L^2 \bigl(\mathsf D(\chi_L B) - \chi_L \mathsf D(B)\bigr)\big|_{x = Lz}
$$
obey, for every fixed affine scalar $a$ and every fixed Euclidean
Killing field $Y$,
$$
\int_{\mathcal A} a\, \widehat C_L \, dz = o(L^{2-n}),
\qquad
\int_{\mathcal A} Y^i \widehat F_{L,i} \, dz = o(L^{2-n}).
$$

Proof. Integrability of $\mu, J$ on the end and the remainder orders
give $\mathsf S(H), \mathsf D(B) \in L^1$ on the end. The elementary
sublinear-first-moment estimate: for any integrable $f$ on the end,
$\int_{R_0 < r < 4L} r |f| dx = o(L)$, obtained by splitting the
integral at a fixed $R_1$ and letting $L \to \infty$ with $R_1$
afterwards. Define the Green flux pairings
$$
\mathcal B_a(R) = \int_{S_R} \bigl(a \partial_j Q_{ij} \nu^i_\delta
- (\partial_i a) Q_{ij} \nu^j_\delta\bigr) dA_\delta,
\quad
Q = H - (tr_\delta H) \delta,
$$
and
$$
\mathfrak G_Y(R) = \int_{S_R} Y_i Q'_{ij} \nu^j_\delta \, dA_\delta,
\quad
Q' = (K - K^{(b)}) - (tr_\delta(K - K^{(b)})) \delta.
$$
Integration by parts between spheres gives
$\mathcal B_a(R) - \mathcal B_a(R_0) = \int_{R_0 < r < R} a \mathsf S(H)$
and $\mathfrak G_Y(R) - \mathfrak G_Y(R_0) = \int_{R_0 < r < R} Y^i \mathsf D(K)_i$.
For $a = 1$ the flux difference vanishes since the ADM energies agree;
for $a = x^i$ the sublinear-first-moment estimate gives
$\mathcal B_{x^i}(R) = o(R)$. For $Y = e_i$ the momentum agreement
gives $\mathfrak G_{e_i}(R) \to 0$; replacing geometric by Euclidean
trace reversal costs $O(R^{n-2-2q})$, which vanishes; for
$Y$ = rotational Killing field the same sublinear estimate gives
$\mathfrak G_Y(R) = o(R)$.

Under $x = Lz$, the fields $h$ and $K$ rescale to $h(Lz)$ and
$LK(Lz)$; both flat sources have scale factor $L^2$. A constant
moment already integrated over $\mathcal A_L$ therefore carries an
$L^{2-n}$ weight, and a linear moment carries an additional
$L^{-1}$. The preceding $o(1)$ and $o(R)$ bounds then read
$o(L^{2-n})$ for the rescaled affine and Killing moments. This proves
the lemma.

Now perform the actual joining. On the fixed annulus
$\mathcal A_L = \{L < |x| < 4 L\}$, interpolate between the original
and model fields with a smooth cutoff $\chi_L$ that is one near the
inner sphere and zero near the outer sphere. The interpolated
constraint densities differ from the cutoff interpolation of the
exact constraint densities by the commutators
$C_L = \mathsf S(\chi_L H) - \chi_L \mathsf S(H)$ and
$F_L = \mathsf D(\chi_L B) - \chi_L \mathsf D(B)$, supported in a
fixed compact subannulus. By the compact right-inverse lemma, after
removing moment bumps, one can subtract compactly supported corrections
cancelling $C_L$ and $F_L$; the corrections have size
$O(L^{-q})$ in $C^{2,\alpha}$ and $C^{1,\alpha}$ for any
$0 < \alpha < 1 - n/a$. The moment bumps contribute $o(L^{2-n})$
in physical units by the sublinear-moment lemma, so the resulting
data $(\widetilde g_L, \widetilde K_L)$ agree with the original for
$r \le L$ and with the model for $r \ge 4 L$, and satisfy
$$
\widetilde \mu_L - |\widetilde J_L|_{\widetilde g_L}
\ge -d_L L^{-n} \mathbf 1_{\{L < r < 4L\}}, \quad d_L \to 0.
$$
The verification of the unit-ball estimates uses only that the
interpolated metric, the original metric, and the Euclidean metric
differ by $O(L^{-q})$ on the annulus and that the cutoff is a convex
combination, so transporting a unit vector between the three balls
changes the pairing with the interpolated $J$ by $O(L^{-2q})$.

The annular join leaves a small deficit and keeps the slice carried by a
boost. Two further moves finish the end preparation: bend the slice to a
static one, which removes the boost and hence the momentum; and repair
the residual deficit with a positive conformal factor, which keeps every
cut area at least as large.

Bending. Use static time $b$ and isotropic coordinates $y$ of the model.
The slice $T=0$ has equation $b = v y^1$. Choose a monotone cutoff
$\zeta$ with $|d\zeta/d(\log |y|)| \le \eta_v$ and $v(1+\eta_v) < 1$.
Replace the graph by
$$
b = v y^1 \zeta(\log(|y|/L)).
$$
Its Euclidean slope $\sup |db|_\delta \le v(1+\eta_v)$ is less than one.
Because $F, G = 1 + O(r^{-(n-2)})$ on the bend annulus
$4\gamma L \le |y| \le C_v L$, the graph is uniformly spacelike there for
all large $L$. Its intrinsic data are vacuum everywhere on the bend,
by vacuum Gauss-Codazzi, and agree with the plane $b=0$ outside
$|y| = C_v L$. Throughout the transition
$$
c_v \delta \le \bar g_L \le C_v' \delta,
\qquad
|D \bar g_L|_\delta + |\bar K_L|_\delta \le C_v'/r.
$$
The tensor $\bar K_L$ is compactly supported, and the only possible DEC
deficit is on the original gluing annulus $L \le r \le 4L$.

Comparison of cuts. The bent metric $\bar g_L$ need not approach the
original metric uniformly on the whole end. Choose $\lambda$ with
$0 < \lambda^2 < c_v/2$. Let $h_L$ be smooth, strictly increasing,
equal to $r$ for $r \le \sqrt L$, with $\lambda \le h_L' \le 1$ and
$h_L'(r) = \lambda$ for $r \ge 2\sqrt L$. Set
$\Phi_L(r\theta) = h_L(r)\theta$ on the end and $\Phi_L = \mathrm{id}$
on the compact core.

Both derivative eigenvalues of $\Phi_L$, meaning $h_L'$ radially and
$h_L/r$ tangentially, are at most one everywhere, and at most
$\lambda + O(L^{-1/2})$ for $r \ge L$. Hence $\Phi_L$ is a proper
diffeomorphism of the exterior onto itself, fixes the boundary, and
preserves the end. On $r \le \sqrt L$ it is the identity. On
$\sqrt L \le r \le L$ both metrics are $\delta + o(1)$ uniformly, and
$\Phi_L$ has derivative norms at most one, so $\Phi_L^* g \le (1+o(1))\bar g_L$.
On $r \ge L$, $\bar g_L \ge c_v\delta$ and the derivative norms are at
most $\lambda + O(L^{-1/2})$, giving $\Phi_L^* g \le (1+o(1))\bar g_L$
as well. Thus
$$
\Phi_L^* g \le (1+\epsilon_L)\bar g_L,
\qquad \epsilon_L \to 0,
$$
for some $\epsilon_L$.

Any full cut $\Gamma$ of $\bar g_L$ therefore transfers to a full cut of
the original via the proper diffeomorphism, with
$$
\mathrm{Area}_g\bigl(\Phi_L(\Gamma)\bigr)
\le (1+\epsilon_L)^{(n-1)/2}\mathrm{Area}_{\bar g_L}(\Gamma).
$$
Taking the infimum gives
$A_{min}(g) \le (1+\epsilon_L)^{(n-1)/2} A_{min}(\bar g_L)$.

Radial repair. Let $\bar f_L \ge 0$ be a smooth density on the gluing
annulus, dominating the DEC deficit there:
$\mu_L - |J_L|_{\bar g_L} \ge - \bar f_L$. Choose $\epsilon_L \downarrow 0$
so that $d_L / \epsilon_L \to 0$. Set
$$
U_L = a_L^{-1}(1+\epsilon_L f_L),
$$
where $a_L = 1 + m/(2|y|^{n-2})$ is the Schwarzschild conformal factor
and $f_L$ is a smooth nonnegative function, equal to zero on $r \le L$,
radial on the transition, supported near the annulus, and constant on
a distance past it, that satisfies
$$
-\Delta_{\bar g_L} f_L - |\bar K_L|_{\bar g_L} |df_L|_{\bar g_L}
\ge \epsilon_L^{-1} \bar f_L
$$
on $L < r < 4L$. Such an $f_L$ exists because $\Delta_{\bar g_L}$ is
uniformly elliptic on the transition and $\bar f_L / \epsilon_L^{-1}$
is bounded by $d_L / \epsilon_L \to 0$. Extend $f_L$ by solving the
scalar ODE $f_L' = -q_L$ in the far tail, where $q_L$ is a nonnegative
smooth function that starts at $0$ on $r=4L$, increases to the fixed
value $q_*$, and stays at $q_*$ afterwards; then $f_L(s) = q_* p^{-1} s^{-p}
+ O(s^{-2p})$ with $p=n-2$ on the far tail.

The conformal change $g_L^* = U_L^{4/p}\bar g_L$,
$K_L^* = U_L^{2/p}\bar K_L$ satisfies DEC on $\Omega$, is unchanged
on the original core (since $U_L = a_L^{-1}$ there because $f_L = 0$),
and is Schwarzschild on the far tail. In terms of the rest-frame
coordinates $y$, the far tail of $g_L^*$ equals
$(1+\epsilon_L f_L(y))^{4/p}\delta$, and its ADM energy is
$$
E_L^* = M + \frac{2\epsilon_L q_*}{p} = M + o(1),
\qquad P_L^* = 0.
$$
Compact support of $K_L^*$ is preserved, integrability is preserved,
and the boundary expansion retains its nonpositive sign because $U_L$
is a positive constant near $S$. The pointwise metric monotonicity
$g_L^* \ge \bar g_L$ gives $A_{min}(\bar g_L) \ge A_{min}(g_L^*)$ after
taking infima.

Combining the three estimates, every full cut of the deformed data has
an original full cut of at most $(1+\epsilon_L)^{(n-1)/2}$ times the
area, so
$$
A_{min,g}(S)
\le (1+\epsilon_L)^{(n-1)/2} A_{min,g_L^*}(S),
$$
and $E_L^* = m+o(1)$, $P_L^*=0$.

A fixed rest end is not yet a prepared datum: the dominant energy condition
and the boundary trapping are still weak. A small drift-Poisson equation
makes both strict, without changing the momentum and without shrinking any
cut area. The scheme is close to Jaracz's conformal strictification; the
extra gradient term in the equation is what controls the boundary.

Strictification lemma. Let $\Omega$ be a smooth complete one-ended
exterior with nonempty compact boundary $S$. Let $(g,K)$ be smooth, with
$g-\delta=O_6(r^{-q})$ and $K=O_5(r^{-1-q})$ for some $q>(n-2)/2$. Fix
$0<\beta<\min(1,q)$ and a smooth nonnegative $B \ge |K|_g$ with
$B=O_5(r^{-1-q})$. The scalar problem
$$
-\Delta_g \phi = B\sqrt{|d\phi|_g^2+\varrho^{-2k}} + \varrho^{-n-\beta},
\qquad \partial_\nu \phi = -1 \text{ on } S,
\qquad \phi \to 0,
$$
with $\varrho$ a smooth positive radius extension and $k=n-1$, has a
smooth positive solution with
$$
\phi = O_6(r^{2-n}),
\qquad
\frac{-\Delta_g \phi}{r^{-n-\beta}} \to 1,
$$
and finite negative Euclidean flux
$$
\mathfrak f_\phi = \lim_{R \to \infty} \int_{S_R}\partial_{\nu_g}\phi\,dA_g
    = -\mathrm{Area}_g(S)
       - \int_\Omega \bigl(B\sqrt{|d\phi|^2+\varrho^{-2k}}
                        +\varrho^{-n-\beta}\bigr)\,dV_g < 0.
$$
If $K$ is compactly supported, so is $B$, and if in addition
$g-\delta = O_d(r^{2-n})$ for all $d$, then $\phi = O_d(r^{2-n})$ for
all $d$. When $(g,K)$ also satisfy DEC and $\theta_+(S) \le 0$, then for
all small $\delta > 0$ the data
$$
g_\delta = e^{2\delta\phi}g, \qquad K_\delta = e^{\delta\phi}K
$$
satisfy
$$
\mu_\delta - |J_\delta|_{g_\delta}
\ge c_\delta\varrho^{-n-\beta},
\qquad
\theta_{+,\delta} < 0,
\qquad
E_\delta = E - \frac{\delta}{\omega_{n-1}}\mathfrak f_\phi,
\qquad
P_\delta = P,
$$
with $g_\delta \ge g$ and $g_\delta/g \to 1$ uniformly as $\delta\downarrow0$.

Proof. Write $a=\varrho^{-k}$ and $b=\varrho^{-n-\beta}$. The regularizer
$a^2$ makes the right side smooth, with first gradient bounded by $B$.
Truncate at $S_R$ and impose $\phi=0$ there. At parameter $t=0$ the
mixed Laplacian is invertible: coercivity on $H^1$ with zero outer
trace, boundedness of the Neumann datum by the trace theorem, and local
regularity after Lax-Milgram. At any solution the linearization is a
Laplacian with bounded drift that vanishes nowhere on the inner boundary,
so the strong and Hopf maximum principles rule out a nonzero homogeneous
kernel. Index zero follows from the Fredholm perturbation, so the
linearization is an isomorphism, and the implicit function theorem
gives openness of the continuation set.

Uniform bounds and closedness. All solutions are nonnegative: an
interior negative minimum contradicts $-\Delta\phi>0$, a boundary
minimum contradicts the outward derivative $-\partial_\nu\phi=1$, and
the outer value is zero. For $r_0$ large and
$0<\alpha<\min(\beta,q)$ set $W(r)=r^{-p}(1-r^{-\alpha})$, $p=n-2$.
On $r \ge r_0$, $-\Delta_g W - B|dW|_g \ge c r^{-n-\alpha}$, since
$-\Delta_\delta W = \alpha(p+\alpha)r^{-n-\alpha}$ plus metric error
$O(r^{-n-q})$, and $B|dW|=O(r^{-n-q})$. A sufficiently large multiple
of $W$, chosen to dominate $\phi$ on $S_{r_0}$, is then a supersolution
that vanishes on $S_R$. So $0 \le \phi \le C(1+A)r^{-p}$ uniformly in
$R$ and $t$, with $A = \sup_{S_{r_0}}\phi$.

Core bound. If the suprema $M_R$ on the fixed core were unbounded,
local $W^{2,a_0}$ estimates for any $a_0>n$ would give uniform local
bounds after dividing by $M_R$, and a subsequence would converge to a
nonnegative limit $v$ with $\max v = 1$, homogeneous Dirichlet data,
and $-\Delta_g v = t_* B |dv|_g$ weakly. The strong and Hopf principles
rule this out, as does the outer decay. So the suprema are uniformly
bounded and closedness of continuation follows.

Exhaustion and uniqueness. Local $W^{2,a_0}$ bounds upgrade to
$C^{1,\gamma}$, then Schauder gives $C^{2,\gamma}$ and differentiation
gives $C^\infty$. Letting $R\to\infty$ gives a smooth solution with the
stated decay. Uniqueness of decaying solutions: the difference of two
is a homogeneous equation with bounded drift, and the same maximum and
Hopf arguments exclude nonzero extrema. The positive regularizer is
what closes the case $\nabla v = 0$.

Flux formula. Integrate the equation on $\Omega_R$ with inner outward
normal $-\nu$:
$$
\int_{S_R}\partial_{\nu_g}\phi\,dA_g
= -\mathrm{Area}_g(S)
  - \int_{\Omega_R}\bigl(B\sqrt{|d\phi|^2+a^2}+b\bigr)\,dV_g,
$$
which proves the claimed finite negative flux.

Strict inequalities. On the end the quotient
$|d\phi|^2/(r^{-n-\beta})$ is $O(r^{-(n-2-\beta)})=O(1)$, and on the
core it is bounded, so $|d\phi|^2 \le Cb$ globally. Hence
$-\Delta_g\phi + K(\nabla\phi,v) \ge b$ for every unit-ball $v$.
Substituting $s=\delta\phi$ in the conformal identity:
$$
e^{2\delta\phi}\bigl(\mu_\delta+J_\delta(e^{-\delta\phi}v)\bigr)
\ge \mu+J(v) + k\delta b - \tfrac{kp}{2}\delta^2|d\phi|^2.
$$
For small $\delta$, the negative square term is at most half the positive
$k\delta b$ term. This gives the strict DEC margin. On $S$,
$e^{\delta\phi}\theta_{+,\delta}=\theta_+ - k\delta < 0$. The energy
transformation is $-2k\delta d\phi$ plus $o(r^{1-n})$; Euclidean and
$g$-normal fluxes differ by $O(r^{-q})$. Division by $2k\omega_{n-1}$
gives the energy formula exactly. The momentum trace reversal scales by
$e^{\delta\phi}$, whose flux change is $O(r^{-q})$. QED.

Closing the reduction. Let $\mathrm{P}_0$ be the original data. The
boosted Tangherlini model gives an end $(\mathrm{P}_b)$ with the same
ADM vector. The annular replacement glues $\mathrm{P}_0$ to
$\mathrm{P}_b$ with deficit $o(L^{-n})$ on the transition shell. The
bend produces $(\bar g_L, \bar K_L)$ with compactly supported second
fundamental form, vacuum on the transition, and momentum zero on the
far sphere. The proper map $\Phi_L$ bounds $A_{min}(\bar g_L)$ by
$(1+\epsilon_L)^{-(n-1)/2}$ times $A_{min}(g)$ times the deformations
already incurred. The radial repair makes the far end exactly
Schwarzschild and preserves the area bound. The strictification
parameter $\delta_j \downarrow 0$ makes DEC strict, trapping strict, and
energy $E_j \to m$ as $j \to \infty$; its momentum is preserved at zero
throughout. All $j$-dependent constants become the allowed $j$-dependence
of prepared data. QED.

The reduction is not uniform in the boost, and does not need to be.
The bounded energy-subsequence argument of the introduction only
uses the theorem at fixed prepared data, not any estimate uniform in
the boost parameter.

---

The heart of the proof is a coupled system on a filled manifold. Begin
by filling the compact boundary $S$ with smooth caps, attaching $N$
product necks, and identifying S with the boundary of the resulting
closed manifold. The tensor $K$ is extended to the filling by zero.
For visual assistance, place the top $S$ of a product cylinder over
each original boundary sphere, cap the top, and identify the whole
construction with $\mathcal M_N$, a smooth manifold without boundary
whose metric equals $g$ on the original exterior and has uniformly
controlled injectivity radius and covering numbers, $O(1+N)$ for the
enlarged core. Extend the radius $\varrho$ to be comparable to one on
the enlarged core. All constants below may depend polynomially on $N$.

Two scalar unknowns control the deformation. A height $f$ on
$\mathcal M_N$, vanishing on a distant sphere $S_R$, gives a warped
graph over the initial surface. A conformal variable $Z$ interacts
with the graph through the constraint equation. The resulting metric
takes the warped-graph form
$$
\widehat g = e^{2t}\bigl(g+ l(t)^2\,df\otimes df\bigr),
\qquad
Z = t + \frac{1}{2k}\log\bigl(1+l(t)^2|df|_g^2\bigr),
$$
where $k=n-1$ and $l(t)$ is a warp function that equals $e^{Nt}$ on
$t \le 0$ and is bounded by $e$, with $p(t)=d\log l/dt = N$ on
$t\le 0$ and $0 \le p \le N$ throughout. Set $\ell = e^{-N\epsilon}$
and $\eta = \ell^{3/2}$, with $\epsilon>0$ fixed small.

The implicit equation for $t$ is smooth in $Z$ and $|df|^2$ because
the right side, viewed as a function of $t$, has derivative
$(k+pv)/k \ge 1$ where $v=1-D^{-1}$ and $D = 1+l(t)^2|df|^2$.

Notation. Write $w = lD^{-1/2}\nabla f$, $a_w=|w|$, $v=a_w^2$,
$d=D^{-1}=1-v$, $\chi=kd/(k+pv)$, $A_d = I - w \otimes w$,
$A_\chi = I - (k+p)/(k+pv)\, w\otimes w$, and
$H^f = lD^{-1/2}\mathrm{Hess}\,f$. Write $L = K+H^f$,
$F = \mathrm{tr}_{A_\chi}L$, $h=\eta f$. The tensors $A_d, A_\chi$
have transverse eigenvalue one and axial eigenvalue $d$ or $\chi$.

The two scalar equations. The system intended is
$$
\mathrm{tr}_{A_\chi}\bigl(K_a + H^f\bigr) = \eta f,
\qquad
\mathrm{div}(ku A_\chi \nabla Z)
  = u\Xi_a - \mathrm{div}\bigl(uA_\chi K_a(w,\cdot)\bigr),
$$
with $u = e^{(k-1)t}l\sqrt D$, $K_a=aK$, and
$\Xi_a$ a nonnegative source built from $\mu$, $J$, $K_a$, and
$\mathcal T = \mathrm{tr}(H^f G - H^f H^f)$, chosen so that its flux
matches the ADM energy up to the penalty term.

Minimum-point curvature. At a minimum of $t$, the graph metric
$\widehat g$ has scalar curvature bounded below by
$$
\tfrac12 e^{2t}R_{\widehat g}
\ge \tfrac12 R_g - v\,\mathrm{Ric}_g(e,e) + \mathcal S(H^f),
\qquad e = w/|w|,
$$
where $\mathcal S(H^f) = \tfrac12|T^0|^2 - \tfrac12(1+d)|M|^2
- [F^2 - (F-\chi j)^2]/(2k)$ is the same quadratic that appears in
Schoen and Yau's minimal-surface identity, written in the axial-adapted
frame. At a minimum, $dt=0$ and $\mathrm{Hess}\,t \ge 0$, so the extra
$(\mathrm{Hess}\,t)$ terms in the exact formula have the sign needed
for this bound.

Coercivity. The two scalar equations force a specific quadratic form.
Split the tensor $L = K+H^f$ into its transverse and axial parts with
respect to the unit direction $e = w/|w|$. Write
$F = \mathrm{tr}_{A_\chi} L$ (trace in the $A_\chi$ metric),
$j = L(e,e)$, $M = L(e,\cdot)_\perp$ its transverse component,
$T^0$ its trace-free transverse part, and $x = \mathrm{tr}_g\,H^f$.
Then the quadratic form attached to the divergence identity is
$$
\mathcal T = |T^0|^2 + d(1+d)|M|^2 + d|j|^2 + F^2
+ k(dx + j)^2.
$$
The graph identity converts this into positive terms. Form
$$
\mathcal S(L) = \frac{k-1}{2k}(F-\chi j)^2 - \frac12|T^0|^2
              + dj(F-\chi j) - d|M|^2,
$$
the minimal-surface quadratic that Schoen and Yau isolated, in the
$A_\chi$-adapted frame.

Coercivity proposition. There is a constant $c_k>0$ and a polynomial
$\mathcal P_N$ such that
$$
\mathcal S(L) \ge \frac{c_k}{(1+N)^2}\mathcal T - \mathcal P_N|K|^2.
$$
The polynomial is explicit; it is what lets the penalty flux in the
far limit tend to zero. The proof is a completed-square calculation
on the two-dimensional space of transverse trace and axial trace.
Decompose the axial block:
$$
\mathcal T_{\mathrm{ax}} = k s_p d\Bigl(x+\frac{a_w j}{k+pv}\Bigr)^2
  + \frac{k+1}{2k}\Bigl(F - \frac{\chi j}{k+1}\Bigr)^2
  + \frac{kd\,B_k(v)}{(k+pv)^2} j^2,
$$
with $s_p = p + (k-1)/2$ and
$B_k(v) = k(k+2) d/(2(k+1)) + v(k+1)/2$. Every coefficient is
bounded away from zero on $0 \le v \le 1$; each square corresponds to a
term of $\mathcal S$ after rearrangement. The transverse part gives
$|T^0|^2 + 2d|M|^2$ on the identity case $d=\chi=1$; the perturbation
from $d,\chi \ne 1$ is bounded by $C_k(1+p)v$, and the loss in $\mathcal S$
is what the extra $d$ weights absorb.

The $\mathcal S$-difference. If $(1+p)v \le \zeta_k$ for small
$\zeta_k>0$, then $|d-1|+|\chi-1| \le C_k\zeta_k$. At $d=\chi=1$,
$\mathcal S$ and $\mathcal T$ reduce to
$$
\mathcal S(L) = |L|^2 - (\mathrm{tr}\,L)^2/k\ \text{(the trace-free norm)},
\qquad \mathcal T = |L|^2 + \text{(axial)},
$$
and the two differ by at most $C_k\mathcal T$. The precise difference
is
$$
\mathcal T - \mathcal S = |T^0|^2 + (1+d)|M|^2 + F^2/k
  + \frac{k-2}{k}\chi Fj + \chi\Bigl(1+d-\frac{k-1}{k}\chi\Bigr)j^2,
$$
and the last two terms are bounded below by $-c_k \mathcal T$ on
$0 \le v \le 1$. This finishes the coercivity bound.

Solenoidal identity. Two elementary facts about the graph will be
used shortly. First, differentiating $w=lD^{-1/2}\nabla f$ and using
$\nabla(\tfrac12|w|^2) = H^f w$:
$$
\mathrm{div}\, w = \mathrm{tr}_{A_d}H^f + p d\,w(t),
\qquad
\sqrt D\, A_d \nabla h = (\eta/l)\,w.
$$
These are the two height identities. Second, the full divergence
identity takes the shape
$$
\mathrm{div}\,V + ku\,\Xi_a
= \text{trace-free part} + \text{(lower-order mass term)},
$$
with $V = uA_\chi(k\nabla Z + K(w,\cdot))$ and
$\Xi_a = (\mathrm{tr}\,K_a)^2 - |K_a|^2 - 2\mu$ a scalar built from
the dominant-energy sources with $K_a = aK$.

The penalty. To close the system at each truncation, subtract a source
$s\,\rho$ from the divergence equation, with $s\in[0,1]$ a homotopy
parameter and $\rho$ small; and add a compensation source
$m_0 C_N \rho_0$ with $m_0(t)=\vartheta(N(t+\epsilon))$, where
$\vartheta$ is a smooth cutoff equal to one on $t \le -\epsilon$ and
zero on $t \ge 0$; $C_N$ a large constant; $\rho_0 \ge 0$ a smooth
compact-source bump, and $s$ a second homotopy parameter. The
resulting penalized system reads
$$
\mathrm{div}(kuA_\chi\nabla Z)
= u\Xi_a + s u\rho - m_0 C_N u\rho_0
  - \mathrm{div}(uA_\chi K_a(w,\cdot)).
$$

When $s=1$ and no floor is present, the penalty keeps $t$ from reaching
$-\epsilon$: at any point where $t=-\epsilon$, the source term
$m_0 = 1$ makes the right side strictly negative, while the left
side at a minimum is nonnegative by the maximum principle. This is
the content of the floor lemma below.

Floor lemma. Fix $\epsilon$, $N$, $R$. Let $(f,Z)$ be a smooth solution
of the penalized system with $t\ge -\epsilon$ (weak sense) on
$\mathcal M_{N,R}$ and $Z=0$ on $S_R$. Then $\min t > -\epsilon$,
strictly.

Proof of floor lemma. On the graph metric $\widehat g = e^{2t}\bar g$
with $\bar g = g + l^2 df \otimes df$. At any minimum of
$t$:
$$
\tfrac12 e^{2t} R_{\widehat g}
\ge \tfrac12 R_g - v\,\mathrm{Ric}_g(e,e) + \mathcal S(H^f).
$$
The constraint identity reads
$$
\tfrac12 R_g - v\,\mathrm{Ric}_g(e,e) + \mathcal S(H^f)
\ge \text{(dominant-energy, from Xi)} + \text{(flux terms)}.
$$
Combining the constraint bound with the coercivity estimate and the
height identities, and integrating against a truncated bump function
concentrated near the minimum of $t$, gives
$$
0 \ge c_* \mathcal T - \mathcal P_N(|K|^2 - \mu_\Xi) + \text{flux}.
$$
Because $\mu_\Xi$ is positive-definite on the original data and the
$K$-term has a coercivity deficit controlled by $\mathcal P_N$, while
the penalty at $t=-\epsilon$ contributes $-m_0 C_N \rho_0 = -\rho_0 C_N$
and dominates all positive terms for $C_N$ large, the inequality
cannot be satisfied. So $t > -\epsilon$ strictly. QED.

Penalty flux and ADM normalization. Fix N, R, and take s=1. On any
above-floor solution of the penalized system with $Z=0$ on $S_R$,
integrate the divergence identity over $\mathcal M_{N,R}$ (which has
no inner boundary) and let $R\to\infty$. The boundary flux of
$V = uA_\chi(k\nabla Z+K(w,\cdot))$ matches the ADM expansion of
$\widehat g$, and the surviving bulk integral is
$$
\widehat E - E_0 = -\frac{1}{k\omega_{n-1}}
\int_{\mathcal M_N} u\,\Xi_a\,dV_g.
$$
The quantity $\widehat E$ is the ADM energy of the graph metric $\widehat g$.
A matching calculation for the ADM momentum shows it is preserved
when $K$ is compactly supported and $N$-large.

Bounding the integral. The height identities
$$
|dh|_{A_d}\le\ell^{1/2},
\qquad
|e^{2t}\Delta_{\widehat g}h| \le \mathcal P_N \ell^{1/2}(1+\sqrt{\mathcal T})
$$
compare $u\,\Xi_a$ with $\eta a_w|df| \le \eta\ell^{1/2}$ plus a
$\mathcal T$-controlled remainder. The upper bound on $Z$ (proved
below via Moser iteration) is what closes the $\mathcal T$-integral.
The result is
$$
\int_{\mathcal M_N}u\Xi_a\,dV_g \ge -\mathcal P_N(\ell+\ell^{1/2}),
$$
and hence
$$
\widehat E - E_0 \le \mathcal P_N(\ell+\ell^{1/2}).
$$

Metric domination. The floor lemma gives $t>-\epsilon$ pointwise, and
$|df|^2 \ge 0$. Hence
$$
\widehat g = e^{2t}(g + l^2 df\otimes df)
           \ge e^{-2\epsilon}g
$$
as quadratic forms. Every full enclosing cut of $\widehat g$ is a
full enclosing cut of $g$ with $\widehat g$-area at least
$e^{-(n-1)\epsilon}$ times its $g$-area. This is what preserves
enclosure during the metric deformation.

---

Analytic existence. Fix prescribed prepared data, $N$, $\epsilon$. The
task at each $R$ is to solve the penalized system in $X_R = \{z \in
C^{1,\gamma}(\mathcal M_{N,R}) : z|_{S_R} = 0\}$ for some
$0<\gamma<1$. All constructions are by degree theory on this Banach
space and its compact maps.

The trace equation. For a trial $z \in X_R$ and $a \in [0,1]$, solve
$$
\mathrm{tr}_{A_\chi}\bigl(K_a + H^f\bigr) = \eta f,
\qquad f|_{S_R} = 0,
\qquad
t + \frac{1}{2k}\log\bigl(1+l(t)^2|df|^2\bigr) = z.
$$
The third line defines $t$ uniquely from $z$ and $|df|^2$. The
operator is uniformly elliptic for each fixed $N$ because $A_\chi$ is
positive definite with lower bound $\chi \ge k/(k+N)$ and the graph
solution $f \in C^{3,\gamma}$ is unique at each input (Lemma
trial-trace, stated later).

The height lemma on the far end. In the tail $r \ge r_1$, the far
end of the metric dominates, $K_a = 0$, and the trace equation
reduces to $A_\chi : \mathrm{Hess}\,f = \eta\sqrt D \cdot f / l$.

On an exact Euclidean metric with $A_\chi \approx \mathrm{diag}(d, 1,
\dots, 1)$ and the axial direction $e$ radial, testing against
$H(r) = C(r^{-b_1}-R^{-b_1})$ for $0 < b_1 < 1$ gives on a contact
point
$$
A_\chi : \mathrm{Hess}\,H
= Cb_1 r^{-b_1-2}\bigl((b_1+1)\chi-k\bigr),
$$
which is strictly negative because $b_1+1<k$. The small metric error
$g-\delta = O_2(r^{2-n})$ adds $O(r^{2-n})$ to the coefficient, which
does not overturn the sign for $r \ge r_1$ independent of $N$.

The comparison principle then gives
$$
|f| \le C(r^{-b_1}-R^{-b_1})
\quad\text{on } r_1 \le r \le R,
\qquad |\eta f| \le C,
$$
with $C$ independent of $R$. Taking the one-sided derivative at
$S_R$ gives
$$
|df|_{S_R} \le C\eta^{-1} R^{-b_1-1},
$$
so the identity $Z = t + (2k)^{-1}\log(1+l^2|df|^2)$ bounds $t$ from
below on $S_R$:
$$
0 \ge t \ge -\frac{1}{2k}\log\bigl(1+e^2C^2\eta^{-2}R^{-2b_1-2}\bigr).
$$
Choosing $R_{\min}(N)$ large enough makes $t > -\epsilon$ on $S_R$ for
all $R \ge R_{\min}(N)$.

Upper bound on Z. Fix $\alpha \in (0,\min(1,k-1))$. On the enlarged
core and on the drift support,
$$
\sup_Q e^Z \le \exp(\mathcal P(N))
$$
on every coordinate ball $Q$ of uniformly bounded geometry, with the
bound independent of $R$ and of the homotopy parameter.

Step 1: pointwise distortion. The implicit relation
$Z = t+(2k)^{-1}\log D$ with $l \ge \ell$ and $t\ge-\epsilon$
implies, whenever $Z$ stays below a fixed threshold $Z_0$, that
$e^{\alpha Z}$ is uniformly bounded. No case analysis is needed there.

Step 2: integral bound above the threshold. Two structural bounds are
available above $Z_0$. First,
$$
\frac{e^{\alpha Z}\sqrt D}{u}
= l^{-1}e^{(\alpha-k+1)t}D^{\alpha/(2k)}
\le \exp(\mathcal P(N))
$$
because $1/2+\alpha/(2k)\le 1$. Second, in the regime $D \ge 2$,
$$
u\eta a_w|df| = \eta e^{(k-1)t}(D-1),
\qquad
e^{\alpha Z}\sqrt D = e^{\alpha t}D^{1/2+\alpha/(2k)},
$$
and the ratio is bounded by $C\eta^{-1}$ times the first expression,
hence by $\exp(\mathcal P(N))$.

Step 3: Sobolev iteration. Testing the trace equation on the graph of
$f$ and combining with the divergence equation gives, on each patch,
$$
\int_Q e^{\alpha Z}\,dV_g \le \exp(\mathcal P(N)).
$$
The standard Moser iteration on the divergence equation with
$q$ increasing by factor $2\kappa_n/( \kappa_n-n)>1$ and geometrically
shrinking patches converges. The bound
$$
\sup_{Q'}e^Z \le \exp(\mathcal P(N))\cdot
\|e^{Z}\|_{2q_0,Q}
$$
follows from finite sums over $O(1+N)$ patches.

Step 4: exclusion of a large maximum outside the drift support. Suppose
the maximum of $Z$ on the core exceeds the bound. At the point of
maximum, $dZ=0$ and $Z$ is at least $Z_0+1$. If the maximum is
outside the source support, then $K_a=0$ there, and
$u^{-1}\mathrm{div}(kuA_\chi\nabla Z)
 = kA_\chi : \mathrm{Hess}\,Z \le 0$. But the source is
$\Xi_a > 0$, contradiction. So maxima lie in the enlarged core or the
drift support, and the polynomial bound applies.

Gradient and Hessian mass on $f$. The trace equation
$$
A_\chi : \mathrm{Hess}_g f = \frac{\sqrt D}{l}\bigl(\eta f - \mathrm{tr}_{A_\chi}K_a\bigr)
$$
is uniformly elliptic with measurable axial coefficient
$\chi \in [k/(k+N),1]$. Its right side is bounded and its gradient
dependence is quadratic. Schauder theory is unavailable because $Z$ is
not yet continuous. The following two-step mechanism supplies both a
H\"older modulus for $Df$ and a local Hessian mass bound.

Step A: strict absorption. Normalize the metric at a point. The
principal matrix
$$
a^{ij} = g^{ij} - (1-\chi) e^i e^j
$$
satisfies $|I-a| \le 1-c$ whenever $\chi \le 1-c$. Then Young's
inequality, with gap $\gamma = 1-(1-c)^2>0$, gives, when
$s=1-|P|_g^2\ge 0$ with $P=\nabla_g f$,
$$
\mathrm{div}_g\bigl(A\nabla_g s + 2GP\bigr) \le -\gamma|H^2_g f|^2 + C_cG^2 + 2|\mathrm{Ric}|_g
$$
where $G$ collects the right side of the trace equation and the
Christoffel terms. Discarding the negative Hessian term and applying
De Giorgi-Nash to $(1-|P|^2)^{1/2}$ gives a weak Harnack bound.

Step B: flux and reflection. Extend the equation oddly at a Dirichlet
face and use the trace-reversed identity
$$
\mathrm{div}_g\mathcal F = |\mathrm{Hess}\,f|^2 - (\Delta_g f)^2 + \mathrm{Ric}_g(P,P),
\qquad
\mathcal F = (H^\sharp - (\Delta_g f)\mathrm{id})P,
$$
to reflect the equation and rule out a large angular component.

Iterating Step A / Step B over dyadic scales yields the estimate
$$
\|Df\|_{C^{\alpha_1}(B_{1/2})} + \int_{B_{1/2}}|D^2f|^2
\le C
$$
for some fixed $\alpha_1 \in (0,1)$ and constant independent of $R$
and of the homotopy parameter.

Regularity proposition. Fix $N$. Any above-floor solution of the
penalized system on $\mathcal M_{N,R}$ has bounds of every fixed
order on coordinate balls of small fixed size, uniform in $R$ and the
two homotopy parameters. In particular, $C^{1,\gamma}$ norms are
bounded on each fixed truncation, for any chosen $0<\gamma<1$.

Proof. Step 1 (f): apply the gradient and Hessian mass estimate above
with the source controlled by $Z$. Step 2 (Z): apply the scalar
divergence lemma with source $|D^2f|^2$; the mass bound
$\int_{B_r}|D^2f|^2 \le C r^{n-2+a}$ satisfies $\nu(B_r)\le Cr^{n-2+a}$
for the natural measure $\nu = (1+|D^2f|^2)\,dV$. The lemma returns an
$C^b$ modulus for $Z$ on smaller patches, for some $b>0$.
Step 3 (coupled): with $Z$ and $Df$ now H\"older, the coefficients of
the trace equation are $C^c$. Interior Schauder and absorbing gradients
give
$$
\|f\|_{C^{2,c}} \le C
$$
with $c = \min(a,b) > 0$. Step 4 (bootstrap): differentiate the
divergence equation to obtain $Z \in C^{2,d}$ for any $d<1$; apply
the Schauder estimate in $C^{2,d}$ and repeat to obtain bounds of
order up to any fixed $K$. All constants depend polynomially on $N$;
none depend on $R$.

The absorption and the mixed-boundary Schauder estimate are the
standard tools of uniformly elliptic theory; once the first round of
estimates has been made, local Schauder theory gives bounds of every
fixed derivative order.

Trial trace lemma. Let $\mathcal D$ be a compact smooth $n$-manifold
with $n\ge 3$ and nonempty smooth boundary. Let $K$ be a smooth
symmetric covariant tensor on $\mathcal D$. Fix $N>k$, $0<\gamma<1$,
positive $\eta$. Then for every $z \in C^{1,\gamma}(\mathcal D)$ and
$a\in[0,1]$, the Dirichlet problem
$$
\mathrm{tr}_{A_\chi}\bigl(K_a + H^f\bigr) = \eta f,
\qquad
f|_{\partial \mathcal D} = 0,
\qquad
t + \frac{1}{2k}\log\bigl(1+l(t)^2|df|^2\bigr) = z
$$
has a unique solution $f = f_a[z]\in C^{3,\gamma}(\mathcal D)$. The
solution map $(a,z)\mapsto f_a[z]$ is continuous from
$[0,1]\times C^{1,\gamma}(\mathcal D)$ to $C^{3,\gamma}(\mathcal D)$
and sends bounded sets to bounded sets. All constants may depend on
$N,\eta,\mathcal D,g,K$ and on the prescribed bound for $z$. No lower
bound on the minimum of $t$ is required by the statement.

Proof. Height bound. Extrema of $\pm f$. At a positive maximum of $f$,
$\nabla f=0$, $A_\chi=I$, so
$$
A_\chi : \mathrm{Hess}\,f \le 0,
\qquad
\frac{\sqrt D}{l}\bigl(\eta f-\mathrm{tr}_{A_\chi}K_a\bigr) \ge \eta f - C.
$$
At the maximum the left side is at most zero, so $\eta f \le C$ and
$f \le C/\eta$. The negative minimum gives $\ge -C/\eta$. The bound
is attained uniformly in $q$ and $a$.

Continuation. For $q\in[0,1]$ set $A_q = (1-q)I + qA_\chi$ and
$c_q = 1-q+q\eta b$, with $b = \sqrt D/l$. Consider the equation
$$
A_q : \mathrm{Hess}\,f - c_q f = -q b\,\mathrm{tr}_{A_\chi}K_a,
\qquad f|_{\partial \mathcal D} = 0.
$$
At $q=0$ this is a Dirichlet Laplacian with positive zeroth-order
term, solvable by Lax-Milgram and Schauder. At any solution,
extrema give $|f|\le C/\eta$; $c_q$ is positive. On that height range,
$$
|G(x, f, df)| \le C(1+|df|),
$$
uniformly in $q$ and $a$. Fixed-$q$ solutions are unique: their
difference satisfies a linear equation with a positive zeroth-order
coefficient, so it has no nonzero extremum. The linearization is a
Fredholm operator of index zero with no kernel, hence an isomorphism.
Implicit function theorem gives openness of the set of $q$ with a
solution, and compactness plus the uniform height bound gives
closedness. So all $q\in[0,1]$ and $a\in[0,1]$ are covered.

Gradient estimate at the Dirichlet face. Choose $K_0>0$ large, then a
distance $d_0$ less than the boundary tubular radius. Set
$$
\psi(d) = K_0^{-1}\log\bigl(1+K_0 M d\bigr),
\qquad
\psi'' = -K_0(\psi')^2,
$$
with $M$ chosen so $\psi'(d_0)\ge 1$ and $\psi(d_0) > 2\|f\|_\infty$.
Comparing with $\pm\psi(d(x,\partial \mathcal D))$ in the collar:
the gradient at contact is purely normal, of magnitude $\sigma=\psi'$.
The tangential Hessian contributes at most $C\sigma$, whereas
$$
\chi_q\psi'' \le -\frac{c K_0\sigma^2}{1+\sigma} \le -\frac{c K_0}{2}\sigma.
$$
For $K_0$ large this is a strict supersolution, dominating $\pm f$ on
the inner edge and equal to zero on the face. Thus
$|f(x)| \le \psi(d(x,\partial \mathcal D))$ near $\partial \mathcal D$.

Schauder. Interior $W^{2,p}$ with $p>n$ on the fixed Sobolev range,
division by the uniform lower bound on $\chi_q$, and $C^{0,\alpha}$-in-$x$
dependence on the coefficients give $C^{1,\alpha_0}$ regularity locally.
The gradient estimate extends this to the Dirichlet face. Schauder
then gives $C^{2,\alpha_0}$, and differentiating the implicit equation
in $t$, one obtains $C^{3,\alpha_0}$. Since $z \in C^{1,\gamma}$ and
$f\in C^{3,\gamma}$ iteratively, the full regularity holds. QED.

Compact fixed-point map. Turn to the actual problem on $\mathcal M_{N,R}$.
Put $X = \{z\in C^{1,\gamma}(\mathcal M_{N,R}) : z|_{S_R} = 0\}$.
For each trial $z$, solve $f = f_a[z]$, evaluate $t$ by the implicit
equation, and define $T_a(z) = Z_{\mathrm{out}}$ as the unique smooth
solution of the Dirichlet problem
$$
\mathrm{div}(ku A_\chi\nabla Z_{\mathrm{out}})
= u\Xi_a - \mathrm{div}(uA_\chi K_a(w,\cdot)),
\qquad
Z_{\mathrm{out}}|_{S_R} = 0.
$$
The source $\Xi_a$ is evaluated at the input $z$; only the gradient of
$Z_{\mathrm{out}}$ is the unknown. On bounded subsets of $X$, the
trace lemma gives uniform $C^{3,\gamma}$ bounds for $f$. The principal
coefficient $kuA_\chi$ is $C^{1,\gamma}$ and uniformly positive
definite; the drift flux is $C^{1,\gamma}$; $u\Xi_a$ is $C^{0,\gamma}$.
By the linear scalar Dirichlet theorem and Schauder estimate, $T_a$
is continuous and compact from $X$ to itself, uniformly in the two
homotopy parameters.

Second homotopy. Take $a=0$, $s\in[0,1]$, with source $s\,u\Xi_0$.
Since $K_0 = 0$, the trace equation becomes
$\mathrm{tr}_{A_\chi}H^f = \eta f$. A positive maximum of $f$ gives
$A_\chi:\mathrm{Hess}\,f \le 0$ and $\eta f >0$ at the maximum,
contradiction; a negative minimum is analogous. So $f\equiv0$, hence
$t=Z$ and $T_{0,0}(z) \equiv 0$ for every $z$.

Degree domain. For a combined homotopy parameter $\lambda$ taking
values in the product $[0,1]\times[0,1]$, put
$$
\mathcal U_\lambda = \bigl\{z\in X : \min t[z, df_\lambda[z]] > -\epsilon,
\ \|z\|_{C^{1,\gamma}} < M\bigr\},
$$
$$
\mathcal U = \{(\lambda, z)\in[0,1]^2\times X : z \in \mathcal U_\lambda\}.
$$
The set is relatively open in the product since $t$ depends
continuously on $(z, f_a[z])$. Choose $M$ above the $C^{1,\gamma}$ bound
for all above-floor fixed points; the norm boundary is then free of
fixed points. The floor lemma excludes the other boundary.
The closure of $\mathcal U$ is compact: the parameter squares are
compact, the norms are bounded, and the trace and $T$ maps are
compact. Excision and the standard Leray-Schauder homotopy invariance
give the same degree at either end of the parameter square.

Base and unit degree. At $(a,s)=(0,0)$, $T_{0,0}(z) = 0$ for every
$z\in X$, so the only fixed point is $z = 0$ and the degree is one.
First follow the second homotopy to $s=1$, then the first to $a=1$.
The degree stays at one, giving a smooth solution of the penalized
system on every sufficiently large truncation.

Exhaustion to $R=\infty$. The regularity estimate gives uniform $C^K$
bounds on each compact subset of $\mathcal M_N$, for each fixed $K$,
with constants depending on $N,\epsilon,K$ but not on $R$. A diagonal
smooth limit as $R\to\infty$ gives a solution of the penalized system
on the whole $\mathcal M_N$, at fixed $s=1$ and $a=1$.

End decay. On the far spherical annulus $r\ge r_1$, the metric
approaches flat Euclidean, coefficients of the divergence equation are
$O(r^{-b_1})$-perturbations, and the trace solution is $O_j(r^{2-n})$
at every fixed derivative order. Interior $W^{2,p}$ and Schauder
estimates on rescaled annuli give
$$
Z,t = O_j(r^{2-n}),
\qquad
|dZ|^2 = O(r^{2-2n})\quad\text{on each end annulus}.
$$
Because $\beta < n-2$, the divergence source $|dZ|^2 = O(r^{-n-\beta})$
is integrable. The end equation then gives the corresponding decay
for $\Delta_g t$, and differentiation recovers every fixed derivative.

Energy flux. On the limiting solution, the flux identity becomes
$$
\widehat E - E_0
= -\frac{1}{k\omega_{n-1}}\int_{\mathcal M_N}u\Xi_a\,dV_g
\ge -\mathcal P_N(\ell+\ell^{1/2}).
$$
This closes the elliptic construction at fixed $N$ and $\epsilon$. The
metric domination $\widehat g \ge e^{-2\epsilon}g$ remains in force
by the floor lemma.

Height separation. Throughout, the data are prepared: $K$ is compactly
supported, the dominant energy condition is strict, every component of
$S$ has strictly negative future expansion. Fix $0<\epsilon<1$. Set
$\ell = e^{-N\epsilon}$, $\eta = \ell^{3/2}$, $h=\eta f$, and take the
graph metric $\widehat g = e^{2t}(g + l^2 df\otimes df)$ as before. The minimizing enclosure will be forced into the
small-height band $|h| \le b_3$ and placed away from the filling.

Height bound. The far-end comparison of the trace equation gives
$|h| \le C$ on $r\ge r_1$, with $C$ independent of $N$. Integrating
the height identities gives $|dh|_{A_d}\le \ell^{1/2}$. So the whole
band $|h|\le b_3$ lies, from the far side, inside a fixed coordinate
radius.

Plateau metric. Choose $0<b_3<b_2/4$ small enough that
$$
\mu - |J|_g - \rho - b_3|\mathrm{tr}_gK| \ge \tfrac12\rho > 0
$$
on the original exterior $\Omega$. Such $b_3$ exists by strictness of
the prepared DEC margin and smallness of $|\mathrm{tr}_gK|$ on the
fixed inner region.

Let $\zeta\in C^\infty(\mathbb R)$ be nonnegative, equal to zero for
$|z| \le b_3/4$, equal to one for $|z| \ge b_3/2$, with values in
$[0,1]$. Set
$$
A_N = (\ell/\eta)^{1/2} = \ell^{-1/4},
\qquad
\psi_N = A_N \zeta(h),
\qquad
\widetilde g_N = e^{2\psi_N}\widehat g.
$$
On the far end, $|h|\le b_3/4$ by the height bound, so $\psi_N=0$ and
$\widetilde g_N$ agrees with $\widehat g$ asymptotically. In
particular, the changes on receding annuli do not affect any ADM term.

Curvature lemma. For every sufficiently large $N$:
$$
R_{\widetilde g_N} \ge 0
\quad\text{on } \Omega\cap\{|h|\le b_3\},
\qquad
\widetilde g_N \ge e^{-2\epsilon}g \text{ on }\Omega,
$$
and
$$
E(\widetilde g_N) \le E + \mathcal P_N(\ell+\ell^{1/2}).
$$

Proof. The conformal law gives
$$
e^{2\psi_N}R_{\widetilde g_N}
= R_{\widehat g} - 2k\,\Delta_{\widehat g}\psi_N
  - k(k-1)|d\psi_N|^2_{\widehat g}.
$$
In the interior $|h| \ge b_3/2$, $\zeta(h)=1$ there, so $\psi_N=A_N$
is locally constant, $D\psi_N=0$, $\Delta\psi_N=0$, and
$e^{2A_N}R_{\widetilde g_N}=R_{\widehat g}$. Since $R_{\widehat g}$ is
bounded below by the constraint inequality for the graph metric and
$A_N\to\infty$, the scaled curvature is at most a factor $e^{-2A_N}$
times a bounded term, hence nonnegative.

In the transition shell $b_3/4\le|h|\le b_3/2$, the height identities
and the trace equation give
$$
|d\psi_N|_{\widehat g} \le A_N\ell^{1/2} = \ell^{1/4},
\qquad
|\Delta_{\widehat g}\psi_N| \le A_N(\mathcal P_N\ell^{1/2}+\ell) \le \mathcal P_N\ell^{1/4}.
$$
Both are perturbatively small. The constraint bound on $R_{\widehat g}$
dominates. On $|h|\le b_3/4$, $\psi_N=0$ by definition, so
$R_{\widetilde g_N}=R_{\widehat g}\ge 0$ there. Combining the three
zones gives the first claim.

For the metric comparison, $\zeta\ge 0$ and $A_N>0$ give
$\psi_N\ge 0$, so $e^{2\psi_N}\ge 1$ and
$\widetilde g_N \ge \widehat g \ge e^{-2\epsilon}g$.

For the energy, the perturbation of the ADM flux caused by $\psi_N$
is supported in the transition shell, where $|\psi_N|\le A_N$,
$|d\psi_N|\le \ell^{1/4}$, and $|D^2\psi_N|\le\mathcal P_N\ell^{1/4}$.
The ADM flux changes by $O(\ell^{1/4})$ from the shell integral plus
$O(\ell^{1/4})$ from the boundary term at $S_R$, so
$E(\widetilde g_N)\le E + \mathcal P_N(\ell+\ell^{1/2})$.

Perimeter minimization. The minimizing set will be built at the level
of sets of finite perimeter in $(\mathcal M_N, \widetilde g_N)$. Fix
$$
C_N^h = \{|h|\ge b_3\},
\qquad
O_N = \{x : \mathrm{dist}_g(x,C_N^h)\le s_N\},
\qquad
s_N = e^{-Q_N},
$$
with $Q_N$ a polynomial large enough that $s_N<r_0/10$ and
$3s_N\sup|dh|_g < b_3/2$. Then the whole $2s_N$-neighborhood of $O_N$
lies in $\{|h|>b_3/2\}$, where $\widetilde g_N$ is a fixed conformal
rescaling of $\widehat g$. The obstacle $O_N$ is the union of the added
filling and a thin distance collar around $C_N^h$. No smoothness of
$\partial O_N$ is asserted.

Existence of a minimizing set. There is a bounded finite-perimeter
set $F_N \supset O_N$, modulo null sets, minimizing
$P_{\widetilde g_N}$ among bounded sets containing $O_N$. One may
take it to contain every other minimizer modulo null sets. Its
perimeter satisfies
$$
P_{\widetilde g_N}(F_N) \le e^{\mathcal P_N}.
$$

Proof. At fixed $N$, the end is smooth and asymptotically Euclidean
with the differentiated falloff already established. Sufficiently
large coordinate spheres have positive mean curvature for
$\widetilde g_N$ toward infinity. The radius may depend arbitrarily
on $N$.

Clipping. On the large end let $Y=\nabla r/|\nabla r|$; so $|Y|=1$
and $\mathrm{div}\,Y=H_r>0$. For a bounded finite-perimeter
competitor $F$ and almost every large $R$, Gauss--Green on $F\cap\{r>R\}$
gives
$$
\int_{F\cap\{r>R\}}H_r\,dV
= \int_{\partial^*F\cap\{r>R\}}\langle Y,\nu_F\rangle\,dA
- \mathcal H^k(F^{(1)}\cap S_R).
$$
The spherical normal is $-Y$. Boundedness of $F$ eliminates the
outer flux; or one cuts $Y$ off beyond its support. The first
boundary integral is at most $P(F;\{r>R\})$. The BV truncation
formula, valid at almost every radius by standard trace and slicing
theory, gives
$$
P_{\widetilde g_N}(F\cap\{r<R\})
\le P_{\widetilde g_N}(F)
- \int_{F\cap\{r>R\}}H_r\,dV_{\widetilde g_N}.
$$
The notation $\{r<R\}$ includes the entire compact core, hence $O_N$
once $R$ is large enough.

Applying this clipping inequality to each term of a minimizing
sequence, with an almost-everywhere-admissible radius in a fixed
interval beyond the obstacle, confines the sequence to one compact
set. BV compactness gives $L^1$ convergence along a subsequence,
and lower semicontinuity gives a minimizer. Containment of $O_N$ is
preserved almost everywhere.

Competitor bound. Choose a fixed original coordinate sphere $S_{R_*}$
beyond $O_N$. On and beyond $S_{R_*}$ one has $|h|<b_3/4$, so
$\psi_N=0$ and, from the floor lemma,
$\widehat g\le e^{\mathcal P_N} g$. The filled side of $S_{R_*}$ is
a competitor, so
$$
P_{\widetilde g_N}(F_N)\le e^{\widetilde{\mathcal P}_N}
P_{\widetilde g_N}(S_{R_*})\le e^{\mathcal P_N}.
$$
The fixed sphere itself need not lie in the region used for
clipping: that was needed only for compactness at fixed $N$.

Detachment lemma. For all sufficiently large $N$, the frontier
$\Sigma_N$ of $F_N$ has positive distance from $O_N$:
$$
\mathrm{dist}_{\widetilde g_N}(\Sigma_N, O_N) > 0.
$$

Proof. Two different density arguments are combined. Suppose $p$
belongs to the frontier of $F_N$ and to $O_N$. It cannot lie in the
interior of $O_N$, because $F_N$ contains $O_N$ almost everywhere.
Thus $\mathrm{dist}_g(p, C_N^h) = s_N$. The complete background
metric has a minimizing geodesic from some $z\in C_N^h$ to $p$, so the
midpoint of that geodesic
is the center of a ball $B^g_{s_N/4}$ contained in both $O_N$ and
$B^g_{s_N}(p)$. Uniform background ball-volume bounds give the
filled-side density
$$
\mathrm{Vol}_g(F_N\cap B^g_{s_N}(p)) \ge c s_N^n.
$$

For the unfilled side, work in $\mathcal M_N\setminus F_N$. Since $p$
is in the support of the perimeter measure, $V(r)=\mathrm{Vol}_g(U\cap
B^g_r(p))$ satisfies $V(r)>0$ for every $r>0$: otherwise $1_{F_N}$
would be constant almost everywhere on a neighborhood of $p$. For
almost every $0<r<s_N$, filling the ball is an admissible competitor
(with the same filled set away from the ball), so the perimeter
truncation formula gives
$$
P_g(F_N;B^g_r(p)) \le \mathcal H^k_g(U^{(1)}\cap \partial B^g_r(p)).
$$
The coarea formula and the pointwise comparison (from the metric
domination lemma: $e^{-\mathcal P_N}dA_g \le dA_{\widetilde g_N}\le
e^{\mathcal P_N}dA_g$ on this ball) then yield
$$
P_g(U;B^g_r(p)) \le e^{\mathcal P_N}V'(r)
\quad\text{for almost every }r\in(0,s_N).
$$

Relative isoperimetry in $B^g_{s_N}(p)$, together with the
filled-side density and the unfilled-side density, gives
$$
P_g(F_N;B^g_{s_N}(p)) \ge e^{-\mathcal P_N}s_N^{n-1}.
$$
Absorbing $s_N^{n-1}=e^{-(n-1)Q_N}$ into a larger polynomial, using
the pointwise comparison, and multiplying by the conformal factor over
the obstacle gives the contact cost
$$
P_{\widetilde g_N}(F_N)\ge \exp\bigl(kA_N-\mathcal P_N\bigr)
= \exp\bigl(k\ell^{-1/4}-\mathcal P_N\bigr).
$$
Since $\ell=e^{-N\epsilon}$, this equals
$\exp(k e^{N\epsilon/4}-\mathcal P_N)$. For fixed $\epsilon>0$,
$e^{N\epsilon/4}$ dominates every polynomial in $N$. The contact cost
therefore contradicts the competitor bound
$P_{\widetilde g_N}(F_N)\le e^{\mathcal P_N}$ for large $N$. So
$\Sigma_N$ is a positive distance from $O_N$.

The exterior. Choose $F_N$ in the existence lemma, with the frontier
$\Sigma_N$. By the detachment lemma $\Sigma_N$ is compact and has
positive distance from $O_N$. Every sufficiently small variation of
$F_N$ near a frontier point remains admissible, so its boundary is
locally perimeter minimizing in the smooth metric $\widetilde g_N$.

Two-sided density. Both insertion and deletion of a sufficiently small
ball are admissible at a frontier point. Applying the detachment
two-density argument separately to $F_N$ and to its complement gives
positive lower volume density on both sides of every point in the
perimeter support. On a ball that meets the support nowhere, the
indicator has zero distributional derivative, hence is constant almost
everywhere. Let $U_N$ be the union of those balls on which the
constant is zero, and let the closed filled representative be
$\mathcal M_N\setminus U_N$. The two density bounds place the
topological frontier exactly on the perimeter support: every
neighbourhood of a support point contains both filled and unfilled
points, and outside it the indicator is constant. The representative
agrees with $F_N$ up to a null set; the regularity theorem makes its
frontier measure null. Its perimeter and all preceding minimizing
comparisons are therefore unchanged.

Completeness. The filled metric is complete because
$\widetilde g_N\ge e^{-2\epsilon}g$ and the background filled metric
is complete. Its restriction to $X_N$, the closed component of
$\mathrm{int}(\mathcal M_N)\setminus F_N$ that contains the distant
end, is complete with the frontier included. For intrinsic length
distance on $X_N$: a Cauchy sequence has a subsequence whose successive
intrinsic distances are below $2^{-j}$. Join successive terms by curves
of length below $2^{1-j}$ in $X_N$. Their concatenation has an ambient
endpoint, by ambient completeness, and that endpoint lies in $X_N$
because $X_N$ is closed. Appending this endpoint gives a curve in
$X_N$ whose tail length tends to zero, proving intrinsic convergence.

Full-cut comparison. Detachment and the height bound put
$\Sigma_N$ in $\mathrm{int}(\Omega)$, separated from a neighbourhood of
the entire original filling, including every component of $S$. There
are bounded smooth open sets $F_j$ in the filled manifold such that
$$
1_{F_j}\to 1_{F_N}\text{ in }L^1,
\qquad
P_g(F_j)\to P_g(F_N),
$$
and each $F_j$ contains a fixed neighbourhood of the whole filling. One
constructs such $F_j$ by smoothing $1_{F_N}$ in finitely many charts
away from the two constant regions; coarea yields a regular level
$t_j$ with the stated perimeter limit and $L^1$ convergence.

For each $j$ retain the component of
$\mathcal M_N\setminus\overline{F_j}$ that contains the distant end,
and let $D_j$ be its closure. Retaining this exterior component merely
selects some entire components of a compact smooth boundary; it
preserves smoothness and only decreases the full boundary area. Since
$F_j$ contains a neighbourhood of the whole filling, $D_j\subset\Omega$,
its interior lies in $\mathrm{int}(\Omega)$, and $D_j$ is closed and
connected with compact smooth intrinsic boundary $\Gamma_j$. It is an
admissible full cut, so $a\le\mathrm{Area}_g(\Gamma_j)\le P_g(F_j)$.
Taking the limit gives $P_g(F_N)\ge a$.

Metric comparison. Since $\widetilde g_N\ge e^{-2\epsilon}g$ on
$\Omega$, comparison on every approximate tangent $k$-plane yields
$$
\mathcal H^k_{\widetilde g_N}(\Sigma_N) = P_{\widetilde g_N}(F_N)
\ge e^{-k\epsilon}P_g(F_N)\ge e^{-k\epsilon}a.
$$
The first equality uses the regularity theorem and the zero
measure of the singular set.

The Riemannian endpoint. The exterior $X_N$ is a complete
enclosing-set exterior, its compact frontier $\Sigma_N$ included, with
metric $\widetilde g_N$ smooth through the frontier. The single
Euclidean end has compact complement and decay
$\widetilde g_N-\delta = O_2(r^{2-n})$ with $n-2>(n-2)/2$. Its scalar
curvature is nonnegative and integrable, with pointwise decay
$O(r^{-n-\beta})$ for $\beta>0$. The whole frontier is outer minimizing
and locally perimeter minimizing as the boundary of the filled side,
its regular part is a smooth embedded minimal hypersurface, and its
singular set has dimension at most $n-8$. The following theorem is
proved in the Riemannian companion.

Riemannian theorem (companion). Let $X^n$ be a connected complete
enclosing-set exterior, its compact frontier $\Sigma$ included, with
metric $\gamma$ smooth through that frontier. Suppose there is one
Euclidean coordinate end with compact complement,
$\gamma-\delta = O_2(r^{-\tau})$ with $\tau>(n-2)/2$, and
$R_\gamma\ge 0$, $R_\gamma\in L^1$, $R_\gamma = O(r^{-n-\beta})$ for
some $\beta>0$. Suppose its full frontier $\Sigma$ is outer minimizing
and locally perimeter minimizing as the boundary of the filled side,
allowing compactly supported variations on either side. Its regular
part is a smooth embedded minimal hypersurface and its singular set
has Hausdorff dimension at most $n-8$. Then
$$
E(\gamma) \ge \frac12
\left(\frac{\mathcal H^{n-1}_\gamma(\Sigma)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$
The assertion includes zero frontier area and imposes neither a spin
condition nor connectedness of the frontier.

Prepared energy inequality. Apply the Riemannian theorem to $X_N$:
$$
E(\widetilde g_N) \ge \frac12
\left(\frac{\mathcal H^{n-1}_{\widetilde g_N}(\Sigma_N)}
   {\omega_{n-1}}\right)^{(n-2)/(n-1)}
\ge \frac12\left(\frac{e^{-(n-1)\epsilon}A_0}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$
Combining with the energy bound,
$$
E_0+\mathcal P_N(\ell+\ell^{1/2})
\ge \frac12\left(\frac{e^{-(n-1)\epsilon}A_0}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$
Let $N\to\infty$ first; the exponential decay $\ell=e^{-N\epsilon}$
dominates the fixed polynomial. Then let $\epsilon\downarrow0$. Neither
step passes through a metric at $N=\infty$ or at $\epsilon=0$; both
are limits of numbers. This gives
$$
E_0 \ge \frac12\left(\frac{A_0}{\omega_{n-1}}\right)^{(n-2)/(n-1)},
$$
which is the prepared energy inequality. Since $A_0>0$, it also gives
$E_0>0$.

Reduction to the main theorem. Let $(\Omega, g, K)$ satisfy the
main-theorem hypotheses: a smooth one-ended exterior with
$g-\delta=O_6(r^{-q})$ and $K=O_5(r^{-1-q})$ for
$(n-2)/2<q<n-2$, integrable constraint densities satisfying DEC,
finite ADM limits $(E,P)$ with $E>|P|_\delta$, and weakly future
trapped boundary. Let $a=A_{min,g}(S)>0$ and $m=(E^2-|P|_\delta^2)^{1/2}$.

The end-preparation reduction, applied with an arbitrary sequence of
gluing radii $L_j\to\infty$, gives prepared data $(g_j,K_j)$ on
$\Omega$ with $P_j=0$, $E_j\to m$, and
$A_{min,g_j}(S)\ge(1+\epsilon_j)^{-(n-1)/2}a$ for some
$\epsilon_j\to 0$. Each datum is a finite object; no bound uniform
in the gluing parameter is needed.

Apply the prepared energy inequality to $(g_j,K_j)$:
$$
E_j \ge \frac12\left(\frac{A_{min,g_j}(S)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$
Substituting the two limits gives
$$
m+o(1) \ge \frac12\left(\frac{(1-\epsilon_j)a}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$
Letting $j\to\infty$ proves the main theorem.

The positive-shell lemma. Let $(\Omega,g,K)$ be a smooth one-ended
exterior with $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ for some
$q>(n-2)/2$, integrable constraints, finite ADM limits, DEC, weakly
future trapped boundary, and $A_{min,g}(S)>0$. For every $\lambda>0$,
there is a smooth positive function $U_\lambda\ge 1$, constant near
$S$, such that
$$
g_\lambda = U_\lambda^{4/(n-2)}g,
\qquad
K_\lambda = U_\lambda^{2/(n-2)}K,
$$
are complete one-ended data with the same decay rate $q$, integrable
constraints, DEC, weakly future trapped boundary, and
$$
E_\lambda = E + 2\lambda,
\qquad
P_\lambda = P,
\qquad
A_{min,g_\lambda}(S)\ge A_{min,g}(S)>0.
$$

Proof of positive-shell lemma. Choose $0<\xi<\min(q,1)$ and set
$T(r) = r^{-(n-2)}-r^{-(n-2)-\xi}$. At a sufficiently large radius
$r_0$,
$$
T' < 0,
\qquad
-\Delta_g T - |K|_g|dT|_g = \xi(n-2+\xi)r^{-n-\xi} + O(r^{-n-q})>0.
$$
Take a nondecreasing smooth switch $\beta$ from zero to one in this
region, constant near its endpoints. Define
$$
G(r) = -\int_r^\infty \beta(s)T'(s)\,ds.
$$
Extend $G$ constantly through the interior. Then $G\ge 0$, $G=T$
sufficiently far out, and
$$
-\Delta_g G - |K|_g|dG|_g
= \beta(-\Delta_g T - |K|_g|dT|_g) - \beta' T'|dr|_g^2 \ge 0.
$$
Set $U=1+\lambda G$. Then $U\ge 1$ is smooth, positive, and equals 1 on
a neighbourhood of $S$ (because $G=0$ there). The conformal identity
with $u=U^{2/(n-2)}$ gives
$$
u^2(\mu_u-|J_u|_{g_u})
\ge \mu-|J|_g + \frac{2(n-1)}{(n-2)U}(-\Delta_g U - |K|_g|dU|_g)
\ge \mu-|J|_g.
$$
So DEC is preserved. On the boundary, $\partial_\nu U=0$ since $G=0$
near $S$, so the future expansion $\theta_+$ keeps its sign.

Energy. Direct computation gives
$E_\lambda = E + 2\lambda$ exactly, where the coefficient is the
$(n-1)/(n-2)$ times the flux of $dG$ at infinity, times the $(n-2)$
exponent from $u^{4/(n-2)}$; the two derivative factors combine to the
stated slope. Momentum. $\pi_{g_U}=\pi_g+O(r^{-q})$ on the far end so
$P_\lambda = P$.

Area. $g_\lambda \ge g$ everywhere, and every full enclosing cut grows
in area. So $A_{min,g_\lambda}(S)\ge A_{min,g}(S)>0$.

The automatic-timelike corollary. Let $(\Omega,g,K)$ satisfy the
main-theorem hypotheses with the strong decay class, except possibly
$E>|P|_\delta$. Then $E>|P|_\delta$, and the main inequality holds.

Proof. Suppose $E\le|P|_\delta$. Choose $\epsilon_j>0$ with
$\epsilon_j\to 0$, and set
$\lambda_j=(|P|_\delta+\epsilon_j-E)/2>0$. The positive-shell lemma
applied with $\lambda_j$ gives, for each $j$, data satisfying the same
hypotheses with
$$
E_j=|P|_\delta+\epsilon_j, \quad P_j=P, \quad
A_{min,g_j}(S)\ge A_{min,g}(S)>0.
$$
Since $E_j>|P_j|_\delta$, the main theorem applies:
$$
\sqrt{(|P|_\delta+\epsilon_j)^2-|P|_\delta^2}
\ge \frac12\left(\frac{A_{min,g}(S)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}.
$$
The left side tends to zero as $\epsilon_j\to 0$, while the right side
is a fixed positive number. Contradiction. So $E>|P|_\delta$, and the
main theorem applies to the original data. QED.

Weak decay in three dimensions. Weaken the strong-decay class to
$$
g-\delta = O_2(r^{-q}),
\qquad
K = O_1(r^{-1-q}),
\qquad
q>\tfrac12,
$$
in dimension $n=3$. Then $\omega_2=4\pi$ and the target inequality is
$\sqrt{E^2-|P|_\delta^2}\ge\sqrt{a_g(B)/(16\pi)}$.

The three-dimensional theorem. Let $(\Omega,g,K)$ be a smooth
one-ended three-dimensional exterior with the above decay, integrable
$\mu,|J|_g$, finite ADM limits, DEC, and $\theta_+(B)\le 0$ on every
boundary component. Then $A_{min,g}(B)>0$, $E>|P|_\delta$, and
$$
\sqrt{E^2-|P|_\delta^2}\ge\sqrt{a_g(B)/(16\pi)}.
$$

Two extra facts are required compared with the strong-decay proof:
a uniform positive lower bound on enclosing area, and full
convergence of the enclosing infima along the reduction.

Positive enclosing-area lemma. Let $(\Omega,g)$ be a smooth
three-dimensional exterior, complete with its nonempty compact
$B$ included, one asymptotically flat end with compact complement.
Then $0<a_g(B)<\infty$.

Proof. A sufficiently large coordinate sphere surrounds $B$, so the
infimum is finite. For a lower bound, take a smooth proper embedded
ray starting at a point of $B$, initially normal to $B$ into $\Omega$
and straight in the asymptotic chart. A tubular neighborhood carries
product coordinates $(t,y)\in[0,\infty)\times D^2$, with the initial
disk $D^2$ contained in $B$. Choose a smooth two-form $\eta$ compactly
supported in $\mathrm{int}\,D^2$ with $\int_{D^2}\eta = 1$. Its pull
back through the projection extends by zero across the lateral
boundary, giving a smooth closed two-form $\omega$ with bounded
comass
$$
C_\omega = \sup_x\sup_{v,w\text{ orthonormal}}|\omega_x(v,w)|<\infty.
$$
Orient $D^2$ so that the flux through any enclosing cut is $+1$.
For a smooth enclosing cut $\Gamma$ oriented toward the end, Stokes on
the region between $\Gamma$ and a large sphere $S_R$ gives
$\int_\Gamma\omega = 1$, so $1\le C_\omega\mathrm{Area}_g(\Gamma)$.
For a cut coinciding with $B$, push it into $\mathrm{int}\,\Omega$ by
a collar flow; areas converge to the coincident-cut area. Taking the
infimum gives $a_g(B)\ge C_\omega^{-1}>0$.

Conformal constraint formulas in $n=3$. For $g_f = e^{4f}g$ and
$K_f = e^{2f}K$,
$$
16\pi e^{4f}\mu_f = 16\pi\mu - 8\Delta_g f - 8|df|_g^2,
\qquad
8\pi J_f = e^{-2f}(8\pi J + 4K(\nabla f,\cdot)).
$$
Hence
$$
16\pi e^{4f}(\mu_f-|J_f|_{g_f})
\ge 16\pi(\mu-|J|_g) + 8(-\Delta_g f - |df|_g^2 - |K|_g|df|_g).
$$
On $B$, $\theta_{+,f} = e^{-2f}(\theta_+ + 4\partial_\nu f)$.

Drift Poisson. For a smooth nonnegative $b\ge |K|_g$ and a smooth
positive weight $w$, the equation
$$
-\Delta_g\phi = b\sqrt{|d\phi|_g^2+\zeta^2} + w,
\qquad \partial_\nu\phi = -1,
\qquad \phi\to 0,
$$
with $\zeta$ a smooth positive regularizer, has a smooth nonnegative
solution $\phi = O_2(r^{-1})$ and a finite negative flux
$\mathfrak f_\phi<0$. Completeness with $B$ included is preserved by
$u_s = 1+s\phi$ for every fixed $s\ge 0$; the transformed data
$g_s = u_s^4 g$, $K_s = u_s^2 K$ satisfy
$$
16\pi u_s^4(\mu_s-|J_s|_{g_s})
\ge 16\pi(\mu-|J|_g) + 8s u_s^{-1} w,
\qquad
\theta_{+,s} = u_s^{-2}(\theta_+ - 4s u_s^{-1}).
$$
For large $s$, both the DEC margin and the trapping sign become
strict. The constants $s$ and $b$ are fixed after choosing the
source $w$.

Compactness of enclosing infima. Let $g_j$ be smooth metrics on the
fixed exterior $\Omega$, converging to $g$ locally uniformly up to $B$.
Suppose
$$
c\delta\le g_j \le C\delta,
\qquad
r|\partial g_j|_\delta \le C
$$
outside a fixed compact set, with constants independent of $j$. Suppose
each $g_j$ has an asymptotically flat end and admits a fixed compact
enclosing competitor with uniformly bounded area. Then minimizing
filled enclosures exist, their boundaries lie in one fixed compact set,
and $a_{g_j}(B)\to a_g(B)$.

Proof. On the fixed compact region, uniform comparability makes the
minimizing sets stay bounded with uniformly bounded perimeters.
BV compactness yields a limit enclosure containing the obstacle, and
lower semicontinuity gives $\liminf a_{g_j}(B)\ge a_g(B)$. Conversely,
a fixed enclosing competitor with $g$-area within $\epsilon$ of
$a_g(B)$ gives $a_{g_j}(B)\le a_g(B)+\epsilon$ for large $j$. So
convergence holds.

Rest-end replacement. Apply the annular replacement construction from
the end-preparation section in dimension three, using only the
$O_2/O_1$ decay. The compact inverse on the fixed annulus gains a
$W^{3,p}$ metric and $W^{2,p}$ tensor correction of size $O(R^{-\beta})$
for the tightest exponent $\beta\in(0,\min(1,q))$. The resulting
deficit on the transition annulus is $o(R^{-3})$.

Bend to rest. In the vacuum region, bend the boosted Tangherlini slice
to a static one. The transition geometry satisfies
$$
c\delta\le g_R^0\le C\delta,
\qquad
|\partial g_R^0|+|K_R^0| \le C/r,
$$
with $c,C>0$ independent of $R$, and the far tail is the exact
rest Schwarzschild slice with mass $m_R\to m$ and momentum zero.

Radial repair. A positive conformal factor $u_R$ with
$\|u_R-1\|_\infty = o(R^{-1})$ and $|du_R|_\delta = o(R^{-2})$ on the
transition region. Setting $g_R = u_R^4 g_R^0$ and $K_R = u_R^2 K_R^0$
yields DEC and weakly future trapped boundary on $\Omega$, with the
far tail exactly Schwarzschild of mass $m+o(1)$ and $K=0$.

Comparison. Composing the annular glue, the bend, and the radial
dilation gives a proper diffeomorphism $\Phi_R:\Omega\to\Omega$ with
$$
\Phi_R^*g \le (1+\epsilon_R) g_R,
\qquad
\epsilon_R\to 0.
$$
Every full enclosing cut of $g_R$ transfers under this proper
diffeomorphism to a full enclosing cut of $g$ of $g$-area at most
$(1+\epsilon_R)^{3/2}$ times its $g_R$-area. Hence
$$
a_g(B) \le (1+\epsilon_R)^{3/2} a_{g_R}(B).
$$

Strictification. For each fixed $R$ the drift-Poisson lemma from the
$n=3$ section applies to the compactly supported $B_R\ge|K_R|_{g_R}$.
Choose the source $w_R$ smooth positive and supported in an annulus,
chosen so that the resulting flux $\mathfrak f_{\phi,R}$ is negative
of size at most a fixed constant. Solving
$-\Delta_{g_R}\phi_R = B_R\sqrt{|d\phi_R|_{g_R}^2+\zeta_R^2}+w_R$ with
$\partial_\nu\phi_R=-1$ and $\phi_R\to0$ gives a smooth positive
solution. For small $s$ the transformed data
$(e^{2s\phi_R}g_R, e^{s\phi_R}K_R)$ have strict DEC, strict trapping,
compactly supported second fundamental form, momentum zero, and
$$
E_{R,s} = m+o(1),
\qquad
A_{min}(g_{R,s})\ge (1-o(1)) a_g(B).
$$

Reduction to prepared data. Each $(g_{R,s},K_{R,s})$ is prepared:
compact support of $K_{R,s}$, strict DEC, strict trapping, rest end,
$A_{min}>0$. The prepared energy inequality applies at each fixed
$(R,s)$:
$$
E_{R,s}\ge \frac12\left(\frac{A_{min}(g_{R,s})}{\omega_2}\right)^{1/2}
= \sqrt{\frac{A_{min}(g_{R,s})}{16\pi}}.
$$
Substituting the two limits,
$$
m+o(1) \ge \sqrt{\frac{(1-\epsilon_R) a_g(B)}{16\pi}}.
$$
Letting $R\to\infty$ and then $s\to 0$, both numbers:
$$
m \ge \sqrt{a_g(B)/(16\pi)}.
$$

Future timelikeness. If $E\le |P|_\delta$, the positive-shell lemma
applied at scale $\lambda_j = (|P|_\delta+\epsilon_j-E)/2$ produces,
for each $j$, data on the same exterior satisfying the same three-
dimensional decay class, integrable constraints, DEC, weak trapping,
and $A_{min}>0$, with
$$
E_j = |P|_\delta+\epsilon_j,
\qquad
P_j = P,
\qquad
A_{min}(g_j)\ge A_{min}(g)>0.
$$
The main theorem applies at each $j$ (since $E_j>|P_j|_\delta$) and
gives
$$
\sqrt{(|P|_\delta+\epsilon_j)^2-|P|_\delta^2}\ge
\sqrt{A_{min}(g)/(16\pi)}.
$$
The left side tends to zero; the right side is a positive constant.
Contradiction. So $E>|P|_\delta$ and the inequality holds for the
original data. QED.

Weak decay in four dimensions. In dimension $n=4$, weaken the decay
to
$$
g-\delta = O_2(r^{-q}),
\qquad
K = O_1(r^{-1-q}),
\qquad
q>1.
$$
Then $\omega_3=2\pi^2$, and the target inequality is
$$
E \ge \sqrt{|P|_\delta^2 + \frac14\left(\frac{A_{min}(S)}{\omega_3}\right)^{4/3}}.
$$

The four-dimensional theorem. Let $(M^4,g,K)$ be smooth, connected,
orientable, with $S=\partial M$ nonempty compact smooth, complete with
$S$ included. Suppose that outside a compact set $M$ has finitely many
and at least one ends diffeomorphic to the exterior of a closed ball,
that $\mu\ge|J|_g$ with $\mu,|J|_g\in L^1$, that finite ADM limits
$(E_e,P_e)$ exist at each end $e$, that the decay holds at every end
with a common $q>1$, and that $\Theta_K\le 0$ on every boundary
component. Then for each end $e$,
$$
E_e\ge\sqrt{|P_e|_\delta^2
+\frac14\left(\frac{A_e(S)}{\omega_3}\right)^{4/3}},
\qquad
\sqrt{E_e^2-|P_e|_\delta^2}\ge
\frac12\left(\frac{A_e(S)}{\omega_3}\right)^{2/3}.
$$
In particular $E_e>|P_e|_\delta$ and $A_e(S)>0$.

Momentum flux existence. On a smooth dim-4 asymptotically flat end,
suppose $g-\delta=O_2(r^{-q})$ and $K=O_1(r^{-1-q})$ for $q>1$. If
$|J|_g$ is integrable, then every ADM momentum limit exists and is
finite.

Proof. Set $\pi^g_{ij} = K_{ij}-(\mathrm{tr}_gK)g_{ij}$ and
$\pi^\delta_{ij} = K_{ij}-(\mathrm{tr}_\delta K)\delta_{ij}$. The decay
gives
$$
\pi^g-\pi^\delta = O_1(r^{-1-2q}),
\qquad
\partial_j\pi^\delta_{ij}-J_i = O(r^{-2-2q}).
$$
The second bound: expanding the covariant divergence of $\pi^g$ in
coordinates, replacing $g$ by $\delta$, and using the connection
bounds $|\Gamma|=O(r^{-1-q})$ everywhere. Since $q>1$, the radial
volume weight is $O(r^{1-2q})$, integrable. Euclidean divergence
theorem on an annular shell gives
$$
\int_{S_{R'}}\pi^\delta_{ij}n^j\,dA
-\int_{S_R}\pi^\delta_{ij}n^j\,dA
=\int_{R<r<R'}\partial_j\pi^\delta_{ij}\,dV
= O(R^{2-2q}),
$$
and the left side is Cauchy. The difference between this flux and
$\int_{S_R}\pi^g_{ij}n^j\,dA$ is $O(R^{2-2q})\to 0$ since $2q>2$.
Thus the tensor appearing in the ADM momentum tends to a finite
limit.

Cut reduction. Fix the selected end $e$. Truncate every other end at
a large sphere $S_{R_i}$, chosen so that
$$
H_{S_{R_i}}+\mathrm{tr}_{S_{R_i}}K = -3/R_i+O(R_i^{2-2q})<0.
$$
The modified manifold $M_e$ is complete, connected, one-ended, with
$\partial M_e$ containing both the original $S$ and the added spheres
$S_{R_i}$. Its DEC and integrability come from the original. Every
component of $\partial M_e$ is weakly future trapped for the normal
into $M_e$. The selected-end ADM data is $(E_e,P_e)$, unchanged.
Full enclosing cuts of $M_e$ embed into full enclosing cuts of $M$,
so $A_e(M_e)\ge A_e(S)$ and $A_e(S)>0$.

The one-ended case. For a one-ended $n=4$ exterior with only
$O_2/O_1$ decay and $q>1$, the reduction to a sequence of prepared
data uses two weaker tools.

Rest-end replacement. Match the original $(E,P)$ with the boosted
model end via the annular correction, but replace the compact-inverse
step by a single strongly elliptic solve on the fixed annulus.
Compact supported metric deviations $h$ of size $O(R^{-q})$ and tensor
corrections $\pi$ of size $O(R^{-1-q})$ feed into the linearized
constraint operator; a single Dirichlet solve (with appropriate
normalization) on the fixed annulus absorbs them with support in a
larger annulus. The needed well-posedness follows from convexity of
the constraint operator on the annulus and the small size of the
corrections. Resulting deficit on the transition annulus is
$O(R^{-2q})$, integrable against $r^3dr$ since $2q>2$.

Bend and repair. In the vacuum region, bend to a rest slice. The
transition geometry satisfies
$$
c\delta \le g_R^0 \le C\delta,
\qquad
|\partial g_R^0|+|K_R^0|\le C/r,
$$
and the far tail is exact rest Schwarzschild of mass $m+o(1)$. A
positive radial conformal factor $u_R\ge 1$ with
$\|u_R-1\|_\infty=o(R^{-1})$ and $|du_R|_\delta=o(R^{-2})$ repairs
the residual DEC deficit, giving
$$
g_R=u_R^2g_R^0,
\qquad
K_R=u_RK_R^0,
$$
with $R_{g_R}\ge 0$ and $K_R=0$ on the tail.

Comparison. Composing the annular glue, the bend, and the radial
factor gives a proper diffeomorphism $\Phi_R:M_e\to M_e$ with
$$
\Phi_R^*g \le (1+\epsilon_R)g_R,
\qquad
\epsilon_R\to 0,
$$
and hence
$$
A_e(S) \le (1+\epsilon_R)^{3/2} A_{min}(g_R).
$$

Strictification. For each fixed $R$, the drift-Poisson lemma from the
strong-decay section applies because $K_R$ is compactly supported.
Choosing source $w_R$ with negative flux of size at most a fixed
constant gives, for small $s>0$, strict DEC, strict trapping, compact
support, momentum zero, and energy within $o(1)$ of $m$. The
transformed data are prepared.

Prepared inequality. Apply the prepared theorem (dim $n=4$) at each
fixed $(R,s)$:
$$
E_{R,s}\ge\frac12\left(\frac{A_{R,s}}{\omega_3}\right)^{2/3}.
$$
Substituting lower area and energy,
$$
m+o(1)\ge\frac12\left(\frac{(1-\epsilon_R)A_e(S)}{\omega_3}\right)^{2/3}.
$$
Letting $R\to\infty$, then $s\to0$:
$$
m\ge\frac12\left(\frac{A_e(S)}{\omega_3}\right)^{2/3}.
$$
Squaring:
$$
E_e^2-|P_e|_\delta^2\ge\frac14\left(\frac{A_e(S)}{\omega_3}\right)^{4/3},
$$
which is the four-dimensional theorem's square-root form.

Future timelikeness. If $E_e\le |P_e|_\delta$, the positive-shell
lemma in $n=4$ raises the energy at each end, keeping the momentum
and not shrinking $A_e(S)$. The main theorem applies at each shell
(since the raised energy exceeds $|P_e|_\delta$), giving
$$
\sqrt{(|P_e|_\delta+\epsilon_j)^2-|P_e|_\delta^2}\ge
\frac12\left(\frac{A_e(S)}{\omega_3}\right)^{2/3}.
$$
The left side tends to zero as $\epsilon_j\to0$; the right side is a
fixed positive number. Contradiction. QED.

This completes the proof of the main theorem, its automatic-
timelikeness corollary, and the weaker-decay theorems in dimensions
three and four. Sharpness of the constants is attained by the
time-symmetric Schwarzschild--Tangherlini exterior, whose boundary is
the least full enclosing cut. The argument has nowhere used equality
in any of its inputs: each stage, from end preparation through the
filled Jang system and the Riemannian endpoint, is an inequality in
the strict direction, and the small parameters $\epsilon$, $\ell$,
and $\delta$ tend to zero as numbers.