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: they 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.

./pdfs/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.

./pdfs/math_9906087.pdf
./pdfs/math_9911173.pdf
./pdfs/0708.0850.pdf
./pdfs/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.

./pdfs/0910.4785.pdf
./pdfs/0905.2622.pdf
./pdfs/2304.09332.pdf
./pdfs/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.

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

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, which is
not the subject of the present text.

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 $O(L^{-2q})$.